MSC Nastran 2012 Implicit Nonlinear (SOL 600) User’s Guide

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MSC Nastran 2012

Implicit Nonlinear (SOL 600) User’s Guide

D R2 Nastran Implicit Nonlinear (SOL 600) User’s Guide

Worldwide Webwww.mscsoftware.com

DisclaimerMSC.Software Corporation reserves the right to make changes in specifications and other information contained in this document without prior notice.

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C o n t e n t sMSC Nastran Implicit Nonlinear (SOL 600) User’s Guide

1 Introduction

MSC.Software Products 2

MSC Nastran Implicit Nonlinear (SOL 600) 3Defining the Model 3Nonlinear Analysis 4Results 5

Feature List 7

How SOL 600 Solves Nonlinear Problems 10

This User’s Guide 12Other MSC Nastran Documentation for SOL 600 12Marc Documentation 12Patran Documentation 13

2 MSC Nastran Bulk Data File and Results Files

The MSC Nastran Bulk Data File 16Input Conventions 17Defaults 18Section Descriptions 18Example 19Running Existing Nonlinear Models 20SOL 600 Executive Control Statement: 20Restart from SOL 600 into SOL 103 or into Another Linear Solution Sequence 24Generating and Editing the Bulk Data File in Patran 25

Output Requests 26Deformations 26Grid Point Force Balance and Element Strain Energy 35

Results Files 40Files Generated During the Analysis 40Analysis Results Files 40Postprocessing with Patran 41

MSC Nastran Implicit Nonlinear (SOL 600) User’s Guide

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3 Solution Methods and Strategies in Nonlinear Analysis

Introduction 44

Linear Static Analysis Procedure 45

Differences Between Linear and Nonlinear Analysis 46

Applying Constraints 48Single Degrees of Freedom 48Multiple Degrees of Freedom 48

Adding Nonlinear Effects 53Sources of Nonlinearity 53Subcases, Load Increments, and Iterations 54Nonlinear Equation Solution 55SOL 600 Analysis Procedure 57

Numerical Methods in Solving Equations 58Direct Methods 58Iterative Methods 59Preconditioners 59Storage Methods 60Nonsymmetric Systems 61Specifying the Solution Procedure 61Other Factors Affecting Performance 61

Iteration Methods 63Full Newton-Raphson Algorithm 63Modified Newton-Raphson Algorithm 64Strain Correction Method 65The Secant Method 66Specifying the Iteration Method 67

Load Increment Size 68Fixed Load Incrementation 68Adaptive Load (AUTO) Incrementation 68Specifying the Load Incrementation Method 77

Convergence Controls 78Specifying Convergence Criteria 80

Singularity Ratio 81

Guidelines for Analysis Methods 83Analysis Methods 83General Tips 83

vCONTENTS

Choosing a Solution Method 84Time Steps or Load Increments 84Nonlinear Dynamics 85Efficiency 85

4 Nonlinearity and Analysis Types

Linear and Nonlinear Analysis 88Linear Analysis 88Nonlinear Analysis 88

Nonlinear Effects and Formulations 90Geometric Nonlinearities 93Material Nonlinearities 102Nonlinear Boundary Conditions 111

Overview of Analysis Types 112

Static Analysis 114Post-Buckling 115Creep, Viscoplastic, and Viscoelastic Behavior 115

Body Approach 116

Buckling Analysis 117Eigenvalue Buckling Prediction 117Bifurcation Approach 118Eigenvalue Extraction Methods 119

Normal Modes 120Eigenvalue Analysis 122Free Vibration Analysis 125Support of Complex Eigenvalue Analysis 126

Transient Dynamic Analysis 128Direct Transient Response 128Technical Background 130Time Step Definition 134Initial Conditions 134Damping 135

Creep 136


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5 Analysis Techniques

Domain Decomposition 140Specifying Domain Decomposition 140Single Input File Parallel Processing for SOL 600 142DDM Results in Patran 142DDM Configuration 142

RESTARTS 144Specifying Restarts and Parameters 144

Inertia Relief with Auto-Support 145Review 145General Formulation 146SUPORT6 Entry 147

Superelements and Modal Neutral Files 149

BRKSQL 150

User Subroutine Support 154

6 Modeling

Coordinate Systems 156Nodal Coordinate Systems 156Element Coordinate Systems 156

Nodes 158

Elements 159

Modeling in Patran 161Creating Geometry in Patran 161Creating Finite Element Meshes in Patran 163

7 Setting Up, Monitoring, and Debugging the Analysis

Solution Type 166Specifying the Solution Type 166SOL 600 Executive Control Statement 166Defining the Solution Type in Patran 166

Analysis Procedures 168Analysis Types 168Specifying the Analysis Type for a Subcase 169

viiCONTENTS

Translation Parameters 172Specifying the Translation Parameters 172

Solution Parameters 175Specifying Solution Parameters 175

Subcases 178Specifying Subcases 178

Subcase Parameters 181Specifying Static Subcase Parameters 181Specifying Normal Modes Subcase Parameters 183Specifying Buckling Subcase Parameters 185Specifying Transient Dynamic Subcase Parameters 186Specifying Creep Subcase Parameters 188Specifying Body Approach Subcase Parameters 190

Execution Procedure for MSC Nastran Implicit Nonlinear from the Command Line 192

Using Patran to Execute MSC Nastran 193How to Tell When the Analysis is Done 193How to Tell if the Analysis Ran Successfully 194

Monitoring the Analysis 195Editing a MSC Nastran Input File 196

Debugging the Analysis 197Resolving Convergence Problems 197Standard Exit Messages 203Using Patran to Debug an Analysis 206

8 Output from the Analysis

Overview 210Input 210.OP2 Data 210

Output Requests 212Specifying Output Requests 212

SOL 600 Results Quantities 220Using Patran to Postprocess Results Quantities 223

MSC Nastran Results Quantities 225Using Patran to Postprocess MSC Nastran Results Quantities 225


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9 Assigned Conditions

Constraints 230Multi-Point Constraints 230Support Conditions 240

Loads and Boundary Conditions 242Using Patran to Apply Loads and Boundary Conditions 244Displacement LBCs 247Force LBCs 248Pressure LBCs 249Temperature LBCs 251Inertial Loads LBCs 254Velocity LBCs 255Acceleration LBCs 255Distributed Load LBCs 256Total Load LBCs 258Contact LBCs 259

Initial Conditions 260Initial Displacement LBCs 260Initial Velocity LBCs 260

10 Materials

Overview 264Constitutive Models 265MSC Nastran Implicit Nonlinear Material Entries 266

Linear Elastic 268Isotropic Materials 269Orthotropic Materials 270Anisotropic Materials 272

Nonlinear Elastic 273Hypoelastic - Isotropic 273Hyperelastic - Isotropic 273Viscoelastic 300Narayanaswamy Model 309

Inelastic 312Yield Conditions 313Work Hardening Rules 318Flow Rules 322Rate Dependent Yield 325

ixCONTENTS

Experimental Stress-Strain Curves 327Temperature-Dependent Behavior 336Temperature-Dependent Stress Strain Curves 337Specifying Elastoplastic Material Entries 339

Failure and Damage Models 344Isotropic/Orthotropic/Anisotropic Failure Models 344Damage Models 354

Creep 363Oak Ridge National Laboratory Laws 366Viscoplasticity (Explicit Formulation) 367Creep (Implicit Formulation) 367Specifying Creep Material Entries 368

Composite 370Specifying Composite Material Entries 371

Gasket 373Specifying Gasket Material Entries 377

Material Damping 379Specifying Material Damping Entries 379

Experimental Data Fitting 381

11 Element Library

Overview 402Element Types 402

Element Selection 404Element Interpolation 404Element Integration 404Incompressible Elements 405Overriding MSC Nastran Element Selections 405

Global Element Controls 406Assumed Strain 406Constant Dilatation 406Setting Global Element Parameters in Patran 406

Mass Elements, Springs, Dampers, and Connector Elements 407Patran FE Application Input Data 407

Gap Elements 409


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Patran FE Application Input Data 409

Line Elements 410Patran FE Application Input Data 410

Membranes, Panels, and Shells 411Patran FE Application Input Data 411

Solid Elements 413Axisymmetric Elements 414Plane Strain Elements 4143-D Solid Elements 415

Beam/Bar and Shell Offsets 417

12 Contact

Overview 420

Contact Methodology 421Contact Bodies 421Numerical Procedures 430Implementation of Constraints 434Separation 436Higher Order Elements 4373-D Beam and Shell Contact 437Friction Modeling 438

Defining Contact Bodies 447Deformable and Rigid Surfaces 447Motion of Surfaces 447Cautions 448Control Variables and Option Flags 448Time Step Control 449Dynamic Contact - Impact 449Two-dimensional Rigid Surfaces 449Specifying Contact Body Entries 462

Selecting and Controlling Contact Behavior 467Contact Parameters 467Contact Table 472Movement of Contact Bodies 476Initial Conditions 477

Heat Transfer and Thermal Contact 479Heat Transfer Examples 481

xiCONTENTS

References 482

13 SOL 600 Example Problems

Introduction 484

Engine Gasket Under Bolt Preload 485Problem Statement 485Model Description 486Solving the Problem 488Inspecting the Results 492

Elastic-Plastic Collapse of a Cylindrical Pipe under External Rigid Body Loading 494

Problem Statement 494Model Description 495Solving the Problem 497Inspecting the Results 500

Rubber Door Seal - Performance Door Closing 503Problem Statement 503Model Description 503Solving the Problem 504Inspecting the Results 507

Brake Forming 509Problem Statement 509Model Description 509Solving the Problem 510Inspecting the Results 513

Panel Buckling 514Problem Statement 514Model Description 514Solving the Problem 515Inspecting the Results 518

Index 519


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Chapter 1: Introduction

1 Introduction

MSC.Software Products

MSC Nastran Implicit Nonlinear (SOL 600)

Feature List

How SOL 600 Solves Nonlinear Problems

This User’s Guide

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideMSC.Software Products

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MSC.Software ProductsMSC.Software Corporation provides an extensive array of software products that make it possible to simulate almost any engineered component with any level of detail you require. MSC is recognized as a leader in finite element analysis software with a product list that includes MSC Nastran, Patran, Marc, Dytran and many others. Each of these codes within themselves are powerful general-purpose analysis codes that can be used to solve structural, heat transfer, and coupled thermal-structural finite element problems. When paired together and supplemented with special purpose application modules and interfaces these software products can be tailor made to suit specific industries and engineering problems unique to those industries.

To keep this user’s guide to a reasonable size, complete descriptions of all the MSC Nastran Bulk Data entries and Case Control commands have not been included. A brief description of the input format is given for entries and commands when it is helpful to understand the material., also in many cases the Patran interface is used to describe the input. You should consult the MSC Nastran Quick Reference Guide for detailed descriptions of MSC Nastran input formats, many have direct links.

This guide contains many highlighted links (in blue) to other MSC Nastran documents and all the documents were delivered together as a collection. If you keep the collection together the links between documents will work. Two suggestions when working with links are 1) “alt ” returns you back in the window your mouse is in and 2) you can open the other “linked to” document in a new window from an Adobe Reader by unselecting the checkbox Edit Preferences Documents Open cross-document links in the same window and select “OK”.

3CHAPTER 1Introduction

MSC Nastran Implicit Nonlinear (SOL 600)MSC Nastran Implicit Nonlinear (SOL 600) is an application module in the MSC Nastran system that pairs the full features of MSC Nastran with the Marc solver to analyze a wide variety of structural problems subjected to geometric and material nonlinearities, and contact. An extensive finite element library for building your simulation model, and a set of solution procedures for the nonlinear analysis, which can handle very large matrix equations, are available in MSC Nastran Implicit Nonlinear (SOL 600).

Defining the ModelA finite element model consists of a geometric description, which is given by the elements and their nodes and a set of properties associated with the elements, describing their attributes. These properties include material definitions, cross-section definitions in the case of structural elements like beams and shells, and other parameters for contact bodies, springs, dashpots, etc. There may also be constraints that must be included in the model - RBE elements, or “multi-point constraints'' or “equations'' (linear or nonlinear equations involving several of the fundamental solution variables in the model), or simple “boundary conditions'' that are to be imposed throughout the analysis. Nonzero initial conditions, such as initial temperatures, displacements, velocities, and even initial stresses and/or plastic strains may also be specified.

The model is described and communicated to MSC Nastran in the form of a text file, called a MSC Nastran Input file. You can generate this file using any text editor, but it must adhere to MSC Nastran conventions for the ordering and format of the model information.

Using Patran with SOL 600

The amount of information that needs to be conveyed in the MSC Nastran Input file is extensive for even a modest size model. The amount of information and the complexity of most models makes it virtually impossible to generate the MSC Nastran Input file with a text editor alone. Typically you benefit from using a preprocessor such as, Patran. Patran is another MSC Software simulation code that provides a graphical user interface and an extensive line of model building tools that you can use to construct and view your model, and generate a MSC Nastran Input file.

If you are using Patran as a preprocessor, you are required to specify an analysis code. Selecting MSC Nastran Implicit Nonlinear (SOL 600) as the analysis code under the Analysis Preference menu, customizes Patran in five main areas:

• Material Library

• Element Library

• Loads and Boundary Conditions

• MPCs

• Analysis forms

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The analysis preference also specifies that the model information be output in the MSC Nastran Input File format.

Throughout this Users Guide, actual examples are described in the context of using Patran. Actual Patran forms and instructions are provided.

Nonlinear AnalysisLinear analysis assumes a linear relationship between the load applied to a structure and the response of the structure. The stiffness of a structure in a linear analysis does not change depending on its previous state. Linear static problems are solved in one step, by a single decomposition of the stiffness matrix. A number of important assumptions and limitations are inherent in linear static analysis. Materials behavior is such that the stress is directly proportional to strain (linear) and to loads that do not take the material beyond its permanent yield point (the material remains elastic). Linear analysis is restricted to small displacements, otherwise the stiffness of the structures changes and must be accounted for by regenerating the stiffness matrix. Lastly, loads are assumed to be applied slowly as to keep the structure in equilibrium.

It becomes necessary to consider nonlinear effects in structures when modeling materials with nonlinear behavior and where large deformations (rotations and/or strains) occur. In addition, contact problems exhibit nonlinear effects due to changes in boundary conditions.

In a nonlinear problem the stiffness of the structure depends on the displacement and the response is no longer a linear function of the load applied. As the structure displaces due to loading, the stiffness changes, and as the stiffness changes the structure’s response changes. As a result, nonlinear problems require incremental solution schemes that divide the problem up into steps calculating the displacement, then updating the stiffness. Each step uses the results from the previous step as a starting point. As a result the stiffness matrix must be generated and decomposed many times during the analysis adding time and costs to the analysis.

Nonlinear problems present many challenges. A nonlinear problem does not always have a unique solution. Sometimes a nonlinear problem does not have any solution, although the problem can seem to be defined correctly.

Nonlinear analysis requires choosing a solution strategy which includes dividing the loading into logical steps, controlling the numerical processing, and planning for the possibility of changing the solution strategy during the analysis using restarts. Which solution method to use depends on the structure itself, the nature of the loading, and the anticipated nonlinear behavior. In some cases, one method can be advantageous over another; in other cases, the converse might be true.

If a solution is obtainable, there is also the issue of efficiency. Each solution procedure, has pros and cons in terms of matrix operations and storage requirements. In addition, a very important variable regarding overall efficiency is the size of the problem. The time required to assemble a stiffness matrix, as well as the time required to recover stresses after a solution, vary roughly linearly with the number of degrees of freedom of the problem. On the other hand, when using a direct solver the time required to go through the solver varies roughly quadratically with the bandwidth, as well as linearly with the number of degrees of freedom.


Applications for Nonlinear Analysis

Early development of nonlinear finite element technology was mostly influenced by the nuclear and aerospace industries. In the nuclear industry, nonlinearities are mainly due to high-temperature behavior of materials. Nonlinearities in the aerospace industry are mainly geometric in nature and range from simple linear buckling to complicated post-bifurcation behavior. Nonlinear finite element techniques have become popular in metal forming manufacturing processes, fluid-solid interaction, and fluid flow. In recent years, the areas of biomechanics and electromagnetics have seen an increasing use of finite elements.

ResultsLike the enormous amount of data needed to define the simulation model to an analysis code, there is a large volume of data returned from the simulation analysis. And just as it is virtually impossible to construct a model with a text editor alone, it is equally as difficult to read and interpret the results by hand. Using a postprocessor with a graphical user interface such as Patran is highly recommended.

Postprocessing Features of Patran

The Patran Results application gives you control of powerful graphical capabilities to display results quantities in a variety of ways:

• Deformed structural plots

• Color banded fringe plots

• Marker plots (vectors, tensors)

• Freebody diagrams

• Graph (XY) plots

• Animations of most of these plot types.

The Results application treats all results quantities in a very flexible and general manner. In addition, for maximum flexibility results can be:

• Sorted

• Reported

• Filtered

• Derived

• Deleted

All of these features help give meaningful insight into results interpretation of engineering problems that would otherwise be difficult at best.

The Results application is object oriented, providing postprocessing plots which are created, displayed, and manipulated to obtain rapid insight into the nature of results data. The imaging is intended to provide graphics performance sufficient for real time manipulation. Performance will vary depending on

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hardware, but consistency of functionality is maintained as much as possible across all supported display devices.

Capabilities for interactive results postprocessing also exist. Advanced visualization capabilities allow creation of many plot types which can be saved, simultaneously plotted, and interactively manipulated with results quantities reported at the click of the mouse button to better understand mechanical behavior. Once defined, the visualization plots remain in the database for immediate access and provide the means for results manipulation and review in a consistent and easy to use manner.


Feature ListThe most important features of MSC Nastran Implicit Nonlinear (SOL 600) are presented in the following list.

1. MSC Nastran Implicit Nonlinear (SOL 600) solves linear and nonlinear (material, contact and/or geometric) static, heat transfer, modal (vibration), buckling, and transient dynamic structural finite element problems.

2. Eigenvalue solutions are available in MSC Nastran Implicit Nonlinear (SOL 600) for solving linear or nonlinear modal or buckling analyses using either Lanczos or Inverse Power Sweep methods of iteration. Through the use of parameters you can control the convergence of the eigenvalues, and the modes to retain. Modal and buckling eigenvalues must be solved in different runs.

3. MSC Nastran Implicit Nonlinear (SOL 600) has a variety of solution procedures and bandwidth optimizers for serial, parallel and/or multi-threaded approaches.

4. MSC Nastran Implicit Nonlinear (SOL 600) supports the following elements/bodies:

• 3 and 6 noded triangular shell/membrane/plane stress/(generalized) plain strain/axisymmetric elements

• 4 and 8 noded quadrilateral shell/membrane/plane stress/(generalized) plain strain/axisymmetric elements

• 6 and 8 noded solid shell elements

• conversion of penta and hexa to solid shell elements

• 4 and 10 noded solid tetrahedral elements

• 6 and 15 noded solid penta (wedge) elements

• 8 and 20 noded solid hexahedral elements

• 2 and 3 noded beam element

• 2 and 3 noded bar element

• 2 and 3 noded axisymmetric shell element

• 2 noded gap element

• 1 and 2 noded spring elements

• 1 and 2 noded damper elements

• Rigid and deformable contact bodies

• Point Mass element

• Connector elements CBUSH, CWELD, CFAST, and RSSCON

• RSPLINE element

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RBE elements and multi-point constraint equations are supported in MSC Nastran Implicit Nonlinear (SOL 600) to tie specific nodes or degrees-of-freedom to each other. Special MPC entities are supported, (e.g. rigid links) which can be used to tie two nodes together or equate the motion of two DOFs. Both small and large rotations are supported.

5. MSC Nastran Implicit Nonlinear (SOL 600) supports the following loads and boundary conditions:

• Constrained nodal displacements (zero displacements at specified degrees-of-freedom).Enforced nodal displacements (non-zero displacements at specified degrees-of-freedom in the nodal coordinate system).

• Forces applied to nodes in any coordinate system.

• Pressures applied to element edges or faces, including strain-rate controlled application for super-plastic forming simulations.

• Temperature applied to nodes. Temperature can be applied as a load in a structural analysis. The reference temperature is user definable.

• Inertial body forces, acceleration and velocity can be applied in the global coordinate system.

• Contact between two bodies can be defined by selecting the contacting bodies and defining the contact interaction properties. Gluing and ungluing are provided. Enforced motion or velocity of rigid contacts surfaces is available.

6. MSC Nastran Implicit Nonlinear (SOL 600) supports isotropic, orthotropic and anisotropic material properties. Temperature dependent isotropic and orthotropic material properties can be defined for elastic, elastic-plastic, hyper-elastic, visco-elastic, and creep constitutive models. Nonlinear elastic-plastic materials can be defined by specifying piecewise linear stress-strain curves, which may be temperature dependent.

7. Physical properties can be associated with MSC Nastran Implicit Nonlinear (SOL 600) elements such as the cross-sectional properties of the beam element, the area of the beam and rod elements, the thickness of shell, plane stress, and membrane elements, spring parameters, masses, gap element parameters, the alternate material coordinate frame for solid elements and material IDs.

8. Fracture Mechanics capabilities include crack propagation and closure, delamination, birth and death of elements, and a large number of failure index criterion.

9. Laminated composite solid and shell elements are supported in MSC Nastran Implicit Nonlinear (SOL 600) through the PCOMP or PCOMPG cards of the materials capability. Each layer has its own material, thickness, and orientation and may represent linear or nonlinear material behavior. Failure index calculations are also supported. Fast integration techniques are available with the PCOMPF entry. Equivalent material models may be incorporated using PSHELL.

10. Analysis jobs consisting of (possibly) complex loading histories (such as would occur in a multi-step manufacturing process) for MSC Nastran Implicit Nonlinear (SOL 600) are defined using subcases. A single subcase may represent the entire analysis, or may be one step in a multi-step simulation. The loads and constraints in each subcase represent the total load at that point in the analysis, making it easy to determine the state of loading at any point in the analysis. The starting point of the current subcase is the ending point of the previous subcase.


11. MSC Nastran Implicit Nonlinear (SOL 600) jobs are submitted using text-based input decks that may be generated manually with a text editor, or by a variety of pre/post processing programs such as Patran. The input file is read in and a number of text files, such as the .f06, .log, .f04 files are generated.

12. Results can be requested in several output formats such as .f06, .t16, .t19, .xdb, .op2, or punch files. These files are typically read back into the pre/post processing programs for the purpose of evaluating the results with plots such as deformed shape plots, contour stress/strain plots, or X-Y history plots.

13. Nodal displacements, velocities and accelerations, mode shapes, element and nodal stresses, element and nodal strains, element and nodal plastic strains, element and nodal creep strains, nodal reaction forces and contact interface stress/force values, shell element stress resultants, element strain energy, strain energy density, and phase angle values can all be requested as output and visualized with the aforementioned results visualization tools such as Patran. Stress functions, for example von Mises, beam stresses, strains, and internal forces, can also be requested as output. Composite element results are returned for each layer of the composite.

14. A restart capability is available in MSC Nastran Implicit Nonlinear (SOL 600). Any analysis can be saved from any point for a possible restart. A new static load case or a buckling analysis can be solved by restarting from the original static analysis.

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How SOL 600 Solves Nonlinear ProblemsThe primary steps in running a MSC Nastran Implicit Nonlinear (SOL 600) analysis are as follows:

1. Read MSC Nastran Input File in IFP (input file processor) as in other MSC Nastran solution sequences.

2. Convert MSC Nastran input to Marc input and write out a Marc input deck (jid.marc.dat in IFP).

3. If there are no input errors, execute Marc.

4. If you request, do any of the following:

5. Translate Marc’s t16 file to obtain MSC Nastran op2 or xdb output (this is done by code in MSC Nastran that creates output op2 data blocks on a file, which we call the f11 file, then generating DMAP on the fly to use inputt2 to placed the f11 datablocks into the MSC Nastran database, and finally use OUTPUT2 to produce an OP2 file which has the geometry datablocks and the f11 output datablocks all in one file (or similar DMAP to generate an xdb file with geometry and output datablocks.

6. As in Step 5, DMAP can be extended to produce printed output in the.f06 file or punched output in the .pch file having the exact formats MSC Nastran uses for all other solution sequences (this is done by generating OFP DMAP on the fly).

7. Copy Marc’s output file (known as the .out file) to the .f06 file with or without changing any text strings. This output will have the Marc formats, but names such as Marc can be changed to any desired user name (for example MSC Nastran Implicit Nonlinear). It is strongly suggested that Steps 6 and 7are not both done in the same run, or the f06 file output could become confusing.

8. Retain or delete the Marc input and output files (which normally consist of jid.marc.dat, jid.marc.out, jid.marc.sts, jid.marc.log, jid.marc.t16 and possibly others which have the name jid.marc.*).

The process of reading input data from a MSC Nastran Input File, translating the model data to a MSC.Marc input file, running a Marc solution, and translating back the results files is shown in the flowchart that follows.


Figure 1-1 MSC Nastran Implicit Nonlinear (SOL 600) Solution Process

MSC Nastran Input Deck

Use std Nast output req -

deck echo and

Write jobname.marc.dat

IFP Processes Input Deck

Stop

SuccessfulTranslation?

Submit MarcAnalysis?

Marc writes.out,.t16,.t19

Is marccpy= 1or 2?

Submit Marc job -see note

Append runtime error

.t16/19 results to Nast db

Nastran.f06,.f04, .log files

error messages

Yes

Yes

Yes

generate std xdb,op2,f06

.sts, etc (these will bedeleted later by Nastranif marccpy = 1 or 3) -.sts

messages to .f06 and .log

and .log may be used byPatran to monitorthe progress of the jobwhile it is running

Note - every attempt will bemade to have the MSC Nastran InputFile Processor (IFP) catch allinput format errors. However,this may not be possiblein all cases and sometimes it is necessary for youto examine the Marc .dat files

No

No

No

for errors.

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This User’s GuideThis manual provides a complete background to SOL 600 and fully describes using SOL 600 within the MSC Nastran environment. The theoretical aspects of nonlinear analysis methods, types, and techniques are included as well as thorough descriptions for nonlinear material models.

Where appropriate, actual Patran forms and menus are shown so you can easily use SOL 600 from the Patran environment. They have not been updated for this edition, but should be very similar as most of the inputs have not changed.

Other MSC Nastran Documentation for SOL 600

MSC Nastran Quick Reference Guide (QRG)

The QRG contains a complete description of all the input entries for MSC Nastran. Within each section, entries are organized alphabetically so they are easy to find. Each entry provides a description, formats, examples, details on options, and general remarks.

You will find the full descriptions for all SOL 600 input entries in the QRG.

MSC Nastran Linear Static Analysis User’s Guide

The Linear Static Analysis User’s Guide provides support information on the basic use of MSC Nastran which can be applied to using SOL 600.

MSC Nastran Reference Manual

The MSC Nastran Reference Manual provides supporting information that relates to the theory of MSC Nastran inputs, element libraries, and loads and boundary conditions.

Marc DocumentationMSC provides extensive documentation covering all aspects of the Marc code. In particular the following manuals are recommended to use in conjunction with SOL 600:

• Marc Volume A: Theory and User Information - explains the capabilities of Marc and gives pertinent background information.

• Marc Volume B: Theory and Information on the extensive Marc element library

• Marc Volume C: Program Input - describes the file format of the Marc input file.\

• Marc Volume D: User subroutines and special routines - describes format for user subroutines.


Patran DocumentationThree key books from the Patran library may be of assistance in running SOL 600:

• Patran User’s Guide - this introductory guide gives you the essential information you need to immediately begin using Patran for SOL 600 projects. Understanding and using the information in this guide requires no prior experience with CAE or finite element analysis.

• Patran Reference Manual -a counterpart to the MSC Nastran Reference Manual, this manual provides complete descriptions of basic functions in Patran, geometry modeling, finite element modeling, material models, element properties, loads and boundary conditions, analysis, and results.

• MSC Nastran Preference Guide - gives specific information that relates to using Patran with MSC Nastran as the intended analysis code. All application forms and required input are tailored to MSC Nastran.

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Chapter 2: MSC Nastran Bulk Data File and Results Files

2 MSC Nastran Bulk Data File and Results Files

The MSC Nastran Bulk Data File

Output Requests

Results Files

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The MSC Nastran Bulk Data FileThe MSC Nastran Input File, referred to here as the Bulk Data File (BDF), (or .dat in the MSC Nastran manuals), is made up of three distinct sections:

1. Executive Control - describes the problem or solution type and optional file management.

2. Case Control - defines the load history and output requests.

3. Bulk Data - defines a detailed model, load and constraint description.

Input data is organized in (optional) blocks. Key words identify the data for each optional block. This form of input enables you to specify only the data for the optional blocks that you need to define your problem. The various blocks of input are “optional” in the sense that many have built-in default values which can be used in the absence of any explicit input from you.

A typical input file setup for the MSC Nastran program is shown below.

• Executive Control StatementsTerminated by an CEND parameter

• Case Control Commands Terminated by the BEGIN BULK option

• Bulk Data EntriesModel data starting with the BEGIN BULK option and terminated by the ENDDATA option

IFP (Input File Processing) Checking

Checking of MSC Nastran Bulk Data entries are done during IFP. When one of these entries has erroneous data entered it is more likely that IFP will flag the entry and issue a FATAL ERROR. In most cases, IFP error checking has been enhanced to point to the field and continuation line where the erroneous data occurs.

Checking of most SOL 600 only Bulk Data entries is done in Marc and any erroneous data entries will be listed later in the output.

17CHAPTER 2MSC Nastran Bulk Data File and Results Files

Input ConventionsMSC Nastran Implicit Nonlinear performs all data conversion internally so that the system does not abort because of data errors made by you. The program reads all input data options alphanumerically and converts them to integer, floating point, or keywords, as necessary. MSC Nastran Implicit Nonlinear issues error messages and displays the illegal option image if it cannot interpret the option data field according to the specifications given in the manual. When such errors occur, the program attempts to scan the remainder of the data file and ends the run with a FATAL ERROR or SEVERE WARNING message.

Two input format conventions can be used: fixed and free format. You can mix fixed and free format options within a file.

The syntax rules for fixed fields are as follows:

• Give floating point numbers with or without an exponent. If you give an exponent, it must be preceded by the character E or D and must be right-justified. If data is double precision, a D must be used.

Bulk Data

Case Control

Executive Control

Element andMaterialProperties,Fixed Displ,

Load Incrementation,Applied Loads, Applied DisplacementsEtc.

Title, Job Control,Solution Sequence,Etc.

pC

om

ple

te In

put D

eck

Con

trol

Info

rmat

ion

Mod

el D

ata

- gr

ids,

elem

ents

, etc

.Etc.


18

The syntax rules for free fields are as follows. See the Format of Bulk Data Entries (p. 1047) in the MSC Nastran Quick Reference Guide for more details.

• Check that each option contains the same number of data items that it would contain under standard fixed-format control. This syntax rule allows you to mix fixed-field and free-field options in the data file because the number of options you need to input any data list are the same in both cases.

• Separate data items on a option with a comma. The comma can be surrounded by any number of blanks. Within the data item itself, no embedded blanks can appear.

• Give floating point numbers with or without an exponent. If you use an exponent, it must be preceded by the character E or D and must immediately follow the mantissa (no embedded blanks).

• Give keywords exactly as they are written in the manual.

• All data can be entered as uppercase or lowercase text.

• Small field format is limited to 8 columns per field. Large field is 16 columns.

DefaultsFor most bulk data entries, SOL 600 does not make the distinction between zero and blank. Thus, if a zero is entered and the default is some other value, the default will normally be used. If you wish to use zero, enter a small number such as 1.0E-12 instead.

Section Descriptions

Executive Control

This group of entries provides overall job control for the problem and sets up initial switches to control the flow of the program through the desired analysis. This set of input must be terminated with an CEND parameter. See Executive Control Statements (Ch. 3) in the MSC Nastran Quick Reference Guide for additional descriptions on input formats.

Case Control

This group of options provides the loads and constraints and load incrementation method and controls the program after the initial elastic analysis. Case Control options also include blocks which allow changes in the initial model specifications. Case Control options can also specify print-out and postprocessing options. Each set of load sets must be begin with a SUBCASE command and be terminated by another SUBCASE or a BEGIN BULK command. If there is only one load case, the SUBCASE entry is not required. The SUBCASE option requests that the program perform another increment or series of increments. See Case Control Command Descriptions (Ch. 4) in the MSC Nastran Quick Reference Guide for additional descriptions on input formats.


Bulk Data Entries

This set of data options enters the initial loading, geometry, and material data of the model and provides nodal point data, such as boundary conditions. Bulk data options are also used to govern the error control and restart capability.This group of options must be terminated with the ENDDATA option. See Bulk Data Entries (Ch. 8) in the MSC Nastran Quick Reference Guide for additional descriptions on input formats. Multiple BEGIN entries and superelements are not allowed in SOL 600.

ExampleThe following text illustrates a simple example of an MSC Nastran Implicit Nonlinear input file. It includes the required Executive Control, Case Control, and Bulk Data Sections that are required for any MSC Nastran analysis.

See Install.dir/Doc/pdf_nastran/user/implicit_nonlinear_examples/example_input_files/impnonsam.dat.

Listing 2-1 Sample Implicit Nonlinear Solutions 600 Input

$ Advanced Nonlinear AnalysisSOL 600,106 OUTR=F06CENDTITLE = MSC.Nastran job $ Direct Text Input for Global Case Control DataSUBCASE 1$ Subcase name : Default SUBTITLE=Default NLPARM = 1 SPC = 2 LOAD = 2 DISPLACEMENT(SORT1,REAL)=ALL SPCFORCES(SORT1,REAL)=ALL STRESS(SORT1,REAL,VONMISES,BILIN)=ALL$ Direct Text Input for this SubcaseBEGIN BULKPARAM PRTMAXIM YESNLPARM 1 10 AUTO 1 25 P YES$ Direct Text Input for Bulk Data$ Elements and Element Properties for region : shell_propsPSHELL 1 1 .253 1 1$ Pset: "shell_props" will be imported as: "pshell.1"CQUAD4 1 1 1 2 5 4CQUAD4 2 1 2 3 6 5CQUAD4 3 1 4 5 8 7CQUAD4 4 1 5 6 9 8$ Referenced Material Records$ Material Record : steel$ Description of Material MAT1 1 3.+7 .3 .0075$ Nodes of the Entire ModelGRID 1 0. 0. 0.GRID 2 5. 0. 0.GRID 3 10. 0. 0.GRID 4 0. 5. 0.GRID 5 5. 5. 0.


20

GRID 6 10. 5. 0.GRID 7 0. 10. 0.GRID 8 5. 10. 0.GRID 9 10. 10. 0.$ Loads for Load Case : DefaultSPCADD 2 1$ Displacement Constraints of Load Set : fix_edgeSPC1 1 123456 1 4 7$ Contact Table for Load Case: Default$ Nodal Forces of Load Set : point_loadFORCE 2 9 0 100. 0. 0. -1.$ Referenced Coordinate FramesENDDATA

Running Existing Nonlinear ModelsSome users may have existing models that have been developed and analyzed using MSC Nastran Nonlinear Solution Sequences 106 or 129. These models may be run through MSC Nastran Implicit Nonlinear (SOL 600) by changing the SOLUTION procedure input to MSC Nastran Implicit Nonlinear (SOL 600) input.

The following is an example of the change required to run existing models through SOL 600. The first line shows an existing MSC Nastran SOL 106 Executive Control Statement and the second shows its revision for MSC Nastran Implicit Nonlinear (SOL 600).

SOL 106SOL 600,106

SOL 600 Executive Control Statement:The executive control statement is as follows:

SOL 600, ID PATH= COPYR= NOERROR MARCEXE=SOLVE NOEXIT OUTR=op2,xdb,pch,f06,eig,dmap,beam, sdrc,pst,cdb=(0, 1, 2, or 3) STOP= CONTINUE= S67OPT= MSGMESH= SCRATCH= TSOLVE= SMEAR PREMGLUE MRENUELE= MRENUGRD= MRENUMBR= SYSabc= S6NEWS=

Some items such as dmap, beam, CONTINUE and S67OPT are explained here. See the SOL 600,ID (p. 142) in the MSC Nastran Quick Reference Guide for a complete discussion. An explanation of these items follows:

dmap The user will enter his own DMAP to create whatever type of output that is desired, such as op2, xdb, punch, f06. For all other options, DMAP as needed is generated internally by MSC Nastran.

beam The beam option must be specified if op2,xdb,pch. or f06 options are specified and beam internal loads are to be placed in any of these files. The beam and eig options are mutually exclusive (you cannot specify both).


CONTINUE=

CONTINUE= An option that specifies how MSC Nastran will continue its analysis after Marc finishes. To continue the analysis, do not enter any STOP or OUTR options. It is possible to perform more than one of these operations if necessary.

0 Nastran will continue the current solution sequence as normal. For example if SOL 600,106 is entered, SOL 106 will continue as normal after Marc finishes. Of course, no 3D contact or materials not supported by SOL 106 may be used.

1 Nastran will switch to SOL 107 to compute complex eigenvalues. Marc will generate DMIG matrices for friction stiffness (and possibly damping) on a file specified by pram,marcfil1,name and time specified by param,marcstif,time. This is accomplished by making a complete copy of the original Nastran input file and spawning off a new job with the SOL entry changed and an include entry for the DMIG file. The user must put CMETHOD and CEIG in the original Nastran input file.

6 Same as option 1 except SOL 110 is run. For this option, the original Nastran input file must contain METHOD=ID1 and CMETHOD=ID2 in the Case Control as well as matching EIGRL (or EIGR) and EIGC entries in the Bulk Data.

7 Same as option 1 except SOL 103 is run for real eigenvalues/eigenvectors. The database can be saved to restart into SOL 110 if desired. This should be done on the command line or in a rc file with scratch=no. For this situation, the original Nastran input file must include METHOD=id in the Case Control command and a matching EIGRL or EIGR entry in the Bulk Data. (CMETHOD and EIGC can also be included.) The actual restart from SOL 103 to 110 must be performed manually at the present time.

101+ Continue options 101 to 400 are used to convert Marc’s initial contact tying constraints to MPC’s and then continue in SOL 101 to 112 as a standard Nastran execution. For example, if CONTINUE=101, a SOL 101 run with all the geometry load cases, etc. from the original run would be conducted with the addition of the initial contact MPC determined from Marc. The continue=101+ options are frequency used to model dissimilar meshes as well as glued contact which does not change throughout the analysis. This option can be used for any standard Nastran sequence where the initial contact condition does not change. In order for initial contact to work, the surfaces must be initially touching. If they are separated by a gap, the MPC’s will be zero until the gap closes and thus the initial MPC’s are zero. This option automatically sets BCPARA INITCON=1.


22

An example of input using the continue=1 option is as follows:

SOL 600,106 path=1 stop=1 continue=1TIME 10000CEND ECHO = sort DISP(print,plot) = ALL STRESS(CORNER,plot) = ALL STRAIN(plot) = ALL SPC = 1 LOAD = 1 NLPARM = 1 cmethod=101BEGIN BULKparam,marcfil1,dmig002param,mrmtxnam,kaaxparam,mrspawn2,Nastranparam,mrrcfile,nast2.rcPARAM,AUTOSPC,YESPARAM,GRDPNT,0EIGC, 101, HESS, , , , ,50 NLPARM 1 10 AUTO 1 P YESPLOAD4 1 121 -800.PLOAD4 1 122 -800.

(rest of deck)

CQUAD4 239 2 271 272 293 292CQUAD4 240 2 272 273 294 293ENDDATA

See Install_dir/doc/pdf_nastran/implicit_nonlinear_examples/example_input_files/continu2.dat for the complete input file.


Critical items are Case Control command, CMETHOD=101, the four parameters after BEGIN BULK and Bulk Data entry, EIGC. An explanation of the parameters follows:

The flow of the run is as follows:

• Create a primary MSC Nastran SOL 600 input file (we will name it jid.dat for this example)

• Submit MSC Nastran in the standard fashion. For this example, the following command is used:

nastran jid rc=nast1.rc

The nast1.rc file contains items such as scratch=yes, memory=16mw, etc.

• The primary MSC Nastran run creates an Marc input file named jid.marc.dat

• The primary MSC Nastran run spawns Marc to perform nonlinear analysis.

• The nonlinear Marc analyses completes and generates standard files.

• Control of the process returns to MSC Nastran. A new MSC Nastran input file named jid.nast.dat will be created from the original input file. This file will contain the CMETHOD Case Control and CEIG commands, all of the original geometry and additional entries to read the dmig002 file.

• A second MSC Nastran job will be spawned from the primary MC.Nastran run using the command

nastran jid.nast rc=nast2.rc

The nast2.rc file can be the same as nast1.rc or can contain different items. Usually memory will need to be larger in nast2.rc than in nast1.rc.

• The second MSC Nastran run computes the complex eigenvalues and finishes.

• Control of the process returns to the primary MSC Nastran run and it finishes.

The first portion of the dmig002 file is as follows:

$2345678 2345678 2345678 2345678 2345678 2345678 2345678 2345678 234567812345DMIG KAAX 0 1 2 0 324DMIG* KAAX 6 1

PARAM,MARCFILi,DMIG002 This means that a file named dmig002 will be used. It contains stiffness matrix terms (possibly from a set of unsymmetric friction stiffness matrices)

PARAM,MRMTXNAM, Name,KAAX This means that in the dmig002 file, use DMIG matrix terms labeled kaax (or KAAX – case does not matter).

PARAM,MRSPAWN2, CMD,TRAN This means that the primary MSC Nastran run will spawn another MSC Nastran run to compute the complex eigenvalues. The name of the command is nastran.

PARAM,MRRCFILE, RCF,NAST2.RC This is the name of the rc file to be used for the second (spawned) MSC Nastran run.


24

* 6 1 3.014712042D+05* 6 2 4.204709763D+08*DMIG* KAAX 6 2* 6 1 1.204709763D+05* 6 2 3.014712042D+05*DMIG* KAAX 6 3* 6 1-4.616527206D+04* 6 2-4.616527206D+04* 6 3 1.308497299D+05DMIG* KAAX 17 1* 6 1 6.239021038D+04* 6 2-2.528344607D+03* 6 3-6.239758760D+03* 17 1 5.939989945D+05*

When the PATH keyword is omitted on the SOL 600 Executive statement, the program will search the following location to find Marc:

MSC_BASE/MSC_VERSD/marc/MSC_ARCHM/marc20xx/tools

If MSC_ARCHM does not exist, MSC_ARCH is used instead. The environmental variables MSC_BASE, MSC_VERSD, MSC_ARCH and/or MSC_ARCHM are set by the MSC Nastran script (see the MSC Nastran Installation and Operations Guide for further details). If Marc is not found on the above path, likely locations near that path are searched. If Marc is still not found, the job will terminate with an appropriate message and the user must determine the correct location of the Marc installation, use the PATH=1 keyword (see the SOL 600,ID (p. 142) in the MSC Nastran Quick Reference Guide for further details).

DMIG-OUT

The option DMIGOUT allows the stiffness, differential stiffness, damping, mass (assembled or element-by-element) and heat transfer matricies to be output for selected output times or at the end of each nonlinear subcase for use in other analyses. This is a less expensive procedure, than using the Bulk Data entry, MDMIOUT (which creates a superelement), but results in a much larger matrix, however both methods can produce very large files.

Restart from SOL 600 into SOL 103 or into Another Linear Solution SequenceFor the purpose of a prestressed normal modes analysis, the old way of restarting from SOL 106 into SOL 103 is no longer necessary; the user can, instead, restart from a SOL 600 run into another SOL 600 run to perform the prestressed normal modes calculation.

Restarts from SOL 600 into linear solution sequences are not recommended to the novice user because of several limitations. The results of the linear restart are incremental values with respect to the preload, not total values. However, some experienced users restart from SOL 600 into SOL 103 to perform


prestressed modal analysis with changing boundary conditions, or restart into another linear solution sequence to perform a perturbed linear solution on a preluded structure.

Generating and Editing the Bulk Data File in PatranPatran offers a MSC Nastran interface that provides a communication link between Patran and MSC Nastran. It provides for the generation of the MSC Nastran Input file as well as customization of certain features in Patran. The interface is a fully integrated part of the Patran system.

Generating the BDF

Selecting MSC Nastran as the analysis code preference in Patran, activates the customization process. These customizations ensure that sufficient and appropriate data is generated for the MSC Nastran interface. Specifically, the Patran forms in these main areas are modified:

• Materials

• Element Properties

• Finite Elements/MPCs and Meshing

• Loads and Boundary Conditions

• Analysis Forms

Using Patran, you can run a MSC Nastran analysis or you may generate the MSC Nastran Input File to run externally. For information on generating the MSC Nastran Input file from within Patran, see Analysis Form (Ch. 3) in the Patran Interface to MSC Nastran Preference Guide.

Editing the BDF

Once the Bulk Data File has been generated, you can edit the file directly from Patran.

1. Click the Analysis Application button to bring up Analysis Application form.

2. On the Analysis form set the Action>Object>Method combination to Analyze>Existing Deck>Full Run and click Edit Input File...

Patran finds the BDF with the current job name and displays the file for editing in a text editing window.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideOutput Requests

26

Output RequestsAs a part of the input, you can request which results quantities you want to be returned from Marc back to MSC Nastran and the formats of the results files.

MSC Nastran Implicit Nonlinear (SOL 600) produces stress and strain results that differ from those results available with SOL 106 and 129. A detailed discussion of the stress and strain measures for SOL 600 is given in the following section. For a complete listing of all possible results quantities that can be returned for a SOL 600 analysis, see Output from the Analysis (Ch. 8).

Any of the results quantities can be placed on Marc’s t16/t19 output files to be postprocessed by Patran. In addition, the more basic types of output (displacements, velocities, accelerations, Cauchy stress tensor and one type of strain tensor) and basic contact information, can be translated to MSC Nastrans’s standard op2, xdb, punch and even f06 files using the OUTR option described above. At present, new datablock definitions have not been created to handle all types of nonlinear output. Therefore, it is strongly suggested that the t16 or t19 file be selected for postprocessing in order to view all types of output. Patran can postprocess nearly all types of output selected by the MARCOUT entry. For a complete description of the outputs available using MARCOUT (p. 2045) in the MSC Nastran Quick Reference Guide. The default output for nodes or elements is ALL, this can be reduced by using the MT16SEL bulk data entry.

Deformations

Consider a three dimensional body in its undeformed and deformed configuration (see Figure 2-1).

With respect to a Cartesian coordinate system , the position vector of a material point in the

undeformed configuration is written as:

(2-1)

In the deformed configuration, the material point has a position vector , given by:

(2-2)

The displacement vector is defined as the difference between the position vector in the deformed and

the undeformed configuration and reads:

(2-3)

B

E1 E2 E3

X X1E1 X2E2 X3E3+ +=

x

x x1E1 x2E2 x3E3+ +=

u

u x X– u1E1 u2E2 u3E3+ += =


Figure 2-1 Body in Undeformed and Deformed Configuration

It will be assumed that there is always a unique relation between the position vector of a point in the deformed and the position vector of this point in the undeformed configuration. This can formally be expressed as:

(2-4)

Based on Equation (2-4), a fundamental deformation measure can be given, namely the deformation

gradient , which is defined by:

(2-5)

Substituting Equation (2-5) into Equation (2-3) shows that the deformation gradient can also be written as a function of the coordinates in the undeformed configuration and the displacement components:

E1

E3

E2

X

x

u

B B Deformed

Undeformed

dFn

dA

N

dA0

d^F

B

x x X =

F

F

x1X1

---------x1X2

---------x1X3

---------

x2X1

---------x2X2

---------x2X3

---------

x3X1

---------x3X2

---------x3X3

---------

=


28

(2-6)

in which is the 3x3 unit tensor:

(2-7)

Starting out from the deformation gradient, several well-known symmetric strain tensors can be defined,

namely the engineering strain tensor , the Green-Lagrange strain tensor and the right Cauchy-

Green strain tensor :

(2-8)

(2-9)

(2-10)

where denotes the transpose of .

Notice that the Green-Lagrange and the right Cauchy-Green strain tensor are related by:

(2-11)

Example

Suppose that the deformation of a body is described by:

, ,

This deformation can be obtained by first stretching a block of material in the -direction and then

rotating it around the -axis (see Figure 2-2). The deformation gradient can easily be evaluated as:

F

1u1X1

---------+u1X2

---------u1X3

---------

u2X1

--------- 1u2X2

---------+u2X3

---------

u3X1

---------u3X2

--------- 1u3X3

---------+

I

u1X1

---------u1X2

---------u1X3

---------

u2X1

---------u2X2

---------u2X3

---------

u3X1

---------u3X2

---------u3X3

---------

+= =

I

I1 0 0

0 1 0

0 0 1

=

e E

C

e12--- F F

T2I–+ =

E12--- F

TF I– =

C FTF=

FT F

E12--- C I– =

x1 4X1 cos12---X2 sin–= x2 4X1 sin

12---X2 cos+= x3

12---X3=

E1

E3


so that the engineering and the right Cauchy-Green strain tensors are given by:

,

From these expressions, it can be concluded that the engineering strain tensor only provides a useful

deformation measure if the angle remains small, so that and . On the other hand, the components of the right Cauchy-Green tensor, and by virtue of Equation (2-11) also the components

of the Green-Lagrange strain tensor, are independent of the value of the angle .

The deformation gradient can be rewritten as:

Figure 2-2 Stretching and Rotating a Body

F

4 cos 12---– sin 0

4 sin12--- cos 0

0 012---

=

e

4 cos 1–72--- sin 0

72--- sin

12--- 1–cos 0

0 0 12---–

= C

16 0 0

014--- 0

0 014---

=

cos 1 sin 0

E1

E2

E3

L3

L2

L1

L1 L1+

L2 L2+

L3 L3+


30

in which is a rotation tensor and is a symmetric stretch tensor, where the stretch tensor and the right Cauchy-Green strain tensor are related by:

It can be proved that in this way any deformation gradient can be uniquely decomposed into a rotation tensor and a stretch tensor.

If there is no rotation of the material the non-zero components of the right Cauchy-Green strain tensor can be expressed in terms of the components of the engineering strain tensor as:

, ,

Instead of , , and , one often uses the principal stretch ratios , and ,

respectively.

A geometrical interpretation of the principal stretch ratios can be given by indicating the initial edge lengths as , , and the changes in edge lengths as , , (see Figure 2-2). Now the

principal stretch ratios can be written as:

, ,

In the example discussed above, the right Cauchy-Green strain tensor only has non-zero terms on its main diagonal, indicating that the deformation consists of a pure stretch. In a general state of deformation, there will also be non-zero off-diagonal terms. Then the principal stretch ratios must be determined based on the eigenvalues of the right Cauchy-Green strain tensor. Denoting these eigenvalues as , , and

, the principal stretch ratios are generally given by:

, , (2-12)

It can be concluded that the principal stretch ratios completely define the stretch of a material, but not the rotation.

Fcos sin– 0

sin cos 0

0 0 1

4 0 0

012--- 0

0 012---

RU= =

R U

C U1 2

=

C11 1 e11+= C22 1 e22+= C33 1 e33+=

1 e11+ 1 e22+ 1 e33+ 1 2 3

L1 L2 L3 L1 L2 L3

1

L1 L1+

L1-----------------------= 2

L2 L2+

L2-----------------------= 3

L3 L3+

L3-----------------------=

C'11 C'22

C'33

1 C'11= 2 C'22= 3 C'33=


Another way to characterize the deformation of a material is based on the invariants of the right Cauchy-Green strain tensor. These invariants are defined as:

(2-13)

(2-14)

(2-15)

Because , , and are invariants of the right Cauchy-Green strain tensor, their values can also be

determined based on the eigenvalues of the right Cauchy-Green strain tensor. Using Equation (2-12), this yields:

(2-16)

(2-17)

(2-18)

It should be noted that incompressibility of the material can be expressed as:

(2-19)

or:

(2-20)

The compressibility can also be expressed in terms of the determinant of the deformation gradient,

. Since , this can be evaluated as:

(2-21)

so that incompressibility of the material yields:

(2-22)

I1 C11 C22 C33+ +=

I2 C11C22 C22C33 C33C11 C122

C232

C312

–––+ +=

I3 C11C22C33 2C12C23C31 C11C232

–+ +=

C22C312

C33C122

––

I1 I2 I3

I1 12 2

2 32

+ +=

I2 122

2 223

2 321

2+ +=

I3 122

232

=

123 1=

I3 1=

det F F RU=

det F det RU det R det U det U det C12---

123= = = = =

det F 1=


32

Stresses

Consider the deformed configuration of body , as indicated in Figure 2-4. On an elemental area

with unit normal vector , an elemental force vector is acting. This force vector is a result of forces

being transmitted from one portion of the body to another. According to the Cauchy stress principle, the stress vector or traction vector is defined as:

(2-23)

Similar to Equation (2-1) to Equation (2-3), the components of , , and are indicated as , , ,

, , , , and . Now the following relation between the components of the stress vector

and the components of the normal vector can be given:

(2-24)

which, by virtue of Equation (2-23), can also be written as:

(2-25)

In Equation (2-24) and Equation (2-25), to are the components of the true or Cauchy stress tensor

. The components , and are called the normal or direct stress components, while the other

components are called shear stress components. The first index of the stress components defines the normal of the plane on which the stress vector acts. The second index indicates the positive direction of the component (see Figure 2-3). It can be shown that the Cauchy stress tensor is symmetric, so ,

and . The physical meaning of the Cauchy stress tensor is that it gives the current

force per unit deformed area.

Another frequently used stress tensor in a large deformation analysis is the second Piola-Kirchhoff stress

tensor. In order to define this tensor, the force vector is transformed using the inverse of the

deformation gradient :

B dA

n dF

t

tdF

dA---------=

t n dF t1 t2 t3

n1 n2 n3 dF1 dF2 dF3

t1

t2

t3

T11 T12 T13

T21 T22 T23

T31 T32 T33

n1

n2

n3

=

dF1

dF2

dF3

T11 T12 T13

T21 T22 T23

T31 T32 T33

n1

n2

n3

dA=

T11 T33

T T11 T22 T33

T12 T21=

T13 T31= T23 T32=

dF

F


Figure 2-3 Interpretation of Stress Components

(2-26)

Assuming that the transformed force vector acts on the elemental area with unit normal vector

in the undeformed configuration (see Figure 2-4), the components to of the symmetric second

Piola-Kirchhoff stress tensor are defined as:

(2-27)

The physical meaning of the second Piola-Kirchhoff stress tensor is not so clear. It can be considered to give the transformed current force per unit undeformed area.

Using the deformation gradient, the Cauchy stress tensor and the second Piola-Kirchhoff stress tensor can be related to another by:

(2-28)

(2-29)

T22

T23

T21

E3

E2E1

dF1

dF2

dF3

F1–

dF1

dF2

dF3

=

dFdA0

N S11 S33

S

dF1

dF2

dF3

S11 S12 S13

S21 S22 S23

S31 S32 S33

N1

N2

N3

dA0=

S det F F 1–T F

1– T

=

T1

det F -----------------FSF

T=


34

Notice that for small deformations and small rotations, , so the differences between the Cauchy stress tensor and the second Piola-Kirchhoff stress tensor vanish. In that case they reduce to the so-called

engineering stress tensor , which is known to give the force per unit undeformed area.

Example

Due to a uniaxial tensile load, the state of deformation of a body is assumed to be given by (see also Figure 2-4):

, ,

The force is assumed to be homogeneously distributed over the cross section in the - -plane.

Evaluating Equation (2-25) for the cases that , , and yields:

Figure 2-4 Uniaxially Loaded Body

F I

x1 4X1= x212---X2–= x3

12---– X3=

A E2 E3

n E1= n E2= n E3=

A0

A

F

F

A0

A

E1

E2E3

F

0

0

T11 T12 T13

T21 T22 T23

T31 T32 T33

1

0

0

A=

0

0

0

T11 T12 T13

T21 T22 T23

T31 T32 T33

0

1

0

A=

0

0

0

T11 T12 T13

T21 T22 T23

T31 T32 T33

0

0

1

A=


so that the only nonzero component of the Cauchy stress tensor is:

Because:

,

it follows from Equation (2-28) that the only non-zero component of the second Piola-Kirchhoff stress tensor is:

Upon rewriting the current cross sectional area in terms of the original cross-sectional area as

, the nonzero component of the second Piola-Kirchhoff stress tensor can also be written as:

in which is recognized as the engineering stress . The differences between the various stress

components can be summarized as:

,

Grid Point Force Balance and Element Strain Energy

Theory

In nonlinear analysis, the strain energy, , for an element is defined by integrating the specific energy rate, the inner product of strain rate and stress, over element volume and time

(2-30)

T11FA---=

F

4 0 0

0 12---– 0

0 0 12---–

= det F 1=

S11116------ F

A---=

A A0

A 14---A0=

S1114--- F

A0------=

F A0 11

T11 411= S1114---11=

E

E ·T Vd d

V

0

t

=


36

The integration over time leads to the following recursive formula using the trapezoidal rule

(2-31)

The steps and are converged solution steps.

By integrating Equation (2-31) over the element volume, we get

(2-32)

For computational convenience, MSC Nastran uses Equation (2-32) to calculate the element strain energy. The internal element forces are readily available in every step because they are needed for the force equilibrium. Note that temperature effects are included in the internal element forces.

When loads from temperature differences or element deformation are present, the default definition of element strain energy for linear elements differ from the definition for nonlinear elements. For linear elements, the element strain energy is defined as

(2-33)

where is the element load vector for temperature loads and element deformation. Equation (2-33)

assumes that the temperatures are constant within a subcase. For nonlinear elements, the definition of Equation (2-30) is used. In the case of linear material and geometry, Equation (2-30) becomes

stress tensor

strain rate

element volume

actual time in the load history

current load step

previous load step

strain energy increment

strain-increment

internal element forces

displacement increment from to

·

V

t

En 1+ En= En 1++ En=12--- n 1+

T n n 1++ Vd+

n 1+

n

E

n n 1+

En 1+ En=12---un 1+

Tfn fn 1++ +

f

u n n 1+

E12---u

TKeu= u

TPet–

Pet


(2-34)

Equation (2-34) assumes that the temperature varies linearly within a subcase. The user may request the definition of Equation (2-34) to be applied to linear elements by adding PARAM,XFLAG,2 to the input file. The default value for XFLAG is 0, meaning that linear elements will use the definition of Equation (2-33).

User Input

The output of grid point force balance and strain energy in nonlinear analysis is requested with the existing GPFORCE and ESE Case Control commands, respectively. The Case Control commands remain unchanged.

Example

The following Nastran input file represents a simplified model of a mechanical clutch that consists of springs, beams, rigid elements, and gap elements. A geometric nonlinear analysis in SOL 106 is performed. Both GPFORCE and ESE output requests are applied above all subcases. This is a good example to show the grid point force balance with both linear and nonlinear elements, that includes:

• applied loads

• element forces

• SPC forces

• MPC forces

The model is shown in Figure 2-5. For clarity, only the elements are displayed. See Install_dir/doc/pdf_nastran/implicit_nonlinear_examples/example_input_files/gpf005bnl.dat for the complete input file.

E 12---= u

TKeu 1

2---– u

TPet


38

Figure 2-5

The gap and beam elements are shown, the rigid elements are not displayed and the Spring elements are not visible in the figure because their connection points are coincident.

Listing 2-2 Grid Point Force Output

LOAD STEP = 2.00000E+00 G R I D P O I N T F O R C E B A L A N C E POINT-ID ELEMENT-ID SOURCE T1 T2 T3 R1 R2 R3 1 APP-LOAD 0.0 0.0 0.0 0.0 0.0 -2.400000E+02 1 F-OF-SPC 0.0 0.0 0.0 0.0 0.0 2.400000E+02 1 10001 BEAM 0.0 0.0 0.0 0.0 0.0 0.0 1 10002 BEAM 0.0 0.0 0.0 0.0 0.0 0.0 1 *TOTALS* 0.0 0.0 0.0 0.0 0.0 0.0 100 F-OF-SPC 7.056885E-24 -2.106726E-23 0.0 2.103107E-22 7.093074E-23 0.0 100 F-OF-MPC -7.056885E-24 2.106726E-23 0.0 -2.103107E-22 -7.093074E-23 2.492302E-24 100 3 ELAS2 0.0 0.0 0.0 0.0 0.0 6.229551E-25 100 *TOTALS* 0.0 0.0 0.0 0.0 0.0 3.115257E-24 101 F-OF-SPC 0.0 0.0 0.0 1.614713E-28 -1.654361E-24 0.0 101 101 BEAM -3.730332E-08 2.927822E-07 0.0 0.0 0.0 -5.301744E-09 101 2004 BEAM -3.896168E-07 -1.146324E-07 0.0 -1.614713E-28 1.654361E-24 -2.646978E-23 101 1 GAP 1.270000E-10 0.0 0.0 0.0 0.0 0.0 101 TOTALS* -4.267931E-07 1.781498E-07 0.0 0.0 0.0 -5.301744E-09 102 101 BEAM -2.731871E-08 -2.938820E-07 0.0 0.0 0.0 -2.468966E-09 102 102 BEAM -1.637746E-08 1.273060E-07 0.0 0.0 0.0 -4.135903E-25 102 2 GAP 1.270000E-10 0.0 0.0 0.0 0.0 0.0 102 *TOTALS* -4.356917E-08 -1.665760E-07 0.0 0.0 0.0 -2.468966E-09


Listing 2-3 Element Strain Energy Output

Remarks

• Linear and nonlinear elements can be mixed. The output for grid point force and element strain energy in nonlinear analysis follow the same format as in linear analysis.

• The reference system for the grid point force output is the grid point global coordinate system. The user cannot specify another output coordinate system. In nonlinear, the grid point forces are not aligned with element edges. Therefore, PARAM,NOELOF and PARAM,NOELOP are ignored in nonlinear analysis.

• In nonlinear analysis, the element strain energy must be calculated for each intermediate load step even if the output is requested only in the last load step. To save computations, the element strain energy is only calculated upon user request. The Case Control commands, GPFORCE must be present to activate grid point force output or element strain energy calculations and output.

LOAD STEP = 1.00000E+00

E L E M E N T S T R A I N E N E R G I E S

ELEMENT-TYPE = BEAM * TOTAL ENERGY OF ALL ELEMENTS IN PROBLEM = 3.069659E+03

SUBCASE 1 * TOTAL ENERGY OF ALL ELEMENTS IN SET -1 = 3.069659E+03

ELEMENT-ID STRAIN-ENERGY PERCENT OF TOTAL STRAIN-ENERGY-DENSITY

101 3.963134E+02 12.9107 1.858459E+00

102 2.560559E+01 0.8342 1.200740E-01

103 2.954497E+00 0.0962 1.385472E-02

104 2.831680E+00 0.0922 1.327879E-02

105 3.055125E+00 0.0995 1.432660E-02

106 2.984272E+00 0.0972 1.399434E-02

107 2.854425E+00 0.0930 1.338544E-02

108 2.822066E+00 0.0919 1.323370E-02

109 3.316133E+00 0.1080 6.220224E-03

110 9.060877E+00 0.2952 2.010873E-02

111 1.350183E+01 0.4398 5.651308E-02

112 1.904008E+01 0.6203 7.969392E-02

113 1.038940E+01 0.3385 4.348574E-02

121 7.140515E+01 2.3262 6.761059E-01

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideResults Files

40

Results FilesWhen a SOL 600 analysis has been completed successfully, a message file and a results file are created and saved. If you request that a print file be saved in addition to the standard results file, or if the analysis aborts prematurely due to an error, a print file is also saved.

Files Generated During the Analysis

Print Files

The print files jobname.f06 and jobname.marc.out contain a complete text output of solution information, including an input summary, solution diagnostics from each processor, a geometry summary, and results if requested.

Because of the potential size of the print file, certain information is optional. Instead of printing out a complete echo of the input deck, a summary can be printed. Stress and strain results, at the nodes of each element, can be printed or not as selected by the user included in the print file.

Analysis Results FilesThe analysis results file contains some all of the numerical results computed in the analysis. This file in MSC Nastran is designated as jobname.op2 or jobname.xdb. Because SOL 600 uses the Marc solver, a Marc results file is also available, designated jobname.t16/t19. If you are using Patran, the full set of stress and strain measures are available in the t16/t19 file while the more basic measures are available in the .op2 and .xdb files. The t19 file is an ASCII file. The t16 file is a binary file and can be moved and used on different platforms.

For more information, see Patran Reference Manual, Part 6: Results Postprocessing.

Message Files

The message files jobname.marc.sts and jobname.msg (if it is run from Patran) contain diagnostic error and warning messages output by MSC Nastran Implicit Nonlinear (SOL 600). The message file is the best way to immediately check an analysis for successful execution if the job is run from Patran. Otherwise, check JID.MARC.OUT and JID.f06.

SOL 600 has five levels of messages:

1. Informative messages.

2. Nonfatal warning message of something that could affect the results.

3. Severe warnings (similar to fatal errors).

4. Fatal errors (all occurrences will be found before aborting).


5. Immediately fatal errors:

• An example of a Level 1 message is a message that indicates that a new processor has begun execution. These messages provide job information.

• An example of a Level 2 message is one indicating that the aspect ratio is greater than 15. This may or may not be a serious problem.

• An example of a Level 3 message is a warning about a highly distorted element or a in Marc that is not in SOL 600.

• An example of a Level 4 message is the warning “undefined node used in rigid element.”

• An example of a Level 5 message is “Unable to open file” message. The job is immediately aborted.

Postprocessing with PatranThe results from an MSC Nastran Implicit Nonlinear Analysis can be read into and postprocessed using Patran. Typically you will get the most complete set of results (i.e. rigid contact body information such as reaction forces, etc.) if you use the .t16 or .t19 results options (see Output from the Analysis (Ch. 8) on how to select which output files will be created), but you can also postprocess using an .xdb or .op2 formatted file.

The Results application in Patran provides the capabilities for creating, modifying, deleting, posting, unposting and manipulating results visualization plots as well as viewing the finite element model. In addition, results can be derived, interpolated, extrapolated, transformed, and averaged in a variety of ways, all controllable by the user.

Control is provided for manipulating the color/range assignment and other attributes for plot tools, and for controlling and creating animations of static and transient results.

Results are selected from the database and assigned to plot tools using simple forms. Results transformations are provided to derive scalars from vectors and tensors as well as to derive vectors from tensors. This allows for a wide variety of visualization tools to be used with all of the available results.

If the job was created within Patran such that a Patran jobname of the same name as the Nastran jobname exists, you only need to use the Results tools and Patran will import or attach the jobname.xxx file without you having to select it. If you did not create the job in Patran you can still import the model and results and postprocess (both are on the t16 file).

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideResults Files

42

Chapter 3: Solution Methods and Strategies in Nonlinear Analysis

3 Solution Methods and Strategies in Nonlinear Analysis

Introduction

Linear Static Analysis Procedure

Differences Between Linear and Nonlinear Analysis

Applying Constraints

Adding Nonlinear Effects

Numerical Methods in Solving Equations

Iteration Methods

Load Increment Size

Convergence Controls

Singularity Ratio

Guidelines for Analysis Methods

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideIntroduction

44

IntroductionThe finite element method is a powerful tool for analyzing complex problems in structural and continuum mechanics. The analysis of a structure using the finite element method has four basic steps:

1. Modeling, in which the structure is subdivided into an assemblage of discrete volumes called finite elements, and properties are assigned to each element.

2. Evaluation of element characteristics, such as stiffness and mass matrices, followed by assembling the element characteristic matrices to obtain the assembled or so-called “global” matrices characteristic of the entire structure. A similar process is followed to obtain the total loads, in vector form, applied to the structure.

3. Solution of the system equations for displacements, natural frequencies and mode shapes, or buckling load factors.

4. Calculating other quantities of interest, such as strains, stresses and strain energy.

MSC Nastran Implicit Nonlinear uses the finite element displacement method, in which a large system of equations is solved to obtain the displacements at all node points of the structure. Strains are then obtained on the element level as derivatives of displacements and stresses are obtained by multiplying a small matrix of material constants by the strains. Comprehensive presentations of the finite element method together with numerous applications are available in textbooks and the research literature.

The main purposes of this chapter are more limited, namely:

1. To give a brief overview of the finite element displacement method for solving linear and nonlinear structural problems in statics.

2. To describe the theory, techniques and algorithms specifically used in MSC Nastran Implicit Nonlinear (SOL 600).

3. Outline some guidelines for selecting appropriate analysis methods.

45CHAPTER 3Solution Methods and Strategies in Nonlinear Analysis

Linear Static Analysis ProcedureIn a linear static analysis there is assumed to be a linear relationship between the applied loads and the response of the structure. Because of the linear relationship you need only calculate the stiffness of the structure once. From this stiffness representation you can find the structure’s response to other applied loads by multiplying the load vectors by the decomposed stiffness matrix. In addition, loads can be combined using the principle of superposition.

A linear analysis is the simplest and most cost effective type of analysis to perform. Because linear analysis is simple and inexpensive to perform and often gives satisfactory results, it is the most commonly used structural analysis. Nonlinearities due to material, geometry, or boundary conditions are not included in this type of analysis. The behavior of an isotropic, linear, elastic material can be defined by two material constants: Young’s modulus, and Poisson’s ratio.

In actuality, linear analysis is merely an approximation to the true behavior of a structure. In some cases the approximation is very close to the true behavior, in other cases linear analysis may provide highly inaccurate results.

The following is a summary of the main steps in a linear static analysis:

1. Input: The problem geometry (nodes and elements), physical and material properties, and loads and boundary conditions are taken from the MSC Nastran Implicit Nonlinear input file and put into the MSC Nastran Implicit Nonlinear database.

2. Bandwidth: Minimization (Optional). The nodes are renumbered internally for minimum bandwidth.

3. Element stiffness matrix and force vector calculation: The element stiffness matrices and equivalent nodal forces for distributed forces are computed. The detailed descriptions in the Element Library (Ch. 11) provide the kinds of forces that each element can support.

4. Global stiffness matrix and load vector assembly: The global stiffness matrix and the combined nodal force vectors are assembled. Boundary and constraint conditions are incorporated by modifying the element stiffness matrices and force vectors.

5. Solution of equations: The nodal displacement vector {} is computed by solving the system of simultaneous Equation (3-6).

6. Strain energy and reaction force calculation: The strain energy and reaction forces (unbalanced grid point forces) are computed using the displacement vector, the element stiffness matrices and the force vectors.

7. Stresses and strains calculation: The strains and stresses are computed at selected points for each element. See the Element Library (Ch. 11) for a detailed descriptions of the stress recovery points for the MSC Nastran Implicit Nonlinear elements.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideDifferences Between Linear and Nonlinear Analysis

46

Differences Between Linear and Nonlinear AnalysisNon-linear analysis is intrinsically a multi-increment load process where the applied loads and/or displacements are solved for, not in a single load increment but in a number of load increments. The multiple-step procedure is necessary for the FE code to update changing conditions in the model during the analysis. This situation is routinely encountered in non-linear analysis because the material properties and/or boundary conditions can change during the analysis e.g. with the onset of plasticity (material non-linearity), or with the occurrence of contact (BC nonlinearity). Below are the steps in a general linear and a non-linear analysis. The presence of an extra loop of iterations (Newton-Raphson iterations) is the unique feature of a nonlinear solution procedure.

A. Steps in Linear Analysis:

1. Set up the model (done by user, before the model is submitted)

• Mesh the part

• Apply Material Properties

• Apply Boundary Conditions

• Submit Job

2. Job Solution (done by FE Code)

• Assembly of stiffness Matrix

• Solution of stiffness matrix

• Compute displacements, strains, stresses (and other results)

3. View Results

The user is guaranteed a solution if the boundary conditions and material properties are set up correctly (and sometimes even incorrectly!). The stiffness matrix is assembled and solved only once in the entire analysis.

B. Steps in Nonlinear Analysis:

1. Set up the model (done by the user, before model set-up)

• Mesh the part

• Apply Material Properties

• Apply Boundary Conditions

• Submit Job

2. Job Solution (done by FE Code):

Newton-Raphson Iteration scheme begins: Apply a portion of the total load to start: (1% in this case):

• Assembly of stiffness Matrix

• Solution of the Stiffness Matrix

• Check for convergence (IMPORTANT step, seen only in non-linear analysis)


If converged, the solution/structure is in equilibrium. Go to step 3 below

If not converged, update information and re-assemble, re-solve stiffness matrix

Keep iterating till convergence is achieved.

3. (After convergence) Get displacements, strains, stresses

4. Apply the next increment of load and go to Step 2. Keep doing this until all the load is applied

5. View Results

The important point to note is that the total load is applied gradually in steps (or increments) and for each load step, the solution is arrived at after one or more iterations. If the behavior of the model is generally linear, few iterations are required to solve that load step. If the model behavior is complex/nonlinear, many iterations might be required. Each iteration involves an assembly and solution of the stiffness matrix. Hence, nonlinear problems inherently take longer than linear models (of the same size) to solve. At the end of each iteration, a check is made to see if the solution has converged. If the convergence check fails, the iteration is re-repeated with the new information; and it is re-assembled and re-solved. This process repeats until convergence is achieved. Following that, the next increment of load is applied. The load increments are applied until the full load of the model is solved.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideApplying Constraints

48

Applying ConstraintsOnce you have constructed a model of your structure, constraints are added that force selected portions of your model to remain fixed or to move by a specified amount. These constraints can be either:

• Single Point Constraints

• Multipoint Constraints

Single Degrees of FreedomA constraint on a single degree of freedom (Single Point Constraint, or SPC) assigns a zero or nonzero value to a single degree of freedom. It can be expressed as:

(3-1)

where is the value of the prescribed displacement on the degree of freedom . The case of u = 0 is the

most common case, and is often used as a boundary condition, to “fix” or “ground” the movement of a point in a certain direction.

Since the value of is known, one could, in principle, eliminate the specified degree of freedom from

the other degrees of freedom to be solved for as unknowns. This would reduce the size of the system of equations to be solved, but on the other hand it would take time to perform the elimination, and this approach adds complexity to the code.

MSC Nastran Implicit Nonlinear uses a different technique. A number which is large compared to the

stiffness coefficients (say, for discussion, ) is added to the diagonal term of the equation for the

degree of freedom to be constrained. Also, if the degree of freedom is to be constrained to a nonzero value

u, then is added to the right hand side of the modified equation. This modified equation is now:

Assuming all to be small with respect to , the solution of the system of equations is obtained with

negligible error.

The modified system of equations remain well conditioned. The value used by MSC Nastran Implicit

Nonlinear for the large number is times the largest stiffness coefficient found on the diagonal of the stiffness matrix.

Multiple Degrees of FreedomA multipoint constraint (MPC) equation is a relationship between several degrees of freedom that must be enforced on the structure.

i u=

u i

i

1020

K11

u 1020

Ki11 Kii 1020i Kinn+ + + + Fi u 1020 +=

Kij 1020

1010


Some examples are as follows:

Consider the structure in Figure 3-1, made up of three plane stress elements. To make node 4 lie exactly on the straight line between nodes 2 and 7, we need to enforce the two MPC equations.

UX(4) = 0.5 x UX(2) + 0.5 x UX(7)

UY(4) = 0.5 x UY(2) + 0.5 x UY(7)

Figure 3-1 Multipoint Equations to Enforce Compatibility of Node 4 Along Line Connecting Nodes 2 and 7

To connect a plate or beam element (which has six degrees of freedom per node — three displacements and three rotations) to a solid element (which has only three translations), it is necessary to relate the rotation of the plate to the displacements of the solid. As an example, consider Figure 3-2 where we see the side view of a plate-to-solid transition. The equation:

where DELTAY is the difference in Y-coordinate between nodes 2 and 6, will enforce the desired compatibility on the rotation about the global Z axis. Other constraints would need to be written to ensure the compatibility of the other displacements and rotations.

Figure 3-2 Side View of a Solid-to-Plate Transition

Compatibility requires that the rotations of the plate element be related to the displacements on the top and bottom of the solid element.

Consider Figure 3-3, showing a rigid link connecting two nodes M and S, each of which has six degrees of freedom and is attached to other elements in the structure.

Y

321

876

1

2

3

4 5X

ROTZ 3 UX 2 UX 6 – DELTAY =

Y

3

21

6

4

5

XZ


50

The displacements at the dependent node S are related to those at the independent node M by the following relationship:

and

where [I] is a 3 x 3 unit matrix and [H] is given by

Figure 3-3 Example of a “Rigid Link” in the Model

If both nodes have six degrees of freedom, then multipoint constraint equations can be written to ensure that, for small displacements, the two nodes move as a rigid body.

Incorporating MPC Equations

Various techniques can be used to incorporate MPC equations into the equilibrium equations. Two of the methods are as follows:

UXs

UYs

UZs UXm

UYm

UZm

H

ROTXm

ROTYm

ROTZm

+=

ROTXs

ROTYs

ROTZs

I

ROTXm

ROTYm

ROTZm

=

H

0

Zm Zs–

Ys Ym–

Zs Zm–

0

Xm Xs–

Ym Ys–

Xs Xm–

0

=

Y

X

Z

S (XS, YS, ZS)

M (XM, YM, ZM)

M = Independent (master) node

S = Dependent (slave) node


Lagrange Multiplier Method. This method uses the mathematical technique of Lagrange multipliers to enforce the constraints. The number of degrees of freedom is increased, and the conditioning and definiteness of the global matrix may be adversely affected.

Transformation Method. Using the constraint equations, this method eliminates dependent variables from the list of unknowns, modifying the stiffness associated with the independent degrees-of-freedom to account for the constraints.

The following describes the multipoint constraint algorithm used in MSC Nastran Implicit Nonlinear.

First, we rewrite the constraint equations in matrix form as

(3-2)

where are the dependent (“slave”) variables and are the independent (“master”) variables.

Then, the global matrix of Equation (3-6) can be rewritten in partitioned form as

(3-3)

where the subscript i refers to those degrees of freedom which are not referenced in any constraint equation.

Substituting from Equation (3-2) into Equation (3-3) and rearranging, we get

(3-4)

and

(3-5)

Premultiplying Equation (3-5) by , we get

(3-6)

s A m–=

s m

Kss Ksm Ksi

Kms Kmm Kmi

Kis Kim Kii

s

m

i Fs

Fm

Fi

=

s

Kmm Kmi

Kim Kii

KmsA 0

KisA 0–

m

i Fm

Fi

=

K smKsi Ks sA 0 – m

i

Fs =

A 0 T

ATKsm ATKsi

0 0

ATKssA 0

0 0–

m

i ATFs

0

=


52

Now subtracting Equation (3-6) from Equation (3-4):

(3-7)

This symmetric Equation (3-7) gives the reduced equations, which can be solved for the variables {}m,{}i. Values of {}s can then be recovered from Equation (3-2). However, this needs

rearrangement of the coefficients in the stiffness matrix. Hence, we do some more manipulation.

First, we combine Equation (3-2) with Equation (3-7):

(3-8)

Then we restore symmetry to Equation (3-8) by first premultiplying the first equation of Equation (3-8)

by and adding the resulting equation to the second part of Equation (3-8), and then,

premultiplying the first equation of Equation (3-8) by . This results in

(3-9)

If we solve this modified Equation (3-9), which is symmetric, the resulting solution vector will satisfy the constraint equations.

For large problems, the modifications implied by Equation (3-9) will take a substantial amount of computer time, if performed on the global stiffness matrix.

Kmm Kmi

Kim Kii

KmsA 0

KisA 0– ATKsm ATKsi

0 0– ATKssA 0

0 0+

m

i Fm ATFs–

Fi

=

I 0

0

A

Kmm KmsA– ATKsm ATKssA+–

Kim KisA–

0

Kmi ATKsi–

Kii

s

m

i 0

Fm ATFs–

Fi

=

A TKss

Kss

Kss

ATKss

0

KssA

Kmm KmsA– ATKsm 2ATKssA+–

Kim KisA–

0

Kmi ATKsi–

Kii

s

m

i 0

Fm ATFs–

Fi

=


Adding Nonlinear EffectsLinear analysis is based on the following assumptions:

• the structure is only subjected to small displacements when loads are applied.

• the materials in the structure exhibit a linear relationship between stress and strain.

• boundary conditions remain constant.

When these assumptions are violated, linear analysis is no longer valid and nonlinear effects must be introduced.

Nonlinear problems are classified into three broad categories: geometric nonlinearity, material nonlinearity, and boundary condition nonlinearity (contact).

Sources of Nonlinearity

Geometric Nonlinearity

Geometrically nonlinear problems involve large displacements; “large” means that the displacements invalidate the small displacement assumptions inherent in the equations of linear analysis. For example, consider a classical thin plate subject to a lateral load; if the deflection of the plate’s midplane is anything close to the thickness of the plate, then the displacement is considered large and a linear analysis is not applicable.

Another aspect of geometric nonlinear analysis involves follower forces. Consider a slender cantilever beam subject to an initially vertical end load. The load is sufficient to cause large displacements.

In the deformed shape plot, the load is no longer vertical; it has “followed” the structure to its deformed state. Capturing this behavior requires the iterative update techniques of nonlinear analysis.

For details on the finite element formulations for geometric nonlinearities, see Geometric Nonlinearities (Ch. 4).

Material Nonlinearity

Recall that linear analysis assumes a linear relationship between stress and strain.

Material nonlinear analysis solution sequences can be used to analyze problems in static analysis where the stress-strain relationship of the material is nonlinear. In addition, large strain situations can be analyzed. Examples of material nonlinearities include metal plasticity, materials such as soils and

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideAdding Nonlinear Effects

54

concrete, or rubbery materials (where the stress-strain relationship is nonlinear elastic). Various plasticity theories such as von Mises or Tresca (for metals), and Mohr-Coulomb or Drucker-Prager (for frictional materials such as soils or concrete) can be selected by the user. Three choices for the definition of subsequent yield surfaces are available in MSC Nastran Implicit Nonlinear. They are isotropic hardening, kinematic hardening, or combined isotropic and kinematic hardening. With such generality, most plastic material behavior, with or without the Bauschinger effect, can be modeled.

For details on the finite element formulations for material nonlinearities, see Material Nonlinearities (Ch. 4).

Contact

Contact problems exhibit nonlinear effects due to changes in boundary conditions. If there is a change in constraints due to contact during loading, the problem may be classified as a boundary nonlinear problem and would require CGAP elements or the BCONTACT, BCBODY, or BSURF option. The use of GAP elements is discouraged in SOL 600.

For details on the finite element formulations for boundary nonlinearities, see Nonlinear Boundary Conditions (Ch. 4).

Subcases, Load Increments, and IterationsIn a nonlinear analysis, loading is typically applied in subcases both to allow for the nonlinear behavior to occur in the numerical processing and to give you control over restarts if problems (divergence, excessive iterations, etc.) occur during the solution. The subcase structure in a nonlinear analysis differs from a typical linear analysis. In a nonlinear analysis, subcases are cumulative; that is, the loads and boundary conditions at the end of a subcase are the initial conditions for the next subcase. Superposition cannot be applied in nonlinear problems. In general, a different loading sequence (reordering of the subcases) requires a complete new analysis.

Function of the Subcase

In a nonlinear static analysis, you first determine the total value of loading to be applied at a particular stage of the analysis. This loading value is selected with the LOAD Case Control command specifying a load set ID that exists in the Bulk Data. In this case, the subcase functions as a type of landmark in the loading history. It may be an expected point or a point at which the nature of the loading changes (for example, first applying an internal pressure loading and then an axial loading on a cylinder). The subcase is a major partition of the loading history. The loading history should be divided into subcases since this provides you with more control over the solution and restart strategy.

Load Increments

In the loading history, the total change of loading applied during a subcase can be subdivided into smaller parts to allow the solution to converge. These subdivisions within a subcase are termed load increments. Load increments are specified in Patran on the Load Increments subform and are defined in the Bulk Data file by the NINC field on the NLPARM entry. Selecting a number of increments divides the total load change applied during the subcase into NINC equal parts for FIXED load incrementation, but only


provides the initial load increment size in the case of adaptive load incrementation. See NLPARM and NLAUTO Bulk Data entries for more details. Load increments may be saved for restart if desired.This is important because sometimes the solution does not converge during a subcase. If the loading is divided into increments and these values are saved to the database, the restart strategy can continue from a loading value closer to the problem value than having to go back to the previous subcase.

Iterations

In the incremental solution process, the unbalanced forces that occur during a load increment are reintroduced internally into the solution until the solution has converged. The process of redistributing the unbalanced force within a load increment is known as an iteration. The iteration is the lowest level of the solution process. Iterations continue within a load increment until the solution converges or any of the specified convergence parameters are exceeded.

Nonlinear Equation SolutionA linear finite element system is expressed as:

(3-10)

And a nonlinear system is expressed as:

(3-11)

where is the elastic stiffness matrix, is the tangent stiffness matrix in a nonlinear system, is

the displacement vector, is the applied load vector, and is the residual.

The linearized system is converted to a minimization problem expressed as:

(3-12)

For linear structural problems, this process can be considered as the minimization of the potential energy. The minimum is achieved when

(3-13)

The function decreases most rapidly in the direction of the negative gradient.

(3-14)

One method to solve both linear and nonlinear problems is to use iterations. The objective of the

iterative techniques is to minimize the function, , without decomposing the stiffness matrix. In the simplest methods,

(3-15)

Ku F=

KTu F R– r= =

K KT u

F r

u 1 2uT Ku u

TF–=

u K1–F=

u F Ku– r= =

uk 1+ uk krk+=

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideAdding Nonlinear Effects

56

where

(3-16)

The problem is that the gradient directions are too close, which results in poor convergence.

An improved method led to the conjugate gradient method, in which

(3-17)

(3-18)

The trick is to choose to be conjugate to . Hence, the name “conjugate gradient

methods. Note the elegance of these methods is that the solution may be obtained through a series of matrix multiplications and the stiffness matrix never needs to be decomposed.

Certain problems which are ill-conditioned can lead to poor convergence. The introduction of a preconditioner has been shown to improve convergence. The next key step is to choose an appropriate preconditioner which is both effective as well as computationally efficient. The easiest is to use the diagonal of the stiffness matrix. The incomplete Cholesky method has been shown to be very effective in reducing the number of required iterations.

k rkT

rk rkT Krk=

uk 1+ uk kPk+=

k PkT

rk 1– PkT KPk=

Pk K P1 P2 Pk 1–


SOL 600 Analysis Procedure Figure 3-4 is a diagram showing the flow sequence of the nonlinear solution sequence of SOL 600. This diagram shows the input phase, equivalent nodal load vector calculation, matrix assembly, matrix solution, stress recovery, and output phase. It also indicates load incrementation and iteration within a load increment.

Figure 3-4 MSC Nastran Advanced Nonlinear Flow Diagram

No

No

Yes

Yes

Equivalent NodalLoad Vector

Matrix Assembly

Input Phase:Read Input DataSpace Allocation

Data Check

Matrix Solution

Stress Recovery

Convergence

Output Phase

NextIncrement

Stop

IncrementalLoads

Tim

e S

tep

Loo

p

Iter

atio

n L

oop

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideNumerical Methods in Solving Equations

58

Numerical Methods in Solving EquationsThe finite element formulation leads to a set of linear equations. The solution is obtained through numerically decomposing the system or obtaining the solution using iterations. Because of the wide range of problems encountered with MSC Nastran Implicit Nonlinear, there are several solution procedures available.

Most analyses result in a system which is real, symmetric, and positive definite. While this is true for linear structural problems, assuming adequate boundary conditions, it is not true for all analyses. MSC Nastran Implicit Nonlinear has two main modes of solvers – direct and iterative. Each of these modes has two families of solvers, based upon the storage procedure. While all of these solvers can be used if there is adequate memory, only a subset uses spill logic for an out-of-core solution. Finally, there are classifications based upon nonsymmetric and complex systems. This is summarized below:

Direct MethodsTraditionally, the solution of a system of linear equations was accomplished using direct solution procedures, such as Cholesky decomposition and the Crout reduction method. These methods are usually reliable, in that they give accurate results for virtually all problems at a predictable cost. For positive definite systems, there are no computational difficulties. For poorly conditioned systems, however, the results can degenerate but the cost remains the same. The problem with these direct methods is that a large amount of memory (or disk space) is required, and the computational costs become very large.

Direct Profile

Iterative Sparse

Multifrontal Sparse(default) CASI

Solver Option 0 2 8 9

Real Symmetric Yes Yes Yes Yes

Real Nonsymmetric Yes No Yes No

Complex Symmetric Yes No Yes No

Complex nonsymmetric No No Yes No

Out-of-core Yes No Yes Yes

Possible problem with poorly conditioned systems

No Yes No Yes

Can be used in Parallel Yes Yes Yes No


Iterative MethodsIterative solvers are a viable alternative for the solution of large systems. These iterative methods are based on preconditioned conjugate gradient methods. The single biggest advantage of these iterative methods is that they allow the solution of very large systems at a reduced computational cost. This is true regardless of the hardware configuration. The disadvantage of these methods is that the solution time is dependent not only upon the size of the problem, but also the numerical conditioning of the system. A poorly conditioned system leads to slow convergence – hence increased computation costs.

When discussing iterative solvers, two related concepts are introduced: fractal dimension, and conditioning number. Both are mathematical concepts, although the fractal dimension is a simpler physical concept. The fractal dimension, the range of which is between 1 and 3, is a measure of the “chunkiness” of the system. For instance, a beam has a fractal dimension of 1, while a cube has a fractal dimension of 3.

The conditioning number is related to the ratio of the lowest to the highest eigenvalues of the system. This number is also related to the singularity ratio, which is reported in MSC Nastran Implicit Nonlinear output when using a direct solution procedure. In problems involving beams or shells, the conditioning number is typically small, because of large differences between the membrane and bending stiffnesses.

PreconditionersThe choice of preconditioner can substantially improve the conditioning of the system, which in turn

reduces the number of iterations required. While all positive definite systems with degrees of freedom

converges in iterations, a well conditioned system typically converges in less than the square root of

iterations.

The available preconditioners available in the sparse iterative solver are

The sparse iterative solver requires an error criteria to determine when convergence occurs. The default is to use an error criteria based upon the ratio between the residuals in the solution and the reaction force.

After obtaining the solution of the linear equations evaluate:

(3-19)

The residual from the solution procedure is:

(3-20)

Preconditioner Sparse

Diagonal Yes

Scaled Diagonal Yes

Incomplete Cholesky Yes

N

N

N

uc

KuC

FC

=

Res FA

FC

– FA

KuC

–= =


60

If the system is linear ( does not change) and exact numerics are preformed, then .

Because this is an iterative method the residual is nonzero, but reduces in size with further iterations. Convergence is obtained when

(3-21)

The tolerance is specified through the NLPARM, TSTEPNL, NLAUTO, and NLSTRAT options.

Iterative Solvers

In MSC Nastran Implicit Nonlinear, two iterative solvers are available: one using a sparse matrix technique and the other an element-based CASI technique. This method is advantageous for different classes of problems.

There exist certain types of analyses for which the iterative solver is not appropriate. These types include:

• Elastic analysis

• Explicit creep analysis

• Eigenvalue analysis

• Use of gap elements

Elastic or explicit creep analysis involves repeated solutions using different load vectors. When a direct solver is used, this is performed very efficiently using back substitution. However, when an iterative solver is used, the stiffness matrix is never decomposed, and the solution associated with a new load vector requires a complete re-solution.

The sparse iterative solver can exhibit poor convergence when shell elements or Herrmann incompressible elements used for hyperelastic analyses are present.

Storage Methods

In general, a system of linear equations with N unknown is represented by a matrix of size N by , or

variables. Fortunately, in finite element or finite difference analyses, the system is “banded” and not all of the entries need to be stored. This substantially reduces the memory (storage) requirements as well as the computational costs.

In the finite element method, additional zeroes often exist in the system, which results in a partially full bandwidth. Hence, the profile (or skyline) method of storage is advantageous. This profile storage method is used in MSC Nastran Implicit Nonlinear to store the stiffness matrix. When many zeroes exist within the bandwidth, the sparse storage methods can be quite advantageous. Such techniques do not store the zeroes, but require additional memory to store the locations of the nonzero values. You can determine the “sparsity” of the system (before decomposition) by examining the statements:

“Number of nodal entries excluding fill in” x“Number of nodal entries including fill in” y

K Res 0=

Res Reac TOL

N

N2


If the ratio ( ) is large, then the sparse matrix storage procedure is advantageous.

Nonsymmetric SystemsThe following analyses types result in nonsymmetric systems of equations:

• Inclusion of convective terms in heat transfer analysis

• Coriolis effects in transient dynamic analysis

• Fluid mechanics

• Soil analysis

• Follower force stiffness

• Frictional contact

The first three always result in a nonsymmetric system. The last three can be solved either fully using the nonsymmetric solver, or (approximately) using a symmetric solver. The nonsymmetric problem uses twice as much memory for storing the stiffness matrix. Approximations using the symmetric solver may require more iterations.

Specifying the Solution ProcedureSelection of the solution procedure is made through the solver related parameters (ISOLVER, ISYMM, NONPOS, MBYTE, MAXITER, PREVITER, PRECOND, CJTOL) on the NLSTRAT Bulk Data entry or the parameter MARCSOLV.

Other Factors Affecting Performance

.Multifrontal Solver Memory Reduction

In order to efficiently run large analyses using scratch files, the out-of-core behavior of the multi-frontal sparse solver (Marc Solver 8) has been updated. These updates include:

1. Adding functionality to use out-of-core assembly of the operator matrix.

2. Utilizing the RAM, which affects both the in-core and out-of-core assembly of the operator matrix, allocated for the solver more efficiently.

3. Rewriting the code applying the multi-point constraint equations such that the amount of scratch file access is tremendously reduced. This is active in solver by default. If needed, it can be switched off by using the parameter feature, 4900.

x y

Note: For very large analyses, it may be advantageous to set the third entry of the OOC parameter to 1, in which case the solver memory is also used to store some nodal vectors, so that the amount of RAM needed for the analysis is decreased considerably. This is activated using Bulk Data PARAM,MARCOOCC,2.


62

Large Models

Translator speed enhancements have been implemented for certain types of large models. These are not necessary for small or medium sized models but can be requested using the following parameters:

Rigid Element Use

The majority of SOL 600 models using MPC's, RBE2, RBE3, RBAR, RSPLINE, RSSCON and RTRPLT to run to completion without difficulty. But should your analysis exit with a Marc exit code 2011, have a very low singularity ratio, or experience convergence problems (for example Marc exit code 3015), then there are several potential workarounds that you can try to attempt to get the problem to run.

First, try the Bulk Data PARAM,MARMPCHK,3. This will cause Marc to attempt to rearrange these entities if possible. If that does not work, and RBE3’s are present, they can be changed to MPC’s using Bulk Data PARAM,MARCRBE3,0. If there are still problems, all rigid elements can be changed to stiff beams using bulk data parameters PARAM,MARCRIGD,1. If the model still does not run, check all rigid elements carefully and run the model using MSC Nastran SOL 101 and/or 106, replace contact with MPCs or springs between the surfaces, determine from the f06 file if any negative or very large terms on the main diagonal of the decomposed stiffness matrix exist, and add CELAS or SPC to ground for these degrees of freedom. If the MSC Nastran run is satisfactory but SOL 600 still fails, the only other solution is to remodel the rigid elements and MPC’s.

An alternative approach is to add the AUTOMSET option, which is triggered by PARAM,AUTOMSET,YES in the bulk data. The use of the AUTOMSET option will increase the simulation cost, so the MARCMPCHK,3 is preferred. If the PARAM,MARMPCHK is also included in the model, it will be ignored.

Improved Contact

MPC's and rigid elements combined with contact and/or the same node in more than one contact body sometimes caused the Marc portion of SOL 600 to fail. If Marc exit 2011 or convergence problems are encountered with such models, you should try optimized contact. To invoke optimized contact from MSC Nastran, set field 6 of each BCBODY entry with flexible contact to 2. In addition, set field 3 of each “SLAVE: continuation line (the next line after all lines with SLAVE) to 2. In turn, this sets Marc's CONTACT entry 4th datablock, 3rd field to 2 and each CONTACT TABLE 3rd datablock 8th entry to 2 respectively.

References

• For selecting the solution procedure in Patran, see Defining the Solution Type in Patran (Ch. 7).

• Detailed discussions and an example of optimized are provided in Chapter 8 of the Marc Theory and User Information Manual (Volume A of the Marc documentation) - see text before and after Figure 8-4.

PARAM,MSPEEDSE,1 speeds up element processing

PARAM,MSPEEDP4,1 speeds up PLOAD4 processing particularly for solids


Iteration MethodsMSC Nastran Implicit Nonlinear (SOL 600) offers four iterative procedures that are employed to solve the equilibrium problem at each load increment: Newton-Raphson, Modified Newton-Raphson, Newton-Raphson with strain correction, and a secant procedure.

Full Newton-Raphson AlgorithmThe basis of the Newton-Raphson method in structural analysis is the requirement that equilibrium must be satisfied. Consider the following set of equations:

(3-22)

where is the nodal-displacement vector, is the external nodal-load vector, is the internal

nodal-load vector (following from the internal stresses), and is the tangent-stiffness matrix. The internal nodal-load vector is obtained from the internal stresses as

(3-23)

In this set of equations, both and are functions of . In many cases, is also a function of (for

example, if follows from pressure loads, the nodal load vector is a function of the orientation of the structure). The equations suggest that use of the full Newton-Raphson method is appropriate.

Suppose that the last obtained approximate solution is termed , where indicates the iteration number. Equation (3-22) can then be written as

(3-24)

This equation is solved for and the next appropriate solution is obtained by

and (3-25)

Solution of this equation completes one iteration, and the process can be repeated. The subscript

denotes the increment number representing the state . Unless stated otherwise, the subscript

is dropped with all quantities referring to the current state.

The full Newton-Raphson method is the default in MSC Nastran Advanced Nonlinear (see Figure 3-5). The full Newton-Raphson method provides good results for most nonlinear problems, but is expensive for large, three-dimensional problems when the direct solver is used. The computational problem is less significant when the iterative solvers are used. It is also the best method for contact problems.

K u u F R u –=

u F R

K

R T vdV

elem=

R K u F u

F

ui i

K un 1+i 1– u

iF R un 1+

i 1– –=

ui

ui u

i 1– ui

+= un 1+i

un 1+i 1– u

i+=

n

t n=

n 1+

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideIteration Methods

64

Figure 3-5 Full Newton-Raphson

Modified Newton-Raphson AlgorithmThe modified Newton-Raphson method is similar to the full Newton-Raphson method, but does not reassemble the stiffness matrix in each iteration.

(3-26)

Figure 3-6 Modified Newton-Raphson

The process is computationally inexpensive because the tangent stiffness matrix is formed and decomposed once. From then on, each iteration requires only forming the right-hand side and a backward

u1

u1 u2 u3

r1

Solution Converged

Incremental Displacements

0

Fn + 1

Fn

Force

K u0 u

iF R u

i 1– –=

r1

u1Solution Converged


Force

Fn + 1

Fn

0 u1 u2 u5


substitution in the solution process. However, the convergence is only linear, and the potential for a very large number of iterations, or even nonconvergence, is quite high.

If contact or sudden material nonlinearities occur, reassembly cannot be avoided. The modified Newton-Raphson method is effective for large-scale, only mildly nonlinear problems. When the iterative solver is employed, simple back substitution is not possible, making this process ineffective. In such cases, the full Newton-Raphson method should be used instead.

If the load is applied incrementally, MSC Nastran Implicit Nonlinear recalculates the stiffness matrix at the start of each increment or at selected increments, as specified.

Strain Correction MethodThe strain correction method is a variant of the full Newton method. This method uses a linearized strain calculation, with the nonlinear portion of the strain increment applied as an initial strain increment in subsequent iterations and recycles. This method is appropriate for shell and beam problems in which rotations are large, but membrane stresses are small.

In such cases, rotation increments are usually much larger than the strain increments, and, hence, the

nonlinear terms can dominate the linear terms. After each displacement update, the new strains are

calculated from and which yield

This expression is linear except for the last term. Since the iteration procedures start with a fully linearized calculation of the displacement increments, the nonlinear contributions yield strain increments inconsistent with the calculated displacement increments in the first iteration. These errors give rise to either incorrect plasticity calculations (when using small strain plasticity method), or, in the case of elastic material behavior, yields erroneous stresses. These stresses, in their turn, have a dominant effect on the stiffness matrix for subsequent iterations or increments, which then causes the relatively poor performance.

The remedy to this problem is simple and effective. The linear and nonlinear part of the strain increments are calculated separately and only the linear part of

is used for calculation of the stresses. The nonlinear part

(3-27)

is used as an “initial strain” in the next iteration or increment, which contributes to the residual load vector defined by

Ei 1+

ui u = u

i

Ei 1+

Ei

=12--- u u u

iu u u

i u u + + + ++

El E

i=

12--- u u + u

iu u u

i+ ++

Enl

i 1+ 1

2---u u =

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideIteration Methods

66

(3-28)

This “strain correction” term is defined by

(3-29)

Since the displacement and strain increments are now calculated in a consistent way, the plasticity and/or equilibrium errors are greatly reduced. The performance of the strain correction method is not as good if the displacement increments are (almost) completely prescribed, which is not usually the case. Finally, note that the strain correction method can be considered as a Newton method in which a different stiffness matrix is used.

The Secant MethodThe Secant method used by MSC Nastran Implicit Nonlinear is based on the Davidon-rank one, quasi-Newton update. The Secant method is similar to the modified Newton-Raphson method in that the stiffness matrix is calculated only once per increment. The residual is modified to improve the rate of convergence. When the iterative solver is employed, simple back substitution is not possible, making this process ineffective. Use the full Newton-Raphson method instead.

Figure 3-7 Secant Newton

The quasi-Newton requirement is that a stiffness matrix for iteration could be found based on the

right-hand sides of iterations, and , as follows

(3-30)

RC X L

Enl

VdV0

=

K un 1+i

ui

F R un 1+i

– RC

–=

r1

Fn + 1

Fn

Force

u1

u1 u4


i

i i 1–

Kiu

iF R un 1+

i – F R un 1+

i 1– – – r

ir

i 1––= =


This problem does not uniquely determine . The Davidon-rank one update uses an additive form on

the inverse of the tangent stiffness matrix as follows:

(3-31)

Specifying the Iteration MethodSelection of the iteration method in MSC Nastran is made through the IKMETH parameter on the NLSTRAT Bulk Data entry and on the NLPARM entry.

References

• For selecting the iteration method in Patran, see Subcase Parameters (Ch. 7).

Ki

Ki

1–K

0 1– u

i 1–K

0 1–

ri

ri 1–

– – ui 1–

K0

1–r

ir

i 1–– –

T

ui 1–

K0

1–r

ir

i 1–– –

Tr

ir

i 1––

------------------------------------------------------------------------------------------------------------------------------------------------------+=

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideLoad Increment Size

68

Load Increment SizeSelecting a proper load step (time step) increment is an important aspect of a nonlinear solution scheme. Large steps often lead to many recycles per increment and, if the step is too large, it can lead to inaccuracies and nonconvergence. On the other hand, using too small steps is inefficient.

Fixed Load IncrementationWhen a fixed load stepping scheme is used, it is important to select an appropriate load step size that captures the loading history and allows for convergence within a reasonable number of recycles. For complex load histories, it is often necessary to break up the analysis into separate load cases with different step sizes. For fixed stepping, there is an option to have the load step automatically cut back in case of failure to obtain convergence. When an increment diverges, the intermediate deformations after each recycle can show large fluctuations and the final cause of program exit can be any of the following: maximum number of recycles reached (exit 3002), elements going inside out (exit 1005 or 1009) or, in a contact analysis, nodes sliding off a rigid contact body (exit 2400). These deformations are normally not visible as post results (there is a feature to allow for the intermediate results to be available on the post file, see the POST option). If the cutback feature is activated and one of these failures occurs, the state of the analysis at the end of the previous increment is restored from a copy kept in memory or disk, and the increment is subdivided into a number of subincrements. The step size is halved until convergence is obtained or the user-specified number of cutbacks has been performed. Once a subincrement is converged, the analysis continues to complete the rest of the original increment. No results are written to the post file during subincrementation. When the original increment is finished, the calculation continues to the next increment with the original increment count maintained. These issues are avoided by using the AUTO increment options (AUTO on the NLPARM entry).

Adaptive Load (AUTO) IncrementationIn many nonlinear analyses, it is useful to have MSC Nastran Implicit Nonlinear figure out the appropriate load step size automatically. The basic scheme for automatic load incrementation is NLAUTO which is appropriate for most applications and is the default in SOL 600. In addition, so-called “arc-length methods” are available which are designed for applications like post buckling and snap-through analysis.

NLAUTO Basic Load Incrementation Scheme

The scheme appropriate for most applications is NLAUTO (Marc AUTO STEP). The primary control of the load step is based upon the number of recycles needed to obtain convergence. There are a number of optional user-specified physical criteria that can be used to additionally control the load step, but they are rarely used or required. The NLAUTO defaults are appropriate for most models and the NLAUTO entry is not required. For the recycle based option, the user specifies a desired number of recycles. This number is used as a target value for the load stepping scheme. If the number of recycles needed to obtain convergence exceeds the desired number, the load step size is reduced, the recycle counter is reset to zero and the increment is performed again with the new load step. The factor with which the time step is cut back defaults to 1.2 and can be specified by you. The load step for the next increment is increased if the


number of recycles required in the current increment is less than the desired number. The same factor that is used for decreasing the time step is used for increasing it. The load step is never increased during an increment. In addition, the same type of cut-back feature for fixed load stepping, as described in Load Increments, 54, is available for this scheme as well.

There are some exceptions to the basic scheme outlined above. If an increment is consistently converging with the original load step and the number of recycles exceeds the desired number, the number of recycles is allowed to go beyond the desired number until convergence or up to the user specified maximum number. The time step is then decreased for the next increment. An increment is determined to be converging if the convergence ratio was decreasing in three previous recycles.

Special rules also apply in a contact analysis. For quasi-static problems, the NLAUTO option is designed to only use the automated penetration check option (see CONTACT option, 7th field of 2nd data block; option 3 is always used). Even if you flag the increment splitting penetration check option, MSC Nastran Implicit Nonlinear internally converts it to the automated penetration check. During the recycles, the contact status can keep changing (new nodes come in contact, nodes slide to new segments, separate etc.). Whenever the contact status changes during an increment, a new set of contact constraints are incorporated into the equilibrium equations and more recycles are necessary in order to find equilibrium. These extra recycles, which are solely due to contact changes, are not counted when the comparison is made to the desired number for determining if the load step needs to be decreased within the increment. Thus, only true Newton-Raphson iterations are taken into account. For the load step of the next increment, the accumulated number of recycles during the previous increment is used. This ensures that the time step is not increased when there are many changes in contact during the previous increment.

In addition to allowing MSC Nastran Implicit Nonlinear to use the number of recycles for automatically controlling the step size for NLAUTO, user-specified physical criteria can be used for controlling the step size. You can specify the maximum allowed incremental change within certain ranges for specific quantities during an increment. The quantities available are displacements, rotations, stresses, strains, strain energy, and temperature (in thermal or thermomechanically coupled analyses). These criteria can be utilized in two ways. By default, they are used as limits, which means that the load step is decreased if a criterion is violated during the current increment, but they do not influence the decision to change the load step for the next increment (that is, only the actual number of recycles versus desired number of recycles controls the load step for the next increment). The criteria can also be used as targets; in which case, they are used as the main means for controlling the time step for the current and next increments. If the calculated values of the criteria are higher than the user-specified values the time step is scaled down. If the obtained values for a converged increment are less than the user-specified, the time step is scaled up. The scale factor used is the ratio between the actual value and the target value and this factor is limited by user-specified minimum and maximum factors (defaults to 0.1 and 10 respectively). If this type of load step control is used together with the recycle based control, the time step can be reduced due to whichever criterion that is violated. The decision to increase the step size for the next increment is based upon the physical criteria.

In many analyses, it is convenient to obtain post file results at specified time intervals. This is naturally obtained with a fixed load stepping scheme but not with an automatic scheme. Traditionally, the post

output frequency is given as every nth increment. With the NLAUTO procedure, you can request post


70

output to be obtained at equally spaced time intervals. In this case, the time step is temporarily modified to exactly reach the time for output. The time step is then restored in the following increment.

The NLAUTO option also has an artificial damping feature available by default for structured statics analyses. If the time step is decreased to below the user-specified minimum time step, MSC Nastran Implicit Nonlinear normally stops with exit number 3015; but if the artificial damping feature is activated, the analysis is continued with a smaller time step. The solution is stabilized by adding a factored lumped mass matrix to the stiffness matrix and modifying the force vector consistently. This artificial stabilization is turned off once the time step increases above the minimum time step. If the feature is used, it might be useful to write post file results at fixed time interval; otherwise, many increments might appear on the post file even for a small time period. The critical parameter for this feature is the (artificial) mass density, is normally selected automatically by the program. Use of the artificial damping feature allows solution of many post-buckling problems without the need to use arc-length methods (see below).

The defaults of the NLAUTO option are carefully chosen to be adequate in a wide variety of applications. There are cases, however, when the settings may need to be modified. Assume that the default settings are used, which means that the recycle based control is active with an initial load of one per cent of the total. If the structure is weakly nonlinear, convergence is obtained in just a few recycles and the for successive increments get progressively larger. This can lead to problems if the initially weakly nonlinear structure suddenly exhibits stronger nonlinearities; for instance, occurrence of plasticity or parts coming into contact. Possible remedies to this problem include:

• decrease the time step scale factor from 1.2 to a smaller number so the step size does not grow so rapidly;

• use a physical criterion like maximum increment of displacements to limit the load step;

• use the maximum time step to limit large steps;

• decrease the desired and maximum number of recycles to make the scheme more prone to decrease the load step if more recycles are needed.

Another situation is if the structure is highly nonlinear and convergence is slow. In this case, it may be necessary to increase the desired number and maximum number of recycles. In general, there is a close connection between the convergence tolerances used and the desired number and maximum number of recycles. In some rare cases, it may be beneficial to use one or more physical criteria; for example, the increment of plastic strain as targets for controlling the load step.

Arc-Length (AUTO INCREMENT) Methods Formulation

The solution methods described above involve an iterative process to achieve equilibrium for a fixed increment of load. None of them have the ability to deal with problems involving snap-through and snap-back behavior except the NLAUTO method with artificial damping. An equilibrium path as shown in Figure 3-8 displays the features possibly involved.


Figure 3-8 Snap-through Behavior

The issue at hand is the existence of multiple displacement vectors, , for a given applied force vector,

. This method provides the means to ensure that the correct displacement vector is found. If you have a load controlled problem, the solution tends to jump from point 2 to 6 whenever the load increment after 2 is applied. If you have a displacement controlled problem, the solution tends to jump from 3 to 5 whenever the displacement increment after 3 is applied. Note that these problems appear essentially in quasi-static analyses. In dynamic analyses, the inertia forces help determine equilibrium in a snap-through problem.

Thus, in a quasi-static analysis sometimes it is impossible to find a converged solution for a particular load (or displacement increment):

This is illustrated in Figure 3-8 where both the phenomenon of snap-through (going from point 2 to 3) and snap-back (going from point 3 to 4) require a solution procedure which can handle these problems without going back along the same equilibrium curve.

As shown in Figure 3-9, assume that the solution is known at point A for load level . For arriving at

point B on the equilibrium curve, you either reduce the step size or adapt the load level in the iteration process. To achieve this end, the equilibrium equations are augmented with a constraint equation expressed typically as the norm of incremental displacements. Hence, this allows the load level to change from iteration to iteration until equilibrium is found.

u

5

4

3

26

F

Force

Displacements

u

F

n 1+ F nF– F=

nF


72

Figure 3-9 Intersection of Equilibrium Curve with Constraining Surface

The augmented equation, , describes the intersection of the equilibrium curve with an auxiliary

surface for a particular size of the path parameter :

(3-32)

Variations of the parameter moves the surface whose intersection with the equilibrium curve generates a sequence of points along the curve. The distance between two intersection points, denoted

with and , denoted by l is the so-called arc-length.

Linearization of equation Equation (3-33) around point A in Figure 3-9 yields:

(3-33)

where:

(3-34)

(3-35)

(3-36)

(3-37)

F

nF

n 1+ F

A

B

r

u

g

c u g

r u F R u – 0= =

c u g u – 0= =

r

0

K P

nT

n0

u

r–

r0–

=

Kru------ : P r

------= =

nT c

u------ : n0

c------= =

r F R–=

r0 g u –=


It can be noted that a standard Newton-Raphson solution procedure is obtained if the constraint condition is not imposed. The use of the constraint equation causes a loss of the banded system of equations which

would have been obtained if only the matrix was used. Instead of solving the set of equations iteratively, the block elimination process is applied.

Consider the residual at iteration to which the fraction of load level corresponds

(3-38)

The residual for some variation of load level, , becomes

(3-39)

which can be written as:

(3-40)

where (3-41)

and (3-42)

Notice that does not depend on the load level. The equation above essentially establishes the

influence of a change in the load level during one iteration on the change in displacement increment for that iteration. After one iteration is solved, this equation is used to determine the change in the load level such that the constraint is followed. There are several arc-length methods corresponding to different constraints.

Among them, the most well-known arc-length method is one proposed by Crisfield, in which the iterative solution in displacement space follows a spherical path centered around the beginning of the increment. This requirement is translated in the formula:

(3-43)

where l is the arc length. The above equation with the help of Equation (3-46) and Equation (3-25) is applied as:

(3-44)

The equation above is interpreted with and in the prediction phase while retaining the

full form of Equation (3-50) in the correction phase. Two solutions for are available. We choose the one that maintains a positive angle of the displacement increment from one iteration to the next.

K N 1+

i i 1–

ri i 1– i 1–

F Ri

ui 1– –=

i

ri i 1– i

+ iF r

i i 1– +=

ui i 1– i

+ ui i 1– iu

*

i+=

ui i 1– K

i 1–r=

u*

iK

i 1–F=

u*

i

i

c l2 u

iui

= =

u*

i

Tu

*

i i

22 u

i 1– ui i 1– +

Tu

*

i i + +

ui 1– u

i i 1– + Tu

i 1– ui i 1– + l

2– 0=

i 1= u1

0=


74

The two roots of this scalar equation are and . To avoid going back on the original

load-deflection curve, the angle between the incremental displacement vectors, and (before

and after the current iteration, respectively) should be positive. Two alternative values of (namely,

and corresponding to and are obtained and the cosine of two corresponding

angles ( and ) are given by

(3-45)

and (3-46)

Once again, the prediction phase is interpreted with and , while Equation (3-51) and

Equation (3-52) retain their full form in the correction phase.

As mentioned earlier, the appropriate root, or is that which gives a positive . In case

both the angles are positive, the appropriate root is the one closest to the linear solution given as:

(3-47)

Crisfield’s solution procedure, generalized to an automatic load incrementation process, has been implemented in MSC Nastran Implicit Nonlinear as one of the options using (NLPCI, NLSTRAT and PARAM,MARCAUTO). Various components of this process are shown in Figure 3-10.

Figure 3-10 Crisfield’s Constant Arc Length

i 1 i 2

ui 1– u

i

ui

ui 1 u

i 2 i 1 i 2

1 2

1cosu

n 1+

i

1

Tun 1+

i 1–

l-----------------------------------------------------=

2cosu

n 1+

i 2

Tun 1+

i 1–

l-----------------------------------------------------=

i 1= un 1+0

un=

i 1 i 2 cos

i ui 1– u

i+ u

i 1– ui

+ l2

–

2 ui 1– u

i+ u

*

i--------------------------------------------------------------------------------------=

u*2

u*1

0u*

1

r1

u2

1 K2

1–f2

=

F

Force

Incremental Displacement


The constraints in Equation (3-49) and Equation (3-50) are imposed at every iteration. Disadvantage of the quadratic equation suggested by Crisfield is the introduction of an equation with two roots and thus the need for an extra equation to solve the system for the calculated roots if two real roots exists. This

situation arises when the contribution (or ) is very large in comparison to the arc-length. This can be avoided in most cases by setting sufficiently small values of the error tolerance on the residual force. In case the above situation still persists despite the reduction of error tolerance, MSC Nastran Implicit Nonlinear has two options to proceed:

1. To attempt to continue the analysis with the load increment used in the initial step of auto increment process.

2. Use the increment resulting from the linear constraint for the load.

This is circumvented in Ramm’s procedure due to the linearization.

Another approach to impose the constraint is due to Ramm, who also makes use of a quadratic equation to impose the constraint giving rise to the Riks-Ramm method. The difference is that while Crisfield imposes the constraint as a quadratic equation, Ramm linearized the constraint.

Geometrically, the difference between the two methods is that the Crisfield method enforces the correction on the curve of the augmented equation introducing no residual for the augmented equation. Ramm takes the intersection between the linearizations of the curves which gives a residual of the augmented equation for the next step. Both methods converge to the same solution, the intersection of the two curves, unless approximations are made.

The Riks-Ramm constraint is linear, in that:

which results in a linear equation for :

Thus, the load parameter predictor is calculated as:

(3-48)

while during the corrector phase it is:

(3-49)

It is noted that in the definition of the constraint, the normalized displacement of the previous step is used

for the normal to the auxiliary surface . Thus, problems can arise if the step size is too big. In

u1 u

1

c l2 unun 1+= =

unTu

i u*i

+ l2

=

n 1+

1 un l un T Ki

1–r

i –

unT

u*1

----------------------------------------------------------------------=

n 1+i

un 1+

i

TK

i 1–r

i

un 1+i

Tu*

i

----------------------------------------------------–=

cu------ n=


76

situations with sharp curvatures in the solution path, the normal to the prediction may not find intersections with the equilibrium curve. Note that the norm of the displacement increment during the iterations is not constant in Riks-Ramm method.

In contact problems, sudden changes of the stiffness can be present (due to two bodies which are initially not in contact suddenly make contact). Hence, a potential problem exists in the Riks-Ramm method if

the inner-product of the displacement due to the load vector and the displacement increment is

small. This could result in a very large value of the load increment for which convergence in the subsequent iterations is difficult to achieve. Therefore, a modified predictor can be used resulting in a modified Riks-Ramm procedure as:

(3-50)

where

(3-51)

This method effectively scales the load increment to be applied in the prediction and is found to be effective for contact problems.

Refinements and Controls

The success of the methods depend on the suitable choice of the arc-length:

The initial value of the arc-length is calculated from the initial fraction of the load specified by you in the following fashion:

(3-52)

(3-53)

In subsequent steps the arc-length can be reduced or increased at the start of a new load step depending on the number of iterations in the previous step. This number of iterations in compared with the desired

number of iterations which is typically set to 3 or 5. The new arc-length is then given by:

(3-54)

Two control parameters exist to limit the maximum enlargement or the minimum reduction in the arc-length.

u*i

un

1 ln 1– u*

1u

*

1

Tu

1–

u*

1 u

*

1------------------------------------------------------------------=

un

Tu

*

i

unTu

*

i--------------------------=

C l2

=

Ku F R–=

lini2

u=

I0

Id

lnew2 Id

I0---- lprev

2=


(3-55)

In addition, the maximum value can be set to the load multiplier during a particular iteration. In general, control on the limiting values with respect to the arc-length multiplier is preferred in comparison with the maximum fraction of the load to be applied in the iteration since a solution is sought for a particular value of the arc-length.

Also, attention must be paid to the following:

1. In order to tract snap-through problems, the method of allowing solution if the stiffness matrix becomes nonpositive needs to be set.

2. The maximum number of iterations must be set larger than the desired number of iterations.

Specifying the Load Incrementation MethodSelection of the load incrementation method in MSC Nastran is made on the NLSTRAT (p. 2679) in the MSC Nastran Quick Reference Guide Bulk Data entry.

References

• For selecting the load incrementation method in Patran, see Subcase Parameters (Ch. 7).

minl2

lini2

------- max

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideConvergence Controls

78

Convergence ControlsThree methods are available for determining if convergence is obtained on any given iteration: residual force, displacement, and strain energy. You can select one of these three criteria for convergence or you may specify a combination of residual and displacement. The AND combination signals that both residual and displacement must be met, while the OR combination specifies that either one can satisfy convergence criteria. If you are using residual there may be cases in which the force residuals are null in which case is it necessary to switch over to displacement. An Autoswitching option (on by default) allows for this switching. In addition you can specify that the convergence measures be in absolute terms, in relative terms, or in both.

The default measure for convergence in MSC Nastran Implicit Nonlinear is residual which is based on the magnitude of the maximum residual load compared to the maximum reaction force. This method is appropriate since the residuals measure the out-of-equilibrium force, which should be minimized. This technique is also appropriate for Newton methods, where zero-load iterations reduce the residual load. The method has the additional benefit that convergence can be satisfied without iteration. You have complete control over how convergence is defined through the Iterations Parameters form in Patran or through the options on the NLSTRAT entry.

The basic procedures are outlined below.

1. RESIDUAL CHECKING

(3-56)

(3-57)

(3-58)

(3-59)

Where is the force vector, and is the moment vector. and are control tolerances. indicates the component of with the highest absolute value. Residual checking has one

drawback. In some special problems, such as free thermal expansion, there are no reaction forces. If the AUTOSW flag on the NLSTRAT entry is ON the program automatically uses displacement checking in this cases.

2. DISPLACEMENT CHECKING

(3-60)

(3-61)

Fresidual Freact ion

-------------------------------- TOL1

Fresidual Freact ion

-------------------------------- TOL1 andMresidual Mreact ion

---------------------------------- TOL2

Fresidual TOL1

Fresidual TOL1 and Mresidual TOL2

F M TOL1 TOL2

F F

u u

---------------- TOL1

u u

---------------- TOL1 and

---------------- TOL2


(3-62)

(3-63)

where is the displacement increment vector, is the displacement iteration vector, is the incremental rotation vector, and is the rotation iteration vector. With this method, convergence is satisfied if the maximum displacement of the last iteration is small compared to the actual displacement change of the increment. A disadvantage of this approach is that it results in at least one iteration, regardless of the accuracy of the solution.

Figure 3-11 Displacement Control

3. STRAIN ENERGY CHECKING

This is similar to displacement testing where a comparison is made between the strain energy of the latest iteration and the strain energy of the increment. With this method, the entire model is checked.

(3-64)

where is the strain energy of the increment and is the correction to incremental strain energy of the iteration. These energies are the total energies, integrated over the whole volume. A disadvantage of this approach is that it results in at least one iteration, regardless of the accuracy of the solution. The advantage of this method is that it evaluates the global accuracy as opposed to the local accuracy associated with a single node.

Different problems require different schemes to detect the convergence efficiently and accurately. To do this, the following combinations of residual checking and displacement checking are also available.

4. RESIDUAL OR DISPLACEMENT CHECKING

This procedure does convergence checking on both residuals (Procedure 1) and displacements (Procedure 2). Convergence is obtained if one converges.

5. RESIDUAL AND DISPLACEMENT CHECKING

u TOL1

u TOL1 and TOL2

u u

F

u

k

1

0

Correction to incremental displacements of ith iterationi

Displacements at increment nun

i

jj 0=

i

--------------- Tolerance

un 1+0

un 1+k 1+

EE------- TOL1

E E

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80

This procedure does a convergence check on both residuals and displacements (Procedure 4). Convergence is achieved if both criteria converge simultaneously.

For problems where maximum reactions or displacements are extremely small (even close to the round-off errors of computers), the convergence check based on relative values could be meaningless if the convergence criteria chosen is based on these small values. It is necessary to check the convergence with absolute values; otherwise, the analysis is prematurely terminated due to a nonconvergent solution. Such situations are not predicable and usually happen at certain stages of an analysis. For example, problems with stress free motion (rigid body motion or free thermal expansion) and small displacements (springback or constraint thermal expansion) may need to check absolute value at some stage of the analysis, as shown in the table below. However, it is also difficult to determine when to check the absolute value and how small the absolute criterion value should be. In order to improve the robustness of an FE analysis, MSC Nastran Implicit Nonlinear allows you to use the AUTOSW option specified on the NLSTRAT entry to switch the convergence check scheme automatically if the above mentioned situation occurs during the analysis. Using the AUTOSW option allows MSC Nastran Implicit Nonlinear to automatically change the convergence check scheme to Procedure 4 if small reactions or displacements are detected. This function can be deactivated by specifying an absolute value check as before..

Specifying Convergence CriteriaSelection of the convergence criteria in MSC Nastran is made through the convergence criteria parameters on the NLPARM and NLSTRAT Bulk Data entries.

References

• For selecting the load incrementation method in Patran, see Solution Parameters (Ch. 7).

Analysis Type

Convergence Variable

Displacement/Rotation

Residual Force/Torque

Strain Energy

Stress-free motion Yes No No

Springback No Yes No

Free Thermal Expansion Yes No No

Constraint Thermal Expansion No Yes Yes

Yes – relative tolerance testing works.No – relative tolerance testing doesn’t work.


Singularity RatioThe singularity ratio, , is a measure of the conditioning of the system of linear equations. is related

to the conditioning number, , which is defined as the ratio between the highest and lowest eigenvalues in the system. The singularity ratio is an upper bound for the inverse of the matrix conditioning number.

(3-65)

and establish the growth of errors in the solution process. If the errors on the right-hand side of the

equation are less than prior to the solution, the errors in the solution will be less than , with

(3-66)

The singularity ratio is a measure that is computed during the Crout elimination process of MSC Nastran Implicit Nonlinear using the direct solver. In this process, a recursive algorithm redefines the diagonal terms

(3-67)

where is a function of the matrix profile. is a diagonal of the kth degree of freedom. The singularity

ratio is defined as

(3-68)

If all and are positive, the singularity ratio indicates loss of accuracy during the Crout

elimination process. This loss of accuracy occurs for all positive definite matrices. The number of digits lost during the elimination process is approximately equal to

(3-69)

The singularity ratio also indicates the presence of rigid body modes in the structure. In that case, the

elimination process produces zeros on the diagonal . Exact zeros never appear because of

numerical error; therefore, the singularity ratio is of the order

(3-70)

where is the accuracy of floating-point numbers used in the calculation. For most versions of MSC

Nastran Implicit Nonlinear, . If rigid body modes are present, is very small or negative. If

either a zero or a negative diagonal is encountered, execution of MSC Nastran Implicit Nonlinear is terminated because the matrix is diagnosed as being singular.

R R

C

1 R C

C R

E

CE

Kkkk

Kkkk 1–

Kmk

m i=

k 1–

–= Kmk 1 i k 1–

i Kkk

R min Kkkk

Kkkk 1– =

Kkkk

Kkkk 1–

nlost log10R–=

Kkkk

0

R O 10ndigi t–

=

ndigit

ndigit 12 Kkkk

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideSingularity Ratio

82

You can force the solution of a nonpositive definite or singular matrix. In this case, MSC Nastran Implicit

Nonlinear does not stop when it encounters a negative or small term on the diagonal. If you use

Lagrangian multiplier elements, the matrix becomes nonpositive definite and MSC Nastran Implicit

Nonlinear automatically disables the test on the sign of . However, it still tests for singular behavior.

MSC Nastran SOL 600 also supports the PARAM,AUTOSPC, in which case the rigid body mechanism is suppressed by putting a large number on the diagonal. The value of EPS on the AUTOSPC Case Control is set to 1.E-8.

Kkkk

Kkkk

Note: The correctness of a solution obtained for a linearized set of equations in a nonpositive definite system is not guaranteed.


Guidelines for Analysis Methods

Analysis MethodsNonlinear analysis is usually more complex and expensive than linear analysis. Also, a nonlinear problem can never be formulated as a set of linear equations. In general, the solutions of nonlinear problems always require incremental solution schemes and sometimes require iterations (or recycles) within each load/time increment to ensure that equilibrium is satisfied at the end of each step. Superposition cannot be applied in nonlinear problems.

General TipsA nonlinear problem does not always have a unique solution. Sometimes a nonlinear problem does not have any solution, although the problem can seem to be defined correctly.

• Nonlinear analysis requires good judgment and uses considerable computing time. Several runs are often required. The first run should extract the maximum information with the minimum amount of computing time. Some design considerations for a preliminary analysis are:

• Minimize degrees of freedom whenever possible.

• Always run a linear static analysis to check the model before attempting a nonlinear analysis.

• Impose a coarse tolerance on convergence to reduce the number of iterations. A coarse run determines the area of most rapid change where additional load increments might be required. Plan the increment size in the final run by the following rule of thumb: there should be as many load increments as required to fit the nonlinear results by the same number of straight lines.

MSC Nastran Implicit Nonlinear solves nonlinear static problems according to one of the following two methods: tangent modulus or initial strain. Examples of the tangent modulus method are elastic-plastic analysis, nonlinear springs, nonlinear foundations, large displacement analysis and gaps. This method requires at least the following three controls:

• A tolerance on convergence.

• A limit to the maximum allowable number of recycles.

• Specification of a minimum number of recycles.

An example of the initial strain method is creep or viscoelastic analysis. Creep analysis requires the following tolerance controls:

• Maximum relative creep strain increment control.

• Maximum relative stress change control.

• A limit to the maximum allowable number of recycles.

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Choosing a Solution MethodWhich solution method to use depends very much on the problem. In some cases, one method can be advantageous over another; in other cases, the converse might be true.

The four iterative procedures available in MSC Nastran Implicit Nonlinear are: Newton-Raphson, Modified Newton-Raphson, Newton-Raphson with strain correction modification, and a Secant procedure.

For Static analysis, MSC Nastran Implicit Nonlinear uses the Newton-Raphson method as the default for solving the nonlinear equilibrium equations. The motivation for this choice is primarily the convergence rate obtained by using Newton’s method compared to the convergence rates obtained by alternate methods (modified Newton or quasi-Newton methods) for the types of nonlinear problems most often studied by MSC Nastran Implicit Nonlinear.

Time Steps or Load IncrementsThe issue of choosing suitable time steps is a difficult problem to resolve. First of all, considerations are quite different in static, dynamic, and heat transfer cases. It is always necessary to model the response as a function of time to some acceptable level of accuracy. In the case of dynamic or heat transfer problems, time is a physical dimension for the problem, and the time-stepping scheme must provide suitable steps to allow accurate modeling in this dimension. Even if the problem is linear, this accuracy requirement imposes restrictions on the choice of the time step. In contrast, most static problems have no imposed time scale, and the only criterion involved in time step choice is accuracy involved in modeling nonlinear effects. In dynamic and heat transfer problems, it is exceptional to encounter discontinuities in the time history, because inertia and viscous effects provide smoothing in the solution (an exception is impact). However, in static cases, sharp discontinuities (such as bifurcation caused by buckling) are common. Softening systems or unconstrained systems require special attention in static cases, but are handled naturally in static or heat transfer cases. Thus, the consideration upon which time step choice is made are quite different for the three problem classes.

MSC Nastran Implicit Nonlinear provides both fixed and automatic time step choice. Fixed time stepping is useful in cases where the problem behavior is well understood (as might occur when the user is carrying out a series of parameter studies), or in cases where the automatic algorithms do not handle the problem well. However, the automatic schemes in MSC Nastran Implicit Nonlinear are based on extensive experience with a wide range of problems, and therefore generally provide a reliable approach.

A fixed-time stepping approach avoids some convergence problems with Marc AUTO LOAD particularly for multiple subcases. Marc AUTO LOAD is still available but the new approach is recommended particularly for multiple subcases. The available methods are selected using PARAM,MARCITER,N where N is the number of fixed time steps desired.

For static problems, MSC Nastran Implicit Nonlinear uses a scheme based predominantly on the maximum force residuals following each iteration. By comparing consecutive values of these quantities, MSC Nastran Implicit Nonlinear determines whether convergence is likely in a reasonable number of iterations. If convergence is deemed unlikely, MSC Nastran Implicit Nonlinear adjusts the load increment; otherwise MSC Nastran Implicit Nonlinear continues with the iteration process. In this way,


excessive iteration is eliminated in cases, where convergence is unlikely, and an increment that appears to be converging is not aborted due to its needing a few more iterations. One other ingredient in this algorithm is that a minimum increment size is specified. This prevents excessive computation in cases where buckling, limit load, or some modeling error causes the problem to stall. Other controls are built into the algorithm, for example, it will cut back the increment size if an element inverts due to excessively large geometry changes.

Nonlinear DynamicsIn dynamic analysis when implicit integration is used, the automatic time stepping is based on the concept of half-step residuals. The basic idea is that the time-stepping operator defines the velocities and accelerations at the end of the step in terms of displacement at the end of the step and conditions at the beginning of the step. Equilibrium is then established at the end of the step. This, then ensures an equilibrium solution at the end of each time step, and thus, at the beginning and end of any individual time step. However, these equilibrium solutions do not guarantee equilibrium throughout the step. The time-step control is based on measuring the equilibrium error (the force residuals) at some point during the time step, by using the integrator operator together with the solution obtained at the end of the step, to interpolate within a time step. This evaluation is performed at the half step. If the maximum entry in this residual vector (the maximum “half step” residual) is greater than a user-specified tolerance, the time step is considered too big and is replaced by an appropriate factor. If the maximum half-step residual is sufficiently below the user-specified tolerance, the time step may be increased by an appropriate factor for the next increment. Otherwise, the time step is deemed adequate.

MSC Nastran Implicit Nonlinear is designed to analyze structural components, by which is meant that the overall dynamic response of a structure is sought, in contrast to wave propagation solutions associated with relatively local response in continua. These are labelled “inertial problems”, classified as problems in which “wave effects such as focusing, reflection, and diffraction are not important.” Structural problems are considered “inertial” because the response time sought is long compared to the time required for waves to traverse the structure. The equilibrium considerations are similar to those for nonlinear statics.

Buckling

In problems which are linear until buckling occurs, due to a sudden development of nonlinearity, it is sometimes necessary for you to guide the arc-length algorithm by making sure that the arc length remains sufficiently small prior to the occurrence of buckling.

EfficiencyEven if a solution is obtainable, there is always the issue of efficiency. The pros and cons of each solution procedure, in terms of matrix operations and storage requirements have been discussed in the previous sections. A very important variable regarding overall efficiency is the size of the problem. The time required to assemble a stiffness matrix, as well as the time required to recover stresses after a solution, vary roughly linearly with the number of degrees of freedom of the problem. On the other hand, the time

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86

required to go through the solver when using the direct method varies roughly quadratically with the bandwidth, as well as linearly with the number of degrees of freedom.

In small problems, where the time spent in the solver is negligible, you can easily wipe out any solver gains, or even of assembly gains, with solution procedures such as a line search which requires a double stress recovery. Also, for problems with strong material or contact nonlinearities, gains obtained in assembly in modified Newton-Raphson can be nullified by increased number of iterations or nonconvergence.

References

1. Zienkiewicz, O. C. and R. L. Taylor. The Finite Element Method (4th ed.) Vol. 1. Basic Formulation and Linear Problems (1989),) Vol. 2. Solid and Fluid Mechanics, Dynamics, and Nonlinearity (1991) McGraw-Hill Book Co., London, U. K.

2. Bathe, K. J. Finite Element Procedures, Prentice-Hall, Englewood Cliffs, NJ, 1995.

3. Hughes, T. J. R. The Finite Element Method–Linear Static and Dynamic Finite Element Analysis, Prentice-Hall, Englewood Cliffs, NJ. 1987.

4. Ogden, R. W. “Large Deformation Isotropic Elasticity: On The Correlation of Theory and Experiment for Incompressible Rubberlike Solids,” Proceedings of the Royal Society, Vol. A (326), pp. 565-584, 1972.

5. Cook, R. D., D. S. Malkus, and M. E. Plesha, Concepts and Applications of Finite Element Analysis (3rd ed.), John Wiley & Sons, New York, NY, 1989.

6. Bathe, K. J. Finite Element Procedures, Prentice Hall, Englewood Cliffs, NJ, 1996.

7. Riks, E. “An incremental approach to the solution of solution and buckling problems”, Int. J. of Solids and Structures, V. 15, 1979.

8. Riks, E. “Some Computational Aspects of the Stability Analysis of Nonlinear Structures”, Comp. Methods in Appl. Mech. and Eng., 47, 1984.

9. Crisfield, M. A. “A fast incremental iterative procedure that handles snapthrough”, Comput. & Structures, V. 13, 1981.

10. Ramm, E. “Strategies for tracing the nonlinear response near limit points,” in K. J. Bathe et al (eds), Europe-US Workshop on Nonlinear Finite Element Analysis in Structural Mechanics, Ruhr University Bochum, Germany, Springer-Verlag, Berlin, pp/ 63-89. Berlin, 1985.


Chapter 4: Nonlinearity and Analysis Types

4 Nonlinearity and Analysis Types

Linear and Nonlinear Analysis

Nonlinear Effects and Formulations

Overview of Analysis Types

Static Analysis

Body Approach

Buckling Analysis

Normal Modes

Transient Dynamic Analysis

Creep

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88

Linear and Nonlinear Analysis In a linear static analysis we implicitly assume that the deflections and strains are very small and the stresses are smaller than the material yield stresses. Consequently, the stiffness can be considered to remain constant (i.e., independent of the displacements and forces) and the finite element equilibrium equations

are linear. Where the stiffness matrix is independent of both , the generalized displacement vector,

and , the generalized force vector. This linearity implies that any increase or decrease in the load will produce proportional increase or decrease in displacements, strains and stresses. Linear static problems are solved in one step-a single decomposition of the stiffness matrix.

However, we know that in many structures the deflections and the stresses do not change proportionately with the loads. In these problems the structure’s response depends upon its current state and the

equilibrium equations reflect the fact that the stiffness of the structure is dependent on both and .

As the structure displaces due to loading, the stiffness changes, and as the stiffness changes the structure’s response changes. As a result, nonlinear problems require incremental solution schemes that divide the problem up into steps calculating the displacement, then updating the stiffness. Each step uses the results from the previous step as a starting point. As a result the stiffness matrix must be generated and inverted many times during the analysis adding time and costs to the analysis.

Linear AnalysisSOL 600 allows you to perform linear elastic analysis using any element type in the program. Various kinematic constraints and loadings can be prescribed to the structure being analyzed; the problem can include both isotropic and anisotropic elastic materials.

The principle of superposition holds under conditions of linearity. Therefore, several individual solutions can be superimposed (summed) to obtain a total solution to a problem.

Linear analysis does not require storing as many quantities as does nonlinear analysis; therefore, it uses the core memory more sparingly. The assembled and decomposed stiffness matrices can be reused to arrive at repeated solutions for different loads.

Nonlinear AnalysisNonlinear analysis, while most complex and expensive, must be used to establish accurate results when a structure is subject to large deformations, when the material behavior falls outside of a linear elastic model, or where the structural interactions include contact.

In nonlinear analysis the stiffness matrix is assembled and decomposed repeatedly throughout the incrementation process. This adds considerable time and cost to the analysis. In addition, because the

P Ku=

K u

P

u P

P K P u u=

89CHAPTER 4Nonlinearity and Analysis Types

response is not proportional to the loads, each load case must be solved separately and the principle of superposition is not applicable.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideNonlinear Effects and Formulations

90

Nonlinear Effects and FormulationsThere are three sources of nonlinearity: material, geometric, and nonlinear boundary conditions. Material nonlinearity results from the nonlinear relationship between stresses and strains. Considerable progress has been made in attempts to derive the continuum or macroscopic behavior of materials from microscopic backgrounds, but, up to now, commonly accepted constitutive laws are phenomenological. Difficulty in obtaining experimental data is usually a stumbling block in mathematical modeling of material behavior. A plethora of models exist for more commonly available materials like elastomers and metals. Other material model of considerable practical importance are: composites, viscoplastics, creep, soils, concrete, powder, and foams. Figure 4-1 shows the elastoplastic, elasto-viscoplasticity, and creep. Although the situation of strain hardening is more commonly encountered, strain softening and localization has gained considerable importance in recent times.

Geometric nonlinearity results from the nonlinear relationship between strains and displacements on the one hand and the nonlinear relation between stresses and forces on the other hand. If the stress measure is conjugate to the strain measure, both sources of nonlinearity have the same form. This type of nonlinearity is mathematically well defined, but often difficult to treat numerically. Two important types of geometric nonlinearity occur:

1. The analysis of buckling and snap-through problems (see Figure 4-2 and Figure 4-3).

2. Large strain problems such as manufacturing, crash, and impact problems. In such problems, due to large strain kinematics, the mathematical separation into geometric and material nonlinearity is nonunique.

Figure 4-1 Material Nonlinearity

Elasto-Plastic Behavior

Elasto-Viscoplastic Behavior

c

tCreep Behavior


Figure 4-2 Buckling

Figure 4-3 Snap-Through

Boundary conditions and/or loads can also cause nonlinearity. Contact and friction problems lead to nonlinear boundary conditions. This type of nonlinearity manifests itself in several real life situations; for example, metal forming, gears, interference of mechanical components, pneumatic tire contact, and crash (see Figure 4-4). Loads on a structure cause nonlinearity if they vary with the displacements of the structure. These loads can be conservative, as in the case of a centrifugal force field (see Figure 4-5); they can also be nonconservative, as in the case of a follower force on a cantilever beam (see Figure 4-6). Also, such a follower force can be locally nonconservative, but represent a conservative loading system when integrated over the structure. A pressurized cylinder (see Figure 4-7) is an example of this.

Figure 4-4 Contact and Friction Problem

P

u

Linear

Stable

Neutral

Unstable

P

Pc

u

P

u

P

u


92

Figure 4-5 Centrifugal Load Problem (Conservative)

Figure 4-6 Follower Force Problem (Nonconservative)

Figure 4-7 Pressurized Cylinder (Globally Conservative)

The three types of nonlinearities are described in detail in the following sections.

P

P


Geometric NonlinearitiesGeometric nonlinearity leads to two types of phenomena: change in structural behavior and loss of structural stability.

There are two natural classes of large deformation problems: the large displacement, small strain problem and the large displacement, large strain problem. For the large displacement, small strain problem, changes in the stress-strain law can be neglected, but the contributions from the nonlinear terms in the strain displacement relations cannot be neglected. For the large displacement, large strain problem, the constitutive relation must be defined in the correct frame of reference and is transformed from this frame of reference to the one in which the equilibrium equations are written.

The collapse load of a structure can be predicted by performing an eigenvalue analysis. If performed after the linear solution (increment zero), the Euler buckling estimate is obtained. An eigenvalue problem can be formulated after each increment of load; this procedure can be considered a nonlinear buckling analysis even though a linearized eigenvalue analysis is used at each stage.

The kinematics of deformation can be described by the following approaches:

• Lagrangian Formulation

• Eularian Formulation

The choice of one over another can be dictated by the convenience of modeling physics of the problem, rezoning requirements, and integration of constitutive equations, and can be specified using PARAM,MARUPDAT.

Lagrangian Formulation

In the Lagrangian method, the finite element mesh is attached to the material and moves through space along with the material. In this case, there is no difficulty in establishing stress or strain histories at a particular material point and the treatment of free surfaces is natural and straightforward.

The Lagrangian approach also naturally describes the deformation of structural elements; that is, shells and beams, and transient problems, such as the indentation problem shown in Figure 4-8.

Figure 4-8 Indentation Problem with Pressure Distribution on Tool

sz

u


94

This method can also analyze steady-state processes such as extrusion and rolling. Shortcomings of the Lagrangian method are that flow problems are difficult to model and that the mesh distortion is as severe as the deformation of the object. Severe mesh degeneration is shown in Figure 4-9b. However, recent advances in adaptive meshing and rezoning available in Marc have alleviated the problems of premature termination of the analysis due to mesh distortions as shown in Figure 4-9c.

Figure 4-9 Rezoning Example

The Lagrangian approach can be classified in two categories: the total Lagrangian method and the updated Lagrangian method. In the total Lagrangian approach, the equilibrium is expressed with the original undeformed state as the reference; in the updated Lagrangian approach, the current configuration acts as the reference state. The kinematics of deformation and the description of motion is given in Table 4-1 and Figure 4-10.

Table 4-1 Kinematics and Stress-Strain Measures in Large Deformation

Configuration Measures Reference (t = 0 or n) Current (t = n + 1)

Coordinates X x

Deformation Tensor C (Right Cauchy-Green) b (Left Cauchy-Green)

Strain Measure E (Green-Lagrange)F (Deformation Gradient)

e (Logarithmic)

Stress Measure S (second Piola-Kirchhoff)P (first Piola-Kirchhoff)

(Cauchy)

(a) Original (b) Deformed Mesh (Undeformed Mesh) Before Rezoning

(c) Deformed MeshAfter Rezoning


Figure 4-10 Description of Motion

Total Lagrangian Procedure

The total Lagrangian procedure can be used for linear or nonlinear materials, in conjunction with static or dynamic analysis. Although this formulation is based on the initial element geometry, the incremental stiffness matrices are formed to account for previously developed stress and changes in geometry.

This method is particularly suitable for the analysis of nonlinear elastic problems (for instance, with the Mooney or Ogden material model). The total Lagrangian approach is also useful for problems in plasticity and creep, where moderately large rotations but small strains occur. A case typical in problems of beam or shell bending. However, this is only due to the approximations involved.

In the total Lagrangian approach, the equilibrium can be expressed by the principle of virtual work as:

(4-1)

Here is the symmetric second Piola-Kirchhoff stress tensor, , is the Green-Lagrange strain, is

the body force in the reference configuration, is the traction vector in the reference configuration, and

is the virtual displacements. Integrations are carried out in the original configuration at . The

strains are decomposed in total strains for equilibrated configurations and the incremental strains

between and as:

(4-2)

while the incremental strains are further decomposed into linear, and nonlinear, parts as:

Reference

t = 0

Current

t = n + 1

Previous

t = n

un + 1

F

f

u

Fn

un

Fn+1 = Fn

SijEij VdV0

bi0i V ti

0i Ad

A0

+dV0

=

Sij Eij bi0

ti0

i t 0=

t n= t n 1+=

Eijn 1+

Eijn

Eij+=

Eij Eijn


96

where is the linear part of the incremental strain expressed as:

(4-3)

The second term in the bracket in Equation (4-3) is the initial displacement effect. is the nonlinear part of the incremental strain expressed as:

(4-4)

Linearization of equilibrium of Equation (4-1) yields:

(4-5)

where is the small displacement stiffness matrix defined as

is the initial displacement stiffness matrix defined as

in the above equations, and are the constant and displacement dependent symmetric shape

function gradient matrices, respectively, and is the material tangent,

and is the initial stress stiffness matrix

in which is the second Piola-Kirchhoff stresses and is the shape function gradient matrix.

Also, is the correction displacement vector. and are the external and internal forces, respectively.

This Lagrangian formulation can be applied to problems if the undeformed configuration is known so that integrals can be evaluated, and if the second Piola-Kirchhoff stress is a known function of the strain. The first condition is not usually met for fluids, because the deformation history is usually unknown. For

Eij Eij Enij+=

E

E12---

ui

Xj------------

uj

Xi------------+

12---

ukn

Xi--------- uk

Xj------------ +=

En

En 1

2---

uk

Xi------------ uk

Xj------------ uk

n

Xj--------- uk

Xi------------ +=

K0 K1 K2+ + u F R–=

K0

K0 i j

imn0

Dmnpqpqj0

VdV0

=

K1

K1 i j

imnu

Dmnpqpqju

imnu

Dmnpqpqj0

imnu

Dmnpqpqju

+ + VdV0

=

imn0

imnu

Dmnpq

K2

K2 i j

Ni k Nj l Skl VdV0

=

Skl Ni k

u F R


solids, however, each analysis usually starts in the stress-free undeformed state, and the integrations can be carried out without any difficulty.

For viscoelastic fluids and elastic-plastic and viscoplastic solids, the constitutive equations usually supply an expression for the rate of stress in terms of deformation rate, stress, deformation, and sometimes other (internal) material parameters. The relevant quantity for the constitutive equations is the rate of stress at a given material point.

It, therefore, seems most obvious to differentiate the Lagrangian virtual work equation with respect to time. The rate of virtual work is readily found as

(4-6)

This formulation is adequate for most materials, because the rate of the second Piola-Kirchhoff stress can be written as

(4-7)

For many materials, the stress rate is even a linear function of the strain rate

(4-8)

Equation (4-6) supplies a set of linear relations in terms of the velocity field. The velocity field can be solved noniteratively and the displacement can be obtained by time integration of the velocities.

The second Piola-Kirchhoff stress for elastic and hyperelastic materials is a function of the Green-Lagrange strain defined below:

(4-9)

If the stress is a linear function of the strain (linear elasticity)

(4-10)

the resulting set of equations is still nonlinear because the strain is a nonlinear function of displacement.

Updated Lagrangian Procedure

The updated Lagrange formulation takes the reference configuration at . True or Cauchy stress and an energetically conjugate strain measure, namely the true strain, are used in the constitutive relationship.

The updated Lagrangian approach is useful in:

• analysis of shell and beam structures in which rotations are large so that the nonlinear terms in the curvature expressions may no longer be neglected, and

S·i jEij Si j

vk

Xi--------

k

Xj------------+ dV

V0

b·iidV t

·iidA

A0

+V0

=

S·i j S

·i j E

·kl Smn Epq =

S·i j Dijk l Smn Epq E

·kl=

Sij Si j Ekl =

Sij Dijk lEkl=

t n 1+=


98

• large strain plasticity analysis, for calculations which the plastic deformations cannot be assumed to be infinitesimal.

In general, this approach can be used to analyze structures where inelastic behavior (for example, plasticity, viscoplasticity, or creep) causes the large deformations. The (initial) Lagrangian coordinate frame has little physical significance in these analyses since the inelastic deformations are, by definition, permanent. For these analyses, the Lagrangian frame of reference is redefined at the last completed iteration of the current increment.

It is instructive to derive the stiffness matrices for the updated Lagrangian formulation starting from the virtual work principle in Equation (4-9).

Direct linearization of the left-hand side of Equation (4-9) yields:

(4-11)

where u and are actual incremental and virtual displacements respectively, and is Cauchy stress

tensor.

(4-12)

denotes the symmetric part of , which represents the gradient operator in the current configuration. Also, in Equation (4-11) and Equation (4-12), three identities are used:

(4-13)

in which represents the material moduli tensor in the reference configuration which is convected

to the current configuration, . This yields:

(4-14)

where is the material stiffness matrix written as

Sij Eij d VdV0

ikkjuij

vdVn 1+

=

kj

Si jEijd VdV0

si jLi jk ls ukl vd

Vn 1+

=

s

i j1J---FimSmnFjn=

Eij FmismnFnj=

Lijk l1J---FimFjnFkpFlqDmnpq=

and

Dmnpq

Lijk

K1 K2+ u F R–=

K1


in which is the symmetric gradient operator-evaluated in the current configuration and is the

Cauchy stresses and is the geometric stiffness matrix written as

while and are the external and internal forces, respectively.

Keeping in view that the reference state is the current state, a rate formulation analogous to Equation (4-6) can be obtained by setting:

(4-15)

where F is the deformation tensor, and d is the rate of deformation. Hence,

(4-16)

in which and is the body force and surface traction, respectively, in the current configuration

In this equation, is the Truesdell rate of Cauchy stress which is essentially a Lie derivative of Cauchy

stress obtained as:

(4-17)

The Truesdell rate of Cauchy stress is materially objective implying that if a rigid rotation is imposed on the material, the Truesdell rate vanishes, whereas the usual material rate does not vanish. This fact has important consequences in the large deformation problems where large rotations are involved. The constitutive equations can be formulated in terms of the Truesdell rate of Cauchy stress as:

Specifying the Geometric Nonlinearity Formulation

Selection of the geometric nonlinearity formulation in MSC Nastran is made on the MARUPDAT parameter entry.

K1 i j

imnLmnpqpqjVn 1+

=

imn kl

K2

K2 i j

klNi k Nj l vdVn 1+

=

F R

Fij i j Eij dij Xi--------

xi------- Sij i j= = = =

i jdi j i j

vk

xi--------

k

xj------------+ dv

Vn 1+

b· iidvVn 1+

t·iidaAn 1+

+=

bi ti

i j

i j

Fin JFnk

1–klFml

1–

·

= Fmj

i j Lijk dk=


100

References

• For selecting the geometric nonlinearity formulation in Patran, see Subcase Parameters (Ch. 7).

More on Using Total Lagrangian

MSC Nastran Implicit Nonlinear will normally determine whether Total Lagrangian or Updated Lagrangian is best for a particular problem. If you wish to exert more control, the parameters discussed in this section and the next may be employed.

For problems (such as centrifugal or pressure load) that require follower forces, use the LGDISP parameter. This parameter forms all distributed loads on the basis of the current geometry.

When the total Lagrangian method is used, the program uses and prints second Piola-Kirchhoff stress and Green-Lagrange strain. These measures are suitable for analysis with large incremental rotations and large incremental strains.

More on Using Updated Lagrangian

You can use the updated procedure with or without MSC Nastran’s LGDISP parameter. When you use the LGDISP parameter, MSC Nastran Implicit Nonlinear takes into account the effect of the internal stresses by forming the initial stress stiffness. MSC Nastran Implicit Nonlinear also calculates the strain increment to second order accuracy to allow large rotation increments.

Another option is to use the MARUPDAT parameter (with or without the LGDISP parameter) to define a new (Lagrangian) frame of reference at the beginning of each increment. This option is suitable for analysis of problems of large total rotation but small strain. If analysis of large plastic strain is required, use PARAM,MRFINITE,1 in addition to the PARAM, MARUPDAT parameter in which case MSC Nastran Implicit.

With MARUPDAT,1 MSC Nastran Implicit Nonlinear uses Cauchy stresses and true strains. This combination of parameters is suitable for analyses with small incremental rotations and small incremental strains. Stress and strain components are printed with respect to the current state.

The Marc plasticity parameter with options 3 or 5 utilize the updated Lagrange procedure for elastic-plastic analysis. The Marc,elasticity parameter with option 2 utilizes the updated Lagrange procedure for large strain elasticity (Mooney or Ogden).

Note: Depending on the type of analysis specified by all entries in the input file, PARAMETER,MARUPDAT will be specified automatically as -1 or 1 unless entered by the user.

Note: Do not use Marc’s CENTROID parameter with this parameter. Always use residual load corrections with this parameter. To input control tolerances for large displacement analysis, use model definition option NLSTRAT.


The combination of PARAM,MARUPDAT,1 and PARAM,MARCDILT (i.e., with constant dilatation) or a MATEP material entry results in a complete large strain plasticity formulation (with B-Bar method) to satisfy incompatibility using the updated Lagrange procedure. The use of MATEP replaces the need of the MARCDILT parameter. The program internally uses true (Cauchy) stress and rotation neutralized strains. In the case of proportional straining, this method leads to logarithmic strains.

Theoretically and numerically, if formulated mathematically correct, the two formulations yield exactly the same results. However, integration of constitutive equations for certain types of material behavior (for example, plasticity) make the implementation of the total Lagrange formulation inconvenient. If the constitutive equations are convected back to the original configuration and proper transformations are applied, then both formulations are equivalent.

Note: For materials exhibiting large strain plasticity with volumetric changes (for example, soils, powder, snow, wood) only Marc’s LARGE DISP, FINITE and UPDATE should be used (these are created automatically for you by the internal Marc translator in MSC Nastran). Use of MARCDILT parameter or MATEP will enforce the incompressibility condition and, in such materials, yield incorrect and nonphysical behavior.

Options Kinematics Formulation

param,MARCPLAS, n,1

Total Lagrange Small strain, mean normal, additive decomposition of strain rates.

param,marcplas,3

default

Updated Lagrange Large strain, mean normal, additive decomposition of strain rates.

param,marcplas,5 Updated Lagrange Large strain, radial return multiplicative decomposition of deformation gradient.

param,marcelas,1 Total Lagrange Large strain.

param,marcelas,2 Updated Lagrange Large strain.


102

Material NonlinearitiesIn a large strain analysis, it is usually difficult to separate the kinematics from the material description. The following table lists the characteristics of some common materials.

A complete description of the material types mentioned in the table is given in Materials (Ch. 10). However, some notable characteristics and procedural considerations of some commonly encountered materials behavior are listed next.

Material Characteristics Examples Models

Composites Anisotropic:

1) layered, 21 constants

2)Fiber reinforced,

one-dimensional strain in fibers

Bearings, aircraft panels

Tires, glass/epoxy

Composite continuum elements

Rebars

Creep Strains increasing with time under constant load.Stresses decreasing with time under constant deformations. Creep strains are non-instantaneous.

Metals at high temperatures, polymide films

ORNLNortonMaxwell

Elastic Stress functions of instantaneous strain only. Linear load-displacement relation.

Small deformation (below yield) for most materials: metals, glass, wood

Hookes Law

Elasto-plasticity

Yield condition flow rule and hardening rule necessary to calculate stress, plastic strain. Permanent deformation upon unloading.

MetalsSoils

von Mises IsotropicCam -ClayHill’s Anisotropic

Hyperelastic Stress function of instantaneous strain. Nonlinear load-displacement relation. Unloading path same as loading.

Rubber MooneyOgdenArruda BoyceGentFoam

Hypoelastic Rate form of stress-strain law Concrete NLELAST

Viscoelastic Time dependence of stresses in elastic material under loads. Full recovery after unloading.

RubberGlass

Simo ModelNarayanaswamy

Viscoplastic Combined plasticity and creep phenomenon

MetalsPowder

Power lawShima Model

dsij Cijk dk=

SE2--- T

tCT 1– =


Inaccuracies in experimental data, misinterpretation of material model parameters and errors in user-defined material law are some common sources of error in the analysis from the materials viewpoint. It is useful to check the material behavior by running a small model with prescribed displacement and load boundary conditions in uniaxial tension and shear (single element tests are not recommended).

Elasticity

Structures composed of elastomers, such as tires and bushings, are typically subjected to large deformation and large strain. An elastomer is a polymer, such as rubber, which shows a nonlinear elastic stress-strain behavior. The large strain elasticity capability in MSC Nastran Implicit Nonlinear deals primarily with elastomeric materials. These materials are characterized by the form of their elastic strain energy function.

For the finite element analysis of elastomers, there are some special considerations that do not apply for linear elastic analysis. These considerations include:

• Mesh Distortion

• Incompressible Behavior

• Instabilities

• Existence of Multiple Solutions

Mesh Distortions

When extremely large deformations occur, the element mesh should be designed so that it can follow these deformations without complete degeneration of elements. This problem is more prevalent when the updated Lagrange procedure is used. For problems involving extreme distortions, the Marc global adaptive remeshing capability should be used.

Incompressible Behavior

One of the most frequent causes of problems analyzing elastomers is the incompressible material behavior. Lagrangian multipliers (pressure variables) are used to apply the incompressibility constraint. The result is that the volume is kept constant in a generalized sense, over an element.

Both the total, as well as updated Lagrange formulations, are implemented with appropriate constraint ratios for lower- and higher-order elements in 2D and 3D. For many practical analysis, the LBB (Ladyszhenskaya-Babuska-Brezzi) condition does not have to be satisfied in the strictest sense; for example, four node quadrilateral based on Herrmann principle.

For elements that satisfy the LBB condition, error estimates of the following form can be established

(4-18)

where and are the orders of displacements and pressure interpolations, respectively. If

, the rate of convergence is said to be optimal, and elements satisfying the LBB condition will not lock.

uh

u– 1 ph

p– 0+ O hmin k 1+ =

k

K min k 1+ =


104

The large strain elasticity formulation may also be used with conventional plane stress, membrane, and shell elements. Because of the plane stress conditions, the incompressibility constraint can be satisfied without the use of Lagrange multipliers.

Instabilities

Under some circumstances, materials can become unstable. This instability can be real or can be due to the mathematical formulation used in the calculations.

Instability can also result from the approximate satisfaction of incompressibility constraints. If the number of Lagrangian multipliers is insufficient, local volume changes can occur. Under some circumstances, these volume changes can be associated with a decrease in total energy. This type of instability usually occurs only if there is a large tensile hydrostatic stress. Similarly, overconstraints give rise to mesh locking and inordinate increase in total energy under large compressive stresses.

Existence of Multiple Solutions

It is possible that more than one stable solution exists (due to nonlinearity) for a given set of boundary conditions. An example of such multiple solutions is a hollow hemisphere with zero prescribed loads. Two equilibrium solutions exist: the undeformed stress-free state and the inverted self-equilibrating state. An example of these solutions is shown in Figure 4-11 and Figure 4-12. If the equilibrium solution remains stable, no problems should occur; however, if the equilibrium becomes unstable at some point in the analysis, problems can occur.

Figure 4-11 Rubber Hemisphere

y

x


Figure 4-12 Inverted Rubber Hemisphere

When incompressible material is being modeled, the basic linearized incremental procedure is used in conjunction with mixed variational principles similar in form to the Herrmann incompressible elastic formulation. These formulations are incorporated in plane strain, axisymmetric, and three-dimensional elements. These mixed elements may be used in combination with other elements in the library (suitable constraint equations may be necessary) and with each other. Where different materials are joined, the pressure variable at the corner nodes must be uncoupled to allow for mean pressure discontinuity. MPC’s must be used to couple the displacements only.

Plasticity

In recent years there has been a tremendous growth in the analysis of metal forming problems by the finite element method. Although an Eularian flow-type approach has been used for steady-state and transient problems, the updated Lagrangian procedure, pioneered by McMeeking and Rice, is most suitable for analysis of large strain plasticity problems. The main reasons for this are: (a) its ability to trace free boundaries, and (b) the flexibility of taking elasticity and history effects into account. Also, residual stresses can be accurately calculated.

The large strain plasticity capability in MSC Nastran Implicit Nonlinear allows you to analyze problems of large-strain, elastic-plastic material behavior. These problems can include manufacturing processes such as forging, upsetting, extension or deep drawing, and/or large deformation of structures that occur during plastic collapse. The analysis involves both material, geometric and boundary nonlinearities.

In addition to the options required for plasticity analysis, the PARAM, MRTABLS1 parameters are needed for large strain plasticity analysis.

y

x


106

In performing finite deformation elastic-plastic analysis, there are some special considerations which do not apply for linear elastic analysis. These considerations include:

• Choice of Finite Element Types

• Nearly Incompressible Behavior

• Treatment of Boundary Conditions

• Severe Mesh Distortion

• Instabilities

Choice of Finite Element Types

Accurate calculation of large strain plasticity problems depends on the selection of adequate finite element types. In addition to the usual criteria for selection, two aspects need to be given special consideration: the element types selected need to be insensitive to (strong) distortion; for plane strain, axisymmetric, and three-dimensional problems, the element mesh must be able to represent nondilatational (incompressible) deformation modes.

Nearly Incompressible Behavior

Most finite element types tend to lock during fully plastic (incompressible) material behavior. A remedy is to introduce a modified variational principle which effectively reduces the number of independent dilatational modes (constraints) in the mesh. This procedure is successful for plasticity problems in the conventional “small” strain formulation. Zienkiewicz pointed out the positive effect of reduced integration for this type of problem and demonstrates the similarity between modified variational procedures and reduced integration. MSC.Software Corporation recommends the use of lower-order elements, invoking the constant dilatation option with certain exceptions such as 4-node tetras. The lower-order elements, which use reduced integration and hourglass control, also behave well for nearly incompressible materials.

Treatment of Boundary Conditions

In many large strain plasticity problems, specifically in the analysis of manufacturing processes, the material slides with or without friction over curved surfaces. This results in a severely nonlinear boundary condition. The MSC Nastran Implicit Nonlinear CONTACT option can model such sliding boundary conditions.

Severe Mesh Distortion

Because the mesh is attached to the deforming material, severe distortion of the element mesh often occurs, which leads to a degeneration of the results in many problems. To avoid this degeneration, generate a new finite element mesh for the problem and then transfer the current deformation state to the new finite element mesh.


Instabilities

Elastic-plastic structures are sometimes unstable due to necking phenomena. Consider a rod of a rigid-

plastic incompressible workhardening material. With the current true uniaxial strain rate and H the

current workhardening, the rate of true uniaxial stress is equal to

(4-19)

The applied force is equal to , where is the current area of the rod. The rate of the force is therefore equal to

(4-20)

On the other hand, conservation of volume requires that

(4-21)

Hence, the force rate can be calculated as

(4-22)

Instability clearly occurs if . For applied loads (as opposed to applied boundary conditions), the stiffness matrix becomes singular (nonpositive definite).

For the large strain plasticity option, the workhardening slope for plasticity is the rate of true (Cauchy) stress versus the true (logarithmic) plastic strain rate. The workhardening curve must, therefore, be entered as the true stress versus the logarithmic plastic strain in a uniaxial tension test.

Computational Procedures for Elastic-Plastic Analysis

For more information on computational procedures, please see the Marc Reference manual.

Creep

Creep is a time-dependent inelastic behavior that can occur at any stress level, either below or above the yield stress of a material. Creep is an important factor at elevated temperatures. In many cases, creep is also accompanied by plasticity, which occurs above the yield stress of the material.

Conventional creep behavior is based on a von Mises creep potential with isotropic behavior described by the equivalent creep law:

The material behavior is therefore described by:

·

·

·

H·

=

F A= A

F·

·

A A·

+=

A·

A·

+ 0=

F·

H – A·

=

H

· cr

f cr

T t,,, =


108

where is the outward normal to the current von Mises stress surface and is the equivalent creep

strain rate.

There are two numerical procedures used in implementing creep behavior. The default is an explicit procedure in which the above relationship is implemented in the program by an initial strain technique. In other words, a pseudo-load vector due to the creep strain increment is added to the right-hand side of the stiffness equation.

where K is the stiffness matrix, and and are incremental displacement and incremental nodal force vectors, respectively. The integral:

is the pseudo-load vector due to the creep strain increment in which is the strain displacement relation

and is the stress-strain relation. When plasticity is also specified through a suitably defined yield criterion and yield stress in MSC Nastran, the plasticity is treated implicitly while the creep is treated explicitly.

As an alternative, an implicit creep procedure can be requested. In this case, the inelastic strain rate has an influence on the stiffness matrix. Using this technique, significantly larger steps in strain space can be used. This option is only to be used for isotropic materials with the creep strain rate defined by a creep constant.

cr · cr

--------t=

--------

· cr

Ku P T

V Dcr

dv+=

u P

T

V Dcr

dv

D


Creep Buckling

MSC Nastran also predicts the creep time to buckling due to stress redistribution under given load or repeated cyclic load. The buckling option solves the following equation for the first eigenvalue

(4-23)

The geometric stiffness matrix, , is a function of the increments of stress and displacement. These

increments are calculated during the last creep time step increment. To determine the creep time to buckle, perform a buckle step after a converged creep increment. Note that the incremental time must be scaled by the calculated eigenvalue, and added to the total (current) time to get an estimate as to when buckling occurs.

Viscoelasticity

In certain problems, structural materials exhibit viscoelastic behavior. Two examples of these problems are quenching of glass and time-dependent deformation of polymeric materials. The viscoelastic material retains linearity between load and deformation; however, this linear relationship depends on time. Consequently, the current state of deformation must be determined from the entire history of loading. Different models consisting of elastic elements (spring) and viscous elements (dashpot) can be used to simulate the viscoelastic material behavior described in Materials (Ch. 10). Both the equation of state and the hereditary integral approaches can be used for viscoelastic analysis.

A special class of temperature dependence known as the Thermo-Rheologically Simple behavior (TRS) is also applicable to a variety of thermal viscoelastic problems. To model the thermo-rheologically simple material behavior, MATTVE can be used to choose the Williams-Landel-Ferry equation or the power series expression or Narayanaswamy model.

In MSC Nastran Implicit Nonlinear, two options are available for small strain viscoelastic analysis. The first option uses the equation of state approach and represents a Kelvin model. The second option is based on the hereditary integral approach and allows the selection of a generalized Maxwell model. The thermo-rheologically simple behavior is also available in the second option for thermal viscoelastic analysis. The Viscoelastic (Ch. 10) discusses these models in detail.

The Simo model for large strain viscoelasticity can be used in conjunction with the damage and hyperelastic Mooney, Ogden, Gent, or Arruda Boyce material model. The large strain viscoelastic material behavior can be simulated by incorporating MATVE.

Nonlinear structural relaxation behavior of materials can be modeled by the Narayanaswamy model which accounts for memory effect. This model allows simulation of evolution of physical properties of glass subjected to complex time temperature histories. The thermal expansion behavior for the Narayanaswamy model is controlled via the MATTVE Bulk Data option.

Viscoplasticity

There are two procedures in MSC Nastran Implicit Nonlinear for viscoplastic analysis: explicit and implicit. A brief description of each procedure follows:

K KG+ 0=

KG


110

Explicit Method

The elasto-viscoplasticity model in MSC Nastran Implicit Nonlinear is a modified creep model to which a plastic element is added. The plastic element is inactive when the stress is less than the yield stress of the material. You can use the elasto-viscoplasticity model to solve time-dependent plasticity and creep as well as plasticity problems with a nonassociated flow law.

The MPCREEP option in MSC Nastran Implicit Nonlinear has been modified to enable solving problems with viscoplasticity. The method is modified to allow solving elastic-plastic problems with nonassociated flow rules which result in nonsymmetric stress-strain relations if the tangent modulus method is used.

The MPCREEP allows you to select the procedure used to perform the time integration.If the explicit method is used, then the time step must be small, but is the material is elastic and small deformation, no reassembly of the stiffness matrix is required. If the implicit method is chosen, then larger time steps may be used, but reassembly occurs at every increment.

In thermal creep simulations it is necessary that the time step be chosen to satisfy accuracy of both the rate independent thermal stress problem and the rate dependent creep problem. To insure that this occurs the MTCREEP bulk data option has been introduced.

The viscoplastic approach converts an iterative elastic-plastic method to one where a fraction of the initial force vector is applied at each increment with the time step controls. The success of the method depends on the proper use of the automatic creep time step controls. This means that it is necessary to select an initial time step that will satisfy the tolerances placed on the allowable stress change.

The allowable stress change is specified in the creep controls. The most highly stressed element usually yields the maximum strain rate. It is also important to select a total time that gives sufficient number of increments to work off the effects of the initial force vector. A total time of 30 times the estimated t is usually sufficient.

MSC Nastran Implicit Nonlinear does not distinguish between viscoplastic and creep strains. A flag is set in the CREEP parameter in order to use the viscoplastic option with a nonassociated flow rule.

The viscoplasticity feature can be used to implement very general constitutive relations with the aid of user subroutines ZERO and YIEL.

Since the viscoplasticity model in MSC Nastran Implicit Nonlinear is a modified creep model, you should familiarize yourself with the creep analysis procedure (see Creep, Viscoplastic, and Viscoelastic Behavior, 115).

Implicit Method

A general viscoplastic material law can be implemented through user subroutine UVSCPL. When using this method, you are responsible for defining the inelastic strain increment and the current stress.

The initial time step t =allowable stress change x 0.7

Maximum viscoplastic strain rate x Young’s modulus


Nonlinear Boundary ConditionsThere are three types of nonlinear boundary conditions: contact, nonlinear support, and nonlinear loading. The contact problem is solved using the BCONTACT Case Control specification. Nonlinear support might involve nonlinear springs and/or foundations. Sometimes nonlinearities due to rigid links that become activated or deactivated during an analysis can be modeled through adaptive linear constraints. Nonlinear loading is present if the loading system is nonconservative, as is the case with follower forces or frictional slip effects.

Discontinuities are inherent in the nature of many of these nonlinearities, making the solution more challenging. Some of the most severe nonlinearities in mechanics are introduced by nonlinear boundary conditions. It is, therefore, very important to be aware of potential problem areas and to have a good understanding of the underlying principles. This awareness and understanding enables you to validate numerical answers and to take alternative approaches if an initial attempt fails.

Nonlinear Loading

When the structure is deformed, the directions and the areas of the surface loads are changed. For most deformed structures, such changes are so small that the effect on the equilibrium equation can be ignored. For some structures, such as flexible shell structure with large pressure loads, the effects on the results can be quite significant so that the surface load effects have to be included in the finite element equations.

MSC Nastran Implicit Nonlinear forms both pressure stiffness and pressure terms based on current deformed configuration with MSC Nastran’s PARAM,LGDISP. The PARAM,MARCCENT should not be included due to the use of the residual load correction. Point forces may also be updated with deformation.

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Overview of Analysis TypesA large class of stress analysis problems can be solved with MSC Nastran Implicit Nonlinear (SOL 600). A fundamental division of stress problems is into static and dynamic response, the distinction being whether or not inertia effects are significant. SOL 600 allows complete flexibility in making this distinction, so that the same analysis may contain several static and dynamic phases. Thus, a static preload might be applied, and then the linear or nonlinear dynamic response computed, (as in the case of vibrations of a component of a rotating machine, or the response of a flexible offshore system which is initially moved to an equilibrium position subject to buoyancy and steady current loads, then is excited by wave loading).

Static

Nonlinear static analysis requires the solution of nonlinear equilibrium equations. Many problems involve history dependent response, so that the solution is usually obtained as a series of increments, with iteration within each increment to obtain equilibrium. For most cases, the automatic incrementation provided by MSC Nastran Implicit Nonlinear is preferred, although direct user control is also provided for those cases where the user has experience with a particular problem.

Static analysis procedures frequently involve post buckling behavior where the load-displacement response shows a negative stiffness, and the structure must release strain energy to remain in equilibrium.

Buckling

Eigenvalue buckling estimates are obtained. Classical eigenvalue buckling analysis (e.g., “Euler” buckling) is often used to estimate the critical (buckling) load of “stiff” structures. “Stiff” structures are those that carry their design loads primarily by axial or membrane action, rather than by bending action. Their response usually involves very little deformation prior to buckling, although nonlinear effects can be accounted for by preceding the buckling calculations with a nonlinear static analysis.

Normal Modes

This solution type uses eigenvalue techniques to extract the frequencies of the current system. The stiffness determined at the end of the previous step is used as the basis for the extraction, so that small vibrations of a preloaded structure or nonlinearly deformed structure can be modeled.

Transient Dynamic

This solution type is used when the transient dynamic response, which includes inertial effects, is being studied. Because all of the equations of motion of the system must be integrated through time, direct integration methods (which can be used for both linear and nonlinear problems) are generally significantly more expensive than modal methods (which can only be used for linear problems). For most cases, the automatic incrementation provided is preferred, although direct user control is also provided for those cases where the user has experience with a particular problem.


Creep

This analysis procedure performs a transient, static, stress/displacement analysis. It is especially provided for the analysis of materials which are described by the MATVP material form.

Viscoelastic (Time Domain)

This is especially provided for the time domain analysis of materials which are described by the MATVE material options. The dissipative part of the material behavior is defined through a Prony series representation of the normalized shear and bulk relaxation moduli.

Contact

This type of problem can be solved by either nonlinear static or nonlinear transient dynamic solution procedures and simultaneous tracks the movement of multiple geometric bodies to detect contact and then uses appropriate boundary conditions to simulate the friction between surfaces. A robust numerical procedure is required to simulate these complex physical problems.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideStatic Analysis

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Static AnalysisStatic stress analysis is used when inertia effects can be neglected. The problem may still have a real time scale, for example when the material has a viscoplastic response, such as rate dependent yield. The analysis may be linear or nonlinear. Nonlinearity may arise from large displacement effects, material nonlinearity and boundary nonlinearity (such as contact and friction).

Linear static analysis involves the specification of load cases and appropriate boundary conditions. Solutions may be combined in a postprocessing mode.

Nonlinear static analysis requires the solution of nonlinear equilibrium equations, for which the program uses Full Newton-Raphson, Modified Newton-Raphson, Newton-Raphson with Strain Correction, or the Secant method. Many problems involve history dependent response, so that the solution is usually obtained as a series of increments, with iteration within each increment to obtain equilibrium. Increments must sometimes be kept small (in the sense that rotation and strain increments must be small) to assure correct modeling of history dependent effects, but most commonly the choice of increment size is a matter of computational efficiency - if the increments are too large, more iteration will be required. Each solution method has a finite radius of convergence, which means that too large an increment can prevent any solution from being obtained because the initial state is too far away from the equilibrium state that is being sought - it is outside the radius of convergence. Thus, there is an algorithmic restriction on the increment size. For most cases, the automatic incrementation scheme is preferred, because it will select increment sizes based on these considerations. Direct user control of increment size is also provided because there are cases when the user has considerable experience with his particular problem and can therefore select a more economic approach.

References

For directions on setting up a Static analysis using Patran, see “Specifying the Analysis Type for a Subcase” and Specifying Static Subcase Parameters (Ch. 7).


Post-BucklingGeometrically nonlinear static problems frequently involve buckling or collapse behavior, where the load-displacement response shows a negative stiffness, and the structure must release strain energy to remain in equilibrium. Several approaches are possible in such cases. One is to treat the buckling response dynamically, thus actually modeling the kinetic response with inertia effects included as the structure snaps. This is easily accomplished by using a transient dynamic procedure to include inertial effects when the solution goes unstable. In some simple cases, displacement control can provide a solution, even when the conjugate load (the reaction force) is decreasing as the displacement increases. More generally, static equilibrium states during the unstable phase of the response can be found by using an arc-length method. This method is for cases where the loading is proportional - that is, where the load magnitudes are governed by a single scalar parameter. The method obtains equilibrium solutions by controlling the path length along the load-displacement curve within each increment (rather than controlling the load or displacement increment), so that the load magnitude becomes an unknown of the system.

The method can provide solutions even in cases of complex or unstable response.

Creep, Viscoplastic, and Viscoelastic BehaviorTime dependent material response in static analysis may involve creep and swelling (generally occurring over fairly long time periods), or rate dependent yield (which is often important in fairly rapid processes, such as metal working problems). For rate dependent yield, the usual static procedure is used and an appropriate time scale must be introduced so that MSC Nastran Implicit Nonlinear will treat the viscoplasticity correctly. The backward difference operator is used to integrate the plastic strains. Creep and swelling problems, as well as hereditary viscoelasticity models, are analyzed by the CREEP procedure (which is specified by including a non-zero time interval on the NLPARM entry). Nonlinear creep problems are often solved efficiently by forward difference integration of the inelastic strains (the “initial strain” method), because the numerical stability limit of this operator is usually sufficiently large to allow the solution to be developed in a small number of time increments. Linear viscoelasticity models are integrated with a simple, implicit, unconditionally stable operator. Automatic time stepping in such cases is governed by an accuracy tolerance parameter specified by the user. This limits the maximum inelastic strain rate change allowed over an increment.

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Body ApproachBody Approach enables you to position rigid bodies to just touch deformable bodies before beginning a subsequent Load Step. No analysis is actually performed during a Body Approach step.It is used commonly in multi-forming simulations where bodies are brought just into contact before the analysis begins.

Body positioning can be synchronized or not, meaning that if Synchronized is ON, then as soon as one rigid body contacts, all others stop at that point also. Otherwise all rigid bodies move until they come into contact. The SOL 600,ID option, APPROACH and SYNCHRONIZE implement these concepts.

References

For directions on setting up a Body Approach analysis using Patran, see “Specifying the Analysis Type for a Subcase” and Specifying Body Approach Subcase Parameters (Ch. 7).


Buckling AnalysisBuckling analysis allows you to determine at what load the structure will collapse. You can detect the buckling of a structure when the structure’s stiffness matrix approaches a singular value. You can extract the eigenvalue in a linear analyses to obtain the linear buckling load.You can also perform eigenvalue analysis for buckling load in a nonlinear problem based on the incremental stiffness matrices.

MSC Nastran Implicit Nonlinear (SOL 600) solves elastic instability problems using the bifurcation approach. Bifurcation buckling analysis predicts the load at which the structure becomes unstable, and it predicts the shape that the structure will tend to have after the onset of instability. It does not make any statement about whether buckling is coincident with overall structural failure. Some structures, including flat plates, retain finite positive stiffness in the post-buckled range; others, such as thin cylinders under external pressure, do not. In general, bifurcation buckling calculates critical loads which are unconservative (i.e., higher than the loads at which the structure actually becomes elastically unstable).

Eigenvalue Buckling PredictionThe approach to buckling prediction with MSC Nastran Implicit Nonlinear is based on the development of a linear perturbation of the structure’s stiffness about an equilibrium solution point, which may be the initial equilibrium under no load, or a preloaded state. At any time a structure’s total elastic stiffness is

(4-24)

where is the stiffness caused by the material stiffness, and is the initial stress and load stiffness

caused by non-zero loading. For a “stiff” elastic system, is almost constant, and the variation of is proportional to the load variation.

During the BUCKLING step there may be a non-zero “dead” load, P, and there must be a linear perturbation load,Q, specified in the BIFURCATION BUCKLING step. We wish to estimate what multiple of Q, combined with P, which causes instability. Since the response is assumed to be “stiff” and elastic, and therefore closely proportional to load, the stiffness at P + Q. is, to a good approximation,

, where is the initial stress and load stiffness caused by Q. Thus, the buckling load estimate is provided by the eigen problem.

(4-25)

The eigenvalue, , is a multiplier of the applied load which added to the preload provides the critical load estimate: the predicted collapse load is P + Q. is the collapse mode.

If no boundary conditions are given in the BIFURCATION BUCKLING step, the boundary conditions of the state at the start of the buckling investigation (that is, of the previous nonlinear step) are used for the buckling modes as well as for the perturbation loading. Since boundary conditions within any linear perturbation step apply only locally within the step, if BIFURCATION BUCKLING steps follow one another, boundary conditions for the buckling modes must be repeated within each of the BIFURCATION BUCKLING steps except in steps where they are the same as those belonging to the state at the start of the buckling investigation.

K 0 K p+

K 0 K p

K 0 K p

K 0K p K q

+ + K q

K 0 K p K q+ + 0 =

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118

If boundary conditions are specified in the BIFURCATION BUCKLING step, the complete set of boundary conditions must be given, since MSC Nastran Implicit Nonlinear assumes complete definition of such an option in any linear perturbation analysis.

Several modes can be extracted simultaneously. This is often useful when the structure has different buckling modes for which the critical loads have about the same magnitude, so that the designer must consider the possibility of collapse in any of these modes. The collapse modes may be plotted with Patran.

Bifurcation ApproachTo illustrate the bifurcation approach, consider Equation (4-3), which shows a flat plate loaded by uniaxial edge compression. Using linear static analysis, we can find the so-called “primary equilibrium path” of the structure, which is always a straight line (denoted A in Figure 4-3). As shown, increasing the loads will produce no out-of-plane deflection.

Figure 4-13 Load vs. Deflection Paths for Central Deflection of a Flat Square Plate Subjected to Uniaxial Edge Compression

UZ

P

ZX

Y

P

P

Pcrit

A

B

C


Eigenvalue Extraction MethodsMSC Nastran Implicit Nonlinear uses either the inverse power sweep or the Lanczos method to extract eigenvalues and eigenvectors. Both of these methods are described in the following section, see Eigenvalue Analysis, 122.

References

For directions on setting up a Buckling analysis, see “Specifying the Analysis Type for a Subcase” and Specifying Buckling Subcase Parameters (Ch. 7).

A = Primary equilibrium path, determined by linear elastic static analysis.

B = Secondary equilibrium path, determined by bifurcation buckling analysis.

C = Actual load deflection path, considering initial imperfections and geometrical nonlinear effects.

Pcrit = Elastic buckling load.

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Normal ModesThe usual first step in performing a dynamic analysis is determining the natural frequencies and mode shapes of the structure with damping neglected. Eigenvalue analysis is used to determine these basic dynamic characteristics. The results of an eigenvalue analysis indicate the frequencies and shapes at which a structure naturally tends to vibrate. These results characterize the basic dynamic behavior of the structure and are an indication of how the structure will respond to dynamic loading.

The natural frequencies of a structure are the frequencies at which the structure naturally tends to vibrate if it is subjected to a disturbance. For example, the strings of a piano are each tuned to vibrate at a specific frequency. The deformed shape of the structure at a specific natural frequency of vibration is termed its normal mode of vibration. Each mode shape is associated with a specific natural frequency.

Natural frequencies and mode shapes are functions of the structural properties and boundary conditions. A cantilever beam has a set of natural frequencies and associated mode shapes (Figure 4-14). If the structural properties change, the natural frequencies change, but the mode shapes may not necessarily change. For example, if the elastic modulus of the cantilever beam is changed, the natural frequencies change but the mode shapes remain the same. If the boundary conditions change, then the natural frequencies and mode shapes both change. For example, if the cantilever beam is changed so that it is pinned at both ends, the natural frequencies and mode shapes change.


Figure 4-14 The First Four Mode Shapes of a Cantilever Beam

Modal quantities can be used to identify problem areas by indicating the more highly stressed elements. Elements that are consistently highly stressed across many or all modes will probably be highly stressed when dynamic loads are applied.

Modal strain energy is a useful quantity in identifying candidate elements for design changes to eliminate problem frequencies. Elements with large values of strain energy in a mode indicate the location of large elastic deformation (energy). These elements are those which most directly affect the deformation in a mode. Therefore, changing the properties of these elements with large strain energy should have more effect on the natural frequencies and mode shapes than if elements with low strain energy were changed.

SOL 600 contains two methods for eigenvalue extraction and three time integration operators. Nonlinear effects, including material nonlinearity, geometric nonlinearity, and boundary nonlinearity, can be incorporated.

In addition to distributed mass, you can also attach concentrated masses associated with each degree of freedom of the system. You can include damping in either the modal superposition or the direct

x

y

z4

x

y

z1

x

y

z2

x

y

z3


122

integration methods. You can also include (nonuniform) displacement and/or velocity as an initial condition, and apply time-dependent forces and/or displacements as boundary conditions.

Eigenvalue AnalysisMSC Nastran Implicit Nonlinear uses either the inverse power sweep method or the Lanczos method to extract eigenvalues and eigenvectors. The inverse power sweep method is typically used for extracting a few modes while the Lanczos method is optimal for a few or many modes.

In dynamic eigenvalue analysis, we find the solution to an undamped linear dynamics problem:

where is the stiffness matrix, is the mass matrix, are the eigenvalues (frequencies) and are

the eigenvectors. In MSC Nastran Implicit Nonlinear, is the tangent stiffness matrix, which can include material and geometrically nonlinear contributions. The mass matrix is formed from both distributed mass and point masses.

Inverse Power Sweep

MSC Nastran Implicit Nonlinear creates an initial trial vector. To obtain a new vector, the program multiplies the initial vector by the mass matrix and the inverse (factorized) stiffness matrix. This process is repeated until convergence is reached according to either of the following criteria: single eigenvalue convergence or double eigenvalue convergence. In single eigenvalue convergence, the program computes an eigenvalue at each iteration. Convergence is assumed when the values of two successive iterations are within a prescribed tolerance. In double eigenvalue convergence, the program assumes that the trial vector is a linear combination of two eigenvectors.

Using the three latest vectors, the program calculates two eigenvalues. It compares these two values with the two values calculated in the previous step; convergence is assumed if they are within the prescribed tolerance.

When an eigenvalue has been calculated, the program either exits from the extraction loop (if a sufficient number of vectors has been extracted) or it creates a new trial vector for the next calculation. If a single eigenvalue was obtained, MSC Nastran Implicit Nonlinear uses the double eigenvalue routine to obtain the best trial vector for the next eigenvalue. If two eigenvalues were obtained, the program creates an arbitrary trial vector orthogonal to the previously obtained vectors.

K 2M– 0=

K M K


After MSC Nastran Implicit Nonlinear has calculated the first eigenvalue, it orthogonalizes the trial vector at each iteration to previously extracted vectors (using the Gram-Schmidt orthogonalization procedure). Note that the power shift procedure is available with the inverse power sweep method.

• To select the power shift, set the following parameters:

• Initial shift frequency – This is normally set to zero (unless the structure has rigid body modes, preventing a decomposition around the zeroth frequency).

• Number of modes to be extracted between each shift – A value smaller than five is probably not economical because a shift requires a new decomposition of the stiffness matrix.

• Auto shift parameter – When you decide to do a shift, the new shift point is set to

Highest frequency2 + scalar x (highest frequency - next highest frequency)2

You can define the value of the scalar through the EIGR/EIGRL option.

The Lanczos Method

The Lanczos algorithm converts the original eigenvalue problem into the determination of the eigenvalues of a tri-diagonal matrix. The method can be used either for the determination of all modes or for the calculation of a small number of modes. For the latter case, the Lanczos method is the most efficient eigenvalue extraction algorithm. A simple description of the algorithm is as follows.

Consider the eigenvalue problem:

(4-26)

Equation (4-26) can be rewritten as:

(4-27)

Consider the transformation:

(4-28)

Substituting Equation (4-30) into Equation (4-29) and premultiplying by the matrix on both sides of the equation, we have

(4-29)

The Lanczos algorithm results in a transformation matrix such that:

(4-30)

(4-31)

where the matrix is a symmetrical tri-diagonal matrix of the form:

2 M u K u+– 0=

1

2------ M u M K 1– M u=

u Q =

QT

1

2------ QT M Q QT M K 1– M Q =

Q

QT M Q I=

QT M K 1– MQ T=

T


124

(4-32)

Consequently, the original eigenvalue problem, Equation (4-26), is reduced to the following new eigenvalue problem:

(4-33)

The eigenvalues in Equation (4-33) can be calculated by the standard QL-method.

You can either select the number of modes to be extracted, or a range of modes to be extracted. The Sturm sequence check can be used to verify that all of the required eigenvalues have been found. In addition, you can select the lowest frequency to be extracted to be greater than zero.

The Lanczos procedure also allows you to restart the analysis at a later time and extract additional roots. It is unnecessary to recalculate previously obtained roots using this option.

Convergence Controls

Eigenvalue extraction is controlled by:

1. The maximum number of iterations per mode in the power sweep method; or the maximum number of iterations for all modes in the Lanczos iteration method,

2. an eigenvalue has converged when the difference between the eigenvalues in two consecutive sweeps divided by the eigenvalue is less than the tolerance, and

3. the Lanczos iteration method has converged when the normalized difference between all eigenvalues satisfies the tolerance. The maximum number of iterations and the tolerance can be specified.

Modal Stresses and Reactions

After the modal shapes (and frequencies) are extracted, stresses and reactions at a specified mode may be recovered if desired. This option can be repeated for any of the extracted modes. The stresses are

computed from the modal displacement vector ; the nodal reactions are calculated from

.

T

1 2 0 0

2 2 3 0

0 3 3 m

0 0 m m

=

1

2------ T =

F K 2M–=


Free Vibration AnalysisIf a structure is not totally constrained in space, it is possible for the structure to displace (move) as a rigid body or as a partial or complete mechanism. For each possible component of rigid-body motion or mechanism, there exists one natural frequency which is equal to zero. The zero-frequency modes are called rigid-body modes. Rigid-body motion of all or part of a structure represents the motion of the structure in a stress-free condition. Stress-free, rigid-body modes are useful in conducting dynamic analyses of unconstrained structures, such as aircraft and satellites. Also, rigid-body modes can be indicative of modeling errors or an inadequate constraint set.

Nastran Implicit Nonlinear, SOL 600 can perform free vibration analysis to compute the natural frequencies and associated mode shapes of linear elastic structures. The structure is assumed to be initially unstressed. A real eigenvalue analysis is performed, which assumes that there is no damping and that the structure is not spinning (i.e., no Coriolis force).

Nastran Implicit Nonlinear, SOL 600 free vibration analysis consists of the following steps:

1. Input. The problem geometry (nodes and elements), physical and material properties, loads and boundary conditions are taken from the Patran Neutral File and put into the MSC Nastran Implicit Nonlinear, SOL 600 file.

2. Bandwidth Minimization (Optional). The FEA nodes are renumbered for minimum bandwidth.

3. Element stiffness matrix and mass matrix. The element stiffness matrices and the consistent mass matrices are computed. See Element Library (Ch. 11) for a detailed description of the MSC Nastran Implicit Nonlinear, SOL 600.

4. Global stiffness matrix and mass matrix assembly. Stiffness matrix and the mass matrix are assembled. Boundary and constraint conditions are incorporated by appropriately modifying the element stiffness and mass matrices.

5. Solution of the generalized eigenvalue problem. The frequencies and mode shape vectors are computed by solving the generalized eigenvalue problem.

Modal strain energy. The modal strain energies are computed using the mode shape vectors.

References

For directions on setting up a Normal Modes analysis, see “Specifying the Analysis Type for a Subcase” and Specifying Normal Modes Subcase Parameters (Ch. 7).


126

Support of Complex Eigenvalue AnalysisSOL 600 supports complex eigenvalue analysis via the CMETHOD Case Control command and the EIGC Bulk Data entry. In addition, four new Bulk Data parameters have been introduced:

The flow of the run is as follows:

• Create a primary MSC Nastran SOL 600 input file (we will name it jid.dat for this example), using CONTINUE option on the command line.

• Submit MSC Nastran in the standard fashion. For this example, the following command is used:

nastran jid rc=nast1.rcThe nast1.rc file contains items such as scratch=yes, memory=16mw, etc.

• The primary MSC Nastran run creates an Marc input file named jid.marc.dat

• The primary MSC Nastran run spawns Marc to perform nonlinear analysis. Marc generates the required DMIG matrices for this example.

• The nonlinear Marc analyses completes and generates standard files.

• Control of the process returns to MSC Nastran. A new MSC Nastran input file named jid.nast.dat will be created from the original input file. This file will contain the CMETHOD Case Control command and EIGC Bulk Data entry, all of the original geometry and additional entries to read the dmig002 file.

• A second MSC Nastran job will be spawned from the primary MSC Nastran run using the command

nastran jid.nast rc=nast2.rcThe nast2.rc file can be the same as nast1.rc or can contain different items. Usually memory will need to be larger in nast2.rc than in nast1.rc.

• The second MSC Nastran run computes the complex eigenvalues and finishes.

• Control of the process returns to the primary MSC Nastran run and it finishes.

param,MARCFILi,dmig002

This means that a file named dmig002 will be used. It contains stiffness matrix terms (possibly from a set of unsymmetric friction stiffness matrices)

param,MRMTXNAM, Name,kaax

This means that in the dmig002 file, use DMIG matrix terms labeled kaax (or KAAX – case does not matter).

param,MRSPAWN2, CMD,tran

This means that the primary MSC Nastran run will spawn another MSC Nastran run to compute the complex eigenvalues. The name of the command is nastran (nas is always used and the characters specified by this parameter are added to the end of nas. Thus, we get nas+tran=nastran).

param,MRRCFILE, RCF,nast2.rc

This is the name of the rc file to be used for the second (spawned) MSC Nastran run.


The first portion of the dmig002 file is as follows:

$2345678 2345678 2345678 2345678 2345678 2345678 2345678 2345678 234567812345DMIG KAAX 0 1 2 0 324DMIG* KAAX 6 1* 6 1 3.014712042D+05* 6 2 4.204709763D+08*DMIG* KAAX 6 2* 6 1 1.204709763D+05* 6 2 3.014712042D+05*DMIG* KAAX 6 3* 6 1-4.616527206D+04* 6 2-4.616527206D+04* 6 3 1.308497299D+05DMIG* KAAX 17 1* 6 1 6.239021038D+04* 6 2-2.528344607D+03* 6 3-6.239758760D+03* 17 1 5.939989945D+05

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Transient Dynamic AnalysisTransient response analysis is the most general method for computing forced dynamic response. The purpose of a transient response analysis is to compute the behavior of a structure subjected to time-varying excitation. The transient excitation is explicitly defined in the time domain. All of the forces applied to the structure are known at each instant in time. Forces can be in the form of applied forces and/or enforced motions.

The important results obtained from a transient analysis are typically displacements, velocities, and accelerations of grid points, and forces and stresses in elements.

Depending upon the structure and the nature of the loading, two different numerical methods can be used for a transient response analysis: direct and modal. The direct method performs a numerical integration on the complete coupled equations of motion. The Direct method can take into account nonlinearities. The modal method is a linear solution and utilizes the mode shapes of the structure to reduce and uncouple the equations of motion (when modal or no damping is used); the solution is then obtained through the summation of the individual modal responses.

Direct Transient ResponseTransient dynamic analysis deals with an initial-boundary value problem. In order to solve the equations of motion of a structural system, it is important to specify proper initial and boundary conditions. You obtain the solution to the equations of motion by direct integration (for linear or nonlinear systems). In direct integration, selecting a proper time step is very important. You can include damping in the system.

The following sections discuss the applicable aspects of transient analysis listed below.

• Direct Integration

• Time Step Definition

• Initial Conditions

• Time-Dependent Boundary Conditions

• Damping


Direct Integration

Direct integration is a numerical method for solving the equations of motion of a dynamic system. It is used for both linear and nonlinear problems. In nonlinear problems, the nonlinear effects can include geometric, material, and boundary nonlinearities. For transient analysis, MSC Nastran Implicit Nonlinear offers three direct integration operators listed below.

• Newmark-beta Operator

• Houbolt Operator

• Single Step Houbolt

Direct integration techniques are imprecise; this is true regardless of which technique you use. Each technique exhibits at least one of the following problems: conditional stability, artificial damping, and phase errors, but all can be minimized so that they are negligible.

Newmark-beta Operator

This operator is probably the most popular direct integration method used in finite element analysis. For linear problems, it is unconditionally stable and exhibits no numerical damping. The Newmark-beta operator can effectively obtain solutions for linear and nonlinear problems for a wide range of loadings. The procedure allows for change of time step, so it can be used in problems where sudden impact makes a reduction of time step desirable. This operator can be used with adaptive time step control. Although this method is stable for linear problems, instability can develop if nonlinearities occur. By reducing the time step and/or adding damping, you can overcome these problems.

Houbolt Operator

This operator has the same unconditional stability as the Newmark-beta operator. In addition, it has strong numerical damping characteristics, particularly for higher frequencies. This strong damping makes the method very stable for nonlinear problems as well. In fact, stability increases with the time step size. The drawback of this high damping is that the solution can become inaccurate for large time steps. Hence, the results obtained with the Houbolt operator usually have a smooth appearance, but are not necessarily accurate. The Houbolt integration operator, implemented in MSC Nastran Implicit Nonlinear uses a fixed time step procedure, is particularly useful in obtaining a rough scoping solution to the problem.

Single Step Houbolt Operator

Two computational drawbacks of the Houbolt operator are the requirement of a special starting procedure and the restriction to fixed time steps. A Single Step Houbolt procedure has been presented [Ref. 1.], being unconditionally stable, second order accurate and asymptotically annihilating. In this way, the algorithm is computationally more convenient compared to the standard Houbolt method, and the numerical damping for this method as implemented in SOL 600 has been significantly improved over the standard Houbolt method. This algorithm is recommended for all dynamic analyses.


130

Technical BackgroundConsider the equations of motion of a structural system:

(4-34)

where , , and are mass, damping, and stiffness matrices, respectively, and , , , and are acceleration, velocity, displacement, and force vectors. Various direct integration operators can be used to integrate the equations of motion to obtain the dynamic response of the structural system. The technical background of the three direct integration operators available in MSC Nastran Implicit Nonlinear is described below.

Newmark-beta Operator

The generalized form of the Newmark-beta operator is

(4-35)

(4-36)

where superscript denotes a value at the nth time step and , , and take on their usual meanings.

The particular form of the dynamic equations corresponding to the trapezoidal rule

results in

(4-37)

where the internal force is

Ma Cv Ku+ + F=

M C K a v u F

U··

t t+

t t

t t+

U··

t U··

t t+

t t+

un 1+ un tvn 1 2 – t2an t2an 1++ + +=

vn 1+ vn 1 – tan tan 1++ +=

n u v a

1 2= 1 4=,

4

t2--------M

2t-----C K+ +

u Fn 1+ Rn

– M an 4t-----vn+

Cvn+ +=

R


(4-38)

Equation (4-39) allows implicit solution of the system

(4-39)

Notice that the operator matrix includes , the tangent stiffness matrix. Hence, any nonlinearity results in a reformulation of the operator matrix. Additionally, if the time step changes, this matrix must be recalculated because the operator matrix also depends on the time step. It is possible to change the values

of and if so desired.

Step by Step Solution Algorithm using Newmark Beta Method

1. Initialize:

2. Select time step size t and calculate integration constants:

3. Form effective stiffness matrix [K]*:

[K]* = [K] + a0[M] + a1[C]

4. Triangularize (reduce) [K]*:

5. Compute effective load vector at time t + t:

(4-40)

6. Solve for displacements at time t + t:

R TdvV=

un 1+ un u+=

K

Uo , U· o , U·· o

ao1t--------- ; a1

t

--------------= =

a2

t -------------- ; a3

12------ 1–= =

a41--- 1 ; a5–

t2-----

--- 2– = =

Rt t+* Rt t+ M a0Ut a2U· t a3U·· t++ +=

+ C a1Ut a4U· t a5U·· t++

K *Ut t+ Rt t+*=


132

7. Compute accelerations at time t + t, using:

8. Compute velocities at time t + t, using:

Repeat steps 5 through 8 for each time step.

Houbolt Operator

The Houbolt operator is based on the use of a cubic fitted through three previous points and the current (unknown) in time. This results in the equations

(4-41)

and

(4-42)

Substituting this into the equation of motion results in

(4-43)

This equation provides an implicit solution scheme. By solving Equation (4-39) for , you obtain

Equation (4-44), and so obtain and .

(4-44)

Equation (4-43) is based on uniform time steps – errors occur when the time step is changed. Also, a

special starting procedure is necessary since and appear in Equation (4-43).

U·· t t+ a0 Ut t+ Ut– a2U· t a3U·· t––=

U· t t+ U· t 1 – tU·· t t U·· t t+++=

vn 1+ 116

------un 1+ 3un–32---un 1– 1

3---un 2––+

t=

an 1+ 2u n 1+ 5un– 4un 1– un 2––+ t2=

2

t2--------M

116t---------C K+ +

u Fn 1+ Rn–1

t2-------- 3un 4un 1–– un 2–+ M + +=

1t----- 7

6---un 3

2---un 1––

13---un 2–+

C

u

vn 1+

an 1+

un 1+ un u+=

un 1–

un 2–


Single Step Houbolt Operator

The Single Step Houbolt operator starts with the following equilibrium equation and expressions for the velocity and acceleration:

(4-45)

(4-46)

(4-47)

Notice that in contrast to the Newmark and the standard Houbolt method, the equilibrium equation also contains terms corresponding to the beginning of the increment. Without loss of generality, the parameter

can be set to 1. Based on asymptotic annihilation and second order accuracy, the remaining parameters can be shown to fulfill:

, , , , ,

, , ,

In this way, the number of unknown parameters has been reduced to two. Based on a Taylor series

expansion of the displacement about the nth time step, and should be related by ,

which finally yields . According to [Ref. 1.], should be set to 3/2 (with )

to minimize the velocity error and to 1/2 (with ) to avoid velocity overshoot. The default values

in MSC Nastran Implicit Nonlinear are and , but the user can modify and if so desired.

Substitution of the velocity and acceleration into the equilibrium equation results in:

(4-48)

m1Ma

n 1+ c1Cv

n 1+ k1Ku

n 1+ mMa

n cCv

n kKu

n+ + + + + =

f1F

n 1+a

fF

n+

un 1+

un tv

n t2a

n 1t2a

n 1++ + +=

vn 1+

vn ta

n 1tan 1+

+ +=

m1

k0= = 1 1

+= m1 2–= k1

1 21=

c2 1

+ 412–= c1

2 31+ 412

= f k= f1 k1

=

1 1+ 1 2=

1 2 1 2 1– = 1 1 2–=

0=

13 2= 1– 2= 1

1

1t2k1

------------------------Mc11

1tk1---------------------C K+ +

u Fn 1+

Kun

–

1

1t2k1

------------------------M tvn t

2a

n+ m

k1---------Ma

n

c1

k1---------C v

n tan 1

1t------------ tvn t

2an+

–+

–

c

k1---------Cv

n–

–

+=


134

Time Step DefinitionIn a transient dynamic analysis, time step parameters are required for integration in time. MSC Nastran’s PARAM,MARCAUTO,-1 option can be used for the Newmark-beta operator and the Single Step Houbolt operator to invoke the adaptive time control. Enter parameters to specify the time step size and period of time for this set of boundary conditions.

When using the Newmark-beta operator, decide which frequencies are important to the response. The time step in this method should not exceed 10 percent of the period of the highest relevant frequency in the structure. Otherwise, large phase errors will occur. The phenomenon usually associated with too large a time step is strong oscillatory accelerations. With even larger time steps, the velocities start oscillating. With still larger steps, the displacement eventually oscillates. In nonlinear problems, instability usually follows oscillation. When using adaptive dynamics, you should prescribe a maximum time step.

As in the Newmark-beta operator, the time step in Houbolt integration should not exceed 10 percent of the period of the highest frequency of interest. However, the Houbolt method not only causes phase errors, it also causes strong artificial damping. Therefore, high frequencies are damped out quickly and no obvious oscillations occur. It is, therefore, completely up to the engineer to determine whether the time step was adequate. The damping problem is alleviated to a large extent with the Single Step Houbolt operator.

In nonlinear problems, the mode shapes and frequencies are strong functions of time because of plasticity and large displacement effects, so that the above guidelines can be only a coarse approximation. To obtain a more accurate estimate, repeat the analysis with a significantly different time step (1/5 to 1/10 of the original) and compare responses.

Initial ConditionsIn a transient dynamic analysis, you can specify initial conditions such as nodal displacements and/or nodal velocities. To enter initial conditions, use the following option: TIC for specified nodal displacements, and Bulk Data nodal velocities.


DampingIn a transient dynamic analysis, damping represents the dissipation of energy in the structural system. It also retards the response of the structural system.

MSC Nastran Implicit Nonlinear allows you to enter two types of damping in a transient dynamic analysis: discrete dampers and Rayleigh damping.

For direct integration damping, you can specify the damping matrix as a linear combination of the mass and stiffness matrices of the system. You can specify damping coefficients on an element basis.

Stiffness damping should not be applied to either Herrmann elements or gap elements because of the presence of Lagrange multipliers.

Numerical damping is used to damp out unwanted high-frequency chatter in the structure. If the time step is decreased (stiffness damping might cause too much damping), use the numerical damping option to make the damping (stiffness) coefficient proportional to the time step. Thus, if the time step decreases, high-frequency response can still be accurately represented. This type of damping is particularly useful in problems where the characteristics of the model and/or the response change strongly during analysis (for example, problems involving opening or closing gaps).

Element damping uses coefficients on the element matrices and is represented by the equation:

(4-49)

where

If the same damping coefficients are used throughout the structure, Equation (4-49) is equivalent to Rayleigh damping.

The damping on elastic foundations is the same as the damping on the element on which the foundation is applied.

References

For directions on setting up a Transient Dynamic analysis, see “Specifying the Analysis Type for a Subcase” and Specifying Transient Dynamic Subcase Parameters (Ch. 7).

= the global damping matrix

= the mass matrix of ith element

= the stiffness matrix of the ith element

= the mass damping coefficient on the ith element

= the usual stiffness damping coefficient on the ith element

= the numerical damping coefficient on the ith element

= the time increment

C iMi i it-----+

Ki+

i 1=

n

=

C

Mi

Ki

ai

i

i

t

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideCreep

136

CreepMSC Nastran Implicit Nonlinear (SOL 600) offers two schemes for modeling creep in conjunction with plasticity. Creep for SOL 600 is described differently than Creep for SOL 106. The CREEP entry used in SOL 106 will not work in SOL 600, and if entered will cause the job to terminate with an appropriate message. Creep in SOL 600 must be described using viscoplastic materials (MATVP). The creep formulations for SOL 600 are:

1. Treating creep strains and plastic strains separately using an explicit procedure (where the creep is treated explicitly) or an implicit procedure (where both creep and plasticity are treated implicitly). These procedures are available with standard options via data input or with user-specified options via user subroutines. More details are provided below.

2. Modeling creep strains and plastic strains in a unified fashion (viscoplasticity). Both explicit and implicit procedures are again available for modeling unified viscoplasticity. More details are provided in the section titled Viscoplasticity, 109. The options offered by MSC Nastran for modeling creep are as follows:

• Creep data can be entered directly through the MATVP Bulk Data entry. The form of the creep is designated with either POWER for empirical creep law or TABLE for a tabular input of creep model parameters.

• An automatic time stepping scheme can be used to maximize the time step size in the analysis.

• Eigenvalues can be extracted for the estimation of creep buckling time. In addition, for explicit creep, the following additional options can be used:

• Creep behavior can be either isotropic or anisotropic.

• The Oak Ridge National Laboratory (ORNL) rules on creep can be activated.

Adaptive Time Control

An automatic creep option takes advantage of the diffusive characteristics of most creep solutions. Specifically, this option controls the transient creep analysis. You specify a period of creep time and a suggested time increment. The program automatically selects the largest possible time increment that is consistent with the tolerance set on stress and strain increments (see Creep Control Tolerances, 137 in this chapter).

The algorithm is: for a given time step, a solution is obtained. The program then finds the largest values of stress change per stress, and creep strain change per elastic strain. It compares these values to the tolerance values, (stress change tolerance) and (strain change tolerance), for this period.

The value is calculated as the larger of:

(4-50)

or

(4-51)

T

cr el T


If , the program resets the time step as:

(4-52)

The time increment is repeated until convergence is obtained or the maximum recycles control is exceeded. In the latter case, the run is ended.

If the first repeat does not satisfy tolerances, the possible causes are:

• Excessive residual load correction

• Strong additional nonlinearities such as creep buckling-creep collapse

• Incorrect coding in user subroutine CRPLAW, VSWELL, or UVSCPL

Appropriate action should be taken before the solution is restarted.

If all is well, the solution is stepped forward and the next step is begun. The time step used in the next increment is chosen as

if (4-53)

if (4-54)

if (4-55)

Since the time increment is adjusted to satisfy the tolerances, it is impossible to predetermine the total number of time increments for a given total creep time.

Creep Control Tolerances

SOL 600 performs a creep analysis under constant load or displacement conditions on the basis of a set of tolerances and controls you provide.These are as follows:

1. Stress change tolerance – This tolerance controls the allowable stress change per time step during the creep solution, as a fraction of the total stress at a point. Stress change tolerance governs the accuracy of the transient creep response. If you need accurate tracking of the transient response, specify a tight tolerance of 1 percent or 2 percent stress change per time step. If you need only the steady-state solution, supply a relatively loose tolerance of 10-20 percent. It is also possible to check the absolute rather than the relative stress.

2. Creep strain increment per elastic strain – SOL 600 uses either explicit or implicit integration of the creep rate equation. When the explicit procedure is used, the creep strain increment per elastic strain is used to control stability. In almost all cases, the default of 50 percent represents the stability limit, so that you need not provide any entry for this value. It is also possible to check the absolute rather than the relative strain.

p 1

tnew 0.8told p=

tnew told= 0.8 p 1

tnew 1.25told= 0.65 p 0.8

tnew 1.5told= p 0.65


138

3. Maximum number of recycles for satisfaction of tolerances – The automatic creep option in SOL 600 chooses its own time step. In some cases, the program recycles to choose a time step that satisfies tolerances, but recycling rarely occurs more than once per step. Excessive recycling can be caused by physical problems such as creep buckling, poor coding of user subroutine CRPLAW, VSWELL, or UVSCPL or excessive residual load correction that can occur when the creep solution begins from a state that is not in equilibrium. The maximum number of recycles allows you to avoid wasting machine time under such circumstances. If there is no satisfaction of tolerances after the attempts at stepping forward, the program stops. The default of five recycles is conservative in most cases.

4. Low stress cut-off – Low stress cut-off avoids excessive iteration and small time steps caused by tolerance checks that are based on small (round off) stress states. A simple example is a beam in pure bending. The stress on the neutral axis is a very small roundoff-number, so that automatic time stepping scheme should not base time step choices on tolerance satisfaction at such points. The default of five percent of the maximum stress in the structure is satisfactory for most cases.

5. Choice of element for tolerance checking – Creep tolerance checking occurs as a default for all integration points in all elements. You might wish to check tolerances in only 1 element or in up to 14 elements of your choice. Usually, the most highly stressed element is chosen.

References

1. Chung, J. and Hulbert, G.M., “A family of single-step Houbolt time integration algorithms for structural dynamics”, Comp. Meth. in App. Mech. Engg., 118, 1994.

Chapter 5: Analysis Techniques

5 Analysis Techniques

Domain Decomposition

RESTARTS

Inertia Relief with Auto-Support

Superelements and Modal Neutral Files

BRKSQL

User Subroutine Support

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideDomain Decomposition

140

Domain DecompositionThe Domain Decomposition Method (DDM) is the ability to subdivide your model into domains. Each domain is then submitted to a separate computer or CPU for parallel processing. With this ability, you can analyze large models with much less over all compute time. A single processor job that might take 30 hours to run, can run in half the time on two processors, or even a quarter of the time with four processors. Jobs that take days to run on a single machine can be run overnight on multiple processors that would otherwise lay idle. With DDM, large models that were once thought impossible to practically optimize, now can be solved.

Specifying Domain DecompositionThe PARAMARC Bulk Data entry controls the domain decomposition process. Domains can now be specified by you or automatically determined.

Defining Domain Decomposition Parameters in Patran


2. Click on Solution Type..., then click Solution Parameters...

3. Select Domain Decomposition... to bring up the subform shown below.

Entry Description

PARAMARC Specifies parallel regions for domain decomposition in nonlinear analysis when Marc is executed from MSC Nastran

141CHAPTER 5Analysis Techniques

Decomposition Method

• Automatic Automatic is recommended because the work is done by MSC Nastran.

• Manual If manual is selected, groups must be defined previously.

Number of Domains Defines the number of domains to be created.

Model/Current Group This switch is not applicable to this release. By default groups from all domains will be translated.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideDomain Decomposition

142

Single Input File Parallel Processing for SOL 600MSC Nastran has the capability to use a Marc feature called the Single File Parallel file. To use this capability specify KIND=0 or blank on the PARAMARC entry as shown below.

Format:

Example: To create 4 parallel processes using a single file input procedure.

A similar option to create a single-file Marc t16 file is also available. This option is selected using Bulk Data PARAM,MARCOUTR,1 which is the default starting with the 2005 r3 version.

DDM Results in PatranThere may be multiple results (post) files from a DDM run just as there may be multiple input files. There is one for each domain by the same names with the .t16 /.t19 file extension plus the master. If the master jobname.marc.t16/t19 file is attached, results from all domains are automatically accessed from each domain post file. If however, you want only results for a particular domain, you must attach that file only.

DDM ConfigurationPlease see the MSC.Marc Parallel Version for Windows NT / UNIX Installation and User Notes for proper configuration. MSC.Marc Parallel must be configured properly in order for DDM to work from MSC.Patran or MSC Nastran. If you have trouble, please check the following:

1 2 3 4 5 6 7 8 9 10

PARAMARC

ID KIND NPROC

PARAMARC

51 4

Field Contents

ID Identification number of the PARAMARC entry -- Not presently used. (Integer)

KIND Designates how parallel domains are created. (Integer > 0, Default = 0)0=Parallel processing is accomplished using Marc’s single file input. The command line to execute Marc is changed from -np N (or -nprocd N) to -nps N where N is the number of processors. The maximum number of processors for is 256. Continuation lines may not be entered for KIND=0.

NPROC Number of processors to be used.


On Windows machines:

1. Make sure MPICH is installed. This can be done automatically by including bulk data PARAM, MARMPICH,1 and a file named mpich.dat in the same directory as the Nastran input deck with 3 lines having the content (book) user name (the name you use to log in the PC) domain name (if you are not a member of the domain, enter local) password (password you enter to login in to the PC). All items must start in column 1.

2. When using a cluster of Windows machines you must have all the input files in a shared directory when you submit the job. The MSC.Marc installation on the master host must be in a shared directory also unless all machines have their own installation of MSC.Marc, and then they must be referenced in the hostfile.

For UNIX you must be able to “rlogin” to all referenced machines in the hostfile without supplying a password. If you cannot, check that your .rhosts file has the name of all the machines in it. Check with a system administrator if you need help.

Only homogeneous clusters of machines are currently supported. They must all be running the same MPI service or daemons. For example a cluster of 64 bit HP machines must all use the HP MPI; a cluster of 32 bit HP machines can use either HP MPI or MPICH, but not a mixture; heterogeneous clusters should work if they all use MPICH; UNIX and Windows clusters are not supported.

More information on running jobs in parallel is provided in the MSC Nastran Implicit SOL 600 Parallel Guide.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideRESTARTS

144

RESTARTSA restart capability is available in MSC Nastran Implicit Nonlinear (SOL 600). Any analysis can be saved from any point for a possible restart. A new static load case or a buckling analysis can be solved by restarting from the original static analysis.

Specifying Restarts and ParametersThe RESTART Bulk Data entry controls a restart for SOL 600.

Specifying a Restart in Patran


2. On the Analysis form, set the Action>Object>Method combination to Analyze>Restart>Full Run.

3. Click on Restart Parameters... to bring up the subform shown below.

Entry Description

RESTART Specifies writing or reading of restart data .


Inertia Relief with Auto-SupportInertia Relief in SOL 600 exceeds that available in other MSC Nastran solution sequences by using the Bulk Data entry, SUPORT6. One method is available for MSC.Nastran 2005 r3.

The “support” method may be used to specify which degrees of freedom should be “supported” for each body. This is an extension of the PARAM,INREL,1 method and may use fewer computer resources than the eigenvalue method for some models.

Inertia Relief may be employed on a subcase-by-subcase basis and can be removed if all previously unsupported bodies merge into the main body (which is supported) either all at once or gradually.

ReviewInertia relief has long been a feature in MSC Nastran SOL 101, which enables applied static loading to an unconstrained structure and the calculation of deformed shape and internal loads within the accelerated structure. Inertia relief calculates the rigid body mass x acceleration loads imparted by the applied loads, and applies them in combination to the flexible body to produce a load-balanced static formulation in the linear acceleration reference frame. The “steady-state” relative structural displacements and internal loads are calculated using support entries (PARAM,INREL,-1) or the auto-support capability (PARAM,INREL,-2).

Inertia relief is commonly used to calculate psuedo-static stresses, strains, and loads of unsupported structures due to static loading.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideInertia Relief with Auto-Support

146

General FormulationConsider common 3D unconstrained structure with six rigid body modes.

Rigid body mechanics loads balance (small motion):

and (5-1)

In finite element matrix notation:

(5-2)

is a (a-dof x 6) geometric rigid body matrix resulting from unit displacements in each basic direction

with respect to GRDPNT or (0,0,0). provides summation and cross-product utilities for loads and

motion at each dof . Rigid body accelerations and are represented by 6 x 1 at

PARAM,GRDPNT. All and are entered into load vector ; and are entered into a-set mass

matrix . Solve for the rigid body accelerations:

(5-3)

is the total 6 x 6 a-set mass, nonsingular for normal 3D models with appropriate mass properties.

Apply the balanced loads to the finite element structure in linear statics formulation. This form is employed by the PARAM,INREL,-2 method:

(Inrel = -2) (5-4)

In contrast, older method INREL = -1 and SOL 111 employ the following:

(Inrel = -1) (5-5)

(SOL 111 free-free RESVEC’s) (5-6)

Each method uses a different representation for the rigid body matrix and accelerations. Stiffness matrix

K is singular (i.e., rank ), and each method likewise employs different techniques to solve for displacement shape U.

Fi miai– 0= Mi ri o+ Fi Iii ri o+ miai – 0=

R T P R T M R u··o – 0 =

R

R

i ai i u··o

Fi Mi P mi Ii

M

u··o R T M R 1–

R T P =

RT

MR

K U P = M R u··o –

K U P = M D U·· r –

K U P = M r q··r –

l a 6–=


SUPORT6 EntrySUPORT6 Inertia relief used in SOL 600 only.

Format:

Example:

1 2 3 4 5 6 7 8 9 10

SUPORT6 SID METH IREMOV IDS1

SUPORT6 0 3 1 101

SUPORT6 4 3 -2

Field Contents

SID Set ID corresponding to a Case Control SUPORT1 entry or zero. (Integer; Default = 0)

0 If this is the only SUPORT6 entry, use this SUPORT6 entry for all subcases. If there are multiple SUPORT6 entries, use the one with SID=0 for Marc increment zero.

N Use this SUPORT6 entry for the subcase specified by Case Control SUPORT1=N.

Different SUPORT6 entries can be used for each subcase if desired and different subcases can use different methods.

If there is only one SUPORT6 entry (with SID=0), no Case Control SUPORT1 entries are necessary.

METH Method to use (Integer; Default = 0)

0 Inertia relief is not active for this subcase.

3 Use the “Support Method”, usually specified using param,inrel,-1 for other solution sequences. (See Remark 1.) Input will come from all SUPORT entries and those SUPORT1 entries with ID=SID.

IREMOV Method to retain or remove inertia relief from a previous subcase (Integer; Default = 1).

1 Retain inertia relief conditions from previous subcase.

-1 Remove inertia relief loads immediately.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideInertia Relief with Auto-Support

148

Remark:

1. The parameter INREL is ignored by SOL 600.

-2 Remove inertia relief loads gradually

IREMOV should be blank or 1 unless METH is 0.

IDS1 ID of SUPORT1 entries to be used if METH=3 and SID=0 (Integer; no Default).

For METH=3, only SUPORT1 entries with ID=IDS1 will be used in Marc increment zero. All SUPORT entries will be used.

(Used for METH=3 when SID=0 ONLY.)

Field Contents


Superelements and Modal Neutral FilesMSC Nastran SOL 600 allows you to create external superelements or to output MSC Adams MNF files.

External Superelements

External superelements are available both for input (generated by previous MSC Nastran jobs) and output. To generated matrices use Bulk Data entry, MDMIOUT to obtain the reduced (or full) stiffness. These matrices can then be used to compute eigenvalues, perform harmonic or random vibration analyses, etc.

For MSC Nastran-generated matrices, use the same procedures that are used by other MSC Nastran external superelement creation runs employing the EXTSEOUT Case Control command. For the analysis that combines the external superelements, use the new Bulk Data entry, MESUPER and include the .asm and .pch files from the superelement creation runs.

Example

An example of the input data for the combination run follows:

SOL 600,101 path=1 stop=1CENDparam,marcbug,0TITLE = 2 SUPERELEMENTS AND THE RESIDUAL -- TEST PROBLEM NO. EXTSE2RSUBTITLE = 8 X 8 MESH OF QUAD4 ELEMENTS; GM-CMS PROJECTparam,mextsee,1SPC = 100LOAD = 1000DISP = ALLK2GG=KAAXM2GG=MAAXBEGIN BULKparam,marcnd99,-1force, 1000, 844, , 0.1, 0., 0., 1.SPC1 100 12346 840 848$2345678 2345678 2345678mesuper 100 extse2a.pchmesuper 200 extse2b.pchinclude 'OUTDIR:extse2a.asm'include 'OUTDIR:extse2b.asm'include 'OUTDIR:extse2a.pch'include 'OUTDIR:extse2b.pch'ENDDATA

MDMIOUT Entry for MNF Files and Stiffness Matrices

You can now create MSC Adams modal neutral files (MNF) using the Bulk Data entry, MDMIOUT. Once read into MSC Adams you can view and animate modal results. You can find more information on the MSC Adams family of motion products by visiting our MSC website.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideBRKSQL

150

BRKSQLBulk Data entry BRKSQL is available for brake squeal simulation which replaces several parameters and MARCIN entries previously used. It is now possible to determine the unstable brake squeal roots using MSC Nastran’s complex eigenvalue solver and unsymmetric friction stiffness matrices form Marc either for an undeformed structure or after a nonlinear subcase. Brake squeal analysis for SOL 600 is accomplished by starting a primary MSC Nastran job, spawning Marc to calculate the unsymmetric friction stiffness matrices either at the beginning or end of a nonlinear subcase, then spawning a second MSC Nastran job to calculate the complex eigenvalues. Unstable roots indicate potential brake squeal. They are designated by positive real roots and negative damping in the f06 output file.

Specifies data for brake squeal calculations using SOL 600.

Format:

Example:

BRKSQL Specifies data for Brake Squeal Calculations using SOL 600

1 2 3 4 5 6 7 8 9 10

BRKSQL METH AVSTIF FACT1 GLUE ICORD

R1 R2 R3 X Y Z

NASCMD

RCFILE

BRKSQL 1 5.34E6 1.0 1.0

0.0 0.0 1.0 2.0 3.0 4.0

tran

nastb


Field Contents

METH Method flag corresponding to the type of brake squeal calculations to be performed. (Integer, Default = 1)

0 = Perform brake squeal calculations before any nonlinear analysis has taken place

1 = Perform brake squeal calculations after all nonlinear load cases

AVSTIF Approximate average stiffness per unit area between the pads and disk. Corresponds to Marc’s PARAMETERS fifth datablock, field 1. This value is also known as the initial friction stiffness in the Marc Volume C documentation. AVSTIF can be obtained by either experiment or numerical simulation. A larger value of AVSTIF corresponds to a higher contact pressure, which usually results in more unstable modes. (Real; no Default; required field)

FACT1 Factor to scale friction stiffness values calculated by Marc. (Real; Default = 1.0)

GLUE Flag specifying whether MPC for non-pad/disk surfaces with glued contact are used or ignored (Integer, Default = 0). A value of 0 means ignore the MPC; a value of 1 means include the MPCs (see Remark 6).

ICORD Flag indicating whether coordinates are updated or not. A value of 0 means coordinates are not updated. A value of 1 means coordinates are updated using the formula Cnew=Corig+Defl where Cnew are updated coordinates, Corig are original coordinates, and Defl are the final displacements from last Marc increment. (Integer; Default = 0)

R1 X direction cosine (basic coord system) of axis of rotation; corresponds to Marc ROTATION A second datablock. (Real; no Default. Required field)

R2 Y direction cosine (basic coord system) of axis of rotation; corresponds to Marc ROTATION A second datablock.

R3 Z direction cosine (basic coord system); corresponds to Marc ROTATION A second datablock. (Real; no Default. Required field)

X X coordinate in basic coord system of a point on the axis of rotation; corresponds to Marc ROTATION A third datablock. (Real; no Default. Required field)

Y Y coordinate in basic coord system of a point on the axis of rotation; corresponds to Marc ROTATION A third datablock. (Real; no Default. Required field)

Z Z coordinate in basic coord system of a point on the axis of rotation; corresponds to Marc ROTATION A third datablock. (Real; no Default. Required field)

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Remarks:

1. This entry is used to calculate complex eigenvalues for brake squeal using unsymmetric stiffness friction matrices calculated by Marc. Options exist to obtain the unsymmetric stiffness matrices using the undeformed geometry (initial contact) or after all specified nonlinear subcases.

2. SOL 600 performs brake squeal calculations. The main (original) MSC Nastran job with input file jid.dat or jid.bdf spawns Marc just as it does for any other SOL 600 job. Marc calculates unsymmetric friction stiffness matrices that1 are saved on a file (jid.marc.bde with associated file jid.marc.ccc). The primary MSC Nastran job then creates input data for a second MSC Nastran job (jid.nast.dat) to use the unsymmetric stiffness matrices in an complex eigenvalue extraction. The primary MSC Nastran job spawns a second MSC Nastran job to calculate the complex eigenvalues. The complex eigenvalues and eigenvectors are found in jid.nast.f06, jid.nast.op2, etc.

NASCMD is the name of the command used to execute the secondary MSC Nastran job. NASCMD can be up to 64 characters long and must be left justified in field 2. The sting as entered will be used as is -- except that it will be converted to lower case regardless of whether it is entered in upper or lower case.

RCFILE is the name of an RC file to be used for the secondary MSC Nastran job. It should be similar to the RC file used for the primary run except that additional memory will usually be necessary to calculate the complex eigenvalues and batch=no should also be specified. RCFILE is limited to 8 characters and an extension of “.rc” will be added automatically. This entry will be converted to upper case in MSC Nastran but will be converted to lower case before spawning the complex eigenvalue run. This RC file must be located in the same directory as the MSC Nastran input file. This entry is the same as specifying PARAM,MRRCFILE. One or the other should be used.

3. MPC are produced for contact surfaces with glued contact. DMIGs are produced for contact surfaces without glued contact. The brakes and drums should not use glued contact; other regions of the structure can used glued contact.

4. The continuation lines may be omitted if defaults are appropriate.

5. When a BRKSQL entry is used, PARAM,MRMTXNAM and PARAM,MARCFIL1 should not be entered.

NASCMD Name of a command to run MSC Nastran (limited to 64 characters) -- used in conjunction with the CONTINUE options on the SOL 600 entry. The full path of the command to execute MSC Nastran should be entered. The string will be converted to lower case. See Remark 2. (Character; Default = nastran)

RCFILE Name of an RC file to be used with a secondary MSC Nastran job (limited to 8 characters) -- used in conjunction with the CONTINUE options on the SOL 600 entry. An extension of “.rc” will automatically be added. See Remark 2. (Character; Default = nastb.rc)

Field Contents


6. When brake squeal matrices are output by Marc, unsymmetric friction stiffness matrices are output for non-glued contact surfaces. For surfaces with glued contact, MPCs are output. The GLUE flag signals SOL 600 to look for these MPCs and combine them with other MPCs that might be in the model using MPCADD, or if no MPCs were originally used, to add the MCPs due to glued contact. Glued contact surfaces may not be used for the disk-rotor interface. If GLUE is zero or blank, the MPC for glued contact in the Marc brake squeal bde file (if any) will be ignored. Sometimes Marc puts out MPCs with only one degree-of-freedom defined. Such MPCs will be ignored; otherwise, MSC Nastran will generate a fatal error.

7. If ICORD=1, an Marc t19 file will automatic.

8. The names NASCMD and RCFILE must be entered in small fixed field and start in column 9 (i.e., left justified in the field).

9. The Nastran input file name used for a brake squeal analysis may only contain lower case letters and the underscore and/or dash characters.

10. Brake squeal is not available with DDM (parallel processing). Do not enter a PARAMARC when using the BRKSQL entry.

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User Subroutine Support User subroutine support has been added through the bulk data entry USRSUB6.

Defines user subroutines used in SOL 600 only.

Format:

Examples:

Notes:

1. All user subroutines must reside in the directory where the Nastran input file resides.

2. All user subroutines on disk must be in lower case and have an extension of .f. The names entered in the bulk data entry may be in upper or lower case. They will be converted to lower case.

3. SOL 600 combines all user subroutines into one large subroutine named u600.f and u600.f is passed to the Marc command line when spawned from Nastran.

4. If only one user subroutine is required, an alternate is to use PARAM,MARCUSUB,name.

USRSUB6 Defines User Subroutines for SOL 600

1 2 3 4 5 6 7 8 9 10

USRSUB6 U1 U2 U3 U4 U5 U6 U7 U8

U9 U10

USRSUB6 UDAMAG uvoid TENSOF

USRSUB6* SEPFORBBC

Field Contents

Ui Name of user subroutine(s) to be included. See Marc Volume D for list of available user subroutines. Do not include the .f extension on this entry, however, the actual file on the disk must have the .f extension. If any user subroutine exceeds 8 characters, use the wide field format for the primary line and all continuation lines. (Character; no Default)

Chapter 6: Modeling

6 Modeling

Coordinate Systems

Nodes

Elements

Modeling in Patran

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Coordinate SystemsThe basic coordinate system in MSC Nastran Nonlinear is a right-handed, rectangular Cartesian system. Your may choose other systems locally for input, for output of nodal variables (displacements, velocities, etc.) and point loads or boundary condition specification, and for material options. In general, all coordinate systems are assumed to be right-handed. MSC Nastran Implicit Nonlinear output is provided in the “Global” Coordinate System. The Global Coordinate System is defined by field 7 of each GRID entry and therefore may refer to a rotated rectangular or cylindrical coordinate system. If field 7 is blank or zero, the output is in the Basic Coordinate System (which is rectangular). MSC Nastran Implicit Nonlinear does not support output in spherical coordinate systems, however input may be specified in spherical coordinate systems.

Nodal Coordinate SystemsIf the GRID CP (Coordinate system ID) is nonzero, it may refer to a rectangular, cylindrical or spherical local coordinate frame. See CORD1Ci Bulk Data entries.

Note the following points regarding nodal coordinate systems:

1. Displacement vectors and unbalanced force vectors are computed and written to the results file in the global coordinate system.

2. Boundary conditions and nodal forces are applied in the global coordinate system.

3. Coupling equations and multipoint constraint (MPC) equation, including the MPC equations that result from rigid links, relate displacement components in the global coordinate system.

4. Rigid link elements will produce erroneous results if a local coordinate system is defined at either end of the element.

5. For 2-D models (i.e., models whose elements have only UX and UY degrees-of-freedom), the Z-axis of the nodal coordinate system must coincide with the Z-axis of the basic coordinate system.

Element Coordinate Systems There is a rectangular coordinate system associated with each element in a MSC Nastran SOL 600 analysis. The default coordinate system and the options available for modifying it, depend on the element type. Descriptions are given in Element Library (Ch. 11) for each element type.

If the element is homogeneous and either isotropic or orthotropic, the components of stress and strain are computed and passed to the results file in the element coordinate system.

If the element is homogeneous and orthotropic, the directions of orthotropy coincide with the element coordinate system, unless an MCID is specified.

For laminated elements each layer has its own coordinate system. The reference line for defining layer orientations is the X-axis of the element coordinate system. Material properties are entered, and stresses and strains are computed in the layer coordinate systems for all layers of laminated elements.

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Depending on the element type and material model, the default element coordinate system can be overridden by referring to a nonzero Coordinate ID when meshing.

Defining Material Axes Orientations

With MSC Nastran’s THETA and MCID fields on the CQUAD4, CTRIA3, etc. elements you specify the orientation of the material axes of symmetry (relationship between the element coordinate system and

the global coordinate system, or the 0o ply angle line, if composite) in one of four different ways:

1. as a specific angle offset from an element edge,

2. as a specific angle offset from the line created by two intersecting planes,

3. as a particular coordinate system specified by user-supplied unit vectors, or

4. as specified by user subroutine ORIENT. This is accomplished by the specification of an orientation type, an orientation angle, or one or two user-defined vectors.

Defining Material Axes in Patran

The orientation of the material axes are defined in Patran using the Element Properties application.

1. Click the Properties application icon to access the Element Properties application.

2. Set the Action>Object>Method combination to create a CQUADi or CTRIAi element.

3. Click Input Properties... and enter the material axes orientation information.

There are three ways to assign the material orientation:

a. reference a coordinate system, which is then projected onto the element,

b. define a vector that will be projected onto the element, or

c. define a constant angle offset from the default element coordinate system.

This defines the setting of the THETA or MCID field on the CQUADi or CTRIAi entry. This scalar value can either be a constant value in degrees, a vector, or a reference to an existing coordinate system. This property is optional.

Note: When used to define an element coordinate system, the Coordinate ID can reference only a rectangular coordinate system.

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NodesModel geometry is defined in MSC Nastran with GRID points. A grid point is a point on or in the structural continuum which is used to define a finite element. A simple model may have only a handful of grid points; a complex model may have many tens of thousands. The structure’s grid points displace with the loaded structure. Each grid point of the structural model has six possible components of displacement: three translations (in the x-, y-, or z-directions) and three rotations (about the x-, y-, or z-axes). These components of displacement are called degrees of freedom (DOFs).

Degrees-of-Freedom

The degrees-of-freedom in MSC Nastran Nonlinear are always referred to as follows:

MSC Nastran Nonlinear only activates those degrees-of-freedom needed at a node. Thus, some of the degrees of freedom listed above may not be used at all nodes in a model, because each element type only uses those degrees of freedom which are relevant. For example, two-dimensional solid (continuum) stress/displacement elements only use degrees of freedom 1 and 2. The degrees of freedom actually used at any node are thus the envelope of those variables needed in each element that uses the node.

1 x-displacement.

2 y-displacement.

3 z-displacement.

4 Rotation about the x-axis.

5 Rotation about the y-axis.

6 Rotation about the z-axis.


ElementsOnce the geometry (grid points) of the structural model has been established, the grid points are used to define the finite elements.

MSC Nastran has an extensive library of finite elements covering a wide range of physical behavior. Some of these elements and their names are shown in figure below. The C in front of each element name stands for “connection.”

• Point Element (not a finite element, but can be included in the finite element model)

• Spring Elements (they behave like simple extensional or rotational springs)

• Line Elements (they behave like rods, bars, or beams)

• Surface Elements (they behave like membranes or thin plates)

• Solid Elements (they behave like bricks or thick plates)

• Rigid Bar (infinitely stiff without causing numerical difficulties in the mathematical model)

Structural elements are defined on Bulk Data connection entries that identify the grid points to which the element is connected. The mnemonics for all such entries have a prefix of the letter “C”, followed by an indication of the type of element, such as CBAR and CROD. The order of the grid point identification defines the positive direction of the axis of a one-dimensional element and the positive surface of a plate element. The connection entries include additional orientation information when required. Some elements allow for offsets between its connecting grid points and the reference plane of the element. The coordinate systems associated with element offsets are defined in terms of the grid point coordinate systems. For most elements, each connection entry references a property definition entry. If many elements have the same properties, this system of referencing eliminates a large number of duplicate entries.

CMASS1 (Scalar mass connection)CONM1 (Concentrated mass)

CELAS2

CROD, CBAR, CBEAM

CTRIA3 CQUAD4

CHEXA CTETRACPENTA

RBE2

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Details for each element type are described in Element Library (Ch. 11).


Modeling in PatranIn Patran, geometric models are the foundation on which most finite element models are built. Geometric curves, surfaces, or solids provide the base for creating nodes, elements, and loads and boundary conditions; the geometric model also serves as the structure to which material properties, as well as element properties, may be assigned even before any mesh is actually generated.

Creating Geometry in PatranModel geometry may be constructed in Patran, accessed directly from a CAD application, or imported in specially formatted translator files. Whatever the source of the geometry, a single geometric model will be maintained throughout all geometric and finite element operations. Geometric entities, even if obtained from external files, retain their original mathematical representation without any approximations or substitutions.

Accessing the Geometry Application

In Patran you can create, modify, and delete points, curves, surfaces, and solids. Patran assigns a default color to the display of all geometric entities.

Pick the Geometry icon in the Patran Main Form to access the Geometry application.

The Geometry form controls all processes in the Geometry application. The top portion of the form contains three keywords, Action, Object, and Method; these remain the same throughout all activities. The rest of the entries will vary depending on the requirements posed by the specified action, object, and method.

There are hundreds of action, object, method combinations available for creating geometric entities in Patran. For complete descriptions on creating geometry models, see the Geometry Modeling - Reference Manual Part 2,.

Action Names the operation that will be performed; for example Create, Edit, or Delete.

Object Identifies the geometric entity upon which the action is performed, for example, Solid. In this case, if the Action is Create, then the command requests that a solid be created.

Method Specifies the procedure used to perform the action. Taking the above example one step further, if the Method is Surface, a solid will be created by one of the techniques that utilize surfaces.

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Utilizing External Geometry (CAD) Files

Patran can make use of geometry created in databases outside of Patran by either accessing geometric data directly from one of several CAD systems, or importing geometry using special files.

Geometry access, performed through the unique Direct Geometry Access (DGA) feature, does not require any translation. Patran accesses the original geometry and uses the geometric definitions of all entities.

On the other hand, when geometry is imported, Patran first evaluates the mathematical definition of entities in their originating CAD system, and then formulates the information to be appropriate for Patran operations.

Imported geometry comes to Patran via IGES, Express Neutral files, or Patran Neutral files.

IGES (Initial Graphic Exchange Specification) is an ANSI standard formatted file that makes it possible to exchange data among most commercial CAD systems. Express Neutral files are intermediate files created during a Unigraphics or CV CAD model access. Patran Neutral files are specially formatted for the purpose of providing a means of importing and exporting model data.

Geometry received into the database, whether through direct access or import, is treated as if it had been built in Patran; meshing, load and boundary condition assignments, element and material properties definitions are all performed as if on Patran’s own “native” geometry.


Creating Finite Element Meshes in PatranFinite elements themselves are defined by both their topology (i.e., their shape) and their properties. For example, the elements used to create a mesh for a surface may be composed of quadrilaterals or triangles. Similarly, one element may be a steel plate modeling structural effects such as displacement and rotation, while another may represent an air mass in an acoustic analysis.Patran provides numerous ways to create a finite element mesh.

At this stage of using Patran, where you are creating a finite element mesh using the Finite Elements application form, elements are defined purely in terms of their topology. Other properties such as materials, thickness and behavior types are then defined for these elements in subsequent applications, and discussed in later chapters of this guide.

The most rudimentary method of creating a finite element mesh is to manually generate individual nodes, and then to create individual elements from previously defined nodes. Individual nodes can be either be generated from the geometry model or directly created using node creation tools that bypass the need for point definitions. A finite element model created manually supports the entire Patran element library and where applicable, Patran automatically generates midedge, midface and midbody nodes.

Patran contains many capabilities to help you manually create the right kind of finite element mesh for your model, and capabilities that automate the process of finite element creation. Patran provides the following capabilities for finite element modeling (FEM):

• Mesh seeding tools to control specific mesh densities in specific areas of your geometry.

• Several highly automated techniques for mesh generation.

• Equivalencing capabilities for joining meshes in adjacent regions.

• Tools to verify the quality and accuracy of your finite element model.

• Capabilities for direct input and editing of finite element data.

Automatic Meshing Tools

There are four basic mesh generation techniques available in Patran: IsoMesh, Paver Mesh, Auto TetMesh, and 2-1/2D Meshing. Selecting the right technique for a particular model must be based on geometry, model topology, analysis objectives, and engineering judgment.

Isomesh

Creates a traditional mapped mesh on regularly shaped geometry via simple subdivision. This method creates Quad and Tria elements on surfaces and brick elements on solids. The resulting mesh supports all element configurations in Patran.

Paver

The Paver is an automated surface meshing technique that you can use with any arbitrary surface region, including trimmed surfaces, composite surfaces, and irregular surface regions. Unlike the IsoMesh approach, the Paver technique creates a mesh by first subdividing the surface boundaries into mesh points, and then operates on these boundaries to construct interior elements

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TetMesh

Arbitrary solid mesher generates tetrahedral elements within Patran solids defined by an arbitrary number of faces or volumes formed by collection of triangular element shells. This method is based on MSC plastering technology.

2-1/2D Mesher

Transforms a planar 2D mesh to produce a 3D mesh of solid elements, using sweep and extrude operations.

Accessing the Finite Element Application

All of Patran’s finite element modeling capabilities are available by selecting the Finite Element button on the main form.

Like the Geometry Application, the top portion of the Finite Element form contains three keywords, Action, Object, and Method; these remain the same throughout all activities. Finite Element (FE) Meshing, Node and Element Editing, Nodal Equivalencing, ID Optimization, Model Verification, FE Show, Modify and Delete, and ID Renumber, are all accessible by setting the Action/Object/Method combination on the Finite Elements form.

For complete descriptions on creating geometry models, see the Patran Reference Manual, Part 3: Finite Element Modeling.

Chapter 7: Setting Up, Monitoring, and Debugging the Analysis

7 Setting Up, Monitoring, and Debugging the Analysis

Solution Type

Analysis Procedures

Translation Parameters

Solution Parameters

Subcases

Subcase Parameters

Execution Procedure for MSC Nastran Implicit Nonlinear from the Command Line

Monitoring the Analysis

Debugging the Analysis

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Solution TypeMSC Nastran can simulate many different types of structural response. In general an analysis type can be either static or dynamic. In a static analysis, loads and boundary conditions are applied to a model and the response is assumed to remain the constant over time. In dynamic analysis the response changes over time. In MSC Nastran, both static and dynamic analysis may simulate linear response or nonlinear response. SOL 600 incorporates the formulations and functionality to simulate nonlinear static and dynamic structural responses. The specific procedure MSC Nastran will run is specified on the Executive Control Statement by the ID entry. SOL 600 represents multiple types of analysis procedures, any of which can be specified by the ID value on the SOL 600 Executive Control Statements.

Specifying the Solution TypeMSC Nastran Implicit Nonlinear (SOL 600) is designated with the following Executive Control Statement in the MSC Nastran Bulk Data file, where the ID entry indicates which analysis procedure is to be run.

SOL 600 Executive Control StatementThe SOL 600,ID Executive control statement is as follows:

SOL 600, ID PATH= COPYR= NOERROR MARCEXE=SOLVE NOEXIT OUTR=op2,xdb,pch,f06,eig,dmap,beam, sdrc,pst,cdb=(0, 1, 2, or 3) STOP= CONTINUE= S67OPT= MSGMESH= SCRATCH= TSOLVE= SMEAR PREMGLUE MRENUELE= MRENUGRD= MRENUMBR= SYSabc= S6NEWS=

See SOL 600 Executive Control Statement: (Ch. 2) for an explanation of some of the options.

Defining the Solution Type in PatranPrior to selecting a Solution Type, check to see that under Analysis Preferences the Analysis Code is set to MSC Nastran, and the Analysis Type is set to Structural.

To set the Solution Type:

1. Click on the Analysis Application button.

2. On the Analysis Application form, click Solution Type... and select Implicit Nonlinear from the list of available Solution Types.

Entry Description

SOL 600, ID Creates Marc input and optionally executes Marc from inside MSC Nastran Implicit Nonlinear (SOL 600).

167CHAPTER 7Setting Up, Monitoring, and Debugging the Analysis

References

• Analyze - Setting Up a File for Analysis (p. 9) in the MSC.Patran Reference Manual)

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Analysis Procedures The subcase is the MSC Nastran mechanism for associating loads and boundary conditions, output requests, and various other parameters to be used during part of a complete run. Each subcase can be designated with one of the analysis types listed below. For each analysis type, you will define the Solution Parameters and Output Requests; these collectively constitute the Analysis Procedures.

In MSC Nastran, Case Control options provide the loads and constraints, and load incrementation method, and controls the program after the initial elastic analysis. Case Control options also include blocks which allow changes in the initial model specifications. Case Control options can also specify print-out and postprocessing options.Each set of load sets must be begin with a SUBCASE command and be terminated by another SUBCASE or a BEGIN BULK command. If there is only one load case, the SUBCASE entry is not required. The SUBCASE option requests that the program perform another increment or series of increments. The input format for these options is described in MSC Nastran Quick Reference Guide.

Analysis TypesAnalysis Types for subcases in SOL 600 include the following:

Type Description

Linear Static Static stress analysis is used when inertia effects can be neglected. During a linear static step, the model’s response is defined by the linear elastic stiffness at the base state, the state of deformation and stress at the beginning of the step. Contact conditions cannot change during the step - they remain as they are defined in the base state.

Nonlinear Static Nonlinear static analysis requires the solution of nonlinear equilibrium equations, for which MSC Nastran Implicit Nonlinear uses Newton’s method. Many problems involve history dependent response, so that the solution is usually obtained as a series of increments, with iteration within each increment to obtain equilibrium. For most cases, the automatic incrementation provided by MSC Nastran Implicit Nonlinear is preferred, although direct user control is also provided for those cases where the user has experience with a particular problem.

Normal Modes This solution type uses eigenvalue techniques to extract the frequencies of the current system, The stiffness determined at the end of the previous step is used as the basis for the extraction, so that small vibrations of a preloaded structure can be modeled.

Transient Dynamic

This solution procedure integrates all of the equations of motion through time. For linear systems, the dynamic method, using the Single-Step Houbolt operator, is unconditionally stable, meaning there is no mathematical limit on the size of the time increment that can be used to integrate a linear system. However, the time step or the maximum allowable error parameter must be small enough to ensure an accurate solution.


Specifying the Analysis Type for a SubcaseThe Analysis Type is designated by specifying an ID as part of the Executive Control Statement in the MSC Nastran Bulk Data file.

For example SOL 600, 101 selects the Linear Statics Solution Sequence.

The following Solution Sequences are available with SOL 600.

Nonlinear Transient Dynamic

This solution type is used when nonlinear dynamic response is being studied. For most cases, the automatic incrementation provided is preferred, although direct user control is also provided for those cases where the user has experience with a particular problem.

Buckling Eigenvalue buckling estimates are obtained. Classical eigenvalue buckling analysis (e.g., “Euler” buckling) is often used to estimate the critical (buckling) load of “stiff” structures. “Stiff” structures are those that carry their design loads primarily by axial or membrane action, rather than by bending action. Their response usually involves very little deformation prior to buckling.

Creep This analysis procedure performs a transient, static, stress/displacement analysis. It is especially provided for the analysis of materials which are described by the CREEP material form.

Viscoelastic (Time Domain)

This is especially provided for the time domain analysis of materials which are described by the VISCOELASTIC, TIME material forms. The dissipative part of the material behavior is defined through a Prony series representation of the normalized shear and bulk relaxation moduli, either specified directly on the VISCOELASTIC, TIME material forms, determined from user input creep test data, or determined from user input relaxation test data.

Body Approach Body Approach enables you to position rigid bodies to just touch deformable bodies before beginning a subsequent Load Step. No analysis is actually performed during a Body Approach step.It is used commonly in multi-forming simulations where bodies are brought just into contact before the analysis begins.

Entry Description

SOL 600,ID Creates Marc input and optionally executes Marc from inside MSC Nastran Implicit Nonlinear (SOL 600).

Type Description

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Defining Analysis Type for a Subcase in Patran

To define the Analysis Type for a subcase:

1. Click on the Analysis Application button to bring up the Analysis Application form.

2. Click Solution Type... and click on the Implicit Nonlinear toggle.

3. Click Subcases... and select an analysis type from the Analysis Type pull-down menu.

Table 7-1 Solution Sequences

SOL Number SOL Name Description

101 SESTATIC Statics

103 SEMODES Normal Modes

105 SEBUCKL Buckling

106 NLSTATIC Nonlinear or Linear Statics

108 SEDFREQ Direct Frequency Response

109 SEDTRAN Direct Transient Response

111 SEMFREQ Modal Frequency Response

112 SEMTRAN Modal Transient Response

129 NLTRAN Nonlinear or Linear Transient Response

153 NLSCSH Static structural and/or steady state heat Transfer analysis with options: Linear or nonlinear analysis

159 NLTCSH Transient structural and/or Transient heat Transfer analysis with options: Linear or nonlinear analysis

Other, as specified in the QRG, are available for special use.

Note: There are two ways to set up SOL 600 analysis jobs in Patran. The first is to select Implicit Nonlinear as the Solution Type. This option gives you access to most of the various analysis capabilities and numerical controls available through SOL 600. The second option uses the same menus as if you were setting up a non-SOL 600 analysis, such as a SOL 106 or 129, but simply changes the executive command line. To use this second option, simply set up your analysis the same way you would a non-SOL 600 job and click on the SOL 600 Run toggle on the Solution Parameters form.


References

• Solution Types (p. 261) in the Patran Interface to MSC Nastran Preference Guide)

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Translation ParametersTranslation parameters define file formats, numerical tolerances, processing options, numbering offsets, and include files.

Specifying the Translation ParametersThere are numerous translation parameters for SOL 600. Most of these parameters start with the letters M, MR or MARC and follow the format MXXXX, MRXXXX, or MARCXXX. See the MSC Nastran Quick Reference Guide

Defining Translation Parameters in Patran

To set translation parameters:


2. On the Analysis Application form, click Translation Parameters...


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References

• Translation Parameters (p. 255) in the Patran Interface to MSC Nastran Preference Guide)

Data Output Defines the type of data output.

• XDB Buffer Size For the XDB results file, defines the buffer size used for accessing results.

Tolerances • Division - prevents divide by zero errors.

• Numerical - determines if two real values are equal.

• Writing determines if a value is approximately zero when generating a Bulk Data entry field.

Bulk Data Format • Sorted Bulk Data -

• Card Format -

• Grid Precision Digits - Specifies where to round off a grid point coordinate before it’s written out to the bdf file. For example if this value is specified as 2 the number 1.3398 will be written out as 1.34.

Node Coordinates Defines which coordinate frame is used when generating the grid coordinates.

Number of Tasks Represents the number of processors to be used to run an analysis. It is assumed that the environment is configured for distributed parallel processing.

Numbering Options... Subform used to indicate offsets for all IDS to be automatically assigned during translation.

Bulk Data Include File... Prompts you for the filename of the include file.


Solution ParametersSolution parameters control a range of functions in the SOL 600 analysis. Functions such as selecting the solver type, establishing a restart, specifying domain decomposition are all part of the solution parameters.

Specifying Solution ParametersSolution Parameters are designated in the Parameters portion of the MSC Nastran Input file with the following entries.

Entry Description

INCLUDE Inserts an external file into the input file. The INCLUDE statement may appear anywhere within the input data file.

NLPARM Selects the parameters used for nonlinear static analysis.

TSTEPNL Selects integration and output time steps for a nonlinear dynamic analysis.

NLSTRAT Defines strategy parameters for nonlinear structural analysis.*

NLAUTO Defines parameters for automatic load/time stepping.*

Note: *NLSTRAT and NLAUTO defaults are appropriate for most analyses and these entries are not normally required.

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Defining Solution Parameters in Patran

To set solution parameters:


2. On the Analysis Application form, click Solution Type..., and select Implicit Nonlinear. Then click Solution Parameters...

Solver Options Specifies the solver to be used in numerically inverting the system of linear equilibrium equations.

Contact Parameters Defines options for detecting and handling contact.

Direct Text Input This subform is used to directly enter entries in the File Management, Executive Control, Case Control, and Bulk Data sections of the MSC Nastran input file.

Restart Parameters Includes a Restart option in the MSC Nastran input file.

Advanced Job Control Sets alternate versions of the solver and alternate formats for the results file.

Domain Decomposition Designates that domain decomposition be done manually, semi-automatically, or automatically.


References

• For more information on Solver Options, see Numerical Methods in Solving Equations (p. 58) in the .

• For more information on Contact parameters, see Contact Parameters Subform (p. 299) in the Patran Interface to MSC Nastran Preference Guide.

• For more information on Restart Parameters, see Restart Parameters Subform (p. 308) in the Patran Interface to MSC Nastran Preference Guide.

• For more information on Domain Decomposition, see Domain Decomposition (p. 311) in the Patran Interface to MSC Nastran Preference Guide.

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SubcasesCreating multiple subcases allows you to efficiently analyze multiple load cases in one run. Each subcase is a collection of loads and boundary conditions, output requests, and other parameters. For nonlinear analysis runs the starting point of each subcase is the ending point of the previous subcase.

Specifying SubcasesEach subcase is designated with the following Case Control Command.

Defining Subcases in Patran

To define a subcase:


Entry Description

SUBCASE Delimits and identifies a subcase.


2. From the Analysis Application form click Subcases...

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Subcase Name Specifies a name for a new subcase.

Available Load Cases Selects one or more available load cases to be applied to the new subcase.

Subcase Options

• Subcase Parameters Controls load increment and iteration parameters for the subcase. Also defines the nonlinear effects for the subcase. See Subcase Parameters, 181.

• Output Requests Defines the nodal and element results quantities and also determines the frequency of results reporting. See Output Requests (Ch. 8).

• Direct Text Input This subform is used to directly enter entries in the File Management, Executive Control, Case Control, and Bulk Data Sections of the MSC Nastran input file.

• Select Superelements Defines which superelements are to be included in the subcase.

• Select Explicit MPCs Selects explicit MPCs to be included in the subcase.


Subcase ParametersThe subcase parameters represent the settings in MSC Nastran Case Control and Bulk Data Section that take effect within a subcase and do not affect the analysis in other subcases. Subcase parameters are dependent on the type of analysis being performed. The set of subcase parameters applicable for each analysis type are described in the following sections. For more information, see Solution Methods and Strategies in Nonlinear Analysis (Ch. 3).

Specifying Static Subcase ParametersFor static nonlinear analysis the subcase parameters control the iteration process and the load incrementation.

Defining Static Subcase Parameters in Patran

1. Click the Analysis Application button to bring up Analysis Application form. Click on Solution Type and check to see that Implicit Nonlinear is the selected Solution Type, then click OK.

2. On the Analysis form select Subcases... and choose Static from the Analysis Type pull-down menu.

Entry Description

NLPARM Nonlinear Static Analysis Parameter Selection.

NLPCI Defines a set of parameters for the arc-length incremental solution strategies in nonlinear static analysis.

NLAUTO Defines parameters for automatic load/time stepping used in SOL 600.

NLSTRAT Defines strategy parameters for nonlinear structural analysis used in SOL 600.

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3. Click Subcase Parameters...

Linearity Prescribes the nonlinear effects for the subcase.

Nonlinear Solution Parameters

• Nonlinear Geometric Effects Defines the type of geometric or material nonlinearity to be included in the subcase.

• Follower Forces Specifies whether forces will follow displacements.

Load Increment Params... Defines whether the load increments will be fixed or adapted in each iteration and the method by which adaptive load increments will be determined.

Iteration Parameters... Sets forth the iterative procedures that are employed to solve the equilibrium problem at each load increment.

Contact Table... Activates, deactivates, and controls the behavior of contact bodies in the analysis.

Active/Deactive Elements... Defines groups of elements to be active or deactive for the subcase.


Specifying Normal Modes Subcase ParametersFor normal modes nonlinear analysis the subcase parameters control the eigenvalue extraction techniques and the range of frequencies to be targeted for extraction.

Defining Normal Modes Subcase Parameters in Patran


2. On the Analysis form select Subcases... and choose Normal Modes from the Analysis Type pull-down menu.


Entry Description

EIGR Defines data needed to perform real eigenvalue analysis.

EIGRL Defines data needed to perform real eigenvalue (vibration or buckling) analysis with the Lanczos method.


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Extraction Method Defines the method to use to extract the real eigenvalues.

Lancozs Parameters

• Number of Modes Indicates an estimate of the number of eigenvalues to be located.

• Lowest/Highest Frequency Defines the lower and upper limits to the range of frequencies to be examined.

Sequence Checking Requests that Sturm sequence checking be performed on the extracted eigenvalues.


Specifying Buckling Subcase ParametersFor nonlinear buckling analysis the subcase parameters control the eigenvalue extraction techniques and the range of frequencies to be targeted for extraction.

A METHOD command is specified in the desired subcase and selects an EIGB, or EIGRL Bulk Data entry.

The linear buckling load analysis is correct when you take a very small load step in increment zero, or make sure the solution has converged before buckling load analysis (if multiple increments are taken).

In a buckling problem that involves material nonlinearity (for example, plasticity), the nonlinear problem must be solved incrementally. During the analysis, a failure to converge in the iteration process or nonpositive definite stiffness signals the plastic collapse.

For extremely nonlinear problems, the EIGB option cannot produce accurate results. In that case, use NLSTRAT options to specify an arc-length method that allows automatic load stepping in a quasi-static fashion for both geometric large displacement and material (elastic-plastic) nonlinear problems. The option can handle elastic-plastic snap-through phenomena. Therefore, the post-buckling behavior of structures can be analyzed.

The eig option must be specified with the OUTR option on SOL 600 Executive Control statement if op2, xdb, pch, or .f06 options are specified and Marc performs natural frequency or buckling eigenvalue analysis. The reason it must be provided on the SOL entry is to enable MSC Nastran to create DMAP on the fly which include the LAMA data block. If the eig option is omitted, eigenvectors will be present in the MSC Nastran output but no eigenvalues will be available.

Defining Buckling Subcase Parameters in Patran


2. On the Analysis form select Subcases... and choose Buckling from the Analysis Type pull-down menu.

Entry Description

METHOD Selects the real eigenvalue extraction parameters.

EIGB Defines data needed to perform buckling analysis.

EIGRL Defines data needed to perform real eigenvalue (vibration or buckling) analysis with the Lanczos method.

MARCRBAL This parameter is used for eigenvalue analysis where natural frequencies or buckling modes need to be calculated using the deformed geometry from a nonlinear analysis.


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Specifying Transient Dynamic Subcase ParametersFor transient dynamic nonlinear analysis the subcase parameters control the iteration process and the load incrementation.

Defining Transient Dynamic Subcase Parameters in Patran


2. On the Analysis form select Subcases... and choose Transient Dynamic from the Analysis Type pull-down menu.

Extraction Method Defines the method to use to extract the real eigenvalues.

Lancozs Parameters

• Max # of Modes Indicates the maximum number of eigenvalues to be located.

• Max # of Modes w/Pos. Eigenvalues

Indicates the maximum number of positive eigenvalues to be located.

Entry Description

TSTEPNL Nonlinear Dynamic Analysis Parameter Selection.

NLAUTO Parameters for automatic load/time stepping.

NLSTRAT Strategy Parameters for nonlinear structural analysis.



Linearity Prescribes the nonlinear effects for the subcase.

Nonlinear Solution Parameters



Load Increment Params... Defines whether the load increments will be fixed or adapted in each iteration and the method by which adaptive load increments will be determined.

Iteration Parameters... Sets forth the iterative procedures that are employed to solve the equilibrium problem at each load increment.


Active/Deactive Elements... Defines groups of elements to be active or deactive for the subcase. Note that this option uses the equivalent of direct text input (MARCIN option) and is not explicitly supported by SOL 600.


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Specifying Creep Subcase ParametersThe creep analysis option is activated in MSC Nastran through the CREEP Bulk Data entry. The creep time period and control tolerance information are input through the MARCAUTO=1 parameter. This option can be used repeatedly to define a new creep time period and new tolerances. These tolerances are defined in the section on Creep Control Tolerances. Alternatively, a fixed time step can also be specified through the MARCAUTO parameter. In this case, no additional tolerances are checked for controlling the time step.

Creep analysis is often carried out in several runs using the RESTART Bulk Data entry. Save restart files for continued analysis. The RESTART entry allows you to reset the parameters defined in MARCAUTO upon restart.

Defining Creep Subcase Parameters in Patran


2. On the Analysis form select Subcases... and choose Creep from the Analysis Type pull-down menu.

Entry Description

NLPARM Nonlinear Static Analysis Parameter Selection.

MATVP Defines creep characteristics based on experimental data or known empirical creep law.

MARCAUTO Determines which Marc’s increment option is used.

RESTART Specifies writing or reading of restart data for Nonlinear Analysis when Marc is executed from MSC Nastran.


3. Click Subcase Parameters...:


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Specifying Body Approach Subcase ParametersFor body approach analysis the subcase parameters control the iteration process and the load incrementation.

Defining Body Approach Subcase Parameters in Patran


2. On the Analysis form select Subcases... and choose Body Approach from the Analysis Type pull-down menu.

Creep Solution Parameters

• Procedure Selects Implicit or Explicit Creep method.



Increment Type Defines a fixed or adaptive increment method.

• Adaptive Increment Parameters...

For adaptive methods, sets boundaries for incrementation.

Iteration Parameters Sets forth the iterative procedures that are employed to solve the equilibrium problem at each load increment.


Active/Deactive Elements... Defines groups of elements to be active or deactive for the subcase.

Entry Description

BCMOVE Specifies movement of rigid surfaces.



Body Approach Parameters

• Total Time Places a time step option in the Load Step.

• Synchronized If ON, specifies that when the first rigid body comes into contact, the rest stop moving.


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Execution Procedure for MSC Nastran Implicit Nonlinear from the Command LineAfter the generation of the input file is complete, it is submitted for execution as a batch process (MSC Nastran is not an interactive program). Once the input file has been submitted, you have no additional interaction with MSC Nastran until the job is complete except that you can terminate the job prior to completion if it becomes necessary and monitor several keys files such as, .f04, .sts, .log, etc.

MSC Nastran is executed with a command called nastran. (Your system manager may assign a different name to the command.) The nastran command permits the specification of keywords used to request options affecting MSC Nastran job execution. The format of the NASTRAN command is:

nastran input_data_file [keyword1 = value1 keyword2 = value2 ...]

where input_data_file is the name of the file containing the input data and keywordi=valuei is one or more

optional keyword assignment arguments. For example, to run an a job using the data file example1.dat, enter the following command:

nastran example1

See The nastran Command (p. 2) in the MSC Nastran Quick Reference Guide.

The details of submitting an MSC Nastran job are specific to your computer system— contact your computer system personnel or your MSC Nastran 2012 Installation and Operations Guide for further information.


Using Patran to Execute MSC NastranThe Analysis Application controls the execution of MSC Nastran.

When the Action is set to Analyze, the Method is set to Full Run, and the Apply button is selected from the Analysis form, a jobname.bdf file is created which contains the analysis model, and the P3TRANS.INI script is spawned by Patran. This script controls the analysis process outside and independent of Patran.

When the analysis is successfully completed, one or more output file is produced. These output files can be directly imported or attached into the Patran database for postprocessing by setting the Action menu to Access Results.

How to Tell When the Analysis is DoneIf you submit the job from the MSC Nastran icon (i.e., outside Patran), as long as the parent window the job was run from is active, the analysis is still running. If you submit the job from within Patran and use -stdout when you execute Patran, you can look in the Patran parent window and it will tell you when it submits the Nastran job, and also when the Nastran job is completed. Of course you can always use the Analysis Manager. Once the job is complete look in the parent window to see what files were generated.

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How to Tell if the Analysis Ran SuccessfullyLook in the working directory and you will see the typical jobname.f06, jobname.f04, jobname.log. If these files are there, you successfully submitted the Nastran job. If you submitted a job with SOL 600, xxx as the executive command, there will also be some jobname.marc.xxx files in the subdirectory. These are the files from the Marc run. To see if the run was successful, open jobname.marc.sts and look for the number at the bottom. For most analyses, 3004 means the run was successful.

For others, such as thermal contact, generation of DMIG matrices for subsequent use, other exit codes such as 3031, 3030, 3031, and 3022 mean a successful run. Exit code 13 signifies a dta input error which must normally be evaluated by MSC development. Exit 3015 means the job diverged and requires additional time steps, changes to convergence controls or other changes to the input. A brief description of most errors is provided at the end of this jid.mar.out file.

If there are no jobname.marc.xxx files, check to make sure you can submit Marc jobs successfully. At the end of the jobname.f06 file, Nastran will tell you what command it used to submit the Marc job. Take this command to a command prompt and enter it to see why the Marc job wasn’t submitted. If you can go to a command window and type in “run_marc jid-jobname“ and it finds the Marc executable and runs the jobname.dat Marc input file, you can just use PATH=2 on the SOL 600 command line.


Monitoring the AnalysisThe nastran command permits the specification of keywords used to request options affecting MSC Nastran job execution. The format of the NASTRAN command is:

nastran input_data_file [keyword1 = value1 keyword2 = value2 ...]

where input_data_file is the name of the file containing the input data and keywordi=valuei is one or

more optional keyword assignment arguments.

Use the keyword xmonast to monitor a MSC Nastran job as described below.

MSC Nastran Implicit Nonlinear provides a status file (jobname.marc.sts) that can be queried periodically to see how the analysis is progressing and if the job is completed. The file will report the information relating to the progress of the analysis, with warning and informative messages.

The file review is especially important when manual or automatic time stepping procedures are being used to step through an analysis procedure. One line is written after each successful increment. An example file output is shown below.

Figure 7-1 information Summary of Job: nas.cant_bmsm.marc

The first column shows the procedural step, while the second column shows the increment number. Note that not every increment size is equal, as can be seen in the “TimeStep of the INC” column. For this example, the third increment size is larger than the first two increments, which means that the procedure is satisfied that equilibrium is being satisfied and that it has increased the time step size to take advantage of the better convergence characteristics. The third column (“cycle# of the Inc) indicates the number of

xmonast xmonast={yes|no|kill Default: No

Indicates if XMONAST is to be run to monitor the MSC Nastran job. If “xmonast=yes” is specified, XMONAST will be automatically started; you must manually exit XMONAST when the MSC Nastran job has completed. If “xmonast=kill” is specified, XMONAST will start and will automatically exit when the MSC Nastran job has completed.

Example: nastran example xmon=kill

This example runs the XMONITOR utility while the MSC Nastran job is running. Once the job completes, the XMONITOR program is automatically terminated.

dip

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Newton Raphson iterations made during this increment. The next three columns show the increment information, while the next six columns show the numbers for the total analysis.

Sepa means number of contact separations, cut refers to the number of time step size cutbacks, and split refers to the number of increment splits due to contact penetration.

If the increment size becomes small, and there are a number of increments of this size, the analysis has probably “stalled”, indicating that convergence is becoming very difficult to attain for the problem. A review of the model is indicated.

The max disp column provides a valuable means to tell if a job is diverging or in some other way not proceeding correctly.

Editing a MSC Nastran Input FileThere may be instances when you want to directly edit the MSC Nastran Bulk Data file. Some experienced MSC Nastran users may want to add options directly to specific Parameters and Bulk Data entries. Patran provides direct access to the Bulk Data file as follows.

To edit an existing Bulk Data File:

1. Click on the Analysis Application icon to bring up the Analysis Application form.

2. Set the Action>Object>Method combination to Analyze>Existing Deck>Full Run.

3. Click Edit Input File...

Patran automatically looks for an existing deck name that matches the current database name and displays the existing deck.


Debugging the AnalysisMSC Nastran generates a substantial amount of information concerning the problem being executed. The .f04 file provides information on the sequence of modules being executed and the time required by each of the modules; the .log file contains system messages.

MSC Nastran may terminate as a result of errors detected by the operating system or by the program. If the DIAG 44 is set, MSC Nastran will produce a dump of several key internal tables when most of these errors occur. Before the dump occurs, there may be a fatal message written to the .f06 file. The general format of this message is

***SYSTEM FATAL ERROR 4276, subroutine-name ERROR CODE n

These messages are SEVERE WARNING, or other text ... issued whenever an interrupt occurs that MSC Nastran is unable to satisfactorily process. The specific reasons for the interrupt are usually printed in the .f06 and/or .log file.

Resolving Convergence Problems There are three major steps in getting a complete solution.

1. get the model input debugged (see section titled “Exit 13 Errors”).

2. establish initial equilibrium (see section titled “Exit 2004 Errors”).

3. getting the analysis to run to completion.

Step 1: De-Bugging the Model Input

See How to Tell if the Analysis Ran Successfully, 194.

To Debug a Failed Analysis

First, check the .sts, .f04, .f06, or .out files for licensing, disk access or format errors. The number a the bottom of the jobname.marc.sts is the Marc Exit Number. Exit 13 means there was a format error in the Marc input. Exit 2004 generally means you have unconstrained degrees of freedom or rigid body modes. 3002 means the analysis got part way through and then stopped. The complete Marc Exit message is given at the end of the jobname.marc.out file, which may also be in jobname.f06 depending on the value of COPYR.

If you get an Exit 13 check your input, make sure everything you need is in the Nastran input deck. Check for elements, grids, contact body creation, etc. An easy way to debug these type of problems is to read the jobname.marc.dat file (the Marc input file Nastran created) in to Mentat or Patran (an empty db with preference set to Marc) and see what is missing. Often you will see that some elements are missing, or that the contact bodies were not created as you would expect them. For more information see Figure 7-2.

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Figure 7-2 Flowchart to Debugging a SOL 600 Run

Submit job

Wait until process finishes.

Does .sts file exist?

Do.f06, .f04, .log files exist?

Isthere a Fatal

Error Message in .f06, .f04,

.log?

Ck Marc submit card, find in .f06 and type in a cmd window to see why Marc job

was not submitted.

no yes no

Nastran job submit from windows

w/Pause=YES, ck error msg

no

ck .sts for Exit #

fix-it - typical Nastran debug

IsExit

#=3004?

yesyes

Do youget Nastran requested

formatted output, such

as .xdb?

yes Successful completion - go on to postprocess

yes

look in jobname.marc.out for Exit # and message.

ck T160P2.exe run, on windows make sure it is in search path.

IsExit #=13?

Follow instructions in exit msg; if its a formatting error ck Marc, Volc, or read the jobname.marc.dat file into Patran’s Marc pref (or Mentat) and compare this model w/original. The problem may be displayed when the file is read in.

IsExit #=3002?

IsExit #=2004?

Some convergence increments exist, postprocess them & look for possible causes of stability loss, such as contact changes. Ck

, you may be encountering buckling modes. Do a buckling solution to see or try an arc-length method.

max

No converged increments, likely unconstrained rigid body motion exists: a.) do modal to ID them, b.) Ck equivalencing of modal integrity, c.) add constraints or weak springs, d.) see section on convergence problems.

no

no

yes

no no

yes

yes


Step 2: Establishing Initial Equilibrium

If you get an Exit 2004 it means that the model was unable to reach equilibrium for the loads and constraints applied in the initial subcase. Options include:

• adding more constraints (or equivalently soft springs) to ground the model.

• run a modal analysis to identify unconstrained rigid body modes.

Step 3: Getting the Analysis to Run to Completion

The first thing to check at this point is: “Has the solution gone as far as it can,” i.e. maybe you already have the complete solution. Sometimes nonlinear loading causes structures to buckle which may take your analysis into the post-buckled region. Depending on the type of buckling, you may be simply trying to drive your analysis farther into the post-buckling range which may not give you the information you are after. For example, if you load a frame structure with a load that exceeds the critical buckling load and the analysis is simply working to drive the structure further into a plastic hinge.

Things to Do to Fix Non-Convergence

• Check the .sts, .log, and .out files for Exit 2004, 3002, or 3003 format error messages.

• If using fixed load incrementation try using a smaller time step, or use the automatic cutback feature, or use adaptive time-stepping.

• Try running an eigenvalue buckling solution to see if you’ve passed a critical buckling load or examine the stresses and strains to see if some portion of the structure has failed (if MATF failure criteria was not included in the modal).

• If using Contact set the Contact Tolerance Bias to 0.9, particularly if doing shell contact (done by default in SOL 600).

• Turn on Quasi-Static inertial damping or Non-Positive Definite to eliminate un-constrained rigid body motions (done by default in SOL 600).

• Try running an eignevalue modal solution to identify unconstrained rigid body modes.

• Try using an arc-length method - you may be encountering local buckling.

• Look at any available results of converged increments.

• If doing contact try a different contact tolerance value.

• Isolate Non-linearities and add them one at a time.

• Making Sure Appropriate Non-linearities are Included.

• Check Material Stability – Make sure the entire strain range is covered by the material data.

For complex models involving multiple forms of nonlinear behavior the “tried and true” approach (particularly if you are new to this type of problem) is to start with a linear model and add non-linearities one at a time. Alternatively, remove the non-linearities one at a time until it runs. This approach helps you determine which type of non-linearity is causing the convergence problem. If you have contact, remove it and let the bodies “pass through” one another or replace the contact condition with an equivalent displacement constraint. If you have nonlinear materials replace them with simple elastic


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ones. Add the non-linearities back one at a time, making sure the behavior is reasonable and correct. Look at reaction forces and displacements of any converged increments to make sure they are of the expected magnitude.

If you run the analysis and it doesn’t run at all, or ends before completing, you will get an error message in the .OUT or .LOG file that will give you an indication of what the problem is. Do a text search on the word “error” in the .OUT file. The first thing to check is to make sure you were able to get a license to run the job. Licensing problems are one of the most common reasons for a run to fail. If you are sure you have a license and submit the job correctly you should get a .OUT file that will end with an Exit # preceded by a description of why the run stopped. Common Exit #’s are: Exit 3004 – means success, i.e. the job ran to completion and did everything you asked it to. Exit 13 – means you have a syntax error in the input file. You should check the input syntax of the line the error message points to, but it is likely that the actual error was in the input block PRIOR to where the message points. Exit 2004 – typically means non-convergence due to rigid body motions. See recommendations for Equilibrium. Exit 3002 – this means the analysis ran into convergence problems part way through and did not complete. Any Exit Message of 3000 or higher means there are converged increments. Plot the converged increments to see what is going on. See Technical Application Note 4575 or Appendix A of Volume C: Program Input for a more complete list with suggested “fixes.”

Things to consider if your model doesn’t converge:

1. Equilibrium - Make sure your model has LBC’s and contact conditions that will ensure force equilibrium at EVERY increment/iteration and for ALL rigid-body modes (typically there are 6). When in doubt eliminate this as the source of non-convergence by intentionally over-constraining the model (or adding soft springs) and then removing constraints one at a time until you figure out the unconstrained rigid body mode. One area that is sometimes overlooked regarding equilibrium is that of the rigid body control. If you don’t specify adequate control information (e.g. you forget to add the zero that fixes the rigid body rotation value) you may have convergence problems.

2. LBC’s - When LBC’s are removed, the forces/pressures (and the reaction forces due to displacement constraints) are removed gradually over the subsequent step. The forces and pressures are always removed gradually, but the reaction forces of displacement constraints may be are removed suddenly at the beginning of the subsequent step. This sudden change in loading can cause convergence problems.

3. Stability and Collapse - Non-convergence will occur when a structural instability (i.e., buckling) mode is encountered. Buckling can occur either locally (in highly stressed area where the stability of individual elements is exceeded) or globally when the critical buckling load of any part of the model is exceeded. You may want to do a linear buckling analysis to determine the load that would buckle the least-stable part of the structure. If you suspect that you are approaching the post-buckled region here are some other things to try: a) try using Quasi-static inertial damping (turn this on under Analysis – Step Create – Solution Parameters) or one of the Arc-length methods. This will help get through the unstable region if doing a snap-through buckling problem, and may help get you past one or two elements of local buckling, but probably not more than that.) try a finer mesh (smaller elements have shorter length and so higher Pcr);


4. Materials - Make sure that the material coefficient values are realistic and that the models will support the stresses and loads developed in the model. For example if you hang a 1000 lb. weight from a perfectly plastic wire with a 0.001 in**2 cross section and a 20 ksi yield stress, the resulting 100 ksi stress cannot be supported by the (20 ksi yield stress) material and the run will not converge. Comparable behavior in bending is referred to as a “plastic hinge.” Units mis-matches will often result in this type of problem (note that this only occurs in non-linear analyses). For example, let’s say you are modeling a cantilever beam and using a perfectly plastic material model and a “follower force” tip load, and you mistakenly add an extra zero to the tip load. A plastic hinge will develop with the beam “winding up” like a spring and the analysis continuing to run until it runs out of increments (which may take a long time). If you suspect this type of problem first run the problem with a small fraction of the load to see if it will converge. If you are using an orthotropic or hyperelastic material it is possible to select combinations of material properties that will result in a non-positive definite material coefficient matrix. Normally the analysis code will warn you if you violate this requirement.

5. Contact - If there is a problem with “chattering” (a condition where a particular node jumps into and out of contact thus preventing the increment from converging), you can go to Translation Parameters – Contact Control Parameters – Separation and set the Chattering toggle to “Suppress”. The parameters which have the biggest effect on contact behavior are “Contact Distance Tolerance”, D (see Fig. 1), “Bias Factor,” B (see Fig. 2) -) and “Separation Force.” The default uses D = 1/20 the of the element edge length. You can find the specific value in the.OUT file and try a larger or smaller value, whichever you feel is most appropriate. The default on the bias value is 0, if having problems with contact one of the first things to try is to over-ride this value on the Analysis – Translation Parameters – Contact Detection – Contact Parameters form with 0.9. Another option would be to increase the separation force (which defaults to 0) to prevent chattering. When considering contact problems look for places (such as corners and other discontinuities) where one contact surface may “slip” off.

Standard Steps to Resolving Convergence Problems:

If your model doesn’t run, or stops pre-maturely: FIRST, READ THE MESSAGES IN The .sts, .f06, .log and .out files. Common causes of the run to fail include:

1. unconstrained rigid body modes.

2. you're in the post-buckled region.

3. problems resolving contact.

4. some part of the model/material is “over-constrained” such that the given displacement solution doesn’t change when the load is increased (i.e. individual elements are buckling locally), this type of non-unique solution can prevent convergence.

After trying the obvious things, talk to other experienced users about possible reasons your run isn’t

working. In one case, a user was using the standard element formulation with = 0.5 and hex/21 elements and his model would not converge even though there were no obvious problems. For this case, using the constant volume formulation should provide a unique solution and allow convergence, unless

= 0.5 causes numerical problems. In that case you should use the Herrmann elements which should


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take care of the numerical problems as well as the non-unique solution problem. If these options don’t work, you could try using reduced integration, which may solve both problems at once, but may have problems with energy-free or “spurious” deformation modes (also called “hour-glassing”), although the analysis has built-in hour glass stabilization. Also, try quasi-static inertial damping or an arc-length methods.

Here are some other things to try:

a. try a finer mesh;

b. modify the material model: if its simple elastic-perfectly plastic with large plastic strains try using constant volume Herrmann elements, if using a hyper-elastic material model try lowering Nu from 0.5 to maybe 0.49 or so (or lower if you have to); make sure its based on test data that includes the type of behavior you are trying to model (i.e. if your test data is from a uni-axial tensile test and you are modeling a pressurized cylinder, which is a bi-axial stress state, try analyzing a simple bi-axial sheet to see if your hyper-elastic material model will successfully handle bi-axial stress states; if not you may have to include some bi-axial test data (hyper-elastic models based on test data should include at least 2 "modes," although the program has a new Arruda-Boyce model which often yields better results with limited experimental data);

c. simplify - if the model you are running is a 3D cylinder made of solid elements, run a 2D axi-symmetric test case to check out the mesh refinement and material model. If not in the post-buckled region try: 1) look at deformed shape to see if it looks reasonable. Remember that static equilibrium must be maintained at every step; 2) check reaction forces to see if the load path is reasonable; 3) look for highly distorted elements, both visually and in the .out file - if you find any you may need to go back and refine your mesh in that area to keep those elements well-behaved, i.e. converging, or use adaptive re-meshing. Although distorted elements will normally just give you bad results but not necessarily prevent convergence. Typically linear elements (i.e. quad/4 instead of quad/8) do better in analyses where severe distortion is expected. 4) if using contact elements you may be able to ease convergence problems by simplifying the contact interaction: a) look at the .sts file for the # of increment splits and # of separations to see if contact is the problem; b) set bias to 0.9, increase (or decrease) the contact tolerance distance, suppress chattering; c) modify the contact table to eliminate suspected trouble areas (at least as a diagnostic measure); d) look for areas where contact bodies may be “sliding off. 5) PAY ATTENTION TO THE MESSAGES IN THE .STS, .LOG, .f06 AND .OUT FILES, they may tell you why the model was not translated or convergence was not reached and the analysis terminated. 6) if non-convergence relates to inelastic behavior of the material, such as in a plasticity analysis, make sure there are no "plastic hinges" formed, where static equilibrium cannot be achieved because the material is not strong enough, in this case all the iterations go to deforming the body around the plastic region and static equilibrium may never be reached. 7) when doing a hyperelastic material analysis the material model may be unpredictable since the coefficients are generally quite unintuitive. The run may not converge simply because the material model, while it may look reasonable, may actually be inherently unstable (things like negative energy behavior, etc.) 8) make sure you aren't stuck at a stability


bifurcation point, (i.e. at a buckling mode), what may be happening is that there are 2 valid (post-buckling in this case) equilibrium paths and the code flips back and forth between them preventing convergence; the way to get past this is to make the problem dynamic and use the inertia of the body to “select” the appropriate equilibrium path.

Again, the “tried and true” method is to start with a linear model and add non-linearities one at a time, or remove nonlinearities.

Consider changing the Contact Distance Tolerance. If you run into contact-related convergence problems this is one of the first things to try.

Standard Exit MessagesPlease refer to the MSC Nastran Quick Reference Guide and the MSC Nastran Reference Manual for exit codes and numbers.

SOL 600 Exit Numbers and their Interpretation:

Most SOL 600 jid.marc.out files end with an EXIT MESSAGE & NUMBER. This exit number is located in the last few lines of the output file (.out file) of the SOL 600 run. If a.out file is not created, that sometimes points to an installation issue or a FATAL ERROR or SEVERE WARNING will be present in the jid.f06 file. EXIT #3004 is GOOD. It indicates a successful run. Most other exit numbers indicate a failed run. Below are some common exit numbers that a user might encounter. Information on more exit numbers are given in Volume C (Program Messages) of SOL 600 documentation, which contains the full list of exit numbers/messages.

1. Exit 13: This exit # indicates an error with the SOL 600 input file. SOL 600 gives an exit 13 if it does not understand some term in the input deck. Causes include either the input deck is written out incorrectly, or because the set-up is inconsistent (user set-up issue) e.g. user may set up a shell model and not provide shell thickness; or the user creates but neglects to assign material properties.

Fixing ‘exit 13’ errors: Go to the.out file and search for the keyword: error (*** error). The error message listed there points to the section/line of the input deck causing the problem. The user has to figure out why that is incorrect, along the lines discussed in the above paragraph or send the MSC Nastran implicit file (2) to MSC.

2. Exit 2004: If the user gets this message as soon as the job is submitted - i.e. no increments/iterations successfully completed, it points to an unconstrained mesh. Exit 2004, from a numerical stand-point, indicates a non-positive definite stiffness matrix. From a model set-up stand-point, this implies that the mesh is not constrained in space, i.e. either incomplete (or no) BCs have been applied to the model. User needs to check the BCs.


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If this exit number comes after the analysis begins i.e. after some increments are done, the reasons are to do with the mechanics of the model e.g. formation of a plastic hinge, or due to buckling, or a deformable body sliding due to lack of friction in the model. The fix for each situation is unique to the underlying physics of the model e.g. adding friction if that reflects the actual situation. Turning on the “Suppress Rigid Body Motion” option on the NonLinear Solution parameters can help in some cases.

3. Exit 1005, Exit 1009: If this exit message (exit 1005) comes as soon as the job is submitted i.e. at the first assembly of the first iteration, it indicates a meshing problem. Re-check mesh and re-mesh. In most cases, this error message(s) comes after an analysis has started. Both exit numbers 1005 & 1009 indicate excessive element deformation during a particular load increment, in a particular iteration. The way to get around this error is to reduce step size. However SOL 600 does that automatically and if the problem still persists it gives exit 3015/3009. See explanations below on exit 3015/3009.

4. Exit 3002: This message indicates that SOL 600 has reached the user-specified upper limit on the number of (Newton-Raphson) iterations within a load increment. The default is set to 25. SOL 600 keeps iterating 25 times, and tries to converge to a solution for that increment. If that does not happen, SOL 600 will cut back the load (by half) and re-solve that increment. Sometimes this is not enough to get convergence, and it will exit with 3002. One way to get around this exit message is to increase the # of iterations, but that may not be the best way, since 25 is a high number to begin with. Treat this exit message the same way you would treat exit 3015/3009 i.e. look at the model to see what is causing it to not converge (i.e. what is causing it difficulty at that stage of the analysis) and make changes accordingly. If this exit message shows up in the first increment of the run, it could happen if the residual loads are very low in the model to begin with. Check the output file to confirm this. Switching from relative to absolute criterion will help, as might switching from load to displacement-based criteria.

5. Exit 3015, Exit 3009:

Exit 3009: This exit number indicates that SOL 600 cuts back to a the time-step size too small for the analysis to continue. The load stepping algorithm has a cut-back feature where the load-step is automatically reduced (halved) if SOL 600 runs into certain problems (exit 3002, or exit 1005). When an increment runs into these exit numbers, it will automatically cut the load-step and re-solve that increment. If the problem continues to persist, it will cut back the load-step again. This happens until the limit of the number of cut-backs is reached. This can result in a very small time-step. In such a case, SOL 600 stops the analysis with an exit 3009. To fix this situation, the user has to look at the results up to the point of failure to understand why SOL 600 cuts back repeatedly at this stage of the analysis. An understanding of the physics of the model and/or run-time issues at this stage of the analysis is important here. The user has to make a determination and modify the model.

Exit 3015: SOL 600’s automatic load-stepping scheme is set up such that the applied load in an increment scales up (or down) depending on how easy (or difficult) the solution was in the previous increment. The degree of difficulty is determined based on the parameter: ‘desired number of recycles’ (default = 3). SOL 600 will scale down the step size until it reaches a lower limit on the step size (default = 0.001% of total time step) and then exit with # 3015. This is an indication to the user that the analysis encountered some difficulty at that stage. As before, the


user has to view the results of the run up to that point and make a determination, based on the physics of the run, as to why the analysis has problems. For more information on SOL 600’s Automatic load-stepping procedure, please see Chapter 11 of Volume A: Considerations for Non-linear Analysis, section: Automatic load-stepping.

6. Exit 2400 (seen only in contact problems):

This is seen in problems where a node on a mesh (deformable body) slides off from a rigid surface that it was in contact with. For this exit # (and for exit numbers 1005, 1009 and 2004), SOL 600 attempts an automatic internal correction if the cut-back option is turned on (this is the default). SOL 600 tries to set it right by cutting back the load step and re-solve. It keeps cutting back and if it cannot resolve the problem, you get an exit 3009. To avoid this situation altogether, during model set-up make sure that the ends of the rigid surface (or curves) are not close to the mesh. If they are, you must extend them to beyond the end of the mesh- in any direction, preferably in a manner satiate a smooth profile is maintained. If the extension of the rigid curve/surface results in a sharp corner in the rigid body, put a fillet at the corner.

General Hints for Starting out with Nonlinear Analysis in Contact:

1. Start your model simple and gradually add complexity. For example, if you have a 4-body 3-D contact problem that you are solving for the first time, initially try to run it as a 2-body contact problem, get that running, and THEN add the 3rd and 4th contact body. In general, for any nonlinear model, try to start with a simpler model and gradually add complexity.

2. A deformable-to-deformable contact problem takes more time/effort than a rigid-to-deformable contact problem. So, wherever possible, use a rigid body in place of a mesh if that is appropriate for the problem.

3. When starting with a new model, set up your model such that you get some initial results or a run- failure within a few minutes. This means that you may need to start with a coarse mesh. Once you know your model runs to completion, you can add refinement and/or complexity. As a general comment, a 1000-2000 node job would fail within few minutes if there are set-up errors. This is what we want: if the job fails, it should fail fast. These initial few runs serve the purpose of testing the set-up parameters to make sure that they work right for this model. One can expect to make a few/several runs to determine that the parameters are OK for that class of problems. Once these parameters are known, the user can apply them to other models in that class of problems. Once the job runs to completion, you can add complexity/refinement. Now the job will take longer, but we know that it will run to completion.

4. Memory issues: Make sure your machine has enough RAM to accommodate the job run. If the job goes out of RAM, the model will slow down significantly. The RAM needed for the run is listed in the.out file (look for the keyword: memory and/or workspace). Open the file, search for the key-words and pick out the largest RAM number you see. The RAM is in 'words'. Multiply by 4 to get it in bytes. If the phrase ‘out-of-core’ appears in the.out file, it means that the job went out of RAM. Typically, for large contact problems, it is recommended that the computer have 1.0+ GB of RAM but you can run non-linear jobs on computers with a fraction of that RAM.


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5. Displacement control: In general, problems with applied displacements are numerically more ‘stable’ than problems with applied forces. For example, if a cantilever beam with a point BC at the end is loaded using a force, formation of a plastic hinge can make the model go non-positive definite. With an applied displacement, this scenario is less likely.

6. Linear problems have a unique solution, but that is not always true of non-linear problems. Solutions to non-linear problems can also be quite sensitive to initial and boundary conditions. Small changes in BCs/Is can sometimes change the solution quite a bit. When tackling nonlinear problems, the user should take these into consideration.

Using Patran to Debug an AnalysisThere are many error or warning messages that may be generated by the Patran MSC Nastran Interface. The following table outlines some of these.

Description

Unable to open a new message file " ". Translation messages will be written to standard output.

If Patran tries to open a message file and cannot, it will write messages to Standard Output. On most systems, messages are written to standard output and never to a separate message file.

Unable to open the specified OUTPUT2 file " ".

The OUTPUT2 file was not found. Check the OUTPUT2 file specification in the translation control file.

The specified OUTPUT2 file " " is not in standard binary format and cannot be translated.

The OUTPUT2 file is not in standard binary format. Check the OUTPUT2 file specification in the translation control file.

Group " " does not exist in the database. Model data will not be translated.

The name of a nonexistent group was specified in the translator control file. No model data will be translated from the OUTPUT2 file.

Needed file specification missing! The full name of the job file must be specified as the first command-line argument to this program.

The Patran control file must be specified as the first on-line argument to the translator.

Unable to open the specified database " ". Writing the OUTPUT2 information to the PCL command file " ".

If Patran cannot communicate directly to the specified database. It will write the results and/or model data to a PCL session file.

Unable to open either the specified database " ", or a PCL command file, " ".

The naspat3 translator is unable to open any output file. Check file specification and directory protection.

Unable to open the NASTRAN input file " ".

Patran was unable to open a file to where the input file information will be written.


Unable to open the specified database, " " .

The forward Patran MSC Nastran translator was unable to open the specified Patran database.

Alter file of the name " " could not be found. No OUPUT2 alter will be written to the NASTRAN input file.

The OUTPUT2 DMAP alter file, for this type of analysis, could not be found. Correct the search path to include the necessary directory if you want the alter files to be written to the input file.

No property regions are defined in the database. No elements or element properties can be translated.

Elements referenced by an element property region in the Patran database will not get translated by the forward Patran MSC Nastran translator. If no element regions are defined, no elements will be translated.

Description


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Chapter 8: Output from the Analysis

8 Output from the Analysis

Overview

Output Requests

SOL 600 Results Quantities

MSC Nastran Results Quantities

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideOverview

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OverviewMSC Nastran Implicit Nonlinear (SOL 600) produces stress and strain results that may differ from those results available with SOL 106 and 129.

The Marc t16 or t19 file can be used to evaluate results. The result quantities included in the t16/t19 files are controlled using the MARCOUT Bulk Data entry. Patran can be used to postprocess nearly all types of output selected by the MARCOUT entry. In general, if the t16 file is saved and brought into Patran, the types of stresses and strains will be labeled correctly and you can easily choose which quantities to plot.

The more basic types of output (displacements, velocities, accelerations, Cauchy stress tensor and one type of strain tensor) can be translated back to standard MSC Nastran op2, xdb, punch and even f06 files using the OUTR option on the SOL 600,ID Executive Control statement. If the stresses and strains are brought back into the MSC Nastran files (op2, xdb, f06 or pch), only one type of stress and strain may be placed on the OP2 file. Generally the Cauchy Stress tensor will be available along with a user selection of one of the following strains: plastic, total or elastic in the strain measure selected for the analysis. The type of stress-strain pair brought back into the Nastran results files is specified using PARAM, MARCEKND.

Input

SOL 600 Statement Default

If SOL 600 with nothing else on the line is entered, the statement will act the same as if the following statement was used:

SOL 600,NLSTATIC OUTR=OP2

.OP2 DataOutputs in the OP2 file (as well as f06, xdb and punch) have been enhanced in the following areas:

• MPC forces are available

• SPC forces are available

• 3-D contact results are available

• Displacement, velocity, acceleration results are available

• Cauchy Stress and one type of strain (total, plastic or elastic) are available

• Beam loads are available

• Output in the MSC Nastran files is controlled the same way as in other MSC Nastran solution sequences

• Set definitions may be used to limit output for any of the above items

• Grid force

211CHAPTER 8Output from the Analysis

You must include Case Control requests such as DISPLACEMENT(PLOT)=ALL in order to obtain output in op2, xdb, punch or f06 files. In addition, OUTR requests on the SOL 600 entry must be made (for example OUTR=OP2,F06). The applicable Case Control requests for SOL 600 are DISPLACEMENT, STRESS, STRAIN, SPCFORCES, MPCFORCES, and BOUTPUT. BOUTPUT maps 3D contact to the older 2D Slideline Contact datablock (see item codes for contact in section 6 of the 2005 Quick Reference Guide).

The output interval for the t16 file (and thus the OP2 file) is controlled by the NLPARM Bulk Data entry or the MARCOTIM entry..


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Output RequestsOutput requests are made in the Case Control Section of the MSC Nastran Input file. Case Control commands are organized into three categories:

1. Output Control - defines how and where the output is delivered.

2. Sets/Grouping of Output - prescribes the set of geometrical or FEM entities for which results are to be returned.

3. Actual Result Quantities - identifies the individual result quantities to be returned.

Specifying Output RequestsFor a complete list of the output requests available via the Case Control commands, see Case Control Command Summary (p. 185) in the MSC Nastran Quick Reference Guide.

Making Output Requests in Patran

The Output Requests form is used to request results from the MSC Nastran analysis for use in postprocessing (post tape) and verification (output file).


2. On the Analysis Application form, select Subcases... and choose Output Requests... from the Subcase Options section.


Although printed output requests can be different from Subcase to Subcase, there are certain aspects of these requests that can only be written once. For those aspects of output requests that must remain constant regardless of the Load Step, that information is extracted from the first Subcase in the Subcase Selection form.

Results (POST) File Options

• Increments between Writing Results

Defines the number of increments between writing results to the MSC Nastran results file after the first increment of the analysis. The default is one (1) for every increment.

• Select Nodal Results... Brings up a subform for selecting nodal results

• Select Element Results... Brings up a subform for selecting elemental results.


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Select Nodal Results

This subform controls which nodal result quantities are returned from the analysis.

Available Result Types Lists all of the available result types for the analysis. The numbers in parentheses are the Marc POST code numbers.

Selected Result Types Shows the set of result types that have been selected to be returned in the analysis.


The following table shows the post codes that may be selected for a SOL 600 structural nonlinear analysis.

Nodal Result Postcode Default(?)

DISPLACEMENT 1 YES

ROTATION 2 YES

EXTERNAL FORCE 3 no

EXTERNAL MOMENT 4 no

REACTION FORCE 5 YES

REACTION MOMENT 6 no

PORE PRESSURE 23 no

VELOCITY 28 no

ROTATIONAL VELOCITY 29 no

ACCELERATION 30 no

ROTATIONAL ACCELERATION 31 no

MODAL MASS 32 no

ROTATION MODAL MASS 33 no

CONTACT NORMAL STRESS 34 no

CONTACT NORMAL FORCE 35 YES

FRICTION STRESS 36 no

FRICTION FORCE 37 YES

CONTACT STATUS 38 no

CONTACT TOUCHED body 39 YES

HERRMANN VARIABLE 40 no

POST CODE, No. -11 -11 thru -16 no

POST CODE, No. -22 -21 thru -23 no

POST CODE, No. -31 -31 no



Note: The POST CODE (<0) are for user-defined quantities via user subroutine UPSTNO.


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Element Output Requests

This subform controls which element result quantities are returned from the analysis.


The following table shows the post codes that may be selected for a SOL 600 structural nonlinear analysis.

Available Result Types Lists all of the available result types for the analysis. The numbers in parentheses are the Marc POST code numbers.

Selected Result Types Shows the set of result types that have been selected to be returned in the analysis.

Element X-section Results Defines the number of layer points to use through the cross section of homogeneous shells, plates and beams. This number must be odd if not a composite.

Note: If no elemental results are selected and no nodal results are selected, no POST option is written.

Elemental Result Postcode Solutions Default(?)

STRAIN, TOTAL COMPONENTS 301 nonlinear only YES

STRAIN, TOTAL COMPONENTS(defined system)

461 nonlinear only no

STRAIN, ELASTIC COMPONENTS 401 any YES

STRAIN, ELASTIC COMPONENTS (global system)

421 any no

STRAIN, ELASTIC EQUIVALENT 127 any no

STRAIN, PLASTIC COMPONENTS 321 nonlinear only YES

STRAIN, PLASTIC COMPONENTS(global system)


STRAIN, PLASTIC EQUIVALENT 27 nonlinear only YES

STRAIN, PLASTIC EQUIVALENT (from rate)

7 nonlinear only YES

STRAIN, CREEP COMPONENTS 331 creep only no

STRAIN, CREEP COMPONENTS(global system)

441 creep only no

STRAIN, CREEP EQUIVALENT 37 creep only no

STRAIN, CREEP EQUIVALENT(from rate)

8 creep only no

STRAIN, THERMAL 371 any no

STRAIN, THICKNESS 49 any no


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STRAIN, VELOCITY 451 nonlinear only no

STRESS, COMPONENTS 311 any no

STRESS, COMPONENTS (defined system)

391 an no

STRESS, COMPONENTS(global system)

411 any no

STRESS, EQUIVALENT YIELD 59 nonlinear only no

STRESS, EQUIVALENT MISES 17 any YES

STRESS, MEAN NORMAL 18 any YES

STRESS, INTERLAMINAR SHEAR No. 1

108 any no

STRESS, INTERLAMINAR SHEAR No. 2

109 any no

STRESS, INTERLAMINAR COMPONENTS

501,511 any no

STRESS, CAUCHY COMPONENTS 341 nonlinear only no

STRESS, CAUCHY EQUIVALENT 47 nonlinear only YES

STRESS, HARMONIC COMPONENTS

351 (real) 361(imag)

harmonic only no

FORCES, ELEMENT 264-269 any no

BIMOMENT 270 any no

STRAIN RATE, PLASTIC 28 nonlinear only no

STRAIN RATE, EQUIVALENTVISCOPLASTIC

175 any no

STATE VARIABLE, SECOND 29 any no

STATE VARIABLE, THIRD 39 any no

TEMPERATURE, ELEMENT TOTAL

9 any no

TEMPERATURE, ELEMENT INCREMENTAL

10 any no

STRAIN ENERGY DENSITY, TOTAL

48 nonlinear only YES

STRAIN ENERGY DENSITY, ELASTIC

58 any no

STRAIN ENERGY DENSITY, PLASTIC


THICKNESS, ELEMENT 20 any no



VOLUME, ELEMENT 78 any no

VOLUME, VOID FRACTION 177 any no

FAILURE, INDEX No. 1-7 91-103 any no

POST CODE, No. 19 19 any no

POST CODE, No. 38 38 any no

POST CODE, No. -11 -11 thru -16 any no

POST CODE, No. -21 -21 thru -23 any no

POST CODE, No. -31 -31 any no




MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideSOL 600 Results Quantities

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SOL 600 Results QuantitiesThe following table indicates all the possible result quantities which can be loaded into the Patran database from the t16 file. The Primary and Secondary Labels are items selected from the postprocessing menus. The Type indicates whether the results are Scalar, Vector, or Tensor. These types will determine which postprocessing techniques will be available in order to view the results quantity. Postcodes indicates which MSC.Marc element postcodes (selected automatically or by Nastran bulk data card MARCOUT) the data comes from. The Description gives a brief discussion about the results quantity. The Output Request forms use the actual primary and secondary labels which will appear in the results. For example, if “Strain, Elastic” is selected on the Element Output Requests form, the “Strain, Elastic” is created for postprocessing.

Primary Label

Secondary Label Type Postcodes Description

Displacement Translation Vector 1 (nodal) Translational displacements at nodes from a structural analysis.

Displacement Rotation Vector 2 (nodal) Rotational displacements at nodes from a structural analysis.

Velocity Translation Vector 28 (nodal) Translational velocities at nodes from a dynamic analysis.

Velocity Rotation Vector 29 (nodal) Rotational velocities at nodes.

Acceleration Translation Vector 30 (nodal) Translational accelerations at nodes from a dynamic analysis.

Acceleration Rotation Vector 31 (nodal) Rotational accelerations at nodes from a dynamic analysis.

Force Nodal External Applied

Vector 3 (nodal) Forces applied to the model in a structural analysis.

Force Nodal Reaction Vector 5 (nodal) Reaction forces at boundary conditions from a structural analysis.

Moment Nodal External Applied

Vector 4 (nodal) Moments applied to the model in a structural analysis.

Moment Nodal Reaction Vector 6 (nodal) Reaction moments at boundary conditions from a structural analysis.

Modal Mass Translation Vector 32 (nodal) Translational modal masses from modal extractions.

Modal Mass Rotation Vector 33 (nodal) Rotational modal masses from modal extractions.

Stress Contact Normal Vector 34 (nodal) Contact Normal Stress

Force Contact Normal Vector 35 (nodal) Contact Normal Force

Stress Friction Vector 36 (nodal) Friction Stress


Force Friction Vector 37 (nodal) Friction Force

Contact Status Scalar 38 (nodal) Contact Status

Contact Touched Body Scalar 39 (nodal) Touched Body Contact

Variable Herrmann Scalar 40 (nodal) Herrmann Variable

Post Code No. -11 through -16

Tensor -11 thru -16, (nodal)

User defined nodal quantities via user subroutine UPSTNO.

Post Code No. -21 through -23

Vector -21 thru -23, (nodal)

User defined nodal quantities via user subroutine UPSTNO.

Post Code No. -31 Scalar -31, (nodal) User defined nodal quantities via user subroutine UPSTNO.



Strain Creep Tensor 31-36 or 331 Creep strain from a nonlinear structural analysis.

Strain Plastic Equivalent Rate

Scalar 28 Equivalent plastic strain rate from a nonlinear structural analysis.

Strain Thermal Tensor 71-76 or 371 Thermal strain from a structural analysis.

Strain Thickness Scalar 49 Thickness strain from a structural analysis.

Strain Total Tensor 1-6 or 301 Total strain from a structural analysis.

Temperature Element Scalar 9 Element temperature from a thermal or structural analysis.

Temperature Element Gradient Vector 181-183 Element temperature gradient from a thermal analysis.

Temperature Element Incremental

Scalar 10 Incremental element temperature from a thermal or structural analysis.

Stress Tensor 11-16 or 311 Stress from a structural analysis.

Stress Cauchy Tensor 41-46 or 341 Cauchy stress from a nonlinear structural analysis.

Stress Cauchy Equivalent Mises

Scalar 47 Equivalent Cauchy stress from a nonlinear structural analysis.

Stress Equivalent Mises Scalar 17 Equivalent (von mises) stress from a structural analysis.

Primary Label



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Stress Hydrostatic Scalar 18 Hydrostatic stress from a structural analysis.

Stress Interlaminar Shear No. 1

Scalar 108 Interlaminar shear in one direction from a structural analysis.

Stress Interlaminar Shear No. 2

Scalar 109 Interlaminar shear in two direction from a structural analysis.

Energy Density

Elastic Scalar 48 Elastic strain energy density from a structural analysis.

Energy Density

Plastic Scalar 58 Plastic strain energy density from a nonlinear structural analysis.

Energy Density

Total Scalar 68 Total strain energy density from a structural analysis.

Flux Element Vector 184-186 Element heat flux from a thermal analysis.

State Variable Second Scalar 29 Second state variable from a nonlinear thermal or structural analysis.

State Variable Third Scalar 39 Third state variable from a nonlinear thermal or structural analysis.

Failure Index No. 1 Scalar 91 Failure index one from a structural analysis.

Failure Index No. 2 Scalar 92 Failure index two from a structural analysis.

Failure Index No. 3 Scalar 93 Failure index three from a structural analysis.

Failure Index No. 4 Scalar 94 Failure index four from a structural analysis.

Failure Index No. 5 Scalar 95 Failure index five from a structural analysis.

Failure Index No. 6 Scalar 96 Failure index six from a structural analysis.

Failure Index No. 7 Scalar 97 Failure index seven from a structural analysis.

Thickness Scalar 20 Element thickness from a thermal or structural analysis.

Volume Scalar 78 Element Volume from a thermal or structural analysis.

Primary Label



In addition to these standard results quantities, several Global Variable results can be created. Global Variables are results quantities where one value is representative of the entire model. The following table defines the Global Variables which may be created.

Using Patran to Postprocess Results Quantities1. Click on the Analysis Application button to bring up the Analysis Application form.

2. Set the Action>Object>Method combination to Access Results>Attach t16/t19>Results Entities.

Global Variable Label Type Description

Increment Scalar Increment of the analysis.

Time Scalar Time of the analysis.

Buckling Mode Scalar Buckling mode number.

Critical Load Factor Scalar Critical load factor for buckling analysis.

Dynamic Mode Scalar Dynamic mode number from modal extraction.

Frequency (radians/time) Scalar Frequency in radians per unit time for modal extraction.


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After selecting a t16 or t19 file, you will need to specify the translation parameters.


MSC Nastran Results QuantitiesIf you wish to return result quantities from Marc back to MSC Nastran you must include the OUTR option on the SOL 600,ID Executive Control statement.

Using Patran to Postprocess MSC Nastran Results Quantities1. Click on the Analysis Application button to bring up the Analysis Application form.

2. Set the Action>Object>Method combination to Access Results>Attach XBD>Results Entities or Access Results>Read Output2>Results Entities.

Entry Description

OUTR Specifies that Marc output results be converted to various types of MSC Nastran formats

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideMSC Nastran Results Quantities

226

After selecting an XDB or op2 file, you will need to specify the translation parameters.


Tolerances

• Division Prevent division by zero errors.

• Numerical Compares real values for equality.

Additional Results to be Imported

• Rotational Nodal Results Indicates whether Rotational Nodal Results are skipped or included in translation.

• Stress/Strain Invariants Indicates whether Stress/Strain Invariants are skipped or included in translation.

• Principal Directions Indicates whether Principal Directions are skipped or included in translation.

• Element Results Positions If an element has results at both the centroid and at the nodes, this filter indicates which results are to be included in the translation.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideMSC Nastran Results Quantities

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Chapter 9: Assigned Conditions

9 Assigned Conditions

Constraints

Loads and Boundary Conditions

Initial Conditions

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideConstraints

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ConstraintsMSC Nastran Implicit Nonlinear allows you to input kinematic constraints through various options that include multi-point constraints, boundary conditions and support conditions.

Multi-Point Constraints

MPCs are special element types which define a rigorous behavior between several specified nodes. The following table lists the MPC types which are supported for MSC Nastran Implicit Nonlinear.

Specifying Explicit MPCs

Explicit MPC’s may be created between a dependent degree of freedom and one or more independent degrees of freedom. The dependent term consists of a node ID and a degree of freedom, while an independent term consists of a coefficient, a node ID, and a degree of freedom. An unlimited number of independent terms can be specified, while only one dependent term can be specified.

Defining Explicit MPCs in Patran

To define an Explicit MPC:

1. Click on the FE Application icon located on the Main form to bring up the Finite Elements Application form.

2. Set the Action>Object>Method combination to Create> MPC>Explicit.

MPC Types

• Explicit • RBE1

• Rigid (Fixed) • RBE2

• Cyclic Symmetry • RBE3

• Sliding Surface • RROD

• RJOINT

• RSPLINE

• RSSCON

• RBAR

• RBAR1

• RTRPLT

• RTRPLT1

Entry Description

MPC Defines a multipoint constraint equation.

231CHAPTER 9Assigned Conditions

3. Click on Define Terms... to define the explicit constraints.

Entry Description

Dependent Terms Dependent terms define the fields for G1 and C1 on the MPC entry. Only one node and DOF combination may be defined for any given explicit MPC. The A1 field on the MPC entry is automatically set to -1.0.

Independent Terms Independent terms define the Gi, Ci, and Ai fields on the MPC entry, where i is greater than one. As many coefficient, node, and DOF combinations as desired may be defined.


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Specifying Rigid MPCs

Rigid MPC’s may be created between one independent node and one or more dependent nodes in which all six structural degrees of freedom are rigidly attached to each other. An unlimited number of dependent terms can be specified, while only one independent term can be specified. Each term consists of a single node. There is no constant term for this MPC type.:

Defining Rigid MPCs in Patran

To define a Rigid MPC:


2. Set the Action>Object>Method combination to Create> MPC>Rigid (Fixed).

3. Click on Define Terms... to define the rigid constraints.

Entry Description

RBE2 Defines a rigid body with independent degrees-of-freedom that are specified at a single grid point and with dependent degrees-of-freedom that are specified at an arbitrary number of grid points.


Specifying Sliding Surface MPCs

Describes the boundary conditions of sliding surfaces, such as pipe sleeves. These boundary conditions are written as explicit MPCs. Be careful, for this option automatically redefines the analysis coordinate references of all affected nodes. This could erroneously alter the meaning of previously applied load and boundary conditions, as well as element properties.

Defining Sliding Surface MPCs in Patran

To define a Sliding Surface MPC:


2. Set the Action>Object>Method combination to Create> MPC>Sliding Surface.

Entry Description

Dependent Terms Dependent terms define the GMi fields on the RBE2 entry. As many nodes as desired may be selected as dependent terms.

Independent Terms Independent terms define the GN field on the RBE2 entry. Only one node may be selected.

Entry Description

Shell Nodes Dependent terms define the ESi fields on the RSSCON entry. One dependent node must be selected for every two independent terms.

Solid Nodes ndependent terms define the EA and EB field on the RSSCON entry. Two independent terms are required.

Entry Description

MPC Defines a multipoint constraint equation.


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3. Click on Define Terms... to define the sliding surface constraints.

Specifying RBAR MPCs

Creates an RBAR element, which defines a rigid bar between two nodes. Up to two dependent and two independent terms can be specified. Each term consists of a node and a list of degrees of freedom. The nodes specified in the two dependent terms must be the same as the nodes specified in the two independent terms. Any combination of the degrees of freedom of the two nodes can be specified as independent as long as the total number of independent degrees of freedom adds up to six. There is no constant term for this MPC type.

Defining RBAR MPCs in Patran

To define a RBAR MPC:

Entry Description

Dependent Region Specifies the dependent nodes on the sliding surface. The same number of unique nodes must be specified in both regions.

Independent Region Specifies the independent nodes on the sliding surface. The same number of unique nodes must be specified in both regions.

Entry Description

RBAR Defines a rigid bar with six degrees-of-freedom at each end.



2. Set the Action>Object>Method combination to Create> MPC>RBAR.

3. Click on Define Terms... to define the RBAR constraints.

Specifying RBE1 MPCs

RBAR1 is not allowed in SOL 600.

Defining RBE1 MPCs in Patran

To define a RBE1 MPC:


Entry Description

Dependent Terms Either one or two nodes may be defined as having dependent terms. The Nodes define the GA and GB fields on the RBAR entry. The DOFs define the CMA and CMB fields.

Independent Terms Either one or two nodes may be defined as having independent terms.The Nodes define the GA and GB fields on the RBAR entry.The DOFs define the CNA and CNB fields.


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2. Set the Action>Object>Method combination to Create> MPC>RBE1.


Creates an RBE2 element, which defines a rigid body between an arbitrary number of nodes. Although the user can only specify one dependent term, an arbitrary number of nodes can be associated to this term. The user is also prompted to associate a list of degrees of freedom to this term. A single independent term can be specified, which consists of a single node. There is no constant term for this MPC type.





3. Click on Define Terms... to define the RBE2 constraints.

Entry Description

RBE2 Defines a rigid body with independent degrees of freedom that are specified at a single grid point and with dependent degrees of freedom that are specified at an arbitrary number of grid points.



Creates an RBE3 element, which defines the motion of a reference node as the weighted average of the motions of a set of nodes. An arbitrary number of dependent terms can be specified, each term consisting of a node and a list of degrees of freedom. The first dependent term is used to define the reference node. The other dependent terms define additional node/degrees of freedom, which are added to the m-set. An arbitrary number of independent terms can also be specified. Each independent term consists of a constant coefficient (weighting factor), a node, and a list of degrees of freedom. There is no constant term for this MPC type.





Entry Description

Dependent Terms Dependent terms define the GMi and CM fields on the RBE2 entry. As many nodes as desired may be selected as dependent terms.

Independent Terms Independent terms define the GN field on the RBE2 entry. Only one node may be selected.

Entry Description

RBE3 Defines the motion at a reference grid point as the weighted average of the motions at a set of other grid points.


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3. Click on Define Terms... to define the RBE3 constraints.

Specifying RROD MPCs

Creates an RROD element, which defines a pinned rod between two nodes that is rigid in extension. One dependent term is specified, which consists of a node and a single translational degree of freedom. One independent term is specified, which consists of a single node. There is no constant term for this MPC type.

.Defining RROD MPCs in Patran

To define a RROD MPC:

Entry Description

Dependent Terms Dependent terms define the GMi and CMi fields on the RBE3 entry. The first dependent term will be treated as the reference node, REFGRID and REFC. The rest of the dependent terms become the GMi and CMi components.

Independent Terms Independent terms define the Gi, j, Ci, and WTi fields on the RBE3 entry.

Entry Description

RROD Defines a pin-ended element that is rigid in translation.



2. Set the Action>Object>Method combination to Create> MPC>RROD.

3. Click on Define Terms... to define the RROD constraints.

Specifying RTRPLT MPCs

Creates an RTRPLT element, which defines a rigid triangular plate between three nodes. Up to three dependent and three independent terms can be specified. Each term consists of a node and a list of degrees of freedom. The nodes specified in the three dependent terms must be the same as the nodes specified in the three independent terms. Any combination of the degrees of freedom of the three nodes can be specified as independent as long as the total number of independent degrees of freedom adds up to six. There is no constant term for this MPC type.

Entry Description

Dependent Terms Dependent terms define the GB and CMB on the RROD entry. Only one translational degree of freedom may be referenced for this entry.

Independent Terms Independent terms define the GA field on the RROD entry. The CMA field is left blank.

Entry Description

RTRPLT Defines a rigid triangular plate.


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Defining RTRPLT MPCs in Patran

To define a RTRPLT MPC:


2. Set the Action>Object>Method combination to Create> MPC>RTRPLT.

3. Click on Define Terms... to define the RTRPLT constraints.

Support ConditionsIn static analysis by the displacement method, the rigid body modes must be restrained in order to remove the singularity of the stiffness matrix. The required constraints may be supplied with single point constraints, multipoint constraints, or free body supports. If free body supports are used, the rigid body characteristics will be calculated and a check will be made on the sufficiency of the supports.

Entry Description

Dependent Terms Dependent terms define the GA, GB, GC, CMA, CMB, and CMC fields of the RTRPLT entry.

Independent Terms The total number of nodes referenced in both the dependent terms and the independent terms must equal three. There must be exactly six independent degrees of freedom, and they must be capable of describing rigid body motion. Defines the GA, GB, GC, CNA, CNB, and CNC fields of the RTRPLT entry.


Free-body supports are defined with a SUPORT6 or SUPORT1 entry. Free-body supports must be defined in the global coordinate system. The SUPORT6 entry must be selected by the SUPORT1 Case Control command.

For more information on Support Conditions, see Rigid Body Supports (p. 297) in the MSC Nastran Reference Manual.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideLoads and Boundary Conditions

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Loads and Boundary ConditionsThe loads for the analysis can be either Static or Time Dependent (dynamic). Time dependency is introduced either through the inclusion of a time dependent field multiplier, or through use of initial condition options (e.g., initial displacements).

Boundary conditions, in terms of fixed displacements at nodes, define one type of kinematic constraint for a structural analysis.

When defining loads on a model it is important to define the following terms and concepts.

Load Sets

A Loads/BC set is comprised of a collection of data (which may include fields) that are associated with both an analysis type and geometric and/or FEM entities. A typical example is displacements associated with nodes in a structural analysis.

Load Cases

A load case contains all the loads and boundary conditions used within a single analysis step. For example, one load case may represent the loads and BC for each time point in a time-dependent nonlinear analysis. Multiple load cases can be applied to the same model for linear analysis to examine how the model reacts to different loading conditions. Load cases are central to the ability to perform complex analyses on an individual model. For nonlinear analysis multiple loadcase runs are used to define the load history on the model. The ending point of the last subcase is the starting point of the next subcase.

Load Steps

A Load Step (or analysis step) is defined by associating a load case, an analysis procedure, output requests, and any associated parameters that guide the solution path for the chosen analysis procedure. Whereas a load case is a collection of loads and boundary conditions for a particular Load Step, a Load Step is a collection of relevant analysis parameters including the associated load case.

The load for a subcase is often subdivided into the number of increments specified for the subcase. The solution strategy in nonlinear analysis is to apply the loads in an incremental fashion until the desired load level is reached.

Load Types

The static loads in nonlinear analysis consist of concentrated loads, distributed loads, and thermal loads as well as applied displacements. Most of the relevant loads data applicable to the linear static analysis are also applicable to nonlinear static analysis. Transient loads define the loadings as functions of time and the location. A load can be applied at a particular degree of freedom, pressure over the surface area, or the body force simulating an acceleration.

The following types of loads are available for MSC Nastran Implicit Nonlinear.


Table 9-1 Bulk Data Entries for Loads

Loads

FORCEi Defines concentrated load at grid point.

MOMENTi Defines moment at a grid point.

NOLIN1i Defines nonlinear transient load.

PLOAD Defines pressure loads on CQUAD4, CTRIA3, CHEXA, CPENTA, and CTETRA. Should not be used for hyperelastic plane elements CQUAD4, CQUAD8, CQUAD, CTRIA3, and CTRIA6 or for hyperelastic CHEXA, CPENTA, CTETRA with midside nodes.

PLOAD2 Defines pressure loads on shell elements, CQUAD4 and CTRIA3. Not available for hyperelastic elements.

PLOAD4 Defines pressure loads on surfaces of CHEXA, CPENTA, CTETRA, CTRIA3 and CQUAD4 elements. Not available for hyperelastic plane elements CQUAD4, CQUAD8, CQUAD, CTRIA3, and CTRIA6.

PLOADX1 Defines pressure loads on axisymmetric elements CQUADX and CTRIAX.

RFORCE Defines load due to centrifugal force field.

TIC Specifies initial values for displacement and velocity.

TLOAD1i Defines loads as a function of time.


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Using Patran to Apply Loads and Boundary ConditionsThe Loads and Boundary Conditions application controls which loads and boundaries and contact information will be created in the MSC Nastran input file.

The Loads and Boundary Conditions application in Patran provides the ability to apply a variety of static and dynamic loads and boundary conditions including contact surfaces to finite element models. Loads/BCs may be associated with geometric entities as well as FEM entities. When associated with geometric entities, they can be transferred to finite elements created on the geometry. Loads and boundary conditions are intended to be created in multiple single purpose groups referred to as load sets. These sets are grouped into load cases in the Load Cases application.

One of the most elegant features in Patran is its ability to create fields that describes the variation of loads and boundary conditions. The way in which Loads and BCs vary may be defined spatially, by previous analysis results, based on time, or associated with material properties.

Sets can be visually displayed on the screen by markers which show the location, type, magnitude, and direction of the applied loads or boundary condition. Only the static portion of a dynamic Loads/BCs set is reflected in the marker display. Sets can also be displayed as tables.

A powerful capability is the display of any set scalar data directly on the model as a fringe plot. For display purposes, data are treated as “results,” with full user control over the spectrum, method, shading, etc. Data display is scalar, but the data can be pressures, vector component magnitudes, and vector resultant magnitudes. Fringe plots can only be displayed on finite elements. Fringes of a dynamic Loads/BCs set may be displayed at user-specified times.

Creating Load Cases

The Load Cases application enables you to combine a large number of individual loads and boundary condition (LBCs) sets into a single coherent case for application to the model. Each load case you create has a unique user-selected descriptive name as well as an associated descriptive statement. Load case information is permanently stored in the database (unless deleted). You can modify it at any time.

Even if you do not create any load cases, your load and boundary conditions will still be placed into a default current load case, named “default.” If you create a special load case and make it the current load case, then all subsequent LBCs will be placed in that load case as long as it is current.

Static Load Cases

Load cases in which none of the constituent loads or boundary conditions sets has a time varying component are called static load cases. Loads and boundary conditions that will make up a static load


case are generated using the Input Data subform. For static load cases, this subform will vary according to the type of load being created, but its general format remains constant.


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Time-Dependent Load Cases

Load cases in which one or more of the loads and boundary conditions sets has a time varying component are called time-dependent, or dynamic load cases Loads and boundary conditions that will make up a time-dependent load case are also generated using the Input Data subform. For time-dependent load cases, this subform incorporates time dependency data fields.

For more information, see Overview of the Loads and Boundary Conditions Application (p. 6) in the Patran Reference Manual.


Displacement LBCsBoundary conditions can be used to specify the value of the displacements at nodes. To create a boundary condition for displacement, you need to specify the node number, the degree of freedom(s), and the magnitude of the displacement.

Displacements can be imposed directly on nodes using SPC1 and SPCD Bulk Data entries. All non blank entries will cause an SPC1 entry to be created. If the specified value is not 0.0, an SCPD entry will also be created to define the non zero enforced displacement or rotation.

Zero or nonzero displacements can also be applied across elements in a uniform or variable fashion. The primary use of this boundary condition is to apply constraints to solid elements.

Patran LBC Application Input Data

Displacement boundary conditions are generated in Patran using the following Object/Type combinations on the LBC Application form.

References

• Loads and Boundary Conditions Form (p. 21) in the Patran Reference Manual.

Object Type Dimension Bulk Data Entries

Displacement Nodal SPC1, SPCD

Element Uniform

Element Variable

2-D/3-D SPC1, SPCD

Entry Description

Translations (T1,T2,T3)

Defines the enforced translational displacement values. These are in model length units.

Rotations (R1,R2,R3) Defines the enforced rotational displacement values. These are in radians.


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Force LBCsConcentrated forces and moments can be applied directly to nodes with the ability to define the direction as well as the magnitude.

Forces and moments are specified with FORCEi and MOMENTi Bulk Data entries, where:


Forces and moments are generated in Patran using the following Object/Type combination on the LBC Application form.

References


Entry Description

FORCE

MOMENT

Defines a static concentrated force/moment at a grid point by specifying the magnitude and direction.

FORCE1

MOMENT1

Defines a static concentrated force/moment at a grid point where the direction of the force/moment is defined to be parallel to a vector between two defined grid points.

FORCE2

MOMENT2

Defines a static concentrated force/moment at a grid point where the direction of the force/moment is parallel to the cross product of vectors from G1 to G2 and G3 to G4.


Force Nodal FORCEi, MOMENTi

Entry Description

Force (F1,F2,F3) Defines the applied forces in the translation degrees of freedom. This defines the N vector and the F magnitude on the FORCE entry.

Moment (M1,M2,M3) Defines the applied moments in the rotational degrees of freedom. This defines the N vector and the M magnitude on the MOMENT entry.


Pressure LBCsPressure loads can be applied to edges or surfaces of 2-D and 3-D elements. Several Bulk Data entries are used to apply pressure loading depending on the element topology.

These pressures are applied to 2-D and 3-D elements only. Pressures for 1-D elements are applied using the Total Load LBCs, 258 object.


Pressures are generated in Patran using the following Object/Type combination on the LBC Application form.

1. Uniform Pressure Loads on 2-D Elements

Entry Description

PLOAD Defines pressure loads on CQUAD4, CTRIA3, CHEXA, CPENTA, and CTETRA. Should not be used for hyperelastic plane elements CQUAD4, CQUAD8, CQUAD, CTRIA3, and CTRIA6 or for hyperelastic CHEXA, CPENTA, CTETRA with midside nodes.

PLOAD2 Defines pressure loads on shell elements, CQUAD4 and CTRIA3.

PLOAD4 Defines pressure loads on surfaces of CHEXA, CPENTA, CTETRA, CTRIA3 and CQUAD4 elements.

PLOADX1 Defines pressure loads on axisymmetric elements CQUADX and CTRIAX.


Pressure Element Uniform 2-D PLOAD4, PLOADX1, or FORCE


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2. Uniform Pressure Loads on 3-D Elements

3. Variable Pressure Loads on 2-D Elements

Entry Description

Top Surf Pressure Defines the top surface pressure load on shell elements using a PLOAD4 entry. The negative of this value defines the P1, P2, P3, and P4 values. These values are all equal for a given element, producing a uniform pressure field across that face.

Bot Surf Pressure Defines the bottom surface pressure load on shell elements using a PLOAD4 entry. This value defines the P1 through P4 values.These values are all equal for a given element, producing a uniform pressure field across that face.

Edge Pressure For Axisymmetric Solid elements (CTRIAX6), defines the P1 through P3 values on the PLOADX1 entry where THETA on that entry is defined as zero. For other 2D elements, this will be interpreted as a load per unit length (i.e. independent of thickness) and converted into equivalent nodal loads (FORCE entries). If a scalar field is referenced, it will be evaluated at the middle of the application region. Edge pressures are not available in SOL 600 prior to the 2006 release.


Pressure Element Uniform 3-D PLOAD4

Entry Description

Pressure Defines the face pressure value on solid elements using a PLOAD4 entry. This defines the P1, P2, P3, and P4 values. If a scalar field is referenced, it will be evaluated once at the center of the applied region.


Pressure Element Variable 2-D PLOAD4, PLOADX1, FORCE


4. Variable Pressure Loads on 3-D Elements

References


Temperature LBCsTemperatures can be defined directly at nodes or temperature fields can be defined across element surfaces.

Temperatures are specified with TEMP, TEMPPi, or TEMPRB Bulk Data entries, where:

Entry Description

Top Surf Pressure Defines the top surface pressure load on shell elements using a PLOAD4 entry. The negative of this value defines the P1, P2, P3, and P4 values. If a scalar field is referenced, it will be evaluated separately for the P1 through P4 values.

Bot Surf Pressure Defines the bottom surface pressure load on shell elements using a PLOAD4 entry. This value defines the P1 through P4 values. If a scalar field is referenced, it will be evaluated separately for the P1 through P4 values.

Edge Pressure For Axisymmetric Solid elements (CTRIAX6), defines the P1 through P3 values on the PLOADX1 entry where THETA on that entry is defined as zero. For other 2D elements, this will be interpreted as a load per unit length (e.g., independent of thickness) and converted into equivalent nodal loads (FORCE entries). If a scalar field is referenced, it will be evaluated independently at each node. Edge pressures are not available in SOL 600 prior to the 2006 release.


Pressure Element Variable 3-D PLOAD4

Entry Description

Pressure Defines the face pressure value on solid elements using a PLOAD4 entry. This defines the P1, P2, P3, and P4 values. If a scalar field is referenced, it will be evaluated separately for each of the P1 through P4 values.

Note: In the current version of SOL 600, a constant pressure is applied on the element face based on the average of P1, P2, P3, and P4.


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Temperatures are generated in Patran using the following Object/Type combinations on the LBC Application form.

1. Grid Point Temperatures

2. Uniform Temperature Fields on 1-D Elements

3. Uniform Temperature Fields on 2-D Elements

Entry Description

TEMP Defines temperature at grid points.

TEMPP1i Defines temperature field for surface elements.

TEMPRB Defines temperature field for line elements.


Temperature Nodal 0D TEMP

Entry Description

Temperature Defines the T fields on the TEMP entry.


Temperature Element Uniform 1-D TEMPRB

Entry Description

Temperature Defines a uniform temperature field using a TEMPRB entry. The temperature value is used for both the TA and TB fields. The T1a, T1b, T2a, and T2b fields are all defined as 0.0.


Temperature Element Uniform 2-D TEMPP1


4. Variable Temperature Fields on 1-D Elements

5. Variable Temperature Fields on -2D Elements

Entry Description

Temperature Defines a uniform temperature field using a TEMPP1 entry. The temperature value is used for the T field. The gradient through the thickness is defined to be 0.0.


Temperature Element Variable 1-D TEMPRB

Entry Description

Centroid Temp Defines a variable temperature file using a TEMPRB entry. A field reference will be evaluated at either end of the element to define the TA and TB fields.

Axis-1 Gradient Defines the temperature gradient in the 1 direction. A field reference will be evaluated at either end of the element to define the T1a and T1b fields.

Axis-2 Gradient Defines the temperature gradient in the 2 direction. A field reference will be evaluated at either end of the element to define the T2a and T2b fields.


Temperature Element Variable 2-D TEMPP1

Entry Description

Top Surf Temp Defines the temperature on the top surface of a shell element. The top and bottom values are used to compute the average and gradient values on the TEMPP1 entry.

Bot Surf Temp Defines the temperature on the bottom surface of a shell element. The top and bottom values are used to compute the average and gradient values on the TEMPP1 entry.


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6. Uniform and Variable Temperature Fields on 3-D Elements

References

• Loads and Boundary Conditions Form (p. 21) in the Patran Reference Manual

Inertial Loads LBCsInertial loads can be applied to the entire model using the GRAV or RFORCE Bulk Data entries.

The IDRF field on the RFORCE entry can be used to allow different portions of the structure to rotate with different angular velocities or in different directions.


Inertial loads are generated in Patran using the following Object/Type combination on the LBC Application form.


Temperature Element Uniform

Element Variable

3-D TEMP

Entry Description

Temperature Defines the temperature or temperature distribution in the element.

Entry Description

GRAV Defines acceleration vectors for gravity or other acceleration loading.

RFORCE Defines load due to centrifugal force field.


Inertial Load Element Uniform Entire Model GRAV or RFORCE

Entry Description

Trans Accel (A1,A2,A3) Defines the N vector and the G magnitude value on the GRAV entry.

Rot Velocity (w1,w2,w3) Defines the R vector and the A magnitude value on the RFORCE entry.


The acceleration and velocity vectors are defined with respect to the input analysis coordinate frame. The origin of the rotational vectors is the origin of the analysis coordinate frame. In generating the GRAV and RFORCE entries, the interface produces one GRAV and/or RFORCE entry image for each Patran load set.

References


Velocity LBCsVelocities can be defined for transient analysis using the TLOAD entry.:


Velocities are generated in Patran using the following Object/Type combination on the LBC Application form.

References


Acceleration LBCsAccelerations can be defined for transient response analysis using the TLOAD entry.:

Entry Description

TLOAD1 Defines a time-dependent dynamic load or enforced motion


Velocity Nodal TLOAD

Entry Description

Trans Veloc (v1,v2,v3) Defines the velocity values for the translational degrees-of-freedom.

Rot Veloc (w1, w2, w3) Defines the velocity values for the rotational degrees-of-freedom.

Entry Description

TLOAD1 Defines a time-dependent dynamic load or enforced motion


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Accelerations are generated in Patran using the following Object/Type combination on the LBC Application form.

References


Distributed Load LBCsDistributed forces and moments can be applied in a uniform or variable fashion to 1D and 2D elements. Several Bulk Data entries are used to apply distributed loading depending on the element topology.


Distributed loads are generated in Patran using the following Object/Type combinations on the LBC Application form.

1. Uniform and Variable Loads on 1-D Elements


Acceleration Nodal TLOAD

Entry Description

Trans Accel (A1,A2,A3) Defines the acceleration values for the translational degrees-of-freedom.

Rot Accel (a1,a2,a3) Defines the acceleration values for the rotational degrees-of-freedom.

Entry Description

PLOAD Defines a uniform static pressure load on a triangular or quadrilateral surface comprised of surface elements and/or the faces of solid elements.

PLOAD1 Defines concentrated, uniformly distributed, or linearly distributed applied loads to the CBAR or CBEAM elements at user-chosen points along the axis. For the CBEND element, only distributed loads over an entire length may be defined

PLOAD2 Defines a uniform static pressure load applied to CQUAD4, CSHEAR, or CTRIA3 two-dimensional elements.

PLOAD4 Defines a pressure load on a face of a CHEXA, CPENTA, CTETRA, CTRIA3, CTRIA6, CTRIAR, CQUAD4, CQUAD8, or CQUADR element.

PLOADX1 Defines surface traction to be used with the CQUADX, CTRIAX, and CTRIAX6 axisymmetric element.


Defines distributed force or moment loading along beam elements using MSC Nastran PLOAD1 entries. The coordinate system in which the load is applied is defined by the beam axis and the Bar Orientation element property. The Bar Orientation must be defined before this Distributed Load can be created. If the Bar Orientation is subsequently changed, the Distributed Load must be updated manually if necessary.

For the element variable type, a field reference is evaluated at each end of the beam to define a linear load variation.

2. Uniform and Variable Loads on 2-D Elements

Defines a distributed force or moment load along the edges of 2-D elements. The coordinate system for the load is defined by the surface or element edge and normal. The x direction is along the edge. Positive x is determined by the element corner node connectivity. See The Patran Element Library (p. 343) in the MSC Patran Reference Manual, Part 3: Finite Element Modeling. For example, if the element is a CQUAD4, with node connectivity of 1, 2, 3, 4. The positive x directions for each edge would be from nodes 1 to 2, 2 to 3, 3 to 4, and 4 to 1. The z direction is normal to the surface or element. Positive z is in the direction of the element normal. The y direction is normal to x and z. Positive y is determined by the cross product of the z and x axes and always points into the element. The MSC Nastran entries generated, depend on the element type.


Distributed Load Element Uniform

Element Variable

1-D PLOAD1

Entry Description

Distributed Load (f1,f2,f3) Defines the FXE, FYE, and FZE fields on three PLOAD1 entries.

Distributed Moment (m1,m2,m3) Defines the MXE, MYE, and MZE fields on three PLOAD1 entries.


Distributed Load Element Uniform

Element Variable

2-D PLOAD, PLOAD2, PLOAD4, PLOADX1


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For the element variable type, a field reference is evaluated at each end of the beam to define a linear load variation.

References


Total Load LBCsThe total load capability is not available directly in MSC Nastran, but is a convenient way to have Patran distribute a force load uniformly over an application area without having to calculate the number of nodes or application area. The total force load is defined for the application region, but equivalent uniform pressures are written to the Bulk Data. The equivalent pressure value is are found by dividing the total load value by the area of the application region.


Total Loads are generated in Patran using the following Object/Type combination on the LBC Application form

Entry Description

Edge Distributed Load (f1,f2,f3)

For axisymmetric solid elements (CTRIAX6), the PA, PB, and THETA fields on the PLOADX1 entry are defined. For other 2-D elements, the input vector is interpreted as load per unit length and converted into equivalent nodal loads (FORCE entries).

Edge Distributed Moment (m1,m2,m3)

For 2-D shell elements, the input vector is interpreted as moment per unit length and converted into equivalent nodal moments (MOMENT entries).

Entry Description

PLOAD4 Defines a pressure load on a face of a CHEXA, CPENTA, CTETRA, CTRIA3, CTRIA6, CTRIAR, CQUAD4, CQUAD8, or CQUADR element.

PLOADX1 Defines surface traction to be used with the CQUADX, CTRIAX, and CTRIAX6 axisymmetric element.

Note: Currently only 1D element types are supported with this Object even though the form allows for other types.


References


Contact LBCsA complete description of Contact loads and boundary conditions is given in Specifying Contact Body Entries (Ch. 12).

References



Total Load Element Uniform 1-D PLOAD4

Entry Description

Load <F1 F2 F3> Defines the total load component values to element nodes.

Analysis Coordinate Frame Defines the coordinate frame for the distributed load.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideInitial Conditions

260

Initial ConditionsInitial conditions provides various ways of initializing the state variables throughout the model.

Initial Displacement LBCsCreates a set of TIC Bulk Data entries..


Initial Displacements are generated in Patran using the following Object/Type combination on the LBC Application form.

References

Loads and Boundary Conditions Form (p. 21) in the Patran Reference Manual.

Initial Velocity LBCsCreates a set of TIC Bulk Data entries..

Entry Description

TIC Defines values for the initial conditions of variables used in structural transient analysis. Both displacement and velocity values may be specified at independent degrees-of-freedom.

Object Type Bulk Data Entries

Initial Displacement Nodal TIC

Entry Description

Translations (T1,T2,T3) Defines the initial translational displacement values. These are in model length units.

Rotations (R1,R2,R3) Defines the initial rotational displacement values. These are in radians.

Entry Description

TIC Defines values for the initial conditions of variables used in structural transient analysis. Both displacement and velocity values may be specified at independent degrees-of-freedom.



Initial velocities are generated in Patran using the following Object/Type combination on the LBC Application form.

References

Loads and Boundary Conditions Form (p. 21) in the Patran Reference Manual.

Object Type Bulk Data Entries

Initial Velocity Nodal TIC

Entry Description

Trans Veloc (v1,v2,v3) Defines the V0 fields for translational degrees of freedom on the TIC entry. A unique TIC entry will be created for each nonblank entry.

Rot Veloc (w1,w2,w3) Defines the V0 fields for rotational degrees of freedom on the TIC entry. A unique TIC entry will be created for each nonblank entry.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideInitial Conditions

262


Chapter 10: Materials

10 Materials

Overview

Linear Elastic

Nonlinear Elastic

Inelastic

Failure and Damage Models

Creep

Composite

Gasket

Material Damping

Experimental Data Fitting


264

OverviewA wide variety of materials are encountered in stress analysis problems, and for any one of these materials a range of constitutive models is available to describe the material’s behavior. We can broadly classify the materials of interest as those which exhibit almost purely elastic response, possibly with some energy dissipation during rapid loading by viscoelastic response (the elastomers, such as rubber or solid propellant); materials that yield, and exhibit considerable ductility beyond yield (such as mild steel and other commonly used metals, ice at low strain rates, and clay); materials that flow by rearrangement of particles which interact generally through some dominantly frictional mechanism (such as sand); and brittle materials (rock, concrete, ceramics).


Composites

(MATi, MATORT, PCOMP)

Anisotropic:

1) Layered,

21 Constants

2)Fiber Reinforced,

One dimensional strain in fibers

Aircraft panels

Tires, glass/epoxy

Composite continuum elements

Creep

(MATVP)

Strains increasing with time under constant load. Stresses decreasing with time under constant deformations. Creep strains are noninstantaneous.

Metals at high temperatures, polymide films, semiconductor materials

ORNL

Norton

Maxwell

Elastic

(MATi, MATORT)

Stress functions of instantaneous strain only. Linear load-displacement relation.

Small deformation (below yield) for most materials: metals, glass, wood

Hookes Law

Elastoplasticity

(MATEP)

Yield condition flow rule and hardening rule necessary to calculate stress, plastic strain. Permanent deformation upon unloading.

Metals

Soils

von Mises Isotropic

Cam -Clay

Hill’s Anisotropic

Hyperelastic

(MATHE)

Stress function of instantaneous strain. Nonlinear load-displacement relation. Unloading path same as loading.

Rubber Mooney

Ogden

Arruda-Boyce

Gent

dsij Cijk ldkl=

SE2--- T

tCT 1– =

265CHAPTER 10Materials

Constitutive ModelsA single material may contain multiple constitutive models. Each constitutive model characterizes distinct ranges of the material’s response. The constitutive models in MSC Nastran Implicit Nonlinear contain a range of linear and nonlinear material models that can address or approximate the material response of most commonly encountered materials. The constitutive models in MSC Nastran Implicit Nonlinear can be accessed by any of the solid or structural elements. The models are assessed independently at each “constitutive calculation point” (i.e., the numerical integration points in the elements). Thus, the constitutive models are concerned only with a single calculation point. The element then provides an estimate of the kinematic solution to the problem at the point under consideration.

Constitutive Models in Patran

In Patran, the constitutive model to be used is defined by the Constitutive Model Status. MSC/PATRAN uses all active constitutive models when the analysis is submitted. Redundant or unneeded constitutive models should be rendered inactive.

Existing constitutive models of an existing material appear in either the active or inactive listbox depending on their active/inactive status. Selection of a model from one listbox will add it to the other one. If you do not wish for a constitutive model to be translated into the MSC Nastran input file, place it in the inactive list box

To view or change the constitutive model status:

1. Click on the Materials Application icon located on the Main form to bring up the Materials Application form.

Hypoelastic Rate form of stress-strain law Concrete Buyukozturk

Viscoelastic

(MATVE)

Time dependence of stresses in elastic material under loads. Full recovery after unloading.

Rubber,

Glass, industrial

plastics

Simo Model

Narayanaswamy

Viscoplastic

(MATVP)

Combined plasticity and creep phenomenon

Metals

Powder

Power law

Shima Model

Shape Memory

(MATSMA)

Thermal - Mechanical Aruchhio’s model

Asaro-Sayeedvafa model



266

2. Select Change Material Status...

MSC Nastran Implicit Nonlinear Material EntriesThe following material bulk data entries are available in SOL 600. Each of these options are overviewed in the sections of this chapter and detailed in the Bulk Data Entries (Ch. 8) in the MSC Nastran Quick Reference Guide. All standard MSC Nastran materials are also available in SOL 600.

Bulk Data Entry Description

MATEP Specifies elasto-plastic material properties.

MATTEP Specifies temperature-dependent elasto-plastic material properties.

MATF Specifies failure model properties for linear elastic materials.

MATG Specifies gasket material properties to be used in MSC Nastran Implicit Nonlinear.

MATTG Specifies gasket material property temperature variation to be used in MSC Nastran Implicit Nonlinear.

MATHE Specifies hyperelastic (rubber-like) material properties for nonlinear (large strain and large rotation) analysis in MSC Nastran Implicit Nonlinear.

MATTHE Specifies temperature-dependent properties of hyperelastic (rubber-like) materials (elastomers) in MSC Nastran Implicit Nonlinear.

MATORT Specifies elastic orthotropic material properties for 3-dimensional and plane strain behavior for linear and nonlinear analyses in MSC Nastran Implicit Nonlinear.


The following sections describe how to model material behavior in MSC Nastran Implicit Nonlinear. Modeling material behavior consists of both specifying the constitutive models used to describe the material behavior and defining the actual material data necessary to represent the material. Directional dependency can be included for materials other than isotropic materials. Data for the materials can be entered into MSC Nastran Implicit Nonlinear either directly through the input file or by user subroutines, or material models may be defined in the Patran Materials Application. Each section of this chapter discusses various options for organizing material data for input. Each section also discusses the constitutive (stress-strain) relation and graphic representation of the models and includes recommendations and cautions concerning the use of the models.

MATTORT Specifies temperature-dependent properties of elastic orthotropic materials for linear and nonlinear analyses used in MSC Nastran Implicit Nonlinear.

MATVE Specifies isotropic visco-elastic material properties in MSC Nastran Implicit Nonlinear.

MATTVE Specifies temperature-dependent visco-elastic material properties in terms of Thermo-Rheologically Simple behavior in MSC Nastran Implicit Nonlinear.

MATVP Specifies viscoplastic or creep material properties to be used for quasi-static analysis in MSC Nastran Implicit Nonlinear.

Bulk Data Entry Description

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideLinear Elastic

268

Linear ElasticMSC Nastran Implicit Nonlinear is capable of handling problems with any combination of isotropic, orthrotropic, or anisotropic linear elastic material behavior.

The linear elastic model is the model most commonly used to represent engineering materials. This model, which has a linear relationship between stresses and strains, is represented by Hooke’s Law. Figure 10-1 shows that stress is proportional to strain in a uniaxial tension test. The ratio of stress to strain is the familiar definition of modulus of elasticity (Young’s modulus) of the material.

E (modulus of elasticity) = (axial stress)/(axial strain) (10-1)

Figure 10-1 Uniaxial Stress-Strain Relation of Linear Elastic Material

Experiments show that axial elongation is always accompanied by lateral contraction of the bar. The ratio for a linear elastic material is:

= (lateral contraction)/(axial elongation) (10-2)

This is known as Poisson’s ratio. Similarly, the shear modulus (modulus of rigidity) is defined as:

(shear modulus) = (shear stress)/(shear strain) (10-3)

A Poisson’s ratio of 0.5, which would be appropriate for an incompressible material, can be used for the following elements: Herrmann, plane stress, shell, truss, or beam. A Poisson’s ratio which is close (but not equal) to 0.5 can be used for constant dilation elements and reduced integration elements in situations which do not include other severe kinematic constraints. Using a Poisson’s ratio close to 0.5 for all other elements usually leads to behavior that is too stiff. A Poisson’s ratio of 0.5 can also be used with the updated Lagrangian formulation in the multiplicative decomposition framework using the standard displacement elements. In these elements, the treatment for incompressibility is transparent.

Stre

ss

Strain

E

1

v

G


Isotropic MaterialsMost linear elastic materials are assumed to be isotropic (their elastic properties are the same in all directions). For an isotropic material, every plane is a plane of symmetry and every direction is an axis of symmetry. It can be shown that for an isotropic material:

(10-4)

The shear modulus can be easily calculated if the modulus of elasticity and Poisson’s ratio are known.

Specifying Isotropic Material Entries

Isotropic material models are designated with the MAT1 Bulk Data entry in the MSC Nastran Input File.

Patran Materials Application Input Data

To define an isotropic material in Patran:

1. From the Materials Application form, set the Action>Object>Method combination to Create>Isotropic>Manual Input.

2. Click Input Properties...

Isotropic linear elastic material models require the following material data via the Input Options subform on the Materials Application form.

The material density, used to define the mass of the structure, and the damping value are used in dynamic loadings, while the expansion coefficient is used to identify the thermal strains.

Entry Description

MAT1 Defines the material properties for linear isotropic materials.

Isotropic-Linear Elastic Description

Elastic Modulus Defines the elastic modulus. This property is generally required. May vary with temperature via a defined material field.

Poisson’s Ratio Defines the Poisson’s ratio. This property is generally required. May vary with temperature via a defined material field.

Density Defines the mass density. This property is optional.

Coefficient of Thermal Expansion

Defines the coefficient of thermal expansion. This property is optional. May vary with temperature via a defined material field.

Reference Temperature Defines the stress free temperature. This property is optional. When defining temperature dependent properties, this is the reference temperature from which values will be extracted or interpolated.

G E 2 1 v+ =

G E v


270

Orthotropic MaterialsAn orthotropic material has three mutually orthogonal planes of symmetry. With respect to a coordinate system parallel to these planes, the constitutive law for this material is given by the following more general form of Hooke’s Law:

3-D Orthotropic

Due to symmetry of the compliance matrix, = , = , and = .

Using these relations, a general orthotropic material has nine independent constants:

, , , , , , , ,

These nine constants must be specified in constructing the material model.

2-D Orthotropic

Orthotropic material models can be used with 2-D elements, such as plane stress, plane strain, and axisymmetric elements. For example, the orthotropic stress-strain relationship for a plane stress element is:

(10-5)

Specifying Orthotropic Material Entries

2-D and 3-D othrotropic materials are characterized in MSC Nastran using the following bulk data entries.

11

22

33

12

23

13

1 E1 12 E1 – 13 E1 – 0 0 0

12– E1 1 E2 23– E2 0 0 0

13– E1 23– E2 1 E3 0 0 0

0 0 0 1 G12 0 0

0 0 0 0 1 G23 0

0 0 0 0 0 1 G13

11

22

33

12

23

13

=

11 21 22 12 22 32 33 23 33 13 11 31

11 22 33 12 23 31 G12 G23 G31

Note: The inequalities E22 > E33, E11 > E22, and E33 > E11 must be satisfied in

order for the orthotropic material to be stable. This is checked by MSC Nastran Implicit Nonlinear.

23 12 31

C1

1 1221– -------------------------------

E1 21E1 0

12E2 E2 0

0 0 1 1221– G

=



To define an orthotropic material in Patran:

1. From the Materials Application form, set the Action>Object>Method combination to Create>2D or 3D Orthotropic>Manual Input.


The required properties for orthotropic linear elastic material models vary based on dimension, element type, and thermal dependencies. 3-D orthotropic material models require the following material data (2-D requires a reduced set) via the Input Properties subform on the Materials Application form.

Entry Description

MAT3 Defines the material properties for linear orthotropic materials used by the CTRIAX6 element entry.

MAT2

MAT8

Defines the material property for an orthotropic material for solids and isoparametric shell elements.

MATORT Specifies elastic orthotropic material properties for three-dimensional and plane strain behavior for linear and nonlinear analyses in MSC Nastran Implicit Nonlinear in a more general way than MAT2 or MAT8.

Orthotropic-Linear Elastic Description

Elastic Modulus 11/22/33 Defines the elastic moduli in the element’s coordinate system. This is required data. May vary with temperature via a defined material field.

Poisson’s Ratio 12/23/31 Defines the Poisson’s ratios relative to the element’s coordinate system. This is required data. May vary with temperature via a defined material field.

Shear Modulus 12/23/31 Defines the shear moduli relative to the element’s coordinate system. This is required data. May vary with temperature via a defined material field.

Coefficient of Thermal Expansion 11/22/33

Defines the coefficients of thermal expansion relative to the element’s coordinate system. These properties are optional. May vary with temperature via a defined material field.

Reference Temperature Defines the stress free temperature which is an optional property. When defining temperature dependent properties, this is the reference temperature from which values will be extracted or interpolated.

Density Defines the mass density which is an optional property.


272

Anisotropic MaterialsAnisotropic material exhibits different elastic properties in different directions. The significant directions of the material are labeled as preferred directions, and it is easiest to express the material behavior with respect to these directions.

The stress-strain relationship for an anisotropic linear elastic material can be expressed as

(10-6)

The values of (the stress-strain relation) and the preferred directions (if necessary) must be defined

for an anisotropic material.

Specifying Anisotropic Material Entries

Anisotropic materials are characterized in MSC Nastran using the following bulk data entries.


To define anisotropic material in Patran:

1. From the Materials Application form, set the Action>Object>Method combination to Create>2D or 3D Anisotropic>Manual Input.


Anisotropic linear elastic material models require the following material data via the Input Properties subform on the Materials Application form.

Entry Description

MAT2 Defines the material properties for linear anisotropic materials for two-dimensional elements.

Anisotropic-Linear Elastic Description

Stress-Strain Matrix, Cij Defines the upper right portion of the symmetric stress-strain matrix relative to the element’s coordinate system.


Defines the coefficients of thermal expansion relative to the element’s coordinate system. They are optional properties.

Reference Temperature Defines the stress free temperature which is an optional property. When defining temperature dependent properties, this is the reference temperature from which values will be extracted or interpolated.


i j Ci jk lkl=

Cijkl


Nonlinear Elastic

Hypoelastic - IsotropicThe hypoelastic model is able to represent a nonlinear elastic (reversible) material behavior. For this constitutive theory, MSC Nastran Implicit Nonlinear assumes that

(10-7)

where is a function of the mechanical strain and is a function of the temperature.

The stress and strains are true stresses and logarithmic strains, respectively, when used in conjunction with the updated Lagrange and large displacement options.

When used in conjunction with the large displacement option only, Equation (10-7) is expressed as

(10-8)

where are the Green-Lagrangian strain and second Piola-Kirchhoff stress, respectively.

This model can be used with any stress element, including Herrmann formulation elements.

The tensors and may be defined by user subroutine HYPELA. In order to provide an accurate

solution, should be a tangent stiffness evaluated at the beginning of the iteration. In addition, the total stress should be defined as its exact value at the end of the increment. This allows the residual load correction to work effectively.

In user subroutine HYPELA2, besides the functionality of HYPELA, additional information is available

regarding the kinematics of deformation. In particular, the deformation gradient ( ), rotation tensor ( ),

and the eigenvalues ( ) and eigenvectors ( ) to form the stretch tensor ( ) are also provided. This information is available only for the continuum elements namely: plane strain, generalized plane strain, plane stress, axisymmetric, axisymmetric with twist, and three-dimensional cases.

Hyperelastic - IsotropicHyperelastic models are specified using either the MATHP or MATHE Bulk Data entries and are used to describe the behavior of materials that exhibit elastic response up to large strains, such as rubber, solid propellant, and other elastomeric materials. These materials are described in terms of a “strain energy potential”, U, which defines the strain energy stored in the material per unit of volume in the initial configuration as a function of the strain at that point in the material.

Elastomeric materials are elastic in the classical sense. Upon unloading, the stress-strain curve is retraced and there is no permanent deformation. Elastomeric materials are initially isotropic. Figure 10-2 shows a typical stress-strain curve for an elastomeric material.

· i j Lijkl·kl

gi j+=

L g

S·i j Lijk lE

·kl

gij+=

E S

L g

L

F R

N U

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideNonlinear Elastic

274

Figure 10-2 A Typical Stress-Strain Curve for an Elastomeric Material

Calculations of stresses in an elastomeric material requires an existence of a strain energy function which is usually defined in terms of invariants or stretch ratios. Significance and calculation of these kinematic quantities is discussed next.

Characteristics of Elastomeric Materials

Most solid rubberlike materials are nearly incompressible: their bulk modulus is several orders of magnitude larger than their shear modulus. For applications where the material is not highly confined, the assumption that the material is fully incompressible is usually a good approximation. In cases where the material is highly confined (such as in an O-ring), modeling the compressibility can be important for obtaining accurate results. In either case, the use of “hybrid” (mixed formulation) elements is recommended for this type of material in all but plane stress cases.

Elastomeric foams on the other hand are elastic but very compressible.

Elastomeric materials are considered to be isotropic in nature with random orientation of the long chain molecules.

Strain Energy Potential and Representative Models

Calculations of stresses in an elastomeric material requires an existence of a strain energy function which is usually defined in terms of invariants or stretch ratios.

In the rectangular block in Figure 10-3, , , and are the principal stretch ratios along the edges of

the block defined by

(10-9)

, S

tres

s

, Strain

100%

1 2 3

i Li ui+ Li=


Figure 10-3 Rectangular Rubber Block

In practice, the material behavior is (approximately) incompressible, leading to the constraint equation

the strain invariants are defined as

(10-10)

Depending on the choice of configurations, for example, reference (at ) or current ( ), you obtain total or updated Lagrange formulations for elasticity. The kinematic measures for the two formulations are discussed next.

Total Lagrangian Formulation

The strain measure is the Green-Lagrange strain defined as:

(10-11)

where is the right Cauchy-Green deformation tensor defined as:

(10-12)

in which is the deformation gradient (a two-point tensor) written as:

(10-13)

L3

2L2

L2

L1

1L1 Undeformed

Deformed

3L3

123 1=

I1 12

22

32+ +=

I2 1

2

2

2

2

2

32

3

2

1

2+ +=

I3 12

22

32

=

t 0= t n 1+=

Eij12--- Cij i j– =

Cij

Cij FkiFkj=

Fkj

Fkj

xk

Xj--------=


276

The Jacobian is defined as:

(10-14)

Thus, the invariants can be written as:

(10-15)

in which is the permutation tensor. Also, using spectral decomposition theorem,

(10-16)

in which the stretches are the eigenvalues of the right Cauchy-Green deformation tensor, and the

eigenvectors are .

Updated Lagrange Formulation

The strain measure is the true or logarithmic measure defined as:

(10-17)

where the left Cauchy-Green or finger tensor is defined as:

(10-18)

Thus, using the spectral decomposition theorem, the true strains are written as:

(10-19)

where is the eigenvectors in the current configuration. It is noted that the true strains can also be

approximated using first Padé approximation, which is a rational expansion of the tensor, as:

(10-20)

where a polar decomposition of the deformation gradient is done into the left stretch tensor and

rotation tensor as:

J

J 123 det Cij

12---

= =

I1 Cii=

I2

CijCij Cii 2– 2

------------------------------------------=

I316--- ei jkepqrCipCjqCkr det Cij = =

(implied sum on i)

eijk

Cij A2

NiA

NjA

=

A2

Cij

Ni

A

i j12--- ln bij=

bij

bi j FikFjk=

i j12--- Aln ni

A njA=

niA

i j 2 Vij i j– Vij i j+ 1–=

Fij Vij

Rij


The Jacobian is defined as:

(10-21)

and the invariants are now defined as:

(10-22)

It is noted that either Equation (10-15) or Equation (10-22) gives the same strain energy since it is scalar and invariant. Also, to account for the incompressibility condition, in both formulations, the strain energy is split into deviatoric and volumertic parts as:

(10-23)

Mooney-Rivlin Model

The generalized Mooney-Rivlin model for nearly-incompressible elastomeric materials is written as:

(10-24)

where and are the first and second deviatoric invariants.

Jamus-Green-Simpson Model

A particular form of the generalized Mooney-Rivlin model, namely the third order deformation (tod) model, is implemented in MSC Nastran Implicit Nonlinear (SOL 600). This is one of the few places where the formulation for SOLs 106 and 129 may be more appropriate because they can use up to fifth order terms. However, the Ogden formulation (below) is usually better for large strain behavior than even the fifth order Mooney-Rivlin.

(10-25)

where is the deviatoric third order deformation form strain energy function,

Fij VikRkj=

J

J 123 det bij

12---

= =

I1 bii=

I212--- bijbij bi i 2– =

I316--- eijkepqrbipbjqbkr det bij = =and

W Wdeviatoric Wvolumetric+=

Wdeviatoricgmr

Cmn I1 3– m

I2 3– n

n 1=

N

m 1=

N

=

I1

I2

Wdevratorictod

C10 I1 3– C01 I2 3– C11 I1 3– I2 3– C20 I1 3– 2 C30 I1 3– 3

+ + + +=

Wdeviatoric

tod


278

are material constants obtained from experimental data.

Simpler and popular forms of the above strain energy function are obtained as:

(10-26)

Ogden Model

The form of strain energy for the Ogden model in MSC Nastran Implicit Nonlinear is,

(10-27)

where are the deviatoric stretch ratios while , , and are the material constants

obtained from the curve fitting of experimental data.

The Ogden model is usually applied to slightly compressible materials. If no bulk modulus is given, it is taken to be virtually incompressible. This model is different from the Mooney model in several respects. The Mooney material model is with respect to the invariants of the right or left Cauchy-Green strain tensor and implicitly assumes that the material is incompressible. The Ogden formulation is with respect to the eigenvalues of the right or left Cauchy-Green strain, and the presence of the bulk modulus implies some compressibility. Using a two-term series results in identical behavior as the Mooney mode if:

, , , and

Arruda-Boyce Model

In the Arruda-Boyce strain energy model, the underlying molecular structure of elastomer is represented by an eight-chain model to simulate the non-Gaussian behavior of individual chains in the network. The

two parameters, and ( is the chain density, is the Botzmann constant, is the temperature,

and is the number of statistical links of length l in the chain between chemical crosslinks) representing initial modules and limiting chain extensibility and are related to the molecular chain orientation thus representing the physics of network deformation.

As evident in most models describing rubber deformation, the strain energy function constructed by fitting experiment data obtained from one state of deformation to another fails to accurately describe that deformation mode. The Arruda-Boyce model ameliorates this defect and is unique since the standard tensile test data provides sufficient accuracy for multiple modes of deformation.

C10 C01 C11 C20 C30

Wdeviatoricmr

C10 I1 3– C01 I2 3– += Mooney-Rivlin

Wdeviatoricnh

C10 I1 3– = Neo-Hookean

Wdeviatoricogden

k

k------ 1

k2

k3

k3–+ +

k 1=

N

=

i

kJ

k3

------–

i

k= Cmn k k

1 2C10= 1 2= 2 2C01–= 2 2–=

nk N n k N


Figure 10-4 Eight Chain Network in Stretched Configuration

The model is constructed using the eight chain network as follows:

Consider a cube of dimension with an unstretched network including eight chains of length ,

where the fully extended chain has an approximate length of Nl. A chain vector from the center of the cube to a corner can be expressed as:

(10-28)

Using geometrical considerations, the chain vector length can be written as:

(10-29)

and

(10-30)

Using statistical mechanics considerations, the work of deformation is proportional to the entropy change on stretching the chains from the unstretched state and may be written in terms of the chain length as:

(10-31)

where is the chain density and is a constant. is an inverse Langevin function correctly accounts for the limiting chain extensibility and is defined as:

20

30

10

C1

k

j

i

0 r0 Nl=

C1

0

2------ 1 i

0

2------ 2 j

0

2------ 3 k+ +=

rchain1

3------- Nl 1

2 22 3

2+ + 1 2

=

chain

rchain

r0-------------

1

3------- I1 1 2

= =

W nkNrchain

Nl------------- ln

sinh

-------------- C–+=

n C


280

(10-32)

where Langevin is defined as:

(10-33)

With Equation (10-30) through Equation (10-33), the Arruda-Boyce model can be written

(10-34)

Gent Model

Also, using the notion of limiting chain extensibility, Gent proposed the following constitutive relation:

(10-35)

where

(10-36)

The constant is independent of molecular length and, hence, of degree of crosslinking. The model is

attractive due to its simplicity, but yet captures the main behavior of a network of extensible molecules over the entire range of possible strains.

The volumetric part of the strain energy is for all the rubber models in MSC Nastran Implicit Nonlinear is:

(10-37)

when is the bulk modulus. It can be noted that the particular form of volumetric strain energy is chosen such that:

1. The constraint condition is satisfied for incompressible deformations only; for example:

L 1–rchain

Nl------------- =

1---–coth=

WdevArruda-Boyce

nk 12--- I1 3– 1

20N---------- I1

2 9– 11

1050N2

------------------- I13 27– + +=

19

7000N3

------------------- I14 81– 519

673750N4

------------------------- I15 243– + +

WdevGent EIm–

6-------------

Im

Im I1*–

-----------------log=

I1* I1 3–=

EIm

Wvolumetric9K2

------- J

13---

1–

2

=

K


(10-38)

2. The constraint condition does not contribute to the dilatational stiffness.

This yields the constraint function as:

(10-39)

upon substitution of Equation (10-39) in Equation (10-35) and taking the first variation of the variational principle, you obtain the pressure variable as:

(10-40)

The equation has a physical significance in that for small deformations, the pressure is linearly related

to the volumetric strains by the bulk modulus .

The discontinuous or continuous damage models discussed in the models section on damage can be included with the generalized Mooney-Rivlin, Ogden, Arruda-Boyce, and Gent models to simulate Mullins effect or fatigue of elastomers when using the updated Lagrangian approach. In the total Lagrangian framework however, this is available for the Ogden model only.

Foam Model

Sometimes elastomeric materials show large volumetric deformations. For this type of behavior, the models discussed above are not appropriate. Instead, the foam model expressed by:

(10-41)

should be used. In contrast to the Ogden model, the first part of the foam strain energy function is not purely deviatoric. The material constants provide additional flexibility to describe the material

behavior also for a large amount of compressibility.

Updated Lagrange Formulation for Nonlinear Elasticity

The Mooney-Rivlin, Ogden, Arruda-Boyce, Gent and Foam models may be used either in the total Lagrange or updated Lagrange framework. This is selected using the PARAM,MARUPDAT. For plane stress analysis the total Lagrange procedure will always be used.

f I3

> 0 if I3 0

0= if I3 1=

< 0 if I3 0

f I3 3 I3

16---

1–

=

p 3K J

13---

1–

=

K

Wn

n------ 1

n 2

n 3

n3–+ +

n

n------ 1 J

n–

n 1=

N

+

n 1=

N

=

n


282

The updated Lagrangian rubber elasticity capability can be used in conjunction with both continuous as well as discontinuous damage models. Thermal, as well as viscoelastic, effects can be modeled with the current formulation. While the Mooney model can account for the temperature dependent material properties, the Ogden model does not support the temperature dependence at this time. The singularity ratio of the system is inversely proportional to the order of bulk modulus of the material due to the condensation procedure.

A consistent linearization has been carried out to obtain the tangent modulus. The singularity for the case of two- or three-equal stretch ratios is analytically removed by application of L’Hospital’s rule. The current framework with an exact implementation of the finite strain kinematics along with the split of strain energy to handle compressible and nearly incompressible response is eminently suitable for implementation of any nonlinear elastic as well as inelastic material models. In fact, the finite

deformation plasticity model based on the multiplicative decomposition, is implemented in the same framework.

To simulate elastomeric materials, incompressible element(s) are used for plane strain, axisymmetric, and three-dimensional problems for elasticity in total Lagrangian framework. These elements can be used with each other or in combination with other elements. For plane stress, beam, plate or shell analysis, conventional elements can be used. For updated Lagrangian elasticity, both conventional elements (as well as Herrmann elements) can be used for plane strain, axisymmetric, and three-dimensional problems.

Experimental Determination of Hyperelastic Material Parameters

In order to determine the material parameters to be used, like Mooney coefficients, Ogden moduli, relaxation times, etc., experiments must be carried out. In this section, the laboratory tests of which data can be used to fit the material parameters will be described. Once the test data is available the Experimental Data Fitting module in Patran can be used to calculate appropriate coefficient values.

For a homogeneous material, homogeneous deformation modes suffice to characterize the material constants. MSC Nastran Implicit Nonlinear accepts test data from the following deformation modes:

• Uniaxial tension and compression.

• Biaxial tension and compression.

• Planar tension and compression (also known as pure shear).

• Simple Shear

• Volumetric tension and compression

F FeFF

p=


Figure 10-5 Test Data

1

2

3

1

2

3

1

2

3

1

2

3

Uniaxial Test Data

Biaxial Test Data

Planar Test Data

Volumetric Test Data


284

Uniaxial Test

Probably the most popular test is the uniaxial test (see Figure 10-6). This test can be used in tension as well as in compression, both for incompressible and (slightly) compressible elastomeric materials. The shape of the specimen used in compression will usually be less slender than the shape used in tension. Within the region indicated by the dashed line, the state of deformation will be homogeneous, where the deformation can be described by:

, (10-42)

while the corresponding engineering stresses are given by:

, (10-43)

in which is the applied force and is the cross sectional area of the undeformed specimen in the -

-plane, within the region indicated by the dashed line.

Figure 10-6 Uniaxial (Tensile) Test

Necessary input for the curve fitting program in Patran consists of at least engineering strain ( ) versus

engineering stress ( ) data points. In case of (slightly) compressible materials, information about the

volume changes is also needed. This data can be given either in terms of the area ratio or the volume ratio.

The area ratio is defined by the current cross sectional area over the original cross sectional area .

Similarly, the volume ratio is defined by the current volume over the undeformed volume . Notice

that the volume ratio and the area ratio are related by:

If, for a particular elastomeric material, both a tensile and a compression test have been performed, all the data points should be collected into one data file. The layout of a data file containing uniaxial test data is given in the figure below. The columns may be separated by either spaces or commas. For (nearly) incompressible material behavior, the third column can be omitted.

1 1 e11+= = 2 3 J = =

11 FA0------= = 22 33 0= =

F A0 E2

E3

E1E3

E2

F F

e11

11

A A0

V V0

VV0------ J

AA0------ 1 e11+ = =


Figure 10-7 Layout of Data File for a Uniaxial Test

Equi-Biaxial Test

The equi-biaxial tensile test outlined in Figure 10-8 can be used to obtain, within the region indicated by the dashed line, a homogeneous state of deformation defined by:

Figure 10-8 Equi-biaxial (Tensile) Test

, (10-44)

with corresponding engineering stresses:

, (10-45)

e11 11 A A0 e11 11 V V0

or

F

F

F F

E1E3

E2

1 2 1 e11+ 1 e22+= = = = 3 J 2=

11 22 FA0------= = = 33 0=


286

with being the original cross sectional area of the elastomeric sheet in the direction perpendicular to

the applied forces, which is assumed to be the same in the - -plane and the - -plane.

For compressible elastomers, volumetric information is needed. For the equi-biaxial test, this can be given in terms of a thickness ratio or, similar to the uniaxial test, a volume ratio. The thickness ratio is

defined as the current sheet thickness over the original sheet thickness . The relation between the

thickness ratio and the volume ratio is:

(10-46)

The layout of a data file for an equi-biaxial tensile test is given in Figure 10-8.

Planar Shear Test

A state of planar shear, also sometimes called pure shear, can be obtained by clamping and stretching an elastomeric rectangular sheet of material, as indicated in Figure 10-9.

Figure 10-9 Planar Shear Test

Except for the vicinity of the free edges and the clamps, the state of strain can be found to be substantially uniform, according to:

, , (10-47)

where the known stress components are given by:

, (10-48)

A0

E1 E3 E2 E3

t t0

VV0------ J

tt0---- 1 e11+ 2= =

FF

E1E3

E2

1 1 e11+= = 2 1= 3J---=

11 FA0------= = 33 0=


in which is the cross sectional area of the undeformed specimen in the - -plane. Notice that the

engineering strain is zero, but that the corresponding engineering stress depends on the material

behavior.

(10-49)

(10-50)

Simple Shear Test

A test which, compared to the above mentioned tests, leads to a more complex kinematic description, is

the simple shear test (see Figure 10-10).Upon introducing the shear strain , the coordinates in the deformed configuration are given by:

, , (10-51)

which yields for the deformation gradient:

(10-52)

Figure 10-10 Simple Shear Test

Notice that , irrespective of the value of , from which it can be concluded that a simple shear test is a constant volume test.

Based on Equation (10-51), Equation (10-52) and Figure 10-10, the engineering strain tensor and the right Cauchy-Green strain tensor can be evaluated as:

A0 E1 E3

e22 22

U TSS=

TS S

U 2 S S3–

–

I1U

I2U+

= =

x1 X1 X2+= x2 X2= x3 X3=

F1 0

0 1 0

0 0 1

=

2F

E1E3

E2 atan

det F 1=


288

(10-53)

(10-54)

According to Equation (10-54), the principal stretch ratios follow from the principal values of and read:

, (10-55)

It can easily be verified that , which again shows that the simple shear test is a constant

volume test. The relevant engineering stress is given by:

(10-56)

with being the cross sectional area of the undeformed specimen in the - -plane.

The layout of a data file containing measurements of a simple shear test is given in Figure 10-11.

Figure 10-11 Layout of Data File for a Simple Shear Test

Volumetric Test

Although a uniaxial, equi-biaxial and planar shear test can be used to obtain information about the volumetric behavior, for compressible materials an additional volumetric test may be preferable. This is especially true for slightly compressible materials, since volumetric data from other tests other than a volumetric one may easily be inaccurate (because most of the deformation is deviatoric). Two commonly used volumetric tests are outlined in Figure 10-12. In Figure 10-12a, a cylindrical specimen is compressed in a cylindrical hole. This test can be successfully applied for slightly compressible materials. In Figure 10-12b, a specimen is deformed by compressing the surrounding fluid. This volumetric test can also be used for highly compressible materials.

e0 2 0

2 0 0

0 0 0

=

C1 0

1 2+ 0

0 0 1

=

C

1 2 12

2----- 1 2

4-----++= 3 1=

123 1=

12FA0------=

A0 E1 E3

2e12 = 12


Figure 10-12 Volumetric Tests

For a volumetric test, the direct true stress components are assumed to be equal to the hydrostatic

pressure and given by:

(10-57)

in which denotes the area of the piston in the - -plane. The deformation can be expressed in terms

of an engineering strain and corresponding stretch ratio , which can be determined from the measured volume change according to:

(10-58)

Based on according to Figure 10-12b, the engineering stress follows from:

(10-59)

Notice that only in the case of Figure 10-12b the engineering strain and the engineering stress are equal to the direct components of the engineering strain and the engineering stress tensor.

The layout of the data file corresponding to a volumetric test is given in Figure 10-13. Notice that because of Figure 10-12b, the entries of the first and the third column are not independent.

F

F

F

(a) (b)

E2E3

E1

p

T11 T22 T33F

Ap

------= = =

Ap

E2 E3

e

e 1– VV0------3 1– J3 1–= = =

T112

=

e


290

Figure 10-13 Layout of Data File for a Volumetri Test

Relaxation Test

The basic feature of a relaxation test is that the force or stress response to a prescribed fixed displacement or deformation is measured as a function of time. A relaxation test for a large strain elastomeric material

is indicated in Figure 10-14. By measuring the force needed for a displacement at different time intervals, the decay of the strain energy as a function of time can be determined. For linear elastic isotropic material, similar tests can be performed to get information about the shear modulus and/or the bulk modulus as a function of time. In order to properly measure the instantaneous values, application of the prescribed displacement should occur sufficiently fast. It should be noted, due to the assumption introduced in equation Equation (10-94), that for large strain visco-elastic materials the magnitude of (the instantaneous value of) the strain energy is not important, since every energy term in the Prony series expansion is related to the instantaneous strain energy using a scalar multiplier. The data does not need to be equispaced in time. Usually, at the beginning of the relaxation experiment the measurements are done at smaller time intervals than at the end of the experiment.

Figure 10-14 Relaxation Test

If, for linear visco-elastic materials, instead of a relaxation test only a creep test can be performed, the creep data must be transformed into relaxation data. Converting creep data into relaxation data can be done using a numerical integration scheme, but is not part of MSC Nastran Implicit Nonlinear.

Hyperelastic Foam Properties

Elastomeric foams are cellular solids that have the following primary mechanical characteristics:

e V V0

u

u


• They can deform elastically up to large strain: up to 90% strain in compression. In most applications, this is the dominant mode of deformation.

• Their porosity permits very large volumetric changes. This is in contrast to solid rubbers, which are approximately incompressible.

• Cellular solids are made up of interconnected networks of solid struts or plates which form the edges and faces of cells. Foams are made up of polyhedral cells that pack in three dimensions. The foam cells can either be open (e.g., sponge) or closed (e.g., flotation foam). Common examples of elastomeric foam materials are cellular polymers such as cushions, padding, and packaging materials which utilize the excellent energy absorption properties of foams - for a certain stress level, the energy absorbed by foams is substantially greater than by ordinary stiff elastic materials.

Figure 10-15 shows a typical compressive stress-strain curve for elastomeric foam.

Figure 10-15 Typical Compressive Stress-Strain Curve

Three stages can be distinguished during compression:

At small strains (< 5%) the foam deforms in a linear elastic manner, due to cell wall bending.

This is followed by a plateau of deformation at almost constant stress, caused by the elastic buckling of the columns or plates which make up the cell edges or walls. In closed cells, the enclosed gas pressure and membrane stretching increase the level and slope of the plateau.

Finally, a region of densification occurs, where the cell walls crush together, resulting in a rapid increase of compressive stress. Ultimate compressive nominal strains of 0.7 to 0.9 are typical.

The tensile deformation mechanisms for small strains are similar to the compression mechanisms but differ for large strains. The figure shows a typical tensile stress-strain curve.

STRAIN

ST

RE

SS

Densification

Plateau: Elastic buckling of cell walls

Cell wall bending


292

Figure 10-16 Typical Tensile Stress-Strain Curve

There are two stages during tension:

At small strains the foam deforms in a linear, elastic manner, due to cell wall bending, similar to that in compression.

The cell walls rotate and align, resulting in rising stiffness. The walls are substantially aligned at a tensile strain of about 1/3. Further stretching results in increased axial strains in the walls.

At small strains for both compression and tension, the average experimentally observed Poisson's ratio, , of foams is 1/3. At larger strains it is commonly observed that Poisson's ratio is effectively zero during compression - the buckling of the cell walls does not result in any significant lateral deformation. However, during tension, is nonzero, which is a result of the alignment and stretching of the cell walls.

The manufacture of foams often results in cells with different principal dimensions. This shape anisotropy results in different loading responses in different directions. However, the foam model does not take this kind of initial anisotropy into account.

Determination of Foam Material Parameters

The response of the material is defined by the parameters in the strain energy function, U, so that it is necessary to determine these parameters to use the foam model. Patran contains a capability for obtaining the i, i and i for the foam model with up to six terms (N=6) directly from test data. It is usually best

to obtain data from several experiments involving different kinds of deformation, over the range of strains of interest in the actual application, and to use all of these data to determine the parameters.

Since the properties of foam materials can vary significantly from one batch to another, all of the experiments should be performed on specimens taken from the same batch of material or to use MSC.Stocastics in combination with SOL 600.

Uniaxial, Equibiaxial and Planar Deformations

The deformation modes are characterized in terms of the principal stretches, i, and the volume ratio, J.

The elastomeric foams are not incompressible, so that J = 123 != 1. The transverse stretches, 2 and/or

STRAIN

ST

RE

SS

Cell wall bending

Cell wall alignment


3, are independently specified in the test data either as individual values from the measured lateral

deformations or through the definition of an effective Poisson’s ratio.

Uniaxial mode: 1=U, 2=3, J=U22

Equibiaxial mode: 1=2=B, J=B23

Planar mode: 1=P, 2=1, J=P3

The three deformation modes above use a single form of the nominal stress-stretch relation,

(10-60)

where TL is the nominal stress and LL is the stretch in the direction of loading. Because of the

compressible behavior, the planar mode does not result in a state of pure shear. In fact, if the effective Poisson’s ratio is zero, planar deformation is identical to uniaxial deformation.

Simple Shear Deformation

Simple shear is described by the deformation gradient

(10-61)

where is the shear strain. For this deformation, J=det F =1. A schematic illustration of simple shear deformation is shown in Figure 10-17.

The nominal shear stress is:

(10-62)

where = are the principal stretches in the plane of shearing, related to the shear strain, , by:

(10-63)

TL LU 2

L-------

ii----- L

iJii–

–

i 1=

N

= =

F1 00 1 00 0 1

=

TS

TS U 2

2 j2

1–

2–

--------------------------------------ii----- j

i1–

i 1=

N

j 1=

2

= =

j

1 2 1 2

2----- 1

2

4-----++=


294

.

Figure 10-17 Simple Shear Test

The stretch in the direction perpendicular to the shear plane is . The transverse (tensile) stress, ,

developed during simple shear deformation due to the Poynting effect, is

(10-64)

Volumetric Deformation

The volumetric deformation mode consists of all principal stretches being equal,

1=2=3=V, J=V3.

The pressure-volumetric ratio relation is

(10-65)

A volumetric compression test is illustrated Figure 10-18.

The pressure exerted on the foam specimen is the hydrostatic pressure of the fluid and the decrease in the specimen volume is equal to the additional fluid entering the pressure chamber. The specimen is sealed against fluid penetration.

2F

E1E3

E2 atan

L3 1= TT

TT U

2 j2

1–

2j4

j22

2+ –---------------------------------------------

ii----- j

i1–

i 1=

N

j 1=

2

= =

p–J

U 2J---

ii----- J

i3-----

Jii–

–

i 1=

N

= =


Figure 10-18 Volumetric Compression Test Setup

Difference in Compression and Tension Deformation

For small strains (< 5%), foams behave similarly for both compression and tension. However, we have seen that at large strains, the deformation mechanisms differ for compression (buckling and crushing) and tension (alignment and stretching). Accurate modeling with the FOAM option therefore requires that the experimental data used to define the material parameters correspond to the dominant deformation modes of the actual problem being analyzed.

If compression dominates in the problem, the pertinent tests are:

• Uniaxial compression.

• Simple shear.

• Planar compression (if Poisson’s ratio ).

• Volumetric compression (if Poisson’s ratio ).

If tension dominates, the pertinent tests are:

• Uniaxial tension.

• Simple shear.

• Biaxial tension (if Poisson’s ratio ).

• Planar tension (if Poisson’s ratio ).

Lateral strain data can also be used to define the compressibility of the foam. Measurement of the lateral strains may make other tests redundant, e.g., providing lateral strains for a uniaxial test eliminates the need for a volumetric test. The foam model may not accurately fit Poisson's ratio if it varies significantly between compression and tension.

F

F

F

(a) (b)

E2E3

E1

0

0

0

0


296


Least Squares Fit

The equations derived above for TU, TB, and TS, with the assumption of material incompressibility, allow

the material parameters Cij and i, i to be determined from the experimentally measured stress-strain

relationships in the uniaxial, equibiaxial, and planar loading tests. A least squares fit, which minimizes the relative error in stress, is used for this purpose. The equation for TS alone will not determine the

constants uniquely. The planar test data input must be augmented by either or both of the other two types of test data to determine the material parameters.

The Ogden potential is linear in the coefficients i but strongly nonlinear in terms of the exponents i,

thus necessitating use of a nonlinear least squares procedure. For the nominal stress-nominal strain data

pairs, the error measure, E, is minimized by E = sum(i=1to n)(1-Tith/Ti

test2), where Titest is a stress value

from the test data and Tith comes from one of the nominal stress expressions derived above.

The foam parameters i, i, i are determined from the experimentally measured stress-strain

relationships in the various loading tests described above. A least squares fit, which minimizes the relative error in stress, is used for this purpose.

The foam potential is linear in the coefficients i but strongly nonlinear in terms of the exponents i and

i thus necessitating use of a nonlinear least squares procedure. For the n nominal stress-nominal strain

data pairs, the error measure E is minimized by E = sum(i=1to n)(1-Tith/Ti

test2, where Titest is a stress

value from the test data and Tith comes from one of the nominal stress expressions derived above.

Minimizing the relative error in stress implies that the error in slope (modulus) is minimized; minimization of the absolute error would decrease the error at larger strains, at the expense of the accuracy at small strains.

Alternative Method for Determination of Constants for Moderate Strains

Since the polynomial form with N=1 is very commonly used for cases where the nominal strain is not too large, an alternative method of finding the material constants, assuming incompressibility, is to use the uniaxial test data as follows. The nominal strain in the direction of loading in the uniaxial test is U=U-1. Expanding the equation for TB in terms of U, using the Mooney-Rivlin form, and neglecting

terms of higher than second-order in U, gives

TU=6U(C10+C01 -(C10+2C01)U).

This is a parabola: the slope of this curve at the origin (the effective Young’s modulus at zero strain) is

6(C10+C01); this slope, together with the second-order term -6(C10+2C01)U2, defines the constants C10

and C01.

If compressibility should be modeled, then, under pure pressure loading, the compressible model with N=1 gives, to first-order in the volumetric strain V=311,


p=-(2 / D1)V,

so that, at small nominal strains, the bulk modulus is defined as:

K=(2 / D1)

Hyperelastic Models in MSC Nastran

Various options are provided for defining the material properties. The first (available in both Patran and MSC Nastran) is to give the parameters of the polynomial form and , or the parameters of the

Ogden form and as functions of the temperature. The second is to give the value of N, and

give experimental stress-data for up to four simple tests: uniaxial, equilibrium, planar and, if the material is compressible for volumetric compression test. MSC Nastran Implicit Nonlinear will then compute the

or and the . This method is available for N = 1 and N=2 for the polynomial form and up to

N = 6 for the Ogden form, and does not allow the properties to be temperature dependent.

In either case, you should be careful about defining the or : especially when N > 1, the

behavior at higher strains is strongly sensitive to the values of the or , and unstable material

behavior may result if these values are not correctly defined. When some of the coefficients are strongly negative, instability at higher strain levels is likely to occur.

Because the properties of rubber-like materials can vary significantly from one sample to another, it is important that test data are taken from experiments on the same sample (or samples cut from the same sheet), regardless whether the or are computed by the user or by the built-in method.

This material option can be used by itself, or can be combined with viscoelasticity to define time dependent hyperelastic behavior. It cannot be combined with other material options such as plasticity or creep. It may be used with the pure displacement formulation elements or with the “hybrid” (mixed formulation) elements. Because elastomeric materials are usually almost completely incompressible, fully integrated pure displacement method elements are not recommended for use with this material, except for plane stress cases. If fully or selectively reduced integration displacement method elements are used with the almost incompressible form of this material model in anything except plane stress analysis, a penalty method is used to impose the incompressibility constraint. This can sometimes lead to numerical difficulties, and the fully or selectively reduced integrated “hybrid” formulation elements are therefore recommended.

Specifying Hyperelastic Material Entries

Nonlinear hyperelastic materials are characterized in MSC Nastran with the following Bulk Data entries:.

Entry Description

MATHP Specifies material properties for use in fully nonlinear (i.e., large strain and large rotation) hyperelastic analysis of rubber-like materials (elastomers).

MATHE Specifies hyperelastic (rubber-like) material properties for nonlinear (large strain and large rotation) analysis in Nonlinear Analysis.

N Aij Di

N i i Di

Aij i i Di

Aij i i

Aij i i

Aij i i


298


To define a hyperelastic material in Patran:


2. Click Input Properties..., and select Hyperelastic from the Constitutive Model pull-down menu.

3. Select Test Data or Coefficients as the Data Type.

4. From the Strain Energy Potential pull-down menu, select a model and enter properties as described below.

Hyperelastic material models require the following material data via the Input Properties subform on the Materials Application form.

Mooney-Rivlin and James-Green-Simpson

Ogden

Hyperelastic -Mooney/ James Description

Strain Energy Function, C10, C01, C11, C20, C30

Strain energy densities as a function of the strain invariants in the material. May vary with temperature via a defined material field. This option consolidates several of the Marc hyperelastic material models.



Defines the instantaneous coefficient of thermal expansion. This property is optional. May vary with temperature via a defined material field.

Bulk Modulus Defines the Bulk Modulus.

Reference Temperature Defines the reference temperature for the thermal expansion coefficient.

Hyperelastic-Ogden Description

Bulk Modulus K Defines the Bulk Modulus.

Density Defines the material mass density.


Defines the instantaneous coefficient of thermal expansion. This property is optional. May vary with temperature via a defined material field


Modulus k in the Ogden equation.

Exponent k in the Ogden equation.

k

k


Foam

Arruda-Boyce

Gent

Hyperelastic-Foam Description

Density Defines the material mass density.




Modulus n in the Foam equation.

Deviatoric Exponent n in the Foam equation.

Volumetric Exponent n in the Foam equation.

Hyperelastic-Arruda- Boyce Description

NKT Chain density times Boltzmann constant times temperature.

Chain Length Average chemical chain cross length.


Density This defines the material mass density.




Hyperelastic-Gent Description

Tensile Modulus Defines standard tension modulus (E).

Maximum 1st Invariant Defines .


Density This defines the material mass density.


Defines the coefficient of thermal expansion.


un

n

n

I1* I1

*I1 3–=


300

ViscoelasticThe material models discussed in previous sections are considered to be time independent. However, rubber materials often show a rate-dependent behavior and can be modeled as viscoelastic materials. Viscoelasticity can be applied:

• To determine the current state of deformation based on the entire time history of loading.

• To characterize small strain and large strain problems.

• With other material options for linear elastic response (small strain) and hyperelastic response (large strain).

• To include temperature dependencies.

• For isotropic, anisotropic, and incompressible materials.

Small Strain Viscoelasticity

In the stress relaxation form, the constitutive relation can be written as a hereditary integral formulation

(10-66)

The functions are called stress relaxation functions. They represent the response to a unit applied

strain and have characteristic relaxation times associated with them. The relaxation functions for materials with a fading memory can be expressed in terms of Prony or exponential series.

(10-67)

in which is a tensor of amplitudes and is a positive time constant (relaxation time). In the current

implementation, it is assumed that the time constant is isotropic. In Equation (10-67), represents the

long term modulus of the material.

The short term moduli (describing the instantaneous elastic effect) are then given by

(10-68)

The stress can now be considered as the summation of the stresses in a generalized Maxwell model (Figure 10-19)

ij t Gijkl t – dkl

d-------------------d

0

t

Gijkl t kl 0 +=

Gijkl

Gijkl t Gijkl

Gijkln

exp t n

–

n 1=

N

+=

Gijkl

nn

Gijkl

Gijkl0

Gijkl 0 Gijkl

Gijkln

n 1=

N

+= =


(10-69)

where

(10-70)

(10-71)

Figure 10-19 The Generalized Maxwell or Stress Relaxation Form

For integration of the constitutive equation, the total time interval is subdivided into a number of subintervals ( ) with time-step . A recursive relation can now be derived expressing

the stress increment in terms of the values of the internal stresses at the start of the interval. With the

assumption that the strain varies linearly during the time interval h, we obtain the increment stress-strain relation as

(10-72)

where

(10-73)

and

(10-74)

ij t ij

t ijn

t

n 1=

N

+=

ij

Gijkl

kl t =

ijn

Gijkln

exp t – n

– dkl

d-------------------d

0

t

=

E1

1

q1

E0

i = i/Ei

E2

2

q2

Ei

i

qiE

tm 1– tm h tm tm 1––=

i jn

ij tm Gijkl

nh Gijkl

n

n 1=

N

+ kl n

h ijn

tm h–

n 1=

N

–=

n h 1 exp h n

– –=

n

h n

h n

h=


302

In MSC Nastran Implicit Nonlinear, the incremental equation for the total stress is expressed in terms of the short term moduli (See Equation (10-68)).

(10-75)

Note that the set of equations given by Equation (10-75) can directly be used for both anisotropic and isotropic materials.

Isotropic Viscoelastic Material

For an isotropic viscoelastic material, MSC Nastran Implicit Nonlinear assumes that the deviatoric and volumetric behavior are fully uncoupled and that the behavior can be described by a time dependent shear and bulk modules. The bulk moduli is generally assumed to be time independent; however, this is an unnecessary restriction of the general theory.

Both the shear and bulk moduli can be expressed in a series

(10-76)

(10-77)

with short term values given by

(10-78)

(10-79)

Let the deviatoric and volumetric component matrices and be given by

ij tm Gijkl0

1 n

h – Gijkln

n 1=

N

– kl tm n

h ijn

n 1=

N

–= tm h–

G t G

Gn

exp t dn

–

n 1=

N

+=

K t K

Kn

exp t vn

–

n 1=

N

+=

G0 G Gn

n 1=

N

+=

K0 K Kn

n 1=

N

+=

d v


The increment set of equations is then given by

(10-80)

and

(10-81)

Note that the deviatoric and volumetric response are fully decoupled.

Note that the algorithm is exact for linear variations of the strain during the increment. The algorithm is implicit; hence, for each change in time-step, a new assembly of the stiffness matrix is required.

d

4 3 2– 3 2– 3 0 0 0

2– 3 4 3 2– 3 0 0 0

2– 3 2– 3 4 3 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 0 0 1

=

v

1 1 1 0 0 0

1 1 1 0 0 0

1 1 1 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

=

tm G0

1 dn

h – Gn

n 1=

Nd

–

d tm =

dn

h dn

tm h–

n 1=

Nd

vn

h vn

tm h–

n 1=

Nv

––

K0

1 vn

h – Kn

n 1=

Nv

–

v tm

dn

tm dn

h Gnd tm d

nh d

ntm h– –=

vn

tm vn

h Knv tm v

nh v

ntm h– –=


304

Anisotropic Viscoelastic Material

Equation (10-75) can be used for the analysis of anisotropic viscoelastic materials. Also, a complete set of moduli (21 components) can be specified in the HOOKVI user subroutine. Referencing a local coordinate system or use of the ORIENT user subroutine can be used to define a preferred orientation

both for the short time moduli and the amplitude functions .

Incompressible Isotropic Viscoelastic Materials

Incompressible elements in MSC Nastran Implicit Nonlinear allow the analysis of incompressible and nearly incompressible materials in plane strain, axisymmetric and three-dimensional problems. The incompressibility of the element is simulated through the use of an perturbed Lagrangian variational principle based on the Herrmann formulation.

The constitutive equation for a material with no time dependence in the volumetric behavior can be expressed as

(10-82)

(10-83)

The hydrostatic pressure term is used as an independent variable in the variational principle. The Herrmann pressure variable is now defined in the same way as in the formulation for time independent elastic materials.

(10-84)

The constitutive Equation (10-82) and Equation (10-83) can then be rewritten

(10-85)

where

(10-86)

Gijkl

0Gijkl

n

i j tm 2 Gijkl0

1 n

h – Gijkln

n 1=

N

–

kl tm 13---pp tm kl–=

n

h ij n

tm

n 1=

N

–13---kkij+

pp tm 3K0pp tm =

Hpp

2G0 1 0+ --------------------------------=

ij tm 2Geij Hij+ n h ij

ntm h–

n 1=

N

–=

Ge G0 1 n h Gn– n 1=

N

–=


(10-87)

Large Strain Viscoelasticity

For an elastomeric time independent material, the constitutive equation is expressed in terms of an

energy function . For a large strain viscoelastic material, Simo generalized the small strain viscoelasticity material behavior to a large strain viscoelastic material. The energy functional then becomes

(10-88)

where are the components of the Green-Lagrange strain tensor, internal variables and the

elastic strain energy density for instantaneous deformations. In MSC Nastran Implicit Nonlinear, it is

assumed that , meaning that the energy density for instantaneous deformations is given by the third order James Green and Simpson form or the Ogden form.

The components of the second Piola-Kirchhoff stress then follow from

(10-89)

The energy function can also be written in terms of the long term moduli resulting in a different set of

internal variables

(10-90)

where is the elastic strain energy for long term deformations. Using this energy definition, the stresses are obtained from

(10-91)

Observing the similarity with the equations for small strain viscoelasticity the internal variables can be obtained from a convolution expression

(10-92)

G0

1 0

+ Ge

1 20

– –

3Ge

--------------------------------------------------------------------=

W

EijQijn

0

Eij Qijn

Eij

n 1=

N

– In

Qijn

n 1=

N

+=

Eij Qij

n0

0W=

SijEij---------- 0

Eij---------- Qij

n

n 1=

N

–= =

Tij

n

Eij Ti jn

Eij Tijn

Eij

n 1=

N

+=

Sij E Eij

-------------------- Tijn

n 1=

N

+=

Tijn

S·i jn

0t exp t –

n– d=


306

where are internal stresses obtained from energy functions.

(10-93)

Let the total strain energy be expressed as a Prony series expansion

(10-94)

If, in the energy function, each term in the series expansion has a similar form, Equation (10-94) can be rewritten

(10-95)

where is a scalar multiplier for the energy function based on the short term values.

The stress-strain relation is now given by

(10-96)

(10-97)

(10-98)

Analogue to the derivation for small strain viscoelasticity, a recursive relation can be derived expressing the stress increment in terms of values of the internal stresses at the start of the increment.

The equations are reformulated in terms of the short time values of the energy function

(10-99)

Sij

n

Sijn n

Eij----------=

nexp t n–

n 1=

N

+=

n0exp t n–

n 1=

N

+=

n

Sij t Sij

t Tijn

n 1=

N

t +=

Sij

Eij----------- 1 n

n 1=

N

–

0

Eij----------= =

Tijn

nSi j

0t exp t –

n– d

0

t

=

Sij tm 1 1 n h –

n 1=

N

– n

Sij0

tm Sij0

tm h– – =

nSi jn

tm h–

n 1=

N

–


(10-100)

It is assumed that the viscoelastic behavior in MSC Nastran Implicit Nonlinear acts only on the deviatoric behavior.

Viscoelastic Models in MSC Nastran

MSC Nastran Implicit Nonlinear has two models that represent viscoelastic materials. The first can be defined as a Kelvin-Voigt model. The latter is a general hereditary integral approach.

Kelvin-Voigt Model

The Kelvin model allows the rate of change of the inelastic strain to be a function of the total stress and previous strain.

The Kelvin material behavior (viscoelasticity) is modeled by assuming an additional creep strain ,

governed by

(10-101)

where and may be defined in the user subroutine CRPVIS and the total strain is

(10-102)

(10-103)

(10-104)

(10-105)

(10-106)

(10-107)

The CRPVIS user subroutine is called at each integration point of each element when the Kelvin model is used.

Use the NLPARM option and set a nonzero time increment to define the time step and to set the tolerance control for the maximum strain in any increment.

This option allows Maxwell models to be included in series with the Kelvin model.

Sij tm n h n Sij0

tm Sijn

tm h– – n h Sijn

tm h– –=

i j

k

ddt-----

i jk Ai jklkl

Bijklklk–=

A B

i j i je

i jp

i jc

i jk

i jth+ + + +=

i jth

= thermal strain components

i je

= elastic strain components (instantaneous response)

i jp

= plastic strain components

i jc = creep strains defined via the CRPLAW and VSWELL user subroutines

i jk = Kelvin model strain components as defined above


308

Hereditary Integral Model

The stress-strain equations in viscoelasticity are not only dependent on the current stress and strain state (as represented in the Kelvin model), but also on the entire history of development of these states. This constitutive behavior is most readily expressed in terms of hereditary or Duhamel integrals. These integrals are formed by considering the stress or strain build-up at successive times. Two equivalent integral forms exist: the stress relaxation form and the creep function form. In MSC Nastran Implicit Nonlinear, the stress relaxation form is used.

The viscoelasticity option in MSC Nastran Implicit Nonlinear can be used for both the small strain and large strain Mooney, Ogden, Arruda Boyce, and Gent material stress-relaxation problems. A description of these models is as follows:

Experimental Determination of Viscoelastic Material Parameters

The free energy function versus time data being used for large strain viscoelasticity can be generated by fitting experimental data provided the following two tests are done:

1. Standard quasi-static tests (tensile, planar-shear, simple-shear, equi-biaxial tension, volumertic) to determine the elastomer free energy constants.

2. Standard relaxation tests to obtain stress versus time.

Temperature Dependence of Viscoelastic Materials

The rate processes in many viscoelastic materials is known to be highly sensitive to temperature changes. Such temperature-dependent properties cannot be neglected in the presence of any appreciable temperature variation. For example, there is a large class of polymers which are adequately represented by linear viscoelastic laws at uniform temperature. These polymers exhibit an approximate translational shift of all the characteristic response functions with a change of temperature, along a logarithmic time axis. This shift occurs without a change of shape. These temperature-sensitive viscoelastic materials are characterized as Thermo-Rheologically Simple.

A “reduced” or “pseudo” time can be defined for the materials of this type and for a given temperature field. This new parameter is a function of both time and space variables. The viscoelastic law has the same form as one at constant temperature in real time. If the shifted time is used, however, the transformed viscoelastic equilibrium and compatibility equations are not equivalent to the corresponding elastic equations.

In the case where the temperature varies with time, the extended constitutive law implies a nonlinear dependence of the instantaneous stress state at each material point of the body upon the entire local temperature history. In other words, the functionals are linear in the strains but nonlinear in the temperature.

The time scale of experimental data is extended for Thermo-Rheologically Simple materials. All characteristic functions of the material must obey the same property. The shift function is a basic property of the material and must be determined experimentally. As a consequence of the shifting of the mechanical properties data parallel to the time axis, the values of the zero and infinite frequency complex moduli do not change due to shifting. Hence, elastic materials with temperature-dependent

W0


characteristics neither belong to nor are consistent with the above hypothesis for the class of Thermo-Rheologically Simple viscoelastic solids.

In addition to the Thermo-Rheologically Simple material behavior variations of initial stress-strain

moduli , the temperature of the other mechanical properties (coefficient of thermal expansion, etc.)

due to changes in temperature can be specified.

Note, however, that only the instantaneous moduli are effected. Hence, the long term moduli given by

(10-108)

can easily become negative if the temperature effects are not defined properly.

The effect of temperature, , on the material behavior is introduced through the dependence of the elastic modulus, G, on temperature, and through a reduced time concept:

(10-109)

where G=G(), and xi(t) is the reduced time, defined by

(10-110)

where A((t)) is the shift function at time t. Often the shift function is approximated by the Williams Landell Ferry (WLF) form:

(10-111)

where C1, C2 and 0 are constants (0 is the “glassy transition” temperature).

Narayanaswamy ModelThe annealing of flat glass requires that the residual stresses be of an acceptable magnitude, while the specification for optical glass components usually includes a homogenous refractive index. The design of heat treated processes can be accomplished using the Narayanaswamy model. This allows you to study the time dependence of physical properties (for example, volumes) of glass subjected to a change in temperature.

For more information pertaining to the Narayanaswamy Model, see Marc Volume A: Theory and User Information, Chapter 7 Material Library.

Gijkl

0

Gijkl

Gijkl0

t Gijkln

n 1=

N

–=

G g t s – · s ds0

t

+

=

t dsA s --------------------

0

t

=

A logC1 0–

C2 0– +----------------------------------–=


310

Specifying Viscoelastic Material Entries

The viscoelastic MATVE and MATTVE material options are provided for cases where dissipative losses caused by “viscous” (internal friction) effects in materials must be modeled. For time domain analysis, this option is used with an elastic model to define classical linear, small strain, viscoelastic behavior, or with hyperelastic or foam models to define finite linear, large deformation, viscoelastic behavior. As described in the previous section, viscoelastic relaxation data can be fit using the experimental data fitting (EDF) capability available in Patran. See Experimental Data Fitting, 381.


To define a viscoelastic material in Patran:


2. Click Input Properties..., and select Viscoelastic from the Constitutive Model pull-down menu.

This input data creates the viscoelastic options. All inputs must have the same number of time points (at the same times) in the referenced fields. The following equations may be useful when creating the Prony

series for the bulk and shear moduli: .

Viscoelastic material models require the following material data via the Input Properties subform on the Materials Application form.

Entry Description

MATVE Specifies isotropic visco-elastic material properties to be used for quasi-static or dynamic analysis in MSC Nastran Implicit Nonlinear.

MATTVE Specifies temperature-dependent visco-elastic material properties in terms of Thermo-Rheologically Simple behavior to be used for quasi-static or transient dynamic analysis in MSC Nastran Implicit Nonlinear.

Isotropic Description

Shear Constant If a material field of time vs. value is supplied, will create a MATVE option. This is valid when MAT1/MATS1 are used.

Energy Function Multiplier Defines the duration effect on the hyperelastic model as a multiplier to the strain energy density function. This is valid when a Hyperelastic constitutive model for Neo-Hookean, Mooney-Rivlin, Jamus-Green-Simpson, Arruda-Boyce, or Gent is present.

Deviatoric Multiplier If a material field of time vs. value is supplied, will create a MATVE option.

K E 3 1 2v– G E 2 1 v+ ==


Dilatational Multiplier Creates a dilatational multiplier.

Solid Coeff. of Thermal

Exp

Creates coefficient of thermal expansion for solids.

Liquid Coeff of Thermal Exp Creates coefficient of thermal expansion for liquids.

Orthotropic Description

Young’s Modulus, E11/E22/E33 Defines the duration effects on the elastic moduli. This information is optional. This is only valid when an elastic and/or plastic constitutive model is present.

Poissons Ratio 12/23/31 Defines the duration effects on the Poisson’s ratios. This information is optional.

Shear Modulus G12/G23/G31 Defines the duration effects on the shear moduli. This information is optional.

Solid Coeff of Thermal

Exp

Same as for Isotropic

Liquid Coeff of Thermal Exp Same as for Isotropic


MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideInelastic

312

InelasticMost materials of engineering interest initially respond elastically. Elastic behavior means that the deformation is fully recoverable, so that, when the load is removed, the specimen returns to its original shape. If the load exceeds some limit (the “yield load”), the deformation is no longer fully recoverable. Some parts of the deformation will remain when the load is removed as, for example, when a paper clip is bent too much, or when a billet of metal is rolled or forged in a manufacturing process. Plasticity theories model the material’s mechanical response as it undergoes such nonrecoverable deformation in a ductile fashion. The theories have been developed most intensively for metals, but they are applied to soils, concrete, rock, ice, and so on. These materials behave in very different ways (for example, even large values of pure hydrostatic pressure cause very little inelastic deformation in metals, but quite small hydrostatic pressure may cause a significant, non-recoverable volume change in a soil sample), but the fundamental concepts of plasticity theories are sufficiently general that models based on these concepts have been successfully developed for a wide range of materials. A number of these plasticity modes are available in the MSC Nastran Implicit Nonlinear material library.

In nonlinear material behavior, the material parameters depend on the state of stress. Up to the proportional limit, i.e., the point at which linearity in material behavior ceases, the linear elastic formulation for the behavior can be used. Beyond that point, and especially after the onset of yield, nonlinear formulations are required. In general, two ingredients are required to ascertain material behavior:

1. an initial yield criterion to determine the state of stress at which yielding is considered to begin

2. mathematical rules to explain the post-yielding behavior.

There are two major theories of plastic behavior that address these criterion differently. In the first, called deformation theory, the plastic strains are uniquely defined by the state of stress. The second theory, called flow or incremental theory, expresses the increments of plastic strain (irrecoverable strains) as functions of the current stress, the strain increments, and the stress increments. Incremental theory is more general and can be adapted in its particulars to fit a variety of material behaviors. The plasticity models in MSC Nastran Implicit Nonlinear are “incremental” theories, in which the mechanical strain rate is decomposed into an elastic part and a plastic (inelastic) part through various assumed flow rules.

The incremental plasticity models are formulated in terms of:

• A yield surface, which generalizes the concept of “yield load” into a test function which can be used to determine if the material will respond purely elastically at a particular state of stress, temperature, etc.;

• A flow rule that defines the inelastic deformation that must occur if the material point is no longer responding purely elastically;

• and some evolution laws that define the hardening - the way in which the yield and/or flow definitions change as inelastic deformation occurs.

The models also need an elasticity definition, to deal with the recoverable part of the strain models divide into those that are rate-dependent and those that are rate-independent.

MSC Nastran Implicit Nonlinear includes the following models of inelastic behavior.


• Metal Plasticity (von Mises or Hill)

• ORNL (Oak Ridge National Laboratory) - characterizes creep behavior and cyclic loading effects on stainless steel materials.

• Porous Metal Plasticity (Gurson) - includes effects of hydrostatic pressure and failure processes in ductile materials.

• Pressure-Dependent models - models the behavior of granular (soil and rock) materials or polymers, in which the yield behavior depends on the equivalent pressure stress.

• Linear Mohr-Coulomb

• Parabolic Morh-Coulomb

• Buyukozturk Concrete

Yield ConditionsThe yield stress of a material is a measured stress level that separates the elastic and inelastic behavior of the material. The magnitude of the yield stress is generally obtained from a uniaxial test. However, the stresses in a structure are usually multiaxial. A measurement of yielding for the multiaxial state of stress is called the yield condition. Depending on how the multiaxial state of stress is represented, there can be many forms of yield conditions. For example, the yield condition can be dependent on all stress components, on shear components only, or on hydrostatic stress. A number of yield conditions are available in MSC Nastran Implicit Nonlinear, and are discussed in this section.

Metal Plasticity

The von Mises yield surface is widely used for plasticity in isotropic metals. It is assumed that the yield and plastic flow describe isotropic metals at low temperatures where creep effects can be ignored. Anisotropic metals and composite materials, can be treated by extensions of von Mises yield function, as described in Hill’s yield function.

von Mises

The success of the von Mises criterion is due to the continuous nature of the function that defines this criterion and its agreement with observed behavior for the commonly encountered ductile materials. The von Mises criterion states that yield occurs when the effective (or equivalent) stress () equals the yield stress (y) as measured in a uniaxial test. Figure 10-20 shows the von Mises yield surface in

two-dimensional and three-dimensional stress space.


314

Figure 10-20 von Mises Yield Surface

For an isotropic material

(10-112)

where 1, 2, and 3 are the principal Cauchy stresses.

can also be expressed in terms of nonprincipal Cauchy stresses.

(10-113)

The yield condition can also be expressed in terms of the deviatoric stresses as:

(10-114)

where is the deviatoric Cauchy stress expressed as

(10-115)

For isotropic material, the von Mises yield condition is the default condition in MSC Nastran Implicit Nonlinear.

YieldSurface

ElasticRegion

(b) -Plane

YieldSurface

ElasticRegion

(a) Two-dimensional Stress Space

3

21

1

2

1 2– 2 2 3– 2 3 1– 2+ + 1 2 2=

x y– 2 y z– 2 z x– 2 6 xy2 yz

2 zx2+ + + + + 1 2= 2

32---

i j

i j=

i j

i j

i j13---kki j–=


Hill’s Yield Function

Hill’s yield surface has been widely used both as a yield surface and as a failure surface for anisotropic and composite materials. Hill’s yield function is a generalization of von Mises as expressed below.

(10-116)

Note the following points about Hill’s surface:

1. It degenerates into von Mises surface when all three direct yield stresses are equal

(Fx = Fy = Fz) and all three shear yield stresses are equal.

2. It is invariant with respect to hydrostatic stress, as is von Mises.

3. Hill's surface, unlike von Mises, is not always an ellipsoid in stress space. When it is not an ellipsoid, it is not appropriate for use as a yield function (since it does not have an inside and an outside, thereby dividing stress space into elastic and plastic regions).

Mohr-Coulomb Material (Hydrostatic Stress Dependence)

MSC Nastran Implicit Nonlinear includes options for elastic-plastic behavior based on a yield surface that exhibits hydrostatic stress dependence. Such behavior is observed in a wide class of soil and rock-like materials. These materials are generally classified as Mohr-Coulomb materials (generalized von Mises materials). Ice is also thought to be a Mohr-Coulomb material. The generalized Mohr-Coulomb model developed by Drucker and Prager is implemented in MSC Nastran Implicit Nonlinear. There are two types of Mohr-Coulomb materials: linear and parabolic. Each is discussed on the following pages.

xx

Fx--------

2 yy

Fy--------

2 zz

Fz--------

2+ +

1

Fx2

------ 1

Fy2

------ 1

Fz2

------–+

xxyy–

1

Fx2

------ 1

Fy2

------– 1

Fz2

------+

xxzz–

1

Fx2

------– 1

Fy2

------ 1

Fz2

------+ +

yyzz–

+xy

Fxy--------

2 yz

Fyz--------

2 zx

Fzx--------

2+ + 1=


316

Linear Mohr-Coulomb Material

The deviatoric yield function, as shown in Figure 10-21, is assumed to be a linear function of the hydrostatic stress.

(10-117)

where

(10-118)

(10-119)

The constants and can be related to and by

(10-120)

where is the cohesion and is the angle of friction.

Figure 10-21 Yield Envelope of Plane Strain (Linear Mohr-Coulomb Material)

Parabolic Mohr-Coulomb Material

The hydrostatic dependence is generalized to give a yield envelope which is parabolic in the case of plane strain (see Figure 10-22).

(10-121)

f J1 J21 2

3-------–+ 0= =

J1 i i=

J212---

i j

i j=

c

c

3 1 122– 1 2

---------------------------------------------- 3

1 32– 1 2----------------------------------; sin= =

c

Yield Envelope

c

R

x + y2

f 3J2 3J1+ 1 2 – 0= =


(10-122)

where is the cohesion.

Figure 10-22 Resultant Yield Condition of Plane Strain (Parabolic Mohr-Coulomb Material

Buyukozturk Criterion (Hydrostatic Stress Dependence)

The Buyukozturk concrete plasticity model is a particular form of the generalized Drucker-Prager plasticity model, which is developed specifically for plane stress cases by Buyukozturk. This yield criterion, which originally has been proposed as a failure criterion, has the general form:

(10-123)

The Buyukozturk criterion reduces to the parabolic Mohr-Coulomb criterion if .

Oak Ridge National Laboratory Options

Oak Ridge National Laboratory (ORNL) has performed a large number of creep tests on stainless and other alloy steels. It has also set certain rules that characterize creep behavior for application in the nuclear structures. A summary of the ORNL rules on creep is discussed in Marc Volume A, Theory and User Information. In MSC Nastran Implicit Nonlinear, the ORNL options are based on the definitions of ORNL-TM- 3602 [1] for stainless steels and ORNL recommendations [2] for 2 1/4 Cr-1 Mo steel.

The initial yield stress should be used for the initial inelastic loading calculations for both the stainless steels and 2 1/4 Cr-1 Mo steel. The 10th-cycle yield stress should be used for the hardened material. The 100th-cycle yield stress must be used in the following circumstances:

1. To accommodate cyclic softening of 2 1/4 Cr-1 Mo steel after many load cycles.

2. After a long period of high temperature exposure.

3. After the occurrence of creep strain.

3 3c2 2– 1 2

---------------------------------------------=2

3 c2 2

3------–

=

c

c2

c

R

x + y2

f 3J1 J12

3J2 2

–+ +=

0=


318

Work Hardening RulesThe work-hardening rule defines the way the yield surface changes with plastic straining. A material is said to be “perfectly plastic” if, upon the stress state touching the yield surface, an infinitesimal increase in stress causes an arbitrarily large plastic strain. MSC Nastran Implicit Nonlinear models all materials as work hardening, and treats perfectly plastic materials as a special case. Because the tangent stiffness method is used, no difficulties arise in setting the work hardening slope equal to zero. Besides perfect plasticity, three possibilities are provided: isotropic hardening and kinematic hardening.

The isotropic workhardening rule assumes that the center of the yield surface remains stationary in the stress space, but that the size (radius) of the yield surface expands, due to workhardening. This type of hardening is appropriate when the straining is the same in all directions.

For many materials, the isotropic workhardening model is inaccurate if unloading occurs (as in cyclic loading problems). For these problems, the kinematic hardening model or the combined hardening model represents the material better.

Isotropic, Kinematic, and Combined Hardening

The isotropic workhardening rule assumes that the center of the yield surface remains stationary in the stress space, but that the size (radius) of the yield surface expands, due to workhardening. The change of the von Mises yield surface is plotted in Figure 10-23b.

A review of the load path of a uniaxial test that involves both the loading and unloading of a specimen will assist in describing the isotropic workhardening rule. The specimen is first loaded from stress free (point 0) to initial yield at point 1, as shown in Figure 10-23a. It is then continuously loaded to point 2. Then, unloading from 2 to 3 following the elastic slope E (Young’s modulus) and then elastic reloading from 3 to 2 takes place. Finally, the specimen is plastically loaded again from 2 to 4 and elastically unloaded from 4 to 5. Reverse plastic loading occurs between 5 and 6.

It is obvious that the stress at 1 is equal to the initial yield stress and stresses at points 2 and 4 are larger

than , due to workhardening. During unloading, the stress state can remain elastic (for example, point

3), or it can reach a subsequent (reversed) yield point (for example, point 5). The isotropic workhardening rule states that the reverse yield occurs at current stress level in the reversed direction.

Original

Hardened

Isotropic Hardening Kinematic Hardening

y

y


Figure 10-23 Schematic of Isotropic Hardening Rule (Uniaxial Test)

Let be the stress level at point 4. Then, the reverse yield can only take place at a stress level of

(point 5).

For many materials, the isotropic workhardening model is inaccurate if unloading occurs (as in cyclic loading problems). For these problems, the kinematic hardening model or the combined hardening model represents the material better.

6

5

12

65

4

3

2 1

0

E

E

E

+4

y

4

4

30

(a) Loading Path

(b) von Mises Yield Surface

3

2

1

4 4–


320

Kinematic Hardening

Under the kinematic hardening rule, the von Mises yield surface does not change in size or shape, but the center of the yield surface can move in stress space. Figure 10-23d illustrates this condition. Ziegler’s law is used to define the translation of the yield surface in the stress space.

The loading path of a uniaxial test is shown in Figure 10-23c. The specimen is loaded in the following order: from stress free (point 0) to initial yield (point 1), 2 (loading), 3 (unloading), 2 (reloading), 4 (loading), 5 and 6 (unloading). As in isotropic hardening, stress at 1 is equal to the initial yield stress ,

and stresses at 2 and 4 are higher than , due to workhardening. Point 3 is elastic, and reverse yield takes

place at point 5. Under the kinematic hardening rule, the reverse yield occurs at the level of , rather than at the stress level of . Similarly, if the specimen is loaded to a higher stress

level (point 7), and then unloaded to the subsequent yield point 8, the stress at point 8 is

. If the specimen is unloaded from a (tensile) stress state (such as point 4 and 7), the

reverse yield can occur at a stress state in either the reverse (point 5) or the same (point 8) direction.

For many materials, the kinematic hardening model gives a better representation of loading/unloading behavior than the isotropic hardening model. For cyclic loading, however, the kinematic hardening model can represent neither cyclic hardening nor cyclic softening.

Combined Hardening

Figure 10-25 shows a material with highly nonlinear hardening. Here, the initial hardening is assumed to be almost entirely isotropic, but after some plastic straining, the elastic range attains an essentially constant value (that is, pure kinematic hardening). The basic assumption of the combined hardening model is that such behavior is reasonably approximated by a classical constant kinematic hardening constraint, with the superposition of initial isotropic hardening. The isotropic hardening rate eventually decays to zero as a function of the equivalent plastic strain measured by

(10-124)

Figure 10-24 Basic Uniaxial Tension Behavior of the Combined Hardening Model

y

y

5 4 2y– = 4–

7

8 7 2y– =

· p

· p

dt23---

·i j

p·i j

p

1 2

dt= =

Fully HardenedPure KinematicRange

CombinedHardeningRange

InitialElasticRange

InitialYield

One-half CurrentElastic Range

32

ddpKinematic Slope,

Stress

Strain


This implies a constant shift of the center of the elastic domain, with a growth of elastic domain around this center until pure kinematic hardening is attained. In this model, there is a variable proportion between the isotropic and kinematic contributions that depends on the extent of plastic deformation (as

measured by ).

The workhardening data at small strains governs the isotropic behavior, and the data at large strains ( ) governs the kinematic hardening behavior. If the last workhardening slope is zero, the behavior is the same as the isotropic hardening model.

Experimental Determination of Work Hardening Slope

In a uniaxial test, the workhardening slope is defined as the slope of the stress-plastic strain curve. The workhardening slope relates the incremental stress to incremental plastic strain in the inelastic region and dictates the conditions of subsequent yielding. A number of workhardening rules (isotropic, kinematic, and combined) are available in MSC Nastran Implicit Nonlinear. A description of these workhardening rules is given below. The uniaxial stress-plastic strain curve can be represented by a piecewise linear function or through the user subroutine WKSLP. This requires the use of MARCIN to specify the MARC WORKHARD option.

Figure 10-25 Workhardening Slopes

p

p 1000

E E E E

Strain

Stress

1p

32

1

2p

3p


322

You enter a table of yield stress, plastic strain points.

The yield stress and the workhardening data must be compatible with the procedure used in the analysis. For small strain analyses, the engineering stress and engineering strain are appropriate. If only PARAM,LGDISP is used, the yield stress should be entered as the second Piola-Kirchhoff stress, and the workhard data be given with respect to plastic Green-Lagrange strains. If PARAM,LGDISP,1 or 2 are used, the yield stress must be defined as a true or Cauchy stress, and the workhardening data with respect to logarithmic plastic strains. Engineering stress and strain may be defined and Bulk Data parameter MRTABLS1 used to provide the program with rules to convert to the proper stress and strain measures.

Flow RulesYield stress and workhardening rules are two experimentally related phenomena that characterize plastic material behavior. The flow rule is also essential in establishing the incremental stress-strain relations for

plastic material. The flow rule describes the differential changes in the plastic strain components as a function of the current stress state. So long as a material point is elastic, Hooke’s law provides a relationship between total stress and strain. After a material becomes plastic, however, there is no longer a unique relationship between total stress and strain. The problem then is usually solved incrementally, following the exact loading path.

For points which are plastic, a flow rule is used to relate increments of stress to plastic strain. MSC Nastran Implicit Nonlinear uses an associated flow rule, which prescribes that increments of plastic strain are computed as a constant times the gradient of the yield function.

In other words, considering the yield function as a surface in stress space, the plastic strain increment is a vector in the direction of the outward normal to the surface at the point where it is touched by the stresses on the loading path.

1

1p

----------

2

2p

----------

3

3p

----------

0.0

1p

1p

2p

+

Slope Breakpoint

Note: The data points should be based on a plot of the stress versus plastic strain for a tensile test. The elastic strain components should not be included.

dp


The equation representing this is:

(10-125)

where is a constant. Writing the six equations explicitly:

(10-126)

These stress vs. plastic strain equations are analogous to the stress vs. total strain equations of elasticity, where elastic strains can be computed as the gradient of a strain energy potential function, namely;

(10-127)

Thus, the yield function F plays the role of a plastic potential. If a theory of plasticity uses something other than the yield function as a plastic potential, a so-called nonassociated flow rule results. Nonassociated flow rules are not available in MSC Nastran Implicit Nonlinear.

For the von Mises and modified Hill yield functions programmed in MSC Nastran Implicit Nonlinear, the derivatives in the yield function are obtained simply by differentiating with respect to individual components of stress. For example, for the modified Hill function, we have:

di jp F

i j----------=

dxxp F

xx------------=

dyyp F

yy------------=

dzzp F

zz-----------=

dxyp F

xy-----------=

dyzp F

yz----------=

dzxp F

xz----------=

di jUi j----------=


324

(10-128)

The constant in these flow rule equations is evaluated automatically by MSC Nastran Implicit Nonlinear on the basis of material stability during plastic flow (i.e., by the requirement that the stress state remain on the yield surface during plastic straining).

The Prandtl-Reuss representation of the flow rule is available in MSC Nastran Implicit Nonlinear. In conjunction with the von Mises yield function, this can be represented as:

(10-129)

where and are equivalent plastic strain increment and equivalent stress, respectively.

The significance of this representation is illustrated in Figure 10-26. This figure illustrates the “stress-space” for the two-dimensional case. The solid curve gives the yield surface (locus of all stress states causing yield) as defined by the von Mises criterion.

Equation (10-139) expresses the condition that the direction of inelastic straining is normal to the yield surface. This condition is called either the normality condition or the associated flow rule.

If the von Mises yield surface is used, then the normal is equal to the deviatoric stress.

dxxp

2xx

Fx2

------------yy

FxFy------------

zz

FxFz------------––=

dyyp

xx

FxFy------------

2yy

Fy2

------------zz

FyFz------------–+–=

dzzp

xx

FxFz------------

yy

FyFz------------

2zz

Fz2

-----------+––=

dxyp

xy

Fxy2

---------=

dyz2

xz

Fyz2

--------=

dzxp

yz

Fzx2

--------=

di jp

dp

i j

------------=

dp


Figure 10-26 Yield Surface and Normality Criterion 2-D Stress Space

Rate Dependent YieldStrain rate effects cause the structural response of a body to change because they influence the material properties of the body. These material changes lead to an instantaneous change in the strength of the material. Strain rate effects become more pronounced for temperatures greater than half the melting temperature ( ), but are sometimes present even at room temperature. The following discussion

explains the effect of strain rate on the size of the yield surface.

Using the von Mises yield condition and normality rule, we obtain an expression for the stress rate of the form

For elastic-plastic response

(10-130)

and

(10-131)

where

(10-132)

Yield Surface

2

d2dp

dp1

1

p

Tm

· i j Lijkl·kl ri j

·· p+=

Lijk l Cijk l Cijmn

mn-------------

pq------------ Cpqkl

D–=

ri j Cijmn

mn-------------

23---

· p

-------- D=

D49--- 2

p--------

i j---------- Cijkl

kl-----------+=


326

As strain rates increase, many materials show an increase in yield strength. The model provided in MSC Nastran Implicit Nonlinear for this purpose is

where:

Yield stress variation with strain rate is given using one of three options:

1. The breakpoints and slopes for a piecewise linear approximation to the yield stress strain rate curve are given. The strain rate breakpoints should be in ascending order, or

2. The Cowper and Symonds model is used. The yield behavior is assumed to be completely determined by one stress-strain curve and a scale factor depending on the strain rate.

Perfectly Plastic

A material is said to be “perfectly plastic” if, upon the stress state touching the yield surface, an infinitesimal increase in stress causes an arbitrarily large plastic strain. The uniaxial stress-strain diagram for an elastic-perfectly plastic material is shown in Figure 10-27. Some materials, such as mild steel, behave in a manner which is close to perfectly plastic.

= the uniaxial equivalent plastic strain rate

= the effective yield stress at a non-zero strain rate

= the static yield stress (which may depend on the equivalent plastic strain, , via isotropic hardening, or on the temperature, .

= are material parameters that may be functions of temperature. and are defined on the input forms. This model is effective in both static and dynamic procedures.

·

D 0------ 1– P

for 0=

· pl

0 plT

pl T

D T p T D p

Note: If multiple material models are used, they must all be expressed as piecewise linear, or as Cowper and Symonds model.


.

Figure 10-27 Perfectly Plastic Material Stress-Strain Relationship

Experimental Stress-Strain Curves

Metals

In uniaxial tension tests of most metals (and many other materials), the following phenomena can be observed. If the stress in the specimen is below the yield stress of the material, the material behaves elastically and the stress in the specimen is proportional to the strain. If the stress in the specimen is greater than the yield stress, the material no longer exhibits elastic behavior, and the stress-strain relationship becomes nonlinear. Figure 10-28 shows a typical uniaxial stress-strain curve. Both the elastic and inelastic regions are indicated.

Figure 10-28 Typical Uniaxial Stress-Strain Curve (Uniaxial Test)

xx

xx

YS

E

1

Note: Stress and strain are total quantities.

Stress InelasticRegion

Elastic Region

YieldStress

Strain


328

Within the elastic region, the stress-strain relationship is unique. As illustrated in , if the stress in the specimen is increased (loading) from zero (point 0) to (point 1), and then decreased (unloading) to

zero, the strain in the specimen is also increased from zero to , and then returned to zero. The elastic

strain is completely recovered upon the release of stress in the specimen.

The loading-unloading situation in the inelastic region is different from the elastic behavior. If the specimen is loaded beyond yield to point 2, where the stress in the specimen is and the total strain is

, upon release of the stress in the specimen the elastic strain, , is completely recovered. However,

the inelastic (plastic) strain, , remains in the specimen. Figure 10-29 illustrates this relationship.

Similarly, if the specimen is loaded to point 3 and then unloaded to zero stress state, the plastic strain

remains in the specimen. It is obvious that is not equal to . We can conclude that in the

inelastic region:

• Plastic strain permanently remains in the specimen upon removal of stress.

• The amount of plastic strain remaining in the specimen is dependent upon the stress level at which the unloading starts (path-dependent behavior).

The uniaxial stress-strain curve is usually plotted for total quantities (total stress versus total strain). The total stress-strain curve shown in Figure 10-29 can be replotted as a total stress versus plastic strain curve, as shown in Figure 10-30. The slope of the total stress versus plastic strain curve is defined as the workhardening slope (H) of the material. The workhardening slope is a function of plastic strain.

Figure 10-29 Schematic of Simple Loading - Unloading (Uniaxial Test)

1

1

2

2 2

e

2

p

3

p

2

p

3

p

Stress

Total Strain = Strain and Elastic Strain

Yield Stress

Strain

2 2p

2e

+=

3 3p

3e

+=

y

3

2

1

321

3

p

3e

2e

2

p

0

1

2

3


Figure 10-30 Definition of Workhardening Slope (Uniaxial Test)

The stress-strain curve shown in Figure 10-29 is directly plotted from experimental data. It can be simplified for the purpose of numerical modeling. A few simplifications are shown in Figure 10-31 and are listed below:

1. Bilinear representation – constant workhardening slope.

2. Elastic perfectly-plastic material – no workhardening.

3. Perfectly-plastic material – no workhardening and no elastic response.

4. Piecewise linear representation – multiple constant workhardening slopes.

5. Strain-softening material – negative workhardening slope.

In addition to elastic material constants (Young’s modulus and Poisson’s ratio), it is essential to include yield stress and workhardening slopes when dealing with inelastic (plastic) material behavior. These quantities can vary with parameters such as temperature and strain rate. Since the yield stress is generally measured from uniaxial tests, and the stresses in real structures are usually multiaxial, the yield condition of a multiaxial stress state must be considered. The conditions of subsequent yield (workhardening rules) must also be studied.

Plastic Strain

H = tan (Workhardening Slope)

= d/dp

p

Total Stress


330

Figure 10-31 Simplified Stress-Strain Curves (Uniaxial Test)

Geological Materials

Data for geological materials are most commonly available from triaxial compression testing. In such a test, the specimen is confined by pressure and an additional compression stress is superposed in one direction. Thus, the principal stresses are all negative, with .

Figure 10-32 Triaxial Compression and Tension

(1) Bilinear Representation

(3) Perfectly Plastic

(5) Strain Softening

(2) Elastic-Perfectly Plastic

(4) Piecewise Linear Representation

0 1 2 3=

-

-


The values of the stress invariants in a uniaxial compression experiment are:

p=-{1/3}(2+)

q=-

r3=-(-)3

so that t=q=-

The triaxial results may thus be plotted in the t-p plane shown above. Fitting the best straight line through the results then provides and d.

Triaxial tension data are also needed to define K. Under triaxial tension, the specimen is again confined by pressure, then the pressure in one direction is reduced. In this case, the principal stresses are

.

The stress invariants are now:

p=-{1/3}(+2),

q=-

r3=(-)3,

so that t={q/K}={1/K}(-)

K may thus be found by plotting these test results as q versus p and again fitting the best straight line. The triaxial compression and tension lines must intercept the p-axis at the same point, and the ratio of values of q for triaxial tension and compression at the same value of p then gives K as shown in Figure 10-33.

Figure 10-33 Triaxial Compression and Tension Data

1 2 3=

q

p

hthc

d

Best fit to triaxialtension data

Best fit to triaxialcompression data


332

Matching Mohr-Coulomb Parameters

Sometimes, experimental data are not directly available. Instead, the user is provided with the friction angle and cohesion values for the Mohr-Coulomb model. We, therefore, need to calculate values for the parameters of the Drucker-Prager model to provide a reasonable match to the Mohr-Coulomb parameters.

The Mohr-Coulomb failure model is based on plotting Mohr’s circle for states of stress at failure in the plane of the maximum and minimum principal stresses. The failure line is the best straight line that touches these Mohr’s circles.

The Mohr-Coulomb model is thus

s+msin-c cos=0,

where s={1/2}(-)

is half of the difference between the maximum and minimum principal stresses (and is, therefore, the maximum shear stress), and

m={1/2}(+)

is the average of the maximum and minimum principal stresses.

We see that the Mohr-Coulomb model assumes that failure is independent of the value of the intermediate principal stress. The Drucker-Prager model does not. The failure of typical geotechnical materials generally includes some small dependence on the intermediate principal stress.

Matching Triaxial Test Response

One approach to matching Mohr-Coulomb and Drucker-Prager model parameters is to make the two models provide the same failure definition in triaxial compression and tension. For this purpose, we can rewrite the Mohr-Coulomb model in terms of principal stresses.

(10-133)

Using the results above (for the stress invariants p, q, and r), in triaxial compression and tension, allows the Drucker-Prager model to be written for triaxial compression as

(10-134)

and, for triaxial tension, as

(10-135)

1 3– 1 3+ 2c cos–sin+ 0=

1 3–tan

213--- tan+

------------------------------- 1 3+

113--- tan–

116--- tan+

-------------------------c0

+ + 0=

1 3–tan

2K---- 1

3---– tan

----------------------------- 1 3+

113--- tan–

1K---- 1

6---– tan

-------------------------c0

+ + 0=


We wish to make the equations for triaxial compression and biaxial tension identical to the general Mohr-Coulomb equation for all values of ,).

Comparing the equations for triaxial compression and triaxial tension requires that:

(10-136)

so that

(10-137)

Comparing the coefficients of (+) in the equation for triaxial compression and that for triaxial

tension provides:

(10-138)

and hence, from the derived equation for K:

(10-139)

Finally, comparing the last terms in the general expression for the Mohr-Coulomb model and the equation for triaxial compression and using the expression for tan provides:

(10-140)

The expression for tan, K, and this last expression and thus provide Drucker-Prager parameters that match the Mohr-Coulomb model in triaxial compression and tension.

The value of K in the Drucker-Prager model is restricted to for the yield surface to remain convex. Rewriting the expression for K as:

(10-141)

shows that this implies . Many real materials have a larger Mohr-Coulomb friction angle than this value. In such circumstances, one approach is to choose K = 0.778 and then to use the expression for

tan to define and the expression for to define , ignoring the expression for K. This matches the

models for triaxial compression only, while providing the closest approximation that the model can provide to failure being independent of the intermediate principal stress. If is significantly larger than 22, this approach may provide a poor Drucker-Prager match of the Mohr-Coulomb parameters. MSC Nastran Implicit Nonlinear uses K=1 by default.

116--- tan+ 1

K----

16--- tan–=

K 1

113--- tan+

-------------------------=

6 sin3 sin–---------------------tan

K 3 sin–3 sin+-----------------------=

c0 2c cos

1 sin–----------------------=

K 0.778

sin 31 K–1 k+-------------- =

22

c0 c

0


334

Matching Plane Strain Response

Plane strain problems are often encountered in geotechnical analysis: examples are long tunnels, footings, and embankments. For this reason, the constitutive model parameters are often matched to provide the same flow and failure response in plane strain.

The Drucker-Prager flow potential defines the plastic strain increment as:

(10-142)

where is the equivalent plastic strain increment.

Since we only wish to match the behavior in one plane we can assume K=1, which implies that t=q. Then:

(10-143)

Writing this expression in terms of principal stresses provides:

(10-144)

with similar expressions for and .

Assume plane strain in the 1-direction. Then, at limit load, we must have =0. From the above

expression, this provides the constraint:

(10-145)

so that:

(10-146)

Using this constraint, we can rewrite q and p in terms of the principal stresses in the plane of deformation,

(10-147)

and

dpld

pl 1

113--- tan–

--------------------------

t p tan–

=

dpl

dpl

dpl 1

113--- tan–

--------------------------

q

p

tan– =

d1pl

dpl 1

113--- tan–

--------------------------

12q------ 21 2 3–– 1

3--- tan+

=

d2pl

d3pl

d1pl

12q------ 21 2– 3– 1

3--- tan+ 0=

112--- 2 3+ 1

3--- qtan–=

q3 3

2 9 tan 2

–

-------------------------------------- 2 3– =


(10-148)

With these expressions, the Drucker-Prager yield surface can be written in terms of 2 and as

(10-149)

The Mohr-Coulomb yield surface in the (2,3) plane is:

(10-150)

By comparison,

(10-151)

(10-152)

Now consider the two extreme cases of flow definition: associated flow, =, and nondilatant flow, when =0.

Assuming associated flow, the last two equations provide:

(10-153)

and

(10-154)

while for nondilatant flow they give and

In either case, is immediately available as:

p12--- 2 3+ –

tan

2 3 9 tan 2

–

----------------------------------------------- 2 3– +=

9 tantan–

2 3 9 tan 2

–

----------------------------------------------- 2 3– 12--- 2 3+ d–tan+ 0=

2 – 3 2 3+ 2c cos–sin+ 0=

sin 3 9 tan

2– tan

9 tantan–--------------------------------------------------------=

c cos3 9 tan

2–

9 tantan–-------------------------------------------d=

tan 3 sin

113--- sin

2+

--------------------------------------=

dc--- 3 cos

113--- sin

2+

--------------------------------------=

tan 3 sin= dc--- cos=

c0


336

(10-155)

The difference between these two approaches increases with the friction angle but, for typical friction angles, the results are not very different, as illustrated in the table below.

Plane strain matching of Drucker-Prager and Mohr-Coulomb models.

As strain rates increase, many materials show an increase in yield strength. This effect often becomes important when the strain rates are in the range of -0.1 to 1 per second, and can be very important if the strain rates are in the range of 10 to 100 per second, as commonly occurs in high energy dynamic events or in manufacturing processes.

Temperature-Dependent BehaviorThis section discusses the effects of temperature-dependent plasticity on the constitutive relation.

The following constitutive relations for thermo-plasticity were developed by Naghdi. Temperature effects are discussed using the isotropic hardening model and the von Mises yield condition.

The stress rate can be expressed in the form

(10-156)

For elastic-plastic behavior, the moduli are

(10-157)

and for purely elastic response

Mohr-Coulomb Friction Angle, Associated Flow Nondilatant Flow

Drucker-Prager friction angle,

d/c Drucker-Prager friction angle,

d/c

10 16.7 1.70 16.7 1.70

20 30.2 1.60 30.6 1.63

30 39.8 1.44 40.9 1.50

40 46.2 1.24 48.1 1.33

50 50.5 1.02 53.0 1.11

c0 1

1 13--- tan–

--------------------------d=

i j·

Lijkl··kl hi jT

·+=

Lijkl

Li jk l Cijk l Cijmn

mn-------------

pq------------ Cpqkl

D–=


(10-158)

The term that relates the stress increment to the increment of temperature for elastic-plastic behavior is

(10-159)

and for purely elastic response

(10-160)

where

(10-161)

and

(10-162)

and are the coefficients of thermal expansion.

Temperature-Dependent Stress Strain CurvesStarting in MSC.Nastran 2005, SOL 600 offers the capability of stress-strain curve dependence as a function of temperature. The user specifies these stress strain curves at different temperatures and then specifies the temperature to use for each subcase. Linear interpolation between the supplied curves is used to determine the appropriate curve at the temperature specified for a particular subcase. Marc’s AF-Flowmat capability is used for this capability; therefore, user subroutines do not have to be supplied. This capability is best explained with an example. See Install_dir/doc/pdf_nastran/implicit_nonlinear_examples/example_input_files/mattep20.dat.

SOL 600,NLSTATIC path=1 stop=1TIME 10000CEND ECHO = NONE DISPLACEMENT(plot) = ALL SPCFORCE(PLOT) = ALL Stress(PLOT) = ALL Strain(PLOT) = ALL SPC = 1 NLPARM = 2 temp(init)=10subcase 1 temp(load)=11 LOAD = 100subcase 2 temp(load)=12 LOAD = 200

Lijk l Cijk l=

hij Xij Cijklkl– Cijklkl----------- pqXpq

23---

T-------–

D–=

Hij Xij Cijklkl–=

D49--- 2

p--------

i j---------- Cijkl

kl-----------+=

Xij

Cijkl

T---------------kl

e=

kl


338

subcase 3 temp(load)=13 LOAD = 300BEGIN BULKparam,mrafflow,mymat0param,mrtabls1,4param,mrtabls2,1NLPARM 2 10 AUTO 1 20 PPARAM,LGDISP,1tempd, 10, 70.tempd, 11, 110.tempd, 12, 700.tempd, 13, 1100.$LOAD, 20, 1.0, 2.0, 1, 1.0, 2load, 100, 1., 1., 1load, 200, 1., -.5, 1load, 300, 1., 1.1, 1PLOAD4 1 1 -15....$ Constraint Set 1 : UntitledSPC 1 1 123456 0.SPC 1 8 123456 0.SPC 1 15 123456 0.SPC 1 22 123456 0.SPC 1 29 123456 0.$ Property 1 : UntitledPSHELL 1 1 0.125 1 1 0.$ Material 1 : AISI 4340 SteelMATEP, 1,TABLE, 35000., 2,CAUCHY,ISOTROP,ADDMEANMAT1 1 2.9E+7 0.327.331E-4 6.6E-6 70. +MT 1+MT 1 215000. 240000. 156000.MAT4 14.861E-4 38.647.331E-4$ 1 2 3 4 5 6 7 8 9$2345678 2345678 2345678 2345678 2345678 2345678 2345678 2345678 2345678MATTEP 1 21 MATT1 1 7 TABLEM1 7+ 70.0 6.6E-6 1000. 6.5E-6 1200. 6.4E-6 1500. 6.3E-6+ 2000. 6.2E-6 ENDT$2345678 2345678 2345678 2345678 2345678 2345678 2345678 2345678 2345678TABLEST 21+ 70.0 31 1000. 32 1200. 33 1500. 34+ 2000. 35 ENDTTABLES1, 31 , 0., 15000., 1.0, 16000., 10., 25000., 100., 30000., , 99999., 40000., ENDTTABLES1, 32 , 0., 13000., 1.0, 14000., 10., 23000., 100., 28000., , 99999., 28000., ENDTTABLES1, 33 , 0., 11000., 1.0, 12000., 10., 21000., 100., 26000., , 99999., 25000., ENDTTABLES1, 34 , 0., 9000., 1.0, 10000., 10., 19000., 100., 22000., , 99999., 24000., ENDTTABLES1, 35 , 0., 5000., 1.0, 7000., 10., 9000., 100., 13000., , 99999., 15000., ENDT


GRID 1 0 0. 0. 0. 0...CQUAD4...ENDDATA

In this input, the stress strain curves are specified by TABLES1 entries. The collection of stress-strain curves to be used is specified in the TABLEST entry and the corresponding temperatures at which they apply is specified in the TABLEM1 entry. The TABLEM1 ID is called out in field 7 of the MATT1 entry and the TABLEST ID is called out in field 5 of the MATTEP entry. TABLEST must list the stress strain TABLES1 IDs in order of increasing temperature and the first ID must be at the lowest temperature specified anywhere in the analysis. In this example, it is a temperature of 70 corresponding to TEMPERATURE(init)=10 in the Case Control. Similarly, the temperatures in the TABLEM1 entry must be in increasing order. The stress-strain curves should cover the entire range of temperatures for the analysis so that no extrapolation is needed. The actual temperatures for each subcase are given by the TEMPERATURE(load) specifications for each subcase.

There is one parameter that is critical to this analysis:

Specifying Elastoplastic Material EntriesEach of the elastoplastic models described in this section can be selected with the MATEP Bulk Data entry.

param,MRAFFLOW, Name,mymat0

Name of the file containing temperature dependent stress versus plastic strain curves in Marc’s AF_flowmat format. This file can be generated from the current MSC Nastran run using TABLEST and TABLES1 entries or a pre-existing file can be used depending on the value of PARAM,MRAFFLOR. The extension “.mat” will be added to Name. If this is a new file, it will be saved in the directory from which the MSC Nastran execution is submitted. If a pre-existing file is to be used, it can either be located in the directory where the MSC Nastran execution is submitted and run or in the Marc AF_flowmat directory.

Entry Description

MATEP Specifies elasto-plastic material properties to be used for large deformation analysis.

MATTEP Specifies temperature-dependent elasto-plastic material properties to be used for static, quasi-static, or transient dynamic analysis.


340


To define an inelastic material in Patran:

1. From the Materials Application form, set the Action>Object>Method combination to Create>Isotropic-or-Orthotropic-or-Anisotropic>Manual Input.

2. Click Input Properties..., and select Elastoplastic from the Constitutive Model pull-down menu.

The required properties for describing elasticplastic behavior vary based on material type, dimension, type of nonlinear data input, hardening rule, yield criteria, strain rate method, and thermal dependencies.

The table below shows the various input options and criteria available to you for defining elastoplastic behavior.

Elastoplastic Model Summary

Constitutive Model

Nonlinear Data Input

Hardening Rule Yield Criteria

Strain Rate Method

• Plastic • Stress/Strain Curve

• Isotropic

• Kinematic

• Combined

• von Mises

• Tresca

• Mohr-Coulomb

• Drucker-Prager

• Parabolic Mohr-Coulomb


• Oak Ridge National Lab

• 2-1/4 Cr-Mo ORNL

• Reversed Plasticity ORNL

• Full Alpha Reset ORNL

• Piecewise Linear

• Cowper-Symonds

• Hardening Slope

• Isotropic

• Kinematic

• Combined

• von Mises

• Tresca

• Mohr-Coulomb

• Drucker-Prager

• None


Nonlinear Data Input

The type of nonlinear data input you choose to use to define elastoplastic material behavior determines the input data required for the Input Properties subform on the Materials Application form.

• Stress/Strain Curve - All stress-strain curves are input as piecewise linear. Patran transfers the stress-strain curve input on the material property field directly to the TABLES1 entry.

Constitutive Model Type

Hardening Rule Yield Criteria

Strain Rate Method

• Plastic • Perfectly Plastic

• None • von Mises

• Linear Mohr-Coulomb

• Parabolic Mohr-Coulomb


• Oak Ridge National Lab

• 2-1/4 Cr-Mo ORNL

• Reversed Plasticity ORNL

• Full Alpha Reset ORNL


• Cowper-Symonds

• Power Law

• Rate Power Law

• Johnson-Cook

• Kumar


• None • Piecewise Linear

• Cowper-Symonds

Elastoplastic Model Summary


342

The number of linear segments used to define the stress-strain curve may be different from one material to another. The same strain breakpoints need not be used for all of the different material’s stress-strain curves. It is recommended to define the stress-strain curves throughout the range of strains which the analysis is likely to predict. If the analysis predicts a plastic strain greater than the last point defined by the user, MSC Nastran Implicit Nonlinear continues the analysis after shifting the last strain breakpoint on that curve to match the predicted value, thereby changing (reducing) the work hardening slope for the last segment of the curve.

• Hardening Slope - The hardening slope and the yield point are required with this Nonlinear Data Input option.

• Perfectly Plastic - Perfect plasticity is described by simply specifying the yield point.

The tables below provide descriptions for the input data for each of the four types of nonlinear input.

Isotropic - Stress/Strain Curve or Perfectly Plastic:All Yield Functions

Property Name Description

Stress /Strain CurveorYield Stress

Defines the Cauchy stress vs. logarithmic strain (also called equivalent tensile stress versus total equivalent strain) by reference to a tabular field. The field is selected from the Field Definition list. The field is created using the Fields application. See Fields Create (Spatial, Tabular Input) (p. 210) in the Patran Reference Manual. For Perfectly Plastic models, only a Yield Stress needs to be entered.

Can also be strain rate dependent if Strain Rate Method is Piecewise Linear. Accepts field of yield stress vs. strain rate.

10th Cycle Yield Stress vs. Plastic Strain

or10th Cycle Yield Stress

When set to ORNL, accepts field of 10th cycle yield stress vs. plastic strain. Can be temperature dependent also. For Perfectly Plastic models, only a 10th Cycle Yield Stress needs to be entered.

Coefficient C Visible if Strain Rate Method is Cowper-Symonds.

Inverse Exponent P Visible if Strain Rate Method is Cowper-Symonds.

Alpha When set to Linear Mohr-Coulomb, defines the slope of the yield surface in square root J2 versus J1 space. This property is required.

Beta When set to Parabolic Mohr-Coulomb, defines the beta parameter in the equation that defines the parabolic yield surface in square root J2 versus J1 space. This property is required.


Hardening Slope - Nonlinear Data Input

Note: 2 1/4 Cr-Mo ORNL, Reversed Plasticity ORNL, Full Alpha Reset ORNL are the same as Oak Ridge National Labs. Generalized Plasticity is the same as von Mises.

Perfectly Plastic is identical to Stress/Strain except that no hardening rules apply.

Anisotropic/Orthotropic - Stress/Strain Curve or Perfectly Plastic:All Yield Functions

Description

Stress vs. StrainorTensile Yield Stress

Same as description for Isotropic Elastic-Plastic. If Strain Rate Method is Piecewise Linear, accepts field of yield stress vs. strain rate.

Or defines an isotropic yield stress. It is a required property when the plasticity type is Perfectly Plastic.

Stress 11/22/33 Yield Ratios Defines the ratios of direct yield stresses to the isotropic yield stress in the element’s coordinate system.

Stress 12/23/31 Yield Ratios Defines the ratios of shear yield stresses to the isotropic shear yield stress (yield divided by square root three) in the element’s coordinate system.

Note: Perfectly Plastic is identical to Elastic-Plastic except that no hardening rules apply. Stress vs Plastic Strain is replaced with Yield Stress data only as is 10th Cycle Yield vs. Strain replaced with 10th Cycle Yield Stress data. Thus no tabular data is necessary.

Isotropic/Anisotropic/Orthotropic - Hardening Slope


Hardening Slope Slope of the stress-strain curve once yielding has started.

Yield Point Defines the stress level at which plastic strain begins to develop.

Internal Friction Angle When yield function is set to Mohr-Coulomb or Drucker-Prager this gives the parameter describing the effect of hydrostatic pressure on the yield stress.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideFailure and Damage Models

344

Failure and Damage ModelsOne of the nonlinear features of a material's behavior is failure. When a certain criterion (failure criterion) is met, the material fails and no longer sustains its loading and breaks. In a finite-element method, this means that the element, where the material reaches the failure limit, cannot carry any stresses anymore. The stress tensor is effectively zero. The element is flagged for failure, and, essentially, is no longer part of the structure.

Failure criteria can be defined for a range of materials and element types. The failure models are referenced from the material definition entries.

Isotropic/Orthotropic/Anisotropic Failure ModelsFor isotropic, 2-D orthotropic, and 2-D anisotropic materials, you can implement one of five failure models in MSC Nastran Implicit Nonlinear (SOL 600). Failure models are based on maximum stress criteria, maximum strain criteria, or one of three composite stress/strain failure theories.

Maximum Stress Criterion

At each integration point, MSC Nastran Implicit Nonlinear calculates six quantities:

(10-163)

(10-164)

Failure Model Applicable Material Type

Maximum Stress Isotropic, 2-D Orthotropic, 2-D Anisotropic

Maximum Strain 2-D Orthotropic

Hill Isotropic, 2-D Orthotropic (stress or strain based), 2-D Anisotropic

Hoffman Isotropic, 2-D Orthotropic (stress or strain based), 2-D Anisotropic

Tsai-Wu Isotropic, 2-D Orthotropic (stress or strain based), 2-D Anisotropic

1

Xt-------

F

1Xc------–

F

1 0

1 0

if

if

1.

2Yt------

F

2Yc------–

F

2 0

2 0

if

if

2.


(10-165)

(10-166)

(10-167)

(10-168)

where

Maximum Strain Failure Criterion

At each integration point, calculates six quantities:

(10-169)

is the failure index (F =1.0).

are the maximum allowable stresses in the 1-direction in tension and compression.

are maximum allowable stresses in the 2-direction in tension and compression.

are maximum allowed stresses in the 3-direction in tension and compression.

is the maximum allowable in-plane shear stress.

is the maximum allowable 23 shear stress.

is the maximum allowable 31 shear stress.

3

Zt------ F

3

Zc------–

F

3 0

3 0

if

if

3.

12

S12---------

F4.

23

S23--------

F5.

31

S31--------

F6.

F

Xt Xc

Yt Yc

Zt Zc

S12

S23

S31

1

e1 t------- F

1

e1c-------–

F

1 0

1 0

if

if

1.


346

(10-170)

(10-171)

(10-172)

(10-173)

(10-174)

where

Hill Failure Criterion

Assumptions:

• Orthotropic materials only

• Incompressibility during plastic deformation

• Tensile and compressive behavior are identical

is the failure index (F=1.0).

are the maximum allowable strains in the 1 direction in tension and compression.



is the maximum allowable shear strain in the 12 plane.



2

e2 t------- F

2

e2c-------–

F

2 0

2 0

if

if

2.

3

e3 t------- F

3

e3c-------–

F

3 0

3 0

if

if

3.

12

g12--------

F4.

23

g23--------

F5.

31

g31--------

F6.

F

e1 t e1c

e2 t e2c

e3 t e3c

g12

g23

g31


At each integration point, MSC Nastran Implicit Nonlinear calculates:

(10-175)

For plane stress condition, it becomes

(10-176)

where

Hoffman Failure Criterion

At each integration point, MSC Nastran Implicit Nonlinear calculates:

(10-177)

with

is the maximum allowable stress in the 1 direction



are as before

12

X2

------2

2

Y2

------3

2

Z2

------ 1

X2

------ 1

Y2

------ 1

Z2

------–+ 12– 1

X2

------ 1

Z2

------ 1

Y2

------–+ 13–+ +

1

Y2

------ 1

Z2

------ 1

X2

------–+ 23–

122

S122

---------13

2

S132

---------23

2

S232

--------- F+ + +

12

X2------

12

X2-------------–

22

Y2------

122

S122

---------+ +

F

X

Y

Z

S12 S23 S31 F

Note: Hoffman criterion is essentially Hill criterion modified to allow unequal maximum allowable stresses in tension and compression.

C1 2 3– 2 C2 3 1– 2 C3 1 2– 2 C41 C52+ + + +

C63 C7232 C813

2 C9122 F+ + + +


348

(10-178)


(10-179)

where: are as before.

Tsai-Wu Failure Criterion

Tsai-Wu is a tensor polynomial failure criterion. At each integration point, MSC Nastran Implicit Nonlinear calculates:

(10-180)

where are as before.

C112--- 1

ZtZc----------- 1

YtYc----------- 1

XtXc-----------–+

=

C212--- 1

XtXc----------- 1

ZtZc----------- 1

YtYc-----------–+

=

C312--- 1

XtXc----------- 1

YtYc----------- 1

ZtZc-----------–+

=

C41Xt----- 1

Xc------–=

C51Yt---- 1

Yc-----–=

C61Zt---- 1

Zc-----–=

C71

S232

--------=

C81

S132

--------=

C91

S122

--------=

1Xt----- 1

Xc------–

11Yt---- 1

Yc-----–

2

12

XtXc-----------

22

YtYc-----------

122

S122

---------12

XtXc-------------–+ + + +

F

Xt Xc Yt Yc Zt Zc S12 S23 S31 F

Note: For small ratios of, for example, , the Hoffman criteria can become negative due to the presence of the linear terms.

1

Xt------

1Xt----- 1

Xc------–

11Yt---- 1

Yc-----–

21Zt---- 1

Zc-----–

3

12

XtXc-----------

22

YtYc-----------

32

ZtZc-----------+ + + + +

122

S122

--------23

2

S232

--------13

2

S132

-------- 2F1212 2F2323 2F1313 F+ + + + + +

Xt Xc Yt Yc Zt Zc S12 S23 S31 F



(10-181)

See Wu, R.Y. and Stachurski, 2, “Evaluation of the Normal Stress Interaction Parameter in the Tensor Polynomial Strength Theory for Anisotropic Materials”, Journal of Composite Materials, Vol. 18, Sept. 1984, pp. 456-463.

Interlaminar Shear for Thick Shell and Beam Elements

Calculation of interlaminar shear stress (a parabolic distribution through the thickness direction) for thick shells and beams is available. These interlaminar shears are printed in the local coordinate system above and below each layer selected for printing. These values are also available for postprocessing. PARAM,MRTSHEAR,1 must be used for activating the parabolic shear distribution calculations.

In MSC Nastran Implicit Nonlinear, the distribution of transverse shear strains through the thickness for thick shell and beam elements was assumed to be constant. From basic strength of materials and the equilibrium of a beam cross section, it is known that the actual distribution is more parabolic in nature. As an additional option, the formulations for certain beam and shell elements have been modified to include a parabolic distribution of transverse shear strain. The formulation is exact for Marc beam element 45, but is approximate for Marc thick shell elements 22, 75, and 140. Nevertheless, the approximation is expected to give improved results from the previous constant shear distribution. Furthermore, interlaminar shear stresses for composite beams and shells can be easily calculated.

With the assumption that the stresses in the and direction are uncoupled, the equilibrium condition through the thickness is given by

(10-182)

where is the layer axial stress; is the layer shear stress. From beam theory, we have

(10-183)

Interactive strength constant for the 12 plane



F12

F23

F13

1Xt----- 1

Xc------–

11

Y2------ 1

Yc-----–

2

12

XtXc-----------

22

YtYc-----------

122

S12--------- 2F1212+ + + + +

F

Note: In order for the Tsai-Wu failure surface to be closed,

F122 1

XtXc------------ 1

YtYc----------- F23

2 1YtYc----------- 1

ZtZc----------- F31

2 1XtXc----------- 1

ZtZc-----------

V1

V2

z z

------------- z x

--------------+ 0=

z t

V Mx--------+ 0=


350

where is the section bending moment and is the shear force. Assuming that

(10-184)

by taking the derivative of Equation (10-184) with respect to x, substituting the result into Equation (10-182), using Equation (10-183) and integrating, we obtain

(10-185)

The function is given from beam theory as

(10-186)

where is the layer initial Young’s modulus, is the location of the neutral axis and is the section

bending moment of inertia. Equation (10-186) and Equation (10-184) express the usual bending relation

(10-187)

except that these two equations are written so that the axis is not necessarily the neutral axis of bending. With respect to this axis, membrane and bending action is, in general, coupled. Note that

(10-188)

and stress at the top and bottom surface of the shell.

Interlaminar Stresses for Continuum Composite Elements

In MSC Nastran Implicit Nonlinear, the interlaminar shear and normal stresses are calculated by averaging the stresses in the stacked layers. The stresses are transformed into a component tangent to the interface and a component normal to the interface. The two components, considered as shear stress and normal stress, respectively, are printed out in the output file.

Progressive Composite Failure

A model has been put into MSC Nastran Implicit Nonlinear to allow the progressive failure of certain types of composite materials. The aspects of this model are defined below:

1. Failure occurs when any one of the failure criteria is satisfied.

2. The behavior up to the failure point is linear elastic.

M V

z f z M=

z f z dz Vz=

f z

f z E0 z

EI-------------- z z– =

E0 z z EI

z MzI

-------–=

z 0=

z

zE z dzz

E z dzz-----------------------=

z 0=


3. Upon failure, the material moduli for orthotropic materials at the integration points are changed such that all of the moduli have the lowest moduli entered.

4. Upon failure, for isotropic materials, the failed moduli are taken as 10% of the original moduli.

5. If there is only one modulus, such as in a beam or truss problem, the failed modulus is taken as 10% of the original one.

6. There is no healing of the material.

Specifying the Failure Criteria

Any of the failure models described above can be selected with the MATF Bulk Data entry.

Defining Failure Models in Patran

To define a Failure Model in Patran:

1. From the Materials Application form, set the Action>Object>Method combination to one of the following:

• Create>Isotropic>Manual Input

• Create>2D Orthotropic>Manual Input

• Create>2D Anisotropic>Manual Input

2. Click on Input Properties.... , and choose Failure from the Constitutive Model pull-down menu.

3. If the failure model applies to a 2-D orthotropic material, you can select Stress or Strain from the Failure Limit pull-down menu.

Entry Description

MATF Specifies failure model properties for linear elastic materials to be used for static, quasi static or transient dynamic analysis in MSC Nastran Implicit Nonlinear.


352

Isotropic Material Input Data

Isotropic materials require the following failure model data via the Input Options subform on the Materials Application form.


2-D Orthotropic Material Input Data

2-D orthotropic materials require the following failure model data via the Input Options subform on the Materials Application form.

2-D Anisotropic Material Input Data

2-D anisotropic materials require the following failure model data via the Input Options subform on the Materials Application form.

Failure Theory : Hill, Hoffman, Tsai-Wu, Maximum Stress


Tension Stress Limit Defines the tension stress (or strain) limits in the element’s coordinate system.

Compression Stress Limit Defines the compression stress (or strain) limits in the element’s coordinate system. Absolute values are used.

Shear Stress Limit Defines the shear stress (or strain) limits.

Failure Theory : Hill, Hoffman, Tsai-Wu, Maximum Stress, Maximum Strain


Tension Stress (Strain) Limit 11

Defines the tension stress (or strain) limits in direction 1 of the element’s coordinate system.

Tension Stress (Strain) Limit 22

Defines the tension stress (or strain) limits in direction 2 of the element’s coordinate system.

Compression Stress (Strain) Limit 11 Defines the compression stress (or strain) limits in direction 1 of the element’s coordinate system. Absolute values are used.

Compression Stress (Strain) Limit 22 Defines the compression stress (or strain) limits in direction 2 of the element’s coordinate system. Absolute values are used.

Shear Stress (Strain) Limit Defines the shear stress (or strain) limits.

Interaction Term Defines the stress interaction parameter.


354

Damage ModelsIn many structural applications, the finite element method is used to predict failure. This is often performed by comparing the calculated solution to some failure criteria, or by using classical fracture mechanics.

Ductile Metals

In ductile materials given the appropriate loading conditions, voids will form in the material, grow, then coalesce, leading to crack formation and potentially, failure. Experimental studies have shown that these processes are strongly influenced by hydrostatic stress. Gurson studied microscopic voids in materials and derived a set of modified constitutive equations for elastic-plastic materials. Tvergaard and Needleman modified the model with respect to the behavior for small void volume fractions and for void coalescence.

In the modified Gurson model, the amount of damage is indicated with a scalar parameter called the void volume fraction f. The yield criterion for the macroscopic assembly of voids and matrix material is given by:

(10-189)

as seen in Figure 10-34.

Failure Theory : Hill, Hoffman, Tsai-Wu, Maximum Stress


Tension Stress Limit Defines the tension stress (or strain) limits in the element’s coordinate system.

Compression Stress Limit Defines the compression stress (or strain) limits in the element’s coordinate system. Absolute values are used.

Shear Stress Limit Defines the shear stress (or strain) limits.

Fy------ 2

2q1fq2kk

2y--------------

1 q1f 2+ –cosh+ 0= =


Figure 10-34 Plot of Yield Surfaces in Gurson Model

The parameter was introduced by Tvergaard to improve the Gurson model at small values of the void

volume fraction. For solids with periodically spaced voids, numerical studies [10] showed that the values of and were quite accurate.

The evolution of damage as measured by the void volume fraction is due to void nucleation and growth. Void nucleation occurs by debonding of second phase particles. The strain for nucleation depends on the particle sizes. Assuming a normal distribution of particle sizes, the nucleation of voids is itself modeled as a normal distribution in the strains, if nucleation is strain controlled. If void nucleation is assumed to be stress controlled in the matrix, a normal distribution is assumed in the stresses. The original Gurson model predicts that ultimate failure occurs when the void volume fraction f, reaches unity. This is too

high a value and, hence, the void volume fraction f is replaced by the modified void volume fraction in the yield function.

The parameter is introduced to model the rapid decrease in load carrying capacity if void coalescence occurs.

(10-190)

where fc is the critical void volume fraction, and is the void volume at failure, and . A safe

choice for would be a value greater than namely, . Hence, you can control the void

volume fraction, , at which the solid loses all stress carrying capability.

Numerical studies show that plasticity starts to localize between voids at void volume fractions as low as 0.1 to 0.2. You can control the void volume fraction , beyond which void-void interaction is modeled

by MSC Nastran Implicit Nonlinear. Based on the classical studies, a value of can be chosen.

The existing value of the void volume fraction changes due to the growth of existing voids and due to the nucleation of new voids.

1.0

0.5

00 1 2 3 4

e M

f* 0=

f* fu* 0.01=

kk 3M

0.90.6

0.30.1

q1

q1 1.5= q2 1=

f

f

f f= if f fc

f fc

fu* fc–

fF fc–---------------

f fc– += if f > fc

fF fu* 1 q1=

fF 1 q1 fF 1.1 q1=

fF

fc

fc 0.2=


356

(10-191)

The growth of voids can be determined based upon compressibility of the matrix material surrounding the void.

(10-192)

As mentioned earlier, the nucleation of new voids can be defined as either strain or stress controlled. Both follow a normal distribution about a mean value.

In the case of strain controlled nucleation, this is given by

(10-193)

where is the volume fraction of void forming particles, the mean strain for void nucleation and

the standard deviation.

In the case of stress controlled nucleation, the rate of nucleation is given by:

(10-194)

If the second phase particle sizes in the solid are widely varied in size, the standard deviation would be larger than in the case when the particle sizes are more uniform. The MSC Nastran Implicit Nonlinear user can also input the volume fraction of the nucleating second phase void nucleating particles in the input deck, as the variable .

A typical set of values for an engineering alloy is given by Tvergaard for strain controlled nucleation as

(10-195)

It must be remarked that the determination of the three above constants from experiments is extremely difficult. The modeling of the debonding process must itself be studied including the effect of differing particle sizes in a matrix. It is safe to say that such an experimental study is not possible. The above three constants must necessarily be obtained by intuition keeping in mind the meaning of the terms.

When the material reaches 90 percent of , the material is considered to be failed. At this point, the

stiffness and the stress at this element are reduced to zero.

Elastomers

Under repeated application of loads, elastomers undergo damage by mechanisms involving chain breakage, multi-chain damage, micro-void formation, and micro-structural degradation due to detachment of filler particles from the network entanglement. Two types of phenomenological models namely, discontinuous and continuous, exists to simulate the phenomenon of damage.

f·

f·growth f

·nucleat ion+=

f·growth 1 f–

··kk

p=

f·nucleat ion

fN

S 2-------------- exp

12---

mp n–

S-------------------

2

– ··mp=

fN n S

f·nucleat ion

fN

S 2--------------exp

12---

13---kk n–+

S------------------------------------

2

– * · 1

3---

·kk+=

fN

n 0.30 ; fN 0.04 ; S 0.01= = =

fF


Discontinuous Damage

The discontinuous damage model simulates the “Mullins’ effect” as shown in Figure 10-35.

Figure 10-35 Discontinuous Damage

This involves a loss of stiffness below the previously attained maximum strain. The higher the maximum attained strain, the larger is the loss of stiffness. Upon reloading, the uniaxial stress-strain curve remains insensitive to prior behavior at strains above the previously attained maximum in a cyclic test. Hence, there is a progressive stiffness loss with increasing maximum strain amplitude. Also, most of the stiffness loss takes place in the few earliest cycles provided the maximum strain level is not increased. This phenomenon is found in both filled as well as natural rubber although the higher levels of carbon black particles increase the hysteresis and the loss of stiffness. The free energy, W, can be written as:

(10-196)

where is the nominal strain energy function, and

(10-197)

determines the evolution of the discontinuous damage. The reduced form of Clausius-Duhem dissipation inequality yields the stress as:

(10-198)

Mathematically, the discontinuous damage model has a structure very similar to that of strain space plasticity. Hence, if a damage surface is defined as:

(10-199)

The loading condition for damage can be expressed in terms of the Kuhn-Tucker conditions:

(10-200)

W K W0=

W0

max W0 =

S 2K , W0

C-----------=

W 0–=

0 ·

0 · 0=


358

The consistent tangent can be derived as:

(10-201)

Continuous Damage

The continuous damage model can simulate the damage accumulation for strain cycles for which the values of effective energy is below the maximum attained value of the past history as shown in Figure 10-36.

Figure 10-36 Continuous Damage

This model can be used to simulate fatigue behavior. More realistic modeling of fatigue would require a departure from the phenomenological approach to damage. The evolution of continuous damage parameter is governed by the arc length of the effective strain energy as:

(10-202)

Hence, accumulates continuously within the deformation process.

The Kachanov factor is implemented in MSC Nastran Implicit Nonlinear through both an additive as well as a multiplicative decomposition of these two effects as:

(10-203)

C 4 K 2W

0

CC---------------

K

W0

----------- W

0

C----------- W

0

C-----------+=

s -------W

0s s d

0

t

=

K

K d

dn

n------–

dn

n-----–

exp

n 1=

2

+exp

n 1=

2

+=


(10-204)

You specify the phenomenological parameters and . If is not defined, it is

automatically determined such that, at zero values of and , the Kachanov factor . If,

according to Equation (10-203) or Equation (10-204) the value of exceeds 1, is set back to 1.

The above damage model is available for deviatoric behavior. In addition, viscoelastic behavior can be included. Finally, the user subroutine, UELDAM available starting in version 2005, can be used to define damage functions different from Equation (10-211) to Equation (10-214).

The parameters required for the continuous or discontinuous damage model can be obtained using the experimental data fitting option in Mentat.

Specifying Hyperelastic Damage Model Entries

The hyperelastic damage model described above can be selected with the MATHED Bulk Data entry.


Patran does not support this option in the current release.

Fracture Mechanics

Capabilities include; calculation of Energy Release Rates and Stress Intensity Factors using VCCT or Lorenzi; Crack Propagation, Delamination, more failure criteria and Birth and Death of Elements.

Virtual Crack Closure Technique (VCCT)

The Marc’s VCCT capability is fully supported by using the VCCT Case Control entry and the VCCT Bulk Data entry. This option defines that the virtual crack closure technique is to be used for evaluating energy release rates. The user defines the node (in 2-D or for shells) or nodes (in 3-D) that define each crack tip. The supported elements are lower- and higher-order 2-D solids and 3-D shells, lower- and higher-order 3-D hexahedral solids and lower order 3-D tetrahedral solids. For 3-D solids it is important that a regular mesh around the crack front is used.

Multiple cracks can be defined and results obtained for each crack separately. Each crack consists of a crack tip node in 2-D and for shells and a list of nodes along the crack front for 3-D solids. Shell elements

Entry Description

MATHED Specifies damage model properties for hyperelastic materials to be used for static, quasi static or transient dynamic analysis in MSC Nastran Implicit Nonlinear.

K d

dn n+

n--------------------–

exp

n 1=

2

+=

dn

dn n n dn n d

d

K 1=

K K


360

can be used for defining a 2-D style line crack and also be connected to the face of another shell or 3-D solid to form a 3-D style surface crack. The different cases are automatically identified.

The VCCT method is advantageous because it may be used with any material model including orthotropic or anisotropic behavior, and because it automatically obtains the mode I, II, and III stress intensity factors. This makes is applicable to composite structures.

For crack propagation, there are two modes of growth: fatigue and direct growth. For fatigue style, the user specifies a load sequence time period. During the load sequence, the largest energy release rate and the corresponding estimated crack growth direction is recorded. At the end of the load sequence, the crack is grown using the specified method. For direct growth, the crack grows as soon as the calculated energy release rate is larger than the user-specified Gc. Note that Gc can be made a function of the accumulated crack growth length to model a crack growth resistance behavior. This release does not support large crack propagation which requires remeshing.

Fracture Mechanics J-Integral (LORENZI)

This option gives an estimation of the J-Integral for a crack configuration using the domain integration method. The domain integration method has the advantage that it can also be used for problems with thermal behavior and for dynamic analysis. This procedure is only available for continuum elements. Only the nodes defining the crack front (crack tip in two dimensions) need to be defined. The program automatically finds integrations paths according to the format below. The complete J-Integral is evaluated and output. For the case of linear elastic material with no external loads on the crack faces, the program automatically separates mode I, mode II, and mode III (3-D only) stress intensity factors from the J-Integral. for isotropic materials.

The Bulk Data entry, LORENZI, is necessary to activate this capability and if entered applies to all subcases in the analysis.

Delamination

An alternative method to model failure is to use the COHESIV bulk data option in conjunction with special delamination or interface elements. Three different models are available along with a user subroutine. The user defines the traction versus the relative separation. The area under the curve is the cohesive energy, often known as the critical energy release rate.


MATEP Extensions

Material description MATEP was extended to add Chaboche, Power Law, Kumar, Johnson Cook and other options.

SOL 600 Failure Description – MATF

For SOL 600 failure indices or actual material failure is only described using the MATF entry. For the R2 and later releases, MATF has been revised to accommodate additional types of failure and improve the input and user understanding of the input. The user should be aware that other Nastran solutions can specify failure index calculation on various MAT entries. These specifications are not available in SOL 600 – only MATF may be used. To activate the MATF entries, the PARAM, MRMATFSB, 1 must also be included.

Bilinear model Exponential Model Linear-Exponential Model

Element Type Number of Nodes Characteristic

186 4 Planar

187 8 Planar

188 8 3-D

189 20 3-D

190 4 Axisymmetric

191 8 Axisymmetric

192 6 3-D

193 15 3-D


362

See Format for MATF - Sol 600.

Element Birth and Death

You can deactivate and re-activate elements in the model that have failed or for some other reason needs to be deactivated or re-activated. This is accomplished using Case Control commands DEACTEL and ACTIVAT as well as matching Bulk Data entries DEACTEL and ACTIVAT. Once an element is deactivated or activated it stays that way during the entire subcase case unless it fails due to a MATF criteria.

Unglue

Frequently in contact analysis it is known beforehand that two surfaces will never separate once they contact. To prevent numerical chattering contact between these surfaces is frequently described using glued contact. In order to perform VCCT analysis of such surfaces it might be necessary to unglue those nodes near a crack. The Bulk Data entry, UNGLUE, is available for such purposes.


CreepCreep is an important factor in elevated-temperature stress analysis. In MSC Nastran Implicit Nonlinear, creep is represented by a Maxwell model. Creep is a time-dependent, inelastic behavior, and can occur at any stress level (that is, either below or above the yield stress of a material). The creep behavior can be characterized as primary, secondary, and tertiary creep, as shown in Figure 10-39. Engineering analysis is often limited to the primary and secondary creep regions. Tertiary creep in a uniaxial specimen is usually associated with geometric instabilities, such as necking. The major difference between the primary and secondary creep is that the creep strain rate is much larger in the primary creep region than it is in the secondary creep region. The creep strain rate is the slope of the creep strain-time curve. The creep strain rate is generally dependent on stress, temperature, and time.

The creep data can be specified in either an exponent form or in a piecewise linear curve.

(10-205)

Figure 10-37 Creep Strain Versus Time (Uniaxial Test at Constant Stress and Temperature)

Forms of Creep Material Law

There are three possible modes of input for creep constitutive data.

1. Express the dependence of equivalent creep strain rate on any independent parameter through a piecewise linear relationship. The equivalent creep strain rate is then assumed to be a piecewise linear approximation to

(10-206)

·

c dc

dt--------=

Note: Primary Creep: Fast decrease in creep strain rate Secondary Creep: Slow decrease in creep strain rate Tertiary Creep: Fast increase in creep strain rate

SecondaryCreep

PrimaryCreep

TertiaryCreep

Creep StrainC

Time (t)

· c

A f g c h T dk t dt

-------------=


364

where A is a constant; is equivalent creep strain rate; and , , , and are equivalent stress, equivalent creep strain, temperature and time, respectively. The functions , , , and are piecewise linear. This representation is shown in Figure 10-40. (Any of the functions ( , , , or

) can be set to unity by setting the number of piecewise linear slopes for that relation to zero on the input data.)

2. The dependence of equivalent creep strain rate on any independent parameter can be given directly in power law form by the appropriate exponent. The equivalent creep strain rate is

(10-207)

This is often adequate for engineering metals at constant temperature where Norton’s rule is a good approximation.

(10-208)

3. Use the MATEP material to activate the ORNL (Oak Ridge National Laboratory rules) capability of the program.

Isotropic creep behavior is based on a von Mises creep potential described by the equivalent creep law

(10-209)

Figure 10-38 Piecewise Linear Representation of Creep Data

· c

c T tf g h k

f g hk

· c

Am · c

nTp qtq 1– =

· c

A= n

·

f · c

T t =

(1) Slope-Break Point Data S1X1 S2X2 S3X3 (2) Function-Variable Data F1X1 F2X2 F3X3 F4X4

Function F (X)

[Such as t ,

g , h (T), k (t)]

c

X1

F1

X2

F2

X3

F3

X4

F4

S1

S2

S3

Variable X (Such as , C, T, t)


The material creep behavior is described by

(10-210)

During creep, the creep strain rate usually decreases. This effect is called creep hardening and can be a function of time or creep strain. The following section discusses the difference between these two types of hardening.

Consider a simple power law that illustrates the difference between time and strain-hardening rules for the calculation of the creep strain rate.

(10-211)

where is the creep strain, and are values obtained from experiments and is time. The creep rate

can be obtained by taking the derivative with respect to time

(10-212)

However, being greater than 0, we can compute the time as

(10-213)

Substituting Equation (10-209) into Equation (10-212) we have

(10-214)

Equation (10-213) shows that the creep strain rate is a function of time (time hardening). Equation (10-214) indicates that the creep strain rate is dependent on the creep strain (strain hardening). The creep strain rates calculated from these two hardening rules generally are different. The selection of a hardening rule in creep analysis must be based on data obtained from experimental results. Figure 10-41 and Figure 10-42 show time and strain hardening rules in a variable state of stress. It is assumed that the stress in a structure varies from to to ; depending upon the model chosen, different creep strain

rates are calculated accordingly at points 1, 2, 3, and 4. Obviously, creep strain rates obtained from the time hardening rule are quite different from those obtained by the strain hardening rule.

· c

i j · c

i j----------

=

c tn=

c n t

c

· c dc

dt-------- ntn 1–= =

t t

tc

-----

1 n/

=

· c

ntn 1– n 1 n c n 1– n = =

1 2 3


366

Figure 10-39 Time Hardening

Figure 10-40 Strain Hardening

Oak Ridge National Laboratory LawsOak Ridge National Laboratory (ORNL) has performed a large number of creep tests on stainless and other alloy steels. It has also set certain rules that characterize creep behavior for application in the nuclear structures. A summary of the ORNL rules on creep is discussed in Marc Volume A, Theory and User Information. The references listed at the end of this section offer a more detailed discussion of the ORNL rules.

c1

0

1

2

3

4

t

2

3

c

1

0

12

3

4

t

2

3


Viscoplasticity (Explicit Formulation)The creep (Maxwell) model can be modified to include a plastic element (as shown in Figure 10-43).

This plastic element is inactive when the stress ( ) is less than the yield stress ( ) of the material. The

modified model is an elasto-viscoplasticity model and is capable of producing some observed effects of creep and plasticity. In addition, the viscoplastic model can be used to generate time-independent plasticity solutions when stationary conditions are reached. At the other extreme, the viscoplastic model can reproduce standard creep phenomena. The model allows the treatment of nonassociated flow rules and strain softening which present difficulties in conventional (tangent modulus) plasticity analyses.

It is recommended that you use the implicit formulation described in the following paragraphs to model general viscoplastic materials.

Figure 10-41 Uniaxial Representation of Viscoplastic Material

Creep (Implicit Formulation)This formulation, as opposed to that described in the previous section, is fully implicit. A fully implicit formulation is unconditionally stable for any choice of time step size; hence, allowing a larger time step than permissible using the explicit method. Additionally, this is more accurate than the explicit method. The disadvantage is that each increment may be more computationally expensive. There are two methods for defining the inelastic strain rate. The creep model definition option can be used to define a Maxwell creep model. The back stress must be specified through the field reserved for the yield stress in the MAT1 or other material definitions. There is no creep strain when the stress is less than the back stress. The equivalent creep strain increment is expressed as

(10-215)

y

ee

evp

Plastic ElementInactive if < y

p vp=

· c

Am

· c

n T

P qtq 1–=


368

and the inelastic deviatoric strain components are

where is the deviatoric stress at the end of the increment and is the back stress. is a function of

temperature, time, etc. Creep only occurs if sigma is greater .

One of three tangent matrices may be formed. The first uses an elastic tangent, which requires more iterations, but can be computationally efficient because re-assembly might not be required. The second uses an algorithmic tangent that provides the best behavior for small strain power law creep. The third uses a secant (approximate) tangent that gives the best behavior for general viscoplastic models.

Specifying Creep Material EntriesEach of the creep models described in this section can be selected with the MATVP Bulk Data entry. MATVP is the only form of creep data material input supported by SOL 600, ie.e., no other MSC Nastran creep data formats are supported by SOL 600.


To define creep behavior in Patran:

1. From the Materials Application form, set the Action>Object>Method combination to Create>Isotropic-or-Orthotropic-or-Anisotropic>Manual Input.

2. Click Input Properties..., and select Creep from the Constitutive Model pull-down menu and MATVP from the Creep Data Input pull-down menu.

Creep material models require the following MATVP material data via the Input Properties subform on the Materials Application form.

Entry Description

MATVP Specifies viscoplastic or creep material properties to be used for quasi-static analysis in MSC Nastran Implicit Nonlinear.

Isotropic-Anisotropic-Orthotropic Description

Coefficient Specifies the coefficient, A.

Exponent of Temperature Defines temperature exponent.

Temperature vs. Creep Strain References a material field of temperature vs. value. Overrides Exponent of Temperature if present.

Exponent of Stress Defines stress exponent

i ji 3

2---

ii j

---------=

i j y A

y


Creep Strain vs. Stress References a material field of stress vs. value. Overrides Exponent of Stress if present.

Exponent of Creep Strain Defines creep strain exponent.

Strain Rate vs. Creep Strain References a material field of strain rate vs. value. Overrides Exponent of Creep Strain if present.

Exponent of Time Defines time exponent.

Time vs. Creep Strain References a material field of time vs. value. Overrides Exponent of Time if present.

Back Stress Defines the back stress for implicit creep

Isotropic-Anisotropic-Orthotropic Description

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideComposite

370

CompositeComposite materials are composed of a mixture of two or more constituents, giving them mechanical and thermal properties which can be significantly better than those of homogeneous metals, polymers and ceramics.

Laminate composite materials are based on layering homogeneous materials using one of several methods. In order to define a laminate composite material, you must define the homogeneous materials that form the layers, the thickness of each layer, and the orientation angle of the layers relative to the standard coordinate axis being used for the model. The orientation is particularly important for orthotropic and anisotropic materials, whose properties vary in different directions. The material in each layer may be either linear or nonlinear. Tightly bonded layers (layered materials) are often stacked in the thickness direction of beam, plate, shell structures, or solids.

Figure 10-42 identifies the locations of integration points through the thickness of beam and shell elements with and without a composite formulation.

Note that when the COMPOSITE option is used, as shown in Figure 10-42, the layer points are positioned midway through each layer. When the COMPOSITE option is not used, the layer points are equidistantly spaced between the top and bottom surfaces. MSC Nastran Implicit Nonlinear performs a numerical integration through the thickness. If the COMPOSITE option is used, the trapezoidal method is employed; otherwise, Simpson’s rule is used.

Each layer is a “ply”, and each ply can have a different material, thickness, or material orientation (angle).


Figure 10-42 Integration Points through the Thickness of Beam and Shell Elements

Figure 10-43 shows the location of integration points through the thickness of continuum elements. MSC Nastran Implicit Nonlinear forms the element stiffness matrix by performing numerical integration based on the standard isoparametric concept.

Figure 10-43 Integration Points through the Thickness of Continuum Elements

Specifying Composite Material EntriesMSC Nastran provides a property definition specifically for performing composite analysis. You specify the material properties and orientation for each of the layers and MSC Nastran produces the equivalent PSHELL and MAT2 entries for shells. This is extended to PSOLID and MATORTH for SOL 600 only.

The stacking direction for 3-D composite solids was added with a new entry, MSTACK.

SOL 600 provids two options for composite analyses:

Entry Description

PCOMP Defines the properties of an n-ply composite material laminate.

PCOMPG Defines global (external) ply IDs and properties for a composite material laminate

****

*****Beams or Shells with

Composite OptionBeams or Shells without

Composite Option

**** *

***

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideComposite

372

1. Complete through the thickness integration at every iteration

2. The “smeared” approach as used in other Nastran solution sequences.

The first approach is more accurate particularly for nonlinear analyses where local buckling takes place and the analysis needs to extend well into the post-bucking regime. The second approach is usually satisfactory for small deformation linear static and dynamic analyses. Method 1 has what is known as “fast integration” techniques and are described by the Bulk Data entry PCOMPF. The limitation is that using these fast integration procedures the material may not exhibit any nonlinear behavior. Large deformation and buckling is supported using these procedures.


GasketEngine gaskets are used to seal the metal parts of the engine to prevent steam or gas from escaping. They are complex (often multi-layer) components, usually rather thin and typically made of several different materials of varying thickness. The gaskets are carefully designed to have a specific behavior in the thickness direction. This is to ensure that the joints remain sealed when the metal parts are loaded by thermal or mechanical loads. The through-thickness behavior, usually expressed as a relation between the pressure on the gasket and the closure distance of the gasket, is highly nonlinear, often involves large plastic deformations, and is difficult to capture with a standard material model. The alternative of modeling the gasket in detail by taking every individual material into account in the finite element model of the engine is not feasible. It requires a lot of elements which makes the model unacceptably large. Also, determining the material properties of the individual materials might be cumbersome.

The gasket material model addresses these problems by allowing gaskets to be modeled with only one element through the thickness, while the experimentally or analytically determined complex pressure-closure relationship in that direction can be used directly as input for the material model. The material must be used together with 2-D or 3-D first-order solid composite element types or 2-D axi-symmetric elements. In that case, these elements consists of one layer and have only one integration point in the thickness direction of the element.

Constitutive Model

The behavior in the thickness direction, the transverse shear behavior, and the membrane behavior are fully uncoupled in the gasket material model. In subsequent sections, these three deformation modes are discussed.

Local Coordinate System

The material model is most conveniently described in terms of a local coordinate system for the integration points of the element (see Figure 10-44). For three-dimensional elements, the first and second directions of the coordinate system are tangential to the midsurface of the element at the integration point. The third direction is the thickness direction of the gasket and is perpendicular to the midsurface. For two-dimensional elements, the first direction of the coordinate system is the direction of the midsurface at the integration point, the second direction is the thickness direction of the gasket and is perpendicular to the midsurface, and the third direction coincides with the global 3-direction.

In a total Lagrange formulation, the orientation of the local coordinate system is determined in the undeformed configuration and is fixed. In an updated Lagrange formulation, the orientation is determined in the current configuration and is updated during the analysis.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideGasket

374

Figure 10-44 The Location of the Integration Points and the Local Coordinate Systems in Two- and Three-dimensional Gasket Elements

Thickness Direction - Compression

In the thickness direction, the material exhibits the typical gasket behavior in compression, as depicted in Figure 10-45. After an initial nonlinear elastic response (section AB), the gasket starts to yield if the pressure p on the gasket exceeds the initial yield pressure py0. Upon further loading, plastic deformation

increases, accompanied by (possibly nonlinear) hardening, until the gasket is fully compressed (section BD). Unloading occurs in this stage along nonlinear elastic paths (section FG, for example). When the gasket is fully compressed, loading and unloading occurs along a new nonlinear elastic path (section CDE), while retaining the permanent deformation built up during compression. No additional plastic deformation is developed once the gasket is fully compressed.

The loading and unloading paths of the gasket are usually established experimentally by compressing the gasket, unloading it again, and repeating this cycle a number of times for increasing pressures. The resulting pressure-closure data can be used as input for the material model. The user must supply the loading path and may specify up to ten unloading paths. In addition, the initial yield pressure py0 must

be given. The loading path should consist of both the elastic part of the loading path and the hardening part, if present. If no unloading paths are supplied or if the yield pressure is not reached by the loading path, the gasket is assumed to be elastic. In that case, loading and unloading occurs along the loading path.

The loading and unloading paths must be defined using the TABLES1 bulk data entries and must relate the pressure on the gasket to the gasket closure. The unloading paths specify the elastic unloading of the gasket at different amounts of plastic deformation; the closure at zero pressure is taken as the plastic closure on the unloading path. If unloading occurs at an amount of plastic deformation for which no path has been specified, the unloading path is constructed automatically by linear interpolation between the two nearest user supplied paths. The unloading path, supplied by the user, with the largest amount of plastic deformation is taken as the elastic path at full compression of the gasket.

For example, in Figure 10-45, the loading path is given by the sections AB (elastic part) and BD (hardening part) and the initial yield pressure is the pressure at point B. The (single) unloading path is curve CDE. The latter is also the elastic path at full compression of the gasket. The amount of plastic closure on the unloading path is cp1. The dashed curve FG is the unloading path at a certain plastic

closure cp that is constructed by interpolation from the elastic part of the loading path (section AB) and

the unloading path CD.

1

2

Midsurface

Integration Point

1

2

3

MidsurfaceIntegration Point


Figure 10-45 Pressure-closure Relation of a Gasket

The compressive behavior in the thickness direction is implemented by decomposing the gasket closure rate into an elastic and a plastic part:

(10-216)

Of these two parts, only the elastic part contributes to the pressure. The constitutive equation is given by the following rate equation:

(10-217)

Here, Dc is the consistent tangent to the pressure-closure curve.

Plastic deformation develops when the pressure p equals the current yield pressure py. The latter is a

function of the amount of plastic deformation developed so far and is given by the hardening part of the loading path (section BD in Figure 10-45).

Initial Gap

The thickness of a gasket can vary considerably throughout the sealing region. Since the gasket is modeled with only one element through the thickness, this can lead to meshing difficulties at the boundaries between thick regions and thin regions. The initial gap parameter can be used to solve this. The parameter basically shifts the loading and unloading curves in the positive closure direction. As long

py0

py1

py

cp0 cp1cp cy0 cy1cyA

B

C

D

E

F

G

Gasket Closure Distance c

Gas

ket P

ress

ure

ploading path

unloading path

c·

c·e

c·p

+=

p·

Dcc·e

Dc c·

c·p

– = =


376

as the closure distance of the gasket elements is smaller than the initial gap, no pressure is built up in the gasket. The sealing region can thus be modeled as a flat sheet of uniform thickness and the initial gap parameter can be set for those regions where the gasket is actually thinner than the elements of the finite element mesh used to model it.

Thickness Direction - Tension

The tensile behavior of the gasket in the thickness direction is linear elastic and is governed by a tensile modulus Dt. The latter is defined as a pressure per unit closure distance (that is, length).

Transverse Shear and Membrane Behavior

The transverse shear is defined in the 2-3 and 3-1 planes of the local coordinate system (for three-dimensional elements) or the 1-2 plane (for two-dimensional elements). It is linear elastic and characterized by a transverse shear modulus Gt.

The membrane behavior is defined in the local 1-2 plane (for three-dimensional elements) or the local 3-1 plane (for two-dimensional elements) and is linear elastic and isotropic. Young’s modulus Em and

Poisson’s ratiom that govern the membrane behavior are taken from an existing material that must be

defined using the MAT1 Bulk Data entry. Multiple gasket material can refer to the same isotropic material for their membrane properties (see also the GASKET model definition option in Marc Volume C: Program Input).

Thermal Expansion

The thermal expansion of the gasket material is isotropic and the thermal expansion coefficient are taken from the isotropic material that also describes the membrane behavior.

Constitutive Equations

As mentioned above, the behavior in the thickness direction of the gasket is formulated as a relation between the pressure p on the gasket and the gasket closure distance c. In order to formulate the constitutive equations of the gasket material, this relation must first be written in terms of stresses and strains. This depends heavily on the stress and strain tensor employed in the analysis. For small strain analyses, for example, the engineering stress and strain are used. In that case, the gasket closure rate and the pressure rate are related to the strain rate and the stress rate by

and (10-218)

in which h is the thickness of the gasket.

The resulting constitutive equation for three-dimensional elements, expressed in the local coordinate system of the integration, now reads

c h–= p –=


(10-219)

in which C = hDc. For two-dimensional elements, the equation is given by

(10-220)

For large deformations in a total Lagrange formulation, in which the Green-Lagrange strains and the second Piola-Kirchhoff stresses are employed (as well as in an updated Lagrange environment) in which the logarithmic strains and Cauchy stresses are being used, similar but more complex relations can be derived.

Specifying Gasket Material EntriesThe MATG provides specifically for modeling gasket materials.

Entry Description

MATG Specifies gasket material properties to be used in MSC Nastran Implicit Nonlinear.

MATTG Specifies gasket material property temperature variation to be used in MSC Nastran Implicit Nonlinear.

11

22

33

12

23

31

Em

1 m2

–-----------------

mEm

1 m2

–----------------- 0 0 0 0

mEm

1 m2

–-----------------

Em

1 m2

–----------------- 0 0 0 0

0 0 C 0 0 0

0 0 0Em

2 1 m+ -------------------------- 0 0

0 0 0 0 Gt 0

0 0 0 0 0 Gt

11

22

33 33p

–

12

23

31

=

11

22

33

12

Em

1 m2

–----------------- 0

mEm

1 m2

–----------------- 0

0 C 0 0

mEm

1 m2

–----------------- 0

Em

1 m2

–----------------- 0

0 0 0 Gt

11

22 22p

–

33

12

=


378

References

• MATG (p. 2438) in the MSC Nastran Quick Reference Guide

• MATTG (p. 2501) in the MSC Nastran Quick Reference Guide


Patran 2005 support this option.


Material DampingIn direct integration analysis, the user very often defines energy dissipation mechanisms as part of the basic model - dashpots, inelastic material behavior, etc. In such cases, there is usually no need to introduce additional “structural” or general damping: it is unimportant compared to these other dissipative effects. However, some models do not have such dissipation sources (an example is a linear system with chattering contact, such as a pipeline in a seismic event). In such cases, it is usually desirable to introduce some general low level of damping. MSC Nastran Implicit Nonlinear provides “Rayleigh” damping for this purpose. The user includes the two Rayleigh damping factors, R for mass proportional

damping and R for stiffness proportional damping on the NLSTRAT Bulk Data entry. In the case of

elements the damping values must be used in conjunction with these property references. For a linear problem, these provide a damping matrix [C] as described above:

[C]=R[M]+R[K].

Since the model may have quite general nonlinear response, the concept of “stiffness proportional damping” must be generalized, since it is possible for the tangent stiffness matrix to have negative eigenvalues (which would imply negative damping). To overcome this problem, R is interpreted as

defining viscous material damping which creates an additional “damping stress,” d, proportional to the

total strain rate:

(10-221)

Here D0el is the material’s initial (virgin) elastic stiffness. This damping stress is added to the stress

caused by the constitutive response at the integration point when the dynamic equilibrium equations are formed, but it is not included in the stress output. This allows damping to be introduced for any nonlinear case, and provides standard Rayleigh damping for linear cases.

Since the R factor introduces damping proportional to the strain rate, this may be thought of as damping

associated with the material itself, while the R factor introduces damping forces caused by the absolute

velocities of the model, and so simulates the idea of the model moving through a viscous “ether” (a permeating, still fluid, so that any motion of any point in the model causes damping).

The R factor is applied to all elements that have mass. The R factor applies to all elastic elements and

to beam and shell elements. The R factor is not applied to spring elements. Discrete dashpot elements

should be used as needed for springs.

Specifying Material Damping EntriesParameters for material damping are input through the NLSTRAT entry.

d D0el·

=

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideMaterial Damping

380


MSC.Patran 2005 does not yet support material damping specified by the NLDAMP entry, but you can specify the transient analysis damping parameters on the NLSTRAT entry under the Load Increment Parameters subform.

Entry Description

NLSTRAT Defines transient analysis damping parameters BETA, GAMMA, GAMMA1, GAMMA2.

NLDAMP Defines damping constants for nonlinear analysis when Marc is executed from MSC Nastran used in SOL 600 only (Not supported in MSC.Patran 2004).


Gamma (Newmark) Mass proportional damping coefficient.

Beta (Newmark) Stiffness proportional damping coefficient.


Experimental Data FittingThis is a very useful tool available under the Tools pull-down menu from the main MSC.Patran form and is available if the Analysis Preference is set to MSC Nastran.

The tool is used to curve fit experimentally derived, raw elastomeric material data and fit a number of material models to the data. The data can then be saved as constitutive hyperelastic and/or viscoelastic models for use in anMSC NastranImplicit Nonlinear or MSC.Marc analysis.

The operation of curve fitting is done in three basic steps corresponding to the actions in the Action pull-down menu.

1. Import the Raw Data - data is read from standard ASCII files and stored in MSC.Patran in the form a field (table).

2. Select the Test Data - the fields from the raw data are associated to a test type.

3. Calculate the Properties - the curve fit is done to the selected test data; coefficients are calculated based on the selected material model; curve fit is graphically displayed and the properties can be saved as a constitutive model for a later analysis.

Import Raw Data

Importing the data is done by following these steps:

1. Enter a New Field Name - this is the name of the raw data table as it will be stored in MSC.Patran as a material field.

2. Select the Independent Variable - this is defaulted to Strain but could be any of Strain, Time, Frequency, Temperature, or Strain Rate.

3. Select the File and press the Apply button.

The following notes are made:

• You can skip any number of header lines in the raw data file by setting the Header Lines to Skip widget.


MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideExperimental Data Fitting

382

• You may edit the raw data file after selecting it by using the Edit File... button. The editor is Notepad on Windows platforms and vi on UNIX platforms unless you change the environment variable P3_EDITOR to reference a different editor. The editor must be in the user’s path or the entire pathname must be referenced.

• Raw data files may have up to three columns of data. By default the first column of data is the independent variable value. The second column is the measured data, and the last column can be the area reduction or volumetric data. More than three columns is not accepted. If the third column is blank, the material is considered incompressible.

• The data may be space, tab, or comma delimited.

• If you have cross sectional area reduction data in the third column, you can give it an optional field name also. If you do not specify that you have this data and a third column is detected, two fields will still be created and a _C1 and _C2 will be appended to the given field name.

• If for some reason the independent and dependent columns need to be interchanged, you can turn this toggle on. Check your imported fields before proceeding to ensure they are correct. This is done in the Fields application.

• When you press the Apply button, you will be taken to the second step. If you need to import more than one file, you will have to reset the Action pull-down.


Import Raw Data Description

New Field The field name under which the raw data will be saved. For an explanation of what the raw data files should contain, see the description on each mode (Uniaxial, Biaxial, etc.) in the table for Select Test Data.

Area Data If volumetric data or cross sectional area reduction data is stored in the third column of a raw data file it will be imported automatically and two field will be created, one with a _C1 and the other with a _C1 appended to the name. Optionally, you can give it it’s own name by turning this toggle ON.

Area Field Name If the Area Data toggle is on, you supply the name to the field of volumetric or area data here.

Independent Variable

This defines the independent variable. The material field created from reading the raw data will be tagged with this independent variable. This is simply a label and has no effect on the actual curve fits.

Header Lines to Skip

If any header lines in the data file are to be skipped before the raw data is processed, you can specify this with this databox.

Select File Select a raw data file. The .dat and .csv file types are filtered by default. csv files are comma separated and created by Microsoft Excel.


Edit File Once a file has been selected you may edit it with Notepad on Windows and vi on UNIX. The editor can be changed by setting the environment variable P3_EDITOR to the editor or choice. The editor command must be in the user’s path or the full path must be provided as part of the P3_EDITOR environment variable.

Apply This command will import the raw data and save it as a field(s) in the name(s) given in New Field (and/or Area Field Name) databoxes. You will also be taken to automatically to the Select Test Data action which is the next step. If you need to import multiple raw data files, you will have to set the Action back to Import Raw Data.

Cancel Closes the Experimental Data Fitting tool.


Import Raw Data Description


384

Select Test Data

Once the raw test data is imported, you must associate with a particular test type or mode by following these steps:

1. Put the cursor in the data field of the appropriate type of test.

2. Select associated field from the Select Material Test Data listbox which should have a list of the imported raw data fields.

3. Repeat this for each test you wish to include in the calculations (curve fit).


• Typical stress-strain data for Deformation Mode tests are referenced in the Primary column. If you have volumetric data, these are entered in the Secondary column databoxes and are optional.

• For Viscoelastic (time relaxation data), you must turn ON the ViscoElastic toggle. Only viscoelastic curve fitting will be done in this case. To return to Deformation Mode, turn this toggle OFF.

• Damage models are not yet supported.

• When you press the Apply button, you will be taken to the third step.



Select Test Data Description

Uniaxial

Biaxial

Planar Shear

Simple Shear

Volumetric

Select the field of raw test data corresponding to each of these tests if they exist. You need to supply at least one. The stress-strain data field is referenced in the Primary column. If you have volumetric or area data fields, they are referenced in the Secondary column. If no Secondary field is supplied, the material models are assumed incompressible.

For time independent elastomeric materials (uniaxial, biaxial, planar shear, simple shear and volumetric tests) the data should be as such in the raw data file:

eng. strain_1, eng. stress_1, volumetric data_1

eng. strain_2, eng. stress_2, volumetric data_2

. . .

eng. strain_n, eng. stress_n, volumetric data_n

If, for uniaxial, biaxial or planar shear data, the third column is left empty, the material is assumed to be incompressible.

Viscoelastic Turn this toggle ON if you wish to do a data fit on viscoelastic relaxation data.

For visco-elastic material behavior (shear relaxation, bulk relaxation and energy relaxation tests) the data should be as such in the raw data file:

time_1, value_1 (shear modulus, bulk modulus or strain energy)

time_2, value_2 (shear modulus, bulk modulus or strain energy)

. . .

time_n, value_n (shear modulus, bulk modulus or strain energy)


386

Damage Not yet supported. Turn this toggle ON if you wish to do a data fit on damage models. For continuous damage (resulting from a constant strain amplitude test) the data in the raw input file should be:

cycle_1, str_energy_d_1

cycle_2, str_energy_d_2

. .

cycle_n, str_energy_d_n

In addition, before fitting the data, the free energy (which is the strain energy density corresponding to the undamaged state) is required. Notice that the data points should not include the range of cycles at which damage did not start to evaluate.

For discontinuous damage (resulting from an increasing strain amplitude test):

str_energy_d_1, strain_energy_d_1/str_energy_d_1_undamaged

str_energy_d_2, strain_energy_d_2/str_energy_d_2_undamaged

. . .

str_energy_d_n, strain_energy_d_n/str_energy_d_n_undamaged

Notice that the data points should not include the range of cycles at which damage did not start to evaluate.

Viscoelastic/Damage In this field, select the Viscoelastic or Damage raw test data field.

Select Material Test Data

From this listbox you select the field corresponding to the Deformation Mode or the Viscoelastic/Damage data.

Apply Once the test data is associated to the respective modes, the Apply button will take you to the Calculate Properties action.



Select Test Data Description


Calculate Properties

Once test data has been associated to a test type or mode, the curve fit is done by following these steps:

1. Select the material Model you wish to do a curve fit for. The available models will depend on the test data selected in the previous step. Hyperelastic models will be available for deformation mode test data. Viscoelastic models will be available for relaxation test data.

2. In general you will leave Use Test Data to All for hyperelastic models. If however you only want the curve fit to use one of the deformation modes, you may set it here.

3. Press the Compute button. The coefficient values will be displayed in the Coefficients spreadsheet.

4. To visually see the curve fit, press the Plot button.

You may repeat the above four steps for as many material models as you wish to curve fit.

5. Select an existing material or type in a New Material Name and press the Apply button to save the material model as either a Hyperelastic or Viscoelastic constitutive model for use in a subsequent analysis.


• The plots are appended to the existing XY Window until you press the Unpost Plot button. You can turn the Append function ON/OFF under the Plot Parameters... form.

• By default, all the deformation modes are plotted along with the raw data even if raw data has not been supplied for those mode. This is very



388

important. These additional modes are predicted for you. You should always know your model’s response to each mode of deformation due to the different types of stress states. For example, a rule of thumb for natural rubber and some other elastomers is that the tensile tension biaxial response should be about 1.5 to 2.5 times the uniaxial tension response.

• You can turn ON/OFF these additional modes or any of the curves under the Plot Parameters button as well as change the appearance of plot. More control and formatting of the plot can be done under the XY Plot application on the MSC.Patran application switch on the main form.

• Viscoelastic constitutive models are useless without a Hyperelastic constitutive model also. Be sure your model has both defined under the same material name if you use viscoelastic properties.

• You may actually change the coefficient values in the Coefficients spread sheet if you wish to see the effect they have on the curve fit. Select one of the cells with the coefficient you wish to change, then type in a new coefficient value in the Coefficient Value data box and press the Return or Enter key. Then press the Plot button again. If you press the Apply button, the new values will be saved in the supplied material name.

• For viscoelastic relaxation data, the Number of Terms used in the data fit should, as a rule of thumb, be as many as there are decades of data.

• A number of optional parameters are available to message the data and control the curve fitting. See the table below for more detailed descriptions.


The following tables more fully describes each widget in the Experimental Data Fitting tool:

Calculate Properties Description

Model:Neo-Hookean* This command is used if experimental data must be fitted using the Neo-Hookean strain energy function W, which is given by:

W = C10*(I1 - 3)

where I1 is the first invariant of the right Cauchy-Green strain tensor and C10 is the material parameter to be determined. For this model, a volumetric test can not be supplied. The user may enter a bulk modulus. If no bulk modulus is given, nearly incompressible material behavior is assumed.

Model:Mooney(2)* This command is used if experimental data must be fitted using the two term Mooney-Rivlin strain energy function W, which is given by:

W = C10*(I1 - 3) + C01*(I2 - 3)

where I1 and I2 are the first and second invariant of the right Cauchy-Green strain tensor and C10 and C01 are the material parameters to be determined. For this model, a volumetric test can not be supplied. The user may enter a bulk modulus. If no bulk modulus is given, nearly incompressible material behavior is assumed.

Model:Mooney(3)* This command is used if experimental data must be fitted using the three term Mooney-Rivlin strain energy function W, which is given by:

W = C10*(I1 - 3) + C01*(I2 - 3) + C11*(I1 - 3)*(I2 - 3)

where I1 and I2 are the first and second invariant of the right Cauchy-Green strain tensor and C10, C01 and C11 are the material parameters to be determined. For this model, a volumetric test can not be supplied. The user may enter a bulk modulus. If no bulk modulus is given nearly incompressible material behavior is assumed.

Model:Signiorini* This command is used if experimental data must be fitted using the Signiorini strain energy function W, which is given by:

W = C10*(I1 - 3) + C01*(I2 - 3) + C20*(I1 - 3)^2

where I1 and I2 are the first and second invariant of the right Cauchy-Green strain tensor and C10, C01 and C20 are the material parameters to be determined. For this model, a volumetric test can not be supplied. The user may enter a bulk modulus. If no bulk modulus is given, nearly incompressible material behavior is assumed.


390

Model:2nd Order Invariant*

This command is used if experimental data must be fitted using the second order invariant strain energy function W, which is given by:

W = C10*(I1 - 3) + C01*(I2 - 3) + C11*(I1 - 3)*(I2-3) +C20*(I1 - 3)^2

where I1 and I2 are the first and second invariant of the right Cauchy-Green strain tensor and C10, C01, C11 and C20 are the material parameters to be determined. For this model, a volumetric test can not be supplied. The user may enter a bulk modulus. If no bulk modulus is given, nearly incompressible material behavior is assumed.

Model:3rd Order Deform*

This command is used if experimental data must be fitted using the third order deformation strain energy function W, which is given by:

W = C10*(I1 - 3) + C01*(I2 - 3)+ C11*(I1 - 3)*(I2 - 3) +C20*(I1 - 3)^2 + C30*(I1 - 3)^3

where I1 and I2 are the first and second invariant of the right Cauchy-Green strain tensor and C10, C01, C11, C20 and C30 are the material parameters to be determined. For this model, a volumetric test can not be supplied. The user may enter a bulk modulus. If no bulk modulus is given, nearly incompressible material behavior is assumed.

Note: The data input for all of these options has been consolidated under the Jamus-Green-Simpson option of the Patran Materials-Input-Data-Hyperelastic form.

Model:Yeoh This command is used if experimental data must be fitted using the Yeoh strain energy function W, which is given by:

W = C10*(I1 - 3) + C20*(I1 - 3)^2 + C30*(I1 - 3)^3

where I1 is the first invariant of the right Cauchy-Green strain tensor and C10, C20 and C30 are the material parameters to be determined. For this model, a volumetric test can not be supplied. The user may enter a bulk modulus. If no bulk modulus is given, nearly incompressible material behavior is assumed.



Model:Ogden This command is used if experimental data must be fitted using the Ogden strain energy function W, which is given by:

where lam1, lam2 and lam3 are the principal stretch ratios, J is the determinant of the deformation gradient, N is the number of terms and mu_n, alpha_n and K are the material parameters to be determined. The maximum number of terms is 10, but it is recommended to use no more terms than necessary to get a sufficiently good fit. This model can be used for incompressible as well as for slightly compressible elastic materials. Compressibility is included based on a constant bulk modulus. In case of compressibility, volumetric information is needed, preferably using a volumetric test, but volumetric data can also be included for uniaxial, biaxial and planar shear tests. In order to perform a plausible extrapolation for the compressible Ogden model, dilatational information is needed beyond the data set. This is achieved using linear extrapolation based on the two start and/or end points of the measured data. This linear extrapolation may restrict the validity of the response outside the range of the measured data. For dual mode plotting (except for simple shear), dilatational information is needed for the compressible Ogden model. For a volumetric test, this readily follows from the strain, but for uniaxial, biaxial and planar shear tests this must be calculated. This calculation is based on the requirement that the stress in perpendicular direction must be zero. If the fitted coefficients do not fulfil this requirement, zero stresses are returned for such a dual mode.

Note: Ogden hyperelastic coefficients are different in MSC.Marc and MSC Nastran. For information on experimental data fitting for the MSC Nastran MATHP and MATHE entries, see Experimental Data Fitting, 296.


{(mu_n/alpha_n) * (J^(-alpha_n/3)) * (lam1âlpha_n + lam2âlpha_n + lam2âlpha_n - 3)} + 4.5 * K * (J^(1/3) - 1)^2

W

N

n 1=

=


392

Model:Foam This command is used if experimental data must be fitted using the foam strain energy function W, which is given by:

where lam1, lam2 and lam3 are the principal stretch ratios, J is the determinant of the deformation gradient, N is the number of terms and mu_n, alpha_n and beta_n are the material parameters to be determined. The maximum number of terms is 10, but it is recommended to use no more terms than necessary to get a sufficiently good fit. This model should be used for highly compressible elastic materials. Except for the simple shear test, volumetric information must be available. In order to perform a plausible extrapolation for the foam model, dilatational information is needed beyond the data set. This is achieved using linear extrapolation based on the two start and/or end points of the measured data. This linear extrapolation may restrict the validity of the response outside the range of the measured data. For dual mode plotting (except for simple shear), dilatational information is needed for the foam model. For a volumetric test, this readily follows from the strain, but for uniaxial, biaxial and planar shear tests this must be calculated. This calculation is based on the requirement that the stress in perpendicular direction must be zero. If the fitted coefficients do not fulfil this requirement, zero stresses are returned for such a dual mode.


{(mu_n/alpha_n)*(lam1âlpha_n + lam2âlpha_n + lam2âlpha_n - 3) +(mu_n/beta_n)*(1 - J^beta_n)}

W

N

n 1=

=


Model:Arruda-Boyce Elastomer Free Energy Function; Number of coefficients 1

Ref: "A Three-Dimensional Constitutive Model For the Large Stretch Behavior of Rubber Elastic Materials"

by: Ellen M. Arruda and Mary C. Boyce

J.Mech.Phys.Solids Vol.41, No.2, pp.389-412

Parameter: N -- The number of mers in a typical polymer chain - Specified by the user.

Coefficient: n,k,T -- Determined by the fitter

n : Number of Polymer chains per unit volume

k : Boltzmann constant

T : Temperature

For this model, the calculation of the Bulk Modulus is not required. A volumetric test need not be supplied.



394

Model:Gent Elastomer Free Energy Function; Number of coefficients 2

Ref: "A new constitutive relation for rubber"

by: A.N. Gent

Rubber Chemistry and Technology,

Vol.79, pp.59-61, 1996

Coefficient: E, I_m -- Determined by the fitter

E : small strain tensile modulus

I_m : maximum value for the first

invariant of deformation (I1)

WARNING: This phenomenological model is designed to exhibit finite extensibility of polymer chains and forces the stresses and tangent to assymptote to infinity as I1 approaches Im. It is conceivable that during the solution stage, a set of trial displacements is evaluated such that I1 > Im. In which case, the convergence ratios can be expected to oscillate, or even worse, the solution may not converge at all. Therefor, if Im < 4, it is recommended that loads be applied in very small increments.

For this model, the calculation of the Bulk Modulus is not required and a volumetric test need not be supplied.



Model:Visco Shear Relax

This command is used if experimental shear relaxation data must be fitted using the following Prony series expansion for the shear modulus G:

where t is the time, N is the number of terms and G_infinity, G_n and tau_n are material parameters to be determined. The data points provided by the user must give the value of the shear modulus at different time stations, which do not need to be equi-spaced.

Model:Visco Bulk Relax

This command is used if experimental bulk relaxation data must be fitted using the following Prony series expansion for the bulk modulus K:

where t is the time, N is the number of terms and K_infinity, K_n and tau_n are material parameters to be determined. The data points provided by the user must give the value of the bulk modulus at different time stations, which do not need to be equi-spaced.

Model:Visco Energy Relax

This command is used if experimental strain energy relaxation data must be fitted using the following Prony series expansion for the strain energy W

:

where t is the time, N is the number of terms, W0 is the instantaneous strain energy and W_infinity, delta_n and tau_n are material parameters to be determined. The data points provided by the user must give the value of the strain energy at different time stations, which do not need to be equi-spaced.

Model:Cont. Damage Not yet supported.

Model:Disc. Damage Not yet supported.


G(t) = G_infinity {G_n*exp(-t/tau_n)} W

N

n 1=

=

K(t) = K_infinity {K_n*exp(-t/tau_n)}

N

n 1=

K(t) = K_infinity {delta_n*W0*exp(-t/tau_n)}

N

n 1=


396

Compute This command starts the data fitting program with the selected data. After fitting, the measured and fitted curves can be displayed and the corresponding material model coefficients and the least squares error are reported. For the Mooney-Rivlin, Ogden, Foam, Arruda-Boyce and Gent models, the response in the modes for which no data is measured, is predicted. Notice that if volumetric data is relevant in order to predict a uniaxial, biaxial or planar shear mode, this is calculated using the constraint of a zero stress component in a direction perpendicular to the direction of the measured stress component. If this calculation fails, the predicted mode will contain zero stresses.

If the data set contains a large number of entries, or if the model is highly non-linear and/or contains many coefficients, then the fitting procedure may take some time.

Since the curve fitting procedure does not use weighting factors per data point, it might be useful to have many data points near regions where an accurate response is desired.

Coefficient Value This option allows the user to manually enter the coefficients of the material model and updates the response curves. You must select the coefficient to be modified from the spread sheet and then enter the value by pressing the Return or Enter key after putting in the new coefficient. Notice that, although no fit is performed, at least one set of test data must be selected. This feature is available for the Mooney-Rivlin, Ogden, Foam, Arruda-Boyce and Gent material models.

Plot This plots the current curve fit. The raw data and the calculated curve fits for the supplied experimental data and the other predicted modes are plotted.

Unpost Plot This clears the plot of all curves. By default additional data fit curves are appended the XY Window.

New Material Name If a new name is supplied here and this material name does not yet exist, it will be created with the appropriate constitutive models when the Apply button is pressed.

Select Material If you want the hyperelastic or viscoelastic data to be saved into an existing material set, select it from this listbox.

Apply This command copies the computed material model coefficients into a material model: it either creates a new one as indicated by the New Material Name or changes an existing one as selected from Select Material.





Optional Parameters Description

Uniaxial Test: A/Ao V/Vo Volumetric data of a uniaxial test can be given as: the ratio of the current cross sectional area and the original cross sectional area (A/A0) or: the ratio of the current volume and the original volume (V/V0). By default, the volumetric data is expected to be in terms of A/A0. If the actual data is in terms of V/V0, use this switch to change the type of input. Not all material models need this information and therefore these widgets may appear dimmed.

Biaxial Test: t/to V/Vo Volumetric data of an equibiaxial test can be given as: the ratio of the current thickness and the original thickness (t/t0) or: the ratio of the current volume and the original volume (V/V0). By default, the volumetric data is expected to be in terms of t/t0. If the actual data is in terms of V/V0, use this switch to change the type of input. Not all material models need this information and therefore these widgets may appear dimmed.

Planar Shear t/to V/Vo Volumetric data of a planar shear test can be given as: the ratio of the current thickness and the original thickness (t/t0) or: the ratio of the current volume and the original volume (V/V0) If the actual data is in terms of V/V0, use this switch to change the type of input. Not all material models need this information and therefore these widgets may appear dimmed.

Mathematical Checks This command activates mathematical checks for Ogden and Foam materials and causes the data fitter to discard the coefficients when one of the mathematical conditions on them is not satisfied. These conditions are considered to be very strict and at times no set of coefficients may be found, or the fit may be very poor.

Positive Coefficients Since curve fitting is a mathematical operation, the fitted material model coefficients may be physically non-realistic. This command forces the fitting procedure to return coefficients which are all positive. The quality of the fit may be worse than that without this restriction. For example Ogden coefficients come in pairs. If each pair of modulus and exponent have the same sign, stability is guaranteed. If one is positive and the other negative, the material might be unstable. Thus you must visually determine the stability range of the model.


398

Extrapolate For Mooney-Rivlin, Ogden, Foam, Arruda-Boyce and Gent models, this command gives the possibility to get the response of the material outside the range of measurements. This might be important if the deformations of the structure to be analyzed exceed those of the experiments. One needs to set the new left and right bounds up to which the extrapolation will be performed. Notice that when volumetric information is provided, outside the range of measurements the volumetric data is calculated based on linear extrapolation using the two closest measured data points.

Error The least squares error to be minimized during data fitting can be based on absolute or relative errors:

Err_Abs = sum {[data_measured(i) - data_calculated(i)]^2} iErr_Rel = sum {[1 -data_calculated(i)/data_measured(i)]^2} i

This command can be used to switch between relative (default) and absolute errors.

Error Limit During curve fitting, an optimal set of material coefficients is searched for using the Downhill Simplex method. If the least squares error corresponding to a set of material coefficients is larger than the error limit, a new minimum will be searched for, unless the maximum number of iterations has been reached. This command sets the error limit. If, upon fitting, the reported error is larger than the error limit, searching for a new minimum has been terminated due to reaching the maximum number of iterations.

Number of Iterations At times, the data fitter may arrive at coefficients which correspond to a minimum in the objective function, not fulfilling the least squares error limit. This command defines how many attempts the program will make to exit the minimum. The maximum number of iterations is 500.

Convergence Tolerance The data fitter uses the Downhill Simplex method to find a (local) minimum. This method uses several sets of material coefficients and calculates the corresponding objective functions. Based on the values of the objective functions, the sets of material coefficients are modified. This process is terminated when:

2*abs{(fh-fl)/(abs(fh)+abs(fl))} < convergence tolerance

in which fh is the highest and fl is the lowest objective function found so far. The process is also terminated after 2000 trials if the convergence tolerance has not been reached. This command sets the convergence tolerance.




Use Fictive Coefficient For Foam models, toggle this value to ON in order to use the fictive Poisson's ratio to create volumetric information.

Fictive Coefficient Supply the fictive Poisson’s ratio here.

OK Closes the Optional Parameters form.


Plot Parameters Description

Append Curves If this toggle is ON, curves keep accumulating on the XY Window. If it is OFF, the curves are cleared each time a new data fit is done and plotted.

X-Axis OptionsY-Axis Options

You can plot the curves in linear or log scales.

Deformation Modes If any of these toggles is OFF, that particular deformation mode will be removed from the XY Window.

OK Closes the Plot Parameters form.



Note: The plotting and deleting of plots in this utility tool is fairly self contained and little, if any, need to use the XY Plot application is necessary. If however, you find it necessary to use the XY Plot application and post/unpost curves, the naming convention is as such:


400

Raw Data Curve Name = R_"mode type"_"field id"_"color"_"marker type

Data Fit Curve Name = "model"_"mode type"_"field id"_"color"_"line type

where:

mode type (string) = "U" : Uniaxial"B" : Biaxial"P" : Planar Shear"S" : Simple Shear"V" : Volumetric"C" : VisCoelastic

model (string)= "N" : Neo-Hookean"M2" : Mooney 2"M3" : Mooney 3"S" : Signiorini"2O" : Second Order Invariant"3O" : Third Order Deformation"Y" : Yeoh"O#" : Ogden # where # is the number of terms, e.g., O2"F#" : Foam # where # is the number of terms, e.g., F3"B" : Boyce"G" : Gent"CD" : Continuous damage"DD" : Discontinuous Damange"SV" : Shear Relaxation Visco"BV" : Bulk Relaxation Visco"EV" : Energy Relaxation Visco

Chapter 11: Element Library

11 Element Library

Overview

Element Selection

Global Element Controls

Mass Elements, Springs, Dampers, and Connector Elements

Gap Elements

Line Elements

Membranes, Panels, and Shells

Solid Elements

Beam/Bar and Shell Offsets


402

OverviewThe heart of a finite element program lies in its element library which allows you to model a structure for analysis. MSC Nastran has a very comprehensive element library which lets you model 1-D, 2-D, or 3-D structures. This section gives some basic definitions of the element types available in MSC Nastran Implicit Nonlinear. Please note, these elements differ from those used in other portions of MSC Nastran.

Element TypesEach element has five definitive characteristics that determine its behavior:

• Class

• Number of Nodes

• Interpolation

• Degrees of Freedom

• Integration Method

Class

The type of geometric domain that an element represents determines the class of the element. Listed below are the classes of elements in the MSC Nastran Implicit Nonlinear element library.

• Beam Elements - is a 3-D bar with axial, bending, and torsional stiffness.

• Shell Elements - is a curved, thin or thick structure with membrane/bending capabilities.

• Plate Elements - is a flat thin structure carrying in-plane and out-of-plane loads.

• Continuum Elements

• Plane stress is a thin plate with in-plane stresses only. All normal and shear stresses associated with the out-of-plane direction are assumed to be zero. (All plane strain elements lie in the global x-y plane.)

• Generalized plane strain is the same as plane strain except that the normal z-strain can be a prescribed constant or function of x and y.

• Axisymmetric elements are describe in 2D, but represent a full 3D structure where the geometry and loading are both axisymmetric.

• 3D solid is a solid structure with only translational degrees of freedom for each node (linear or quadratic interpolation functions).

• Truss Elements- is a 3D rod with axial stiffness only (no bending).

• Membrane Elements -is a thin sheet with in-plane stiffness only (no bending resistance).

• Gap Elements

• Points/Springs/Damper Elements

• Rigid Constraints

403CHAPTER 11Element Library

Number of Nodes

The number of nodes for an element define where the displacements are calculated in the analysis. Elements with only corner nodes are classified as first order elements and the calculation of displacements at locations within the element are made by linear interpolation. Elements that contain midside nodes are second order elements and quadratic interpolations are made for calculating displacements.

In MSC Nastran the number of nodes is designated at the end of the element name. For example, a CQUAD4 has 4 nodes.

Interpolation

Interpolation (shape) function is an assumed function relating the displacements at a point inside an element to the displacements at the nodes of an element. In MSC Nastran, three types of shape functions are used: linear, quadratic, and cubic. Certain types of enhancements, such as Assumed Strain, shape functions, may increase the elements ability to capture accurately certain types of deformation states.

Degrees of Freedom

Degrees of freedom is the number of unknowns at a node. In the general case, there are six degrees of freedom at a node in structural analysis (three translations, three rotations). In special cases, the number of degrees of freedom is two (translations) for plane stress, plane strain, and axisymmetric elements; three (translations) for 3-D truss element; six (three translations, three rotations) for a 3-D beam element).

Integration

Numerical integration is a method used for evaluating integrals over an element. Element quantities – such as stresses, strains, and temperatures – are calculated at each integration point of the element. Full integration (quadrature) requires, for every element, 2d integration points for linear interpolation and 3d points for quadratic interpolation, where scalar “d” is the number of geometric dimensions of an element (that is, d = 2 for a quad; d = 3 for a hexahedron). Reduced integration uses a lower number of integration points than necessary to integrate exactly. For example, for an 8-node quadrilateral, the number of integration points is reduced from 9 to 4 and, for a 20-node hexahedron, from 27 to 8. For some elements, an “hourglass” control method is used to insure an accurate solution.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideElement Selection

404

Element SelectionThe MSC Nastran Implicit Nonlinear element library provides a complete finite element modeling capability. Selecting elements to use for a SOL 600 analysis can be very different than elements you might use for other solution sequences. In particular, the large strains encountered in nonlinear analysis have implications for the element formulations.

SOL 600 selects appropriate element formulations based on the analysis type, the elements you use in constructing your model, element properties, and global parameters. While most of the element formulation decisions made by SOL 600 require no additional user input, some of the formulation principles are discussed in this section. Certain analyses may require element types other than the default which can be specified using

Element InterpolationAll of the elements in MSC Nastran Implicit Nonlinear are formulated in “element” coordinate systems described in the Marc Volume B: Element Library. For almost all elements, primary vector quantities (such as displacements) are defined in terms of nodal values with scalar interpolation functions. When the interpolation function for these vector quantities are the same interpolation functions used to define the geometry (i.e., the position vector) the elements are called isoparametric. Such elements are guaranteed to be able to exactly represent all rigid body modes and homogeneous deformation modes, a necessary condition for convergence to the exact solution as the mesh is refined (i.e. the patch test).

Element IntegrationAll elements are integrated numerically. MSC Nastran Implicit Nonlinear normally uses “full” integration elements but “reduced” integration elements are also available. For full integration, the number of integration points is sufficient to integrate the governing virtual work expression exactly, at least for linear material behavior. All triangular and tetrahedral elements in MSC Nastran Implicit Nonlinear use full integration. Reduced integration can be used for quadrilaterals and hexahedral elements; in this procedure, the number of integration points is sufficient to exactly integrate the contributions of the strain field that are one order less then the order of the interpolation. The (incomplete) higher-order contributions to the strain field present in these elements will not be integrated.

Hourglassing

The advantage of the reduced integration elements is that the strains and stresses are calculated at the location that provide optimal accuracy, the so-called Barlow points. The reduced integration elements also tend to underestimate the stiffness of the element which often gives better results in a typically overly-stiff finite element analysis displacement method. An additional advantage is that the reduced number of integration points decreases CPU time and storage requirements. The disadvantage is that the reduced integration procedure may admit deformation modes that cause no straining at the integration points. These zero-energy modes cause a phenomenon called “hourglassing,” where the zero energy mode starts propagating through the mesh, leading to inaccurate solutions. This problem is particularly severe in first-order quadrilaterals and hexahedrals. To prevent these excessive deformations, an


additional artificial stiffness is added to these elements. In this so-called hourglass control procedure, a small artificial stiffness is associated with the zero-energy modes. This procedure is used in many of the alternate solid and shell elements in MSC Nastran Implicit Nonlinear All primary elements translated from the standard MSC Nastran finite elements are fully integrated. Reduced integration elements may be selected using PARAM,MRALIAS ID (MALIAS02, MALIAS03, etc.)..

Figure 11-1 Hourglassing

Incompressible ElementsSometimes fully integrated solid elements are unsuitable for the analysis of (approximately) incompressible material behavior. The reason for this is that the material behavior forces the material to deform (approximately) without volume changes. Fully integrated solid element meshes, and in particular lower-order element meshes, do not allow such deformations (other than purely homogeneous deformation). For fully incompressible behavior, another complication occurs: the bulk modulus and hence the stiffness matrix becomes infinitely large. For this case, a mixed (Herrmann) formulation is required, where the displacement field is augmented with a hydrostatic pressure field. In this formulation, only the inverse of the bulk modulus appears, and consequently the contribution of the operator matrix vanishes. In this formulation, the hydrostatic pressure field play the role of a Lagrange multiplier enforcing the incompressibility constraints.

Overriding MSC Nastran Element SelectionsIn most cases MSC Nastran selects an equivalent Marc element to use in the analysis. The selection is based on the large number of analysis correlations between SOL 600 and SOL 100, SOL 106, or 129 results. In some cases an experienced Marc user may want to use alternate elements and may do so by adding a Parameter to the MSC.nastran Input File as follows.

PARAM,MRALIAS ID (MALIAS02, MALIAS03, etc.)

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideGlobal Element Controls

406

Global Element Controls

Assumed Strain Conventional isoparametric four-node plane stress and plane strain, and eight-node brick elements behave poorly in bending. The reason is that these elements do not capture a linear variation in shear strain which is present in bending when a single element is used in the bending direction. As a default in MSC Nastran Implicit Nonlinear, the element interpolation functions have been modified such that shear strain variation can be better represented. For elastic isotropic bending problems, this allows the exact displacements to be obtained with only a single element through the thickness. Use PARAM, MARCASUM, 1 to activated this option.

Constant DilatationWhen performing nearly incompressible analysis with displacement based elements, the conventional isoparametric interpolation methods result in poor behavior for lower order elements. To address this case, an integration scheme option is included (default) which makes the dilatational strain constant throughout the element. Constant dilatational element formulation is preferred in approximately incompressible, inelastic analysis, such as large strain plasticity, because conventional elements can produce volumetric locking due to overconstraints for nearly incompressible behavior. This option is also the formulation of choice for elastic-plastic analysis and creep analysis because of the potentially nearly incompressible behavior. Use PARAM, MARCDILT, 1 to activate this option.

Setting Global Element Parameters in PatranAssumed strain, constant dilatation, plane stress, and reduced integration options, are turned ON and OFF for the entire model.

1. On the Analysis Application form, select Solution Type..., then click Solution Parameters...

2. Use the check boxes to turn OFF and ON Assumed Strain and Constant Dilatation.


Mass Elements, Springs, Dampers, and Connector ElementsThe following 0-D and 1-D special purpose elements allow you to model very specific types of behavior.

PBUSHT support has been added for nonlinear springs. TABLED1 can be used to specify the load-deflection curve. The PBUSHT/TBLED1 data is mapped to Marc’s SPRINGS option with table-driven force-deflection curves.

The connector elements (CBUSH,CFAST, and CWELD) in SOL 600 are truly nonlinear and support large deformation and rotation, while in the rest of Nastran they are linear, thus the results may differ.

Bolt elements were added to support Marc both outside the USA and within the USA. MSC Nastran Bulk Data entries, MBOLT and MBOLTUS reflect these additions.

Patran FE Application Input DataThese elements are generated in Patran using the following Object/Type combination on the Element Properties Application form.

Entry Description

CMASS1i Defines scalar mass elements.

CONM1, CONM2

Defines a concentrated mass at a grid point.

CELAS1 Defines a scalar spring element.

CDAMP1i Defines a scalar damper element.

CVISC Defines a viscous damper element.

CBUSH Defines a generalized spring-and-damper structural element that may be nonlinear or frequency dependent.

CBUSH1D Defines the connectivity of a one-dimensional spring and viscous damper element.

CFAST Defines a fastener with material orientation connecting two surface patches.

CWELD Defines a weld or fastener connecting two surface patches or points.

Object Type Options Bulk Data Entries

0D Mass Coupled, Grounded, Lumped CONM1, CONM2

1D Mass CMASSi

0D Grounded Spring CELAS1

1D Spring CELAS1

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0D Grounded Damper

1D Damper Scalar, Viscous CDAMPi, CVISC

OD Grounded Bush CBUSH

1D Bush CBUSH1D



Gap ElementsGap elements are an obsolete method to model contact problems. These elements allow for gap opening and closing as well as friction along a surface. Gap elements should normally be replaced with 2D or 3D contact. Gaps for SOL 600 are only offered for compatibility with other MSC Nastran solution sequences and cannot be simulated exactly the same in SOL 600 as in the other solution sequences. There use is not recommended.

Patran FE Application Input DataThese elements are generated in Patran using the following Object/Type combination on the Element Properties Application form.

Entry Description

CGAP Defines a gap or friction element.


1D Gap Adaptive, Nonadaptive CGAP

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Line ElementsMSC Nastran offers a wide variety of line elements that include beams, bars, rods. Beams can be defined using standard cross sections supplied in a library or general cross sections that are user defined.

Patran FE Application Input DataLine elements are generated in Patran using the following Object/Type combination on the Element Properties Application form.

Entry Description

CBAR Defines a simple beam element.

CBEAM Defines a beam element.

CBEND Defines a curved beam, curved pipe, or elbow element.

CROD Defines a tension-compression-torsion element.

CONROD Defines a rod element without reference to a property entry.

CTUBE Defines a tension-compression-torsion tube element.


1D Beam General Section CBAR

Curved w/ General Section CBEND

Curved w/Pipe Section CBEND

Lumped Section CBEAM/PBCOMP

Tapered Section CBEAM

General Section (CBEAM) CBEAM

1D Rod General Section CROD/CONROD

Pipe Section CTUBE


Membranes, Panels, and ShellsMSC Nastran includes standard triangular and quadrilateral elements as well as a special purpose shear element. By specifying element properties these standard elements can represent anticipated membrane, bending, and shearing responses.

PSHELL with the same bending-membrane coupling as in other solution sequences has been added into the 2005 r3 release. This will allow you to analyze composite structures using the smeared approach (such as done in other MSC Nastran solution sequences) or through-the-thickness integration (which is more accurate and presently the only way to analyze composite structures) will be offered. If material nonlinearity occurs in the element, then the PSHELL smeared approach should not be used. The choice is activated by using PARAM,MRPSHELL,1.

Patran FE Application Input DataThese 2-D elements are generated in Patran using the following Object/Type combination on the Element Properties Application form.

Entry Description

CTRIA3 Defines an isoparametric membrane-bending or plane strain triangular plate element.

CTRIA6 Defines a curved triangular shell element or plane strain with six grid points.

CQUAD4 Defines an isoparametric membrane-bending or plane strain quadrilateral plate element.

CQUAD8 Defines a curved quadrilateral shell or plane strain element with eight grid points.

CTRIAR Defines an isoparametric membrane-bending triangular plate element.

CQUADR Defines an isoparametric membrane and bending quadrilateral plate element.

CSHEAR Defines a shear panel element.

Object Type Option 1 Option 2 Bulk Data Entries

2-D Shell Homogeneous Standard CTRIA3, CTRIA6, CQUAD4, CQUAD8

w/PSHELL

Revised CTRIAR, CQUADR

w/PSHELL

Laminate Standard CTRIA3, CTRIA6, CQUAD4, CQUAD8

w/PCOMP

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w/PCOMP

Equlvalent Section Standard CTRIA3, CTRIA6, CQUAD4, CQUAD8

w/PSHELL


w/PSHELL

2-D Bending Panel Standard CTRIA3, CTRIA6, CQUAD4, CQUAD8

w/PSHELL


w/PSHELL

2-D Membrane Standard CTRIA3, CTRIA6, CQUAD4, CQUAD8

w/PSHELL


w/PSHELL

2-D Shear Panel CSHEAR w/PSHEAR

Note: For SOL 600 CQUADR and CQUAD4 are the same and CTRIAR is the same as CTRIA3.

Object Type Option 1 Option 2 Bulk Data Entries


Solid ElementsMSC Nastran Advanced Nonlinear (SOL 600) contains continuum elements that can be used to model plane stress, plane strain, generalized plane strain, axisymmetric and three-dimensional solids. These elements have only displacement degrees of freedom. As a result, solid elements are not efficient for modeling thin structures dominated by bending. Either beam or shell elements should be used in these cases. The CHEXA and CPENTA elemments can be converted to solid shells by using the CSSHLH or CSSHLP bulk data entries.

The solid elements that are available in MSC Nastran Implicit Nonlinear have either linear or quadratic interpolation functions.

They include

4-, 6-, and 8-node plane stress elements (create plane stress elements using PARAM, MRALIAS ID (MALIAS02, MALIAS03, etc.))

3-, 4-, 6-, and 8-node plane strain elements

3-, 4-, 6-, and 8-node axisymmetric ring elements

8-, 10-, and 20-node brick elements

4- and 10-node tetrahedron

6 and 8 node solid shell

In general, the elements in MSC Nastran Implicit Nonlinear use a full-integration procedure. Some elements use reduced integration. The lower-order reduced integration elements include an hourglass stabilization procedure to eliminate the singular modes.

Continuum elements are widely used for thermal stress analysis. For each of these elements, there is a corresponding element available for heat transfer analysis in MSC Nastran Implicit Nonlinear. As a result, you can use the same mesh for the heat transfer and thermal stress analyses.

MSC Nastran Implicit Nonlinear has no singular element for fracture mechanics analysis. The simulation of stress singularities can be accomplished by moving the midside nodes of 8-node quadrilateral and 20-node brick elements to quarter-point locations near the crack tip. Many fracture mechanics analyses have used this quarter-point technique successfully.

The 4- and 8-node quadrilateral elements can be degenerated into triangles, and the 8-and 20-node solid brick elements can be degenerated into wedges and tetrahedra by collapsing the appropriate corner and midside nodes. The number of nodes per element is not reduced for degenerated elements. The same node number is used repeatedly for collapsed sides or faces. When degenerating incompressible elements, exercise caution to ensure that a proper number of Lagrange multipliers remain. You are advised to use the higher-order triangular or tetrahedron elements wherever possible, as opposed to using collapsed quadrilaterals and hexahedra.

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Axisymmetric ElementsSolids of revolution (axisymmetric solids) subjected to axially symmetric loading can be modeled using the elements described in this section. For these problems, the coordinate convention is a cylindrical system. Because of symmetry, the stress components are independent of the angular coordinate so the components uq, grq, gqz, trq, and tqz are zero. The nonzero components are sr, sq, sz, and trz.

Patran FE Application Input Data

Axisymmetric elements are generated in Patran using the following Object/Type combination on the Element Properties Application form.

Plane Strain ElementsPlane strain problems involve a long body whose geometry and loading do not vary significantly in the longitudinal direction. In these problems, the dependent variables can be assumed to be functions of only the x and y coordinates, provided we consider a cross section at some distance away from the ends. If it is further assumed that the displacement component in the z direction is zero at every cross section then the strain components z, yz, and zx vanish and the remaining non-zero strain components are x, y

and xy. Also, since z is assumed zero, the stress z can be expressed in terms of x and y for the linear

elastic case as:

(11-1)

Entry Description

CTRIAX6 Defines an isoparametric and axisymmetric triangular cross section ring element with midside grid points.

CTRIAX Defines an axisymmetric triangular element with up to 6 grid points.

CQUADX Defines an axisymmetric quadrilateral element with up to nine grid points.


2-D 2-D Solid Axisymmetric

• Standard CTRIAX6,

• Hyperelastic CTRIAX, CQUADX

• PLPLANE CTRIAX, CQUADX

z x y+ =



These elements are generated in Patran using the following Object/Type combination on the Element Properties Application form.

3-D Solid ElementsThese are all isoparametric solid elements. These elements may have either a homogeneous material definition, meaning that the element is made of a single material, or a laminated material definition.

Entry Description

CTRIA3 Defines an isoparametric membrane-bending or plane strain triangular plate element.

CTRIA6 Defines a curved triangular shell element or plane strain with six grid points.

CQUAD4 Defines an isoparametric membrane-bending or plane strain quadrilateral plate element.

CQUAD8 Defines a curved quadrilateral shell or plane strain element with eight grid points.

CTRIAR Defines an isoparametric membrane-bending triangular plate element.

CQUADR Defines an isoparametric membrane and bending quadrilateral plate element.


2-D 2-D Solid Plane Strain

• Standard CTRIA3, CTRIA6, CQUAD4, CQUAD8

w/PSHELL

• Revised CTRIAR, CQUADR

w/PSHELL

• Hyperelastic CTRIA3, CTRIA6, CQUAD4, CQUAD8

w/PLPLANE

Entry Description

CHEXA Defines the connections of the six-sided solid element with eight to twenty grid points.

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These elements are generated in Patran using the following Object/Type combination on the Element Properties Application form.

CTETRA Defines the connections of the four-sided solid element with four to ten grid points.

CPENTA Defines the connections of a five-sided solid element with six to fifteen grid points.

CSSHL Defines a connection for a Solid Shell with 6 or 8 grid points in SOL 600 only.

Object Type Options Options Bulk Data Entries

3-D Solid • Homogeneous • Standard CHEXA, CTETRA, CPENTA w/PSOLIDCSSHL w/PSSHL

• Hyperelastic w/PLSOLID

• Laminate CHEXA, CTETRA, CPENTA w/PCOMPor PCOMPG

CSSHL w/PCOMP or PCOMPG

Entry Description


Beam/Bar and Shell OffsetsThere are two methods to model beam and shell offsets. The choice is determined using bulk data parameter, PARAM,MAROFSET with values of zero or 1. Method Zero uses extra grid points and RBE2 elements and method 1 (default) does not. Method 1 can be used with large deformation analysis.

Pin flags have been incorporated within the SOL 600 translator (they are not yet available directly in Marc). Pin flags are simulated by adding extra grids at the same location and connecting them with MPC’s. Pin flags and offsets cannot both be defined for a particular end of a beam, and if entered will generate a error.

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Chapter 12: Contact

12 Contact

Overview

Contact Methodology

Defining Contact Bodies

Selecting and Controlling Contact Behavior

Heat Transfer and Thermal Contact

References


420

OverviewThe simulation of many physical problems requires the ability to model the contact phenomena. This includes analysis of interference fits, rubber seals, tires, crash, and manufacturing processes among others. The analysis of contact behavior is complex because of the requirement to accurately track the motion of multiple geometric bodies, and the motion due to the interaction of these bodies after contact occurs. This includes representing the friction between surfaces and heat transfer between the bodies if required. The numerical objective is to detect the motion of the bodies, apply a constraint to avoid penetration, and apply appropriate boundary conditions to simulate the frictional behavior and heat transfer. Several procedures have been developed to treat these problems including the use of Perturbed or Augmented Lagrangian methods, penalty methods, and direct constraints. Furthermore, contact simulation has often required the use of special contact or gap elements. MSC Nastran Implicit Nonlinear allows contact analysis to be performed automatically without the use of special contact elements. A robust numerical procedure to simulate these complex physical problems has been implemented in MSC Nastran Implicit Nonlinear.

Contact problems can be classified as one of the following types of contact.

• Deformable-Deformable contact between two- and three-dimensional deformable bodies.

• Rigid - Deformable contact between a deformable body and a rigid body, for two- or three-dimensional cases.

• Glued contact in two and three dimensions. This is a general capability for tying (bonding) two deformable bodies, or a deformable body and a rigid body, to each other.

Contact problems involve a variety of different geometric and kinematic situations. Some contact problems involve small relative sliding between the contacting surfaces, while others involve large sliding. Some contact problems involve contact over large areas, while others involve contact between discrete points. The general Contact Body approach adopted by MSC Nastran Implicit Nonlinear to model contact can be used to handle most contact problem definitions.

The contact body approach provides two formulations for modeling the interaction between surfaces of structures. One formulation is a small-sliding formulation, in which the surfaces can only undergo small sliding relative to each other, but may undergo arbitrary rotation. An example of this type of application is the classical Hertz contact problem. The second formulation is a large-sliding formulation, where separation and sliding of large amplitude, and arbitrary rotation of the surfaces, may arise. An example is the modeling of a rubber tire rolling on the ground. Currently, the contact pair approach does not support large-sliding contact between two three-dimensional deformable surfaces.

A special case of the small-sliding formulation is glued contact, in which the surfaces are unable to penetrate each other, separate from each other, or slide relative to each other. This feature is useful for mesh refinement purposes.

421CHAPTER 12Contact

Contact MethodologyThis section describes how contact is implemented in MSC Nastran Implicit Nonlinear (SOL 600).

Contact BodiesThere are two types of contact bodies in MSC Nastran Implicit Nonlinear – deformable and rigid. Deformable bodies are simply a collection of finite elements as shown below.

Figure 12-1 Deformable Body

This body has three key aspects to it:

1. The elements which make up the body.

2. The nodes on the external surfaces which might contact another body or itself. These nodes are treated as potential contact nodes.

3. The edges (2-D) or faces (3-D) which describe the outer surface which a node on another body (or the same body) might contact. These edges/faces are treated as potential contact segments.

Note that a body can be multiply connected (have holes in itself). It is also possible for a body to be composed of both triangular elements and quadrilateral elements in 2-D or tetrahedral elements and brick elements in 3-D. Beam elements and shells are also available for contact.

Each node and element should be in, at most, one body. The elements in a body are defined using the BCBODY option. It is not necessary to identify the nodes on the exterior surfaces as this is done automatically. The algorithm used is based on the fact that nodes on the boundary are on element edges or faces that belong to only one element. Each node on the exterior surface is treated as a potential contact node. In many problems, it is known that certain nodes never come into contact; in such cases, the BCHANGE option can be used to identify the relevant nodes. As all nodes on free surfaces are considered contact nodes, if there is an error in the mesh generation such that internal holes or slits exist, undesirable results can occur.

The potential segments composed of edges or faces are treated in potentially two ways. The default is that they are considered as piece-wise linear (PWL). As an alternative, a cubic spline (2-D) or a Coons

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surface (3-D) can be placed through them. The BCBODY option is used to activate this procedure. This improves the accuracy of the calculation of the normal.

Rigid bodies are composed of curves (2-D) or surfaces (3-D) or meshes with only thermal elements in coupled problems. The most significant aspect of rigid bodies is that they do not distort. Deformable bodies can contact rigid bodies, but contact between rigid bodies is not considered.

They can be created either in CAD systems and transferred through Patran or some other GUI into MSC Nastran Implicit Nonlinear, created within Patran, or created directly through the MSC Nastran Implicit Nonlinear input. There are several different types of curves and surfaces that can be entered including:

Within Patran, all contact curves or surfaces are mathematically treated as NURB surfaces. This allows the greatest level of generality. Within the analysis, these rigid surfaces can be treated in two ways – discrete piecewise linear lines (2-D) or patches (3-D), or as analytical NURB surfaces. When the discrete approach is used, all geometric primitives are subdivided into straight segments or flat patches. You have control over the density of these subdivisions to approximate a curved surface within a desired degree of accuracy. This subdivision is also relevant when determining the corner conditions ( see Corner Conditions, 433). The treatment of the rigid bodies as NURB surfaces is advantageous because it leads to greater accuracy in the representation of the geometry and a more accurate calculation of the surface normal. Additionally, the variation of the surface normal is continuous over the body which leads to a better calculation of the friction behavior and a better convergence.

To create a rigid body, you can either read in the curve and surface geometry created from a CAD system or create the geometry in Patran, or directly enter it into the MSC Nastran Implicit Nonlinear Bulk Data. You then use the BCBODY option to select which geometric entities are to be a part of the rigid body. An important consideration for a rigid body is the definition of the interior side and the exterior side. For two-dimensional analysis, the interior side is formed by the right-hand rule when moving along the body.

2-D 3-D

line 4-node patch

circular arc ruled surface

spline surface of revolution

NURB Bezier

poly-surface

cylinder

sphere

NURB

trimmed NURB


Figure 12-2 Orientation of Rigid Body Segments

For three-dimensional analysis, the interior side is formed by the right-hand rule along a patch. The interior side is visualized in Patran as the side with markers, the exterior side is visualized in Patran as the side without markers.

It is not necessary for rigid bodies to define the complete body. Only the bounding surface needs to be specified. You should take care, however, that the deforming body cannot slide out of the boundary curve in 2D (Figure 12-3). This means that it must always be possible to decompose the displacement increment into a component normal and a component tangential to the rigid surface.

Figure 12-3 Deformable Surface Sliding Out of Rigid Surface

Numbering of Contact Bodies

When defining contact bodies for a deformable-to-deformable analysis, it is important to define them in the proper order. As a general rule, a body with a finer mesh should be defined before a body with a coarser mesh..

If one has defined a body numbering which violates the general rule, then a BCTABLE definition option can be used to modify the order in which contact will be established. This order can be directly user-defined or decided by the program. In the latter case, the order is based on the rule that if two deformable bodies might come into contact, searching is done for nodes of the body having the smallest element edge length. It should be noted that this implies single-sided contact for this body combination, as opposed to the default double-sided contact.

123

Interior1

4

3

2

Interior Side

Side

Incorrect Correct


424

Contact Detection

During the incremental procedure, each potential contact node is first checked to see whether it is near a contact segment. The contact segments are either edges of other 2-D deformable bodies, faces of 3-D deformable bodies, or segments from rigid bodies. By default, each node could contact any other segment including segments on the body that it belongs to. This allows a body to contact itself. To simplify the computation, it is possible to use the BCTABLE entry to indicate that a particular body will or will not contact another body. This is often used to indicate that a body will not contact itself. During the iteration process, the motion of the node is checked to see whether it has penetrated a surface by determining whether it has crossed a segment.

Because there can be a large number of nodes and segments, efficient algorithms have been developed to expedite this process. A bounding box algorithm is used so that it is quickly determined whether a node is near a segment. If the node falls within the bounding box, more sophisticated techniques are used to determine the exact status of the node.

During the contact process, it is unlikely that a node exactly contacts the surface. For this reason, a contact tolerance is associated with each surface.

Figure 12-4 Contact Tolerance

If a node is within the contact tolerance, it is considered to be in contact with the segment. The contact tolerance is calculated by the program as the smaller of 5% of the smallest element side or 25% of the smallest (beam or shell) element thickness. It is also possible for you to define the contact tolerance through the input.

During an increment, if node A moves from to , where is beyond the contact tolerance, the node is considered to have penetrated. In such a case, either the increment is divided into subincrements as discussed in Marc Volume A: Theory and User Information under the Mathematical Aspects of Contact” section or the increment is reduced in size.

2 x Tolerance

At

Atrial

t t+ Atrial

t t+


Figure 12-5 Trial Displacement with Penetration

The size of the contact tolerance has a significant impact on the computational costs and the accuracy of the solution. If the contact tolerance is too small, detection of contact is difficult, leading to higher costs. Also many nodes are more likely to be considered penetrating leading to increase in increment splitting, therefore, increasing the computational costs. If the contact tolerance is too large, nodes are considered in contact prematurely, resulting in a loss of accuracy. Furthermore, nodes might “penetrate” the surface by a large amount.

An effective compromise is to bias the tolerance area so that a smaller distance is on the outside surface than on the inside surface. This is done by entering a bias factor. The bias factor should have a value between 0.0 and 1.0. The default in MSC Nastran Implicit Nonlinear is 0.9. This results in good accuracy and reasonable computational costs. In analyses involving frictional contact, a bias (recommended value: 0.95 - 0.99) to the contact core is also found beneficial to facilitate convergence.

In some instances, you might wish to influence the decision regarding the deformable segment a node contacts (or does not contact). This can be done using the EXCLUDE variable on the BCBODY Bulk Data entry.

Figure 12-6 Biased Contact Tolerance

A(t)

Atrial (t + t)

(1 - Bias)* tolerance

(1 + Bias)* tolerance


426

Shell Contact

A node on a shell makes contact when the position of the node plus or minus half the thickness projected with the normal comes into contact with another segment. In 2-D, this can be shown as:

Figure 12-7 Default Shell Contact

If point x or y falls within the contact tolerance distance of segment S, node A is considered in contact with the segment S. Here and are the position vectors of a point on the surfaces 1 and 2 on the shell,

A is the position vector of a point (node in a discretized model) on the midsurface of the shell, is the

normal to the midsurface, and is the shell thickness.

As the shell has finite thickness, the node (depending on the direction of motion) can physically contact either the top surface, bottom surface, or mathematically contact can be based upon the midsurface. You can control whether detection occurs with either both surfaces, the top surface, the bottom surface, or the middle surface. In such cases, either two or one segment will be created at the appropriate physical location. Note that these segments will be dependent, not only on the motion of the shell, but also the current shell thickness.

are segments associated with shell consisting of node 1 and 2.

x1 A n t 2+=

x2 A n t 2–=

x

x2x tolerance A

S

ShellMidsurface

t

1

2

x1 x2

n

t

S1 S2


Figure 12-8 Selective Shell Contact

Neighbor Relations

When a node is in contact with a rigid surface, it tends to slide from one segment to another. In 2-D, the segments are always continuous and so are the segment numbers. Hence, a node in contact with segment

n slides to segment or to segment . This simplifies the implementation of contact.

Figure 12-9 Neighbor Relationship (2-D)

In 3-D, the segments are often discontinuous. This can be due to the subdivision of matching surfaces or, more likely, the CAD definition of the under lying surface geometry.

n

1

2

S1

S2

1

n

2

S1

S2

2

1

n

1

S12

Include Both Segments Top Segment Only

Bottom Segments Only Ignore Shell Thickness

n 1– n 1+

n - 1n

n + 1


428

Figure 12-10 Neighbor Relationship (3-D)

Continuous surface geometry is highly advantageous as a node can slide from one segment to the next with no interference (assuming the corner conditions are satisfied). Discontinuous surface geometry results in additional operations when a node slides off a patch and cannot find an adjacent segment. Hence, it is advantageous to use geometry clean-up tools to eliminate small sliver surfaces and make the surfaces both physically continuous and topologically contiguous.

Dynamic Impact

The Newmark-beta and the Single Step Houbolt procedure have the capability to allow variable time steps and, when using the user-defined fixed time step procedure, the time step is split by the algorithm to satisfy the contact conditions.

For most dynamic impact problems, the Single Step Houbolt method is recommended, as this procedure possesses high-frequency dissipation. This is often necessary to avoid numerical problems by contact-induced high-frequency oscillations. If the other dynamic operators are used, it is recommended that numerical damping be used during the analysis.

In dynamic analysis, the requirement of energy conservation is supplemented with the requirement of momentum conservation. In addition to the constraints placed upon the displacements, additional constraints are placed on the velocity and acceleration of the nodal points in contact, except for the Single Step Houbolt method.

When a node contacts a rigid surface, it is given the velocity and acceleration of the rigid surface in the normal direction. The rigid surfaces are treated as if they have infinite mass, hence, infinite momentum.

Nonmatching Segments

Continuous Surface Segments Discontinuous Surface Geometry


Results Evaluation

The MSC Nastran Implicit Nonlinear post files t16 and t19 contain the results for both the deformable bodies and the rigid bodies. In performing a contact analysis, you can obtain three types of results. The first is the conventional results from the deformable body. This includes the deformation, strains, stresses, and measures of inelastic behavior such as plastic and creep strains. In addition to reaction forces at conventional boundary conditions, you can obtain the contact forces and friction forces imparted on the body by rigid or other deformable bodies. By examining the location of these forces, you can observe where contact has occurred, and MSC Nastran Implicit Nonlinear also allows you to select the contact status.

It is also possible to obtain the resultant force following from contact on the deformable bodies and the resultant force and moment on the rigid bodies. The moment is taken about the user-defined centroid of the rigid body. The time history of these resultant forces are of significant issues in many engineering analysis. Of course, if there is no resultant force on a rigid body, it implies that body is not in contact with any deformable body.

Finally, if the additional print is requested using PARAM,MARCPRN,1 or 2, the output file reflects information showing when a node comes into contact, what rigid body/segment is contacted, when separation occurs, when a node contacts a sharp corner, the displacement in the local coordinate system, and the contact force in the local coordinate system. For large problems, this can result in a significant amount of output.

The motion of the rigid bodies can be displayed in Patran as well as the deformable bodies. Rigid bodies which are modeled using the piecewise linear approach are displayed as line segments for flat patches. When the rigid surfaces are modeled as analytical surfaces, the visualization appears as trimmed NURBS.

Tolerance Values

Five tolerances can be set for determination of the contact behavior. Not entering any values here means that MSC Nastran Implicit Nonlinear calculates values based on the problem specification.

Relative Sliding Velocity Between Surfaces Below Which Friction Forces Drop

As discussed in Friction Modeling, 438, the equations of friction are smoothed internally in the program to avoid numerical instabilities. The equations are inequalities whenever two contacting surfaces stick to each other and equalities whenever the surfaces slide (or slip). Thus, the character of contact constraints change depending on whether there is sticking or slipping. The smoothing procedure consist of modifying it in such a way, that there is always slip; the amount is a function of the relative velocity and a constant RVcnst. The value of this constant must be specified. It actually means, that if we specify a

small value in comparison to the relative velocity, the jump behavior is better approximated, but numerical instabilities can be expected. A large value means, that we need a large relative velocity before we get the force at which the slip occurs.

It is suggested to use values between 0.1 and 0.01 times a typical surface velocity.


430

Distance Below Which a Node is Considered Touching a Surface

In each step, it is checked whether a (new) node is in contact with other surfaces. This is determined by the distance between the nodes and the surfaces. Since the distance is a calculated number, there are always roundoff errors involved. Therefore, a contact tolerance is provided such that if the distance calculated is below this tolerance, a node is considered in contact. A too large value means that a high number of body nodes are considered to be in contact with the surface and are consequently all moved to the surface, which can be unrealistic in some applications. A too small value of this number means that the applied deformation increment is split into a high number of increments, thus increasing the cost of computation.

The tolerance must be provided by the analyst or can be calculated by MSC Nastran Implicit Nonlinear. In general, the contact tolerance should be a small number compared to the geometrical features of the configuration being analyzed. The value calculated by MSC Nastran Implicit Nonlinear is determined as 1/20 of the smallest element size for solid elements or 1/4 of the thickness of shell elements. If both shell and continuum elements are present, the default is based upon the smaller of the two values.

Tolerance on Nodal Reaction Force on Nodal Stress Before Separation Occurs

If a tensile force occurs at a node which is in contact with a surface, the node should separate from the surface. Rather than using any positive value, a threshold value can be specified. This number should theoretically be zero. However, because a small positive reaction might be due only to errors in equilibrium, this threshold value avoids unnecessary separations. A too small value of this force results in alternating separation and contact between the node and the surface. A too large value, of course, results in unrealistic contact behavior.

MSC Nastran Implicit Nonlinear calculates this value as the maximum residual force in the structure. The default for this value, is 10 percent of the maximum reaction force. Consequently, if locally high reaction forces at a particular point are present, the separation force is large as well. In most cases, however, the default value is a good measure.

If you indicate that separation is to be based upon stresses, a value of the separation stress is used. The default value is the maximum residual force at node n divided by the contact area of node n.

Numerical Procedures

Lagrange Multipliers

In performing contact analyses, you are solving a constrained minimization problem where the constraint is the ‘no penetration’ constraint. The Lagrange multiplier technique is the most elegant procedure to apply mathematical constraints to a system. Using this procedure, if the constraints are properly written, overclosure or penetration does not occur. Unfortunately, Lagrange multipliers lead to numerical difficulties with the computational procedure as their inclusion results in a nonpositive definite mathematical system. This requires additional operations to insure an accurate, stable solution which leads to high computational costs. Another problem with this method is that there is no mass associated with the Lagrange multiplier degree of freedom. This results in a global mass matrix which cannot be decomposed. This precludes the used of Lagrange multiplier techniques in explicit dynamic calculations


which are often used in crash simulations. The Lagrange multiplier technique has often been implemented in contact procedures using special interface elements such as the MSC Nastran Implicit Nonlinear gap element. This facilitates the correct numerical procedure, but puts a restriction on the amount of relative motion that can occur between bodies. The use of interface elements requires an apriori knowledge of where contact occurs. This is unachievable in many physical problems such as crash analysis or manufacturing simulation.

Penalty Methods

The penalty method or its extension, the Augmented Lagrangian method, is an alternative procedure to numerically implement the contact constraints. Effectively, the penalty procedure constrains the motion by applying a penalty to the amount of penetration that occurs. The penalty approach can be considered as analogous to a nonlinear spring between the two bodies. Using the penalty approach, some penetration occurs with the amount being determined by the penalty constant or function. The choice of the penalty value can also have a detrimental effect on the numerical stability of the global solution procedure. The penalty method is relatively easy to implement and has been extensively used in explicit dynamic analysis although it can result in an overly stiff system for deformable-to-deformable contact since the contact pressure is assumed to be proportional to the pointwise penetration. The pressure distribution is generally oscillatory.

Hybrid and Mixed Methods

In the hybrid method, the contact element is derived from a complementary energy principle by introducing the continuity on the contact surface as a constraint and treating the contact forces as additional elements. Mixed methods, based on perturbed Lagrange formulation, usually consist of pressure distribution interpolation which is an order less than the displacement field, have also been used to alleviate the difficulties associated with the pure Lagrange method.

Direct Constraints

Another method for the solution of contact problems is the direct constraint method. In this procedure, the motion of the bodies is tracked, and when contact occurs, direct constraints are placed on the motion using boundary conditions – both kinematic constraints (MPC and SPC) on transformed degrees of freedom and nodal forces. This procedure can be very accurate if the program can predict when contact occurs. This is the procedure that is implemented in MSC Nastran Implicit Nonlinear through the BCBODY option. No special interference elements are required in this procedure and complex changing contact conditions can be simulated since no apriori knowledge of where contact occurs is necessary.

Mathematical Aspects of Contact

Please refer to Marc Volume A, Theory and User Information, section 8 for the complete description.

Automatic Penetration Checking Procedure

To detect contact between bodies whose boundaries are moving towards each other, an automatic penetration checking procedure is available. This procedure significantly increases accuracy and stability for models in which boundary nodes are displacing significantly. Typical examples include


432

metal forming processes (sheet forming and forging), highly deformable elastomeric models (rubber boots), and snap-fit problems (inserting a key into a lock).

The automatic penetration checking procedure is automatically activated if the adaptive loading procedure is selected. If the automatic penetration checking procedure is selected for these two options, a different procedure, as described below, is used instead.

From a computational perspective, the automatic penetration checking procedure detects penetration each time displacements are updated.

For implicit analysis, this typically happens after a matrix solution which produces a change in the displacements due to a change in applied loads and internal forces. The procedure detects nodes traversing a contact boundary due to the change in displacements. If at least one node penetrates a contact surface, a scale factor is applied to the change in displacements such that the penetrating nodes are moved back to the contact surface.

The automatic penetration checking procedure can, therefore, be considered to be a type of a line search. The procedure also looks at the magnitude of the change in displacement of nodes which already are contacting and not necessarily penetrating. Using stability considerations, the scale factor calculated above may be further modified. In addition, for nodes on a contact boundary which are not yet contacting, a similar procedure is followed to enhance stability.

Because the procedure can reduce the change in displacements, it may require more iterations to complete an increment. It is important to ensure that the maximum allowable number of iterations to complete an increment is set to a sufficiently large value. When the adaptive loading procedure is used, or when the fixed time stepping procedure is used with automatic restarting, the increment automatically restarts if the maximum allowable number of iterations is exceeded. In the case of the adaptive loading procedure, the time step is modified.

When dynamics or the arc length control method is used, the above procedure is not available. Instead, penetration is checked for when convergence is achieved, usually after multiple iterations.

Contact Tolerance

A node comes into contact with another body when it enters the contact tolerance zone. This area is dependent upon the value of ERROR and BIAS entered on the BCPARA Bulk Data entry. When BIAS is zero (the default is 0.9), the tolerance is equidistant from the actual surface as shown in Figure 12-11(a); otherwise, the situation shown in Figure 12-11(b) is used. If a node would have moved past line B, then an additional iteration is required.

Note: The automatic penetration checking procedure is always used with the default time stepping procedures in MSC Nastran Implicit Nonlinear.


Figure 12-11 Contact Tolerances

Separation

A node on a body separates from another surface when a tensile load is required to keep it on the surface. The procedure used is either based upon the nodal force or an effective nodal stress. The default separation force is the maximum residual force (separation based upon nodal force method) or the maximum stress at reaction nodes times the convergence tolerance (nodal stress method).

Corner Conditions

When a node slides along a surface composed of multiple segments, three conditions can occur based on the angle that the segments make. This is true for both two-dimensional and three-dimensional problems. The Figure 12-12 shows the two-dimensional case for simplicity. If the angle between the two segments is between 180 - < < 180� + , the node smoothly slides between the segments. If the angle is such that 0 < < 180 - , the node sticks in the sharp concave corner. If the angle is such that > 180 + , the node separates. The value of is 8.625� for two-dimensional problems and 20� for three-dimensional problems.

Figure 12-12 Corner Conditions

ERROR

ERROR

B

ERROR*(1-BIAS)

ERROR*(1+BIAS)

(a) Equidistant Default (b) Biased

Smooth Sharp Concave

Sharp Convex


434

Implementation of ConstraintsFor contact between a deformable body and a rigid surface, the constraint associated with no penetration is implemented by transforming the degrees of freedom of the contact node and applying a boundary condition to the normal displacement. This can be considered solving the problem:

where represents the nodes in contact which have a local transformation, and represents the nodes not in contact and, hence, not transformed. Of the nodes transformed, the displacement in the normal direction is then constrained such that is equal to the incremental normal displacement of the rigid

body at the contact point.

Figure 12-13 Transformed System (2-D)

As a rigid body can be represented as either a piecewise linear or as an analytical (NURB) surface, two procedures are used. For piecewise linear representations, the normal is constant until node P comes to the corner of two segments as shown in Figure 12-14. During the iteration process, one of three

circumstances occur. If the angle is small , the node P slides to the next segment.

In such a case, the normal is updated based upon the new segment. If the angle is large ( or

) the node separates from the surface if it is a convex corner, or sticks if it is a concave corner.

The value of is important in controlling the computational costs. A larger value of

reduces the computational costs, but might lead to inaccuracies. The default values are 8.625� for 2-D and 20� for 3-D. These can be reset using the ANG2D or ANG3D fields on the NLSTRAT entry for each subcase.

Kaa Kab

Kba Kbb

ua

ub fa

fb

=

a b

uan

n

tP

smooth smooth –

smooth

smooth–

smooth smooth


Figure 12-14 Corner Conditions (2-D)

In 3-D, these corner conditions are more complex. A node (P) on patch A slides freely until it reaches the intersection between the segments. If it is concave, the node first tries to slide along the line of intersection before moving to segment B. This is the natural (lower energy state) of motion.

These corner conditions also exist for deformable-to-deformable contact analysis. Because the bodies are continuously changing in shape, the corner conditions (sharp convex, smooth or sharp concave) are continuously being re-evaluated.

When a rigid body is represented as an analytical surface, the normal is recalculated at each iteration based upon the current position. This leads to a more accurate solution, but can be more costly because of the NURB evaluation.

Figure 12-15 Corner Conditions (3-D)

When a node of a deformable body contacts a deformable body, a multipoint constraint (MPC) is automatically imposed. Recalling that the exterior edges (2-D) or faces (3-D) of the other deformable bodies are known, a constraint expression is formed. For 2-D analysis, the number of retained nodes is three – two from the edge and the contacting node itself. For 3-D analysis, the number of retained nodes is five – four from the patch and the contacting node itself. The constraint equation is such that the contacting node should be able to slide on the contacted segment, subject to the current friction conditions. This leads to a nonhomogeneous, nonlinear constraint equation. In this way, a contacting node is forced to be on the contacted segment. This might introduce undesired stress changes, since a small gap or overlap between the node and the contacted segment will be closed. During initial detection of contact (increment 0), the stress-free projection option avoids those stress changes for deformable contact by adapting the coordinates of the contacting nodes such that they are positioned on the contacted

Convex Corner Concave Corner

P

A

B

P

P


436

segment. This stress-free projection can be activated using the BCTABLE entry. A similar option exists for glued contact; however, in this case, overlap will not be removed.

During the iteration procedure, a node can slide from one segment to another, changing the retained nodes associated with the constraint. A recalculation of the bandwidth is automatically made. Because the bandwidth can radically change, the bandwidth optimization is also automatically performed.

A node is considered sliding off a contacted segment if is passes the end of the segment over a distance more than the contact tolerance. As mentioned earlier, the node separates from the contacted body if this happens at a convex corner. For deformable contact, this tangential tolerance at convex corners can be enlarged by using the delayed sliding off option activated via the BCTABLE Bulk Data entry.

SeparationAfter a node comes into contact with a surface, it is possible for it to separate in a subsequent iteration or increment. Mathematically, a node should separate when the reaction force between the node and surface becomes tensile or positive. Physically, you could consider that a node should separate when the tensile force or normal stress exceeds the surface tension. Rather than use an exact mathematical definition, you can enter the force or stress required to cause separation.

Separation can be based upon either the nodal forces or the nodal stresses. The use of the nodal stress method is recommended as the influence of element size is eliminated.

In many analysis, contact occurs but the contact forces are small; for example, laying a piece of paper on a desk. Because of the finite element procedure, this could result in numerical chattering. MSC Nastran Implicit Nonlinear has some additional contact control parameters that can be used to minimize this problem. As separation results in additional iterations (which leads to higher costs), the appropriate choice of parameters can be very beneficial.

When contact occurs, a reaction force associated with the node in contact balances the internal stress of the elements adjacent to this node. When separation occurs, this reaction force behaves as a residual force (as the force on a free node should be zero). This requires that the internal stresses in the deformable body be redistributed. Depending on the magnitude of the force, this might require several iterations.

You should note that in static analysis, if a deformable body is constrained only by other bodies (no explicit boundary conditions) and the body subsequently separates from all other bodies, it would then have rigid body motion. For static analysis, this would result in a singular or nonpositive definite system. This problem can be avoided by appropriate boundary conditions.

Release

A special case of separation is the intentional release of all nodes from a rigid body. This is often used in manufacturing analysis to simulate the removal of the workpiece from the tools. After the release occurs in such an analysis, there might be a large redistribution of the loads. It is possible to gradually reduce the residual force to zero, which improves the stability, and reduces the number of iterations required. The BCMOVE Bulk Data entry allows the release (separation) of all the nodes in contact with a particular surface at the beginning of the increment. The rigid body should be moved away using the BCMOVE Bulk


Data entry or deactivated using the BCTABLE entry to ensure that the nodes do not inadvertently recontact the surface they were released from.

Higher Order ElementsMSC Nastran Implicit Nonlinear allows contact with almost all of the available elements, but the use of certain elements has a consequence on the analysis procedure. Contact analysis can be performed with all of the structural continuum elements, either lower order or higher order, including those of the Herrmann (incompressible) formulation, except axisymmetric elements with twist. Friction modeling is available in all of these elements except the semi-infinite elements. Traditionally, higher order isoparametric shape functions have interpolation functions which lead to the equivalent nodal forces that oscillate between the corner and midside nodes. As this has a detrimental effect on both contact detection and determining contact separation, two procedures have been implemented to eliminate this problem.

1. On the exterior surfaces, the midside nodes are constrained (tied) to the corner nodes automatically. This effectively results in a linear variation of the displacement along this edge. Hence, the element does not behave as a full bi-quadratic (2-D) or tri-quadratic element (3-D). All elements in the interior of the body behave in the conventional higher-order manner. In many manufacturing and rubber analyses, the lower-order elements behave better than the higher-order elements because of their ability to represent the large distortion; hence, these lower-order elements are recommended.

2. (Default for parabolic elements, LINQUAD=1). This is a new method that has the added advantage of giving an accurate interface pressure distribution.

The constraints imposed on the nodal degrees of freedom are dependent upon the type of element.

1. When a node of a continuum element comes into contact, the translational degrees of freedom are constrained.

2. When a node of a shell element comes into contact, the translational degrees of freedom are constrained and no constraint is places on the rotational degrees of freedom. The exception to this is when a shell contacts a symmetry surface. In this case, the rotation about the element edge is also constrained.

3-D Beam and Shell ContactAdditionally, beams and shells contact is governed by the rules outlined below.

2-D Beams

All nodes on beams are potential contact nodes. Beam elements can be used in contact in two modes.

1. The two-dimensional beams can come into contact with rigid bodies composed of curves in the same x-y plane. The normal is based upon the normal of the rigid surface.


438

2. The two-dimensional beams can come into contact with deformable bodies either of continuum elements or other beam elements. As the beams are in two dimensions, they do not intersect one another.

3-D Beams

Three-dimensional beam elements can be used in contact in three modes.

1. The nodes of the beams can come into contact with rigid bodies composed of surfaces. The normal is based upon the normal of the rigid surface.

2. Nodes of the three-dimensional beams can also come into contact with the faces of three-dimensional continuum elements or shell elements.

3. The three-dimensional elastic beams can also contact other elastic beams. In this case, we can consider beams crossing one another. In such cases, the beams are automatically subdivided such that four beams are created. As the beams slide upon each other, they are adaptively changed in length.

Figure 12-16 Beam-to-Beam Contact

Shell Elements

All nodes on shell elements are potential contact nodes. As the midside nodes of shell elements are automatically tied, the high-order shell elements have no benefit. Shell elements can contact either rigid bodies, continuum elements, or other shell elements. Shell-shell contact involves a more complex analysis because it is necessary to determining which side of the shell contact occurs.

Friction ModelingThe regularized form of the Coulomb friction model can be written as:

Contact Occurs New Beams Created Adaptive Meshing of Sliding Beam

ft

2fn

-----------arctan

vr

RVCNST----------------------- =


is a nonlinear relation between the relative sliding velocity and the friction force. Implementation in MSC Nastran Implicit Nonlinear has been done using a nonlinear spring model. Noting that the behavior of a nonlinear spring, as shown in Figure 12-17, is given by the equation:

Figure 12-17 Spring Model

in which is the spring stiffness and , , , and are displacements and forces of points 1 and

2, the equivalent in terms of velocities is readily seen to read

Since is a nonlinear function of the relative velocity, the above equation is solved incrementally,

where within each increment a number of iterations may be necessary. For a typical iteration , the equation to be solved looks like

(12-1)

where and are used to update and by

(12-2)

Notice that and correspond to the beginning of the iteration. For deformable-rigid contact, it

is easy to see that

, (12-3)

since the motion of a rigid body (to which node 2 belongs) is exactly prescribed by you. In a static analysis, MSC Nastran Implicit Nonlinear provides no direct information about velocities, so they

have to be calculated from the displacement and time increments. Denoting a time increment by , we can write

K K–

K– K

u1

u2

F1

F2

=

u, F u, F

1 2

K u1 u2 F1 F2

K K–

K– K

v1

v2

F1 t

F2 t

=

K

i

Ki

Ki

–

Ki

– Ki

v1i

v2i

F1 ti

F2 ti

=

v1i

v2i

v1i

v2i

v1i

v1i 1–

v1i

+=

v2i

v2i 1–

= v2i

+

v1i 1–

v2i 1–

v2i

0=

t


440

, (12-4)

in which represents the correction of the incremental displacement for iteration like (see

also Equation (12-2)).

(12-5)

Substituting Equation (12-3) and Equation (12-6) into Equation (12-1) yields

(12-6)

For the first iteration of an increment, an improvement of Equation (12-6) can be achieved by taking into

account the velocity at the end of the previous increment. Then Equation (12-3) can be rewritten as

, (12-7)

so that Equation (12-6) can be modified like

(12-8)

For the subsequent iterations,

(12-9)

In Equation (12-9), denotes the relative velocity between the points 1 and 2 at the end of the previous

increment. It must be noted that the additional term in Equation (12-9) is especially important if the velocity of the rigid body differs much from the relative velocity. This is usually the case in rolling processes, when the roll has been modeled as a rigid body. For this reason, this improved friction model is called friction for rolling.

Friction is a complex physical phenomena that involves the characteristics of the surface such as surface roughness, temperature, normal stress, and relative velocity. The actual physics of friction continues to be a topic of research. Hence, the numerical modeling of the friction has been simplified to two idealistic models.

The most popular friction model is the Adhesive Friction or Coulomb Friction model. This model is used for most applications with the exception of bulk forming such as forging. The Coulomb model is:

where

v1i u1

i

t---------=

u1i

u1i 1–

i

u1i

u1i 1–

= u1i

+

1t-----K

iu1i

F1i

=

v1p

v11 u1

1

t---------- v1

p–=

1t-----K

1u11 F1

1K vr

pv2

p– –=

1t-----K

iu1i

F1i

=

vri

fr – n t


is the relative sliding velocity.

The Coulomb model is also often written with respect to forces

where

Quite often in contact problems, neutral lines develop. This means that along a contact surface, the material flows in one direction in part of the surface and in the opposite direction in another part of the surface. Such neutral lines are, in general, not known a priori.

For a given normal stress, the friction stress has a step function behavior based upon the value of

or .

Figure 12-18 Coulomb Friction Model

is the normal stress

is the tangential (friction) stress

is the friction coefficient

is the tangential vector in the direction of the relative velocity

is the tangential force

is the normal reaction

n

fr

t

tvr

vr

--------=

vr

ft fn t–

ft

fn

vr

u

vr

ft or fr

Stick

Slip


442

This discontinuity in the value of can result in numerical difficulties so a modified Coulomb friction

model is implemented:

Physically, the value of RVCNST is the value of the relative velocity when sliding occurs. The value of RVCNST is important in determining how closely the mathematic model represents the step function. A very large value of RVCNST results in a reduced value of the effective friction. A very small value results in poor convergence. It is recommended that the value of RVCNST be 1% or 10% of a typical relative sliding velocity, . Because of this smoothing procedure, a node in contact always has some slipping.

Besides the numerical reasons, this ‘ever slipping node’ model has a physical basis. Oden and Pires pointed out that for metals, there is an elasto-plastic deformation of asperities at the microscopic level (termed as ‘cold weld’) which leads to a nonlocal and nonlinear frictional contact behavior. The arctan representation of the friction model is a mathematical idealization of this nonlinear friction behavior.

When the Coulomb model is used with the stress based model, the integration point stresses are first extrapolated to the nodal points and then transformed so a direct component is normal to the contacted surface. The tangential stress is then evaluated and a consistent nodal force is calculated.

For shell elements, since a nodal force based Coulomb model is used:

Figure 12-19 Stick-slip Approximation ( )

fr

fr – n 2---arctan

vr

RVCNST----------------------- t

vr

n 0

ft fn2---arctan

vr

RVCNST----------------------- t–=

ft

1C = 0.01

C = 0.1C = 1

C = 10

C = 100

-10 r

-1

10

fn 1 C RVCNST= =


This nodal forced based model should not be used if a nonlinear friction coefficient is to be used, as this nonlinearity is, in general, dependent upon the stress, not the force. This model can also be used for continuum elements.

The Coulomb friction model can also be utilized as a true stick-slip model. In this procedure, a node completely sticks to a surface until the tangential force reaches the critical value . Also, to model the

differences in static versus dynamic friction coefficients, an overshoot parameter, , can be used.

The stick-slip model is always based upon the nodal forces. When using the stick-slip procedure, the program flow is:

Note that this procedure requires additional computations to determine if the stick-slip condition has converged. It requires that

fn

Initial Contact

ut 0

Assume SlippingMode

Assume StickingMode

Determine Solutionof Next Iteration

Remain in Slipping Mode if:

ft ut 0 and ut Remain in Sticking Mode if:

ft fn

Change to Sticking Mode if:

ft ut 0 and ut

or if ut

Change to Slipping Mode if:

ft fn

No Yes

1 eft

ftp

---- 1 e+ –


444

where is the tangential force in the previous iteration.

This additional testing on the convergence of the friction forces is not required when the smooth/continuous model is used.

The friction model can be represented as shown in Figure 12-20.

Figure 12-20 Stick-Slip Friction Parameters

Coulomb friction is a highly nonlinear phenomena dependent upon both the normal force and relative velocity. Because the Coulomb friction model is an implicit function of the velocity or displacement increment, the numerical implementation of friction has two components: a force contribution and a contribution to the stiffness matrix. The stiffness is calculated based upon:

This later contribution leads to a nonsymmetric system. Because of the additional computational costs – both in terms of memory and CPU costs, the contribution to the stiffness matrix is symmetrized. For the calculation of the instabilities associated with brake squeal, the nonsymmetric friction contribution to the stiffness is made.

When the stress based friction model is used, the following steps are taken.

1. Extrapolate the physical stress, equivalent stress, and temperature from the integration points to the nodes using the conventional element shape functions.

ftp

fnf

n

ft

2

2

t

= 1.05 (default; can be user-defined)

= 1 x 10-6 (default; can be user-defined)

= 1 x 10-6 (fixed; so that 0)

e = 5 x 10-2 (default; can be user-defined)

Kij

fti

vrj

---------=


2. Calculate the normal stress.

3. Calculate the relative sliding velocity. At the beginning of an increment, the previously calculated relative sliding velocity is used as the starting point. When a node first comes into contact, it is assumed that it is first sticking, so the relative sliding velocity is zero.

4. Numerically integrate the friction forces and the stiffness contribution.

For the case of deformable-deformable contact, loads equal in magnitude and opposite in direction are applied to the body that is contacted. Each of these loads is extrapolated to the closest boundary nodes. With this procedure, it is guaranteed that all friction forces applied are in self equilibrium.

The Coulomb friction model often does not correlate well with experimental observations when the normal force/stress becomes large. If the normal stress becomes large, the Coulomb model might predict that the frictional shear stresses increase to a level that can exceed the flow stress or the failure stress of the material. As this is not physically possible, the choices are either to have a nonlinear coefficient of friction or to use the cohesive, shear based friction model.

Figure 12-21 Linear Coulomb Model Versus Observed Behavior

The shear based model states that the frictional stress is a fraction of the equivalent stress in the material:

Again, this model is implemented using an arctangent function to smooth out the step function:

This model is available for all elements using the distributed load approach.

When a node contacts a rigid body, the coefficient of friction associated with the rigid body is used. When a node contacts a deformable body, the average of the coefficients for the two bodies are used. Various BCTABLE options can be used if complex situations occur.

fr

Linear Coulomb Model

Observed Behavior

n

fr m3

------- t–

fr m3

------- 2---arctan

vr

RVCNST----------------------- t–


446

Recalling that friction is a complex physical phenomena, due to variations in surface conditions, lubricant distribution, and lubricant behavior, relative sliding, temperature, geometry, and so on.

The above two friction models may be extend, if necessary, by means of user subroutine UFRIC. In such a routine, you provide the friction coefficient or the friction factor as

or

Glue Model

A special type of friction model is the glue option, which imposes that there is no relative tangential motion. The glue motion is activated through the BCTABLE Bulk Data entry.

A novel application of contact is to join two dissimilar meshes. In such a case, by specifying a very large separation force and that the glue motion is activated, the constraint equations are automatically written between the two meshes.

position of the point at which friction is being calculated

normal force at the point at which friction is being calculated

temperature at the point at which friction is being calculated

relative sliding velocity between point at which friction is being calculated and surface

flow stress of the material

x fn T vr y =

m m x fn T vry =

x

fn

T

vr

y


Defining Contact BodiesThis section describes surface geometry definition, motion definition, and friction description in automatic two- and three-dimensional contact applications. The basic philosophy behind these applications is the existence of one or more bodies that might or might not come into contact with one another, or even contact with themselves during an analysis. As far as the contact is concerned, it is the surface associated with the body that plays a role.

There is a limit of 999 bodies in an analysis. Bodies may be combined if the 999 body limit is not exceeded. Some can be deformable, others can be rigid. Deformable surfaces must always be declared in the input file before rigid surfaces.

Deformable and Rigid SurfacesA deformable surface is simply defined by the set of elements that constitute the body to which it is associated. When a node of another body or the same body (in self contact) comes into contact with a deformable surface, information regarding the contacted surface is obtained. This is based upon the coordinates of the nodes on the face of the element or the coordinates and an averaged normal if the BCBODY option is used. This can improve the accuracy of the solution.

A rigid surface does not deform. There are two modes to describe the geometric profile of a rigid surface. In the first, labelled the PieceWise Linear approach (PWL), the profile is defined by sets of geometrical data which can be comprised of straight lines, circles and splines, ruled surfaces, surfaces of revolution and patches, etc. These sets have to be given in a proper sequence around the rigid body they define, even if it is not necessary that the full enclosure be defined.

In the second method, labelled Analytical, the geometric profile is defined by prescribing 2-D NURB curves, 3-D NURB surfaces, or exact quadratic descriptions. Using this method, the surface is divided into line segments or patches The contact condition is based on the true surface geometry. This method is more accurate for curved surfaces, and might reduce the number of iterations, especially if friction is present.

In coupled thermal-stress contact, it is possible to have a surface defined strictly by thermal elements with a rigid body motion applied to it.

Motion of SurfacesDeformable surfaces can move either because of contact with other surfaces, or because of directly applied displacement boundary conditions or loads.

To each surface, we associate a point (center of rotation) that can be anywhere in space. A translative velocity and a rotational velocity around that point define the instantaneous motion of the surface. These velocities are integrated forward in time to define the motion of the surfaces. It is also possible to directly prescribe the location of the rigid body. As an alternative, you can prescribe a force or SPCD which is consider a special type of force to the rigid body.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideDefining Contact Bodies

448

The BCBODY model definition option can be used for the input of constant rigid body motions which do not change with time during the analysis. However, changes in rigid body motion (time dependent motion) can be simulated either applying the proper motion to the GRID point at the CG of the rigid body or by the user subroutine MOTION activated through the model definition option UMOTION.

CautionsIn static analysis, it is necessary to artificially connect (for instance, by very low stiffness springs) deformable bodies that during an analysis might be completely separated from other deformable bodies and have no kinematic boundary conditions applied. This is to avoid rigid body motion (PARAM, MRSPRING).

A debug printout parameter (PARAM,MARCPRN) is available in contact analysis, it produces information on when any node on the boundary comes into contact or separates from any surface. It also produces information on whether a contact node is fixed to a surface or is free to slide along it. In addition to the printed contact information, the incremental displacement and the reaction forces for those nodes in contact with rigid surfaces are printed in a local coordinate system.

There are three implied loops in the portion of the program dealing with contact: the outermost loop is over the number of surfaces; the next loop is over the number of sets of geometrical data for each surface; and the innermost loop is over the number of points comprised in each set. In case of deformable surfaces, the two inner loops reduce to the list of elements.

Control Variables and Option FlagsThe variable RVCNST on the Bulk Data BCPARA entry allows the system to self-adaptively search for sticking zones. RVCNST should be a relative sliding velocity very small compared to the typical sliding velocities in the model, but not so small that it would be overcome by changes between iterations. It is suggested you use values between 10-1 and 10-2 times a typical relative surface velocity. MSC Nastran Implicit Nonlinear default is 1.0.

The variable ERROR on the BCPARA entry determines the tolerance for contact. A too small tolerance might provoke too many increment splits. A too coarse tolerance produces unrealistic behavior. If left blank, the code calculates ERROR as the smallest nonzero element dimension divided by 20 or the shell thickness divided by 4. If there are splines in surface definitions, a value should be entered.

The variable FNTOL (BCPARA entry) is used for the input of a separation force in a contact analysis. If the contact force of a node, calculated by MSC Nastran Implicit Nonlinear, is greater than the prescribed separation force (FNTOL), the node is to be separated from the contact surface.

You can control the type of friction in a contact analysis. Either shear friction, Coulomb friction or a frictionless condition can be assumed in the analysis. The friction behavior is either continuous or true-stick slip behavior.

The computation of Coulomb friction in a contact problem can be based on either nodal stresses or nodal forces.


During each load increment, separations can occur. You can control the maximum number of nodal separations allowed in each increment to reduce computational costs. During each load increment, if the contact of a node (or a group of nodes) is detected, iteration occurs in order to accommodate the contact condition. Depending on the occurrence of further contact, the load increment recursively split until the total incremental load is reached.

Time Step ControlThe automatic contact procedure is controlled by the TSTEPNL Bulk Data entry for dynamic problems or the NLPARM entry (actually load steps) for static analyses. This is used to determine the motion of rigid surfaces and to control the splitting of increments if penetration occurs. Even in a quasi-static analysis, a “time step” must be defined by you. Several procedures can be used to enter this data. Additional control is achieved using the NLAUTO and/or NLSTRAT Bulk Data entries. It is highly recommended that at least 100 increments be specified for all contact problems.

• The NLAUTO and NLSTRAT subcase definition options can be used to define several time steps, each of the same magnitude.

• The NLAUTO and NLSTRAT subcase definition options can also be used to define a time period which is divided into equal time steps.

• The NLAUTO or NLSTRAT subcase options can also be used to define a total time period which is divided into variable size time steps.

Dynamic Contact - ImpactThe automatic contact procedure can also be used in dynamic analyses to model impact problems. This can be used with the implicit single step Houbolt or Newmark-beta operator and vibration. The TSTEPNL, NLAUTO, and NLSTRAT Bulk Data entries are used to control the choice. High frequency vibration or impact where wave propagation is important should use SOL 700.

Two-dimensional Rigid SurfacesIn a two-dimensional problem, the rigid surfaces can be represented by any of or a combination of the following geometric entities: (1) straight line segments, (2) circular arcs, and (3) spline.

Note that the normal vector of the geometric entities (line segments, circular arc, and the spline) always points into the rigid-body. The normal vector direction is determined from the direction of the geometric entity, following a right-handed rule. Care must be taken in entering the coordinates (x, y) data, in a correct direction, for rigid-surfaces.

Line Segments

When the Line Segment option is chosen, the number NPOINT and the coordinates (x, y) of (NPOINT) points must be entered for the definition of the rigid surface. MSC Nastran Implicit Nonlinear


450

automatically creates a rigid surface consisting of (NPOINT -1) linear segments for the contact problem. A two-dimensional rigid surface consisted of line segments is shown in Figure 12-22.

This entity supports analytic description/procedure.

Figure 12-22 Two-dimensional Rigid Surface (Line Segment, ITYPE = 1)

Circular Arc

When the Circular Arc option is chosen, one circular segment is created by MSC Nastran Implicit Nonlinear. There are five different methods available to define a circular arc in two dimensions. Each method requires four data blocks with the following type of data may be used to describe the arcs:

Starting Point of Arc(SP)Ending Point of Arc(EP)Center of Circle(C)Radius of Circle(R)Tangent Angles(TA)Swept Angle(SA)Number of Subdivisions(NS)

Clearly, not all of this information is required for each method. The table below describes which data is required. The default number of subdivisions is 10. If the analytical approach is used, the number of subdivisions does not influence the accuracy, but is only used for visualization purposes.

For methods 1 and 3, a positive radius means the center of the circle is on the surface side. A negative radius means the center of the circle is on the outside.

Data Block

Method

0 1 2 3 4

1 SP SP SP SP SP

2 EP EP EP EP blank

3 C C C TA1, TA2 C

4 R, NS R, NS R, NS R, NS SA, NS

y

x

Start point

End point

1 2 3

4

56 7 8

Rigid body


For method 2, the first coordinate of the center is taken into account, determining whether the center is above (>0) or below (<0) the segment defined by the end points.

For planar problems, SP, EP and C are X, Y data.

For axisymmetric problems, SP, EP and C are Z, R data.

For methods 0, 1 and 2, if R is zero, it is calculated as distance from the center to the starting point.

This entity supports analytical description/procedure.

A two-dimensional rigid surface represented by a circular arc is shown in Figure 12-23 and Figure 12-24.

Figure 12-23 Two-dimensional Rigid Surface (Circular Arc, ITYPE = 2, METHOD = 0)

Figure 12-24 Two-dimensional Rigid Surface (Circular Arc)

Spline

When the Spline option is chosen, MSC Nastran Implicit Nonlinear creates a spline by passing from the second point through to the second to last point entered. The first and the last points entered are used to

Start point

End point

1

+

Center

Radius

Note: For additional circular arc definitions, see body 12-24

SP

C

SA

Method 4 Positive R

+

SP

EP

C

RSP

EP

C

R

Method 0 Positive R Method 1 Negative R

SP

EP

C

R

X

TA2

TA1

Method 3 Positive R

+

++


452

define the tangents at the beginning and end of the spline. If a nonanalytical approach is used, then the spline is internally split into linear segments in such a way that the maximum difference between any of them and the spline is less than the contact tolerance ERROR. This operation is done before the automatic tolerance calculation; therefore, a value for ERROR must be entered whenever a spline is used. Figure 12-25 shows a two-dimensional rigid surface defined by a spline.

Figure 12-25 Two-dimensional Rigid Surface (Spline, ITYPE = 3)

This entity supports analytical description/procedure if only one spline is used in a particular rigid body.

Three-dimensional Rigid Surfaces

In a three-dimensional problem, the rigid surfaces are represented by any of or a combination of the following three-dimensional surface entities:

The variable ITYPE defines the type of surface entity to be used for a rigid surface. Since most of the three-dimensional surfaces can be easily and adequately represented by a finite element mesh of 4-node plate (patch) elements, the option ITYPE = 7 is a very convenient way of representing three-dimensional rigid surfaces. Both the connectivities and the coordinates of the 4-node patches can be generated using Patran, or entered through user subroutine DIGEOM.

The three-dimensional surface entities mentioned above, except 4-node patches, can in turn be generated from three-dimensional geometric entities. Available three-dimensional geometric entities are:

Surface Entity Type Type Identification (ITYPE)

Ruled surface 4

Surface of revolution 5

Bezier surface 6

4-node patch 7

Poly-surface 8

NURB 9

Cylinder 10

Sphere 11

Start point

End point

1234

56

Rigid body

Note: The normal vector is pointed into the rigid body.


The variable JTYPE defines the type of geometric entities to be used for the generation of three-dimensional rigid surfaces.

For the (PWL) approach, note that all geometrical data in 3-D space is reduced to 4-node patches. The four nodes will probably not be on the same plane. The error in the approximation is determined by the number of subdivisions of the defined surfaces. Note that the normal to a patch is defined by the right-hand rule, based on the sequence in which the four points are entered.

Ruled Surface

When the Ruled surface option is chosen, a ruled surface is created by MSC Nastran Implicit Nonlinear based on the input of two surface generators, defined by straight line segment (JTYPE = 1), 3D circular arc (JTYPE = 2), spline (JTYPE = 3) or Bezier curve (JTYPE = 4). If the surface generator is not a 3D circular arc, the number NPOINT1 (NPOINT2) and the coordinates (x, y, z) of these NPOINT1 (NPOINT2) points must be entered for the definition of the surface generators. In case the surface generator is a 3D circular arc, a method (METH) must be selected for the definition of the circular arc. A 3D circular arc is defined by four points. In addition, the number of subdivisions, NDIV1, along the first (surface generator) and the NDIV2 along the second (from the first surface generator to second surface generator) direction must also be entered. For a (PWL) approach, MSC Nastran Implicit Nonlinear creates (NDIV1) x (NDIV2) 4-node patches automatically to represent the prescribed ruled surface. For analytical approach, (NDIV1 + 1) x (NDIV2 + 1) points are created and a NURB surface is general which passes exactly through these points. The accuracy in general is controlled by the number of points. Figure 12-26 shows a typical ruled surface.

Geometric Entity Type Type Identification (JTYPE)

Straight line segment 1

3-D circular arc 2

Spline 3

Bezier Curve 4

Poly line 5

Note: Patran produces a nurbs description for all 3-D rigid surfaces, even when patches or other geometrical shapes are specified. If rigid bodies made of patches are desired then the geometry should be meshed, and the elements specified as the application region.


454

Figure 12-26 Three-dimensional Rigid Surface (Ruled Surface, ITYPE = 4)

Surface of Revolution

When the Surface of revolution option is chosen, a surface of revolution is created by MSC Nastran Implicit Nonlinear based on the input of one surface generator, defined by straight line segment (JTYPE = 1), 3-D circular arc (JTYPE = 2), spline (JTYPE = 3) or Bezier curve (JTYPE = 4). If the surface generator is not a 3-D circular arc, the number NPOINT and the coordinates (x, y, z) of these NPOINT points must be entered for the definition of the surface generator. In case the surface generator is a 3-D circular arc, a method (METH) must be selected for the definition of the circular arc. A 3-D circular arc is defined by four points. In addition, the number of subdivisions NDIV1 along the surface generator and NDIV2 along the second (circumferential) direction must also be entered.

MSC Nastran Implicit Nonlinear then creates (NDIV1 x NDIV2) four-node patches automatically, to represent the prescribed surface of revolution. The axis of revolution is defined by the coordinates (x, y, z) of two points in space, and an angle of rotation from the initial position is also needed for the definition of the surface of revolution. A positive rotation is about the axis formed from point 1 to point 2. Figure 12-27 shows a typical surface of revolution.

x

y

z

Start point

End point

Start point

End point1st Geometric entity

2nd Geometric entity

1

2

1: first direction2: second direction: normal direction into the rigid body

NDIV2 = 3

NDIV1 = 4

NDIV1 = number of divisions in the first directionNDIV2 = number of divisions in the second direction


Figure 12-27 Three-dimensional Rigid Surface (Surface of Revolution, ITYPE = 5)

Bezier Surface

When the Bezier Surface option is chosen, a Bezier surface is defined by the coordinates (x, y, z) of NPOINT1 x NPOINT2 control points. NPOINT1 points are entered along the first direction and then repeated NPOINT2 times to fill through the second direction of the surface. NPOINT1 and NPOINT2 have to be at least equal to 4. Number of subdivisions (NDIV1, NDIV2) entered has to be equal or greater than NPOINT1 and NPOINT2 for Bezier surface. (NPOINT1-1) x (NPOINT2-1) 4-node patches are created by MSC Nastran Implicit Nonlinear for the definition of a Bezier surface. Figure 12-28 shows a typical Bezier surface. If it can be treated as an analytical surface, an exact conversion to NURBS is performed.

x y

z

Start point

End point

1

2

Point 2

Angle ofrotation

Point 1

Axis of revolution definedby the coordinates ofpoints 1 and 2

Surface generation(initial position)

1: First direction

2: Second direction

: Normal direction into the rigid body


456

Figure 12-28 Three-dimensional Rigid Surface (Bezier Surface, ITYPE = 6)

Four-node patches

When the Four-Node Patches option is chosen, you enter directly all the 4-node patches that comprise this surface. They are entered following the same format MSC Nastran Implicit Nonlinear would use to specify connectivities and coordinates of a mesh of CQUAD4 elements. In this way, a finite element preprocessor can be used to create surfaces. Alternatively, this data can be entered via the user subroutine DIGEOM further permitting you to read by yourself from any data you have access to. Figure 12-29 shows a typical 4-node patch surface. It cannot be used as an analytical surface.

x

y

z

2

NDIV2 = 4NPOINT2 = 4

1

NDIV1 = 4NPOINT1 = 4r00

r30

r31

r32

r33

r23r13

r03

r02

r01

r12

r22

r21

r20

r10

r11

1: First direction

2: Second direction



Figure 12-29 Three-dimensional Rigid Surface (4-Node Patch, ITYPE = 7)

Poly-surface

When the Poly-Surface option is chosen, a poly-surface is defined by the coordinates (x, y, z) of NPOINT1 x NPOINT2 control points. NPOINT1 points are entered along the first direction and then repeated NPOINT2 times to fill through the second direction of the surface. NPOINT1 and NPOINT2 have to be at least equal to 4 for a poly-surface and there is no need to divide it. A typical poly-surface is shown in Figure 12-30.

x

y

z

12

7 13

8

Number of patches = 12

Number of nodes = 20

Nodal coordinates can be enteredusing user subroutine DIGEOM

7

12

1

13

2

8

Rigid body

7

12

1

13

28

Rigid body

1: First direction

2: Second direction

: Normal vector (right-hand rule) into the rigid body


458

Figure 12-30 Three-dimensional Rigid Surface (Poly Surface, ITYPE = 8)

In a three-dimensional contact problem, as in a two-dimensional situation, the surface generators can be represented in a variety of ways. It can be treated as an analytical surface. Approximate conversion to NURBS.

Nonuniform Rational Bspline Surface, NURBS

When the NURBS option is chosen, NURBS are defined by the coordinates (x, y, z) of NPOINT1 x NPOINT2 control points, NPOINT1 x NPOINT2 homogeneous coordinates and (NPOINT1+NORDER1) + (NPOINT2+NORDER2) normalized knot vectors. If only the control points are entered, the interpolation scheme is used such that the surface passes through all of control points. The homogeneous coordinates and knot vectors are calculated by MSC Nastran Implicit Nonlinear. NPOINTS and NPOINT2 have to be at least equal to 3 for the interpolation scheme. A typical surface described by NURBS is shown in Figure 12-31.

2

NPOINT2 = 5

1

NPOINT1 = 5

55

11

12

13 14 15

21

22

23 24 2531

32

33 3435

44

41

4243

45

51

5253

54

x

y

z

1: First direction

2: Second direction



Figure 12-31 Nonuniform Rational Bspline Surface, NURBS (ITYPE = 9)

Cylinder (Cone) Surface

When the Cylinder (Cone) Surface option is chosen, a cylinder or cone is defined by the coordinates (x, y, z) of the center, C1, with radius, R1, in top face and the coordinate (x, y, z) of center, C2, with radius, R2, in bottom face. The normal vector of cylinder is inwards. If a negative value of R1 is entered, the normal vector is outwards. A typical cylinder is shown in Figure 12-32.

Figure 12-32 Cylinder (Cone) Surface (ITYPE = 10)

+5

+6

+10

+8

+9

+7

+1

+3

+4

+2

Y

Z

X

X

Z

Y

R1

R2

C2

C1


460

Sphere Surface

When the Spherical Surface option is chosen, a sphere is defined by the coordinates (x, y, z) of the center, C1, with radius, R1. The normal vector of sphere is inwards. If a negative value of R1 is entered, the normal vector is outwards. A typical sphere is shown in Figure 12-33.

Figure 12-33 Sphere Surface (ITYPE = 11)

3-D Circular Arc

When JTYPE = 2 is chosen, a circular arc is created by MSC Nastran Implicit Nonlinear. There are three different methods available to define a circular arc in three dimensions. Circular arcs are denoted using the following type of data:

Starting point of arc(SP)Ending point of arc(EP)Enter of circle(C)Radius of circle(R)Swept angle(SA)Swept angle flag(SAF)Middle point(MP)Arbitrary point (lying in plane of circle)(AP)

X

Z

Y

R1

C1


For Method 1, a positive radius means the center of the circle is on the surface side. A negative radius means the center of the circle is on the outside.

For Method 2, SAF that is positive means an angle less than 180, a negative value an angle greater than 180.

For Method 3, the starting point, arbitrary point and center define the plane in which the circular arc lies. SP, EP, C, MP and AP are X, Y, Z data.

For an arc with 180 degrees, either Method 1 or Method 2 is recommended.

A three-dimensional rigid surface represented by a circular arc is shown in Figure 12-34.

Figure 12-34 Three-dimensional Rigid Surface (Circular Arc)

Data Block

Method

0 1 2

1 SP SP SP

2 EP MP AP

3 C EP C

4 R SAF SA

SP

EP

+

SP

EP

C

R

Method 0

+

SP

C

SA

Method 2

MP

Method 1

AP


462

Spline

When JTYPE = 3 is chosen, the spline passes by all NPOINT declared, and has zero curvature at the ends (enter at least 4 points).

Bezier curve

When JTYPE = 4 is chosen, a Bezier curve is defined by NPOINT control points (enter at least 4 points).

Poly-line

When JTYPE = 5 is chosen, a poly-line defined by NPOINT control points.

Selective Contact Surfaces

In both the two- and three-dimensional contact problems, contact is always detected between nodes on the surface of a deformable body and the geometrical profile of another surface. There are two modes of the order in which a node checks contact with other bodies. The default version is the double-sided contact procedure. In the single-sided contact procedure, the nodes on a lower numbered body can come into contact with equally or higher numbered surfaces. For instance, the boundary nodes of body number 1 are checked against the surface profiles of bodies 1, 2, 3, .... The boundary nodes of body number 2, however, are only checked against surface profiles of bodies 2, 3, ... It is possible, therefore, that due to surface discretization, a node of body 2 slightly penetrates the surface of body 1.

The double-sided contact option checks possible contact between any two surfaces (surface i is checked for contact with surface j, and surface j is also checked for contact with surface i, where i, j = 1, 2, 3, ..., total number of surfaces in the problem).

In addition, the BCTABLE entry is provided to you for the selection of contact surfaces. Through this option, you can choose, for instance, the surface no. 1 to be in contact with surfaces 3, 5, 6, 7, but not with surfaces 2 and 4. This option can repeatedly be used during an analysis by specifying different BCTABLE entries for different subcases.

You can further restrict the potential contact by using the BCHANGE Bulk Data entry.

Specifying Contact Body Entries

Entry Description

BCBODY Defines a flexible or rigid contact body in 2-D or 3-D.

GMNURB Defines a 3-D contact region made up of NURBS using the Marc style.

BSURF Defines a contact body or surface defined by Element IDs.

BCBOX Defines a 3-D contact region -- all elements within the region define a contact body.


Defining Contact Bodies in Patran

The Create>Contact>Element Uniform combination on the Loads/Boundary Application form defines slideline, deformable, and rigid contact bodies. This form is used to define certain data for the MSC Nastran Input entries. Other data entries are defined under the Analysis Application when setting up a job for nonlinear static or nonlinear transient dynamic analysis. A contact table is also supported; by default, all contact bodies initially have the potential to interact with all other contact bodies and themselves. This default behavior can be modified under the Contact Table form, located on the Solution Parameters subform in the Analysis Application when creating a Load Step. See Contact Parameters, 467 and Contact Table, 472.

The Application Region form for contact is used to select the contact bodies whether they be deformable or rigid. Deformable contact bodies are always defined as a list of elements, the boundary of which defines the contact surface. Rigid bodies are translated as ruled surfaces (2D) or straight line segments (1D) if a mesh or geometry with an associated mesh is selected. Otherwise, if no mesh is associated with the selected geometry, the contact definition will be written as geometric NURB surfaces during translation.

Deformable Body

Defining a deformable contact body requires the following data via the Input Properties subform on the Loads/Boundary Conditions Application form.

BCPROP Defines a 3-D contact region by element properties. All elements with the specified properties define a contact body.

BCMATL Defines a 3-D contact region by element material. All elements with the specified materials define a contact body.

BCHANGE Changes definitions of contact bodies.

Entry Description


464

Rigid Body

Defining a rigid contact body requires the following data via the Input Properties subform on the Loads/Boundary Conditions Application form. The input data form differs for 1-D and 2-D rigid bodies. One dimensional rigid surfaces are defined as beam elements, or as curves (which may be meshed with beam elements prior to translation) and used in 2-D problems. The lines or beams must be in the global X-Y plane. Two dimensional rigid surfaces must be defined as Quad/4 or Tri/3 elements, or as surfaces (which may be meshed with Quad/4 or Tri/3 elements prior to translation) and are used in 3-D problems. The elements will be translated as 4-node patches if meshed or as NURB surfaces if not meshed.

Description

Friction Coefficient (MU)

Coefficient of static friction for this contact body. For contact between two bodies with different friction coefficients, the average value is used.

Heat Transfer Coefficient to Environment

Heat transfer coefficient (film) to environment. This is only necessary for coupled analysis (not available until version 2006).

Environment Sink Temperature

Environment sink temperature. This is only necessary for coupled analysis.

Contact Heat Transfer Coefficient

Contact heat transfer coefficient (film). This is only necessary for coupled analysis.

Boundary By default a deformable contact body boundary is defined by its elements (Discrete). However, you can use an Analytic surface to represent the deformable body. This improves the accuracy for deformable-deformable contact analysis by describing the outer surface of a contact body by a spline (2-D) or Coons surface (3-D) description.

Exclusion Region This is an optional input. The Analytic surface of a deformable body can be described by a spline (2-D) or Coons surface (3-D) and by default the entire outer surface will be included unless an Exclusion Region is selected. For instance, you may not want to represent locations of a body that never come in contact with the SPLINE option. Select either Geometry entities of the contact body that have element associated to them, or select individual FEM nodes along the outer surface. Care should be take when selecting Exclusion Regions that actual outer surface or edge geometry is selected. If nodes are being selected that describe a 3-D edge of a solid, the nodes must be in order (it is safer to select a geometric entity in this case as the nodes could get reordered incorrectly).


Input Description

Flip Contact Side Upon defining each rigid body, MSC.Patran displays normal vectors or tic marks. These should point inward to the rigid body. In other words, the side opposite the side with the vectors is the side of contact. Generally, the vector points away from the body in which it wants to contact. If it does not point inward, then use the modify option to turn this toggle ON. The direction of the inward normal will be reversed.

Symmetry Plane This specifies that the surface or body is a symmetry plane. It is OFF by default.

Null Initial Motion This toggle is enabled only for Velocity and Position type of Motion Control. If it is ON, the initial velocity, position, and angular velocity/rotation are set to zero in the CONTACT option regardless of their settings here (for increment zero).

Motion Control Motion of rigid bodies can be controlled in a number of different ways: velocity, position (displacement), or forces/moments.

Velocity (vector) For velocity controlled rigid bodies, define the X and Y velocity components for 2-D problems or X, Y, and Z for 3-D problems.

Angular Velocity (rad/time)

For velocity controlled rigid bodies, if the rigid body rotates, give its angular velocity in radians per time (seconds usually) about the center of rotation (global Z axis for 2-D problems) or axis of rotation (for 3-D problems).

Velocity vs Time Field If a rigid body velocity changes with time, its time definition may be defined through a non-spatial field, which can then be selected via this widget. It will be scaled by the vector definition of the velocity as defined in the Velocity widget. The Angular Velocity will also be scaled by this time field.

Friction Coefficient (MU) Coefficient of static friction for this contact body. For contact between two bodies with different friction coefficients the average value is used.

Rotation Reference Point This is a point or node that defines the center of rotation of the rigid body. If left blank the rotation reference point will default to the origin.


466

Slideline

Slideline contact is not supported by SOL 600.

Axis of Rotation For 2D rigid surfaces in a 3-D problem, aside from the rotation reference point, if you wish to define rotation you must also specify the axis in the form of a vector.

First Control Node This is for Force or SPCD controlled rigid motion. It is the node to which the force or SPCD is applied. A separate LBC must be defined for the force, but the application node must also be specified here. If both force and moment are specified, they must use different control nodes even if they are coincident. If only 1 control node is specified the rigid body will not be allowed to rotate.

Second Control Node This is for Moment controlled rigid motion. It is the node to which the moment is applied. A separate LBC must be defined for the moment, but the application node must also be specified here. It also acts as the rotation reference point. If both force and moment are specified, they must use different control nodes even if they are coincident.

Input Description

Note: After defining rigid bodies in your model, you can preview the rigid body motion by selecting Preview Rigid Body Motion...


Selecting and Controlling Contact BehaviorA series of MSC Nastran entries can be used to implement and control the contact behavior in an analysis. If you have subcases where some have contact and others don’t, use the BCONTACT=NONE and BCONTACT=N options, instead of the default BCONTACT=ALLBODY.

Contact ParametersThis section describes the general parameters available in SOL 600 for detecting contact, controlling separation, and modeling friction.

Defining Contact Control Parameters in Patran

To define the Contact Control Parameters for an analysis:


2. Click Solution Type... and select Solution Parameters...

3. Select Contact Parameters... to bring up the Contact Control Parameters subform shown below.

Entry Description

BCONTACT Requests contact to be included in the analysis.

BCPARA Defines contact parameters used in MSC Nastran Implicit Nonlinear (SOL 600)

BOUTPUT Request 2-D or 3-D contact output

Note: For all solution sequences other than SOL 600, BOUTPUT request line contact output. For SOL 600, BOUTPUT request 2-D or 3-D contact output. Only SORT1 output is available for SOL 600.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideSelecting and Controlling Contact Behavior

468

Contact Detection Parameters

4. On the Contact Control Parameters subform, select Contact Detection... This form controls general contact parameters for contact detection.

Deformable-Deformable Method

In Double-Sided method, for each contact body pair, nodes of both bodies will be checked for contact. In Single-Sided method, for each contact body pair, only nodes of the lower-numbered body will be checked for contact. Results are dependent upon the order in which contact bodies are defined.

Penetration Check This controls contact penetration checking. sometimes referred to as the increment splitting option. Available options are: Per Increment, Per Iteration (default), Suppressed (Fixed), Suppressed (Adaptive. Per Increment means penetration is checked at the end of a load increment. Per Iteration means that penetration is checked at the end of every iteration within an increment. If penetration is detected, increments are split. Suppress is to suppress this feature for Fixed and Adaptive load stepping types.

Reduce Printout of Surface Definition

This controls reduction of printout of surface definition.


Distance Tolerance Distance below which a node is considered touching a body (error). Leave the box blank to have MSC Nastran calculate the tolerance.

Bias on Distance Tolerance Contact tolerance BIAS factor. The value should be within the range of zero to one. Models with shell elements seem to be sensitive to this parameter. You may need to experiment with this value if you have shell element models that will not converge.

Suppress Bounding Box Turn ON this button if you want to suppress bounding box checking. This might eliminate penetration, but slows down the solution.

Check Layers For contact bodies composed of shell elements, this option menu chooses the layers to be checked. Available options are: Top and Bottom, Top Only, Bottom Only.


470

Separation

5. On the Contact Control Parameters subform, select Separation... This form controls general contact parameters for contact separation.

Ignore Thickness Turn this button ON to ignore shell thickness.

Activate 3D Beam-Beam Contact

Turn this button ON to activate 3D beam-beam contact.

Quadratic Contact Turn this toggle ON to activate the new quadratic contact algorithm that gives significantly improved interface pressure distribution results when modeling contact using higher order elements.

Maximum Separations Maximum number of separations allowed in each increment. Maximum Separations is entered in the 6th field of the 2nd data block. Default is 9999.

Retain Value on NCYCLE

Turn ON this button if you do not want to reset NCYCLE to zero when separation occurs. This speeds up the solution, but might result in instabilities. You can not set this and Suppress Bounding Box simultaneously.

Increment Specifies whether chattering is allowed or not. Increment and Chattering enters the appropriate flag in the 9th field of the 2nd data block.


Friction Parameters

6. On the Contact Control Parameters subform, select Friction Parameters...

Chattering Specifies the separation criterion (forces or stresses) and the critical value at which the separation will take place.

Separation Criterion Specifies in which increment (current or next) the separation is allowed to occur.

Force Value Stress Value

Force/Stress Value is placed in the 5th field of the 3rd data block.

Friction Type Available options for friction Type are: None (default), Shear (for metal forming), Coulomb (for normal contact - default), Shear for Rolling, Coulomb for Rolling, Stick-Slip.

Note: Unless this pulldown is changed from None, no friction will be active.

Method For Coulomb type of friction models, there are two methods for computing friction: Nodal Stress (by default), Nodal Forces.

Relative Sliding Velocity Critical value for sliding velocity below which surfaces will be simulated as sticking.

Transition Region Slip-to-Stick transition region.

Multiplier to Friction Coefficient

Friction coefficient multiplier.

Friction Force Tolerance

Friction Force Tolerance.


472

Contact TableThis option is useful for controlling or activating contacting bodies and individual contact pairs. To avoid unnecessary detection of contact between bodies, you can control which bodies potentially may come into contact with other bodies. By default Patran writes BCONTACT=ALLBODY which specifies is that every body detects the possibility of contact relative to all other bodies and itself if it is a flexible body. When the BCTABLE option is entered (Patran writes this entry only if you change something on the contact table form), the default of detection for every body is overridden. Instead, you specify the relationship of detection between bodies for contact. The touching body does not contact itself unless you request it. Whenever the touched body is a flexible one, by default, the capability of double-sided contact is applied between the contacting bodies. This can be switched off by selecting single-sided contact or by setting the searching order in the BCTABLE entry. A positive value of the interference closure implies that there is an overlap between the bodies; a negative value implies that a gap exists.

The following control variables of contact between bodies can be modified throughout the table: contact tolerance, separation force, friction coefficient, interference closure and contact heat transfer coefficient (for coupled thermal-stress-contact analysis starting in version 2006). In addition, you can invoke the glue option, delayed slide off a deformable body, and stress-free initial contact. The previous value of those control variables is not overridden unless nonzero values are entered here.

In the glue option, when a node contacts a rigid body, the relative tangential displacement is zero. When a node contacts a deformable body, all the translational degrees of freedom are tied.

By default, if a node slides off the boundary of a deformable body at a sharp corner by a distance more than the contact tolerance, contact between the node and the contacted body is lost. By invoking the delayed sliding off option, the tangential contact tolerance is increased by a factor of 10.

In any static contact analysis, a node contacting a body will be projected onto the contacted segment of this body. Due to inaccuracies in the finite element model, this might introduce undesired stress changes, since an overlap or a gap between the node and the contacted segment will be closed. The option for stress-free initial contact forces a change of the coordinates of a node contacting a deformable body, thus avoiding the stress changes. In combination with the glue option, a similar effect can be obtained; however, the overlap or gap will remain.

Specifying a Contact Table Entry

Defining a Contact Table in Patran

To define a Contact Table:


2. Click Subcases... ,select Subcase Parameters..., and click Contact Table.

Entry Description

BCTABLE Defines a contact table used in MSC Nastran Implicit Nonlinear (SOL 600)



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.

Note: Patran will only write out the contact table if something on the contact table form is changed. The default is to write out BCONTACT=ALLBODY which does not require that the BCTABLE entry be written out

Input Description

Contact Detection • Default (by body #) -This is the default where contact is checked in the order the bodies are written to the input deck. In this scenario, the most finely meshed bodies should be listed first. There will be contact checks first for nodes of the first body with respect to the second body and then for nodes of the second body with respect to the first body. If Single Sided contact is activated on the Contact Parameters subform, then only the first check is done.

• Automatic -Unlike the default, the contact detection is automatically determined and is not dependent on the order they are listed but determined by ordering the bodies starting with those having the smallest edge length. Then there will be only a check on contact for nodes of the first body with respect to the second body and not the other way around.

• First ->Second - Blanks the lower triangular section of the table matrix such that no input can be accepted. Only the contact bodies from the upper portion are written, which forces the contact check of the first body with respect to the second body.

• Second-> First - Blanks the upper triangular section of the table matrix such that no input can be accepted. Only the contact bodies from the lower portion are written. Contact detection is done opposite of First->Second.

• Double-Sided -Writes both upper and lower portions of the table matrix. This overrules the Single Sided contact parameter set on the Contact Parameters subform.

Touch All Places a T to indicate touching status for all deformable-deformable or rigid-deformable bodies.

Glue All Places a G to indicate glued status for all deformable-deformable or rigid-deformable bodies.

Deactivate All Blanks the spreadsheet cells.

Body Type Lists the body type for each body; either deformable or rigid.


Release This cell can be toggled for each body to Y or N (yes or no). If Y, this indicates that the particular contact body is to be removed from this subcase. The forces associated with this body can be removed immediately in the first increment or gradually over the entire Load Step with the Force Removal switch described below.

Touching BodyTouched Body

These are informational or convenience list boxes to allow you to see which bodies an active cell references and to see what settings are active for Distance Tolerance and other related parameters below. You must click on the touched/touching bodies to see what values, if any, have been set for the pair combination.

Distance Tolerance Set the Distance Tolerance for this pair of contact bodies. You must press the Enter or Return key to accept the data in this data box. A nonspatial field can be referenced that will write this data in TABLE format, if this parameter varies with time, temperature, or some other independent variable. This overrides any other settings for Distance Tolerance.

Separation Force Set the Separation Force for this pair of contact bodies. You must press the Enter or Return key to accept the data in this data box. A nonspatial field can be referenced that will write this data in TABLE format, if this parameter varies with time, temperature, or some other independent variable. This overrides any other settings for Separation Force.

Friction Coefficient Set the Friction Coefficient for this pair of contact bodies. You must press the Enter or Return key to accept the data in this data box. A nonspatial field can be referenced that will write this data in TABLE format, if this parameter varies with time, temperature, or some other independent variable. This overrides any other settings for Friction Coefficient.

Interference Closure

Set the Interference Closure for this pair of contact bodies. You must press the Enter or Return key to accept the data in this data box. A nonspatial field can be referenced that will write this data in TABLE format, if this parameter varies with time, temperature, or some other independent variable. This overrides any other settings for Interference Closure.

Heat Transfer Coefficient

Set the Heat Transfer Coefficient for this pair of contact bodies. You must press the Enter or Return key to accept the data in this data box. A nonspatial field can be referenced that will write this data in TABLE format, if this parameter varies with time, temperature, or some other independent variable. This overrides any other settings for Heat Transfer Coefficient. This is only used in Coupled analysis.

Input Description


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Movement of Contact BodiesThe motion of deformable bodies is prescribed using the conventional methods of applying displacements, forces, or distributed loads to the bodies. Symmetry surfaces are treated as a special type of bodies which have the property of being frictionless and where the nodes are not allowed to separate.

There are three ways to prescribe the motion of rigid surfaces:

• Prescribed velocity

• Prescribed position

• Prescribed load or displacement of (a) control node(s).

Associated with the rigid body is a point labeled the centroid. When the first two methods are chosen, you define the translational motion of this point, and the angular motion about an axis through this point. The direction of the axis can be defined for three-dimensional problems. For two-dimensional problems, it is a line normal to the plane. For complex time-dependent behavior, the MOTION user subroutine can be used to prescribe the motion as an alternative to the input. The motion during a time increment is considered to be linear. The position is determined by an explicit, forward integration of the velocities based upon the current time step. A time increment must always be defined even if a static, rate-independent analysis is performed.

When load controlled (a more accurate name would be “control node” rigid bodies) rigid bodies are used, two additional nodes, called the control nodes, are associated with each rigid body. In 2-D problems, the first node has two translational degrees of freedom (corresponding to the global x- and y-direction) and the second node has one rotational degree of freedom (corresponding to the global z-direction). In 3-D problems, the first node has three translational degrees of freedom (corresponding to global x-, y-, and z-direction) and the second node has three rotational degrees of freedom (corresponding to the global x-, y-, and z-direction). In this way, both forces and moments can be applied to a body for the control nodes. Alternatively, one may prescribe one or more degrees of freedom of the control nodes by using the SPCD Bulk Data entries. Generally speaking, load-controlled bodies can be considered as rigid bodies with three (in 2-D) or six (in 3-D) degrees of freedom. The prescribed position and prescribed velocity

Retain Gaps/Overlaps

This is only applicable for the Glued option. Any initial gap or overlap between the node and the contacted body will not be removed (otherwise the node is projected onto the body which is the default). For deformable-deformable contact only.

Stress-free Initial Contact

This is only applicable for initial contact in increment zero, where coordinates of nodes in contact can be adapted such that they cause stress-free initial contact. This is important if, due to inaccuracies during mesh generation, there is a small gap/overlap between a node and the contacted element edge/face. For deformable-deformable contact only.

Delayed Slide Off By default, at sharp corners, a node will slide off a contacted segment as soon as it passes the corner by a distance greater than the contact error tolerance. This extends this tangential tolerance. For deformable-deformable contact only.

Input Description


methods (see Figure 12-35) have less computational costs than the prescribed load method (see Figure 12-36), however it is possible to change the loads and constraints on the control node from one subcase to the next to prescribe more complex motion of the rigid body.

Figure 12-35 Velocity Controlled Rigid Surface

Figure 12-36 Load Controlled Rigid Surface

If the second control node is not specified, the rotation of the body is prescribed to be zero.

Specifying a Contact Movement Entry

This option is used in the Body Approach step which is created by Patran.

Initial ConditionsAt the beginning of the analysis, bodies should either be separated from one another or in contact. Bodies should not penetrate one another at the start of the analysis unless the objective is to perform an

Entry Description

BCMOVE Defines movement of bodies in contact used in MSC Nastran Implicit Nonlinear.

1

2

3

1

2

V

Centroid

Extra NodeFx

Fy

Mz


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interference fit calculation. Rigid body profiles are often complex, making it difficult for you to determine exactly where the first contact is located.

Unlike other MSC Nastran solution sequences, before a SOL 600 analysis begins, some calculations take place. This is defined as increment zero. During increment zero, if a rigid body has nonzero motion, the initialization procedure brings it into first contact with a deformable body. No motion or distortion occurs in the deformable bodies during this process. In a coupled thermal mechanical analysis, no heat transfer occurs during this process. If more than one rigid body exists in the analysis, each one with a nonzero initial velocity is moved until it comes into contact. Because increment zero is used to bring the rigid bodies into contact only, you should not prescribe any loads (distributed or point) or prescribed displacements initially. For multistage contact analysis (often needed to simulate manufacturing processes), the BCMOVE Bulk Data entry in conjunction with the BCTABLE Bulk Data entry allow you to model contact bodies so that they just come into contact with the workpiece. This procedure is called a Body Approach subcase in Patran.


Heat Transfer and Thermal ContactFor heat transfer, most of the capabilities in Nastran SOL 153 and 159 are supported by SOL 600 with the exception of CHBDYP and forced convection, the equivalents of which are not currently available in Marc. The main advantage of using SOL 600 for heat transfer over SOL 153 or 159 is that thermal contact is available directly and that radiation view factors may possibly be calculated faster. The user needs to weight the drawbacks of not having CHBDYP and forced convection. Because of these alternatives, SOL 600 offers two ways to perform a heat transfer analysis. The direct method uses Marc to perform all of the calculations and can support thermal contact that varies during the run. The indirect method is to calculate the thermal contact conditions (if they are needed) at the start of the run and perform the rest of the calculations using Nastran SOL 153/159. This option is addressed using an option on the SOL 600 entry TSOLVE=M or TSOLVE=N respectively. A typical SOL 600 Executive Control statement for heat transfer using the direct method would be:

SOL 600,153 TSOLVE=M

A typical Executive Control statement for heat transfer using the indirect method would be:

SOL 600,153 TSOLVE=N

When using the “Thermal Contact” capability, either TSOLVE=N should be used or the TSOLVE option should be left blank (which will support most existing input decks).

SOL 600 heat transfer addresses conduction, free convection, radiation to space, cavity radiation, thermal contact and latent heat. Steady state or transient heat transfer calculation may be obtained. All material properties may be temperature dependent, and the material may be isotropic, orthotropic or anisotropic. For the direct method, Marc’s table input is used for all applicable input items. The direct method requires postprocessing using the Marc t16 file. All standard output forms (op2, xdb, f06 and/or punch) are available using the indirect method.

The temperature history obtained may then be used in a subsequent thermal stress simulation by using the MINSTAT and MCHSTAT bulk data options to read the temperatures off the t16 file. When used in conjunction with the MTHERM bulk data option the time steps will be either subdivided or merged to satisfy the accuracy and convergence requirements of the nonlinear mechanical analysis.

For the directly solutions, when CTRIA3 or CQUAD4 elements are used, the thermal conduction can be based upon either two methods which is selected on the PSHELL option. Similar to conventional Nastran, the thermal behavior may be membrane like only, in which case there is no thermal gradient through the thickness. To support this, the heat transfer elements are used when appropriate. The elements are:

Element 196 — Three-node, Bilinear Heat Transfer Membrane), 904

Element 197 — Six-node, Biquadratic Heat Transfer Membrane), 907

Element 198 — Four-node, Isoparametric Heat Transfer Element), 911

Element 199 — Eight-node, Biquadratic Heat Transfer Membrane, 915

The second method is that the element has a thermal gradient through the thickness, which may be required for composite simulation or thermal shock type problems. This is activated by specifying a

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nonzero MID2 entry. The MPHEAT options is used to specify, whether the temperature gradient is linear or quadratic through the total thickness of the shell, or linear or quadratic variation is specified per layer basis. In the later case if a composite shell has n layers the number of degrees of freedom per grid is n+1 or 2*n+1 for the quadratic case.

The MHEATSHL parameter may also be used to control this behavior.

Additions/changes to a standard Nastran SOL 153 or SOL 159 heat transfer input file are as follows:

Executive Control

Change SOL statement as described above.

Case Control

No changes

Bulk Data

BCBODY, BCTABLE (BCPARA if necessary to change defaults) if there is thermal contact

MPHEAT – entry that maps to Marc’s HEAT “parameter”.

NLHEATC - Defines numerical analysis parameters for SOL 600 Heat Transfer Analysis.

MCHSTAT - Option to change state variables for SOL 600.

MINSTAT - Option to define initial state variables for SOL 600.

NLSTRAT - Strategy parameters for SOL 600, including Heat Transfer Analysis

Bulk Data Parameters

PARAM,MARHTPRT (Integer) Control heat transfer output in the Marc .out file

0 = Do not print any output except for summary tables

1 = Print the nodal temperatures

2 = Print all possible nodal heat transfer output

PARAM,MRADUNIT (Integer) Controls the units used in radiation heat transfer for SOL 600

1 = Degrees Celsius

2 = Degrees Kelvin (default if parameter not entered)

3 = Degrees Fahrenheit

Remark: Degrees Rankin are not available


Heat Transfer ExamplesThe following heat transfer examples are located in the tpl1 directory:

PARAM,MHEMIPIX (Integer) Controls the number of pixels used in radiation heat transfer for SOL 600 using the hemi-cube method. The default, if this parameter is not entered is 500.

PARAM,MARVFCUT (Real) Controls the fraction of the maximum view factor that is to be used as a cutoff. View factors calculated below this cutoff are ignored. Default is 0.0001 if this parameter is not entered (Used in SOL 600 radiation heat transfer only)

PARAM,MRVFIMPL (Real) Controls the fraction of the maximum view factor that is to be treated implicitly (contribute to operator matrix). View factor values smaller than this cutoff are treated explicitly. Default is 0.01 if this parameter is not entered using this parameter reduces the size of the heat transfer operator matrix, which reduces the computational costs associated with decomposition. (Used in SOL 600 radiation heat transfer only)

PARAM,MRSTEADY (Integer) Controls the solution method for SOL 600 steady state heat transfer

1 Marc STEADY STATE is used with TIME STEP of 1.0 (default if parameter not entered) The specific heat matrix is not formed.

2 AUTO STEP is used.

Remark: Requires that a sufficiently large time period to be simulated for the solution to reach steady state.

Conduction mhqbd1, mhqbd1a, mhqbd1c, mhqbd1s, mhqbd2, mhqbd2c, mhqbd2s, mhbc01, mhbc02, mhtepe

Free Convection mhcbv1, mhcbv1a, mhcbv1b, mhcbv1c, mhcbv1d, mhcbv1e

Radiation to Space mhrad1, mhrad2, mhrad3

Cavity Radiation mhrcv1, mhrcv1a, mhrcv2, mhrhx0, mhrdhx, mhrhx4, mhrc1t

Thermal Contact mhcnoc, mhtc07, mhtc7a

Latent Heat mtlat1, mtlat2

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideReferences

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References1. Oden, J. T. and Pires, E. B. “Nonlocal and Nonlinear Friction Laws and Variational Principles for

Contact Problems in Elasticity,” J. of Applied Mechanics, V. 50, 1983.

2. Ju, J. W. and Taylor, R. L. “A perturbed Lagrangian formulation for the finite element solution of nonlinear frictional contact problems,” J. De Mechanique Theorique et Appliquee, Special issue, Supplement, 7, 1988.

3. Simo, J. C. and Laursen, T. A. “An Augmented Lagranian treatment of contact problems involving friction,” Computers and Structures, 42, 1002.

4. Peric, D. J. and Owen, D. R. J. “Computational Model for 3D contact problems with friction based on the Penalty Method,” Int. J. of Meth. Engg., V. 35, 1992.

5. Taylor, R. L., Carpenter, N. J., and Katona, M. G. “Lagrange constraints for transient finite element surface contact,” Int. J. Num. Meth. Engg., 32, 1991.

6. Wertheimer, T. B. “Numerical Simulation Metal Sheet Forming Processes,” VDI BERICHET, Zurich, Switzerland, 1991

MD/MSC Nastran Implicit Nonlinear (SOL 600) User’s Guide

Chapter 13: SOL 600 Example Problems

13 SOL 600 Example Problems

Introduction

Engine Gasket Under Bolt Preload

Elastic-Plastic Collapse of a Cylindrical Pipe under External Rigid Body Loading

Rubber Door Seal - Performance Door Closing

Brake Forming

Panel Buckling

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IntroductionThe instructions on how to run all the examples discussed in this chapter can be found on the delivery under “Install_dir/doc/pdf_nastran/user/implicit_nonlinear_examples” as PDF files in a directory with a name similar to the example.

485CHAPTER 13SOL 600 Example Problems

Engine Gasket Under Bolt Preload

Problem StatementThis problem illustrates the modeling and analysis of an automotive engine block assembly to evaluate the pressure distribution over the head gasket surface. The simulation highlights special features within SOL 600 for modeling gasket materials via the direct input of pressure/closure loading and unloading curves and for applying bolt preloads via prescribed force values or imposed displacements. Contact capabilities of SOL 600 are also illustrated.

The Structure and its Application

The head gasket in an automotive engine is a critical component that forms a seal between the engine block and the cylinder head. During each combustion cycle this seal must withstand extreme operating conditions where pressures inside the cylinder reach 1000 psi and temperatures exceed 2000F. In addition, the seal must be highly resistant over time to fuel mixtures, combustion gases, oil, and coolant. Any compromise in the seal at any time over the entire life of the engine will eventually result in failure.

Engine block/cylinder head gaskets use a combination gasket body\O-ring seal. The gasket is designed as a thin cut or molded panel that combines a structural gasket body with an inset elastomer O-ring. The gasket body is commonly manufactured using multilayer steel or composite materials and is usually coated with a high temperature elastomer to promote sealing. The O-ring, generally a highly deformable rubber like material, sits taller than the gasket body and under compression conforms to the block and cylinder head surfaces forming a seal.

During manufacturing the gasket is placed between the engine block and cylinder head and then secured in place by torquing the bolts that connect the block to the head. The applied torque compresses the gasket between the block and head forming the seal. In creating this type of seal, it is critical to achieve the correct balance between enough compression and elastomer stress to create a proper seal and too much stress which may lead to damage and premature failure of the O-ring. Thus, there is particular

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideEngine Gasket Under Bolt Preload

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interest in the pressures generated on the face and through the thickness of the gasket during the assembly process.

SOL 600 Features Demonstrated

This example explores the pressures and gap closure introduced on the head gasket as a result of applying specified torques on the bolts that connect the engine block to the cylinder head.

The disparity in the material behavior and geometry of the gasket body and O-ring produce a complex relationship between the pressures generated on the gasket as the gap closes and opens. Values for pressure vs. gap closure are typically derived experimentally and then used as the basis for a material model. Accommodating the complexity of the pressure/closure relationship requires special gasket material modeling features available in SOL 600 with the MATG material entry. The MATG entry provides for nonlinear properties in the thickness direction for compression, incorporating a nonlinear elastic range, a yield pressure, followed by a strain hardening slope in the plastic range. The MATG entry can be referenced for solid elements only and in-plane properties are assumed linear and isotropic. The experimentally derived displacement/pressure curves which define the loading path are supplied using a table entry. In addition experimentally derived unloading curves can be supplied on the same MATG entry. Up to 10 independent unloading curves are available. These features are supported in Patran through the use of fields for defining loading and unloading curves.

The ability to apply a preload via the connecting bolts is a tailor-made feature for SOL 600. Bolt preloads are defined using special MBOLT entries, where forces or displacements can be applied to specified control grids. This feature is fully supported in Patran allowing for easy selection of the control grids and application of forces or displacements.

All components comprising the engine assembly are modeled as deformable bodies for this analysis. The Contact Table feature in Patran provides a convenient means for activating gluing and touching parameters between surfaces.

Model DescriptionThe model consists of the engine block, the cylinder head, two connecting bolts, and the gasket. Due to symmetry, one half of the block-cylinder head assembly is simulated with approximately 2000 3-D elements.

Geometry and Contact Regions

The three-dimensional model for this analysis includes symmetric sections of four separate structures: the cylinder head, the engine block, the gasket, and the connecting bolts. A plane of symmetry is introduced vertically through the two connecting bolts.

The primary structure of interest is the head gasket sandwiched between the contacting surfaces of the cylinder head and engine block. The gasket measures 89mm across and 1 mm high with cutouts for the cylinder and connecting bolts. An O-ring concentric to the cylinder is embedded within the gasket body. The O-ring is 3mm wide and lies 2mm from the inside cylinder boundary.


The head, block, and bolts are modeled to fully reflect the contact between surfaces of all four structures. Extending areas of the block and head panels are modeled with appropriate boundary conditions. The contact surface between the gasket and the cylinder head and block is modeled as a glued surface. There is an initial gap between the gasket body and the head and block of .0909mm and no gap between the gasket ring and the head and block. The overall model measures 93mm length, by 355mm depth, by 44 mm height. The diameter of the cylinder is 48mm.

Finite Element and Contact Model

A total of 2138 hex and wedge elements comprise the finite element model.

The contact surface between the gasket and the cylinder head and block are modeled as glued and defined in a contact table. The surface between the bolts and the block, head, and gasket are modeled as touching surfaces.

Material

The head gasket body is constructed using a multi-layer steel material and for this example modeled as an isotropic material, the in-plane properties are Elastic Modulus of 120 MPa and a Shear Modulus of 60 MPa. The gasket material is highly nonlinear in the thickness direction with a yield pressure of 52 MPa and exhibits different loading and unloading behavior. The gasket ring is softer with a yield pressure of 42 MPa and also exhibits different loading and unloading curves. The loading and unloading curves for both gasket body and ring are read in from an external file using the Fields capability in Patran.

All other components, including the cylinder head, the block, and the two connecting bolts are modeled as a standard steel material with a elastic modulus 210000 MPa, and a Poisson’s Ratio of 0.3.

Loading

The focus of this example is on the pressures introduced on the head gasket as the gap between the cylinder head and block is closed during bolt preloading. Using the Bolt Preload feature for SOL 600, a prescribed initial displacement is applied to a cross-section of the connecting bolts. In turn, enforced displacement entries are generated at the control grids.

Table 13-1 Gasket Properties

Property Gasket Body Gasket Ring

Elastic Modulus (MPa) 120 120

Shear Modulus (MPa) 60 60

Poisson’s Ratio 0.0 0.0

Yield Pressure (MPa) 52 42

Tensile Modulus (MPa) 72 64

Transverse Shear Modulus (MPa) 35 35


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Solution Procedure and ControlsThe analysis is a single step procedure wherein the initial displacement on the bolts is evaluated to determine the closure and pressure on the gasket body and ring. A default initial load increment of 1% is used.

Solving the ProblemThe gasket example is designed to be constructed and setup using Patran and analyzed using MSC Nastran SOL 600. The geometry and gasket properties are imported from external files. Results can be accessed in text form or from inside Patran and processed for viewing the gasket pressures and closure.

Input Files Required

• bolt_n_gasket.bdf - MSC Nastran Bulk Data File containing the nodes, elements, and some element properties.

• body_loading.csv - Defines the loading curve of pressure vs. closure for the gasket body material.

• body_unloading.csv - Defines the unloading curve of pressure vs. closure for the gasket body material.

• ring_loading.csv - Defines the loading curve of pressure vs. closure for the gasket ring material.

• ring_unloading.csv - Defines the unloading curve of pressure vs. closure for the gasket ring material.

Recreating the Problem using Patran

Step-by-step instructions for creating and running the analysis on the engine/block/gasket model can be accessed by linking to the gasket_instructions file below. Once you are done creating the model and running the SOL 600 analysis, proceed by accessing the results file to generate visual displays of the simulated gasket pressures and closure.

Click here for Engine Gasket Model Instructions

Highlighted Sections of the MSC Nastran Bulk Data File

The following sections of the MSC.Nastan Bulk Data file highlight the SOL 600 entries generated from the problem setup described above.


Output Requests

The MARCOUT entry specifies the individual output quantities requested. If nothing is specified, MSC Nastran produces a set of default output according to the type of analysis. In this example, N38 is specifically requested to provide the Nodal Contact Status, E241 to provide Element Gasket Pressures, and E242 to provide Element Gasket Closure.

Output Requests

Gasket Material


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Gasket Material

The MATG entry defines the gasket material properties. Highlighted in the circle, “1” is the table ID for the loading path and “2” is the table ID for the unloading path. Additional properties including yield pressure, tensile modulus, transverse shear modulus, and the initial gap, follow the table IDs.


Loading & Unloading Curves

Bolt Preload


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Loading & Unloading Curves

The TABLES1 entries define the loading and unloading paths for the gasket body and gasket ring.

Bolt Preload

The two MBOLTUS entries define a set of grids through the cross-section of each bolt and identify a control grid which will be associated with enforced displacements.

Enforced Displacements

The FORCE and SPCD entries are used to enforce a displacement on the control grids that were specified in the MBOLTUS entries above.

Inspecting the ResultsThis example examines the gap closure and pressures generated around the circumference of the gasket due to an imposed initial displacement on the two connecting bolts.

Results Files

As an alternative to recreating the analysis model and running the analysis, you may access the generated results file as follows:

Enforced Displacements


• gasket.marc.out - results file that can be opened in a text editor.

• gasket.marc.t16 - accessible in Patran from the gasket.db.

Results Plots

The fringe plot of pressure shows an uneven pressure distribution around the circumference of the gasket ring with larger pressures concentrated around the bolts.

The simulated gap closure

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Elastic-Plastic Collapse of a Cylindrical Pipe under External Rigid Body Loading

Problem StatementA model of a cylindrical pipe is subjected to crushing as rigid bodies above and below the pipe move inward toward each other. The model is created using 2D shell elements to model the pipe, and rigid surfaces above and below the pipe. This exercise illustrates several SOL 600 capabilities including large displacement analysis, contact analysis between rigid and deformable bodies, and plasticity modeled with an elastic - perfectly plastic model.


This problem examines a section of steel pipe eight inches in diameter and 24 inches long with a wall thickness of 0.4 inches. Steel pipes of this size are often used in land-based and offshore oil and gas industries. Individual pipe sections are assembled with threaded connectors to form large pipeline distribution systems. The pipe sections and their connections must withstand installation and operational loads. The effects of these loads can be buckling, torque, bending, axial separation, external pressure, and internal pressure. In addition to environmental loads, surrounding structures undergoing displacement can introduce contact conditions. Forces from these external structures can also lead to intolerable failure scenarios.

Pipe design typically looks at all foreseeable failure scenarios and then a design is measured against the failure mode that is most critical to structural integrity. In most cases that failure mode is bending combined with axial forces while under external pressure (if submerged) and internal pressure (fluid flow). However, in this case we examine failure due to external loading from adjacent structures.

In this problem the pipe section is subjected to offset lateral loading from external structures on the top and bottom. The problem attempts to quantify whether the movement of these external structures cause


the plastic collapse of the pipe. Initial contact with the external structures is expected to cause elastic deformation of the steel pipe. Additional incremental movement potentially subjects the structure to stresses beyond the proportional limit of the material. The yield stress defines the onset of plastic strains that may initiate the collapse of the structure walls.

This example illustrates several of the capabilities in SOL 600 including large displacement analysis, contact analysis of a deformable body by rigid body movement, and material plasticity modeled with an elastic - perfectly plastic model. The resulting deformation data can establish the conditions for catastrophic collapse of the pipe section and define allowable limits to contact of external structures.

Model DescriptionThe model includes a 3-D representation of the deformable pipe structure and two semi-circular sections of rigid pipes sections.


The primary structure is a continuous linear section of pipe 24 inches long with a diameter of 8 inches and a homogeneous wall thickness of 0.4 inches. The pipe surface is void of attachments, holes, bends, or other discontinuities and the internal volume is empty space. Two external pipe structures are oriented in a perpendicular direction and lie directly on the top and underneath the primary pipe. The bottom pipe measures 6 inches in diameter with a length of 10 inches and the top pipe measures 8 inches in diameter also with a length of 10 inches.

The external pipe structures that lie directly on top and bottom surface of the primary pipe impose no initial displacement or force, and no conditions are assumed to exist that might cause friction or slippage at the interface. These external structures move laterally into the pipe section creating a “sandwiching” effect. The pipe movement causes deformations on both the upper and lower side of the pipe.


Actual sections of oil and gas pipes can extend past 30 feet. Lengths beyond the 24-inch section modeled in this problem are assumed be irrelevant in the local collapse of the pipe wall. Each end of the pipe model is assigned boundary conditions to represent the extension of the pipe in both directions. In addition, the pipe wall is sufficiently thin so that it is modeled with thick shell elements in this problem.

The finite element mesh for the primary pipe structure contains 18 elements around the circumference and 18 elements along the length for a total of 324 elements. MSC Nastran CQUAD4 elements are selected along with the PSHELL entry that accounts for standard membrane, bending and transverse shear behavior. All shell elements are assigned a thickness of 0.4 inches, designated with 5 layers, and defined to be the steel material.

In defining the contact model, the primary pipe section is modeled as a deformable body and the two external pipe structures are modeled as rigid bodies. Elements comprising the deformable pipe structure are flagged so that contact by a rigid body creates forces that are used to calculate the deformation of the pipe. The rigid body structures do not deform and their movement is described with the BCBODY


496

option. The geometry profile of the rigid surface is defined using 3D NURB surfaces that describe the true surface geometry and most accurately represent the curved surfaces.

Material

The entire pipe section is made of steel and modeled as an elastic-plastic material using the MAT1 and MATEP entries for large displacement analysis. The linear elastic behavior model is effective up to a yield stress of 36000psi. In this range imposed stresses induce linear proportional strains. The stress-strain relationship is defined using an elastic modulus of 3.0E+6 and a Poisons ratio of 0.3. Beyond the yield point perfect plasticity causes complete yielding of the material upon any incremental stress.

Solution Procedure and Controls

A static analysis is performed to determine the pipe deformations that result from contacting rigid bodies. The rigid body movement and material stiffness specified for this problem require that large displacement effects be included (PARAM LGDISP).

Rigid body movement is spread over multiple load increments to allow for the history-dependent nature of the material response. Thus the equilibrium state and material state at the end of the first increment constitutes the beginning state for the second increment. The automatic load increment option (NLAUTO) is used to control the magnitude of the load increment. The initial time increment is set at .01 sec. The total load is 2 inches of imposed rigid body lateral movement on the top and bottom of the pipe occurring over 1 sec.

Within each increment an iterative process obtains an equilibrium state. Iterations are based on the Full Newton-Raphson method with a desired number of iterations per increment being 3, and a maximum set at 2500. Convergence for each increment is defined based on relative residual forces of less than 0.1.

Output requests are made for four stress-strain components using the MARCOUT entry. Total strains tensor, stress tensor, plastic strain tensor, and Cauchy stress tensor are requested to be included in the t16 file.

Loading and Boundary Conditions

Because only a small section of the pipe is modeled, the ends of the pipe are constrained in all translations. This boundary condition effectively represents the pipe extending a substantial distance in both directions in which case no translation movement would occur at this section’s boundary.

To model the contact, the top and bottom rigid surfaces are given velocity vectors of –2 in/sec., and +2 in/sec. respectively in the y-direction (lateral). This causes the upper structure to be pushed onto the top of the pipe section and the lower structure to be pushed up into the bottom of the pipe section at a rate of 2 in/sec. for a total time of 1 second.


Solving the ProblemThe pipe crush example is designed to be constructed and setup using Patran and analyzed using MSC Nastran SOL 600. The geometry is constructed and then property/load assignments are made in Patran. A MSC.Nastan Bulk Data file (bdf) is exported and then run with MSC Nastran. Results can be accessed in text form or from inside Patran and processed for viewing the pipe deformations and stresses.


• None


Step-by-step instructions for creating and running the analysis on the pipe crush model can be accessed by linking to the pipe_crush instructions file below. Once you are done creating the model and running the SOL 600 analysis, proceed by accessing the results file to generate visual displays of the simulated deformations and stresses.

Click here for Pipe Crush Modeling Instructions


The following sections of the MSC.Nastan Bulk Data file highlight the SOL 600entries generated from the problem setup described above.

Move 2 in the Y

R=4

R=3

Move 2 in the -Y

Pipe

Rigid Body 2

Rigid Body 1


498

Output Requests

This section of the MSC Nastran bdf specifies the type of output requested. The PARAM MARCSLHT 5 entry defines the number of shell layers of output requested. The MARCOUT entry that follows lists the individual output quantities. If nothing is specified, MSC Nastran produces a set of default output according to the type of analysis.

Output Requests


Material Entries

This section of the MSC Nastran bdf describes the material behavior for the pipe. The MAT1 entry defines the elastic part of the material and the MATEP entry describes the perfectly plastic behavior.

Contact Bodies

The BCBODY entry describes the rigid surface as defined by Nurbs geometry surfaces. The value of “analytical form” is 1 (used with all Nurbs rigid geometry). Note the -2 describing the rigid body motion.

Material Entries

Contact Bodies


500

Inspecting the Results

Results Files

The .sts file shown below reports on the status of the run. In this problem the run completed without errors. Using the adaptive load incrementation algorithm the solution was obtained with 27 load increments. Two increments required 22 and 24 iterations to converge.

Adaptive Load Incrementation


Adaptive Load Incrementation

The adaptive load incrementation algorithm adjusts the time step size according to the number of iterations required to achieve convergence on the preceding increment. In this example the 10th increment required 22 iterations to obtain convergence. The time step between the 10th and 11th increment was subsequently adjusted downward and convergence on the 11th increment was reduced to nine iterations. A similar situation arose in the 16th increment.

Results Plots

Figure 13-1 shows the final deformed shape of the pipe section after 1 sec. of loading. Maximum total strains of 6.0E-2 inches develop in rigid body contact areas. Slightly higher total strains are evident on the lower pipe section in this region due to the difference in radius of the external rigid body structures.

Plasticity has occurred over most of the model as shown in Figure 13-2. Maximum stresses reaching 4.25E+4 psi occur as expected in the contact region. Also evident from this figure are high stresses equivalent to those in the contact region occurring at the constrained boundary condition. This is a consequence of the boundary condition. In a full-length pipe section we would not expect to see a stress concentrated in this area.

Figure 13-1 Total Strains imposed on the Deformed Shape


502

Figure 13-2 Deformed Shape and Stress Distributions


Rubber Door Seal - Performance Door Closing

Problem StatementThis problem illustrates the nonlinear analysis of a trunk door rubber seal. The simulation highlights several SOL 600 capabilities including hyperelastic material, large displacement, large strain, and contact between rigid and deformable bodies.


This problem examines the behavior of a rubber seal under a closing trunk door. The rubber seal has an outer diameter of 1.25” and a wall thickness of 0.125”. It is modeled using plane strain elements with hyperelastic material formulation. While the door is closing, the trunk door comes in contact with the rubber seal and deforms it. The deformation of the seal needs to be examined to make sure it properly seals the door when the door is closed.

Model DescriptionThis model consists of a planar representation of the rubber seal and a curve representation of the door. The rubber seal and door are initially located approximately 0.1” apart.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideRubber Door Seal - Performance Door Closing

504

Geometry

This three-dimensional problem is idealized as a plane strain problem. The rubber seal geometry is represented by several surfaces. The trunk door is represented by several curves.


The surfaces representing the seal are meshed with MSC Nastran CQUD4 shell elements. The element property is defined using the nonlinear hyperelastic plane strain property entry PLPLANE. All the seal CQUAD4 elements are defined as a single deformable body using the BCBODY and BSURF entries.

The curves representing the trunk door are used to define a rigid contact body using the BCBODY entry. The NURBS2D option is used to describe the rigid body geometry.

Material

The rubber seal material is defined using the Mooney Rivlin material model. The MSC Nastran hyperelastic material property entry MATHP is used with two material constant terms: A10=80 and A01=20.


Nodes along the bottom edge of the seal are fixed. A displacement of (-0.1, -0.7, 0.) is specified on the rigid body to control the motion of the rigid body.


A nonlinear static analysis is performed to determine the seal deformations that result from contact with the rigid trunk door. The rigid body movement and material stiffness specified for this problem require that large displacement effects be included (PARAM,LGDISP,1).

The NLPARM entry is used to specify the nonlinear analysis iteration strategy. 10 uniform time increments are used to solve this problem. The Full Newton-Raphson method is specified.

Output requests are made for stress and strain using the MARCOUT entry. These results are included in the t16 output file.

Solving the ProblemThe rubber seal problem is designed to be constructed and setup using Patran and analyzed using MSC Nastran SOL 600. First, the geometry is created in Patran by running a session file. Next the seal geometry is meshed in Patran to generate shell elements. Then property/loads/BC assignments are made in Patran. A MSC Nastan Bulk Data file (bdf) is exported and then run with MSC Nastran. Analysis results can be accessed in text format or viewed from inside Patran.


• rubber_seal.ses - Patran session file containing commands to build the rubber seal and trunk door geometry.



Step-by-step instructions for creating and running the analysis on the rubber door seal model can be accessed by linking to the Rubber Door Seal Modeling Instructions file below. Once you are done creating the model and running the SOL 600 analysis, proceed by accessing the results file to generate visual displays of the analysis results.

Click here for Rubber Door Seal Modeling Instructions


506


Large Displacement

Hyperelastic Material Entries

Contact Definition



Results Files

The .sts file shown below reports on the status of the run. Using the fixed load incrementation algorithm, the job was completed in 10 equal time increments

.

Results Plots

The plot below shows the final deformation and strain results.


508


Brake Forming

Problem StatementThis problem illustrates the nonlinear analysis of a flat steel plate being formed into an angled bracket. A cylindrical punch is used to bend the plate by pushing it into a die, then retracting away. This forming process is commonly known as brake forming.

This simulation highlights several SOL 600 capabilities including multiple load steps, material plasticity, large displacement, large strain, and contact between rigid and deformable bodies.


This problem examines the behavior of a flat steel plate undergoing metal forming. The plate is 1.8” wide and 0.1” thick. The punch has a 2.0” diameter. The plate is modeled using plane strain elements. Since the plate will be deformed beyond its material yield point, an elastic-plastic material model will be used to simulate the material yielding behavior. The forming process consists of two steps. In the first step, the punch pushes the plate into the die, causing it to deform plastically. In the second step, the punch is retracted away from the plate.

Model DescriptionThis model consists of a planar representation of the plate and curve representations of the punch and die.

Geometry

This three-dimensional problem is idealized as a plane strain problem. The plate is represented by a surface. The punch is represented by a circular curve. The die is represented by a composite curve.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideBrake Forming

510


The surface representing the plate is meshed with MSC Nastran CQUAD4 shell elements. The element property is defined using the PSHELL entry with MID2 set to -1 to specify the plane strain option. All the plate CQUAD4 elements are defined as a single deformable body using the BCBODY and BSURF entries.

The circular curve representing the punch is used to define a rigid contact body using the BCBODY entry. The composite curve representing the die table is used to define a second rigid contact body using the BCBODY entry. The NURBS2D option is used to describe the geometry for both rigid bodies.

Material

The plate is made of steel and modeled as an elastic-plastic material using the MAT1 and MATEP entries. The linear elastic behavior model is effective up to a yield stress of 50000 psi. In this linear range the stress-strain relationship is defined using an elastic modulus of 30E+6 and a poisson ratio of 0.3. The plastic stress-strain relationship is defined by 7 pairs of stress-strain values using the TABLES1 entry.


The nodes along the centerline of plate are fixed in the lateral direction to prevent the plate from drifting sideways. A one-G gravitational acceleration is applied to the plate to hold the plate against the die table.

The die table rigid body is held stationary. An enforced vertical displacement of -0.3” is applied to the control node of the punch rigid body during the first load step. In the second load step, an enforced vertical displacement of zero is applied to the same control node to return the punch to its initial position.


A nonlinear static analysis is performed to determine the steel plate deformations and stresses that result from the brake forming process. The rigid body movement and material stiffness specified for this problem require that large displacement effects be included (PARAM,LGDISP,1).

The adaptive load increment option is used with an initial time size of 0.01. Output requests are made for stress and strain using the MARCOUT entry. These results are included in the t16 output file.

Solving the ProblemThe brake forming problem is designed to be constructed and setup using Patran and analyzed using MSC Nastran SOL 600. First, an IGES geometry file is imported into Patran. Next the plate geometry is meshed in Patran to generate shell elements. Then property/loads/BC assignments are made in Patran. A MSC.Nastan Bulk Data file (bdf) is exported and then run with MSC Nastran. Analysis results can be accessed in text format or viewed from inside Patran.


• brake_forming_s600.igs - IGES file containing plate, punch, and die geometry.



Step-by-step instructions for creating and running the analysis on the brake forming model can be accessed by linking to the Brake Forming Modeling Instructions file below. Once you are done creating the model and running the SOL 600 analysis, proceed by accessing the results file to generate visual displays of the analysis results.

Click here for Brake Forming Modeling Instructions

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuideBrake Forming

512


Defining Subcases for Multiple Load Steps

Large Displacement

Plasticity

Contact



Results Plots

The plot below shows the analysis results at the end of the forming operation.

MSC Nastran Implicit Nonlinear (SOL 600) User’s GuidePanel Buckling

514

Panel Buckling

Problem StatementThis problem illustrates the nonlinear post-buckling analysis of a curved panel. This simulation highlights several SOL 600 capabilities including large displacement, material plasticity, and contact between multiple deformable bodies.


Panels constructed from thin sheets reinforced with stiffeners are commonly used in many industries. In order to keep these panels light, the thin sheet (skin) is often designed to be thin and allowed to buckle elastically at relatively low levels of compressive loading. Once the skin buckles, the compressive load distribution changes in the panel, and any additional load is carried mostly by the stiffeners. The final failure load is reached when the overall panel fails due to a combination of material yielding and instability.

MSC Nastran SOL 600 provides the capabilities required to simulate this panel post-buckling behavior.

Model DescriptionThe model consists of a stiffened panel approximately 14” by 22”. The skin is constructed from aluminum and has a slight curvature. A longitudinal Z stiffener is attached to the skin by a single row of rivets. The stiffener has a joggle in it, and a doubler is used to fill the gap between the stiffener and the skin. Two additional longitudinal doublers are attached to the outside edges of the skin.

This model was originally constructed in metric units and was converted to English units. As a result of the conversion, some of the dimensions are rounded off.



The skin, stiffener, and doublers are all modeled as shell elements located at the mid-surface of these components. The gap between two components is equal to half the thickness of one component plus half the thickness of another component. These components are allowed to contact each other.


The skin, stiffener, and doublers are modeled using CQUAD4 elements. The element properties are defined using PSHELL entries.

The rivets are modeled using CBAR elements. The element property is defined using the PBAR entry.

The three doublers are modeled as three deformable contact bodies. The stiffener lower flange is defined as a deformable contact body. The skin is in contact with several components. Each contact area within the skin is defined as a deformable contact body.

An RBE2 rigid body element is used to connect all the nodes on one end of the panel to a single node. This node is used to apply the panel loading.

Material

The skin and doublers are made of aluminum alloy 2024 which is modeled as an elastic-plastic material using MAT1 and MATEP entries. The stiffener is made of aluminum alloy 7349 and is also modeled as an elastic-plastic material using MAT1 and MATEP entries. The plastic stress-strain relationships are defined by pairs of stress-strain values using TABLES1 entries.


All the nodes on one end of the panel are fixed in all six degrees of freedom. At the other end of the panel, an RBE2 rigid body element connects the nodes to a single node, and an enforced displacement of 0.28” in the longitudinal direction is applied to this node.

The longitudinal edges of the panel are fixed in the lateral directions but are free to slide in the longitudinal direction.


A nonlinear static analysis is performed to determine the capability of the panel to support a compressive load. PARAM,LGDISP,1 is specified to activate the large displacement capability.

The NLPARM and NLSTRAT entries are used to specify the nonlinear analysis iteration strategy. 40 fixed time increments are specified for this problem. Output requests are made for stress and strain using the MARCOUT entry. These results are included in the t16 output file.

Solving the ProblemThe panel post-buckling problem is designed to be constructed and setup using Patran and analyzed using MSC Nastran SOL 600. A Patran database is provided which contains the complete finite element


516

model. A MSC.Nastan Bulk Data file (bdf) is exported and then run with MSC Nastran. Analysis results can be accessed in text format or viewed from inside Patran.


• stiffened_panel.db - Patran database containing the panel finite element model.


Step-by-step instructions for creating and running the analysis on the panel buckling model can be accessed by linking to the Panel Buckling Modeling Instructions file below. Once you are done creating the model and running the SOL 600 analysis, proceed by accessing the results file to generate visual displays of the analysis results.

Click here for Panel Buckling Modeling Instructions



Plasticity

Contact

Large Displacement


518


Results Plots

The plot below shows the final deformation of the panel.

The plot below shows a graph of the total longitudinal reaction force vs. time increments.

MSC.Fatigue Quick Start Guide

I n d e xMSC Nastran Implicit Nonlinear SOL 600 User’s Guide

Aadaptive load incrementation 68analysis

eigenvalue 122free vibration 125linear 88nonlinear 88, 90post-buckling 115static 114types 112, 168

defining in MSC.Patran 170analysis procedures 57, 83, 168

linear and nonlinear 88static 45, 114

anisotropic materials 272applying constraints

single degrees-of-freedom 48arc-length incrementation 70Arruda-Boyce model 278AUTO 68AUTO INCREMENT 70AUTOSPC 22axisymmetric elements 414

Bbasic load incrementation 68bifurcation approach 118body approach 116

subcase parameters 190defining in MSC.Patran 190

boundary conditions 244nonlinear 111treatment of 106

brake squeal 150, 151, 153BRKSQL 150buckling analysis 85, 117


Bulk Dataentries 19file 16

Bulk Data EntriesEIGC 23MDMIOUT 149MESUPER 149

Buyukozturk criterion 317

CCase Control Commands 18CASI 58, 60Cauchy Stress 210CBAR 159CELAS2 159CHEXA 159combined hardening 320composite 102, 157, 163, 217, 264, 313,

315, 344, 349, 350, 370, 371, 373, 411, 485, 509, 510

conditioning number 59CONROD 159constant dilatation 406constitutive material models 265

defining in MSC.Patran 265constraints 48

multi-point 48, 230single point 48

I I

MSC Nastran Implicit Nonlinear SOL 600 User’s Guide

520

contact 244, 463automatic penetration checking 431beams 437constraints 434controlling 467corner conditions 433deformable 463detection 424dynamic impact 428, 449glue model 446neighbor relations 427parameters 467

defining in MSC.Patran 467penetration 468rigid 464separation 433, 436shell 426, 437tolerance 432

contact bodies 421, 447defining in MSC.Patran 463deformable surfaces 447

defining in MSC.Patran 463movement 476rigid surface

defining in MSC.Patran 464rigid surfaces 447, 449

3D 452circular arc 450line segment 449spline 451

slidelinedefining in MSC.Patran 466

contact table 472defining in MSC.Patran 472

convergencecontrols 78, 124

coordinate systems 156CPENTA 159CQUAD4 159creep 107, 115, 136, 363

defining in MSC.Patran 368implicit formulation 367subcase parameters 188

defining in MSC.Patran 188Crisfield method 75CTETRA 159

CTRIA3 159

Ddamping 135debugging 197

in MSC.Patran 206deformable surfaces 447degrees-of-freedom 158direct linear transient 168DMIG 23, 24, 126, 127, 194domain decomposition 140

defining in MSC.Patran 140ductile metals 354

Eeigenvalue

analysis 122buckling prediction 117extraction 119

eigenvectorstrain energy 121

elasticity 103elastomers 273, 356element strain energy 39

521INDEX

elements3D solid

defining in MSC.Patran 416axisymmetric 414

defining in MSC.Patran 414bush

defining in MSC.Patran 407damper

defining in MSC.Patran 407gap 409

defining in MSC.Patran 409line 410

defining in MSC.Patran 410mass

defining in MSC.Patran 407membrane 411

defining in MSC.Patran 411overriding MSC.Nastran selections 405panel 411

defining in MSC.Patran 411plane strain 414

defining in MSC.Patran 415selection 404shell 411

defining in MSC.Patran 411solid 413spring

defining in MSC.Patran 407types 402

Executive Control Statements 18SOL 600 210

existing models 20experimental data fitting 381external superelements 149

Ffailure models 344

defining in MSC.Patran 351Hill criterion 346Hoffman criterion 347maximum strain 345maximum stress 344Tsai-Wu criterion 348

filesBulk Data 16message 40print 40results 40

fixed load incrementation 68flow rules 322foam model 281

parameters 292free vibration analysis 125friction modeling 438

GGent model 280geometric nonlinearity 53, 93global element controls 406

defining in MSC.Patran 406glue model, friction 446GPFORCE 37, 39grid point force 37, 38, 39Gurson 313, 354, 355

Hhereditary integral model 308Hill’s failure criterion 346Hill’s yield function 315Hoffman failure criterion 347Houbolt operator 129, 132hourglassing 404hyperelastic damage models 354

defining in MSC.Patran 359hyperelastic materials 273hyperelastic properties

least squares fit 296parameters 308

hypoelastic materials 273

IIFP (Input File Processing) Checking 16incompressible materials 103inertia relief 145initial conditions 134, 260INREL 145, 146inverse power sweep 122


522

isotropic hardening 318isotropic materials 269iteration methods 63

Newton-Raphson 63secant 66strain correction 65

iterations 55

JJamus-Green-Simpson model 277, 298

KKachanov factor 358Kelvin-Voigt model 307kinematic hardening 320

LLagrange multipliers 430Lagrangian formulation 93

total 95, 275updated 97, 276

nonlinear elasticity 281Lancozs 123LGDISP 100, 111, 322, 338, 496, 504, 510,

515line elements 410linear analysis 88linear elastic materials 268load increments 54, 84

AUTO 68AUTO INCREMENT 70NLAUTO 68size 68

loads and boundary conditions 242, 244acceleration 255

defining in MSC.Patran 256displacement 247

defining in MSC.Patran 247distributed loads 256

defining in MSC.Patran 256force 248

defining in MSC.Patran 248inertial loads 254

defining in MSC.Patran 254initial displacement 260

defining in MSC.Patran 260initial velocity

defining in MSC.Patran 261pressure 249

defining in MSC.Patran 249temperature 251

defining in MSC.Patran 252total loads 258

defining in MSC.Patran 258velocity 255

defining in MSC.Patran 255

Mmaterial

damping 379defining in MSC.Patran 380

instabilities 107nonlinearity 53, 102

523INDEX

materials2d anisotropic

failure 3442d orthotropic

failure 344anisotropic 272

defining in MSC.Patran 272composite 370ductile metals 354elastomers 274, 356gasket 373

defining in MSC.Patran 378geological 330hyperelastic 273, 282

defining in MSC.Patran 298hypoelastic 273inelastic

defining in MSC.Patran 340isotropic 269

defining in MSC.Patran 269plastic 340

linear elastic 268metals 327Mohr-Coulomb 315nonlinear elastic 273orthotropic 270

defining in MSC.Patran 271viscoelastic

defining in MSC.Patran 310MATF 199, 351MATG 377, 486, 490MATHE 264, 391MATTVE 109, 310maximum strain criterion 345maximum stress criterion 344membrane elements 411mesh

severe distortion 106message files 40metal plasticity 313METHOD 185modal

stresses and reactions 124modal neutral files (MNF) 149mode shape 120

modified Drucker-Prager modelmatching plane strain response 334matching triaxial test response 332

Mohr-Coulomblinear material 316parabolic material 316parameters 332

Mooney-Rivlin model 277, 298MSC.Adams 149MSC.Marc 23, 126Mullin’s effect 357multi-point constraints (MPCs) 230

explicit 230defining in MSC.Patran 230

RBAR 234defining in MSC.Patran 234

RBE1 235defining in MSC.Patran 235



rigid 232defining in MSC.Patran 232

RROD 238defining in MSC.Patran 238

RTRPLT 239defining in MSC.Patran 240

sliding surface 233defining in MSC.Patran 233

NNarayanaswamy model 109, 309nastran command 192natural frequency 120nearly incompressible materials 106neighbor relations 427Neo-Hookean 278Newmark beta method

solution algorithm 131Newmark-beta operator 129, 130Newton-Raphson

full 63modified 64

NLAUTO 68


524

NLSTRAT 60nodes 158nonlinear analysis 88, 90

equation solution 55geometric nonlinearity 53guidelines 83material nonlinearity 53numerical methods 58

normal modes analysis 120subcase parameters 183

defining in MSC.Patran 183numerical methods 58

direct 58NURB 422

OOak Ridge National Laboratory criterion 317,

366Ogden model 278operator

Houbolt 129Newmark-beta 129single step Houbolt 129

orthotropic materials 270output requests 212

defining in MSC.Patran 212form 212

OUTR 225

Ppanel elements 411parallel processing 140, 142PARAM

NOELOF 39NOELOP 39XFLAG 37

penaltymethods 431

penetration 468perfectly plastic 326pin flags 417plane strain elements 414plasticity 105post-buckling 115preconditioners 59

preconditioners, iterative solvers 59prestressed normal mode 24print files 40progressive composite failure 350

Rrate-dependent yield 325RBE2 159restarts 144

defining in MSC.Patran 144results

element 216files 40MSC.Marc quantities 220

defining in MSC.Patran 223MSC.Nastran quantities 225

defining in MSC.Patran 225nodal 214postprocessing with MSC.Patran 41types 220

rigid contact surfaces 447Riks-Ramm method 75, 76

Ssecant method 66shell

contact 426elements 411

single stepHoubolt operator 129, 133

singularity ratio 81SOL 600 166

beam 20CONTINUE 20dmap 20PATH keyword 24procedure 57

solid elements 413solution

parameters 175defining in MSC.Patran 176

procedures 61

525INDEX

solution types 166bifurcation buckling 169creep 169defining in MSC.Patran 166linear static 168nonlinear static 168nonlinear transient dynamic 169viscoelastic (time domain) 169

SPC1 247SPCD 247static analysis 114


stick-slip model, friction 443storage methods 60strain correction method 65strain energy 36stress-strain

curves 327subcases 54, 178

defining in MSC.Patran 178parameters 181

Tt16 file 220Temperature-Dependent Stress Strain Curves

337Thermo-Rheologically Simple material 308TIC 260time dependent

plasticity-creepcreep material law 363

time step 134transient dynamic analysis 128


translation parameters 172defining in MSC.Patran 172

Tsai-Wu failure criterion 348

Vviscoelastic material

anisotropic 304incompressible isotropic 304isotropic 302Thermo-Rheologically Simple 308

viscoelasticity 109, 115viscoplasticity 109, 115, 367

explicit formulation 367explicit method 110implicit method 110

von Mises 313, 319

Wwork hardening

combined 320kinematic 320rules 318slope 328

Xxdb files 225


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