FINITE ELEMENT METHODS I

Application NX-6/7 and NX-NASTRAN

Ortwin Ohtmer

( Volume I )

Disc Enclosed : FE CD Book I

Disc (contents) : Software FEBEAM.exe , and SPRIMOVE (10-25-09)

Disc (contents) : File Dr. Ohtmer , ME Presentation (05-23-08 with ANI.ppt)

(Microsoft Office Power Point 97-2...)

All NX-IDEAS-Solutions are solved in the textbook via NX 6/7 .

Presentation at the PLM-SIEMENS Conference :

Title : Restructuring Engineering Education via Solid Modeling using PLM-Siemens

Software .

115 slides are shown via Power Point 97-2...

At the left bottom use the arrows to go forward or backward . A „CamStudio‟ Menu

on a slide indicates that an animation can be performed . Click on the Menu to start

the animation . To stop the animation , press „Esc‟ on the keyboard .

FINITE ELEMENTS METHODS I (VOLUME 1)

Copyright © 2011 All Rights Reserved.

Printed in the United Stated States of America. No part

of this publication may be reproduced, stored in a retrieval system,

or transmitted in any form or by any means, electronic, mechanical,

photocopying, recording, or otherwise, without the prior written permission

of the publisher.

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Phone: 562.431.9974/562.430.5023

Fax: 562.493.4970

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Thank You very much

The book is devoted to my children : SABINE , MARTIN , STEFAN

I regret , having not spent more time with them.

Many students detected mistakes or typed parts of the book ,

I really appreciate their contributions , I am very grateful to

all of them .

Two Individuals I would like to name for their expertise ,

involvement , and help .

Verne Koenig Martin Ohtmer

(562) 6211941

Dr. Ortwin Ohtmer received his doctorate in Engineering at the Technical

University of Braunschweig , Germany . He served as Assistant Professor

at the University of Braunschweig in Germany from 1963 to 1968 .

Dr. Ohtmer was appointed Department Head for Applied Mechanics at the

Messerschmitt-Boelkow-Blohm Company in Munich from 1968 to 1984 .

During this time , he was actively involved in large international projects

such as the TORNADO Fighter and AIRBUS Airplane .

Dr. Ohtmer was the Head of the Development and Maintenance Team for

the Finite Element Software System MBB-MAN-ICES-STRUDL . In this

capacity he published the ICES-STRUDL- Operation Manual , and ICES

-STRUDL Application Manuals , Volume I , and Volume II .

In 1984 , Dr. Ohtmer joined the faculty of the Mechanical Engineering

Department at California State University Long Beach where he is a

Professor , and served as Department Chair from 1992 to 1998 . During

This time Dean Williams received an eight million Grant from National

Science Foundation in Washington to coordinate the activities of the

Universities in Southern California . Dr. Ohtmer was one of the team

leaders , and in this capacity he joined Dean Williams on many trips to

Washington . He taught a Finite Element Class for USC via Distance

Learning , and wrote the only technical report . A Rapid Prototyping,-

Four Axis CNC- , and other expensive equipment were purchased ,and

a new undergraduate and graduate curriculum developed. Several new

classes were created due to the increasing importance of technology , or

outdated class contents deleted .

Dr. Ohtmer is a member of the international CAD –Committee , selecting

Papers to be presented at the annual CAD-Conference . In 2011 the CAD

Conference is in Taipei, Taiwan .

Dr. Ohtmer is an international known expert in Finite Element Methods ,

3D-Computer-Aided-Design (CAD) , and Optimization Techniques . He

has 80 scholarly publications , and many papers presented at conferences

published in proceedings . (Awards received without application) .

Dr. Ohtmer‟s main objective is to incorporate Rapid Prototyping , 3D –

Scanning , Solid Modeling , FE-Methods , and Optimization Techniques

in the integrated design, and advanced manufacturing chain of processes.

I.1

PREFACE

Dr.-Ing. Ortwin Ohtmer, Professor

Mechanical and Aerospace Engineering Department

California State University, Long Beach

Since my student years, Mechanical Engineering and Applied Mathematics were always

my favorite subjects. Later in my professional life , I selected positions related to the

named topics. Therefore , I publish my first Volume „Finite Element Methods I „ after a

life-long teaching and research experiences. Six volumes were prepared over the years

as manuscripts for different classes :

ME 305, Numerical Methods in Engineering

ME 409A, Finite Element Methods I

ME 563, Linear Finite Element Analysis II

ME 663/763, Nonlinear Optimized Structures and Mechanisms (Nonlinear Finite

Element Analysis III)

ME 490A, CAD/CAM and ME 495/595, Rapid Product Development

ME 677/777, Digital Simulation in Engineering

(Finite Element Method, Finite Volume Method, Boundary Element Method)

Those manuscripts have been upgraded every year since 1990. The modification and

development of source codes for the Finite Element Program Systems STRUDL,

NASTRAN and the CAD/CAM Program System CATIA started already in Germany

during my tenure at Messerschmitt-Boelkow-Blohm (MBB) in Munich for sixteen years,

and developing codes for “Numerical Methods in Engineering” at the University of

Braunschweig for six years as Assistant Professor was a very important learning

experience. Therefore many publications and handbooks from that time were published in

german language. On that platform I have continued learning, teaching, and researching

at CSULB since 1984. My students have been very helpful in correcting the class

manuscripts every semester.

I am happy that I waited so long to publish my books because now major developments

and changes which have occurred in Finite Element Analysis , Optimization , and

CAD/CAM over the last ten years will be incorporated . In 1990 , I wrote (see course

description for ME 677/777- Digital Simulation - in the catalog): “…optimization of heat

transfer-, fluids-, electrodynamics-, and structural problem solutions .” Due to standards

today , the mentioned simulations were not really integrated , but they are completely

integrated design tools today . The p and h versions in Finite Element Meshing and

adaptive meshing are important new developments in fracture mechanics, but automatic

meshing based on Solid Modeling as the core communication cornerstone of concurrent

engineering represents the real integration of CAD/CAM, Animation and Simulation. The

integrated design process is graphically represented below. The analysis techniques of

I.2

Mechanical engineering are now more closely related to the whole design process , as

depicted below, for the solid model of a brake system . NX-IDEAS , NX-NASTRAN

And NX-6/7 form now an integrated Software System to solve the CAD/CAM , Finite

Element and Finite Volume problems , using the same menu-driven syntax . In special

cases , the STRUDL- , NASTRAN- , ANSYS -, and ABAQUS-Input can be created

automatically based on Solid Modeling and Meshing using the NX- Software.

It is well known that for 2-D and 3-D elements the approximation of the Finite Element

Method converges with the refinement of the FE-Mesh against the unknown exact

solution (Ritz Method, 1910).For Structures composed of l-D elements (members, beams,

trusses, and frames) the Finite Element Formulation represents the simplest exact solution

method because of banded and symmetric linear systems of ordinary differential

equations. The so-called Structure Stiffness Matrix (Coefficient Matrix) is banded and

symmetric. The computing time for the solution of linear systems of equations (Gauss

and Cholesky) via elimination increases linearly with the size of the Stiffness Matrix but

increases quadratically with the bandwidth of the Stiffness Matrix. The “breakthrough” in

Finite Element Technology is that in NX- IDEAS the boundary conditions, loadings,

material constants and physical properties are specified for vertices or edges or faces of

the Solid Model before meshing. The graphical results are obtained for one mesh size

then the solution set is deleted and the automatic meshing of the solid is repeated with a

refined mesh. The files for the boundary conditions, loading etc. are always automatically

adapted to joints, edges, surfaces, and elements. Now it is very easy for a design-team to

estimate the quality of the approximations for displacements and stresses. With a

refinement of the mesh the solution converges against the unknown exact solution if

“standard” conditions are specified and no “numerically ill-conditioning effects” are

occurring.

In the past Fluid Flow Analysis was not integrated due to several difficulties. The Finite

Volume Method was not applied or created . Recent publications however indicate that

any complex analysis will use Finite Volume Approximation with implicit “Time

Differencing” as the Common Analysis Tools. The second integration tool is represented

via NURBS (Non-Uniform Rational B-Splines), which is currently the standard tool for

Solid Modeling. Curved beam and shell-elements have to be modified based on NURBS

,to match the created NURBS-Mesh-Surface. Similarly, the Finite Volume 3-D cells,

blocks with plane surfaces, 2-D cells, and quadrilateral planes with straight edges have to

be rewritten based on NURBS. These elements need to be rewritten in order to match the

created NURBS edges or surfaces as a boundary condition (the surface of a Fillet for

example). (NURBS specified as curved shell Finite Elements).

Today physical properties are specified for the solid model prior to meshing. Since the

accuracy of the Finite Element Analysis and Finite Volume Analysis improves with the

refinement of the mesh, the analysis must be performed for several mesh sizes.

I.3

Now this process, including the associated graphical representations and the parameter

checks, can be executed nearly automatically. For example the solid model of a torus can

be subtracted from a block to get the mold for a fluid flow analysis via the Finite Volume

Method. After meshing the mold, the flow analysis menus are applied in NX-6. The mesh

can also be converted to an MSC-NASTRAN-DYTRAN input for analysis. At the MSC

World Users Conference in 1995 I presented the paper “Extending MSC/DYTRAN for

the Numerical Solution of the NAVIER STOKES EQUATIONS”. With NX-6 the

software is now available , to solve Thermodynamic and

Fluid Flow Problems.

The Figure demonstrates that Solid Modeling is the Core Communication

rneCornerstone of Integrated Design , Simulation , Animation , and Advanced Manufacturing in Engineering .

Figure : Solid Model of a Brake System

INTRODUCTION - SOFTWARE – FEBEAM

For 1-D structures, trusses, beams and frames the Finite Element Method represents the

simplest and most precise numerical solution method. For 2-D and 3-D structures (plates,

shells, and solids) the Finite Element Method converges with the refinement of the mesh

against the unknown exact solution. Therefore, it makes sense to introduce the Finite

Element Method for 1-D frame structures. In three years my son Martin developed the

software „FEBEAM‟ and ‟SPRI-MOVE‟, which are now part of the documentation for

MAE 409A, Finite Element Methods I. Since my son has graduated and received his

diploma from the Mechanical and Aerospace Engineering department at CSULB he has

been attempting to streamline the studies in Mechanical Engineering. The CD for

FEBEAM is distributed with this book.

In analytical mechanics classes (static and deformable bodies), many different procedures

are taught to solve mechanical engineering problems. Such a problem could consist of

two elastic elements with three applied forces. The student integrates the associated

differential equations for hours to find the equations for the unknown variables.

Although it is interesting and important to learn and understand the conventional

procedures, these tools are basically of no practical use in the daily life of a mechanical

engineer. A real-life problem might be the analysis of a free-form geometry or tower,

with possibly hundreds of beams.

Before the advent of computers, special methods were applied for different groups of

problems. There are graphical solutions, or the method of superposition, or for example, a

beam can be analyzed by replacing the related fourth order differential equation by two

second order differential equations. Although those methods do not represent a platform

for more advanced studies in structural engineering they are still taught in detail at most

universities. Nearly forty years ago there was an interesting development of a new

method based on matrix notation to solve frame structures called the reduction

procedure, developed by S. Falk. For six years I was his Assistant Professor. At first

glance this numerical procedure seemed to have great potential for computer applications.

A cut-vector at the end of one frame structure served as the cut-vector at the start of the

next adjacent frame member, and so on. The same parameters were carried over several

members (in a row) with different properties. In the end only a maximum of three

equations with three unknown parameters had to be solved. The unknown parameters

could represent deflections, forces, or moments at the start of many frame members in a

row. To fulfill the boundary conditions at the end of the frame structure caused numerical

problems. To avoid ill-conditioning of the numerical procedure while going through

several frame members, new free parameters must replace the previous ones. The

physical explanation is simple: free parameters at one side of the structure are not

influencing the frame boundary conditions at the other side of the structure. Due to this

numerical instability, and the fact that the reduction procedure could not be extended to

structures such as plates, shells, and solids, developers moved away from this procedure.

The replacement was the finite element

I.4

I.5

procedure which had just been developed in the 1960s. Today engineering software uses

the finite element method for simulation and analysis in very different areas.

FEBEAM, the content of this research, is such software. It is not a commercial program

to compete with industry, but is limited to a frame of five nodes and four elements in two

dimensions. The intent is to help teach this finite element method. This software was

written by the author, my son MARTIN OHTMER, in Visual BASIC during the period

January 2000 to March 2003.

FEBEAM contains 480 subroutines and is about eight hundred pages long . FEBEAM

used along with the teacher‟s lecture is a finite element walk-through for the mechanical

engineering student . The student can become familiar with and learn the finite element

method by “playing” with the different steps and trying out various scenarios. FEBEAM

is intended to be self-explanatory software. It activates dialog boxes to allow actions and

grays out dialog boxes to disallow actions . The software corrects all syntax errors and

prompts the user to correct the input. Error dialog boxes or warning messages appear to

guide the user and inform about the correct use of the software . Thus , just like

professional software should be , it is not possible to “crash” the program . Although

FEBEAM contains a Help menu , it is simple enough to use without an instruction

manual.

The solutions of FEBEAM are exact , including shear deformation . Many textbook

solutions lead to different results avoiding shear deformation .

I.6

ME 409A, Finite Element Methods I

Prerequisite: Senior standing or consent of instructor. Finite

element methods for beam and truss elements. Systems of

ordinary differential equations and their exact solution in a

finite elemen t formulation . Variational formulation of the

finite element method . Static and dynamic analysis of

complex structures idealized by truss-beam and plane stress

elements. Rigid elements in an elastic environment.

Automatic mesh generation for 1D, 2D, 3D structures via

Solid Modeling using NX-6/7 (Lecture-problems 3 hours)

Traditional grading only.

Textbook: Instructor‟s Manual, Finite Element Methods I, Application

FE Software , FEBEAM, NX-6, automatic input generation

for NX-NASTRAN, ANSYS, STRUDL, ABAQUS.

Reference: List of References at the end of the book

Coordinator: Ortwin Ohtmer, Professor, Mechanical Engineering

Goals: The topics of the class are specified in the catalog as a

summary and in more detail within the syllabus. Due to the

fact that every problem related to the class-content can be

specified and solved with available menu-driven software,

class instruction is subdivided into three parts. The first part

is devoted to theoretical aspects (matrix notations, numerical

analysis, finite element (finite volume) method, and related

numerical processes. Secondly, static and dynamic problems

are solved in class using only a table calculator. Within a

third section a small amount of class time is spent on the

introduction of related menu-driven software and solving

problems via the computer. The same problems previously

solved with a table calculator, or available in the literature, or

calculated with competing software systems are then solved

with the introduced software (NX-6). Matching the results of

the same problem via a table calculator and solving via a

computer using a software system is a very powerful

assessment tool. All homework assignments are structured

the same way. The hand calculations must match the

I.7

software solutions. This goal is always required as an

assessment of the student‟s performance.

Prerequisites by Topic: Senior standing, completion of all junior (300-level) courses.

Week Chapter Topic 1 1 Fundamental Matrix Procedures

2 2 Idealization of structures by 1-, 2-, and 3-dimensional finite

elements

3-5 3 Systems of ordinary differential equations and their solution

in a finite element formulation

6-7 4 Static and dynamic analysis of truss and beam structures

8 5 Variational formulation of the finite element method,

Generation Plane Stress Elements

11 6 Static and Dynamic Analysis of complex structures with

Kinematic Constraints , Velocities , Accelerations

12-13 7 Generation of complex structures by 1-, 2-, and 3-

Dimensional Finite Elements via Solid Modeling ,Meshing,

and Simulation, using NX-6/7, and FEBEAM Software

14 Graphical representations of un-deformed and deformed

complex structures. Eigenvectors, Response, Displacements

Force-Moment-Distributions, Stress Contour Lines,

Animations

15 8 Pre- and Post-Processing of Finite Element Analysis,

Automatic Input and Graphical Output for NX-NASTRAN,

or ANSYS, or ABAQUS

9 Self-Documenting Free Format Engineering Languages and

Menu Driven Languages

15 10 Miscellanies , what is an Eigenvector ?, Adaptive Meshing.

Laboratory Projects:

Assessment projects, comparing (matching) hand

table calculations with the table calculator and the computer -

software software NX-6/7, providing menu-driven Output.

Estimated Content:

Units % Engineering Science 3.0 100

I.8

ME 563, Linear Finite Element Analysis (3) II

Prerequisite: ME 409A. Finite element (FE) forms of

differential equations. Boundary value problems, energy

theorems, matrix displacement method, and finite difference

method. Generation of FE stiffness, mass, and damping

matrices. Isoparametric Concepts. Dynamic Response of

Damped Elastic Structures, Modal-, and Direct Integration

Analysis. Automatic Mesh Generation via Solid Modeling ,

and Solution using NX-6, Automatic Adaptation to popular

software such as: STRUDL, NASTRAN, ANSYS, and

ABAQUS. FE Fluid Flow and Heat Transfer Analysis via the

Finite Volume Method (NX-6) (Lecture-problems 3 hours)

Traditional grading only.

Textbook: Instructors manual, Finite Element Method II, Application

FE software (STRUDL, NX-NASTRAN, ANSYS, NX-6 )

Reference: List of References at the end of the Book

Coordinator: Ortwin Ohtmer, Professor

Goals: The topics of the class are specified in the catalog as a

summary and in more detail within the syllabus. Due to the

fact that today every problem related to the class-content can

be specified and solved with menu-driven software

available, class instruction is subdivided into three parts. The

first part is devoted to theoretical aspects (matrix notations,

numerical analysis, finite element (finite volume), method,

and related numerical processes. Secondly, static and

dynamic problems are solved in class using only a table

calculator. Within a third section a small amount of class

time is spent on the introduction of related menu-driven

software and solving problems via the computer. The same

problems previously solved with a table calculator, or

available in the literature, or calculated with competing

software systems, are then solved with the introduced

software (NX-6), and NX-NASTRAN. Matching the

I.9

results of the same problem via a table calculator and solving

via a computer using a software system ,is a very power- full

assessment tool . All home work assignments are structured

structured the same way . The hand calculations must match

the software solutions . This goal is always required as

an assessment of the students performance.

Prerequisites by Topic: Elementary finite element analysis and graduate level

engineering mathematics.

Week Chapter Topic

1 11 FE formulation for the solution of systems of ordinary

differential equations.

2 12 Basic equations of elasticity in two and three dimensions

3-5 13 Variational formulation of boundary-value problems,

Euler Differential Equations ,Energy Theorems, Rayleigh-

Ritz Method , Matrix Displacement-, Matrix Force

. Method .Generation of FE Mass, and Damping Matrices,

Isoparametric Concept , Finite Difference Method .

6-7 14 Generation of 2-D and 3-D FE Stiffness Matrices defined

as Displacement Models,

8-9 15 Dynamic Response of Damped Elastic Structures, Modal

Analysis, Direct Integration Analysis, Random Analysis

10 16 Finite Element Input, Analysis, and Output formulation in

a Menu-Driven Language, using NX-6, automatic

d adaptation to STRUDL,NX- NASTRAN, ANSYS, and

ABAQUS.

11,12 17 Fluid Flow Analysis, applying the Finite Element and

Finite Volume method, using NX-6, and Post -Processing

13 18 Heat Transfer Analysis, applying the Finite Element and

Finite Volume method, using NX-6 and Post-Processing.

14 19 Hybrid Element Concept, Equilibrium Models

15 20 Special Elements, Substructure Procedure, Condensation

Procedure, Orthotropic Material, Laminated Plates, Rigid

Bodies in an Elastic Environment.

Laboratory Projects:

Assessment projects , comparing (matching) hand

I.10

calculations using a table calculator and the computer

software software NX-6/7, providing menu driven input and output.

menu-driven input Assessment projects are also comparing competing software

competing software systems (NX-6 , STRUDL, MSC- NASTRAN , or ANSYS ,

or MSC-NASTRAN or ABAQUS)

Estimated Content:

Units _%_

Engineering Science 3.0 100

ME 663/763, Nonlinear Optimized Structures and Mechanisms (3) III

Prerequisite: ME 563. Analysis and optimization of frame,

p plate-, and shell structures with automatic mesh generation

via solid modeling using NX-6, with automatic adaptation

a to popular software such as: STRUDL, NX- NASTRAN ,

N ANSYS , and ABAQUS. Sensitivity analysis ; Generation

Id and idealization of complex structures and mechanisms ;

m Buckling Analysis ; Strength of Structural Elements ;

E Theory of Yield , Ultimate Failure ; Stress Concentrations

;; Non-Linear Stress Analysis ; Non-Linear Material ; Large

M Deflections ; Plastic Deformations ; Non-Linear Buckling

B Composite Structures ; Thermo-elasticity ; Non- Linear

L Dynamic Analysis ; Flutter Analysis ; Random Analysis ;

A Required topics for Ph.D. students : Advanced Numerical

N Methods for Flutter , and Random Analysis.

(L (Lecture-problems 3 hours) Traditional grading only.

Textbook: Instructor‟s manual , Finite Element Method Ill ,

F FE Software (NX-6, STRUDL, NX-NASTRAN, ANSYS ,

and ABAQUS).

Reference: List of References at the end of the book

Coordinator: Ortwin Ohtmer, Professor, Mechanical Engineering

Goals : Due to the fact that today every problem related to the class-

content can be specified and solved with available menu-

sodriven Software , class instruction is subdivided into three

paparts :

The first part is devoted to theoretical aspects (matrix –

notations , numerical analysis , finite element (finite volume)

method , and related numerical processes .

S Secondly , static and dynamic problems are solved in class

u using only a table calculator. Within a third section a small

a amount of class time is spent on the introduction of related

m menu-driven software and solving problems via the

c computer . The same problems previously solved with a table

c calculator , or available in the literature , or calculated with

c competing software systems , are then solved with the

I introduced software (NX-6)/7 . Matching the results of

t the same problem via a table calculator and solving via a

c computer using a software system , is a very powerful

a assessment too l. All homework assignments are structured

t the same way . The hand calculations must match the

software solutions . This goal is always required as an

assessment of the students performance .

Week Chapter Topic

1-2 21 Standard Engineering Command and Menu-Driven Language

using NX-6/7, automatic adaptation to STRUDL,NX-

N NASTRAN, ANSYS , and ABAQUS .

3-5 22 Optimization of frame, plate, and shell structures, sensitivity

analysis.

6 23 Idealization of complex structures, generation of wings and

fuselages of flight vehicles, generation of automobile

t structures via solid modeling using NX-6/7 .

Generation of mechanisms, animations .

7-8 24 Buckling analysis of substructures and complex structures.

9-10 25 Geometric non-linear analysis, non-linear buckling.

11 26 nonlinear stress analysis of structures with nonlinear elastic

material, and large deflections.

12 27 Nonlinear stress analysis of structures with nonlinear elastic

material, large deflections and plastic deformations.

13 28 Thermo-elasticity, composite structures in engineering.

14 29 Analysis of air forces on wing, lift, and drag components of

resultant air force.

15 30 Linear and nonlinear dynamic analysis, nonlinear vibrations,

calculation of transient air forces and flutter analysis, random

analysis.

Laboratory Projects: Assessment projects, comparing (matching) hand calculations

u using a table calculator and the computer software NX-6 /7,

providing menu-driven input and output. Assessment projects

comparing competing software systems (NX-6/7 , ANSYS ,

STRUDL , or MSC-NASTRAN , or ABAQUS ) .

I.11

I.12

TABLE OF CONTENTS

CHAPTER 1

FUNDAMENTAL MATRIX PROCEDURES

1.1 Definition of a Linear System of Equations 1

1.2 Multiplication of Matrices or Multiplication of a Matrix and a Vector 2

1.3 Linear System of Equation – Inverse of a Matrix 2

1.4 Eigenvalue Problem of a Matrix 3

1.5 Solution of Linear Algebraic Equations 3

1.5.1 Gauss‟ Method 3

1.5.2 Gauss‟ Method for Large Systems of Equations (Out of Core Solution) 8

1.5.3 Cholesky Method 9

1.6 Norms and Condition Number of a Matrix 15

1.7 Units – Conversion – Factors 16

CHAPTER 2

IDEALIZATION OF STRUCTURES BY ONE-, TWO-, AND THREE

DINENSIONAL FINITE ELEMENTS

2.1 Truss Element (Plane Truss or Space Truss) 18

2.2 Plane Stress Element 19

2.3 Tridimensional Element 19

2.4 Plane Beam Element 20

2.5 Plate Bending Element 21

2.6 Assembling Truss Element and Beam Element = Plane Frame Element 22

2.7 Space Frame Element 22

2.8 Assembling Plane Stress Element and Plate Bending Element =

Plane Shell Element 23

CHAPTER 3

SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS AND THEIR

SOLUTIONS IN A FINITE ELEMENT FORMULATION

3.1 Truss Structures 26

3.1.1 Generation of the Element Stiffness Matrix 26

3.1.2 Transformation of the Element Stiffness Matrix to Global Coordinates and

Assembling to the Global Stiffness Matrix of the Structure 37

3.1.3 Modification of the Linear System of Equations due to given boundary

Conditions (Reduced Linear System of Equations) 49

3.1.4 Solving the Reduced Linear System of Equations 51

3.1.5 Back Substitution (Computation of Reactions, Member Forces and Stresses 54

3.1.6 Sequence of Commands, Solutions with STRUDL, NX-IDEAS, FEBEAM 57

3.1.7 Solution of Torsion Problems 80

3.1.8 Summary : Finite Element Method 81

3.2.1 Beam Structures (Moment- Curvature Relation) 82

3.2.2 Generation of the Element Stiffness Matrix for Beams and Equivalent Forces 87

3.3 Finite Element Method for Truss Structures with all Degrees in One Direction

120

3.3.1 Truss Member Stiffness Matrices of the Structures 120

3.3.2 Simplified Assembling of the Structure Stiffness Matrix 121

3.3.3 Reduced Linear System of Equations 121

3.3.4 Solution of the Reduced Linear System of Equations 122

3.3.5 Back-substitution 122

3.3.6 Specifying the Subscripts of the Springs 123

3.4 Finite Element Method for Structures with all ID -Elements Positioned in the

same Direction , Element Catalog 125

3.5 Solved Examples to Practice 129 to 158

CHAPTER 4

STATIC AND DYNAMIC ANALYSIS OF FRAME STRUCTURES

4.1 Static Analysis of Frame – Structures 159

4.1.1 Plane Frame – Stiffness Matrix and Examples 191

4.1.2 Transformation of the Element Stiffness Matrix to Global Coordinates and

Assembling to the Structure Stiffness Matrix 181

4.1.3 Bending and Shear Deformation 195

4.1.4 FEBEAM as a Teaching Tool 204

4.2 Dynamic Analysis of Frame – Structures 239

4.2.1 Solution of Eigenvalue Problems 239

4.2.2 Generation of Lumped Truss and Beam Mass Matrices 247

4.3 Creating the Beam Cross Section using Standard Shapes in NX-6/7 249

4.4 Vibration of Elastic Trusses and Beams with Continuous Mass Distribution 249

4.4.1 Truss Structures 249

4.4.2 „Torsion‟ Structures 253

4.4.3 Beam Structures 255

4.5 Solved Examples to Practice 266 to 292

CHAPTER 5

VARIATIONAL FORMULATION OF THE FINITE ELEMENT METHOD

5.1 Ritz Method / Displacement Method 293

5.2 Finite Element Form of the Ritz Method 300

5.3 Verification of the Finite Element Method 313

I.13

5.4 Approximation of the Local Stiffness Matrix for Plates , Shells, and Solids

via the Ritz Procedure 316

5.4.1 Triangular Plate Elements ( In Plane Forces ) 316

5.4.2 Rectangular Plate Element ( In Plane Forces ) 322

5.5 Assembling of Plate-, and Solid Elements 327

5.6 Average Von Mises Stresses 329

5.7 Examples Solving Plate and Shell Problems 330

Plate Stretching , NX-Input 338

Plate Bending, NX-6 Input) 343

Converging of Stresses via Mesh Refinement 351

Animation of Eigenvectors of a Plate 352

Stress Contour Lines 353

Solution of axi-symmetrical Shells via NX-6 355

5.8 Verifying the Ritz Procedure , Using Special Shape Functions 359

5.9 Vibration of a Cantilever with End Mass 365

5.9.1 Solution via 3D Modeling in NX-6 365

5.9.2 Calculated and Animated Eigenvectors via NX-6 compared with exact

Beam Solutions 368

CHAPTER 6

STATIC AND DYNAMIC ANALYSIS OF COMPLEX STRUCTURES WITH

KINEMATIC CONSTRAINTS

6.1 Assembling of Continuous Structures with Kinematic Constraints

( Rigid Elements in Elastic Environment) 373

6.2 Transcendental Equations as Characteristics for Mechanisms with Hinges

and Rigid Elements 381

6.2.1 The Engine System and Characteristics 388

6.2.1.1 General Kinematic Solution of the Engine System 393

6.3 Matrix Notation for a Hinge / Software Code FEBEAM 400

6.4 Rigid Element/ Software Code FEBEAM 408

6.5 Solved Examples to Practice 416 to 464

6.6 Transmission Structures 465

6.7 Four Bar Mechanism/ two Hinges , three Rigid Elements

( Generating the Freudenstein Transcendental Equation as Main

Characteristic) 474

6.8 Four Bar Mechanism , Engine System , Modifications ,

(Geneva Mechanism) 487

CHAPTER 7

GENERATION OF COMPLEX STRUCTURES VIA SOLID MODELING

AND ANIMATION

7.1 Solid Modeling as the Basis for Automatic Adaptive Meshing

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for the Finite Element- , and Finite Volume Numerical Procedure 495

7.2 Solid Modeling and Meshing 495

7.2.1 Mold of a Torus 495

7.2.2 Intersecting Cylinders with Fillets and Meshing 499

7.2.3 Generation of a Spherical Dome with a Man-Hole 503

7.2.4 Mapped Meshing of a Dome with a Man-Hole 506

7.2.5 Design of a Spherical Roof via the Boolean Operation „Intersect‟ 510

7.2.6 Creating a Wing via Lofting 513

7.2.7 Wheel Assembling 513

7.2.8 Elbow (Tube , Disc Copied) 514

7.2.9 Complex Part , Automatically Created Dimensioned Blue Print 519

7.3 Animation 520

7.3.1 Introduction 520

7.3.2 Creating a Mold of a Torus 520

7.3.3 Engine System 521

7.3.4 Four Bar Mechanism 521

7.3.5 Bevel Gear with Curved Teeth 522

7.3.6 Drilling Holes in a Base Plate 522

7.3.7 Intersecting Cylinders with Fillets 523

7.3.8 Conical Dome with Bulge 523

7.3.9 Cantilever Frame with End Mass

(Seven Bending Eigenvectors , Three Axial Eigenvectors) 524

7.3.10 Corner supported Shell

(First Three Eigenvectors , One Shell Stretching-, Two Shell Bending

Eigenvectors) 524

CHAPTER 8

NX-ADVANCED SIMULATION 525

8.1 Advanced Solid Modeling 525

8.2 Advanced Post-Processing 528

8.3 Finite Element Solution of the circular fixed supported Plate (Pressure Load),

(Comparison with „exact‟ Solution) . 529

8.4 Composite Structures (NX-Input File ) 534

8.5 NX-6/7 CAST Online Library

(Linear and Nonlinear Geometric Analysis , NX-Input file) 544

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CHAPTER 9

SELF DOCUMENTING FREE FORMAT STANDARD ENGINEERING

COMMAND LANGUAGE FOR FINITE ELEMENT ANALYSIS

/COMMAND DRIVEN INPUT, BASIC FINITE ELEMENT PROCEDURE

(APPLICATION GTSTRUDL AND NX-NASTRAN)

MENU DRIVEN SOLID MODELING AND FINITE ELEMENT ANALYSIS

(APPLICATION NX-6/7) / SIEMENS 561

9.1 Introduction 561

9.2 GTSTRUDL 561

9.2.1 Standard Engineering Commands Language (SECL) , Input- Output

STRUDL Input –Output 562

9.2.2 Application GTSTRUDL 562

9.3 Application NX-NASTRAN 567

9.3.1 NASTRAN (NX-NASTRAN Quick Reference Guide)

(General Description of Data Deck) 567

9.3.2 NX-NASTRAN Examples, (Equivalent NASTRAN-STRUDL Inputs) 578

9.4 NX-Menu Driven Software 588

9.4.1 Introduction 588

9.4.2 Solution and Solution Processes in the CAST Online Library 590

CHAPTER 10

MISCELLANEOUS 591

10.1 What is an Eigenvector ? 591

10.2 Adaptive Meshing 594

INDEX 595

REFERENCES 599

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.Some prerequisites should be discussed before reading the book .

Matrix Notations and Operations

Polynomials and Solution of Simultaneous Algebraic Equations

Systems of Ordinary Differential Equations

Eigenvalue- , and Eigenvector Analysis

Approximate Numerical-, Differentiation and Integration

Approximation of Curves via NURBS (Non-Uniform Rational B-Splines)

Approximation of Surfaces via NURBS

Application of the Boolean Operations UNITE , SUBTRACT , INTERSECT

in Solid Modeling

Strains , Stresses , and Displacements in Trusses , Beams , Plates , and

Shells

Materials , Elastic , and Plastic Deformations

Mainly an Introduction to Solid Modeling is helpful to understand the Creation of

Solids in the book . Those Models can then easily be meshed , a Finite Element

Analysis performed , including Post-Processing , Animations executed and

automatically real parts manufactured.

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