Abaqus Example Problems Manual Vol1

Abaqus Example Problems Manual

Abaqus

Example Problems Manual

Volume I

Version 6.7

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CONTENTS

Contents Volume I 1. Static Stress/Displacement Analyses Static and quasi-static stress analyses

Axisymmetric analysis of bolted pipe ange connections Elastic-plastic collapse of a thin-walled elbow under in-plane bending and internal pressure Parametric study of a linear elastic pipeline under in-plane bending Indentation of an elastomeric foam specimen with a hemispherical punch Collapse of a concrete slab Jointed rock slope stability Notched beam under cyclic loading Hydrostatic uid elements: modeling an airspring Shell-to-solid submodeling and shell-to-solid coupling of a pipe joint Stress-free element reactivation Transient loading of a viscoelastic bushing Indentation of a thick plate Damage and failure of a laminated composite plate Analysis of an automotive boot seal Pressure penetration analysis of an air duct kiss seal Self-contact in rubber/foam components: jounce bumper Self-contact in rubber/foam components: rubber gasket Submodeling of a stacked sheet metal assembly Axisymmetric analysis of a threaded connection Direct cyclic analysis of a cylinder head under cyclic thermal-mechanical loadings Erosion of material (sand production) in an oil wellbore Submodel stress analysis of pressure vessel closure hardware Using a composite layup to model a yacht hullBuckling and collapse analyses

1.1.1 1.1.2 1.1.3 1.1.4 1.1.5 1.1.6 1.1.7 1.1.8 1.1.9 1.1.10 1.1.11 1.1.12 1.1.13 1.1.14 1.1.15 1.1.16 1.1.17 1.1.18 1.1.19 1.1.20 1.1.21 1.1.22 1.1.23 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6

Snap-through buckling analysis of circular arches Laminated composite shells: buckling of a cylindrical panel with a circular hole Buckling of a column with spot welds Elastic-plastic K-frame structure Unstable static problem: reinforced plate under compressive loads Buckling of an imperfection-sensitive cylindrical shellForming analyses

Upsetting of a cylindrical billet: quasi-static analysis with mesh-to-mesh solution mapping (Abaqus/Standard) and adaptive meshing (Abaqus/Explicit)

1.3.1

v

CONTENTS

Superplastic forming of a rectangular box Stretching of a thin sheet with a hemispherical punch Deep drawing of a cylindrical cup Extrusion of a cylindrical metal bar with frictional heat generation Rolling of thick plates Axisymmetric forming of a circular cup Cup/trough forming Forging with sinusoidal dies Forging with multiple complex dies Flat rolling: transient and steady-state Section rolling Ring rolling Axisymmetric extrusion: transient and steady-state Two-step forming simulation Upsetting of a cylindrical billet: coupled temperature-displacement and adiabatic analysis Unstable static problem: thermal forming of a metal sheetFracture and damage

1.3.2 1.3.3 1.3.4 1.3.5 1.3.6 1.3.7 1.3.8 1.3.9 1.3.10 1.3.11 1.3.12 1.3.13 1.3.14 1.3.15 1.3.16 1.3.17 1.4.1 1.4.2 1.4.3 1.4.4 1.4.5 1.4.6 1.5.1 1.5.2

A plate with a part-through crack: elastic line spring modeling Contour integrals for a conical crack in a linear elastic innite half space Elastic-plastic line spring modeling of a nite length cylinder with a part-through axial aw Crack growth in a three-point bend specimen Analysis of skin-stiffener debonding under tension Failure of blunt notched ber metal laminatesImport analyses

Springback of two-dimensional draw bending Deep drawing of a square box2. Dynamic Stress/Displacement Analyses Dynamic stress analyses

Nonlinear dynamic analysis of a structure with local inelastic collapse Detroit Edison pipe whip experiment Rigid projectile impacting eroding plate Eroding projectile impacting eroding plate Tennis racket and ball Pressurized fuel tank with variable shell thickness Modeling of an automobile suspension Explosive pipe closure Knee bolster impact with general contact

2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6 2.1.7 2.1.8 2.1.9

vi

CONTENTS

Crimp forming with general contact Collapse of a stack of blocks with general contact Cask drop with foam impact limiter Oblique impact of a copper rod Water sloshing in a bafed tank Seismic analysis of a concrete gravity dam Progressive failure analysis of thin-wall aluminum extrusion under quasi-static and dynamic loadsMode-based dynamic analyses

2.1.10 2.1.11 2.1.12 2.1.13 2.1.14 2.1.15 2.1.16 2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.2.7 2.3.1

Analysis of a rotating fan using substructures and cyclic symmetry Linear analysis of the Indian Point reactor feedwater line Response spectra of a three-dimensional frame building Eigenvalue analysis of a structure using the parallel Lanczos eigensolver Brake squeal analysis Dynamic analysis of antenna structure utilizing residual modes Steady-state dynamic analysis of a vehicle body-in-white modelCo-simulation analyses

Closure of an air-lled door seal

vii

CONTENTS

Volume II 3. Tire and Vehicle Analyses Tire analyses

Symmetric results transfer for a static tire analysis Steady-state rolling analysis of a tire Subspace-based steady-state dynamic tire analysis Steady-state dynamic analysis of a tire substructure Coupled acoustic-structural analysis of a tire lled with air Import of a steady-state rolling tire Analysis of a solid disc with Mullins effect Tread wear simulation using adaptive meshing in Abaqus/Standard Dynamic analysis of an air-lled tire with rolling transport effects Acoustics in a circular duct with owVehicle analyses

3.1.1 3.1.2 3.1.3 3.1.4 3.1.5 3.1.6 3.1.7 3.1.8 3.1.9 3.1.10 3.2.1 3.2.2 3.2.3 3.2.4 3.3.1 3.3.2

Inertia relief in a pick-up truck Substructure analysis of a pick-up truck model Display body analysis of a pick-up truck model Continuum modeling of automotive spot weldsOccupant safety analyses

Seat belt analysis of a simplied crash dummy Side curtain airbag impactor test4. Mechanism Analyses

Resolving overconstraints in a multi-body mechanism model Crank mechanism Snubber-arm mechanism Flap mechanism Tail-skid mechanism Cylinder-cam mechanism Driveshaft mechanism Geneva mechanism Trailing edge ap mechanism Substructure analysis of a one-piston engine model Application of bushing connectors in the analysis of a three-point linkage

4.1.1 4.1.2 4.1.3 4.1.4 4.1.5 4.1.6 4.1.7 4.1.8 4.1.9 4.1.10 4.1.11

viii

CONTENTS

5.

Heat Transfer and Thermal-Stress Analyses

Thermal-stress analysis of a disc brake A sequentially coupled thermal-mechanical analysis of a disc brake with an Eulerian approach Exhaust manifold assemblage Coolant manifold cover gasketed joint Radiation analysis of a plane nned surface Thermal-stress analysis of a reactor pressure vessel bolted closure6. Electrical Analyses Piezoelectric analyses

5.1.1 5.1.2 5.1.3 5.1.4 5.1.5 5.1.6

Eigenvalue analysis of a piezoelectric transducer Transient dynamic nonlinear response of a piezoelectric transducerJoule heating analyses

6.1.1 6.1.2 6.2.1

Thermal-electrical modeling of an automotive fuse7. Mass Diffusion Analyses

Hydrogen diffusion in a vessel wall section Diffusion toward an elastic crack tip8. Acoustic and Shock Analyses

7.1.1 7.1.2

Fully and sequentially coupled acoustic-structural analysis of a mufer Coupled acoustic-structural analysis of a speaker Response of a submerged cylinder to an underwater explosion shock wave Convergence studies for shock analyses using shell elements UNDEX analysis of a detailed submarine model Coupled acoustic-structural analysis of a pick-up truck Long-duration response of a submerged cylinder to an underwater explosion9. Soils Analyses

8.1.1 8.1.2 8.1.3 8.1.4 8.1.5 8.1.6 8.1.7

Plane strain consolidation Calculation of phreatic surface in an earth dam Axisymmetric simulation of an oil well Analysis of a pipeline buried in soil Hydraulically induced fracture in a well bore10. Abaqus/Aqua Analyses

9.1.1 9.1.2 9.1.3 9.1.4 9.1.5

Jack-up foundation analyses Riser dynamics

10.1.1 10.1.2

ix

CONTENTS

11.

Design Sensitivity Analyses Overview

Design sensitivity analysis: overviewExamples

11.1.1 11.2.1 11.2.2 11.2.3 11.2.4

Design sensitivity analysis of a composite centrifuge Design sensitivities for tire ination, footprint, and natural frequency analysis Design sensitivity analysis of a windshield wiper Design sensitivity analysis of a rubber bushing12. Postprocessing of Abaqus Results Files

User postprocessing of Abaqus results les: overview Joining data from multiple results les and converting le format: FJOIN Calculation of principal stresses and strains and their directions: FPRIN Creation of a perturbed mesh from original coordinate data and eigenvectors: FPERT Output radiation viewfactors and facet areas: FRAD Creation of a data le to facilitate the postprocessing of elbow element results: FELBOW

12.1.1 12.1.2 12.1.3 12.1.4 12.1.5 12.1.6

x

INTRODUCTION

1.0.1

INTRODUCTION

This is the Example Problems Manual for Abaqus. It contains many solved examples that illustrate the use of the program for common types of problems. Some of the problems are quite difcult and require combinations of the capabilities in the code. The problems have been chosen to serve two purposes: to verify the capabilities in Abaqus by exercising the code on nontrivial cases and to provide guidance to users who must work on a class of problems with which they are relatively unfamiliar. In each worked example the discussion in the manual states why the example is included and leads the reader through the standard approach to an analysis: element and mesh selection, material model, and a discussion of the results. Many of these problems are worked with different element types, mesh densities, and other variations. Input data les for all of the analyses are included with the Abaqus release in compressed archive les. The abaqus fetch utility is used to extract these input les for use. For example, to fetch input le boltpipeange_3d_cyclsym.inp, type abaqus fetch job=boltpipeflange_3d_cyclsym.inp Parametric study script (.psf) and user subroutine (.f) les can be fetched in the same manner. All les for a particular problem can be obtained by leaving off the le extension. The abaqus fetch utility is explained in detail in Execution procedure for fetching sample input les, Section 3.2.12 of the Abaqus Analysis Users Manual. It is sometimes useful to search the input les. The ndkeyword utility is used to locate input les that contain user-specied input. This utility is dened in Execution procedure for querying the keyword/problem database, Section 3.2.11 of the Abaqus Analysis Users Manual. To reproduce the graphical representation of the solution reported in some of the examples, the output frequency used in the input les may need to be increased. For example, in Linear analysis of the Indian Point reactor feedwater line, Section 2.2.2, the gures that appear in the manual can be obtained only if the solution is written to the results le every increment; that is, if the input les are changed to read *NODE FILE, ..., FREQUENCY=1 instead of FREQUENCY=100 as appears now. In addition to the Example Problems Manual, there are two other manuals that contain worked problems. The Abaqus Benchmarks Manual contains benchmark problems (including the NAFEMS suite of test problems) and standard analyses used to evaluate the performance of Abaqus. The tests in this manual are multiple element tests of simple geometries or simplied versions of real problems. The Abaqus Verication Manual contains a large number of examples that are intended as elementary verication of the basic modeling capabilities. The verication of Abaqus consists of running the problems in the Abaqus Example Problems Manual, the Abaqus Benchmarks Manual, and the Abaqus Verication Manual. Before a version of Abaqus is released, it must run all verication, benchmark, and example problems correctly.

1.0.11

STATIC STRESS/DISPLACEMENT ANALYSES

1.

Static Stress/Displacement AnalysesStatic and quasi-static stress analyses, Section 1.1 Buckling and collapse analyses, Section 1.2 Forming analyses, Section 1.3 Fracture and damage, Section 1.4 Import analyses, Section 1.5

STATIC AND QUASI-STATIC STRESS ANALYSES

1.1

Static and quasi-static stress analyses

Axisymmetric analysis of bolted pipe ange connections, Section 1.1.1 Elastic-plastic collapse of a thin-walled elbow under in-plane bending and internal pressure, Section 1.1.2 Parametric study of a linear elastic pipeline under in-plane bending, Section 1.1.3 Indentation of an elastomeric foam specimen with a hemispherical punch, Section 1.1.4 Collapse of a concrete slab, Section 1.1.5 Jointed rock slope stability, Section 1.1.6 Notched beam under cyclic loading, Section 1.1.7 Hydrostatic uid elements: modeling an airspring, Section 1.1.8 Shell-to-solid submodeling and shell-to-solid coupling of a pipe joint, Section 1.1.9 Stress-free element reactivation, Section 1.1.10 Transient loading of a viscoelastic bushing, Section 1.1.11 Indentation of a thick plate, Section 1.1.12 Damage and failure of a laminated composite plate, Section 1.1.13 Analysis of an automotive boot seal, Section 1.1.14 Pressure penetration analysis of an air duct kiss seal, Section 1.1.15 Self-contact in rubber/foam components: jounce bumper, Section 1.1.16 Self-contact in rubber/foam components: rubber gasket, Section 1.1.17 Submodeling of a stacked sheet metal assembly, Section 1.1.18 Axisymmetric analysis of a threaded connection, Section 1.1.19 Direct cyclic analysis of a cylinder head under cyclic thermal-mechanical loadings, Section 1.1.20 Erosion of material (sand production) in an oil wellbore, Section 1.1.21 Submodel stress analysis of pressure vessel closure hardware, Section 1.1.22 Using a composite layup to model a yacht hull, Section 1.1.23

1.11

BOLTED PIPE JOINT

1.1.1

AXISYMMETRIC ANALYSIS OF BOLTED PIPE FLANGE CONNECTIONS

Product: Abaqus/Standard

A bolted pipe ange connection is a common and important part of many piping systems. Such connections are typically composed of hubs of pipes, pipe anges with bolt holes, sets of bolts and nuts, and a gasket. These components interact with each other in the tightening process and when operation loads such as internal pressure and temperature are applied. Experimental and numerical studies on different types of interaction among these components are frequently reported. The studies include analysis of the bolt-up procedure that yields uniform bolt stress (Bibel and Ezell, 1992), contact analysis of screw threads (Fukuoka, 1992; Chaaban and Muzzo, 1991), and full stress analysis of the entire pipe joint assembly (Sawa et al., 1991). To establish an optimal design, a full stress analysis determines factors such as the contact stresses that govern the sealing performance, the relationship between bolt force and internal pressure, the effective gasket seating width, and the bending moment produced in the bolts. This example shows how to perform such a design analysis by using an economical axisymmetric model and how to assess the accuracy of the axisymmetric solution by comparing the results to those obtained from a simulation using a three-dimensional segment model. In addition, several three-dimensional models that use multiple levels of substructures are analyzed to demonstrate the use of substructures with a large number of retained degrees of freedom. Finally, a three-dimensional model containing stiffness matrices is analyzed to demonstrate the use of the matrix input functionality.Geometry and model

The bolted joint assembly being analyzed is depicted in Figure 1.1.11. The geometry and dimensions of the various parts are taken from Sawa et al. (1991), modied slightly to simplify the modeling. The inner wall radius of both the hub and the gasket is 25 mm. The outer wall radii of the pipe ange and the gasket are 82.5 mm and 52.5 mm, respectively. The thickness of the gasket is 2.5 mm. The pipe ange has eight bolt holes that are equally spaced in the pitch circle of radius 65 mm. The radius of the bolt hole is modied in this analysis to be the same as that of the bolt: 8 mm. The bolt head (bearing surface) is assumed to be circular, and its radius is 12 mm. The Youngs modulus is 206 GPa and the Poissons ratio is 0.3 for both the bolt and the pipe hub/ange. The gasket is modeled with either solid continuum or gasket elements. When continuum elements are used, the gaskets Youngs modulus, E, equals 68.7 GPa and its Poissons ratio, , equals 0.3. When gasket elements are used, a linear gasket pressure/closure relationship is used with the effective normal stiffness, , equal to the material Youngs modulus divided by the thickness so that 27.48 GPa/mm. Similarly a linear shear stress/shear motion relationship is used with an effective shear stiffness, , equal to the material shear modulus divided by the thickness so that 10.57 GPa/mm. The membrane behavior is specied with a Youngs modulus of 68.7 GPa and a Poissons ratio of 0.3. Sticking contact conditions are assumed in all contact areas: between the bearing surface and the ange and between the gasket and the hub. Contact between the bolt shank and the bolt hole is ignored.

1.1.11

BOLTED PIPE JOINT

The nite element idealizations of the symmetric half of the pipe joint are shown in Figure 1.1.12 and Figure 1.1.13, corresponding to the axisymmetric and three-dimensional analyses, respectively. The mesh used for the axisymmetric analysis consists of a mesh for the pipe hub/ange and gasket and a separate mesh for the bolts. In Figure 1.1.12 the top gure shows the mesh of the pipe hub and ange, with the bolt hole area shown in a lighter shade; and the bottom gure shows the overall mesh with the gasket and the bolt in place. For the axisymmetric model second-order elements with reduced integration, CAX8R, are used throughout the mesh of the pipe hub/ange. The gasket is modeled with either CAX8R solid continuum elements or GKAX6 gasket elements. Contact between the gasket and the pipe hub/ange is modeled with contact pairs between surfaces dened on the faces of elements in the contact region or between such element-based surfaces and node-based surfaces. In an axisymmetric analysis the bolts and the perforated ange must be modeled properly. The bolts are modeled as plane stress elements since they do not carry hoop stress. Second-order plane stress elements with reduced integration, CPS8R, are employed for this purpose. The contact surface denitions, which are associated with the faces of the elements, account for the plane stress condition automatically. To account for all eight bolts used in the joint, the combined cross-sectional areas of the shank and the head of the bolts must be calculated and redistributed to the bolt mesh appropriately using the area attributes for the solid elements. The contact area is adjusted automatically. Figure 1.1.14 illustrates the cross-sectional views of the bolt head and the shank. Each plane stress element represents a volume that extends out of the xy plane. For example, element A represents a volume calculated as ( ) ( ). Likewise, element B represents a volume calculated as ( ) ( ). The sectional area in the xz plane pertaining to a given element can be calculated as

where R is the bolt head radius, , or the shank radius, (depending on the element location), and and are x-coordinates of the left and right side of the given element, respectively. If the sectional areas are divided by the respective element widths, and , we obtain representative element thicknesses. Multiplying each element thickness by eight (the number of bolts in the model) produces the thickness values that are found in the *SOLID SECTION options. Sectional areas that are associated with bolt head elements located on the models contact surfaces are used to calculate the surface areas of the nodes used in dening the node-based surfaces of the model. Referring again to Figure 1.1.14, nodal contact areas for a single bolt are calculated as follows:

1.1.12

BOLTED PIPE JOINT

where through are contact areas that are associated with contact nodes 19 and through are sectional areas that are associated with bolt head elements CF. Multiplying the above areas by eight (the number of bolts in the model) provides the nodal contact areas found under the *SURFACE INTERACTION options. A common way of handling the presence of the bolt holes in the pipe ange in axisymmetric analyses is to smear the material properties used in the bolt hole area of the mesh and to use inhomogeneous material properties that correspond to a weaker material in this region. General guidelines for determining the effective material properties for perforated at plates are found in ASME Section VIII Div 2 Article 49. For the type of structure under study, which is not a at plate, a common approach to determining the effective material properties is to calculate the elasticity moduli reduction factor, which is the ratio of the ligament area in the pitch circle to the annular area of the pitch circle. In this model the annular area of the pitch circle is given by 6534.51 mm2 , and 2 the total area of the bolt holes is given by 1608.5 mm . Hence, the reduction factor is simply 0.754. The effective in-plane moduli of elasticity, and , are obtained by multiplying the respective moduli, and , by this factor. We assume material isotropy in the rz plane; thus, The modulus in the hoop direction, , should be very small and is chosen such that 106 . The in-plane shear modulus is then calculated based on the effective elasticity modulus: The shear moduli in the hoop direction are also calculated similarly but with set to zero (they are not used in an axisymmetric model). Hence, we have 155292 MPa, 0.155292 MPa, 59728 MPa, and 0.07765 MPa. These elasticity moduli are specied using *ELASTIC, TYPE=ENGINEERING CONSTANTS for the bolt hole part of the mesh. The mesh for the three-dimensional analysis without substructures, shown in Figure 1.1.13, represents a 22.5 segment of the pipe joint and employs second-order brick elements with reduced integration, C3D20R, for the pipe hub/ange and bolts. The gasket is modeled with C3D20R elements or GK3D18 elements. The top gure shows the mesh of the pipe hub and ange, and the bottom gure shows both the gasket and bolt (in the lighter color). Contact is modeled by the interaction of contact surfaces dened by grouping specic faces of the elements in the contacting regions. For three-dimensional contact where both the master and slave surfaces are deformable, the SMALL SLIDING parameter must be used on the *CONTACT PAIR option to indicate that small relative sliding occurs between contacting surfaces. No special adjustments need be made for the material properties used in the three-dimensional model because all parts are modeled appropriately. Four different meshes that use substructures to model the ange are tested. A rst-level substructure is created for the entire 22.5 segment of the ange shown in Figure 1.1.13, while the gasket and the bolt are meshed as before. The nodes on the ange in contact with the bolt cap form a node-based surface, while the nodes on the ange in contact with the gasket form another node-based surface. These node-based surfaces will form contact pairs with the master surfaces on the bolt cap and on the gasket, which are dened with *SURFACE as before. The retained degrees of freedom on the substructure include all three degrees of freedom for the nodes in these node-based surfaces as well as for the nodes on the 0 and 22.5 faces of the ange. Appropriate boundary conditions are specied at the substructure usage level.

1.1.13

BOLTED PIPE JOINT

A second-level substructure of 45 is created by reecting the rst-level substructure with respect to the 22.5 plane. The nodes on the 22.5 face belonging to the reected substructure are constrained in all three degrees of freedom to the corresponding nodes on the 22.5 face belonging to the original rst-level substructure. The half-bolt and the gasket sector corresponding to the reected substructure are also constructed by reection. The retained degrees of freedom include all three degrees of freedom of all contact node sets and of the nodes on the 0 and 45 faces of the ange. MPC-type CYCLSYM is used to impose cyclic symmetric boundary conditions on these two faces. A third-level substructure of 90 is created by reecting the original 45 second-level substructure with respect to the 45 plane and by connecting it to the original 45 substructure. The remaining part of the gasket and the bolts corresponding to the 4590 sector of the model is created by reection and appropriate constraints. In this case it is not necessary to retain any degrees of freedom on the 0 and 90 faces of the ange because this 90 substructure will not be connected to other substructures and appropriate boundary conditions can be specied at the substructure creation level. The nal substructure model is set up by mirroring the 90 mesh with respect to the symmetry plane of the gasket perpendicular to the y-axis. Thus, an otherwise large analysis ( 750,000 unknowns) when no substructures are used can be solved conveniently ( 80,000 unknowns) by using the third-level substructure twice. The sparse solver is used because it signicantly reduces the run time for this model. Finally, a three-dimensional matrix-based model is created by replacing elements for the entire 22.5 segment of the ange shown in Figure 1.1.13 with stiffness matrices, while the gasket and the bolt are meshed as before. Contact between the ange and gasket and the ange and bolt cap is modeled using node-based slave surfaces just as for the substructure models. Appropriate boundary conditions are applied as in the three-dimensional model without substructures.Loading and boundary conditions

The only boundary conditions are symmetry boundary conditions. In the axisymmetric model 0 is applied to the symmetry plane of the gasket and to the bottom of the bolts. In the three-dimensional model 0 is applied to the symmetry plane of the gasket as well as to the bottom of the bolt. The 0 and 22.5 planes are also symmetry planes. On the 22.5 plane, symmetry boundary conditions are enforced by invoking suitable nodal transformations and applying boundary conditions to local directions in this symmetry plane. These transformations are implemented using the *TRANSFORM option. On both the symmetry planes, the symmetry boundary conditions 0 are imposed everywhere except for the dependent nodes associated with the C BIQUAD MPC and nodes on one side of the contact surface. The second exception is made to avoid overconstraining problems, which arise if there is a boundary condition in the same direction as a Lagrange multiplier constraint associated with the *FRICTION, ROUGH option. In the models where substructures are used, the boundary conditions are specied depending on what substructure is used. For the rst-level 22.5 substructure the boundary conditions and constraint equations are the same as for the three-dimensional model shown in Figure 1.1.13. For the 45 secondlevel substructure the symmetry boundary conditions are enforced on the 45 plane with the constraint equation 0. A transform could have been used as well. For the 90 third-level substructure the face 90 is constrained with the boundary condition 0.

1.1.14

BOLTED PIPE JOINT

For the three-dimensional model containing matrices, nodal transformations are applied for symmetric boundary conditions. Entries in the stiffness matrices for these nodes are also in local coordinates. A clamping force of 15 kN is applied to each bolt by using the *PRE-TENSION SECTION option. The pre-tension section is identied by means of the *SURFACE option. The pre-tension is then prescribed by applying a concentrated load to the pre-tension node. In the axisymmetric analysis the actual load applied is 120 kN since there are eight bolts. In the three-dimensional model with no substructures the actual load applied is 7.5 kN since only half of a bolt is modeled. In the models using substructures all half-bolts are loaded with a 7.5 kN force. For all of the models the pre-tension section is specied about halfway down the bolt shank. Sticking contact conditions are assumed in all surface interactions in all analyses and are simulated with the *FRICTION, ROUGH and *SURFACE BEHAVIOR, NO SEPARATION options.Results and discussion

All analyses are performed as small-displacement analyses. Figure 1.1.15 shows a top view of the normal stress distributions in the gasket at the interface between the gasket and the pipe hub/ange predicted by the axisymmetric (bottom) and three-dimensional (top) analyses when solid continuum elements are used to model the gasket. The gure shows that the compressive normal stress is highest at the outer edge of the gasket, decreases radially inward, and changes from compression to tension at a radius of about 35 mm, which is consistent with ndings reported by Sawa et al. (1991). The close agreement in the overall solution between axisymmetric and three-dimensional analyses is quite apparent, indicating that, for such problems, axisymmetric analysis offers a simple yet reasonably accurate alternative to three-dimensional analysis. Figure 1.1.16 shows a top view of the normal stress distributions in the gasket at the interface between the gasket and the pipe hub/ange predicted by the axisymmetric (bottom) and three-dimensional (top) analyses when gasket elements are used to model the gasket. Close agreement in the overall solution between the axisymmetric and three-dimensional analyses is also seen in this case. The gasket starts carrying compressive load at a radius of about 40 mm, a difference of 5 mm with the previous result. This difference is the result of the gasket elements being unable to carry tensile loads in their thickness direction. This solution is physically more realistic since, in most cases, gaskets separate from their neighboring parts when subjected to tensile loading. Removing the *SURFACE BEHAVIOR, NO SEPARATION option from the gasket/ange contact surface denition in the input les that model the gasket with continuum elements yields good agreement with the results obtained in Figure 1.1.16 (since, in that case, the solid continuum elements in the gasket cannot carry tensile loading in the gasket thickness direction). The models in this example can be modied to study other factors, such as the effective seating width of the gasket or the sealing performance of the gasket under operating loads. The gasket elements offer the advantage of allowing very complex behavior to be dened in the gasket thickness direction. Gasket elements can also use any of the small-strain material models provided in Abaqus including user-dened material models. Figure 1.1.17 shows a comparison of the normal stress distributions in the gasket at the interface between the gasket and the pipe hub/ange predicted by the axisymmetric (bottom) and three-dimensional (top) analyses when isotropic material properties are prescribed for gasket elements.

1.1.15

BOLTED PIPE JOINT

The results in Figure 1.1.17 compare well with the results in Figure 1.1.15 from analyses in which solid and axisymmetric elements are used to simulate the gasket. Figure 1.1.18 shows the distribution of the normal stresses in the gasket at the interface in the plane 0. The results are plotted for the three-dimensional model containing only solid continuum elements and no substructures, for the three-dimensional model with matrices, and for the four models containing the substructures described above. An execution procedure is available to combine model and results data from two substructure output databases into a single output database. For more information, see Execution procedure for combining output from substructures, Section 3.2.17 of the Abaqus Analysis Users Manual. This example can also be used to demonstrate the effectiveness of the quasi-Newton nonlinear solver. This solver utilizes an inexpensive, approximate stiffness matrix update for several consecutive equilibrium iterations, rather than a complete stiffness matrix factorization each iteration as used in the default full Newton method. The quasi-Newton method results in an increased number of less expensive iterations, and a net savings in computing cost.Input files

boltpipeange_axi_solidgask.inp boltpipeange_axi_node.inp boltpipeange_axi_element.inp boltpipeange_3d_solidgask.inp boltpipeange_axi_gkax6.inp boltpipeange_3d_gk3d18.inp boltpipeange_3d_substr1.inp boltpipeange_3d_substr2.inp boltpipeange_3d_substr3_1.inp boltpipeange_3d_substr3_2.inp boltpipeange_3d_gen1.inp

boltpipeange_3d_gen2.inp

Axisymmetric analysis containing a gasket modeled with solid continuum elements. Node denitions for boltpipeange_axi_solidgask.inp and boltpipeange_axi_gkax6.inp. Element denitions for boltpipeange_axi_solidgask.inp. Three-dimensional analysis containing a gasket modeled with solid continuum elements. Axisymmetric analysis containing a gasket modeled with gasket elements. Three-dimensional analysis containing a gasket modeled with gasket elements. Three-dimensional analysis using the rst-level substructure (22.5 model). Three-dimensional analysis using the second-level substructure (45 model). Three-dimensional analysis using the third-level substructure once (90 model). Three-dimensional analysis using the third-level substructure twice (90 mirrored model). First-level substructure generation data referenced by boltpipeange_3d_substr1.inp and boltpipeange_3d_gen2.inp. Second-level substructure generation data referenced by boltpipeange_3d_substr2.inp and boltpipeange_3d_gen3.inp.

1.1.16

BOLTED PIPE JOINT

boltpipeange_3d_gen3.inp

boltpipeange_3d_node.inp

boltpipeange_3d_cyclsym.inp boltpipeange_3d_missnode.inp

boltpipeange_3d_isomat.inp

boltpipeange_3d_ortho.inp

boltpipeange_axi_isomat.inp

boltpipeange_3d_usr_umat.inp

boltpipeange_3d_usr_umat.f boltpipeange_3d_solidnum.inp boltpipeange_3d_matrix.inp boltpipeange_3d_stiffPID4.inp

boltpipeange_3d_stiffPID5.inp

Third-level substructure generation data referenced by boltpipeange_3d_substr3_1.inp and boltpipeange_3d_substr3_2.inp. Nodal coordinates used in boltpipeange_3d_substr1.inp, boltpipeange_3d_substr2.inp, boltpipeange_3d_substr3_1.inp, boltpipeange_3d_substr3_2.inp, boltpipeange_3d_cyclsym.inp, boltpipeange_3d_gen1.inp, boltpipeange_3d_gen2.inp, and boltpipeange_3d_gen3.inp. Same as le boltpipeange_3d_substr2.inp except that CYCLSYM type MPCs are used. Same as le boltpipeange_3d_gk3d18.inp except that the option to generate missing nodes is used for gasket elements. Same as le boltpipeange_3d_gk3d18.inp except that gasket elements are modeled as isotropic using the *MATERIAL option. Same as le boltpipeange_3d_gk3d18.inp except that gasket elements are modeled as orthotropic and the *ORIENTATION option is used. Same as le boltpipeange_axi_gkax6.inp except that gasket elements are modeled as isotropic using the *MATERIAL option. Same as le boltpipeange_3d_gk3d18.inp except that gasket elements are modeled as isotropic with user subroutine UMAT. User subroutine UMAT used in boltpipeange_3d_usr_umat.inp. Same as le boltpipeange_3d_gk3d18.inp except that solid element numbering is used for gasket elements. Three-dimensional analysis containing matrices and a gasket modeled with solid continuum elements. Matrix representing stiffness of a part of the ange segment for three-dimensional analysis containing matrices. Matrix representing stiffness of the remaining part of the ange segment for three-dimensional analysis containing matrices.

1.1.17

BOLTED PIPE JOINT

boltpipeange_3d_qn.inp

Same as le boltpipeange_3d_gk3d18.inp except that the quasi-Newton nonlinear solver is used.

References

Bibel, G. D., and R. M. Ezell, An Improved Flange Bolt-Up Procedure Using Experimentally Determined Elastic Interaction Coefcients, Journal of Pressure Vessel Technology, vol. 114, pp. 439443, 1992. Chaaban, A., and U. Muzzo, Finite Element Analysis of Residual Stresses in Threaded End Closures, Transactions of ASME, vol. 113, pp. 398401, 1991. Fukuoka, T., Finite Element Simulation of Tightening Process of Bolted Joint with a Tensioner, Journal of Pressure Vessel Technology, vol. 114, pp. 433438, 1992. Sawa, T., N. Higurashi, and H. Akagawa, A Stress Analysis of Pipe Flange Connections, Journal of Pressure Vessel Technology, vol. 113, pp. 497503, 1991.

1.1.18

BOLTED PIPE JOINT

Top View

= 4

centerline Side View 15 47 r=8 20 d = 50 d = 105 d = 130 d = 165 26

Gasket d = 50 d = 105

2.5

Bolt

24 10 80

16

Figure 1.1.11

Schematic of the bolted joint. All dimensions in mm.

1.1.19

BOLTED PIPE JOINT

2 3 1

2 3 1

Figure 1.1.12

Axisymmetric model of the bolted joint.

1.1.110

BOLTED PIPE JOINT

2

3

1

2

3

1

Figure 1.1.13

22.5 segment three-dimensional model of the bolted joint.

1.1.111

BOLTED PIPE JOINT

TOP VIEW area A

area B Rbolthead Rshank y x

WB WA z C D E F

12 3 4 5 6 7 8 9

contact nodes

HA

element A

HB element B

FRONT VIEW

Figure 1.1.14

Cross-sectional views of the bolt head and the shank.

1.1.112

BOLTED PIPE JOINT

S22 1 2 3 4 5 6 7 8 9 10 11 12

VALUE -1.00E+02 -8.90E+01 -7.81E+01 -6.72E+01 -5.63E+01 -4.54E+01 -3.45E+01 -2.36E+01 -1.27E+01 -1.81E+00 +9.09E+00 +2.00E+01 9 10 11 11 12 12 12 12 12 11 12 11 12 11 9 8 7 4 3 10 9 8 7 6 5 4 11 11 10 3 10 8 9 7 6 5 4 3 10 10 9 8 9 22 3 8 7 5 6

6

5

4 3 2 1

7 8 7

6

5

4

6

5

4

3

1 2 2 1 3 1 23 1

2 2

2 3

1

S22 1 2 3 4 5 6 7 8 9 10 11 12

VALUE -1.00E+02 -8.90E+01 -7.81E+01 -6.72E+01 -5.63E+01 -4.54E+01 -3.45E+01 -2.36E+01 -1.27E+01 -1.81E+00 +9.09E+00 +2.00E+01 10 10 11 12 11 12 12 12 12 12 12 11 12 11 9 8 2 4 3 10 11 10 11 10 11 9 11 10 9 8 7 8 7 6 5 9 8 7 6 5 10 9 4 10 9 8 9 8 9 8 7 6 5 7 6 5 4

5 6 7 6 5 4 3 2

7 8 7

6

5

4

3

2

3

2

4

3

2

3

2

4

32

2 4 3 6 5

2 3

1

Figure 1.1.15 Normal stress distribution in the gasket contact surface when solid elements are used to model the gasket: three-dimensional versus axisymmetric results.

1.1.113

BOLTED PIPE JOINT

S11 1 2 3 4 5 6 7 8 9 10 11 12

VALUE -2.00E+01 -9.09E+00 +1.82E+00 +1.27E+01 +2.36E+01 +3.45E+01 +4.55E+01 +5.64E+01 +6.73E+01 +7.82E+01 +8.91E+01 +1.00E+02

4 3 3 3 3 3 3 3

11 11 10 910 11 1 10 5 910 1 78 4 4 6 7 11 5 11 910 4 4 6 78 7 5 4 11 10 9 11 8 6 7 89 7 5 4 5 6 78 7 5 5 6 6 10 9 7 8 10 7 89 10

7

8

3 3

4

5

6

2 3

1

S11 1 2 3 4 5 6 7 8 9 10 11 12

VALUE -2.00E+01 -9.09E+00 +1.82E+00 +1.27E+01 +2.36E+01 +3.45E+01 +4.55E+01 +5.64E+01 +6.73E+01 +7.82E+01 +8.91E+01 +1.00E+02

11 910 6 4 4 3 4 3 4 3 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 6 6 6 6 5 6 5 6 7 8 910 1 1 7 8 9 10 11 7 89 10 1 1 7 8 9 10 1 1 7 8 9 10 11 7 8 7 8 910 1 1 5 6 7 8 910 11

2 3

1

Figure 1.1.16 Normal stress distribution in the gasket contact surface when gasket elements are used with direct specication of the gasket behavior: three-dimensional versus axisymmetric results.

1.1.114

BOLTED PIPE JOINT

S11 1 2 3 4 5 6 7 8 9 10 11 12

VALUE -1.00E+02 -8.91E+01 -7.82E+01 -6.73E+01 -5.64E+01 -4.55E+01 -3.45E+01 -2.36E+01 -1.27E+01 -1.82E+00 +9.09E+00 +2.00E+01

7 8 9 9 9 9 9 8 8 9 9 8 8 8 8 7

5 6 5 6

2 4 3 2 3 2 4 3

5 6 5 6

10 10 10 10 11 11 10 10 10 10

7

7

2 3 2 4 3 5 6 5 6 2 3 4 3 2 6 5 5 6 2 3 2 4 5 4 3 5 2 2

12 12 12 12 12 12 12 12 11 12 12 12 12 12 11 10 10 8 8 9 9 8 8 7 6 11 11 10 10 8 8 7 6 6

9 9

3 4 5 4 3 5

6

2 3

1

11

10

9 9

7

6

5

3 4 4 3

S11 1 2 3 4 5 6 7 8 9 10 11 12

VALUE -1.00E+02 -8.91E+01 -7.82E+01 -6.73E+01 -5.64E+01 -4.55E+01 -3.45E+01 -2.36E+01 -1.27E+01 -1.82E+00 +9.09E+00 +2.00E+01

5 7 8 8 9 9 8 10 10 11 10 11 11 11 11 12 11 12 12 12 11 10 11 11 10 10 9 8 10 9 10 8 9 10 8 10 9 7 9 8 9 7 9 8 7 7 7 6 6 6

2 4 3 2 4 3

5

4 3 2

5 6

4 3 2

5 6 5 6

4 3 2

12 12 12 12

4 3 2

5 6 5 7 8 6 5 6 5 5

4 3 2

4 3 2

7

4 3 2

2 3

1

12 12

9

8

7

6

4 3 2

Figure 1.1.17 Normal stress distribution in the gasket contact surface when gasket elements are used with isotropic material properties: three-dimensional versus axisymmetric results.

1.1.115

BOLTED PIPE JOINT

22.5_matrix 22.5_no_sup 22.5_sup 45_sup 90_sup 90r_sup

Figure 1.1.18

Normal stress distribution in the gasket contact surface along the line 0 for the models with and without substructures.

1.1.116

ELASTIC-PLASTIC COLLAPSE

1.1.2

ELASTIC-PLASTIC COLLAPSE OF A THIN-WALLED ELBOW UNDER IN-PLANE BENDING AND INTERNAL PRESSURE


Elbows are used in piping systems because they ovalize more readily than straight pipes and, thus, provide exibility in response to thermal expansion and other loadings that impose signicant displacements on the system. Ovalization is the bending of the pipe wall into an ovali.e., noncircularconguration. The elbow is, thus, behaving as a shell rather than as a beam. Straight pipe runs do not ovalize easily, so they behave essentially as beams. Thus, even under pure bending, complex interaction occurs between an elbow and the adjacent straight pipe segments; the elbow causes some ovalization in the straight pipe runs, which in turn tend to stiffen the elbow. This interaction can create signicant axial gradients of bending strain in the elbow, especially in cases where the elbow is very exible. This example provides verication of shell and elbow element modeling of such effects, through an analysis of a test elbow for which experimental results have been reported by Sobel and Newman (1979). An analysis is also included with elements of type ELBOW31B (which includes ovalization but neglects axial gradients of strain) for the elbow itself and beam elements for the straight pipe segments. This provides a comparative solution in which the interaction between the elbow and the adjacent straight pipes is neglected. The analyses predict the response up to quite large rotations across the elbow, so as to investigate possible collapse of the pipe and, particularly, the effect of internal pressure on that collapse.Geometry and model

The elbow conguration used in the study is shown in Figure 1.1.21. It is a thin-walled elbow with elbow factor

and radius ratio 3.07, so the exibility factor from Dodge and Moore (1972) is 10.3. (The exibility factor for an elbow is the ratio of the bending exibility of an elbow segment to that of a straight pipe of the same dimensions, for small displacements and elastic response.) This is an extremely exible case because the pipe wall is so thin. To demonstrate convergence of the overall moment-rotation behavior with respect to meshing, the two shell element meshes shown in Figure 1.1.22 are analyzed. Since the loading concerns in-plane bending only, it is assumed that the response is symmetric about the midplane of the system so that in the shell element model only one-half of the system need be modeled. Element type S8R5 is used, since tests have shown this to be the most cost-effective shell element in Abaqus (input les using element types S9R5, STRI65, and S8R for this example are included with the Abaqus release). The elbow element meshes replace each axial division in the coarser shell element model with one ELBOW32 or two ELBOW31 elements and use 4 or 6 Fourier modes to model the deformation around the pipe. Seven integration points are used through the pipe wall in all the analyses. This is usually adequate to

1.1.21


provide accurate modeling of the progress of yielding through the section in such cases as these, where essentially monotonic straining is expected. The ends of the system are rigidly attached to stiff plates in the experiments. These boundary conditions are easily modeled for the ELBOW elements and for the xed end in the shell element model. For the rotating end of the shell element model the shell nodes must be constrained to a beam node that represents the motion of the end plate. This is done using the *KINEMATIC COUPLING option as described below. The material is assumed to be isotropic and elastic-plastic, following the measured response of type 304 stainless steel at room temperature, as reported by Sobel and Newman (1979). Since all the analyses give results that are stiffer than the experimentally measured response, and the mesh convergence tests (results are discussed below) demonstrate that the meshes are convergent with respect to the overall response of the system, it seems that this stress-strain model may overestimate the materials actual strength.Loading

The load on the pipe has two components: a dead load, consisting of internal pressure (with a closed end condition), and a live in-plane bending moment applied to the end of the system. The pressure is applied to the model in an initial step and then held constant in the second analysis step while the bending moment is increased. The pressure values range from 0.0 to 3.45 MPa (500 lb/in2 ), which is the range of interest for design purposes. The equivalent end force associated with the closed-end condition is applied as a follower force because it rotates with the motion of the end plane.Kinematic boundary conditions

The xed end of the system is assumed to be fully built-in. The loaded end is xed into a very stiff plate. For the ELBOW element models this condition is represented by the NODEFORM boundary condition applied at this node. In the shell element model this rigid plate is represented by a single node, and the shell nodes at the end of the pipe are attached to it by using a kinematic coupling constraint and specifying that all degrees of freedom at the shell nodes are constrained to the motion of the single node.Results and discussion

The moment-rotation responses predicted by the various analysis models and measured in the experiment, all taken at zero internal pressure, are compared in Figure 1.1.23. The gure shows that the two shell models give very similar results, overestimating the experimentally measured collapse moment by about 15%. The 6-mode ELBOW element models are somewhat stiffer than the shell models, and those with 4 Fourier modes are much too stiff. This clearly shows that, for this very exible system, the ovalization of the elbow is too localized for even the 6-mode ELBOW representation to provide accurate results. Since we know that the shell models are convergent with respect to discretization, the most likely explanation for the excessive stiffness in comparison to the experimentally measured response is that the material model used in the analyses is too strong. Sobel and Newman (1979) point out that the stress-strain curve measured and used in this analysis, shown in Figure 1.1.21, has a 0.2% offset yield that is 20% higher than the Nuclear Systems Materials Handbook value for type 304 stainless

1.1.22


steel at room temperature, which suggests the possibility that the billets used for the stress-strain curve measurement may have been taken from stronger parts of the fabrication. If this is the case, it points out the likelihood that the elbow tested is rather nonuniform in strength properties in spite of the care taken in its manufacture. We are left with the conclusion that discrepancies of this magnitude cannot be eliminated in practical cases, and the design use of such analysis results must allow for them. Figure 1.1.24 compares the moment-rotation response for opening and closing moments under 0 and 3.45 MPa (500 lb/in2 ) internal pressure and shows the strong inuence of large-displacement effects. If large-displacement effects were not important, the opening and closing moments would produce the same response. However, even with a 1 relative rotation across the elbow assembly, the opening and closing moments differ by about 12%; with a 2 relative rotation, the difference is about 17%. Such magnitudes of relative rotation would not normally be considered large; in this case it is the coupling into ovalization that makes geometric nonlinearity signicant. As the rotation increases, the cases with closing moment loading show collapse, while the opening moment curves do not. In both cases internal pressure shows a strong effect on the results, which is to be expected in such a thin-walled pipeline. The level of interaction between the straight pipe and the elbows is well illustrated by the strain distribution on the outside wall, shown in Figure 1.1.25. The strain contours are slightly discontinuous at the ends of the curved elbow section because the shell thickness changes at those sections. Figure 1.1.26 shows a summary of the results from this example and Uniform collapse of straight and curved pipe segments, Section 1.1.5 of the Abaqus Benchmarks Manual. The plot shows the collapse value of the closing moment under in-plane bending as a function of internal pressure. The strong inuence of pressure on collapse is apparent. In addition, the effect of analyzing the elbow by neglecting interaction between the straight and curved segments is shown: the uniform bending results are obtained by using elements of type ELBOW31B in the bend and beams (element type B31) for the straight segments. The importance of the straight/elbow interaction is apparent. In this case the simpler analysis neglecting the interaction is conservative (in that it gives consistently lower values for the collapse moment), but this conservatism cannot be taken for granted. The analysis of Sobel and Newman (1979) also neglects interaction and agrees quite well with the results obtained here. For comparison the small-displacement limit analysis results of Goodall (1978), as well as his largedisplacement, elastic-plastic lower bound (Goodall, 1978a), are also shown in this gure. Again, the importance of large-displacement effects is apparent from that comparison. Detailed results obtained with the model that uses ELBOW31 elements are shown in Figure 1.1.27 through Figure 1.1.29. Figure 1.1.27 shows the variation of the Mises stress along the length of the piping system. The length is measured along the centerline of the pipe starting at the loaded end. The gure compares the stress distribution at the intrados (integration point 1) on the inner and outer surfaces of the elements (section points 1 and 7, respectively). Figure 1.1.28 shows the variation of the Mises stress around the circumference of two elements (451 and 751) that are located in the bend section of the model; the results are for the inner surface of the elements (section point 1). Figure 1.1.29 shows the ovalization of elements 451 and 751. A nonovalized, circular cross-section is included in the gure for comparison. From the gure it is seen that element 751, located at the center of the bend section, experiences the most severe ovalization. These three gures were produced with the aid of the elbow element postprocessing program felbow.f (Creation of a data le to facilitate the postprocessing of elbow element results: FELBOW, Section 12.1.6), written in FORTRAN. The postprocessing programs

1.1.23


felbow.C (A C++ version of FELBOW, Section 9.15.6 of the Abaqus Scripting Users Manual) and felbow.py (An Abaqus Scripting Interface version of FELBOW, Section 8.10.12 of the Abaqus Scripting Users Manual), written in C++ and Python, respectively, are also available for generating the data for gures such as Figure 1.1.28 and Figure 1.1.29. The user must ensure that the output variables are written to the output database to use these two programs.Shell-to-solid submodeling

One particular case is analyzed using the shell-to-solid submodeling technique. This problem veries the interpolation scheme in the case of double curved surfaces. A solid submodel using C3D27R elements is created around the elbow part of the pipe, spanning an angle of 40. The ner submodel mesh has three elements through the thickness, 10 elements around half of the circumference of the cylinder, and 10 elements along the length of the elbow. Both ends are driven from the global shell model made of S8R elements. The time scale of the static submodel analysis corresponds to the arc length in the global Riks analysis. The submodel results agree closely with the shell model. The *SECTION FILE option is used to output the total force and the total moment in a cross-section through the submodel.Shell-to-solid coupling

A model using the shell-to-solid coupling capability in Abaqus is included. Such a model can be used for a careful study of the stress and strain elds in the elbow. The entire elbow is meshed with C3D20R elements, and the straight pipe sections are meshed with S8R elements (see Figure 1.1.210). At each shell-to-solid interface illustrated in Figure 1.1.210, an element-based surface is dened on the edge of the solid mesh and an edge-based surface is dened on the edge of the shell mesh. The *SHELL TO SOLID COUPLING option is used in conjunction with these surfaces to couple the shell and solid meshes. Edge-based surfaces are dened at the end of each pipe segment. These surfaces are coupled to reference nodes that are dened at the center of the pipes using the *COUPLING option in conjunction with the *DISTRIBUTING option. The loading and xed boundary conditions are applied to the reference points. The advantage of using this method is that the pipe cross-sectional areas are free to deform; thus, ovalization at the ends is not constrained. The moment-rotation response of the shell-to-solid coupling model agrees very well with the results shown in Figure 1.1.24.Input files

In all the following input les (with the exception of elbowcollapse_elbow31b_b31.inp, elbowcollapse_s8r5_ne.inp, and elbowcolpse_shl2sld_s8r_c3d20r.inp) the step concerning the application of the pressure load is commented out. To include the effects of the internal pressure in any given analysis, uncomment the step denition in the appropriate input le. elbowcollapse_elbow31b_b31.inp elbowcollapse_elbow31_6four.inp elbowcollapse_elbow32_6four.inp elbowcollapse_s8r.inp ELBOW31B and B31 element model. ELBOW31 model with 6 Fourier modes. ELBOW32 model with 6 Fourier modes. S8R element model.

1.1.24


elbowcollapse_s8r5.inp elbowcollapse_s8r5_ne.inp elbowcollapse_s9r5.inp elbowcollapse_stri65.inp elbowcollapse_submod.inp elbowcolpse_shl2sld_s8r_c3d20r.inp

S8R5 element model. Finer S8R5 element model. S9R5 element model. STRI65 element model. Submodel using C3D27R elements. Shell-to-solid coupling model using S8R and C3D20R elements.

References

Dodge, W. G., and S. E. Moore, Stress Indices and Flexibility Factors for Moment Loadings on Elbows and Curved Pipes, Welding Research Council Bulletin, no. 179, 1972. Goodall, I. W., Lower Bound Limit Analysis of Curved Tubes Loaded by Combined Internal Pressure and In-Plane Bending Moment, Research Division Report RD/B/N4360, Central Electricity Generating Board, England, 1978. Goodall, I. W., Large Deformations in Plastically Deforming Curved Tubes Subjected to In-Plane Bending, Research Division Report RD/B/N4312, Central Electricity Generating Board, England, 1978a. Sobel, L. H., and S. Z. Newman, Elastic-Plastic In-Plane Bending and Buckling of an Elbow: Comparison of Experimental and Simplied Analysis Results, Westinghouse Advanced Reactors Division, Report WARDHT940002, 1979.

1.1.25


407 mm (16.02 in) 1.83 m (72.0 in)

10.4 mm (0.41 in) thickness

Moment applied here 610 mm (24.0 in) 70

60 400 50 300 Stress, MPa

40

200

30

Young's modulus: 193 GPa (28 x 106 lb/in2 ) Poisson's ratio: 0.2642

20 100 10

0 0 1 2 3 Strain, % 4 5

0

Figure 1.1.21

MLTF elbow: geometry and measured material response.

1.1.26

Stress, 103 lb/in2


Figure 1.1.22

Models for elbow/pipe interaction study.

1.1.27


1.0 Line variable 1 Experiment 2 S8R5 3 S8R5-finer mesh 4 ELBOW32 - 6 mode 5 ELBOW32 - 4 mode 6 ELBOW31 - 6 mode 7 ELBOW31 - 4 mode 8 ELBOW31 - Coarse 6 9 ELBOW31 - Coarse 4 10 ELBOW31B - 6 mode 11 ELBOW31B - 4 mode 200

End rotation, deg 4.0 7.0 10.0 9 6 8 4 1 10

13.0 5,7 11 2 3

2.0

Moment, kN-m

150

100

1.0

50

0 0.04 0.08 0.12 0.16 End rotation, rad 0.20

0 0.24

Figure 1.1.23

Moment-rotation response: mesh convergence studies.

ELBOW31closing/0 ELBOW31closing/500 ELBOW31opening/0 ELBOW31opening/500 S8R5closing/0 S8R5closing/500 S8R5opening/0 S8R5opening/500

[x10 ] 4.0 3.5 3.0

6

Moment, lb-in

2.5 2.0 1.5 1.0 0.5 0.0 0.00

0.04

0.08

0.12

0.16

0.20

0.24

End Rotation, rad

Figure 1.1.24

Moment-rotation response: pressure dependence.

1.1.28

Moment, 106 lb-in


E22

VALUE -1.56E-02 -1.35E-02 -1.14E-02 -9.40E-03 -7.33E-03 -5.26E-03 -3.19E-03 -1.12E-03 +9.47E-04 +3.01E-03 +5.08E-03 +7.15E-03 +9.22E-03 +1.12E-02

Hoop strain

E11

VALUE -1.36E-02 -1.02E-02 -6.82E-03 -3.43E-03 -4.61E-05 +3.34E-03 +6.73E-03 +1.01E-02 +1.35E-02 +1.69E-02 +2.02E-02 +2.36E-02 +2.70E-02 +3.04E-02

Axial strain

Figure 1.1.25

Strain distribution on the outside surface: closing moment case.

1.1.29


Internal pressure, lb/in2 0 275 2.4 250 Collapse moment, kN-m 225 200 175 150 125 ELBOW31B Sobel and Newman (1979), uniform bending analysis 1 2 3 4 5 6 Goodall(1978), small displacement limit analysis Goodall (1978a), large displacement 31 OW elastic-plastic B EL R5 lower bound S8 2.2 2.0 1.8 1.6 1.4 1.2 1.0 Internal pressure, MPa Collapse moment, 106 lb-in 250 500 750 1000

Figure 1.1.26

In-plane bending of an elbow, elastic-plastic collapse moment results.

MISES_I MISES_O

50.

[ x10 3 ]40.

Mises stress, psiXMIN XMAX YMIN YMAX 1.500E+00 1.322E+02 4.451E+03 5.123E+04

30.

20.

10.

0.

50. Length along pipe, in

100.

Figure 1.1.27

Mises stress distribution along the length of the piping system.

1.1.210


60.

[ x10 3 ]MISES451 MISES751 55.

50. Mises stress, psiXMIN XMAX YMIN YMAX 0.000E+00 4.892E+01 2.635E+04 5.778E+04

45.

40.

35.

30.

25. 0.

5.

10.

15.

20.

25.

30.

35.

40.

45.

50.

Length around element circumference, in

Figure 1.1.28

Mises stress distribution around the circumference of elements 451 and 751.

10.

CIRCLE OVAL_451 OVAL_751 5.

Local y-axis

0.

-5.

XMIN -7.805E+00 XMAX 7.805E+00 YMIN -8.732E+00 YMAX 8.733E+00

-10. -10.

-5.

0. Local x-axis

5.

10.

Figure 1.1.29

Ovalization of elements 451 and 751.

1.1.211


solid elements

shell elements

shell-to-solid interface

3 2

1

Figure 1.1.210

Shell-to-solid coupling model study.

1.1.212

LINEAR ELASTIC PIPELINE

1.1.3

PARAMETRIC STUDY OF A LINEAR ELASTIC PIPELINE UNDER IN-PLANE BENDING


Elbows are used in piping systems because they ovalize more readily than straight pipes and, thus, provide exibility in response to thermal expansion and other loadings that impose signicant displacements on the system. Ovalization is the bending of the pipe wall into an ovali.e., noncircularconguration. The elbow is, thus, behaving as a shell rather than as a beam. This example demonstrates the ability of elbow elements (Pipes and pipebends with deforming cross-sections: elbow elements, Section 23.5.1 of the Abaqus Analysis Users Manual) to model the nonlinear response of initially circular pipes and pipebends accurately when the distortion of the cross-section by ovalization is signicant. It also provides some guidelines on the importance of including a sufcient number of Fourier modes in the elbow elements to capture the ovalization accurately. In addition, this example illustrates the shortcomings of using exibility knockdown factors with simple beam elements in an attempt to capture the effects of ovalization in an ad hoc manner for large-displacement analyses.Geometry and model

The pipeline conguration used in the study is shown in Figure 1.1.31. It is a simple model with two straight pipe sections connected by a 90 elbow. The straight pipes are 25.4 cm (10.0 inches) in length, the radius of the curved section is 10.16 cm (4.0 inches), and the outer radius of the pipe section is 1.27 cm (0.5 inches). The wall thickness of the pipe is varied from 0.03175 cm to 0.2032 cm (0.0125 inches to 0.08 inches) in a parametric study, as discussed below. The pipe material is assumed to be isotropic linear elastic with a Youngs modulus of 194 GPa (28.1 106 psi) and a Poissons ratio of 0.0. The straight portions of the pipeline are assumed to be long enough so that warping at the ends of the structure is negligible. Two loading conditions are analyzed. The rst case is shown in Figure 1.1.31 with unit inward displacements imposed on both ends of the structure. This loading condition has the effect of closing the pipeline in on itself. In the second case the sense of the applied unit displacements is outward, opening the pipeline. Both cases are considered to be large-displacement/small-strain analyses. A parametric study comparing the results obtained with different element types (shells, elbows, and pipes) over a range of exibility factors, k, is performed. As dened in Dodge and Moore (1972), the exibility factor for an elbow is the ratio of the bending exibility of the elbow segment to that of a straight pipe of the same dimensions, assuming small displacements and an elastic response. When the internal (gauge) pressure is zero, as is assumed in this study, k can be approximated as

where

1.1.31


R is the bend radius of the curved section, r is the mean radius of the pipe, t is the wall thickness of the pipe, and is Poissons ratio. Changes in the exibility factor are introduced by varying the wall thickness of the pipe. The pipeline is modeled with three different element types: S4 shell elements, ELBOW31 elbow elements, and PIPE31 pipe elements. The S4 shell element model consists of a relatively ne mesh of 40 elements about the circumference and 75 elements along the length. This mesh is deemed ne enough to capture the true response of the pipeline accurately, although no mesh convergence studies are performed. Two analyses are conducted with the shell mesh: one with automatic stabilization using a constant damping factor (see Automatic stabilization of static problems with a constant damping factor in Solving nonlinear problems, Section 7.1.1 of the Abaqus Analysis Users Manual), and one with adaptive automatic stabilization (see Adaptive automatic stabilization scheme in Solving nonlinear problems, Section 7.1.1 of the Abaqus Analysis Users Manual). The pipe and elbow element meshes consist of 75 elements along the length; the analyses with these element types do not use automatic stabilization. The results of the shell element model with automatic stabilization using a constant damping factor are taken as the reference solution. The reaction force at the tip of the pipeline is used to evaluate the effectiveness of the pipe and elbow elements. In addition, the ovalization values of the pipeline crosssection predicted by the elbow element models are compared. The elbow elements are tested with 0, 3, and 6 Fourier modes, respectively. In general, elbow element accuracy improves as more modes are used, although the computational cost increases accordingly. In addition to standard pipe elements, tests are performed on pipe elements with a special exibility knockdown factor. Flexibility knockdown factors (Dodge and Moore, 1972) are corrections to the bending stiffness based upon linear semianalytical results. They are applied to simple beam elements in an attempt to capture the global effects of ovalization. The knockdown factor is implemented in the PIPE31 elements by scaling the true thickness by the exibility factor; this is equivalent to scaling the moment of inertia of the pipe element by .Results and discussion

The results obtained with the shell element model with automatic stabilization using a constant damping factor are taken as the reference solution. Very similar results are obtained with the same mesh using the adaptive automatic stabilization scheme. The tip reaction forces due to the inward prescribed displacements for the various analysis models are shown in Figure 1.1.32. The results are normalized with respect to those obtained with the shell model. The results obtained with the ELBOW31 element model with 6 Fourier modes show excellent agreement with the reference solution over the entire range of exibility factors considered in this study. The remaining four models generally exhibit excessively stiff response for all values of k. The PIPE31 element model, which uses the exibility knockdown factor, shows a relatively constant error of about

1.1.32


20% over the entire range of exibility factors. The 0-mode ELBOW31 element model and the PIPE31 element model without the knockdown factor produce very similar results for all values of k. The normalized tip reaction forces due to the outward unit displacement for the various analysis models are shown in Figure 1.1.33. Again, the results obtained with the 6-mode ELBOW31 element model compare well with the reference shell solution. The 0-mode and 3-mode ELBOW31 and the PIPE31 (without the exibility knockdown factor) element models exhibit overly stiff response. The PIPE31 element model with the knockdown factor has a transition region near k = 1.5, where the response changes from being too stiff to being too soft. Figure 1.1.34 and Figure 1.1.35 illustrate the effect of the number of included Fourier modes (0, 3, and 6) on the ability of the elbow elements to model the ovalization in the pipebend accurately in both load cases considered in this study. By denition, the 0-mode model cannot ovalize, which accounts for its stiff response. The 3-mode and the 6-mode models show signicant ovalization in both loading cases. Figure 1.1.36 compares the ovalization of the 6-mode model in the opened and closed deformation states. It clearly illustrates that when the ends of the pipe are displaced inward (closing mode), the height of the pipes cross-section gets smaller, thereby reducing the overall stiffness of the pipe; the reverse is true when the pipe ends are displaced outward: the height of the pipes cross-section gets larger, thereby increasing the pipe stiffness. These three gures were produced with the aid of the elbow element postprocessing program felbow.f (Creation of a data le to facilitate the postprocessing of elbow element results: FELBOW, Section 12.1.6), written in FORTRAN. The postprocessing programs felbow.C (A C++ version of FELBOW, Section 9.15.6 of the Abaqus Scripting Users Manual) and felbow.py (An Abaqus Scripting Interface version of FELBOW, Section 8.10.12 of the Abaqus Scripting Users Manual), written in C++ and Python, respectively, are also available for generating the data for these gures. The user must ensure that the output variables are written to the output database to use these two programs.Parametric study

The performance of the pipe and elbow elements investigated in this example is analyzed conveniently in a parametric study using the Python scripting capabilities of Abaqus (Scripting parametric studies, Section 15.1.1 of the Abaqus Analysis Users Manual). We perform a parametric study in which eight analyses are executed automatically for each of the three element types (S4, ELBOW31, and PIPE31) discussed above; these parametric studies correspond to wall thickness values ranging from 0.03175 cm to 0.2032 cm (0.0125 inches to 0.08 inches). The Python script le elbowtest.psf is used to perform the parametric study. The function customTable (shown below) is an example of advanced Python scripting (Lutz and Ascher, 1999), which is used in elbowtest.psf. Such advanced scripting is not routinely needed, but in this case a dependent variable such as k cannot be included as a column of data in an XYPLOT le. customTable is designed to overcome this limitation by taking an XYPLOT le from the parametric study and converting it into a new le of reaction forces versus exibility factors (k). ############################################################### # def customTable(file1, file2):

1.1.33


for line in file1.readlines(): print line nl = string.split(line,',') disp = float(nl[0]) bend_radius = float(nl[1]) wall_thick = float(nl[2]) outer_pipe_radius = float(nl[3]) poisson = float(nl[4]) rf = float(nl[6]) mean_rad = outer_pipe_radius - wall_thick/2.0 k = bend_radius*wall_thick/mean_rad**2 k = k/sqrt(1.e0 - poisson**2) k = 1.66e0/k outputstring = str(k) + ', ' + str(rf) + '\n' file2.write(outputstring) # #############################################################Input files

elbowtest_shell.inp elbowtest_shell_stabil_adap.inp elbowtest_elbow0.inp elbowtest_elbow3.inp elbowtest_elbow6.inp elbowtest_pipek.inp elbowtest_pipe.inp elbowtest.psfReferences

S4 model. S4 model with adaptive stabilization. ELBOW31 model with 0 Fourier modes. ELBOW31 model with 3 Fourier modes. ELBOW31 model with 6 Fourier modes. PIPE31 model with the exibility knockdown factor. PIPE31 model without the exibility knockdown factor. Python script le for the parametric study.

Dodge, W. G., and S. E. Moore, Stress Indices and Flexibility Factors for Moment Loadings on Elbows and Curved Pipes, Welding Research Council Bulletin, no. 179, 1972. Lutz, M., and D. Ascher, Learning Python, OReilly, 1999.

1.1.34


a R

u

a r t pipe cross-sectionFigure 1.1.31

u

Pipeline geometry with inward prescribed tip displacements.

ELBOW31 0 modes ELBOW31 3 modes ELBOW31 6 modes PIPE31 PIPE31 with knockdown Shell S4

Figure 1.1.32

Normalized tip reaction force: closing displacement case.

1.1.35


ELBOW31 0 modes ELBOW31 3 modes ELBOW31 6 modes PIPE31 PIPE31 with knockdown Shell S4

Figure 1.1.33

Normalized tip reaction force: opening displacement case.

close-0 close-3 close-6

Figure 1.1.34 Ovalization of the ELBOW31 cross-sections for 0, 3, and 6 Fourier modes: closing displacement case.

1.1.36


open-0 open-3 open-6

Figure 1.1.35 Ovalization of the ELBOW31 cross-sections for 0, 3, and 6 Fourier modes: opening displacement case.

close-6 open-6

Figure 1.1.36 Ovalization of the ELBOW31 cross-sections for 6 Fourier modes: opening and closing displacement cases.

1.1.37

ELASTOMERIC FOAM INDENTATION

1.1.4

INDENTATION OF AN ELASTOMERIC FOAM SPECIMEN WITH A HEMISPHERICAL PUNCH

Products: Abaqus/Standard

Abaqus/Explicit

Abaqus/Design

In this example we consider a cylindrical specimen of an elastomeric foam, indented by a rough, rigid, hemispherical punch. Examples of elastomeric foam materials are cellular polymers such as cushions, padding, and packaging materials. This problem illustrates a typical application of elastomeric foam materials when used in energy absorption devices. The same geometry as the crushable foam model of Simple tests on a crushable foam specimen, Section 3.2.7 of the Abaqus Benchmarks Manual, is used but with a slightly different mesh. Design sensitivity analysis is carried out for a shape design parameter and a material design parameter to illustrate the usage of design sensitivity analysis for a problem involving contact.Geometry and model

The axisymmetric model (135 linear 4-node elements) analyzed is shown in Figure 1.1.41. The mesh renement is biased toward the center of the foam specimen where the largest deformation is expected. The foam specimen has a radius of 600 mm and a thickness of 300 mm. The punch has a radius of 200 mm. The bottom nodes of the mesh are xed, while the outer boundary is free to move. A contact pair is dened between the punch, which is modeled by a rough spherical rigid surface, and a slave surface composed of the faces of the axisymmetric elements in the contact region. The friction coefcient between the punch and the foam is 0.8. A point mass of 200 kg representing the weight of the punch is attached to the rigid body reference node. The model is analyzed in both Abaqus/Standard and Abaqus/Explicit.Material

The elastomeric foam material is dened with the *HYPERFOAM option using experimental test data. The uniaxial compression and simple shear data whose stress-strain curves are shown in Figure 1.1.42 are dened with the *UNIAXIAL TEST DATA and *SIMPLE SHEAR TEST DATA options. Other available test data options are *BIAXIAL TEST DATA, *PLANAR TEST DATA and *VOLUMETRIC TEST DATA. The test data are dened in terms of nominal stress and nominal strain values. Abaqus performs a nonlinear least-squares t of the test data to determine the hyperfoam coefcients and . Details of the formulation and usage of the hyperfoam model are given in Hyperelastic behavior in elastomeric foams, Section 17.5.2 of the Abaqus Analysis Users Manual; Hyperelastic material behavior, Section 4.6.1 of the Abaqus Theory Manual; and Fitting of hyperelastic and hyperfoam constants, Section 4.6.2 of the Abaqus Theory Manual. Fitting of elastomeric foam test data, Section 3.1.5 of the Abaqus Benchmarks Manual, illustrates the tting of elastomeric foam test data to derive the hyperfoam coefcients.

1.1.41


For the material used in this example, is zero, since the effective Poissons ratio, , is zero as specied by the POISSON parameter. The order of the series expansion is chosen to be 2 since this ts the test data with sufcient accuracy. It also provides a more stable model than the 3 case. The viscoelastic properties in Abaqus are specied in terms of a relaxation curve (shown in Figure 1.1.43) of the normalized modulus , where is the shear or bulk modulus as a function of time and is the instantaneous modulus as determined from the hyperfoam model. This requires the use of the TIME=RELAXATION TEST DATA parameter on the *VISCOELASTIC option. The relaxation data are specied with the *SHEAR TEST DATA option but actually apply to both shear and bulk moduli when used in conjunction with the hyperfoam model. Abaqus performs a nonlinear least-squares t of the relaxation data to a Prony series to determine the coefcients, , and the relaxation periods, . A maximum order of 2 is used for tting the Prony series. If creep data are available, the TIME=CREEP TEST DATA parameter is used to specify normalized creep compliance data. A rectangular material orientation is dened for the foam specimen, so stress and strain are reported in material axes that rotate with the element deformation. This is especially useful when looking at the stress and strain values in the region of the foam in contact with the punch in the direction normal to the punch (direction 22). The rough surface of the punch is modeled by specifying a friction coefcient of 0.8 for the contact surface interaction with the *FRICTION option under the *SURFACE INTERACTION denition.Procedure and loading definitions

Two cases are analyzed. In the rst case the punch is displaced statically downward to indent the foam, and the reaction force-displacement relation is measured for both the purely elastic and viscoelastic cases. In the second case the punch statically indents the foam through gravity loading and is then subjected to impulsive loading. The dynamic response of the punch is sought as it interacts with the viscoelastic foam.Case 1

In Abaqus/Standard the punch is displaced downward by a prescribed displacement boundary condition in the rst step, indenting the foam specimen by a distance of 250 mm. The NLGEOM parameter is specied on the *STEP option, since the response involves large deformation. In the second step the punch is displaced back to its original position. Two analyses are performedone using the *STATIC procedure for both steps and the other using the *VISCO procedure for both steps. During a *STATIC step the material behaves purely elastically, using the properties specied with the hyperfoam model. The *VISCO, *DYNAMIC, or *COUPLED TEMPERATURE-DISPLACEMENT procedure must be used to activate the viscoelastic behavior. In this case the punch is pushed down in a period of one second and then moved back up again in one second. The accuracy of the creep integration in the *VISCO procedure is controlled by the CETOL parameter and is typically calculated by dividing an acceptable stress error tolerance by a typical elastic modulus. In this problem we estimate a stress error tolerance of about 0.005 MPa and use the initial elastic modulus, E 2 0.34, to determine a CETOL of 0.01.

1.1.42


In Abaqus/Explicit the punch is also displaced downward by a prescribed displacement boundary condition, indenting the foam by a depth of 250 mm. The punch is then lifted back to its original position. In this case the punch is modeled as either an analytical rigid surface or a discrete rigid surface dened with RAX2 elements. The entire analysis runs for 2 seconds. The actual time period of the analysis is large by explicit dynamic standards. Hence, to reduce the computational time, the mass density of the elements is increased articially to increase the stable time increment without losing the accuracy of the solution. The mass scaling factor is set to 10 using the *FIXED MASS SCALING option, which corresponds to a speedup factor of . The reaction force-displacement relation is measured for both the elastic and viscoelastic cases.Case 2

The Abaqus/Standard analysis is composed of three steps. The rst step is a *VISCO step, where gravity loading is applied to the point mass of the punch. The gravity loading is ramped up in two seconds, and the step is run for a total of ve seconds to allow the foam to relax fully. In the second step, which is a *DYNAMIC step, an impulsive load in the form of a half sine wave amplitude with a peak magnitude of 5000 N is applied to the punch over a period of one second. In the third step, also a *DYNAMIC step, the punch is allowed to move freely until the vibration is damped out by the viscoelastic foam. For a dynamic analysis with automatic time incrementation, the value of the HAFTOL (half-increment residual tolerance) parameter for the *DYNAMIC procedure controls the accuracy of the time integration. For systems that have signicant energy dissipation, such as t

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