abaqus 6.11, example problems, vol. 1

Abaqus Example Problems Manual

Abaqus Version 6.11 ID:Printed on:

Abaqus 6.11Example Problems ManualVolume I: Static and Dynamic Analyses

Abaqus

Example Problems Manual

Volume I


Legal NoticesCAUTION: This documentation is intended for qualified users who will exercise sound engineering judgment and expertise in the use of the AbaqusSoftware. The Abaqus Software is inherently complex, and the examples and procedures in this documentation are not intended to be exhaustive or to applyto any particular situation. Users are cautioned to satisfy themselves as to the accuracy and results of their analyses.

Dassault Systmes and its subsidiaries, including Dassault Systmes Simulia Corp., shall not be responsible for the accuracy or usefulness of any analysisperformed using the Abaqus Software or the procedures, examples, or explanations in this documentation. Dassault Systmes and its subsidiaries shall notbe responsible for the consequences of any errors or omissions that may appear in this documentation.

The Abaqus Software is available only under license from Dassault Systmes or its subsidiary and may be used or reproduced only in accordance with theterms of such license. This documentation is subject to the terms and conditions of either the software license agreement signed by the parties, or, absentsuch an agreement, the then current software license agreement to which the documentation relates.

This documentation and the software described in this documentation are subject to change without prior notice.

No part of this documentation may be reproduced or distributed in any form without prior written permission of Dassault Systmes or its subsidiary.

The Abaqus Software is a product of Dassault Systmes Simulia Corp., Providence, RI, USA.

Dassault Systmes, 2011

Abaqus, the 3DS logo, SIMULIA, CATIA, and Unified FEA are trademarks or registered trademarks of Dassault Systmes or its subsidiaries in the UnitedStates and/or other countries.

Other company, product, and service names may be trademarks or service marks of their respective owners. For additional information concerningtrademarks, copyrights, and licenses, see the Legal Notices in the Abaqus 6.11 Release Notes.


LocationsSIMULIA Worldwide Headquarters Rising Sun Mills, 166 Valley Street, Providence, RI 029092499, Tel: +1 401 276 4400,

Fax: +1 401 276 4408, [email protected], http://www.simulia.comSIMULIA European Headquarters Stationsplein 8-K, 6221 BT Maastricht, The Netherlands, Tel: +31 43 7999 084,

Fax: +31 43 7999 306, [email protected]

Dassault Systmes Centers of Simulation ExcellenceUnited States Fremont, CA, Tel: +1 510 794 5891, [email protected]

West Lafayette, IN, Tel: +1 765 497 1373, [email protected], MI, Tel: +1 248 349 4669, [email protected], MN, Tel: +1 612 424 9044, [email protected], OH, Tel: +1 216 378 1070, [email protected] Chester, OH, Tel: +1 513 275 1430, [email protected], RI, Tel: +1 401 739 3637, [email protected], TX, Tel: +1 972 221 6500, [email protected]

Australia Richmond VIC, Tel: +61 3 9421 2900, [email protected] Vienna, Tel: +43 1 22 707 200, [email protected] Maarssen, The Netherlands, Tel: +31 346 585 710, [email protected] Toronto, ON, Tel: +1 416 402 2219, [email protected] Beijing, P. R. China, Tel: +8610 6536 2288, [email protected]

Shanghai, P. R. China, Tel: +8621 3856 8000, [email protected] Espoo, Tel: +358 40 902 2973, [email protected] Velizy Villacoublay Cedex, Tel: +33 1 61 62 72 72, [email protected] Aachen, Tel: +49 241 474 01 0, [email protected]

Munich, Tel: +49 89 543 48 77 0, [email protected] Chennai, Tamil Nadu, Tel: +91 44 43443000, [email protected] Lainate MI, Tel: +39 02 334306150, [email protected] Tokyo, Tel: +81 3 5442 6300, [email protected]

Osaka, Tel: +81 6 7730 2703, [email protected], Kanagawa, Tel: +81 45 470 9381, [email protected]

Korea Mapo-Gu, Seoul, Tel: +82 2 785 6707/8, [email protected] America Puerto Madero, Buenos Aires, Tel: +54 11 4312 8700, [email protected] Vsters, Sweden, Tel: +46 21 150870, [email protected] Kingdom Warrington, Tel: +44 1925 830900, [email protected]

Authorized Support CentersCzech & Slovak Republics Synerma s. r. o., Psry, Prague-West, Tel: +420 603 145 769, [email protected] 3 Dimensional Data Systems, Crete, Tel: +30 2821040012, [email protected] ADCOM, Givataim, Tel: +972 3 7325311, [email protected] WorleyParsons Advanced Analysis, Kuala Lumpur, Tel: +603 2039 9000, [email protected] Zealand Matrix Applied Computing Ltd., Auckland, Tel: +64 9 623 1223, [email protected] BudSoft Sp. z o.o., Pozna, Tel: +48 61 8508 466, [email protected], Belarus & Ukraine TESIS Ltd., Moscow, Tel: +7 495 612 44 22, [email protected] WorleyParsons Advanced Analysis, Singapore, Tel: +65 6735 8444, [email protected] Africa Finite Element Analysis Services (Pty) Ltd., Parklands, Tel: +27 21 556 6462, [email protected] & Portugal Principia Ingenieros Consultores, S.A., Madrid, Tel: +34 91 209 1482, [email protected] Simutech Solution Corporation, Taipei, R.O.C., Tel: +886 2 2507 9550, [email protected] WorleyParsons Advanced Analysis, Singapore, Tel: +65 6735 8444, [email protected] A-Ztech Ltd., Istanbul, Tel: +90 216 361 8850, [email protected]

Complete contact information is available at http://www.simulia.com/locations/locations.html.


PrefaceThis section lists various resources that are available for help with using Abaqus Unified FEA software.

Support

Both technical engineering support (for problems with creating a model or performing an analysis) andsystems support (for installation, licensing, and hardware-related problems) for Abaqus are offered througha network of local support offices. Regional contact information is listed in the front of each Abaqus manualand is accessible from the Locations page at www.simulia.com.

Support for SIMULIA productsSIMULIA provides a knowledge database of answers and solutions to questions that we have answered,as well as guidelines on how to use Abaqus, SIMULIA Scenario Definition, Isight, and other SIMULIAproducts. You can also submit new requests for support. All support incidents are tracked. If you contactus by means outside the system to discuss an existing support problem and you know the incident or supportrequest number, please mention it so that we can consult the database to see what the latest action has been.

Many questions about Abaqus can also be answered by visiting the Products page and the Supportpage at www.simulia.com.

Anonymous ftp siteTo facilitate data transfer with SIMULIA, an anonymous ftp account is available on the computerftp.simulia.com. Login as user anonymous, and type your e-mail address as your password. Contact supportbefore placing files on the site.

Training

All offices and representatives offer regularly scheduled public training classes. The courses are offered ina traditional classroom form and via the Web. We also provide training seminars at customer sites. Alltraining classes and seminars include workshops to provide as much practical experience with Abaqus aspossible. For a schedule and descriptions of available classes, see www.simulia.com or call your local officeor representative.

Feedback

We welcome any suggestions for improvements to Abaqus software, the support program, or documentation.We will ensure that any enhancement requests you make are considered for future releases. If you wish tomake a suggestion about the service or products, refer to www.simulia.com. Complaints should be addressedby contacting your local office or through www.simulia.com by visiting the Quality Assurance section ofthe Support page.


CONTENTS

Contents

Volume I

1. Static Stress/Displacement AnalysesStatic and quasi-static stress analysesAxisymmetric analysis of bolted pipe flange connections 1.1.1

Elastic-plastic collapse of a thin-walled elbow under in-plane bending and internal

pressure 1.1.2

Parametric study of a linear elastic pipeline under in-plane bending 1.1.3

Indentation of an elastomeric foam specimen with a hemispherical punch 1.1.4

Collapse of a concrete slab 1.1.5

Jointed rock slope stability 1.1.6

Notched beam under cyclic loading 1.1.7

Uniaxial ratchetting under tension and compression 1.1.8

Hydrostatic fluid elements: modeling an airspring 1.1.9

Shell-to-solid submodeling and shell-to-solid coupling of a pipe joint 1.1.10

Stress-free element reactivation 1.1.11

Transient loading of a viscoelastic bushing 1.1.12

Indentation of a thick plate 1.1.13

Damage and failure of a laminated composite plate 1.1.14

Analysis of an automotive boot seal 1.1.15

Pressure penetration analysis of an air duct kiss seal 1.1.16

Self-contact in rubber/foam components: jounce bumper 1.1.17

Self-contact in rubber/foam components: rubber gasket 1.1.18

Submodeling of a stacked sheet metal assembly 1.1.19

Axisymmetric analysis of a threaded connection 1.1.20

Direct cyclic analysis of a cylinder head under cyclic thermal-mechanical loadings 1.1.21

Erosion of material (sand production) in an oil wellbore 1.1.22

Submodel stress analysis of pressure vessel closure hardware 1.1.23

Using a composite layup to model a yacht hull 1.1.24

Buckling and collapse analysesSnap-through buckling analysis of circular arches 1.2.1

Laminated composite shells: buckling of a cylindrical panel with a circular hole 1.2.2

Buckling of a column with spot welds 1.2.3

Elastic-plastic K-frame structure 1.2.4

Unstable static problem: reinforced plate under compressive loads 1.2.5

Buckling of an imperfection-sensitive cylindrical shell 1.2.6

i

Abaqus ID:exa-toc

Printed on: Fri March 25 -- 17:19:54 2011

CONTENTS

Forming analysesUpsetting of a cylindrical billet: quasi-static analysis with mesh-to-mesh solution

mapping (Abaqus/Standard) and adaptive meshing (Abaqus/Explicit) 1.3.1

Superplastic forming of a rectangular box 1.3.2

Stretching of a thin sheet with a hemispherical punch 1.3.3

Deep drawing of a cylindrical cup 1.3.4

Extrusion of a cylindrical metal bar with frictional heat generation 1.3.5

Rolling of thick plates 1.3.6

Axisymmetric forming of a circular cup 1.3.7

Cup/trough forming 1.3.8

Forging with sinusoidal dies 1.3.9

Forging with multiple complex dies 1.3.10

Flat rolling: transient and steady-state 1.3.11

Section rolling 1.3.12

Ring rolling 1.3.13

Axisymmetric extrusion: transient and steady-state 1.3.14

Two-step forming simulation 1.3.15

Upsetting of a cylindrical billet: coupled temperature-displacement and adiabatic

analysis 1.3.16

Unstable static problem: thermal forming of a metal sheet 1.3.17

Inertia welding simulation using Abaqus/Standard and Abaqus/CAE 1.3.18

Fracture and damageA plate with a part-through crack: elastic line spring modeling 1.4.1

Contour integrals for a conical crack in a linear elastic infinite half space 1.4.2

Elastic-plastic line spring modeling of a finite length cylinder with a part-through axial

flaw 1.4.3

Crack growth in a three-point bend specimen 1.4.4

Analysis of skin-stiffener debonding under tension 1.4.5

Failure of blunt notched fiber metal laminates 1.4.6

Debonding behavior of a double cantilever beam 1.4.7

Debonding behavior of a single leg bending specimen 1.4.8

Postbuckling and growth of delaminations in composite panels 1.4.9

Import analysesSpringback of two-dimensional draw bending 1.5.1

Deep drawing of a square box 1.5.2

2. Dynamic Stress/Displacement AnalysesDynamic stress analysesNonlinear dynamic analysis of a structure with local inelastic collapse 2.1.1

Detroit Edison pipe whip experiment 2.1.2

ii

Abaqus ID:exa-toc


CONTENTS

Rigid projectile impacting eroding plate 2.1.3

Eroding projectile impacting eroding plate 2.1.4

Tennis racket and ball 2.1.5

Pressurized fuel tank with variable shell thickness 2.1.6

Modeling of an automobile suspension 2.1.7

Explosive pipe closure 2.1.8

Knee bolster impact with general contact 2.1.9

Crimp forming with general contact 2.1.10

Collapse of a stack of blocks with general contact 2.1.11

Cask drop with foam impact limiter 2.1.12

Oblique impact of a copper rod 2.1.13

Water sloshing in a baffled tank 2.1.14

Seismic analysis of a concrete gravity dam 2.1.15

Progressive failure analysis of thin-wall aluminum extrusion under quasi-static and

dynamic loads 2.1.16

Impact analysis of a pawl-ratchet device 2.1.17

High-velocity impact of a ceramic target 2.1.18

Mode-based dynamic analysesAnalysis of a rotating fan using substructures and cyclic symmetry 2.2.1

Linear analysis of the Indian Point reactor feedwater line 2.2.2

Response spectra of a three-dimensional frame building 2.2.3

Brake squeal analysis 2.2.4

Dynamic analysis of antenna structure utilizing residual modes 2.2.5

Steady-state dynamic analysis of a vehicle body-in-white model 2.2.6

Eulerian analysesRivet forming 2.3.1

Impact of a water-filled bottle 2.3.2

Co-simulation analysesClosure of an air-filled door seal 2.4.1

Dynamic impact of a scooter with a bump 2.4.2

iii

Abaqus ID:exa-toc


CONTENTS

Volume II

3. Tire and Vehicle AnalysesTire analysesSymmetric results transfer for a static tire analysis 3.1.1

Steady-state rolling analysis of a tire 3.1.2

Subspace-based steady-state dynamic tire analysis 3.1.3

Steady-state dynamic analysis of a tire substructure 3.1.4

Coupled acoustic-structural analysis of a tire filled with air 3.1.5

Import of a steady-state rolling tire 3.1.6

Analysis of a solid disc with Mullins effect and permanent set 3.1.7

Tread wear simulation using adaptive meshing in Abaqus/Standard 3.1.8

Dynamic analysis of an air-filled tire with rolling transport effects 3.1.9

Acoustics in a circular duct with flow 3.1.10

Vehicle analysesInertia relief in a pick-up truck 3.2.1

Substructure analysis of a pick-up truck model 3.2.2

Display body analysis of a pick-up truck model 3.2.3

Continuum modeling of automotive spot welds 3.2.4

Occupant safety analysesSeat belt analysis of a simplified crash dummy 3.3.1

Side curtain airbag impactor test 3.3.2

4. Mechanism AnalysesResolving overconstraints in a multi-body mechanism model 4.1.1

Crank mechanism 4.1.2

Snubber-arm mechanism 4.1.3

Flap mechanism 4.1.4

Tail-skid mechanism 4.1.5

Cylinder-cam mechanism 4.1.6

Driveshaft mechanism 4.1.7

Geneva mechanism 4.1.8

Trailing edge flap mechanism 4.1.9

Substructure analysis of a one-piston engine model 4.1.10

Application of bushing connectors in the analysis of a three-point linkage 4.1.11

Gear assemblies 4.1.12

5. Heat Transfer and Thermal-Stress AnalysesThermal-stress analysis of a disc brake 5.1.1

iv

Abaqus ID:exa-toc


CONTENTS

A sequentially coupled thermal-mechanical analysis of a disc brake with an Eulerian

approach 5.1.2

Exhaust manifold assemblage 5.1.3

Coolant manifold cover gasketed joint 5.1.4

Conductive, convective, and radiative heat transfer in an exhaust manifold 5.1.5

Thermal-stress analysis of a reactor pressure vessel bolted closure 5.1.6

6. Fluid Dynamics and Fluid-Structure InteractionConjugate heat transfer analysis of a component-mounted electronic circuit board 6.1.1

7. Electromagnetic AnalysesPiezoelectric analysesEigenvalue analysis of a piezoelectric transducer 7.1.1

Transient dynamic nonlinear response of a piezoelectric transducer 7.1.2

Joule heating analysesThermal-electrical modeling of an automotive fuse 7.2.1

8. Mass Diffusion AnalysesHydrogen diffusion in a vessel wall section 8.1.1

Diffusion toward an elastic crack tip 8.1.2

9. Acoustic and Shock AnalysesFully and sequentially coupled acoustic-structural analysis of a muffler 9.1.1

Coupled acoustic-structural analysis of a speaker 9.1.2

Response of a submerged cylinder to an underwater explosion shock wave 9.1.3

Convergence studies for shock analyses using shell elements 9.1.4

UNDEX analysis of a detailed submarine model 9.1.5

Coupled acoustic-structural analysis of a pick-up truck 9.1.6

Long-duration response of a submerged cylinder to an underwater explosion 9.1.7

Deformation of a sandwich plate under CONWEP blast loading 9.1.8

10. Soils AnalysesPlane strain consolidation 10.1.1

Calculation of phreatic surface in an earth dam 10.1.2

Axisymmetric simulation of an oil well 10.1.3

Analysis of a pipeline buried in soil 10.1.4

Hydraulically induced fracture in a well bore 10.1.5

Permafrost thawingpipeline interaction 10.1.6

v

Abaqus ID:exa-toc


CONTENTS

11. Structural Optimization AnalysesTopology optimization analysesTopology optimization of an automotive control arm 11.1.1

Shape optimization analysesShape optimization of a connecting rod 11.2.1

12. Abaqus/Aqua AnalysesJack-up foundation analyses 12.1.1

Riser dynamics 12.1.2

13. Design Sensitivity AnalysesOverviewDesign sensitivity analysis: overview 13.1.1

ExamplesDesign sensitivity analysis of a composite centrifuge 13.2.1

Design sensitivities for tire inflation, footprint, and natural frequency analysis 13.2.2

Design sensitivity analysis of a windshield wiper 13.2.3

Design sensitivity analysis of a rubber bushing 13.2.4

14. Postprocessing of Abaqus Results FilesUser postprocessing of Abaqus results files: overview 14.1.1

Joining data from multiple results files and converting file format: FJOIN 14.1.2

Calculation of principal stresses and strains and their directions: FPRIN 14.1.3

Creation of a perturbed mesh from original coordinate data and eigenvectors: FPERT 14.1.4

Output radiation viewfactors and facet areas: FRAD 14.1.5

Creation of a data file to facilitate the postprocessing of elbow element results:

FELBOW 14.1.6

vi

Abaqus ID:exa-toc


INTRODUCTION

1.0 INTRODUCTION

This is the Example Problems Manual for Abaqus. It contains many solved examples that illustrate the use ofthe program for common types of problems. Some of the problems are quite difficult and require combinationsof the capabilities in the code.

The problems have been chosen to serve two purposes: to verify the capabilities in Abaqus by exercisingthe code on nontrivial cases and to provide guidance to users whomust work on a class of problems with whichthey are relatively unfamiliar. In each worked example the discussion in the manual states why the exampleis 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 elementtypes, mesh densities, and other variations.

Input data files for all of the analyses are included with the Abaqus release in compressed archivefiles. The abaqus fetch utility is used to extract these input files for use. For example, to fetch input fileboltpipeflange_3d_cyclsym.inp, type

abaqus fetch job=boltpipeflange_3d_cyclsym.inpParametric study script (.psf) and user subroutine (.f) files can be fetched in the same manner. All files fora particular problem can be obtained by leaving off the file extension. The abaqus fetch utility is explainedin detail in Fetching sample input files, Section 3.2.13 of the Abaqus Analysis Users Manual.

It is sometimes useful to search the input files. The findkeyword utility is used to locate input filesthat contain user-specified input. This utility is defined in Querying the keyword/problem database,Section 3.2.12 of the Abaqus Analysis Users Manual.

To reproduce the graphical representation of the solution reported in some of the examples, the outputfrequency used in the input files may need to be increased. For example, in Linear analysis of the IndianPoint reactor feedwater line, Section 2.2.2, the figures that appear in the manual can be obtained only if thesolution is written to the results file every increment; that is, if the input files 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 suiteof test problems) and standard analyses used to evaluate the performance of Abaqus. The tests in thismanual are multiple element tests of simple geometries or simplified versions of real problems. The AbaqusVerification Manual contains a large number of examples that are intended as elementary verification of thebasic modeling capabilities.

The qualification process for new Abaqus releases includes running and verifying results for all problemsin the Abaqus Example Problems Manual, the Abaqus Benchmarks Manual, and the Abaqus VerificationManual.

1.01


STATIC STRESS/DISPLACEMENT ANALYSES

1. Static Stress/Displacement Analyses

Static 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 flange 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 Uniaxial ratchetting under tension and compression, Section 1.1.8 Hydrostatic fluid elements: modeling an airspring, Section 1.1.9 Shell-to-solid submodeling and shell-to-solid coupling of a pipe joint, Section 1.1.10 Stress-free element reactivation, Section 1.1.11 Transient loading of a viscoelastic bushing, Section 1.1.12 Indentation of a thick plate, Section 1.1.13 Damage and failure of a laminated composite plate, Section 1.1.14 Analysis of an automotive boot seal, Section 1.1.15 Pressure penetration analysis of an air duct kiss seal, Section 1.1.16 Self-contact in rubber/foam components: jounce bumper, Section 1.1.17 Self-contact in rubber/foam components: rubber gasket, Section 1.1.18 Submodeling of a stacked sheet metal assembly, Section 1.1.19 Axisymmetric analysis of a threaded connection, Section 1.1.20 Direct cyclic analysis of a cylinder head under cyclic thermal-mechanical loadings, Section 1.1.21 Erosion of material (sand production) in an oil wellbore, Section 1.1.22 Submodel stress analysis of pressure vessel closure hardware, Section 1.1.23 Using a composite layup to model a yacht hull, Section 1.1.24

1.11


BOLTED PIPE JOINT

1.1.1 AXISYMMETRIC ANALYSIS OF BOLTED PIPE FLANGE CONNECTIONS

Product: Abaqus/StandardA bolted pipe flange connection is a common and important part of many piping systems. Such connectionsare typically composed of hubs of pipes, pipe flanges 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 internalpressure and temperature are applied. Experimental and numerical studies on different types of interactionamong these components are frequently reported. The studies include analysis of the bolt-up procedurethat 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). Toestablish an optimal design, a full stress analysis determines factors such as the contact stresses that governthe sealing performance, the relationship between bolt force and internal pressure, the effective gasket seatingwidth, and the bending moment produced in the bolts. This example shows how to perform such a designanalysis by using an economical axisymmetric model and how to assess the accuracy of the axisymmetricsolution by comparing the results to those obtained from a simulation using a three-dimensional segmentmodel. In addition, several three-dimensional models that use multiple levels of substructures are analyzedto demonstrate the use of substructures with a large number of retained degrees of freedom. Finally, athree-dimensional model containing stiffness matrices is analyzed to demonstrate the use of the matrix inputfunctionality.

Geometry and model

The bolted joint assembly being analyzed is depicted in Figure 1.1.11. The geometry and dimensionsof the various parts are taken from Sawa et al. (1991), modified slightly to simplify the modeling. Theinner wall radius of both the hub and the gasket is 25 mm. The outer wall radii of the pipe flange and thegasket are 82.5 mm and 52.5 mm, respectively. The thickness of the gasket is 2.5 mm. The pipe flangehas eight bolt holes that are equally spaced in the pitch circle of radius 65 mm. The radius of the bolthole is modified 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 pipehub/flange. The gasket is modeled with either solid continuum or gasket elements. When continuumelements are used, the gaskets Youngs modulus, E, equals 68.7 GPa and its Poissons ratio, , equals0.3.

When gasket elements are used, a linear gasket pressure/closure relationship is used with theeffective normal stiffness, , equal to the material Youngs modulus divided by the thickness sothat 27.48 GPa/mm. Similarly a linear shear stress/shear motion relationship is used with aneffective shear stiffness, , equal to the material shear modulus divided by the thickness so that

10.57 GPa/mm. The membrane behavior is specified with a Youngs modulus of 68.7 GPa and aPoissons ratio of 0.3. Sticking contact conditions are assumed in all contact areas: between the bearingsurface and the flange and between the gasket and the hub. Contact between the bolt shank and the bolthole is ignored.

1.1.11


BOLTED PIPE JOINT

The finite element idealizations of the symmetric half of the pipe joint are shown in Figure 1.1.12and 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/flange and gasket and aseparate mesh for the bolts. In Figure 1.1.12 the top figure shows the mesh of the pipe hub and flange,with the bolt hole area shown in a lighter shade; and the bottom figure shows the overall mesh with thegasket and the bolt in place.

For the axisymmetric model second-order elements with reduced integration, CAX8R, are usedthroughout the mesh of the pipe hub/flange. The gasket is modeled with either CAX8R solid continuumelements or GKAX6 gasket elements. Contact between the gasket and the pipe hub/flange is modeledwith contact pairs between surfaces defined on the faces of elements in the contact region or between suchelement-based surfaces and node-based surfaces. In an axisymmetric analysis the bolts and the perforatedflange must be modeled properly. The bolts are modeled as plane stress elements since they do not carryhoop stress. Second-order plane stress elements with reduced integration, CPS8R, are employed for thispurpose. The contact surface definitions, which are associated with the faces of the elements, accountfor the plane stress condition automatically. To account for all eight bolts used in the joint, the combinedcross-sectional areas of the shank and the head of the bolts must be calculated and redistributed to thebolt mesh appropriately using the area attributes for the solid elements. The contact area is adjustedautomatically.

Figure 1.1.14 illustrates the cross-sectional views of the bolt head and the shank. Each plane stresselement represents a volume that extends out of the xy plane. For example, element A represents avolume 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 elementlocation), 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 obtainrepresentative element thicknesses. Multiplying each element thickness by eight (the number of boltsin 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 surfacesare used to calculate the surface areas of the nodes used in defining 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 throughare sectional areas that are associated with bolt head elements CF. Multiplying the above areas byeight (the number of bolts in the model) provides the nodal contact areas found under the *SURFACEINTERACTION options.

A common way of handling the presence of the bolt holes in the pipe flange in axisymmetricanalyses is to smear the material properties used in the bolt hole area of the mesh and to useinhomogeneous material properties that correspond to a weaker material in this region. Generalguidelines for determining the effective material properties for perforated flat plates are found in ASMESection VIII Div 2 Article 49. For the type of structure under study, which is not a flat plate, acommon approach to determining the effective material properties is to calculate the elasticity modulireduction factor, which is the ratio of the ligament area in the pitch circle to the annular area of thepitch circle. In this model the annular area of the pitch circle is given by 6534.51 mm2, andthe total area of the bolt holes is given by 1608.5 mm2. Hence, the reduction factor issimply 0.754. The effective in-plane moduli of elasticity, and , are obtainedby multiplying the respective moduli, and , by this factor. We assume material isotropy in therz plane; thus, The modulus in the hoop direction, , should be very smalland is chosen such that 106 . The in-plane shear modulus is then calculated based on theeffective elasticity modulus: The shear moduli in the hoop direction are alsocalculated similarly but with set to zero (they are not used in an axisymmetric model). Hence, we have

155292MPa, 0.155292MPa, 59728MPa, and 0.07765MPa.These elasticity moduli are specified using *ELASTIC, TYPE=ENGINEERING CONSTANTS for thebolt 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 reducedintegration, C3D20R, for the pipe hub/flange and bolts. The gasket is modeled with C3D20R elementsor GK3D18 elements. The top figure shows the mesh of the pipe hub and flange, and the bottomfigure shows both the gasket and bolt (in the lighter color). Contact is modeled by the interaction ofcontact surfaces defined by grouping specific faces of the elements in the contacting regions. Forthree-dimensional contact where both the master and slave surfaces are deformable, the SMALLSLIDING parameter must be used on the *CONTACT PAIR option to indicate that small relative slidingoccurs between contacting surfaces. No special adjustments need be made for the material propertiesused in the three-dimensional model because all parts are modeled appropriately.

Four different meshes that use substructures to model the flange are tested. A first-level substructureis created for the entire 22.5 segment of the flange shown in Figure 1.1.13, while the gasket and thebolt are meshed as before. The nodes on the flange in contact with the bolt cap form a node-basedsurface, while the nodes on the flange in contact with the gasket form another node-based surface. Thesenode-based surfaces will form contact pairs with the master surfaces on the bolt cap and on the gasket,which are defined with *SURFACE as before. The retained degrees of freedom on the substructureinclude all three degrees of freedom for the nodes in these node-based surfaces as well as for the nodeson the 0 and 22.5 faces of the flange. Appropriate boundary conditions are specified at the substructureusage level.

1.1.13


BOLTED PIPE JOINT

A second-level substructure of 45 is created by reflecting the first-level substructure with respectto the 22.5 plane. The nodes on the 22.5 face belonging to the reflected substructure are constrainedin all three degrees of freedom to the corresponding nodes on the 22.5 face belonging to the originalfirst-level substructure. The half-bolt and the gasket sector corresponding to the reflected substructureare also constructed by reflection. The retained degrees of freedom include all three degrees of freedomof all contact node sets and of the nodes on the 0 and 45 faces of the flange. MPC-type CYCLSYM isused to impose cyclic symmetric boundary conditions on these two faces.

A third-level substructure of 90 is created by reflecting the original 45 second-level substructurewith respect to the 45 plane and by connecting it to the original 45 substructure. The remaining partof the gasket and the bolts corresponding to the 4590 sector of the model is created by reflection andappropriate constraints. In this case it is not necessary to retain any degrees of freedom on the 0 and90 faces of the flange because this 90 substructure will not be connected to other substructures andappropriate boundary conditions can be specified at the substructure creation level.

The final substructure model is set up by mirroring the 90 mesh with respect to the symmetryplane 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-levelsubstructure twice. The sparse solver is used because it significantly reduces the run time for this model.

Finally, a three-dimensional matrix-basedmodel is created by replacing elements for the entire 22.5segment of the flange shown in Figure 1.1.13 with stiffness matrices, while the gasket and the boltare meshed as before. Contact between the flange and gasket and the flange and bolt cap is modeledusing node-based slave surfaces just as for the substructure models. Appropriate boundary conditionsare 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 isapplied 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 and22.5 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 directionsin this symmetry plane. These transformations are implemented using the *TRANSFORM option. Onboth the symmetry planes, the symmetry boundary conditions 0 are imposed everywhere except forthe 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 boundarycondition 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 specified depending onwhat substructure is used. For the first-level 22.5 substructure the boundary conditions and constraintequations are the same as for the three-dimensional model shown in Figure 1.1.13. For the 45 second-level substructure the symmetry boundary conditions are enforced on the 45 planewith the constraintequation 0. A transform could have been used as well. For the 90 third-level substructurethe 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 forsymmetric boundary conditions. Entries in the stiffness matrices for these nodes are also in localcoordinates.

A clamping force of 15 kN is applied to each bolt by using the *PRE-TENSION SECTIONoption. The pre-tension section is identified by means of the *SURFACE option. The pre-tension isthen prescribed by applying a concentrated load to the pre-tension node. In the axisymmetric analysisthe actual load applied is 120 kN since there are eight bolts. In the three-dimensional model with nosubstructures the actual load applied is 7.5 kN since only half of a bolt is modeled. In the models usingsubstructures all half-bolts are loaded with a 7.5 kN force. For all of the models the pre-tension sectionis specified about halfway down the bolt shank.

Sticking contact conditions are assumed in all surface interactions in all analyses and are simulatedwith 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/flange predicted by the axisymmetric (bottom) andthree-dimensional (top) analyses when solid continuum elements are used to model the gasket. Thefigure shows that the compressive normal stress is highest at the outer edge of the gasket, decreasesradially inward, and changes from compression to tension at a radius of about 35 mm, which is consistentwith findings reported by Sawa et al. (1991). The close agreement in the overall solution betweenaxisymmetric 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 theinterface between the gasket and the pipe hub/flange predicted by the axisymmetric (bottom) andthree-dimensional (top) analyses when gasket elements are used to model the gasket. Close agreementin the overall solution between the axisymmetric and three-dimensional analyses is also seen in thiscase. The gasket starts carrying compressive load at a radius of about 40 mm, a difference of 5 mm withthe previous result. This difference is the result of the gasket elements being unable to carry tensileloads in their thickness direction. This solution is physically more realistic since, in most cases, gasketsseparate from their neighboring parts when subjected to tensile loading. Removing the *SURFACEBEHAVIOR, NO SEPARATION option from the gasket/flange contact surface definition in the inputfiles that model the gasket with continuum elements yields good agreement with the results obtainedin Figure 1.1.16 (since, in that case, the solid continuum elements in the gasket cannot carry tensileloading in the gasket thickness direction).

Themodels in this example can bemodified to study other factors, such as the effective seatingwidthof the gasket or the sealing performance of the gasket under operating loads. The gasket elements offerthe advantage of allowing very complex behavior to be defined in the gasket thickness direction. Gasketelements can also use any of the small-strain material models provided in Abaqus including user-definedmaterial models. Figure 1.1.17 shows a comparison of the normal stress distributions in the gasket atthe interface between the gasket and the pipe hub/flange predicted by the axisymmetric (bottom) andthree-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 whichsolid 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 plane0. 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 containingthe substructures described above.

An execution procedure is available to combine model and results data from two substructureoutput databases into a single output database. For more information, see Combining output fromsubstructures, Section 3.2.18 of the Abaqus Analysis Users Manual.

This example can also be used to demonstrate the effectiveness of the quasi-Newton nonlinearsolver. This solver utilizes an inexpensive, approximate stiffness matrix update for several consecutiveequilibrium iterations, rather than a complete stiffness matrix factorization each iteration as used in thedefault full Newton method. The quasi-Newton method results in an increased number of less expensiveiterations, and a net savings in computing cost.

Input files

boltpipeflange_axi_solidgask.inp Axisymmetric analysis containing a gasket modeled withsolid continuum elements.

boltpipeflange_axi_node.inp Node definitions for boltpipeflange_axi_solidgask.inpand boltpipeflange_axi_gkax6.inp.

boltpipeflange_axi_element.inp Element definitions forboltpipeflange_axi_solidgask.inp.

boltpipeflange_3d_solidgask.inp Three-dimensional analysis containing a gasket modeledwith solid continuum elements.

boltpipeflange_axi_gkax6.inp Axisymmetric analysis containing a gasket modeled withgasket elements.

boltpipeflange_3d_gk3d18.inp Three-dimensional analysis containing a gasket modeledwith gasket elements.

boltpipeflange_3d_substr1.inp Three-dimensional analysis using the first-levelsubstructure (22.5 model).

boltpipeflange_3d_substr2.inp Three-dimensional analysis using the second-levelsubstructure (45 model).

boltpipeflange_3d_substr3_1.inp Three-dimensional analysis using the third-levelsubstructure once (90 model).

boltpipeflange_3d_substr3_2.inp Three-dimensional analysis using the third-levelsubstructure twice (90 mirrored model).

boltpipeflange_3d_gen1.inp First-level substructure generation data referenced byboltpipeflange_3d_substr1.inp andboltpipeflange_3d_gen2.inp.

boltpipeflange_3d_gen2.inp Second-level substructure generation data referenced byboltpipeflange_3d_substr2.inp andboltpipeflange_3d_gen3.inp.

1.1.16


BOLTED PIPE JOINT

boltpipeflange_3d_gen3.inp Third-level substructure generation data referenced byboltpipeflange_3d_substr3_1.inp andboltpipeflange_3d_substr3_2.inp.

boltpipeflange_3d_node.inp Nodal coordinates used inboltpipeflange_3d_substr1.inp,boltpipeflange_3d_substr2.inp,boltpipeflange_3d_substr3_1.inp,boltpipeflange_3d_substr3_2.inp,boltpipeflange_3d_cyclsym.inp,boltpipeflange_3d_gen1.inp,boltpipeflange_3d_gen2.inp, andboltpipeflange_3d_gen3.inp.

boltpipeflange_3d_cyclsym.inp Same as file boltpipeflange_3d_substr2.inp except thatCYCLSYM type MPCs are used.

boltpipeflange_3d_missnode.inp Same as file boltpipeflange_3d_gk3d18.inp except thatthe option to generate missing nodes is used for gasketelements.

boltpipeflange_3d_isomat.inp Same as file boltpipeflange_3d_gk3d18.inp except thatgasket elements are modeled as isotropic using the*MATERIAL option.

boltpipeflange_3d_ortho.inp Same as file boltpipeflange_3d_gk3d18.inp except thatgasket elements are modeled as orthotropic and the*ORIENTATION option is used.

boltpipeflange_axi_isomat.inp Same as file boltpipeflange_axi_gkax6.inp except thatgasket elements are modeled as isotropic using the*MATERIAL option.

boltpipeflange_3d_usr_umat.inp Same as file boltpipeflange_3d_gk3d18.inp except thatgasket elements are modeled as isotropic with usersubroutine UMAT.

boltpipeflange_3d_usr_umat.f User subroutine UMAT used inboltpipeflange_3d_usr_umat.inp.

boltpipeflange_3d_solidnum.inp Same as file boltpipeflange_3d_gk3d18.inp except thatsolid element numbering is used for gasket elements.

boltpipeflange_3d_matrix.inp Three-dimensional analysis containing matrices and agasket modeled with solid continuum elements.

boltpipeflange_3d_stiffPID4.inp Matrix representing stiffness of a part of the flangesegment for three-dimensional analysis containingmatrices.

boltpipeflange_3d_stiffPID5.inp Matrix representing stiffness of the remaining part of theflange segment for three-dimensional analysis containingmatrices.

1.1.17


BOLTED PIPE JOINT

boltpipeflange_3d_qn.inp Same as file boltpipeflange_3d_gk3d18.inp except thatthe quasi-Newton nonlinear solver is used.

References

Bibel, G. D., and R. M. Ezell, An Improved Flange Bolt-Up Procedure Using ExperimentallyDetermined Elastic Interaction Coefficients, Journal of Pressure Vessel Technology, vol. 114,pp. 439443, 1992.

Chaaban, A., and U. Muzzo, Finite Element Analysis of Residual Stresses in Threaded EndClosures, 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, Journalof Pressure Vessel Technology, vol. 113, pp. 497503, 1991.

1.1.18


BOLTED PIPE JOINT

80

= 4

d = 50d = 105d = 130d = 165

r = 8

15

d = 105 d = 50

centerline

Top View

Side View

2.5Gasket

Bolt 24 16

10

26

47

20

Figure 1.1.11 Schematic of the bolted joint. All dimensions in mm.

1.1.19


BOLTED PIPE JOINT

1

2

3 1

2

3

1

2

3 1

2

3

Figure 1.1.12 Axisymmetric model of the bolted joint.

1.1.110


BOLTED PIPE JOINT

1

2

3 1

2

3

1

2

3 1

2

3

Figure 1.1.13 22.5 segment three-dimensional model of the bolted joint.

1.1.111


BOLTED PIPE JOINT

y

1 2 3 4 5 6 7 8 9

C D E F

z

x

WB

WA

HA

HB

contact nodes

area A

area B

element A

element B

TOP VIEW

FRONT VIEW

Rbolthead

Rshank

Figure 1.1.14 Cross-sectional views of the bolt head and the shank.

1.1.112


BOLTED PIPE JOINT

12

3

12

12

3

111

23

7

8

9

12

11

10

8

11

9

9

9

11

8

9

9 8

12

8

1011

7

12

123

45

6

123

456

8

7

7

1

23

4

56

7

12

12

123

456

11

10

10

11 10

10

7

12

2

34

56

23

4

56

12

3

S22 VALUE

1 -1.00E+02 2 -8.90E+01 3 -7.81E+01 4 -6.72E+01 5 -5.63E+01 6 -4.54E+01 7 -3.45E+01 8 -2.36E+01 9 -1.27E+0110 -1.81E+0011 +9.09E+0012 +2.00E+01

12

3

12

11

10

10

10

10

10

10

11

10

9

9

9

911

9

8

8

8

8

8

11

8

8

8

11

11

11

9234

56

12

12

7

7

7

7

234

56

7

7

7

12

23

4

56

12

234

56

12

9

7

10

12

9

12

11

23

4

56

234

56

234

56

234

56

12

3

S22 VALUE

1 -1.00E+02 2 -8.90E+01 3 -7.81E+01 4 -6.72E+01 5 -5.63E+01 6 -4.54E+01 7 -3.45E+01 8 -2.36E+01 9 -1.27E+0110 -1.81E+0011 +9.09E+0012 +2.00E+01

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

1.1.113


BOLTED PIPE JOINT

12

3

3

7 8

91011

5

6

4

4

4

4

4

4

3

7 8

91011

7 8

91011

7 8

91011

5

6

5

6

5

6

4

5

6

3

5

65

6

3

3

3

3

3 7 8

91011

4

7 8

91011

7 8

91011

7 8

91011

5

6

4

4

4

4

4

4

4

5

6

5

65

65

65

65

6

5

6

3

4

5

67 8

910117 8

910117 8

910117 8

910117 8

910117 8

91011

7 8

91011

4

7 8

91011

3

3

3

3

3

3

3

12

3

S11 VALUE

1 -2.00E+01

2 -9.09E+00

3 +1.82E+00

4 +1.27E+01

5 +2.36E+01

6 +3.45E+01

7 +4.55E+01

8 +5.64E+01

9 +6.73E+01

10 +7.82E+01

11 +8.91E+01

12 +1.00E+02

12

3

7 89

3

7 891011

7 8911

89101143

3

34

6

5

5 6

6

8

7 8 10

7 8910

711

7 811

7 8 10

5

643 5

6

3

34

56

4

5

6

56

56

3

7 8

91011

78

910

78

91011

3

78

91011

78

910

78

91011

3

4

4

4

12

3

S11 VALUE

1 -2.00E+01

2 -9.09E+00

3 +1.82E+00

4 +1.27E+01

5 +2.36E+01

6 +3.45E+01

7 +4.55E+01

8 +5.64E+01

9 +6.73E+01

10 +7.82E+01

11 +8.91E+01

12 +1.00E+02

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

1.1.114


BOLTED PIPE JOINT

12

3

23456

99 7710

1010

2345623456

88

823456

7

23456

7

788 23456

23456

9

712

712 1111 8

2345611 23456

8

11

1010

21212

99

881010

11

117799

10101212

912

911

11

1212

1112 345610 79

11

8

773456 34561212

8

11

10109

11

9 8

23

4

56

89

710 23

4

56

8

10

9

78

23

4

56

7

9

10

122

34

5611

7

11

12

89

10

12

23

4

56

10

7

12

11

11

8

9

11

12

34

56

7

89

101112

12

3

S11 VALUE

1 -1.00E+02

2 -8.91E+01

3 -7.82E+01

4 -6.73E+01

5 -5.64E+01

6 -4.55E+01

7 -3.45E+01

8 -2.36E+01

9 -1.27E+01

10 -1.82E+00

11 +9.09E+00

12 +2.00E+01

12

3

234

5

8

9 6

7

10

234

5

8

10

96

7

8 234

5

6

7

234

5

12

911 234

51012

8

6

7

811 9

1012

11

11 6

710

8

12

11

9

9 234

512 10

6

7

11

11

10

9 8 234

5

6

7

12

11

12

10

9

12

8 6

7

234

5

2345

10

9

8

67

11

12

234512 11 10 9 8 67

234

5

8

6

7

9

10

234

5

8

10

96

7

8 234

5

6

7

234

5

12

911

10

234

512

6

7

8

811 9

12 10

11

96

710

8

12

11

11

9 234

512 10

6

7

234

5

11

10

9 8 6

7

234

5

12

11

12

10

9 8

12

6

7

11

12

3

S11 VALUE

1 -1.00E+02

2 -8.91E+01

3 -7.82E+01

4 -6.73E+01

5 -5.64E+01

6 -4.55E+01

7 -3.45E+01

8 -2.36E+01

9 -1.27E+01

10 -1.82E+00

11 +9.09E+00

12 +2.00E+01

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

1.1.115


BOLTED PIPE JOINT

22.5_matrix22.5_no_sup22.5_sup45_sup90_sup90r_sup

Figure 1.1.18 Normal stress distribution in the gasket contact surface along the line0 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-PLANEBENDING AND INTERNAL PRESSURE

Product: Abaqus/StandardElbows are used in piping systems because they ovalize more readily than straight pipes and, thus, provideflexibility in response to thermal expansion and other loadings that impose significant displacements on thesystem. Ovalization is the bending of the pipe wall into an ovali.e., noncircularconfiguration. The elbowis, thus, behaving as a shell rather than as a beam. Straight pipe runs do not ovalize easily, so they behaveessentially as beams. Thus, even under pure bending, complex interaction occurs between an elbow and theadjacent straight pipe segments; the elbow causes some ovalization in the straight pipe runs, which in turntend to stiffen the elbow. This interaction can create significant axial gradients of bending strain in the elbow,especially in cases where the elbow is very flexible. This example provides verification of shell and elbowelement modeling of such effects, through an analysis of a test elbow for which experimental results havebeen 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 forthe straight pipe segments. This provides a comparative solution in which the interaction between the elbowand the adjacent straight pipes is neglected. The analyses predict the response up to quite large rotations acrossthe elbow, so as to investigate possible collapse of the pipe and, particularly, the effect of internal pressure onthat collapse.

Geometry and model

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

and radius ratio 3.07, so the flexibility factor from Dodge and Moore (1972) is 10.3. (Theflexibility factor for an elbow is the ratio of the bending flexibility of an elbow segment to that of astraight pipe of the same dimensions, for small displacements and elastic response.) This is an extremelyflexible case because the pipe wall is so thin.

To demonstrate convergence of the overall moment-rotation behavior with respect to meshing, thetwo shell element meshes shown in Figure 1.1.22 are analyzed. Since the loading concerns in-planebending only, it is assumed that the response is symmetric about the midplane of the system so that inthe shell element model only one-half of the system need be modeled. Element type S8R5 is used, sincetests have shown this to be the most cost-effective shell element in Abaqus (input files using elementtypes S9R5, STRI65, and S8R for this example are included with the Abaqus release). The elbowelement meshes replace each axial division in the coarser shell element model with one ELBOW32or 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, whereessentially monotonic straining is expected.

The ends of the system are rigidly attached to stiff plates in the experiments. These boundaryconditions are easily modeled for the ELBOW elements and for the fixed 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 thatrepresents the motion of the end plate. This is done using the *KINEMATIC COUPLING option asdescribed below.

The material is assumed to be isotropic and elastic-plastic, following the measured response of type304 stainless steel at room temperature, as reported by Sobel and Newman (1979). Since all the analysesgive 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 overallresponse of the system, it seems that this stress-strain model may overestimate the materials actualstrength.

Loading

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

Kinematic boundary conditions

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


Themoment-rotation responses predicted by the various analysismodels andmeasured in the experiment,all taken at zero internal pressure, are compared in Figure 1.1.23. The figure shows that the two shellmodels give very similar results, overestimating the experimentally measured collapse moment by about15%. The 6-mode ELBOW element models are somewhat stiffer than the shell models, and those with4 Fourier modes are much too stiff. This clearly shows that, for this very flexible system, the ovalizationof 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 likelyexplanation for the excessive stiffness in comparison to the experimentally measured response is thatthe material model used in the analyses is too strong. Sobel and Newman (1979) point out that thestress-strain curve measured and used in this analysis, shown in Figure 1.1.21, has a 0.2% offsetyield 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 curvemeasurement may have been taken from stronger parts of the fabrication. If this is the case, it pointsout the likelihood that the elbow tested is rather nonuniform in strength properties in spite of the caretaken in its manufacture. We are left with the conclusion that discrepancies of this magnitude cannot beeliminated 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 under0 and 3.45 MPa (500 lb/in2) internal pressure and shows the strong influence of large-displacementeffects. If large-displacement effects were not important, the opening and closing moments wouldproduce the same response. However, even with a 1 relative rotation across the elbow assembly, theopening and closing moments differ by about 12%; with a 2 relative rotation, the difference is about17%. Such magnitudes of relative rotation would not normally be considered large; in this case it is thecoupling into ovalization that makes geometric nonlinearity significant. As the rotation increases, thecases with closing moment loading show collapse, while the opening moment curves do not. In bothcases internal pressure shows a strong effect on the results, which is to be expected in such a thin-walledpipeline. The level of interaction between the straight pipe and the elbows is well illustrated by the straindistribution on the outside wall, shown in Figure 1.1.25. The strain contours are slightly discontinuousat 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 straightand curved pipe segments, Section 1.1.5 of the Abaqus Benchmarks Manual. The plot shows thecollapse value of the closing moment under in-plane bending as a function of internal pressure. Thestrong influence of pressure on collapse is apparent. In addition, the effect of analyzing the elbowby neglecting interaction between the straight and curved segments is shown: the uniform bendingresults 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 thesimpler analysis neglecting the interaction is conservative (in that it gives consistently lower values forthe collapse moment), but this conservatism cannot be taken for granted. The analysis of Sobel andNewman (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 large-displacement, elastic-plastic lower bound (Goodall, 1978a), are also shown in this figure. Again, theimportance of large-displacement effects is apparent from that comparison.

Detailed results obtained with the model that uses ELBOW31 elements are shown in Figure 1.1.27through Figure 1.1.29. Figure 1.1.27 shows the variation of the Mises stress along the length of thepiping system. The length is measured along the centerline of the pipe starting at the loaded end. Thefigure compares the stress distribution at the intrados (integration point 1) on the inner and outer surfacesof the elements (section points 1 and 7, respectively). Figure 1.1.28 shows the variation of the Misesstress around the circumference of two elements (451 and 751) that are located in the bend section ofthe model; the results are for the inner surface of the elements (section point 1). Figure 1.1.29 showsthe ovalization of elements 451 and 751. A nonovalized, circular cross-section is included in the figurefor comparison. From the figure it is seen that element 751, located at the center of the bend section,experiences the most severe ovalization. These three figures were produced with the aid of the elbowelement postprocessing program felbow.f (Creation of a data file to facilitate the postprocessing ofelbow element results: FELBOW, Section 14.1.6), written in FORTRAN. The postprocessing programs

1.1.23



felbow.C (A C++ version of FELBOW, Section 10.15.6 of the Abaqus Scripting Users Manual)and felbow.py (An Abaqus Scripting Interface version of FELBOW, Section 9.10.12 of the AbaqusScripting Users Manual), written in C++ and Python, respectively, are also available for generating thedata for figures such as Figure 1.1.28 and Figure 1.1.29. The user must ensure that the output variablesare 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 verifies theinterpolation scheme in the case of double curved surfaces. A solid submodel using C3D27R elementsis created around the elbow part of the pipe, spanning an angle of 40. The finer submodel mesh hasthree elements through the thickness, 10 elements around half of the circumference of the cylinder, and10 elements along the length of the elbow. Both ends are driven from the global shell model made ofS8R elements. The time scale of the static submodel analysis corresponds to the arc length in the globalRiks analysis. The submodel results agree closely with the shell model. The *SECTION FILE option isused 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 usedfor a careful study of the stress and strain fields in the elbow. The entire elbow is meshed with C3D20Relements, and the straight pipe sections are meshed with S8R elements (see Figure 1.1.210). At eachshell-to-solid interface illustrated in Figure 1.1.210, an element-based surface is defined on the edgeof the solid mesh and an edge-based surface is defined on the edge of the shell mesh. The *SHELLTO SOLID COUPLING option is used in conjunction with these surfaces to couple the shell and solidmeshes.

Edge-based surfaces are defined at the end of each pipe segment. These surfaces are coupled toreference nodes that are defined at the center of the pipes using the *COUPLING option in conjunctionwith the *DISTRIBUTING option. The loading and fixed boundary conditions are applied to thereference points. The advantage of using this method is that the pipe cross-sectional areas are freeto deform; thus, ovalization at the ends is not constrained. The moment-rotation response of theshell-to-solid coupling model agrees very well with the results shown in Figure 1.1.24.

Input files

In all the following input files (with the exception of elbowcollapse_elbow31b_b31.inp,elbowcollapse_s8r5_fine.inp, and elbowcolpse_shl2sld_s8r_c3d20r.inp) the step concerning theapplication of the pressure load is commented out. To include the effects of the internal pressure in anygiven analysis, uncomment the step definition in the appropriate input file.

elbowcollapse_elbow31b_b31.inp ELBOW31B and B31 element model.elbowcollapse_elbow31_6four.inp ELBOW31 model with 6 Fourier modes.elbowcollapse_elbow32_6four.inp ELBOW32 model with 6 Fourier modes.elbowcollapse_s8r.inp S8R element model.

1.1.24



elbowcollapse_s8r5.inp S8R5 element model.elbowcollapse_s8r5_fine.inp Finer S8R5 element model.elbowcollapse_s9r5.inp S9R5 element model.elbowcollapse_stri65.inp STRI65 element model.elbowcollapse_submod.inp Submodel using C3D27R elements.elbowcolpse_shl2sld_s8r_c3d20r.inp 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 onElbows and Curved Pipes, Welding Research Council Bulletin, no. 179, 1972.

Goodall, I. W., Lower Bound Limit Analysis of Curved Tubes Loaded by Combined InternalPressure and In-Plane Bending Moment, Research Division Report RD/B/N4360, CentralElectricity Generating Board, England, 1978.

Goodall, I. W., Large Deformations in Plastically Deforming Curved Tubes Subjected to In-PlaneBending, 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 Simplified Analysis Results, Westinghouse Advanced ReactorsDivision, Report WARDHT940002, 1979.

1.1.25



400

300

200

100

00 1 2 3 4 5

0

10

20

30

40

50

60

70

Strain, %

Str

ess,

MP

a

Str

ess,

103

lb/in

2

1.83 m(72.0 in)

Moment applied here

610 mm(24.0 in)

407 mm(16.02 in)

10.4 mm (0.41 in)thickness

Young's modulus:

Poisson's ratio:

193 GPa (28 x 106 lb/in2 )

0.2642

Figure 1.1.21 MLTF elbow: geometry and measured material response.

1.1.26



Figure 1.1.22 Models for elbow/pipe interaction study.

1.1.27



0.04 0.08 0.12 0.16 0.20 0.240

2

0.04 0.08 0.12 0.16 0.20 0.240

3

0.04 0.08 0.12 0.16 0.20 0.240

4

0.04 0.08 0.12 0.16 0.20 0.240

5,7

0.04 0.08 0.12 0.16 0.20 0.240

6

0.04 0.08 0.12 0.16 0.20 0.240

8

0.04 0.08 0.12 0.16 0.20 0.240

10

0.04 0.08 0.12 0.16 0.20 0.240

0.04 0.08 0.12 0.16 0.20 0.240

0.04 0.08 0.12 0.16 0.20 0.240

1.0

2.01.0 4.0 7.0 10.0 13.0

11

0

1

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 410 ELBOW31B - 6 mode11 ELBOW31B - 4 mode

End rotation, rad

End rotation, deg

9

Mom

ent,

kN-m

200

150

100

50

Mom

ent,

106

lb-in

Figure 1.1.23 Moment-rotation response: mesh convergence studies.

0.00 0.04 0.08 0.12 0.16 0.20 0.24

[x10 ]6

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

End Rotation, rad

Mom

ent,

lb-in

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

Figure 1.1.24 Moment-rotation response: pressure dependence.

1.1.28



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



250 500 750 1000

1.2

1.4

1.6

1.8

2.0

2.2

2.4

1.0

0

1 2 3 4 5

Internal pressure, MPa

275

250

225

200

175

150

125

S8R5ELB

OW31

ELBOW31B

Goodall (1978a), large displacementelastic-plasticlower bound

Sobel and Newman (1979),uniform bending analysis

Internal pressure, lb/in2

Colla

pse

mom

ent,

kN-m

Colla

pse

mom

ent,

106

lb-in

Goodall(1978), smalldisplacement limit analysis

6

Figure 1.1.26 In-plane bending of an elbow, elastic-plastic collapse moment results.

0. 50. 100.

Length along pipe, in

10.

20.

30.

40.

50.

Mises stress, psi

[x10 3]

XMIN 1.500E+00XMAX 1.322E+02YMIN 4.451E+03YMAX 5.123E+04

MISES_IMISES_O

Figure 1.1.27 Mises stress distribution along the length of the piping system.

1.1.210



0. 5. 10. 15. 20. 25. 30. 35. 40. 45. 50.

Length around element circumference, in

25.

30.

35.

40.

45.

50.

55.

60.

Mis

es s

tres

s, p

si

[x10 3]

XMIN 0.000E+00XMAX 4.892E+01YMIN 2.635E+04YMAX 5.778E+04

MISES451MISES751

Figure 1.1.28 Mises stress distribution around the circumference of elements 451 and 751.

-10. -5. 0. 5. 10.

Local x-axis

-10.

-5.

0.

5.

10.

Local y-axis

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

CIRCLE

OVAL_451

OVAL_751

Figure 1.1.29 Ovalization of elements 451 and 751.

1.1.211



1 2

3

shell elements

solid elements

shell-to-solidinterface

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-PLANEBENDING

Products: Abaqus/Standard Abaqus/Explicit

Elbows are used in piping systems because they ovalize more readily than straight pipes and, thus, provideflexibility in response to thermal expansion and other loadings that impose significant displacements on thesystem. Ovalization is the bending of the pipe wall into an ovali.e., noncircularconfiguration. The elbowis, 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 28.5.1 of the AbaqusAnalysis Users Manual) to model the nonlinear response of initially circular pipes and pipebends accuratelywhen the distortion of the cross-section by ovalization is significant. It also provides some guidelines onthe importance of including a sufficient number of Fourier modes in the elbow elements to capture theovalization accurately. In addition, this example illustrates the shortcomings of using flexibility knockdownfactors with simple beam elements in an attempt to capture the effects of ovalization in an ad hoc mannerfor large-displacement analyses. Similar analyses involving pipe elements in Abaqus/Explicit are included.

Geometry and model

The pipeline configuration used in the study is shown in Figure 1.1.31. It is a simple model with twostraight 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 1.27cm (0.5 inches). The wall thickness of the pipe is varied from 0.03175 cm to 0.2032 cm (0.0125 inches to0.08 inches) in a parametric study, as discussed below. The pipe material is assumed to be isotropic linearelastic with a Youngs modulus of 194 GPa (28.1 106 psi) and a Poissons ratio of 0.0. The straightportions of the pipeline are assumed to be long enough so that warping at the ends of the structure isnegligible.

Two loading conditions are analyzed. The first case is shown in Figure 1.1.31 with unit inwarddisplacements imposed on both ends of the structure. This loading condition has the effect of closing thepipeline in on itself. In the second case the sense of the applied unit displacements is outward, openingthe 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, andpipes) over a range of flexibility factors, k, is performed. As defined in Dodge and Moore (1972), theflexibility factor for an elbow is the ratio of the bending flexibility of the elbow segment to that of astraight pipe of the same dimensions, assuming small displacements and an elastic response. When theinternal (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 ofthe pipe, and is Poissons ratio. Changes in the flexibility factor are introduced by varying the wallthickness of the pipe.

The pipeline is modeled with three different element types: S4 shell elements, ELBOW31 elbowelements, and PIPE31 pipe elements. The S4 shell element model consists of a relatively fine meshof 40 elements about the circumference and 75 elements along the length. This mesh is deemed fineenough to capture the true response of the pipeline accurately, although no mesh convergence studies areperformed. Two analyses are conducted with the shell mesh: one with automatic stabilization using aconstant damping factor (see Automatic stabilization of static problems with a constant damping factorin Solving nonlinear problems, Section 7.1.1 of the Abaqus Analysis Users Manual), and one withadaptive automatic stabilization (see Adaptive automatic stabilization scheme in Solving nonlinearproblems, Section 7.1.1 of the Abaqus Analysis Users Manual). The pipe and elbow element meshesconsist of 75 elements along the length; the analyses with these element types do not use automaticstabilization.

The results of the shell element model with automatic stabilization using a constant damping factorare taken as the reference solution. The reaction force at the tip of the pipeline is used to evaluate theeffectiveness of the pipe and elbow elements. In addition, the ovalization values of the pipeline cross-section predicted by the elbow element models are compared.

The elbow elements are tested with 0, 3, and 6 Fourier modes, respectively. In general, elbowelement accuracy improves as more modes are used, although the computational cost increasesaccordingly. In addition to standard pipe elements, tests are performed on pipe elements with a specialflexibility knockdown factor. Flexibility knockdown factors (Dodge andMoore, 1972) are corrections tothe bending stiffness based upon linear semianalytical results. They are applied to simple beam elementsin an attempt to capture the global effects of ovalization. The knockdown factor is implemented in thePIPE31 elements by scaling the true thickness by the flexibility factor; this is equivalent to scaling themoment of inertia of the pipe element by .


The results obtained with the shell element model with automatic stabilization using a constant dampingfactor are taken as the reference solution. Very similar results are obtained with the same mesh using theadaptive automatic stabilization scheme.

The tip reaction forces due to the inward prescribed displacements for the various analysis modelsare shown in Figure 1.1.32. The results are normalized with respect to those obtained with the shellmodel. The results obtained with the ELBOW31 element model with 6 Fourier modes show excellentagreement with the reference solution over the entire range of flexibility factors considered in this study.The remaining four models generally exhibit excessively stiff response for all values of k. The PIPE31element model, which uses the flexibility knockdown factor, shows a relatively constant error of about

1.1.32



20% over the entire range of flexibility factors. The 0-mode ELBOW31 element model and the PIPE31element 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 analysismodels are shown in Figure 1.1.33. Again, the results obtained with the 6-mode ELBOW31 elementmodel compare well with the reference shell solution. The 0-mode and 3-mode ELBOW31 and thePIPE31 (without the flexibility knockdown factor) element models exhibit overly stiff response. ThePIPE31 element model with the knockdown factor has a transition region near k = 1.5, where the responsechanges from being too stiff to being too soft. The results in Abaqus/Explicit for pipe elements areconsistent with those obtained in Abaqus/Standard.

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 bothload cases considered in this study. By definition, the 0-mode model cannot ovalize, which accounts forits stiff response. The 3-mode and the 6-mode models show significant ovalization in both loading cases.Figure 1.1.36 compares the ovalization of the 6-mode model in the opened and closed deformationstates. It clearly illustrates that when the ends of the pipe are displaced inward (closing mode), theheight of the pipes cross-section gets smaller, thereby reducing the overall stiffness of the pipe; thereverse is true when the pipe ends are displaced outward: the height of the pipes cross-section getslarger, thereby increasing the pipe stiffness. These three figures were produced with the aid of the elbowelement postprocessing program felbow.f (Creation of a data file to facilitate the postprocessing ofelbow element results: FELBOW, Section 14.1.6), written in FORTRAN. The postprocessing programsfelbow.C (A C++ version of FELBOW, Section 10.15.6 of the Abaqus Scripting Users Manual)and felbow.py (An Abaqus Scripting Interface version of FELBOW, Section 9.10.12 of the AbaqusScripting Users Manual), written in C++ and Python, respectively, are also available for generating thedata for these figures. The user must ensure that the output variables are written to the output databaseto use these two programs.

Parametric study

The performance of the pipe and elbow elements investigated in this example is analyzed convenientlyin a parametric study using the Python scripting capabilities of Abaqus (Scripting parametric studies,Section 19.1.1 of the Abaqus Analysis Users Manual). We perform a parametric study in which eightanalyses 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 cmto 0.2032 cm (0.0125 inches to 0.08 inches).

The Python script file elbowtest.psf is used to perform the parametric study. The functioncustomTable (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 thiscase a dependent variable such as k cannot be included as a column of data in an XYPLOT file.customTable is designed to overcome this limitation by taking an XYPLOT file from the parametricstudy and converting it into a new file of reaction forces versus flexibility factors (k).

################################################################def customTable(file1, file2):

1.1.33



for line in file1.readlines():print linenl = 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/2.0k = bend_radius*wall_thick/mean_rad**2k = k/sqrt(1.e0 - poisson**2)k = 1.66e0/k

outputstring = str(k) + ', ' + str(rf) + '\n'file2.write(outputstring)

##############################################################

Input files

elbowtest_shell.inp S4 model.elbowtest_shell_stabil_adap.inp S4 model with adaptive stabilization.elbowtest_elbow0.inp ELBOW31 model with 0 Fourier modes.elbowtest_elbow3.inp ELBOW31 model with 3 Fourier modes.elbowtest_elbow6.inp ELBOW31 model with 6 Fourier modes.elbowtest_pipek.inp PIPE31 model with the flexibility knockdown factor.elbowtest_pipek_xpl.inp PIPE31 model with the flexibility knockdown factor in

Abaqus/Explicit.elbowtest_pipe.inp PIPE31 model without the flexibility knockdown factor.elbowtest_pipe_xpl.inp PIPE31 model without the flexibility knockdown factor

in Abaqus/Explicit.elbowtest.psf Python script file for the parametric study.

References

Dodge, W. G., and S. E. Moore, Stress Indices and Flexibility Factors for Moment Loadings onElbows and Curved Pipes, Welding Research Council Bulletin, no. 179, 1972.

Lutz, M., and D. Ascher, Learning Python, OReilly, 1999.

1.1.34



r

t

a

a

Ru

upipe cross-section

Figure 1.1.31 Pipeline geometry with inward prescribed tip displacements.

ELBOW31 0 modesELBOW31 3 modesELBOW31 6 modesPIPE31PIPE31 with knockdownShell S4

Figure 1.1.32 Normalized tip reaction force: closing displacement case.

1.1.35



ELBOW31 0 modesELBOW31 3 modesELBOW31 6 modesPIPE31PIPE31 with knockdownShell S4

Figure 1.1.33 Normalized tip reaction force: opening displacement case.

close-0close-3close-6

Figure 1.1.34 Ovalization of the ELBOW31 cross-sections for0, 3, and 6 Fourier modes: closing displacement case.

1.1.36



open-0open-3open-6

Figure 1.1.35 Ovalization of the ELBOW31 cross-sections for0, 3, and 6 Fourier modes: opening displacement case.

close-6open-6

Figure 1.1.36 Ovalization of the ELBOW31 cross-sections for6 Fourier modes: opening and closing displacement cases.

abaqus 6.11, example problems, vol. 1

Documents