Abaqus Example Problems Manual

CATIA is a registered trademark of Dassault Systémes.

C-MOLD is a registered trademark of Advanced CAE Technology, Inc., doing business as C-MOLD. Compaq Alpha is registered in the U.S. Patent and Trademark Office.

FE-SAFE is a trademark of Safe Technology, Ltd.

Fujitsu, UXP, and VPP are registered trademarks of Fujitsu Limited.

Hewlett-Packard, HP-GL, and HP-GL/2 are registered trademarks of Hewlett-Packard Co. Hitachi is a registered trademark of Hitachi, Ltd.

IBM RS/6000 is a trademark of IBM.

Intel is a registered trademark of the Intel Corporation. NEC is a trademark of the NEC Corporation.

PostScript is a registered trademark of Adobe Systems, Inc.

Silicon Graphics is a registered trademark of Silicon Graphics, Inc. SUN is a registered trademark of Sun Microsystems, Inc.

TEX is a trademark of the American Mathematical Society.

UNIX and Motif are registered trademarks and X Window System is a trademark of The Open Group in the U.S. and other countries.

Windows NT is a registered trademark of the Microsoft Corporation.

ABAQUS/CAE incorporates portions of the ACIS software by SPATIAL TECHNOLOGY INC. ACIS is a registered trademark of SPATIAL TECHNOLOGY INC.

This release of ABAQUS on Windows NT includes the diff program obtained from the Free Software Foundation. You may freely distribute the diff program and/or modify it under the terms of the GNU Library General Public License as published by the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA.

This release of ABAQUS/CAE includes lp_solve, a simplex-based code for linear and integer

programming problems by Michel Berkelaar of Eindhoven University of Technology, Eindhoven, the Netherlands.

Python, copyright 1991-1995 by Stichting Mathematisch Centrum, Amsterdam, The Netherlands. All Rights Reserved. Permission to use, copy, modify, and distribute the Python software and its

documentation for any purpose and without fee is hereby granted, provided that the above copyright notice appear in all copies and that both that copyright notice and this permission notice appear in supporting documentation, and that the names of Stichting Mathematisch Centrum or CWI or CATIA is a registered trademark of Dassault Systémes.

C-MOLD is a registered trademark of Advanced CAE Technology, Inc., doing business as C-MOLD. Compaq Alpha is registered in the U.S. Patent and Trademark Office.

FE-SAFE is a trademark of Safe Technology, Ltd.

Fujitsu, UXP, and VPP are registered trademarks of Fujitsu Limited.

Hewlett-Packard, HP-GL, and HP-GL/2 are registered trademarks of Hewlett-Packard Co. Hitachi is a registered trademark of Hitachi, Ltd.

IBM RS/6000 is a trademark of IBM.

Intel is a registered trademark of the Intel Corporation. NEC is a trademark of the NEC Corporation.

PostScript is a registered trademark of Adobe Systems, Inc.

Silicon Graphics is a registered trademark of Silicon Graphics, Inc. SUN is a registered trademark of Sun Microsystems, Inc.

TEX is a trademark of the American Mathematical Society.

UNIX and Motif are registered trademarks and X Window System is a trademark of The Open Group in the U.S. and other countries.

Windows NT is a registered trademark of the Microsoft Corporation.

ABAQUS/CAE incorporates portions of the ACIS software by SPATIAL TECHNOLOGY INC. ACIS is a registered trademark of SPATIAL TECHNOLOGY INC.

This release of ABAQUS on Windows NT includes the diff program obtained from the Free Software Foundation. You may freely distribute the diff program and/or modify it under the terms of the GNU Library General Public License as published by the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA.

This release of ABAQUS/CAE includes lp_solve, a simplex-based code for linear and integer

programming problems by Michel Berkelaar of Eindhoven University of Technology, Eindhoven, the Netherlands.

Python, copyright 1991-1995 by Stichting Mathematisch Centrum, Amsterdam, The Netherlands. All Rights Reserved. Permission to use, copy, modify, and distribute the Python software and its

documentation for any purpose and without fee is hereby granted, provided that the above copyright notice appear in all copies and that both that copyright notice and this permission notice appear in supporting documentation, and that the names of Stichting Mathematisch Centrum or CWI or

All other brand or product names are trademarks or registered trademarks of their respective companies or organizations.

All other brand or product names are trademarks or registered trademarks of their respective companies or organizations.

1 ft 0.30480 m 1 mile 1609.3 m Area 1 in2 _{0.64516 ´ 10}-3 _m2 1 ft2 _{0.092903 m}2 1 acre 4046.9 m2 Volume 1 in3 _{0.016387 ´ 10}-3 _m3 1 ft3 _{0.028317 m}3 1 US gallon 3.7854 ´ 10-3 _m3

Conversion factors for stress analysis

Quantity U.S. unit SI equivalent

Density 1 slug/ft3 _{= 1 lbf s}2_/ft4 _{515.38 kg/m}3 1 lbf s2_/in4 _{10.687 ´ 10}6 _kg/m3 Energy 1 ft lbf 1.3558 J (N m) Force 1 lbf 4.4482 N (kg m/s2₎ Mass 1 slug = 1 lbf s2_{/ft 14.594 kg (N s}2_/m) 1 lbf s2_{/in 175.13 kg} Power 1 ft lbf/s 1.3558 W (N m/s) Pressure, Stress 1 psi (lbf/in2_{) 6894.8 Pa (N/m}2₎

Conversion factors for heat transfer analysis

Quantity U.S. unit SI equivalent

Conductivity 1 Btu/ft hr °F 1.7307 W/m °C 1 Btu/in hr °F 20.769 W/m °C Density 1 lbm/in3 _{27680. kg/m}3

Energy 1 Btu 1055.1 J

Heat flux density 1 Btu/in2 _{hr 454.26 W/m}2

Power 1 Btu/hr 0.29307 W

Specific heat 1 Btu/lbm °F 4186.8 J/kg °C

Temperature 1 °F 5/9 °C

Temp °F 9/5 ´ Temp °C + 32° 9/5 ´ Temp °K - 459.67° Important constants

Constant U.S. unit SI unit

Absolute zero -459.67 °F -273.15 °C Acceleration of gravity 32.174 ft/s2 _{9.8066 m/s}2

Atmospheric pressure 14.694 psi 0.10132 ´ 106 _Pa

Stefan-Boltzmann constant

0.1714 ´ 10-8 _{Btu/hr ft}2

°R4 5.669 ´ 10-8 W/m2 °K4

where °R = °F + 459.67 where °K = °C + 273.15 Approximate properties of mild steel at room temperature

Quantity U.S. unit SI unit

Conductivity 28.9 Btu/ft hr °F 50 W/m °C 2.4 Btu/in hr °F Density 15.13 slug/ft3 _{(lbf s}2_/ft4_{) 7800 kg/m}3 0.730 ´ 10-3 _{lbf s}2_/in4 1 ft 0.30480 m 1 mile 1609.3 m Area 1 in2 _{0.64516 ´ 10}-3 _m2 1 ft2 _{0.092903 m}2 1 acre 4046.9 m2 Volume 1 in3 _{0.016387 ´ 10}-3 _m3 1 ft3 _{0.028317 m}3 1 US gallon 3.7854 ´ 10-3 _m3

Conversion factors for stress analysis

Quantity U.S. unit SI equivalent

Density 1 slug/ft3 _{= 1 lbf s}2_/ft4 _{515.38 kg/m}3 1 lbf s2_/in4 _{10.687 ´ 10}6 _kg/m3 Energy 1 ft lbf 1.3558 J (N m) Force 1 lbf 4.4482 N (kg m/s2₎ Mass 1 slug = 1 lbf s2_{/ft 14.594 kg (N s}2_/m) 1 lbf s2_{/in 175.13 kg} Power 1 ft lbf/s 1.3558 W (N m/s) Pressure, Stress 1 psi (lbf/in2_{) 6894.8 Pa (N/m}2₎

Conversion factors for heat transfer analysis

Quantity U.S. unit SI equivalent

Conductivity 1 Btu/ft hr °F 1.7307 W/m °C 1 Btu/in hr °F 20.769 W/m °C Density 1 lbm/in3 _{27680. kg/m}3

Energy 1 Btu 1055.1 J

Heat flux density 1 Btu/in2 _{hr 454.26 W/m}2

Power 1 Btu/hr 0.29307 W

Specific heat 1 Btu/lbm °F 4186.8 J/kg °C

Temperature 1 °F 5/9 °C

Temp °F 9/5 ´ Temp °C + 32° 9/5 ´ Temp °K - 459.67° Important constants

Constant U.S. unit SI unit

Absolute zero -459.67 °F -273.15 °C Acceleration of gravity 32.174 ft/s2 _{9.8066 m/s}2

Atmospheric pressure 14.694 psi 0.10132 ´ 106 _Pa

Stefan-Boltzmann constant

0.1714 ´ 10-8 _{Btu/hr ft}2

°R4 5.669 ´ 10-8 W/m2 °K4

where °R = °F + 459.67 where °K = °C + 273.15 Approximate properties of mild steel at room temperature

Quantity U.S. unit SI unit

Conductivity 28.9 Btu/ft hr °F 50 W/m °C 2.4 Btu/in hr °F

Density 15.13 slug/ft3 _{(lbf s}2_/ft4_{) 7800 kg/m}3

Specific heat 0.11 Btu/lbm °F 460 J/kg °C Yield stress 30 ´ 103 _{psi 207 ´ 10}6 _Pa

Specific heat 0.11 Btu/lbm °F 460 J/kg °C Yield stress 30 ´ 103 _{psi 207 ´ 10}6 _Pa

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Hibbitt, Karlsson & Sorensen (West), Inc.

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Inc.

1080 Main Street 14500 Sheldon Road, Suite 160 Pawtucket, RI 02860-4847 Plymouth, MI 48170-2408 Tel: 401 727 4200 Tel: 734 451 0217 Fax: 401 727 4208 Fax: 734 451 0458 E-mail: [email protected], [email protected] E-mail: [email protected] http://www.abaqus.com

Hibbitt, Karlsson & Sorensen (West), Inc.

ABAQUS Solutions Northeast, LLC 39221 Paseo Padre Parkway, Suite F Summit Office Park, West Building Fremont, CA 94538-1611 300 Centerville Road, Suite 209W Tel: 510 794 5891 Warwick, RI 02886-0201

Fax: 510 794 1194 Tel: 401 739 3637 E-mail: [email protected] Fax: 401 739 3302

E-mail: [email protected] AC Engineering, Inc. 1440 Innovation Place West Lafayette, IN 47906-1000 Tel: 765 497 1373 Fax: 765 497 4444 E-mail: [email protected] ARGENTINA AUSTRALIA

KB Engineering S. R. L. Compumod Pty. Ltd. Florida 274, Of. 37 Level 13, 309 Pitt Street (1005) Buenos Aires, Argentina Sydney 2000

Tel: +54 11 4393 8444 P.O. Box A807 Fax: +54 11 4326 2424 Sydney South 1235 E-mail: [email protected] Tel: 02 9283 2577

Fax: 02 9283 2585

E-mail: [email protected] http://www.compumod.com.au

AUSTRIA BENELUX

VOEST-ALPINE STAHL LINZ GmbH ABAQUS Benelux BV Department WFE Huizermaatweg 576

Postfach 3 1276 LN Huizen

A-4031 Linz The Netherlands Tel: 0732 6585 9919 Tel: +31 35 52 58 424 Fax: 0732 6980 4338 Fax: +31 35 52 44 257 E-mail: [email protected] E-mail: [email protected]

CHINA CZECH REPUBLIC AND SLOVAK

REPUBLIC Advanced Finite Element Services ASATTE

Department of Engineering Mechanics Technická 4, 166 07 Praha 6 Tsinghua University Czech Republic

Beijing 100084, P. R. China Tel: 420 2 24352654 Tel: 010 62783986 Fax: 420 2 33322482

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ITALY JAPAN

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Hibbitt, Karlsson & Sorensen, Inc. Viale Certosa, 1 3rd Floor, Akasaka Nihon Building 20149 Milano 5-24, Akasaka 9-chome

Tel: 02 39211211 Minato-ku Fax: 02 39211210 Tokyo, 107-0052 E-mail: [email protected] Tel: 03 5474 5817

Fax: 03 5474 5818 E-mail: [email protected]

KOREA MALAYSIA

Hibbitt, Karlsson & Sorensen Korea, Inc. Compumod Sdn Bhd Suite 306, Sambo Building #33.03 Menara Lion 13-2 Yoido-Dong, Youngdeungpo-ku 165 Jalan Ampang Seoul, 150-010 50450 Kuala Lumpur Tel: 02 785 6707/8 Tel: 3 466 2122 Fax: 02 785 6709 Fax: 3 466 2123

E-mail: [email protected] E-mail: [email protected]

NEW ZEALAND POLAND

Matrix Applied Computing Ltd. BudSoft Sp. z o.o. P.O. Box 56-316, Auckland 61-807 Pozna Courier: Unit 2-5, 72 Dominion Road,

Mt Eden,

Sw. Marcin 58/64

Auckland Tel: 61 852 31 19

Tel: +64 9 623 1223 Fax: 61 852 31 19

Fax: +64 9 623 1134 E-mail: [email protected] E-mail: [email protected]

SINGAPORE SOUTH AFRICA

Compumod (Singapore) Pte Ltd Finite Element Analysis Services (Pty) Ltd. #17-05 Asia Chambers Suite 20-303C, The Waverley

20 McCallum Street Wyecroft Road Singapore 069046 Mowbray 7700 Tel: 223 2996 Tel: 021 448 7608 Fax: 226 0336 Fax: 021 448 7679 E-mail: [email protected] E-mail: [email protected] SPAIN SWEDEN

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E-mail: [email protected] Fax: 01925 810178

Support

HKS offers both technical (engineering) support and systems support for ABAQUS. Technical and systems support are provided through the nearest local support office. You can contact our offices by telephone, fax, electronic mail, or regular mail. Information on how to contact each office is listed in the front of each ABAQUS manual. Support information is also available by visiting the ABAQUS Home Page on the World Wide Web (details are given below). When contacting your local support office, please specify whether you would like technical support (you have encountered problems performing an ABAQUS analysis) or systems support (ABAQUS will not install correctly, licensing does not work correctly, or other hardware-related issues have arisen).

We welcome any suggestions for improvements to the support program or documentation. We will ensure that any enhancement requests you make are considered for future releases. If you wish to file a complaint about the service or products provided by HKS, refer to the ABAQUS Home Page.

Technical support

HKS technical support engineers can assist in clarifying ABAQUS features and checking errors by giving both general information on using ABAQUS and information on its application to specific analyses. If you have concerns about an analysis, we suggest that you contact us at an early stage, since it is usually easier to solve problems at the beginning of a project rather than trying to correct an

analysis at the end.

Please have the following information ready before calling the technical support hotline, and include it in any written contacts:

· The version of ABAQUS that are you using.

- The version numbers for ABAQUS/Standard and ABAQUS/Explicit are given at the top of the data (.dat) file.

- The version numbers for ABAQUS/CAE and ABAQUS/Viewer can be found by selecting Help->On version from the main menu bar.

- The version number for ABAQUS/CAT is given at the top of the input (.inp) file as well as the data file.

- The version numbers for ABAQUS/ADAMS and ABAQUS/C-MOLD are output to the screen.

- The version number for ABAQUS/Safe is given under the ABAQUS logo in the main window.

· The type of computer on which you are running ABAQUS.

Support

HKS offers both technical (engineering) support and systems support for ABAQUS. Technical and systems support are provided through the nearest local support office. You can contact our offices by telephone, fax, electronic mail, or regular mail. Information on how to contact each office is listed in the front of each ABAQUS manual. Support information is also available by visiting the ABAQUS Home Page on the World Wide Web (details are given below). When contacting your local support office, please specify whether you would like technical support (you have encountered problems performing an ABAQUS analysis) or systems support (ABAQUS will not install correctly, licensing does not work correctly, or other hardware-related issues have arisen).

We welcome any suggestions for improvements to the support program or documentation. We will ensure that any enhancement requests you make are considered for future releases. If you wish to file a complaint about the service or products provided by HKS, refer to the ABAQUS Home Page.

Technical support

HKS technical support engineers can assist in clarifying ABAQUS features and checking errors by giving both general information on using ABAQUS and information on its application to specific analyses. If you have concerns about an analysis, we suggest that you contact us at an early stage, since it is usually easier to solve problems at the beginning of a project rather than trying to correct an

analysis at the end.

Please have the following information ready before calling the technical support hotline, and include it in any written contacts:

· The version of ABAQUS that are you using.

- The version numbers for ABAQUS/Standard and ABAQUS/Explicit are given at the top of the data (.dat) file.

- The version numbers for ABAQUS/CAE and ABAQUS/Viewer can be found by selecting Help->On version from the main menu bar.

- The version number for ABAQUS/CAT is given at the top of the input (.inp) file as well as the data file.

- The version numbers for ABAQUS/ADAMS and ABAQUS/C-MOLD are output to the screen.

- The version number for ABAQUS/Safe is given under the ABAQUS logo in the main window.

· Workarounds or tests that you have already tried.

When calling for support about a specific problem, any available ABAQUS output files may be helpful in answering questions that the support engineer may ask you.

The support engineer will try to diagnose your problem from the model description and a description of the difficulties you are having. Frequently, the support engineer will need model sketches, which can be faxed to HKS or sent in the mail. Plots of the final results or the results near the point that the analysis terminated may also be needed to understand what may have caused the problem.

If the support engineer cannot diagnose your problem from this information, you may be asked to send the input data. The data can be sent by means of e-mail, tape, or disk. Please check the ABAQUS Home Page at www.abaqus.com for the media formats that are currently accepted.

All support calls are logged into a database, which enables us to monitor the progress of a particular problem and to check that we are resolving support issues efficiently. If you would like to know the log number of your particular call for future reference, please ask the support engineer. If you are calling to discuss an existing support problem and you know the log number, please mention it so that we can consult the database to see what the latest action has been and, thus, avoid duplication of effort. In addition, please give the receptionist the support engineer's name (or include it at the top of any e-mail correspondence).

Systems support

HKS systems support engineers can help you resolve issues related to the installation and running of ABAQUS, including licensing difficulties, that are not covered by technical support.

You should install ABAQUS by carefully following the instructions in the ABAQUS Site Guide. If you encounter problems with the installation or licensing, first review the instructions in the ABAQUS Site Guide to ensure that they have been followed correctly. If this does not resolve the problems, look on the ABAQUS Home Page under Technical Support for information about known installation problems. If this does not address your situation, please contact your local support office. Send whatever information is available to define the problem: error messages from an aborted analysis or a detailed explanation of the problems encountered. Whenever possible, please send the output from the abaqus info=env and abaqus info=sys commands.

ABAQUS Web server

For users connected to the Internet, many questions can be answered by visiting the ABAQUS Home Page on the World Wide Web at

http://www.abaqus.com

The information available on the ABAQUS Home Page includes: · Frequently asked questions

· ABAQUS systems information and machine requirements · Workarounds or tests that you have already tried.

When calling for support about a specific problem, any available ABAQUS output files may be helpful in answering questions that the support engineer may ask you.

The support engineer will try to diagnose your problem from the model description and a description of the difficulties you are having. Frequently, the support engineer will need model sketches, which can be faxed to HKS or sent in the mail. Plots of the final results or the results near the point that the analysis terminated may also be needed to understand what may have caused the problem.

If the support engineer cannot diagnose your problem from this information, you may be asked to send the input data. The data can be sent by means of e-mail, tape, or disk. Please check the ABAQUS Home Page at www.abaqus.com for the media formats that are currently accepted.

All support calls are logged into a database, which enables us to monitor the progress of a particular problem and to check that we are resolving support issues efficiently. If you would like to know the log number of your particular call for future reference, please ask the support engineer. If you are calling to discuss an existing support problem and you know the log number, please mention it so that we can consult the database to see what the latest action has been and, thus, avoid duplication of effort. In addition, please give the receptionist the support engineer's name (or include it at the top of any e-mail correspondence).

Systems support

HKS systems support engineers can help you resolve issues related to the installation and running of ABAQUS, including licensing difficulties, that are not covered by technical support.

You should install ABAQUS by carefully following the instructions in the ABAQUS Site Guide. If you encounter problems with the installation or licensing, first review the instructions in the ABAQUS Site Guide to ensure that they have been followed correctly. If this does not resolve the problems, look on the ABAQUS Home Page under Technical Support for information about known installation problems. If this does not address your situation, please contact your local support office. Send whatever information is available to define the problem: error messages from an aborted analysis or a detailed explanation of the problems encountered. Whenever possible, please send the output from the abaqus info=env and abaqus info=sys commands.

ABAQUS Web server

For users connected to the Internet, many questions can be answered by visiting the ABAQUS Home Page on the World Wide Web at

http://www.abaqus.com

The information available on the ABAQUS Home Page includes: · Frequently asked questions

· Error status reports

· ABAQUS documentation price list · Training seminar schedule

· Newsletters

Anonymous ftp site

For users connected to the Internet, HKS maintains useful documents on an anonymous ftp account on the computer ftp.abaqus.com. Simply ftp to ftp.abaqus.com. Login as user anonymous, and type your e-mail address as your password. Directions will come up automatically upon login.

Writing to technical support

Hibbitt, Karlsson & Sorensen, Inc. 1080 Main Street

Pawtucket, RI 02860-4847, USA Attention: Technical Support

Addresses for other offices and representatives are listed in the front of each manual.

Support for academic institutions

Under the terms of the Academic License Agreement we do not provide support to users at academic institutions unless the institution has also purchased technical support. Please see the ABAQUS Home Page, or contact us for more information.

Training

All HKS offices offer regularly scheduled public training classes.

The Introduction to ABAQUS/Standard and ABAQUS/Explicit seminar covers basic usage and nonlinear applications, such as large deformation, plasticity, contact, and dynamics. Workshops provide as much practical experience with ABAQUS as possible.

The Introduction to ABAQUS/CAE seminar discusses modeling, managing simulations, and viewing results with ABAQUS/CAE. "Hands-on" workshops are complemented by lectures.

Advanced seminars cover topics of interest to customers with experience using ABAQUS, such as engine analysis, metal forming, fracture mechanics, and heat transfer.

We also provide training seminars at customer sites. On-site training seminars can be one or more days in duration, depending on customer requirements. The training topics can include a combination of material from our introductory and advanced seminars. Workshops allow customers to exercise ABAQUS on their own computers.

· Error status reports

· ABAQUS documentation price list · Training seminar schedule

· Newsletters

Anonymous ftp site

For users connected to the Internet, HKS maintains useful documents on an anonymous ftp account on the computer ftp.abaqus.com. Simply ftp to ftp.abaqus.com. Login as user anonymous, and type your e-mail address as your password. Directions will come up automatically upon login.

Writing to technical support

Hibbitt, Karlsson & Sorensen, Inc. 1080 Main Street

Pawtucket, RI 02860-4847, USA Attention: Technical Support

Addresses for other offices and representatives are listed in the front of each manual.

Support for academic institutions

Under the terms of the Academic License Agreement we do not provide support to users at academic institutions unless the institution has also purchased technical support. Please see the ABAQUS Home Page, or contact us for more information.

Training

All HKS offices offer regularly scheduled public training classes.

The Introduction to ABAQUS/Standard and ABAQUS/Explicit seminar covers basic usage and nonlinear applications, such as large deformation, plasticity, contact, and dynamics. Workshops provide as much practical experience with ABAQUS as possible.

The Introduction to ABAQUS/CAE seminar discusses modeling, managing simulations, and viewing results with ABAQUS/CAE. "Hands-on" workshops are complemented by lectures.

Advanced seminars cover topics of interest to customers with experience using ABAQUS, such as engine analysis, metal forming, fracture mechanics, and heat transfer.

We also provide training seminars at customer sites. On-site training seminars can be one or more days in duration, depending on customer requirements. The training topics can include a combination of material from our introductory and advanced seminars. Workshops allow customers to exercise ABAQUS on their own computers.

Documentation

The following documentation and publications are available from HKS, unless otherwise specified, in printed form and through our online documentation server. For more information on accessing the online books, refer to the discussion of execution procedures in the user's manuals.

In addition to the documentation listed below, HKS publishes two newsletters on a regular schedule: ABAQUS/News and ABAQUS/Answers. ABAQUS/News includes topical information about program releases, training seminars, etc. ABAQUS/Answers includes technical articles on particular topics related to ABAQUS usage. These newsletters are distributed at no cost to users who wish to subscribe. Please contact your local ABAQUS support office if you wish to be added to the mailing list for these publications. They are also archived in the Reference Shelf on the ABAQUS Home Page.

Training Manuals

Getting Started with ABAQUS/Standard: This document is a self-paced tutorial designed to help new users become familiar with using ABAQUS/Standard for static and dynamic stress analysis simulations. It contains a number of fully worked examples that provide practical guidelines for performing structural analyses with ABAQUS.

Getting Started with ABAQUS/Explicit: This document is a self-paced tutorial designed to help new users become familiar with using ABAQUS/Explicit. It begins with the basics of modeling in ABAQUS, so no prior knowledge of ABAQUS is required. A number of fully worked examples provide practical guidelines for performing explicit dynamic analyses, such as drop tests and metal forming simulations, with ABAQUS/Explicit.

Lecture Notes: These notes are available on many topics to which ABAQUS is applied. They are used in the technical seminars that HKS presents to help users improve their understanding and usage of ABAQUS (see the "Training" section above for more information about these seminars). While not intended as stand-alone tutorial material, they are sufficiently comprehensive that they can usually be used in that mode. The list of available lecture notes is included in the

Documentation Price List.

User's Manuals

ABAQUS/Standard User's Manual: This volume contains a complete description of the

elements, material models, procedures, input specifications, etc. It is the basic reference document for ABAQUS/Standard.

ABAQUS/Explicit User's Manual: This volume contains a complete description of the elements, material models, procedures, input specifications, etc. It is the basic reference document for

ABAQUS/Explicit.

Documentation

The following documentation and publications are available from HKS, unless otherwise specified, in printed form and through our online documentation server. For more information on accessing the online books, refer to the discussion of execution procedures in the user's manuals.

In addition to the documentation listed below, HKS publishes two newsletters on a regular schedule: ABAQUS/News and ABAQUS/Answers. ABAQUS/News includes topical information about program releases, training seminars, etc. ABAQUS/Answers includes technical articles on particular topics related to ABAQUS usage. These newsletters are distributed at no cost to users who wish to subscribe. Please contact your local ABAQUS support office if you wish to be added to the mailing list for these publications. They are also archived in the Reference Shelf on the ABAQUS Home Page.

Training Manuals

Getting Started with ABAQUS/Standard: This document is a self-paced tutorial designed to help new users become familiar with using ABAQUS/Standard for static and dynamic stress analysis simulations. It contains a number of fully worked examples that provide practical guidelines for performing structural analyses with ABAQUS.

Getting Started with ABAQUS/Explicit: This document is a self-paced tutorial designed to help new users become familiar with using ABAQUS/Explicit. It begins with the basics of modeling in ABAQUS, so no prior knowledge of ABAQUS is required. A number of fully worked examples provide practical guidelines for performing explicit dynamic analyses, such as drop tests and metal forming simulations, with ABAQUS/Explicit.

Lecture Notes: These notes are available on many topics to which ABAQUS is applied. They are used in the technical seminars that HKS presents to help users improve their understanding and usage of ABAQUS (see the "Training" section above for more information about these seminars). While not intended as stand-alone tutorial material, they are sufficiently comprehensive that they can usually be used in that mode. The list of available lecture notes is included in the

Documentation Price List.

User's Manuals

ABAQUS/Standard User's Manual: This volume contains a complete description of the

elements, material models, procedures, input specifications, etc. It is the basic reference document for ABAQUS/Standard.

ABAQUS/Explicit User's Manual: This volume contains a complete description of the elements, material models, procedures, input specifications, etc. It is the basic reference document for

generation, analysis, and results evaluation.

ABAQUS/Viewer User's Manual: This basic reference document for ABAQUS/Viewer includes an introductory tutorial as well as a complete description of how to use ABAQUS/Viewer to display your model and results.

ABAQUS/ADAMS User's Manual: This document describes how to install and how to use ABAQUS/ADAMS, an interface program that creates ABAQUS models of ADAMS components and converts the ABAQUS results into an ADAMS modal neutral file that can be used by the ADAMS/Flex program. It is the basic reference document for the ABAQUS/ADAMS program.

ABAQUS/CAT User's Manual: This document describes how to install and how to use ABAQUS/CAT, an interface program that creates an ABAQUS input file from a CATIA model and postprocesses the analysis results in CATIA. It is the basic reference document for the ABAQUS/CAT program.

ABAQUS/C-MOLD User's Manual: This document describes how to install and how to use ABAQUS/C-MOLD, an interface program that translates finite element mesh, material property, and initial stress data from a C-MOLD analysis to an ABAQUS input file.

ABAQUS/Safe User's Manual: This document describes how to install and how to use ABAQUS/Safe, an interface program that calculates fatigue lives and fatigue strength reserve factors from finite element models. It is the basic reference document for the ABAQUS/Safe program. The theoretical background to fatigue analysis is contained in the Modern Metal Fatigue Analysis manual (available only in print).

Using ABAQUS Online Documentation: This online manual contains instructions on using the ABAQUS online documentation server to read the manuals that are available online.

ABAQUS Release Notes: This document contains brief descriptions of the new features available in the latest release of the ABAQUS product line.

ABAQUS Site Guide: This document describes how to install ABAQUS and how to configure the installation for particular circumstances. Some of this information, of most relevance to users, is also provided in the user's manuals.

Examples Manuals

ABAQUS Example Problems Manual: This volume contains more than 75 detailed examples designed to illustrate the approaches and decisions needed to perform meaningful linear and nonlinear analysis. Typical cases are large motion of an elastic-plastic pipe hitting a rigid wall; inelastic buckling collapse of a thin-walled elbow; explosive loading of an elastic, viscoplastic thin ring; consolidation under a footing; buckling of a composite shell with a hole; and deep drawing of a metal sheet. It is generally useful to look for relevant examples in this manual and to review them when embarking on a new class of problem.

generation, analysis, and results evaluation.

ABAQUS/Viewer User's Manual: This basic reference document for ABAQUS/Viewer includes an introductory tutorial as well as a complete description of how to use ABAQUS/Viewer to display your model and results.

ABAQUS/ADAMS User's Manual: This document describes how to install and how to use ABAQUS/ADAMS, an interface program that creates ABAQUS models of ADAMS components and converts the ABAQUS results into an ADAMS modal neutral file that can be used by the ADAMS/Flex program. It is the basic reference document for the ABAQUS/ADAMS program.

ABAQUS/CAT User's Manual: This document describes how to install and how to use ABAQUS/CAT, an interface program that creates an ABAQUS input file from a CATIA model and postprocesses the analysis results in CATIA. It is the basic reference document for the ABAQUS/CAT program.

ABAQUS/C-MOLD User's Manual: This document describes how to install and how to use ABAQUS/C-MOLD, an interface program that translates finite element mesh, material property, and initial stress data from a C-MOLD analysis to an ABAQUS input file.

ABAQUS/Safe User's Manual: This document describes how to install and how to use ABAQUS/Safe, an interface program that calculates fatigue lives and fatigue strength reserve factors from finite element models. It is the basic reference document for the ABAQUS/Safe program. The theoretical background to fatigue analysis is contained in the Modern Metal Fatigue Analysis manual (available only in print).

Using ABAQUS Online Documentation: This online manual contains instructions on using the ABAQUS online documentation server to read the manuals that are available online.

ABAQUS Release Notes: This document contains brief descriptions of the new features available in the latest release of the ABAQUS product line.

ABAQUS Site Guide: This document describes how to install ABAQUS and how to configure the installation for particular circumstances. Some of this information, of most relevance to users, is also provided in the user's manuals.

Examples Manuals

ABAQUS Example Problems Manual: This volume contains more than 75 detailed examples designed to illustrate the approaches and decisions needed to perform meaningful linear and nonlinear analysis. Typical cases are large motion of an elastic-plastic pipe hitting a rigid wall; inelastic buckling collapse of a thin-walled elbow; explosive loading of an elastic, viscoplastic thin ring; consolidation under a footing; buckling of a composite shell with a hole; and deep drawing of a metal sheet. It is generally useful to look for relevant examples in this manual and to review them when embarking on a new class of problem.

ABAQUS; the tests are multiple element tests of simple geometries or simplified versions of real problems. The NAFEMS benchmark problems are included in this manual.

ABAQUS Verification Manual: This online-only volume contains more than 5000 basic test cases, providing verification of each individual program feature (procedures, output options, MPCs, etc.) against exact calculations and other published results. It may be useful to run these problems when learning to use a new capability. In addition, the supplied input data files provide good starting points to check the behavior of elements, materials, etc.

Reference Manuals

ABAQUS Keywords Manual: This volume contains a complete description of all the input options that are available in ABAQUS/Standard and ABAQUS/Explicit.

ABAQUS Theory Manual: This volume (available online and, if requested, in print) contains detailed, precise discussions of all theoretical aspects of ABAQUS. It is written to be understood by users with an engineering background.

ABAQUS Command Language Manual: This online manual provides a description of the ABAQUS Command Language and a command reference that lists the syntax of each command. The manual describes how commands can be used to create and analyze ABAQUS/CAE models, to view the results of the analysis, and to automate repetitive tasks. It also contains information on using the ABAQUS Command Language or C++ as an application programming interface (API).

ABAQUS Input Files: This online manual contains all the input files that are included with the ABAQUS release and referred to in the ABAQUS Example Problems Manual, the ABAQUS Benchmarks Manual, and the ABAQUS Verification Manual. They are listed in the order in which they appear in the manuals, under the title of the problem that refers to them. The input file

references in the manuals hyperlink directly to this book.

Quality Assurance Plan: This document describes HKS's QA procedures. It is a controlled document, provided to customers who subscribe to either HKS's Nuclear QA Program or the Quality Monitoring Service.

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 difficult 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

ABAQUS; the tests are multiple element tests of simple geometries or simplified versions of real problems. The NAFEMS benchmark problems are included in this manual.

ABAQUS Verification Manual: This online-only volume contains more than 5000 basic test cases, providing verification of each individual program feature (procedures, output options, MPCs, etc.) against exact calculations and other published results. It may be useful to run these problems when learning to use a new capability. In addition, the supplied input data files provide good starting points to check the behavior of elements, materials, etc.

Reference Manuals

ABAQUS Keywords Manual: This volume contains a complete description of all the input options that are available in ABAQUS/Standard and ABAQUS/Explicit.

ABAQUS Theory Manual: This volume (available online and, if requested, in print) contains detailed, precise discussions of all theoretical aspects of ABAQUS. It is written to be understood by users with an engineering background.

ABAQUS Command Language Manual: This online manual provides a description of the ABAQUS Command Language and a command reference that lists the syntax of each command. The manual describes how commands can be used to create and analyze ABAQUS/CAE models, to view the results of the analysis, and to automate repetitive tasks. It also contains information on using the ABAQUS Command Language or C++ as an application programming interface (API).

ABAQUS Input Files: This online manual contains all the input files that are included with the ABAQUS release and referred to in the ABAQUS Example Problems Manual, the ABAQUS Benchmarks Manual, and the ABAQUS Verification Manual. They are listed in the order in which they appear in the manuals, under the title of the problem that refers to them. The input file

references in the manuals hyperlink directly to this book.

Quality Assurance Plan: This document describes HKS's QA procedures. It is a controlled document, provided to customers who subscribe to either HKS's Nuclear QA Program or the Quality Monitoring Service.

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 difficult 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

mesh densities, and other variations. This results in a relatively large number of input data files for some of the problems. Only a few of the input files are listed in the printed manual. The selection has been made to provide the most guidance to the user.

All input files, both the ones that are listed in the printed manual and the ones that are referenced, are included with the ABAQUS release. The ABAQUS/Fetch utility is used to extract these input files from the compressed archive files provided with the ABAQUS release. For example, to fetch input file

boltpipeflange_3d_cyclsym.inp, type

abaqus fetch job=boltpipeflange_3d_cyclsym.inp

Parametric study script (.psf) and user subroutine (.f) files can be fetched in the same manner. All files for a particular problem can be obtained by leaving off the file extension. The ABAQUS/Fetch execution procedure is explained in detail in ``Execution procedure for ABAQUS/Fetch,'' Section 3.2.9 of the ABAQUS/Standard User's Manual and the ABAQUS/Explicit User's Manual.

It is sometimes useful to search the input files. The findkeyword utility is used to locate input files that contain user-specified input. This utility is defined in ``Execution procedure for querying the keyword/problem database,'' Section 3.2.8 of the ABAQUS/Standard User's Manual and the

ABAQUS/Explicit User's Manual.

In addition, all the input files included with the ABAQUS release can be accessed through the

ABAQUS Input Files electronic book. This book is part of the ABAQUS online documentation collection and, as such, is fully searchable (with the exception of numeric strings and

ABAQUS-specific terms). When reading the online version of the ABAQUS Benchmarks Manual, the

ABAQUS Example Problems Manual, or the ABAQUS Verification Manual, the user can click on an input file name; the ABAQUS Input Files book will open to that file in a separate window.

To reproduce the graphical representation of the solution reported in some of the examples, the output frequency used in the input files may need to be increased. For example, in ``Linear analysis of the Indian Point reactor feedwater line,'' Section 2.2.2, the figures that appear in the manual can be obtained only if the solution 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 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 simplified versions of real problems. The ABAQUS Verification Manual contains a large number of examples that are intended as

elementary verification of the basic modeling capabilities.

The verification of ABAQUS consists of running the problems in the ABAQUS Example Problems Manual, the ABAQUS Benchmarks Manual, and the ABAQUS Verification Manual. Before a version mesh densities, and other variations. This results in a relatively large number of input data files for some of the problems. Only a few of the input files are listed in the printed manual. The selection has been made to provide the most guidance to the user.

All input files, both the ones that are listed in the printed manual and the ones that are referenced, are included with the ABAQUS release. The ABAQUS/Fetch utility is used to extract these input files from the compressed archive files provided with the ABAQUS release. For example, to fetch input file

boltpipeflange_3d_cyclsym.inp, type

abaqus fetch job=boltpipeflange_3d_cyclsym.inp

Parametric study script (.psf) and user subroutine (.f) files can be fetched in the same manner. All files for a particular problem can be obtained by leaving off the file extension. The ABAQUS/Fetch execution procedure is explained in detail in ``Execution procedure for ABAQUS/Fetch,'' Section 3.2.9 of the ABAQUS/Standard User's Manual and the ABAQUS/Explicit User's Manual.

It is sometimes useful to search the input files. The findkeyword utility is used to locate input files that contain user-specified input. This utility is defined in ``Execution procedure for querying the keyword/problem database,'' Section 3.2.8 of the ABAQUS/Standard User's Manual and the

ABAQUS/Explicit User's Manual.

In addition, all the input files included with the ABAQUS release can be accessed through the

ABAQUS Input Files electronic book. This book is part of the ABAQUS online documentation collection and, as such, is fully searchable (with the exception of numeric strings and

ABAQUS-specific terms). When reading the online version of the ABAQUS Benchmarks Manual, the

ABAQUS Example Problems Manual, or the ABAQUS Verification Manual, the user can click on an input file name; the ABAQUS Input Files book will open to that file in a separate window.

To reproduce the graphical representation of the solution reported in some of the examples, the output frequency used in the input files may need to be increased. For example, in ``Linear analysis of the Indian Point reactor feedwater line,'' Section 2.2.2, the figures that appear in the manual can be obtained only if the solution 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 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 simplified versions of real problems. The ABAQUS Verification Manual contains a large number of examples that are intended as

elementary verification of the basic modeling capabilities.

The verification of ABAQUS consists of running the problems in the ABAQUS Example Problems Manual, the ABAQUS Benchmarks Manual, and the ABAQUS Verification Manual. Before a version

1. Static Stress/Displacement Analyses

1.1 Static and quasi-static stress analyses

1.1.1 Axisymmetric analysis of bolted pipe flange connections

A bolted pipe flange connection is a common and important part of many piping systems. Such connections are 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 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 superelements are analyzed to demonstrate the use of superelements with a large number of retained degrees of freedom.

Geometry and model

The bolted joint assembly being analyzed is depicted in Figure 1.1.1-1. The geometry and dimensions of the various parts are taken from Sawa et al. (1991), modified 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 flange and the gasket are 82.5 mm and 52.5 mm, respectively. The thickness of the gasket is 2.5 mm. The pipe flange has eight bolt holes that are equally spaced in the pitch circle of radius 65 mm. The radius of the bolt hole 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 Young's modulus is 206 GPa and the Poisson's ratio is 0.3 for both the bolt and the pipe

hub/flange. The gasket is modeled with either solid continuum or gasket elements. When continuum elements are used, the gasket's Young's modulus, E, equals 68.7 GPa and its Poisson's ratio, º, equals 0.3.

When gasket elements are used, a linear gasket pressure/closure relationship is used with the effective "normal stiffness," Sn, equal to the material Young's modulus divided by the thickness so that Sn =

27.48 GPa/mm. Similarly a linear shear stress/shear motion relationship is used with an effective shear stiffness, St, equal to the material shear modulus divided by the thickness so that St = 10.57 GPa/mm.

The membrane behavior is specified with a Young's modulus of 68.7 GPa and a Poisson's ratio of 0.3. Sticking contact conditions are assumed in all contact areas: between the bearing surface and the flange and between the gasket and the hub. Contact between the bolt shank and the bolt hole is

1. Static Stress/Displacement Analyses

1.1 Static and quasi-static stress analyses

1.1.1 Axisymmetric analysis of bolted pipe flange connections

A bolted pipe flange connection is a common and important part of many piping systems. Such connections are 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 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 superelements are analyzed to demonstrate the use of superelements with a large number of retained degrees of freedom.

Geometry and model

The bolted joint assembly being analyzed is depicted in Figure 1.1.1-1. The geometry and dimensions of the various parts are taken from Sawa et al. (1991), modified 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 flange and the gasket are 82.5 mm and 52.5 mm, respectively. The thickness of the gasket is 2.5 mm. The pipe flange has eight bolt holes that are equally spaced in the pitch circle of radius 65 mm. The radius of the bolt hole 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 Young's modulus is 206 GPa and the Poisson's ratio is 0.3 for both the bolt and the pipe

hub/flange. The gasket is modeled with either solid continuum or gasket elements. When continuum elements are used, the gasket's Young's modulus, E, equals 68.7 GPa and its Poisson's ratio, º, equals 0.3.

When gasket elements are used, a linear gasket pressure/closure relationship is used with the effective "normal stiffness," Sn, equal to the material Young's modulus divided by the thickness so that Sn =

27.48 GPa/mm. Similarly a linear shear stress/shear motion relationship is used with an effective shear stiffness, St, equal to the material shear modulus divided by the thickness so that St = 10.57 GPa/mm.

The membrane behavior is specified with a Young's modulus of 68.7 GPa and a Poisson's ratio of 0.3. Sticking contact conditions are assumed in all contact areas: between the bearing surface and the flange and between the gasket and the hub. Contact between the bolt shank and the bolt hole is

ignored.

The finite element idealizations of the symmetric half of the pipe joint are shown in Figure 1.1.1-2 and Figure 1.1.1-3, 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 a separate mesh for the bolts. In Figure 1.1.1-2the 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 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/flange. The gasket is modeled with either CAX8R solid

continuum elements or GKAX6 gasket elements. Contact between the gasket and the pipe hub/flange is modeled with contact pairs between surfaces defined 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 flange 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 definitions, 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 1.1.1-4 illustrates the cross-sectional views of the bolt head and the shank. Each plane stress element represents a volume that extends out of the x-y plane. For example, element A represents a volume calculated as (HA) ´ (AreaA). Likewise, element B represents a volume calculated as (HB) ´

(AreaB). The sectional area in the x-z plane pertaining to a given element can be calculated as

Area = 2 Z X2 X1 [(R2_{¡ x}2)12_{]dx = [x(R}2 ¡ x2)12 _+R2_{arcsin (} x jRj)] ¯ ¯X2 X1;

where R is the bolt head radius, Rbolthead, or the shank radius, Rshank (depending on the element

location), and X1 and X2 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, WA and WB, 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 model's contact surfaces are used to calculate the surface areas of the nodes used in defining the node-based surfaces of the model. Referring again to Figure 1.1.1-4, nodal contact areas for a single bolt are calculated as follows: A1 = AC 4 ; A9 = AF 4 ; ignored.

The finite element idealizations of the symmetric half of the pipe joint are shown in Figure 1.1.1-2 and Figure 1.1.1-3, 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 a separate mesh for the bolts. In Figure 1.1.1-2the 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 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/flange. The gasket is modeled with either CAX8R solid

continuum elements or GKAX6 gasket elements. Contact between the gasket and the pipe hub/flange is modeled with contact pairs between surfaces defined 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 flange 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 definitions, 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 1.1.1-4 illustrates the cross-sectional views of the bolt head and the shank. Each plane stress element represents a volume that extends out of the x-y plane. For example, element A represents a volume calculated as (HA) ´ (AreaA). Likewise, element B represents a volume calculated as (HB) ´

(AreaB). The sectional area in the x-z plane pertaining to a given element can be calculated as

Area = 2 Z X2 X1 [(R2_{¡ x}2)12_{]dx = [x(R}2 ¡ x2)12 _+R2_{arcsin (} x jRj)] ¯ ¯X2 X1;

where R is the bolt head radius, Rbolthead, or the shank radius, Rshank (depending on the element

location), and X1 and X2 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, WA and WB, 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 model's contact surfaces are used to calculate the surface areas of the nodes used in defining the node-based surfaces of the model. Referring again to Figure 1.1.1-4, nodal contact areas for a single bolt are calculated as follows: A1 =

AC

4 ; A9 = AF

A2 = AC 2 ; A4 = AD 2 ; A6 = AE 2 ; A8 = AF 2 ; A3 = (AC +AD)=4; A5 = (AD +AE)=4; A7 = (AE +AF)=4;

where A1 through A9 are contact areas that are associated with contact nodes 1-9 and Ac through AF

are sectional areas that are associated with bolt head elements C-F . 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 flange 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 flat plates are found in ASME Section VIII Div 2 Article 4-9. For the type of structure under study, which is not a flat 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 AA = 6534.51 mm2_{, and the total area of the bolt}

holes is given by AH = 8¼82 _{= 1608.5 mm}2_{. Hence, the reduction factor is simply 1 ¡ AH=AA =}

0.754. The effective in-plane moduli of elasticity, E10 _{and E2}0_{, are obtained by multiplying the}

respective moduli, E1 and E2, by this factor. We assume material isotropy in the r-z plane; thus, E10 =E20 =E0: The modulus in the hoop direction, E30, should be very small and is chosen such that E0_=E30 _{= 10}6_{. The in-plane shear modulus is then calculated based on the effective elasticity}

modulus: G0

12 =E0=2(1 + º): The shear moduli in the hoop direction are also calculated similarly but

with º set to zero (they are not used in an axisymmetric model). Hence, we have E10 _=E20 _{= 155292}

MPa, E30 _{= 0.155292 MPa, G}0

12 = 59728 MPa, and G013 =G023 = 0.07765 MPa. These elasticity

moduli are specified using *ELASTIC, TYPE=ENGINEERING CONSTANTS for the bolt hole part of the mesh.

The mesh for the three-dimensional analysis without superelements, shown in Figure 1.1.1-3, represents a 22.5° segment of the pipe joint and employs second-order brick elements with reduced integration, C3D20R, for the pipe hub/flange and bolts. The gasket is modeled with C3D20R elements or GK3D18 elements. The top figure shows the mesh of the pipe hub and flange, and the bottom figure shows both the gasket and bolt (in the lighter color). Contact is modeled by the interaction of contact surfaces defined by grouping specific 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 superelements to model the flange are tested. A first-level superelement is created for the entire 22.5° segment of the flange shown in Figure 1.1.1-3, while the gasket and the bolt are meshed as before. The nodes on the flange in contact with the bolt cap form a node-based surface, while the nodes on the flange in contact with the gasket form another node-based surface. A2 = AC 2 ; A4 = AD 2 ; A6 = AE 2 ; A8 = AF 2 ; A3 = (AC +AD)=4; A5 = (AD +AE)=4; A7 = (AE +AF)=4;

where A1 through A9 are contact areas that are associated with contact nodes 1-9 and Ac through AF

are sectional areas that are associated with bolt head elements C-F . 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 flange 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 flat plates are found in ASME Section VIII Div 2 Article 4-9. For the type of structure under study, which is not a flat 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 AA = 6534.51 mm2_{, and the total area of the bolt}

holes is given by AH = 8¼82 _{= 1608.5 mm}2_{. Hence, the reduction factor is simply 1 ¡ AH=AA =}

0.754. The effective in-plane moduli of elasticity, E10 _{and E2}0_{, are obtained by multiplying the}

respective moduli, E1 and E2, by this factor. We assume material isotropy in the r-z plane; thus, E10 =E20 =E0: The modulus in the hoop direction, E30, should be very small and is chosen such that E0_=E30 _{= 10}6_{. The in-plane shear modulus is then calculated based on the effective elasticity}

modulus: G0

12 =E0=2(1 + º): The shear moduli in the hoop direction are also calculated similarly but

with º set to zero (they are not used in an axisymmetric model). Hence, we have E10 _=E20 _{= 155292}

MPa, E30 _{= 0.155292 MPa, G}0

12 = 59728 MPa, and G013 =G023 = 0.07765 MPa. These elasticity

moduli are specified using *ELASTIC, TYPE=ENGINEERING CONSTANTS for the bolt hole part of the mesh.

The mesh for the three-dimensional analysis without superelements, shown in Figure 1.1.1-3, represents a 22.5° segment of the pipe joint and employs second-order brick elements with reduced integration, C3D20R, for the pipe hub/flange and bolts. The gasket is modeled with C3D20R elements or GK3D18 elements. The top figure shows the mesh of the pipe hub and flange, and the bottom figure shows both the gasket and bolt (in the lighter color). Contact is modeled by the interaction of contact surfaces defined by grouping specific 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 superelements to model the flange are tested. A first-level superelement is created for the entire 22.5° segment of the flange shown in Figure 1.1.1-3, while the gasket and the bolt are meshed as before. The nodes on the flange in contact with the bolt cap form a node-based surface, while the nodes on the flange in contact with the gasket form another node-based surface.