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Structural Mechanics MODULE V ERIFICATION M ANUAL COMSOL Multiphysics Structural Mechanics MODULE V ERIFICATION M ANUAL V ERSION 3.5 How to contact COMSOL: Germany United Kingdom COMSOL Multiphysics GmbH COMSOL Ltd. Benelux Berliner Str. 4 UH Innovation Centre COMSOL BV D-37073 Göttingen College Lane Röntgenlaan 19 Phone: +49-551-99721-0 Hatfield 2719 DX Zoetermeer Fax: +49-551-99721-29 Hertfordshire AL10 9AB The Netherlands [email protected] Phone:+44-(0)-1707 636020 Phone: +31 (0) 79 363 4230 www.comsol.de Fax: +44-(0)-1707 284746 Fax: +31 (0) 79 361 4212 [email protected] [email protected] Italy www.uk.comsol.com www.comsol.nl COMSOL S.r.l. Via Vittorio Emanuele II, 22 United States Denmark 25122 Brescia COMSOL, Inc. COMSOL A/S Phone: +39-030-3793800 1 New England Executive Park Diplomvej 376 Fax: +39-030-3793899 Suite 350 2800 Kgs. 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Robert Schuman Fax: +46 8 412 95 10 [email protected] F-38000 Grenoble [email protected] www.comsol.com Phone: +33 (0)4 76 46 49 01 www.comsol.se Fax: +33 (0)4 76 46 07 42 For a complete list of international [email protected] Switzerland representatives, visit www.comsol.fr FEMLAB GmbH www.comsol.com/contact Technoparkstrasse 1 CH-8005 Zürich Company home page Phone: +41 (0)44 445 2140 www.comsol.com Fax: +41 (0)44 445 2141 [email protected] COMSOL user forums www.femlab.ch www.comsol.com/support/forums Structural Mechanics Module Verification Manual © COPYRIGHT 1994–2008 by COMSOL AB. All rights reserved Patent pending The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or reproduced in any form without prior written consent from COMSOL AB. COMSOL, COMSOL Multiphysics, COMSOL Script, COMSOL Reaction Engineering Lab, and FEMLAB are registered trademarks of COMSOL AB. Other product or brand names are trademarks or registered trademarks of their respective holders. Version: September 2008 COMSOL 3.5 Part number: CM021104 CONTENTS Chapter 1: Introduction Benchmark Model Guide . 2 Comparison With Theoretical and Benchmark Results . 4 COMSOL Software Verification and Quality Assurance Programs . 4 Typographical Conventions . 4 Chapter 2: Benchmark Models Wrapped Thick Cylinder Under Pressure and Thermal Loading 8 Introduction . 8 Model Definition . 8 Modeling in COMSOL Multiphysics . 9 Results and Discussion. 11 References . 12 Modeling Using the Graphical User Interface . 12 Large Deformation Beam 19 Model Definition . 19 Results and Discussion. 20 Reference . 22 Modeling Using the Graphical User Interface . 23 Thick Plate Stress Analysis 27 Introduction . 27 Model Definition . 27 Results. 28 Reference . 29 Modeling Using the Graphical User Interface . 29 Kirsch Infinite Plate Problem 37 Introduction . 37 Model Definition . 37 CONTENTS | i Results. 38 Reference . 39 Modeling Using the Graphical User Interface . 39 Thick Wall Cylinder Benchmark Problem 47 Model Definition . 47 Results. 48 Reference . 48 Modeling Using the Graphical User Interface . 49 In-Plane Framework with Discrete Mass and Mass Moment of Inertia 55 Introduction . 55 Model Definition . 55 Results and Discussion. 56 3D Thermally Loaded Beam 64 Introduction . 64 Model Definition . 64 Results and Discussion. 65 Reference . 67 Modeling Using the Graphical User Interface . 67 In-Plane Truss 73 Introduction . 73 Model Definition . 73 Results and Discussion. 74 Reference . 75 Modeling Using the Graphical User Interface . 75 Scordelis-Lo Roof Shell Benchmark 85 Model Definition . 85 Results and Discussion. 87 Reference . 89 Modeling Using the Graphical User Interface . 89 Cylinder Roller Contact 98 Introduction . 98 ii | CONTENTS Model Definition . 98 Results and Discussion. 99 References . 101 Modeling Using COMSOL Multiphysics . 101 Modeling Using the Graphical User Interface . 101 Eigenfrequency Analysis of a Free Cylinder 110 Introduction . 110 Model Definition . 110 Results. 112 Reference . 112 Modeling Using the Graphical User Interface . 113 Frequency Response Analysis of a Simply Supported Plate 118 Model Definition . 118 Results. 119 Reference . 120 Modeling Using the Graphical User Interface . 120 Single Edge Crack 129 Introduction . 129 Model Definition . 130 Modeling in COMSOL Multiphysics . 131 Results. 132 Reference . 133 Modeling Using the Graphical User Interface . 133 Elasto-Plastic Plate 139 Introduction . 139 Model Definition . 139 Results and Discussion. 140 Reference . 142 Modeling Using the Graphical User Interface . 142 Elastoacoustic Effect in Rail Steel 145 Introduction . 145 Model Definition . 145 Results and Discussion. 147 CONTENTS | iii Reference . 149 Modeling Using the Graphical User Interface . 150 INDEX 155 iv | CONTENTS 1 Introduction This Structural Mechanics Module Verification Manual consists of a set of benchmark models from various areas of structural mechanics and solid mechanics engineering. These are models with a theoretical solution or an solution from an established benchmark. Their purpose is to show the close agreement between the numerical solution obtained in COMSOL Multiphysics and the established benchmark data, so that you can gain confidence in the solutions provided when using the Structural Mechanics Module. The models illustrate the use of the various structural-mechanics specific application modes and analysis types. We have tried to cover a wide spectrum of the capabilities in the Structural Mechanics Module. Note that the model descriptions in this book do not contain details on how to carry out every step in the modeling process. Before tackling these models, we urge you to first read the Structural Mechanics Module User’s Guide. This book introduces you to the functionality in the module, reviews new features in the version 3.5 release, and covers basic modeling techniques with tutorials and example models. Another book, the Structural Mechanics Module Model Library, contain a large number of examples models from important application areas such as automotive applications, dynamics and vibration, fluid-structure interaction, fatigue analysis, and piezoelectric applications. Finally, the Structural Mechanics 1 Module Reference Guide contains reference material about command-line programming and functions. It is available in HTML and PDF format from the COMSOL Help Desk. For more information on how to work with the COMSOL Multiphysics graphical user interface, please refer to the COMSOL Multiphysics User’s Guide or the COMSOL Multiphysics Quick Start and Quick Reference manual. An explanation on how to model with a programming language is available in yet another book, the COMSOL Multiphysics Scripting Guide. The book in your hands, the Structural Mechanics Module Verification Manual, provides details about a large number of ready-to-run models that provide numerical solutions to benchmark problems and textbook examples with theoretical closed-form solutions. Each entry comes with theoretical background, a discussion about the results with a comparison to the benchmark data or the analytical solution, as well as instructions that illustrate how to set it up. The documentation for all models contains references to the textbook or technical publication from which we have collected the benchmark data or other verification data. Finally note that we supply these models as COMSOL Multiphysics model files so you can open them in COMSOL Multiphysics for immediate access, allowing you to follow along with these examples every step along the way. Benchmark Model Guide The table below summarizes key information about the available benchmark models. The solution time is the elapsed time measured on a machine running Windows Vista with a 2.6 GHz AMD Athlon X2 Dual Core 500 CPU and 2 GB of RAM. For models with a sequential solution strategy, the Solution Time column shows the total elapsed time for the combined solution steps. The Application Mode column contains the application modes (such as Plane Stress) we used to solve the model. The following 2 | CHAPTER 1: INTRODUCTION columns indicate the analysis type (such as eigenfrequency). The rightmost column contains the key features in the model. MODEL PAGE SOLUTION APPLICATION MODE KEY FEATURES TIME STATIC EIGENFREQUENCY FREQUENCY RESPONSE PARAMETRIC BENCHMARK MODELS Wrapped cylinder 8 2 s Solid, Stress-Strain √ Composite-materials analysis; user-defined coordinate systems Large deformation beam 19 10 s Plane Stress √ √ Large deformation analysis; linear buckling Thick plate 27 4 s Solid, Stress-Strain √ 3D solid model, use of symmetries Kirsch plate 37 1 s Plane Stress √ Use of symmetries Thick wall cylinder 47 1 s Plane Strain √ Multiphysics In-plane frame 55 1 s In-Plane Euler Beam √ Beam elements Thermally loaded beam 64 1 s 3D Euler Beam √ Multiphysics; beam elements; thermal expansion In-plane truss 73 1 s In-Plane
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