Mechanics of Materials – Formulasand Problems

Dietmar Gross • Wolfgang EhlersPeter Wriggers • Jörg SchröderRalf Müller

Mechanics ofMaterials – Formulasand ProblemsEngineering Mechanics 2

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Dietmar GrossDivision of Solid MechanicsTU DarmstadtDarmstadtGermany

Wolfgang EhlersInstitute of Applied MechanicsUniversität StuttgartStuttgartGermany

Peter WriggersInstitute of Continuum MechanicsLeibniz Universität HannoverHannoverGermany

Jörg SchröderInstitute of MechanicsUniversität Duisburg-EssenEssenGermany

Ralf MüllerInstitute of Applied MechanicsTU KaiserslauternKaiserslauternGermany

ISBN 978-3-662-53879-1 ISBN 978-3-662-53880-7 (eBook)DOI 10.1007/978-3-662-53880-7

Library of Congress Control Number: 2016956827

© Springer-Verlag GmbH Germany 2017This work is subject to copyright. All rights are reserved by the Publisher, whetherthe whole or part of the material is concerned, specifically the rights of translation,reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilmsor in any other physical way, and transmission or information storage and retrieval,electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks,etc. in this publication does not imply, even in the absence of a specific statement,that such names are exempt from the relevant protective laws and regulations andtherefore free for general use.The publisher, the authors and the editors are safe to assume that the advice andinformation in this book are believed to be true and accurate at the date ofpublication. Neither the publisher nor the authors or the editors give a warranty,express or implied, with respect to the material contained herein or for any errors oromissions that may have been made.

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Preface

This collection of problems results from the demand of students for sup-plementary problems and support in the preparation for examinations.With the present collection ’Engineering Mechanics 2 - Formulas andProblems, Mechanics of Materials’ we provide more additional exercisematerial.

The subject ’Mechanics of Materials’ is commonly taught in the se-cond course of Engineering Mechanics classes at universities. The pro-blems analyzed within these courses use equilibrium conditions and ki-nematic relations in conjunction with constitutive relations. As we wantconcentrate more on basic concepts and solution procedures the focuslies on linear elastic material behavior and the small strain regime. Ho-wever, this covers a wide range of elasto-static problems with relevancyin engineering applications. Special attention is given to structural ele-ments like bars, beams and shafts as well as plane stress and strainsituations.

Following the warning in the first collection, we would like to makethe reader aware that pure reading and trying to comprehend the pre-sented solutions will not provide a deeper understanding of mechanics.Neither does it improve the problem solving skills. Using this collec-tion wisely, one has to try to solve the problems independently. Theproposed solution should only be considered when experiencing majorproblems in solving an exercise.

Obviously this collection cannot substitute a full-scale textbook. Ifnot familiar with the formulae, explanations, or technical terms the rea-der has to consider his or her course material or additional textbookson mechanics of materials. An incomplete list is provided on page IX.

Darmstadt, Hannover, Stuttgart, Essen and D. GrossKaiserslautern, Summer 2016 P. Wriggers

W. EhlersJ. SchroderR. Muller

Table of Contents

Literature, Notation .................................................. IX

1 Stress, Stain, Hooke’s Law .......................................... 1

2 Tension and Compression in Bars ................................. 29

3 Bending of Beams ..................................................... 57

4 Torsion ................................................................... 111

5 Energy Methods ....................................................... 143

6 Buckling of Bars ....................................................... 181

7 Hydrostatics ............................................................ 195

Literature

Textbooks

Gross, D., Hauger, W., Schroder, J., Wall, W., Bonet, J., EngineeringMechanics 2: Mechanics of Materials, 1st edition, Springer 2011

Gross, D., Hauger, W., Wriggers, P., Technische Mechanik, vol 4:Hydromechanics, Elements of Advanced Mechanics, NumericalMethods (in German), 9th edition, Springer 2014

Beer, F.P., Johnston, E.R., DeWolf, J.T., Mazurek, D.F., Mechanics ofMaterials, 7th edition, McGraw-Hill Education 2012

Hibbeler, R.C., Mechanics of Materials, 10th edition, Pearson 2016

Geer, J.M., Goodno, B.J., Mechanics of Materials, 8th edition, CengageLearning 2013

Ghavami, P., Mechanics of Materials, 1st edition, Springer 2015

Collection of Problems

Schaum’s Outlines Strength of Materials, 6th edition, McGraw-HillEducation 2013

Beer, F.P., Johnston, E.R., DeWolf, J.T., Mazurek, D.F., Mechanics ofMaterials, 7th edition, McGraw-Hill Education 2012

Hibbeler, R.C., Mechanics of Materials, 10th edition, Pearson 2016

Notation

The following symbols are used in the solutions to the problems:

↑ : short notation for sum of all forces in the direction of the

arrow equals zero.�

A : short notation for sum of all moments with reference to

point A equals zero.

� short notation for it follows.