Springer-Textbook Prof. Dietmar Gross received his Engineering Diploma in Applied Mechanics and his Doctor of Engineering degree at the University of Rostock. He was Research Associate at the University of Stuttgart and since 1976 he is Professor of Mechanics at the University of Darmstadt. His research interests are mainly focused on modern solid mechanics on the macro and mi- cro scale, including advanced materials. Prof. Werner Hauger studied Applied Mathematics and Mechanics at the Univer- sity of Karlsruhe and received his Ph.D. in Th eoretical and Applied Mechanics from Northwestern University in Evan- ston. He worked in industry for several years, was a Pro- fessor at the Helmut-Schmidt-University in Hamburg and went to the University of Darmstadt in 1978. His research interests are, among others, theory of stability, dynamic plas- ticity and biomechanics. Prof. Jörg Schröder studied Civil Engineering, received his doctoral degree at the University of Hannover and habilitated at the University of Stuttgart. He was Professor of Mechanics at the University of Darmstadt and went to the University of Duisburg-Essen in 2001. His fi elds of research are theoretical and computer- oriented continuum mechanics, modeling of functional materials as well as the further development of the fi nite element method. Prof. Wolfgang A. Wall studied Civil Engineering at Innsbruck University and re- ceived his doctoral degree from the University of Stuttgart. Since 2003 he is Professor of Mechanics at the TU München and Head of the Institute for Computational Mechanics. His research interests cover broad fi elds in computational mechanics, including both solid and fl uid mechanics. His recent focus is on multiphysics and multiscale problems as well as computational biomechanics. Prof. Javier Bonet studied Civil Engineering at the Universitat Politecnica de Catalunya in Barcelona and received his Doctorate from Swansea University in the UK. He is Professor of Compu- tational Mechanics and Head of the School of Engineer- ing at Swansea University where he has taught Strength of Materials, Structural Mechanics and Nonlinear Mechanics for over 20 years. His research interests are computational mechanics and fi nite element methods. Dietmar Gross · Werner Hauger Jörg Schröder · Wolfgang A. Wall Javier Bonet Engineering Mechanics 2 Mechanics of Materials 1 3 Prof. Dr. Dietmar Gross Prof. Dr. Werner Hauger TU Darmstadt TU Darmstadt Division of Solid Mechanics Hochschulstr. 1 Hochschulstr. 1 64289 Darmstadt 64289 Darmstadt Germany Germany [email protected] [email protected] Prof. Dr. Wolfgang A.Wall Prof. Dr. Jörg Schröder TU München Universität Duisburg-Essen Institute for Computational Institute of Mechanics Mechanics Universitätsstr. 15 Boltzmannstr. 15 45141 Essen 85747 Garching Germany Germany [email protected] [email protected] Prof. Javier Bonet Head of School School of Engineering Swansea University Swansea, SA2 8PP United Kingdom [email protected] ISBN 978-3-642-12885-1 e-ISBN 978-3-642-12886-8 DOI 10.1007/978-3-642-12886-8 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011922991 © Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifi cally the rights of translation, reprinting, reuse of illustra- tions, recitation, broadcasting, reproduction on microfi lm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publica- tion does not imply, even in the absence of a specifi c statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) Preface Mechanics ofMaterials is the second volume of a three-volume textbook on Engineering Mechanics. Volume 1 deals with Statics while Volume 3 contains Dynamics. The original German version of this series has been the bestselling textbook on mechanics for more than two decades; its 11th edition is currently being publis- hed. It is our intention to present to engineering students the basic concepts and principles of mechanics in the clearest and simp- lest form possible. A major objective of this book is to help the studentstodevelopproblemsolvingskillsinasystematicmanner. The book has been developed from the many years of teaching experience gained by the authors while giving courses on engi- neering mechanics to students of mechanical, civil and electrical engineering. The contents of the book correspond to the topics normally covered in courses on basic engineering mechanics, also known in some countries as strength of materials, at universities and colleges. The theory is presented in as simple a form as the subject allows without becoming imprecise.This approachmakes the text accessible to students from different disciplines and al- lows for their different educational backgrounds. Another aim of thebookistoprovidestudentsaswellaspractisingengineerswith a solid foundation to help them bridge the gaps between under- graduatestudiesandadvancedcoursesonmechanicsandpractical engineering problems. A thorough understanding of the theory cannot be acquired by merely studying textbooks. The application of the seemingly simple theory to actual engineering problems can be mastered only if the student takes an active part in solving the numerous examples in this book.It is recommended that the reader tries to solve the problems independently without resorting to the given solutions.Inordertofocusonthefundamentalaspectsofhowthe theoryisapplied,wedeliberatelyplacednoemphasisonnumerical solutions and numerical results. VI We gratefully acknowledge the support and the cooperation of the staffoftheSpringerVerlagwhowereresponsivetoourwishes and helped to create the present layout of the books. Darmstadt, Essen, Munich and Swansea, D. Gross December 2010 W. Hauger J. Schr¨oder W.A. Wall J. Bonet Table of Contents Introduction............................................................... 1 1 TensionandCompressioninBars 1.1 Stress.............................................................. 7 1.2 Strain.............................................................. 13 1.3 Constitutive Law................................................ 14 1.4 Single Bar under Tension or Compression.................. 18 1.5 Statically Determinate Systems of Bars.................... 29 1.6 Statically Indeterminate Systems of Bars.................. 33 1.7 SupplementaryExamples...................................... 40 1.8 Summary......................................................... 46 2 Stress 2.1 Stress Vector and Stress Tensor ............................. 49 2.2 Plane Stress...................................................... 52 2.2.1 Coordinate Transformation.................................... 53 2.2.2 Principal Stresses............................................... 56 2.2.3 Mohr’s Circle .................................................... 62 2.2.4 The Thin-Walled Pressure Vessel............................ 68 2.3 Equilibrium Conditions......................................... 70 2.4 SupplementaryExamples...................................... 73 2.5 Summary......................................................... 75 3 Strain,Hooke’sLaw 3.1 State of Strain................................................... 79 3.2 Hooke’s Law..................................................... 84 3.3 Strength Hypotheses........................................... 90 3.4 SupplementaryExamples...................................... 92 3.5 Summary......................................................... 95 4 BendingofBeams 4.1 Introduction...................................................... 99 4.2 Second Moments of Area..................................... 101 4.2.1 Definitions........................................................ 101 4.2.2 Parallel-Axis Theorem.......................................... 108 VIII 4.2.3 Rotation of the Coordinate System, Principal Moments of Inertia.......................................................... 113 4.3 Basic Equations of OrdinaryBending Theory ............ 117 4.4 Normal Stresses................................................. 121 4.5 Deflection Curve................................................ 125 4.5.1 Differential Equation of the Deflection Curve............. 125 4.5.2 Beams with one Region of Integration...................... 129 4.5.3 Beams with several Regions of Integration ................ 138 4.5.4 Method of Superposition...................................... 140 4.6 Influence of Shear............................................... 151 4.6.1 Shear Stresses................................................... 151 4.6.2 Deflection due to Shear........................................ 161 4.7 Unsymmetric Bending.......................................... 162 4.8 Bending and Tension/Compression.......................... 171 4.9 Core of the Cross Section..................................... 174 4.10 ThermalBending ............................................... 176 4.11 SupplementaryExamples...................................... 180 4.12 Summary......................................................... 187 5 Torsion 5.1 Introduction...................................................... 191 5.2 Circular Shaft.................................................... 192 5.3 Thin-Walled Tubes with Closed Cross Sections........... 203 5.4 Thin-Walled Shafts with Open Cross Sections............ 212 5.5 SupplementaryExamples...................................... 220 5.6 Summary......................................................... 228 6 EnergyMethods 6.1 Introduction...................................................... 231 6.2 Strain Energy and Conservation of Energy................. 232 6.3 Principle of Virtual Forces and Unit Load Method....... 242 6.4 Influence Coefficients and Reciprocal Displacement Theorem........................................ 261 6.5 Statically Indeterminate Systems............................ 265 6.6 SupplementaryExamples...................................... 279 6.7 Summary......................................................... 286 IX 7 BucklingofBars 7.1 Bifurcation of an EquilibriumState......................... 289 7.2 Critical Loads of Bars, Euler’s Column..................... 292 7.3 SupplementaryExamples...................................... 302 7.4 Summary......................................................... 305 Index........................................................................ 307