Table Of ContentDietmar Gross · Werner Hauger
Jörg Schröder · Wolfgang A. Wall
Javier Bonet
Engineering
Mechanics 2
itnee 2Bxdtenibdt sEioton-onsgkel linlsiohnw g
Mechanics of Materials
Second Edition
123
Engineering Mechanics 2
Dietmar Gross · Werner Hauger
Jörg Schröder · Wolfgang A. Wall
Javier Bonet
Engineering Mechanics 2
Mechanics of Materials
2nd Edition
Dietmar Gross Wolfgang A. Wall
Solid Mechanics Computational Mechanics
TU Darmstadt TU München
Darmstadt Garching
Germany Germany
Werner Hauger Javier Bonet
Continuum Mechanics University of Greenwich
TU Darmstadt London
Darmstadt UK
Germany
Jörg Schröder
Institute of Mechanics
Universität Duisburg-Essen
Essen
Germany
ISBN 978-3-662-56271-0 ISBN 978-3-662-56272-7 (eBook)
https://doi.org/ 10.1007/978-3-662-56272-7
Library of Congress Control Number: 2018933018
1st edition: © Springer-Verlag Berlin Heidelberg 2011
2nd edition: © Springer-Verlag GmbH Germany, part of Springer Nature 2018
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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
ofthisseriesisthebestsellingtextbookonEngineeringMechanics
in German speaking countries; its 13th edition is currently being
published.
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
In the second edition, the text has been thoroughly revised
and a number of additions were made. In particular, the number
of supplementary examples has been increased. We would like to
thank all readers who contributed to the improvements through
their feedback.
We gratefully acknowledge the support and the cooperation
of the staff of the Springer Verlag who were complaisant to our
wishes and helped to create the present layout of the book.
Darmstadt, Essen, Munich and Greenwich, D. Gross
December 2017 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......................................................... 47
2 Stress
2.1 Stress Vector and Stress Tensor ............................. 51
2.2 Plane Stress...................................................... 54
2.2.1 Coordinate Transformation.................................... 55
2.2.2 Principal Stresses............................................... 58
2.2.3 Mohr’s Circle .................................................... 64
2.2.4 The Thin-Walled Pressure Vessel............................ 70
2.3 Equilibrium Conditions......................................... 72
2.4 SupplementaryExamples...................................... 75
2.5 Summary......................................................... 78
3 Strain,Hooke’sLaw
3.1 State of Strain................................................... 81
3.2 Hooke’s Law..................................................... 86
3.3 Strength Hypotheses........................................... 92
3.4 SupplementaryExamples...................................... 94
3.5 Summary......................................................... 98
4 BendingofBeams
4.1 Introduction...................................................... 101
4.2 Second Moments of Area..................................... 103
4.2.1 Definitions........................................................ 103
4.2.2 Parallel-Axis Theorem.......................................... 110
VIII
4.2.3 Rotation of the Coordinate System, Principal Moments
of Inertia.......................................................... 115
4.3 Basic Equations of OrdinaryBending Theory ............ 119
4.4 Normal Stresses................................................. 123
4.5 Deflection Curve................................................ 127
4.5.1 Differential Equation of the Deflection Curve............. 127
4.5.2 Beams with one Region of Integration...................... 131
4.5.3 Beams with several Regions of Integration ................ 140
4.5.4 Method of Superposition...................................... 142
4.6 Influence of Shear............................................... 153
4.6.1 Shear Stresses................................................... 153
4.6.2 Deflection due to Shear........................................ 163
4.7 Unsymmetric Bending.......................................... 164
4.8 Bending and Tension/Compression.......................... 173
4.9 Core of the Cross Section..................................... 176
4.10 ThermalBending ............................................... 178
4.11 SupplementaryExamples...................................... 182
4.12 Summary......................................................... 190
5 Torsion
5.1 Introduction...................................................... 193
5.2 Circular Shaft.................................................... 194
5.3 Thin-Walled Tubes with Closed Cross Sections........... 205
5.4 Thin-Walled Shafts with Open Cross Sections............ 214
5.5 SupplementaryExamples...................................... 222
5.6 Summary......................................................... 230
6 EnergyMethods
6.1 Introduction...................................................... 233
6.2 Strain Energy and Conservation of Energy................. 234
6.3 Principle of Virtual Forces and Unit Load Method....... 244
6.4 Influence Coefficients and Reciprocal
Displacement Theorem........................................ 263
6.5 Statically Indeterminate Systems............................ 267
6.6 SupplementaryExamples...................................... 281
6.7 Summary......................................................... 288
IX
7 BucklingofBars
7.1 Bifurcation of an Equilibrium State......................... 291
7.2 Critical Loads of Bars, Euler’s Column..................... 294
7.3 SupplementaryExamples...................................... 304
7.4 Summary......................................................... 308
Index........................................................................ 309
Introduction
Volume1(Statics)showedhowexternalandinternalforcesacting
on structures can be determined with the aid of the equilibrium
conditions alone. In doing so, real physical bodies were appro-
ximated by rigid bodies. However, this idealisation is often not
adequatetodescribethebehaviourofstructuralelementsorwho-
lestructures.Inmanyengineeringproblemsthedeformationsalso
have to be calculated, for example in order to avoid inadmissibly
large deflections. The bodies must then be considered as being
deformable.
It is necessary to define suitable geometrical quantities to de-
scribe the deformations. These quantities are the displacements
andthe strains.The geometryofdeformationisgivenby kinema-
ticequations; they connect the displacements and the strains.
Inadditiontothedeformations,thestressingofstructuralmem-
bers is of great practical importance. In Volume 1 we calculated
the internal forces (the stress resultants). The stress resultants
alone, however, allow no statement regarding the load carrying
ability of a structure: a slender rod or a stocky rod, respectively,
madeofthesamematerialwillfailunderdifferentloads.Therefo-
re, the concept of the stateofstress is introduced. The amount of
loadthatastructurecanwithstandcanbeassessedbycomparing
the calculated stress with an allowable stress which is based on
experiments and safety requirements.
The stressesandstrains areconnectedin the constitutiveequa-
tions. These equationsdescribe the behaviourofthe materialand
can be obtained only from experiments. The most important me-
tallic or non-metallic materials exhibit a linear relationship bet-
ween the stress and the strain provided that the stress is small
enough. Robert Hooke (1635–1703) first formulated this fact in
the language of science at that time: uttensiosicvis (lat., asthe
extension,sotheforce). A material that obeys Hooke’slaw is cal-
led linearelastic; we will simply refer to it as elastic.
Inthepresenttextwewillrestrictourselvestothestaticsofela-
stic structures. We will always assume that the deformations and
thusthestrainsareverysmall.Thisassumptionissatisfiedinma-
