ENGINEERING MECHANICS

Engineering MechanicsVolume 2: Stresses, Strains, Displacements

by

C. HARTSUIJKERDelft University of Technology, Delft, The Netherlandsand

J.W. WELLEMANDelft University of Technology, Delft, The Netherlands

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-1-4020-4123-5 (HB)ISBN 978-1-4020-5763-2 (e-book)

Published by Springer,P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

www.springer.com

This is a translation of the original Dutch work Toegepaste Mechanica, Deel 2: Spanningen, Vervormingen, Verplaatsingen, 2001,Academic Service, The Hague, The Netherlands.

Printed on acid-free paper

All Rights Reserved 2007 SpringerNo part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming,recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered andexecuted on a computer system, for exclusive use by the purchaser of the work.

Table of Contents

Preface ix

Foreword xiii

1 Material Behaviour 11.1 Tensile test 11.2 Stress-strain diagrams 51.3 Hookes Law 11

2 Bar Subject to Extension 152.1 The fibre model 162.2 The three basic relationships 182.3 Strain diagram and normal stress diagram 242.4 Normal centre and bar axis 262.5 Mathematical description of the extension problem 302.6 Examples relating to changes in length and displacements 342.7 Examples relating to the differential equation for extension 452.8 Formal approach and engineering practice 522.9 Problems 54

3 Cross-Sectional Properties 713.1 First moments of area; centroid and normal centre 743.2 Second moments of area 913.3 Thin-walled cross-sections 1213.4 Formal approach and engineering practice 1323.5 Problems 135

4 Members Subject to Bending and Extension 1514.1 The fibre model 1534.2 Strain diagram and neutral axis 1554.3 The three basic relationships 1574.4 Stress formula and stress diagram 1684.5 Examples relating to the stress formula for bending with

extension 1714.6 Section modulus 1844.7 Examples of the stress formula related to bending without

extension 1864.8 General stress formula related to the principal directions 1984.9 Core of the cross-section 203

vi ENGINEERING MECHANICS. VOLUME 2: STRESSES, DEFORMATIONS, DISPLACEMENTS4.10 Applications related to the core of the cross-section 2084.11 Mathematical description of the problem of bending with

extension 2194.12 Thermal effects 2234.13 Notes for the fibre model and summary of the formulas 2284.14 Problems 234

5 Shear Forces and Shear Stresses Due to Bending 2715.1 Shear forces and shear stresses in longitudinal direction 2725.2 Examples relating to shear forces and shear stresses in

the longitudinal direction 2825.3 Cross-sectional shear stresses 3005.4 Examples relating to the shear stress distribution in a

cross-section 3105.5 Shear centre 3675.6 Other cases of shear 3775.7 Summary of the formulas and rules 3825.8 Problems 385

6 Bar Subject to Torsion 4116.1 Material behaviour in shear 4126.2 Torsion of bars with circular cross-section 4156.3 Torsion of thin-walled cross-sections 4266.4 Numerical examples 4456.5 Summary of the formulas 4686.6 Problems 471

7 Deformation of Trusses 4837.1 The behaviour of a single truss member 4847.2 Williot diagram 487

7.3 Williot diagram with rigid-body rotation 5047.4 WilliotMohr diagram 5147.5 Problems 521

8 Deformation Due to Bending 5418.1 Direct determination from the moment distribution 5438.2 Differential equation for bending 5578.3 Forget-me-nots 5768.4 Moment-area theorems 5988.5 Simply supported beams and the M/EI diagram 6338.6 Problems 648

9 Unsymmetrical and Inhomogeneous Cross-Sections 6799.1 Sketch of the problems and required assumptions 6799.2 Kinematic relationships 6829.3 Curvature and neutral axis 6869.4 Normal force and bending moments centre of force 6909.5 Constitutive relationships for unsymmetrical and/or

inhomogeneous cross-sections 6959.6 Plane of loading and plane of curvature neutral axis 7019.7 The normal centre NC for inhomogeneous cross-sections 7069.8 Stresses due to extension and bending a straightforward

method 7149.9 Applications of the straightforward method 7159.10

method 7349.11 Transformation formulae for the bending stiffness tensor 7369.12 Application of the alternative method based on the

principal directions 7529.13 Displacements due to bending 761

Stresses in the principal coordinate system alternative

Table of Contents vii9.14 Maxwells reciprocal theorem 7739.15 Core of a cross-section 7779.16 Thermal effects 7919.17 Shear flow and shear stresses in arbitrary cross-sections

shear centre 8099.18 Problems 845

Index 865

Preface

This Volume is the second of a series of two:

Volume 1: Equilibrium Volume 2: Stresses, deformations and displacements

These volumes introduce the fundamentals of structural and continuummechanics in a comprehensive and consistent way. All theoretical devel-opments are presented in text and by means of an extensive set of figures.Numerous examples support the theory and make the link to engineeringpractice. Combined with the problems in each chapter, students are givenample opportunities to exercise.

The book consists of distinct modules, each divided into sections which areconveniently sized to be used as lectures. Both formal and intuitive (engi-neering) arguments are used in parallel to derive the important principles.The necessary mathematics is kept to a minimum however in some partsbasic knowledge of solving differential equations is required.

The modular content of the book shows a clear order of topics concerningstresses and deformations in structures subject to bending and extension.Chapter 1 deals with the fundamentals of material behaviour and the intro-

duction of basic material and deformation quantities. In Chapter 2 the fibremodel is introduced to describe the behaviour of line elements subject to ex-tension (tensile or compressive axial forces). A formal approach is followedin which the three basic relationships (the kinematic, constitutive and staticrelationships) are used to describe the displacement field with a secondorder differential equation. Numerous examples show the influence of theboundary conditions and loading conditions on the solution of the displace-ment field. In Chapter 3 the cross-sectional quantities such as centre ofmass or centre of gravity, centroid, normal (force) centre, first moments ofarea or static moments, and second moments of area or moments of inertiaare introduced as well as the polar moment of inertia. The influence of thetranslation of the coordinate system on these quantities is also investigated,resulting in the parallel axis theorem or Steiners rule for the static momentsand moments of inertia. With the definitions of Chapters 1 to 3 the completetheory for bending and extension is combined in Chapter 4 which describesthe fibre model subject to extension and bending (EulerBernoulli theory).The same framework is used as in Chapter 2 by defining the kinematic,constitutive and static relationships, in order to obtain the set of differentialequations to describe the combined behaviour of extension and bending. By

x ENGINEERING MECHANICS. VOLUME 2: STRESSES, DEFORMATIONS, DISPLACEMENTS

choosing a specific location of the coordinate system through the normal(force) centre, we introduce the uncoupled description of extension andbending. The strain and stress distribution in a cross-section are introducedand engineering expressions are resolved for cross-sections with at leastone axis of symmetry. In this chapter also some special topics are coveredlike the core of a cross-section, and the influence of temperature effects.For non-constant bending moment distributions, beams have to transfershear forces which will lead to shear stresses in longitudinal and transversalsection planes. Based on the equilibrium conditions only, expressions forthe shear flow and the shear stresses will be derived. Field of applicationsare (glued or dowelled) interfaces between different materials in a compos-ite cross-section and the stresses in welds. Special attention is also given tothin-walled sections and the definition of the shear (force) centre for thin-walled sections. This chapter focuses on homogeneous cross-sections withat least one axis of symmetry. Shear deformation is not considered.Chapter 6 deals with torsion, which is treated according to the same conceptas in the previous chapters; linear elasticity is assumed. The elementarytheory is used on thin-walled tubular sections. Apart from the deformationsalso shear stress distributions are obtained. Special cases like solid circularsections and open thin-walled sections are also treated.Structural behaviour due to extension and or bending is treated in Chap-ters 7 and 8. Based on the elementary behaviour described in Chapters 2 and4 the structural behaviour of trusses is treated in Chapter 7 and of beams inChapter 8. The deformation of trusses is treated both in a formal (analytical)way and in a practical (graphical) way with aid of a relative displacementgraph or so-called Williot diagram. The deflection theory for beams iselaborated in Chapter 8 by solving the differential equations and the in-troduction of (practical) engineering methods to obtain the displacementsand deformations based on the moment distribution. With these engineering

formulae, forget-me-nots and moment-area theorems, numerous examplesare treated. Some special cases like temperature effects are also treated inthis chapter.

Chapter 9 shows a comprehensive description of the fibre model on un-symmetrical and or inhomogeneous cross-sections. Much of the earlierpresented derivations are now covered by a complete description usinga two letter symbol approach. This formal approach is quite unique andoffers a fast and clear method to obtain the strain and stress distributionin arbitrary cross-sections by using an initially given coordinate systemwith its origin located at the normal centre of the cross-section. Although acomplete description in the principal coordinate system is also presented, itwill become clear that a description in the initial coordinate system is to bepreferred. Centres of force and core are also treated in this comprehensivetheory, as well as the full description of the shear flow in an arbitrary cross-section. The last part of this chapter shows the application of this theoryon numerous examples of both inhomogeneous and unsymmetrical cross-sections. Special attention is also given to thin-walled sections as well asthe shear (force) centre of unsymmetrical thin-walled sections which is ofparticular interest in steel structures design.

This latter chapter is not necessarily regarded as part of a first introductioninto stresses and deformations but would be more suitable for a second orthird course in Engineering Mechanics. However, since this chapter offersthe complete and comprehensive description of the theory, it is an essentialpart of this volume.

We do realise, however, that finding the right balance between abstractfundamentals and practical applications is the prerogative of the lecturer.He or she should therefore decide on the focus and selection of the topicstreated in this volume to suit the goals of the course in question.

Preface xiThe authors want to thank especially the reviewer Professor Graham M.L.Gladwell from the University of Waterloo (Canada) for his tedious job toimprove the Dutch-English styled manuscript into readable English. Wealso thank Jolanda Karada for her excellent job in putting it all togetherand our publisher Nathalie Jacobs who showed enormous enthusiasm andpatience to see this series of books completed and to have them publishedby Springer.

Coenraad HartsuijkerHans WellemanDelft, The NetherlandsJuly 2007

Foreword

Structural or Engineering Mechanics is one of the core courses for newstudents in engineering studies. At Delft University of Technology a jointeducational program for Statics and Strength of Materials has been devel-oped by the Koiter Institute, and has subsequently been incorporated inthe curricula of faculties like Civil Engineering, Aeronautical Engineering,Architectural Engineering, Mechanical Engineering, Maritime Engineeringand Industrial Design.

In order for foreign students also to be able to benefit from this pro-gram an English version of the Dutch textbook series written by CoenraadHartsuijker, which were already used in most faculties, appeared to be nec-essary. It is fortunate that in good cooperation between the writers, Springerand the Koiter Institute Delft, an English version of two text books couldbe realized, and it is believed that this series of books will greatly help thestudent to find his or her way into Engineering or Structural Mechanics.

Indeed, the volumes of this series offer some advantages not foundelsewhere, at least not to this extent. Both formal and intuitive approachesare used, which is more important than ever. The books are modular and canalso be used for self-study. Therefore, they can be used in a flexible manner

and will fit almost any educational system. And finally, the SI system isused consistently. For these reasons it is believed that the books form avery valuable addition to the literature.

Ren de BorstScientific Director, Koiter Institute Delft