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ENGINEERING MECHANICS Engineering Mechanics Volume 2: Stresses, Strains, Displacements by C. HARTSUIJKER Delft University of Technology, Delft, The Netherlands and J.W. WELLEMAN Delft 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 Springer No 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 and executed on a computer system, for exclusive use by the purchaser of the work. Table of Contents Preface Foreword ix xiii 1 1.1 1.2 1.3 Material Behaviour Tensile test Stress-strain diagrams Hooke’s Law 1 1 5 11 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 Bar Subject to Extension The fibre model The three basic relationships Strain diagram and normal stress diagram Normal centre and bar axis Mathematical description of the extension problem Examples relating to changes in length and displacements Examples relating to the differential equation for extension Formal approach and engineering practice Problems 15 16 18 24 26 30 34 45 52 54 3 3.1 3.2 3.3 3.4 3.5 Cross-Sectional Properties First moments of area; centroid and normal centre Second moments of area Thin-walled cross-sections Formal approach and engineering practice Problems 71 74 91 121 132 135 4 4.1 4.2 4.3 4.4 4.5 Members Subject to Bending and Extension The fibre model Strain diagram and neutral axis The three basic relationships Stress formula and stress diagram Examples relating to the stress formula for bending with extension Section modulus Examples of the stress formula related to bending without extension General stress formula related to the principal directions Core of the cross-section 151 153 155 157 168 4.6 4.7 4.8 4.9 171 184 186 198 203 vi ENGINEERING MECHANICS. VOLUME 2: STRESSES, DEFORMATIONS, DISPLACEMENTS 4.10 Applications related to the core of the cross-section 4.11 Mathematical description of the problem of bending with extension 4.12 Thermal effects 4.13 Notes for the fibre model and summary of the formulas 4.14 Problems 208 219 223 228 234 271 272 5.5 5.6 5.7 5.8 Shear Forces and Shear Stresses Due to Bending Shear forces and shear stresses in longitudinal direction Examples relating to shear forces and shear stresses in the longitudinal direction Cross-sectional shear stresses Examples relating to the shear stress distribution in a cross-section Shear centre Other cases of shear Summary of the formulas and rules Problems 6 6.1 6.2 6.3 6.4 6.5 6.6 Bar Subject to Torsion Material behaviour in shear Torsion of bars with circular cross-section Torsion of thin-walled cross-sections Numerical examples Summary of the formulas Problems 411 412 415 426 445 468 471 5 5.1 5.2 5.3 5.4 7 7.1 7.2 Deformation of Trusses The behaviour of a single truss member Williot diagram 282 300 310 367 377 382 385 483 484 487 7.3 7.4 7.5 Williot diagram with rigid-body rotation Williot–Mohr diagram Problems 504 514 521 8 8.1 8.2 8.3 8.4 8.5 8.6 Deformation Due to Bending Direct determination from the moment distribution Differential equation for bending Forget-me-nots Moment-area theorems Simply supported beams and the M/EI diagram Problems 541 543 557 576 598 633 648 9 9.1 9.2 9.3 9.4 9.5 Unsymmetrical and Inhomogeneous Cross-Sections Sketch of the problems and required assumptions Kinematic relationships Curvature and neutral axis Normal force and bending moments – centre of force Constitutive relationships for unsymmetrical and/or inhomogeneous cross-sections Plane of loading and plane of curvature – neutral axis The normal centre NC for inhomogeneous cross-sections Stresses due to extension and bending – a straightforward method Applications of the straightforward method Stresses in the principal coordinate system – alternative method Transformation formulae for the bending stiffness tensor Application of the alternative method based on the principal directions Displacements due to bending 679 679 682 686 690 9.6 9.7 9.8 9.9 9.10 9.11 9.12 9.13 695 701 706 714 715 734 736 752 761 Table of Contents Maxwell’s reciprocal theorem Core of a cross-section Thermal effects Shear flow and shear stresses in arbitrary cross-sections – shear centre 9.18 Problems 809 845 Index 865 9.14 9.15 9.16 9.17 773 777 791 vii 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 continuum mechanics in a comprehensive and consistent way. All theoretical developments are presented in text and by means of an extensive set of figures. Numerous examples support the theory and make the link to engineering practice. Combined with the problems in each chapter, students are given ample opportunities to exercise. The book consists of distinct modules, each divided into sections which are conveniently sized to be used as lectures. Both formal and intuitive (engineering) arguments are used in parallel to derive the important principles. The necessary mathematics is kept to a minimum however in some parts basic knowledge of solving differential equations is required. The modular content of the book shows a clear order of topics concerning stresses 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 fibre model is introduced to describe the behaviour of line elements subject to extension (tensile or compressive axial forces). A formal approach is followed in which the three basic relationships (the kinematic, constitutive and static relationships) are used to describe the displacement field with a second order differential equation. Numerous examples show the influence of the boundary conditions and loading conditions on the solution of the displacement field. In Chapter 3 the cross-sectional quantities such as centre of mass or centre of gravity, centroid, normal (force) centre, first moments of area or static moments, and second moments of area or moments of inertia are introduced as well as the polar moment of inertia. The influence of the translation of the coordinate system on these quantities is also investigated, resulting in the parallel axis theorem or Steiner’s rule for the static moments and moments of inertia. With the definitions of Chapters 1 to 3 the complete theory for bending and extension is combined in Chapter 4 which describes the fibre model subject to extension and bending (Euler–Bernoulli 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 differential equations 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 and bending. The strain and stress distribution in a cross-section are introduced and engineering expressions are resolved for cross-sections with at least one axis of symmetry. In this chapter also some special topics are covered like the core of a cross-section, and the influence of temperature effects. For non-constant bending moment distributions, beams have to transfer shear forces which will lead to shear stresses in longitudinal and transversal section planes. Based on the equilibrium conditions only, expressions for the shear flow and the shear stresses will be derived. Field of applications are (glued or dowelled) interfaces between different materials in a composite cross-section and the stresses in welds. Special attention is also given to thin-walled sections and the definition of the shear (force) centre for thinwalled sections. This chapter focuses on homogeneous cross-sections with at least one axis of symmetry. Shear deformation is not considered. Chapter 6 deals with torsion, which is treated according to the same concept as in the previous chapters; linear elasticity is assumed. The elementary theory is used on thin-walled tubular sections. Apart from the deformations also shear stress distributions are obtained. Special cases like solid circular sections and open thin-walled sections are also treated. Structural behaviour due to extension and or bending is treated in Chapters 7 and 8. Based on the elementary behaviour described in Chapters 2 and 4 the structural behaviour of trusses is treated in Chapter 7 and of beams in Chapter 8. The deformation of trusses is treated both in a formal (analytical) way and in a practical (graphical) way with aid of a relative displacement graph or so-called Williot diagram. The deflection theory for beams is elaborated in Chapter 8 by solving the differential equations and the introduction of (practical) engineering methods to obtain the displacements and deformations based on the moment distribution. With these engineering formulae, forget-me-nots and moment-area theorems, numerous examples are treated. Some special cases like temperature effects are also treated in this chapter. Chapter 9 shows a comprehensive description of the fibre model on unsymmetrical and or inhomogeneous cross-sections. Much of the earlier presented derivations are now covered by a complete description using a two letter symbol approach. This formal approach is quite unique and offers a fast and clear method to obtain the strain and stress distribution in arbitrary cross-sections by using an initially given coordinate system with its origin located at the normal centre of the cross-section. Although a complete description in the principal coordinate system is also presented, it will become clear that a description in the initial coordinate system is to be preferred. Centres of force and core are also treated in this comprehensive theory, as well as the full description of the shear flow in an arbitrary crosssection. The last part of this chapter shows the application of this theory on numerous examples of both inhomogeneous and unsymmetrical crosssections. Special attention is also given to thin-walled sections as well as the shear (force) centre of unsymmetrical thin-walled sections which is of particular interest in steel structures design. This latter chapter is not necessarily regarded as part of a first introduction into stresses and deformations but would be more suitable for a second or third course in Engineering Mechanics. However, since this chapter offers the complete and comprehensive description of the theory, it is an essential part of this volume. We do realise, however, that finding the right balance between abstract fundamentals and practical applications is the prerogative of the lecturer. He or she should therefore decide on the focus and selection of the topics treated in this volume to suit the goals of the course in question. Preface The authors want to thank especially the reviewer Professor Graham M.L. Gladwell from the University of Waterloo (Canada) for his tedious job to improve the Dutch-English styled manuscript into readable English. We also thank Jolanda Karada for her excellent job in putting it all together and our publisher Nathalie Jacobs who showed enormous enthusiasm and patience to see this series of books completed and to have them published by Springer. Coenraad Hartsuijker Hans Welleman Delft, The Netherlands July 2007 xi Foreword Structural or Engineering Mechanics is one of the core courses for new students in engineering studies. At Delft University of Technology a joint educational program for Statics and Strength of Materials has been developed by the Koiter Institute, and has subsequently been incorporated in the curricula of faculties like Civil Engineering, Aeronautical Engineering, Architectural Engineering, Mechanical Engineering, Maritime Engineering and Industrial Design. In order for foreign students also to be able to benefit from this program an English version of the Dutch textbook series written by Coenraad Hartsuijker, which were already used in most faculties, appeared to be necessary. It is fortunate that in good cooperation between the writers, Springer and the Koiter Institute Delft, an English version of two text books could be realized, and it is believed that this series of books will greatly help the student to find his or her way into Engineering or Structural Mechanics. Indeed, the volumes of this series offer some advantages not found elsewhere, at least not to this extent. Both formal and intuitive approaches are used, which is more important than ever. The books are modular and can also 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 is used consistently. For these reasons it is believed that the books form a very valuable addition to the literature. René de Borst Scientific Director, Koiter Institute Delft
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