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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 1.2 1.3 Material Behaviour Tensile test Stress-strain diagrams Hooke’s Law 1 11 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 3.1 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 4.1 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 6.1 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 5.1 5.2 5.3 5.4 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 8.1 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 9.1 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 deals with the fundamentals of material behaviour and the intro- duction of basic material and deformation quantities In Chapter 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 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 to the complete theory for bending and extension is combined in Chapter which describes the fibre model subject to extension and bending (Euler–Bernoulli theory) The same framework is used as in Chapter 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 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 and Based on the elementary behaviour described in Chapters and the structural behaviour of trusses is treated in Chapter and of beams in Chapter 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 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 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 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 Unsymmetrical and Inhomogeneous Cross-Sections The coordinates ey and ez of the centre of force (the eccentricity of the normal force N) follow from1 ey = My Mz and ez = N N For the bending moments in the xy and xz plane we now can write My = N · ey and Mz = N · ez Substitute these expressions in (9.42), and we find Ney y N Nez z + + EA EIyy EIzz EA EA N 1+ = ey y + ez z EA EIyy EIzz ε(y, z) = (9.71) The radius of inertia r is used to relate the second moment of area I to the cross-sectional area A, according to2 I = A · r For inhomogeneous cross-sections, we use the radius of inertia to relate the bending stiffness EI to the axial stiffness EA: ry2 = EIyy EIzz and rz2 = EA EA See Section 9.4 See also Section 3.2.1 (9.72) 775 776 ENGINEERING MECHANICS VOLUME 2: STRESSES, DEFORMATIONS, DISPLACEMENTS Since y and z are principal coordinate axes, ry and rz are the principal radii of inertia They have the dimension of a length Comment: Although the notation of the radii of inertia ry and rz suggest that they are components of a vector, this is not the case Upon rotation of the coordinate system they not transform like the components of a vector Using (9.7.2) we can further simplify expression (9.71) for the strain distribution: ey y N ez z (9.73) 1+ + ε(y, z) = EA ry rz This expression shows the strain ε at point (y, z) for a normal force N with its point of application at (ey , ez ) As an experiment of mind we can think of a force N acting at (y, z) and observing the strain ε at (ey , ez ) It appears to result in exactly the same strain This is due to the equivalence of ey and y in the strain formula, and of ez and z We can summarise this phenomenon as follows The strain ε at P due to a force N at Q is equal to the strain ε at Q due to a force N at P (see Figure 9.79) Figure 9.79 Maxwell’s reciprocal theorem: the strain ε at P due to a force N at Q is equal to the strain ε at Q due to a force N at P This is also known as Maxwell’s reciprocal theorem and is general applicable to linear elastic systems for which the superposition theorem holds We will make use of this theorem in the next section on the core of a cross-section Unsymmetrical and Inhomogeneous Cross-Sections 9.15 Core of a cross-section When the neutral axis intersects the cross-section, both tensile and compressive zones will occur on either side of the neutral axis Some materials can hardly sustain tensile stresses, e.g brick walls and unreinforced concrete For these materials, the cross-section should be loaded in such a way that only compression occurs The neutral axis should then be outside the cross-section or just at its boundary With this requirement we can determine the area in which the centre of force should be positioned in order to prevent sign changes in the stress distribution This area is called the core or kern of the cross-section In other words: the core of a cross-section is the set of centres of force for which the neutral axis is outside the cross-section In Section 4.9 the core was introduced for a rectangular cross-section with dimensions b × h, as shown in Figure 9.80 The core appeared to be a diamond with the corner points on the y and z axis with a distance to the NC of b/6 and h/6 respectively After discussing some properties of the core in Section 9.15.1, we will in Section 9.15.2 outline a general method to find the core of (in)homogeneous and unsymmetrical cross-sections Some examples are given in Section 9.15.3 9.15.1 Properties of the core In the following we will make use of two important properties: • For a neutral axis tangent to the cross-section, the associated centre of force is located on the edge of the core • Cross-sections for which all valid boundary positions of the neutral axis form a polygon, also have a polygon as core The number of b b /3 h y h z Figure 9.80 The core of a homogeneous rectangular cross-section 777 778 ENGINEERING MECHANICS VOLUME 2: STRESSES, DEFORMATIONS, DISPLACEMENTS corners of the core is equal to the number of valid boundary positions of the neutral axis A centre of force within the core corresponds with a stress distribution in the cross-section that does not exhibit a change in sign So the total crosssection is in tension or in compression, which implies that the neutral axis is outside the cross-section Figure 9.81 There are six valid boundary positions of the neutral axis: AB, AH, HF, FE, ED and DB For each of the six valid boundary positions of the neutral axis, the associated centre of force is a corner of the core For the cross-section in Figure 9.81 there are six valid boundary positions of the neutral axis: AB, AH, HF, FE, ED and DB Note that a neutral axis which coincides with boundary BC cannot be valid since the neutral axis then intersects the cross-section For each of the six valid boundary positions of the neutral axis, the associated centre of force is a corner of the core, and is called core point The second property states that when all valid boundary positions of the neutral axis form a polygon, the core has straight edges and is also a polygon We will explain this for the simple rectangular cross-section in Figure 9.82 From the two boundary lines 1-1 and 2-2 the associated centres of force (core points) are denoted as and The quest now is to determine the boundary of the core between these points Therefore we use Maxwell’s reciprocal theorem: the strain ε at P due to a force N at Q is equal to the strain ε at Q due to a force N at P (see Figure 9.79) Figure 9.82 Each neutral axis passing through B and not intersecting the cross-section, relates to a centre of force on the straight line between the points and The valid boundary positions 1-1 and 2-2 of the neutral axis correspond with the centres of force and respectively; they are corners of the core, and are called core points When the centre of force is chosen at core point 1, then the neutral axis is the line 1-1 along the upper edge of the cross-section So there is a zero strain at B In reverse, according to Maxwell’s reciprocal theorem: when the centre of force is chosen at B, there will be zero strain at core point When the centre of force is chosen at core point 2, then the neutral axis is the line 2-2 along the left edge of the cross-section Again there is a zero strain at B In reverse, according to Maxwell’s reciprocal theorem: when the centre of force is chosen at B, there will be zero strain at core point 779 Unsymmetrical and Inhomogeneous Cross-Sections Therefore the neutral axis associated with the centre of force at B is a (straight) line which passes through the core points and Conclusion: • For a force at B there is a zero strain at all points on the straight line through the core points and Maxwell’s reciprocal theorem implies the following: • For all centres of force on the straight line between the core points and there is a zero strain at B Each neutral axis passing through B and not intersecting the cross-section, relates to a centre of force on the straight line between the points and (see Figure 9.82) The valid boundary positions 1-1 and 2-2 of the neutral axis correspond with the centres of force and 2; they are corners of the core and are called core points This proves that the boundary of the core for cross-sections with straight edges is built up by straight lines Figure 9.83 shows two cross-sections for which not all edges are straight In Figure 9.83a the four valid boundary positions of the neutral axis form a polygon, so the core of the cross-section is also a polygon, with four sides Figure 9.83b shows a cross-section for which edge AB is not straight The core points (centres of force) and are associated with the neutral axes 2-2 through A and 3-3 through B respectively Any neutral axis tangent to the curved edge AB is associated with a centre of force on the boundary of the core between the points and This results in a curved boundary of the core between these points The determination of this part of the core is quite laborious 4 A 1 3 4 2 1 3 (a ) B (b ) Figure 9.83 Cross-sections with curved edges (a) The four valid boundary positions of the neutral axis form a polygon, so the core of the cross-section is also a polygon, with four sides (b) Any neutral axis tangent to the curved edge AB is associated with a centre of force on the boundary of the core This results in a curved boundary of the core between the points and 3 780 ENGINEERING MECHANICS VOLUME 2: STRESSES, DEFORMATIONS, DISPLACEMENTS General method to find the core 9.15.2 The neutral axis in a cross-section is defined by ε(y, z) = ε + κy · y + κz · z = When the yz coordinate system is chosen in such a way that its origin coincides with the normal centre NC, then extension and bending can be treated separately The strain ε at the origin of the coordinate system is caused by extension and the curvatures κy and κz are caused by bending The cross-sectional constitutive relationships are N = EAε My Mz = EIyy EIyz EIzy EIzz κy κz (extension), (9.26) (bending) (9.40) For a cross-section with non-zero normal force (and therefore ε = 0), the equation for the neutral axis can be written as 1+ Figure 9.84 A neutral axis na which is bounding the cross-section This neutral axis intersects the y and z coordinate axes in the points (y1 , 0) and (0, z1 ) respectively κy κz y+ z = ε ε (9.71) Consider a neutral axis na which is bounding the cross-section (see Figure 9.84) Assume this neutral axis intersects the y and z coordinate axes in the points (y1 , 0) and (0, z1 ) respectively Using (9.71) we can relate these points of intersection to the three cross-sectional deformation quantities: y1 = − ε ε and z1 = − κy κz (9.72) Unsymmetrical and Inhomogeneous Cross-Sections Between the components My and Mz of the bending moment and the components ey and ez of the eccentricity of the normal force N, there is the following relationship:1 My Mz =N ey ez (9.18) Substitute in (9.18) the constitutive relationship for extension: My Mz = EAε ey ez (9.73) Substitute (9.73) in the constitutive relationship for bending, EAε ey ez = EIyy EIyz EIzy EIzz