MODELLING OF MECHANICAL SYSTEMS VOLUME MODELLING OF MECHANICAL SYSTEMS VOLUME Structural Elements Franỗois Axisa and Philippe Trompette AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Elsevier Butterworth-Heinemann Linacre House, Jordan Hill, Oxford OX2 8DP 30 Corporate Drive, Burlington, MA 01803 First published in France 2001 by Hermes Science, entitled ‘Modélisation des systèmes mécaniques, systèmes continus, Tome 2’ First published in Great Britain 2005 Copyright © 2005, Elsevier Ltd All rights reserved The right of Franỗois Axisa and Philippe Trompette to be identified as the authors of this Work has been asserted in accordance with the Copyright, Designs And Patents Act 1988 No part of this publication may be reproduced in any material form (including photocopying or storing in any medium by electronic means and whether or not transiently or incidentally to some other use of this publication) without the written permission of the copyright holder except in accordance with the provisions of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London, England W1T 4LP Applications for the copyright holder’s written permission to reproduce any part of this publication should be addressed to the publisher Permissions may be sought directly from Elsevier’s Science and Technology Rights Department in Oxford, UK: phone: (+44) (0) 1865 843830; fax: (+44) (0) 1865 853333; e-mail: permissions@elsevier.co.uk You may also complete your request on-line via the Elsevier homepage (http://www.elsevier.com), by selecting ‘Customer Support’ and then ‘Obtaining Permissions’ British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data A catalogue record for this book is available from the Library of Congress ISBN 7506 6846 For information on all Elsevier Butterworth-Heinemann publications visit our website at http://books.elsevier.com Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed and bound in Great Britain Working together to grow libraries in developing countries www.elsevier.com | www.bookaid.org | www.sabre.org Contents Preface xvii Introduction xix Chapter Solid mechanics 1.1 Introduction 1.2 Equilibrium equations of a continuum 1.2.1 Displacements and strains 1.2.2 Indicial and symbolic notations 1.2.3 Stresses 1.2.4 Equations of dynamical equilibrium 1.2.5 Stress–strain relationships for an isotropic elastic material 1.2.6 Equations of elastic vibrations (Navier’s equations) 1.3 Hamilton’s principle 1.3.1 General presentation of the formalism 1.3.2 Application to a three-dimensional solid 1.3.2.1 Hamilton’s principle 1.3.2.2 Hilbert functional vector space 1.3.2.3 Variation of the kinetic energy 1.3.2.4 Variation of the strain energy 1.3.2.5 Variation of the external load work 1.3.2.6 Equilibrium equations and boundary conditions 1.3.2.7 Stress tensor and Lagrange’s multipliers 1.3.2.8 Variation of the elastic strain energy 1.3.2.9 Equation of elastic vibrations 1.3.2.10 Conservation of mechanical energy 1.3.2.11 Uniqueness of solution of motion equations 1.4 Elastic waves in three-dimensional media 1.4.1 Material oscillations in a continuous medium interpreted as waves 1.4.2 Harmonic solutions of Navier’s equations 3 11 13 16 17 18 19 20 20 20 21 21 23 23 24 25 27 28 29 31 31 32 vi Contents 1.4.3 Dilatation and shear elastic waves 1.4.3.1 Irrotational, or potential motion 1.4.3.2 Equivoluminal, or shear motion 1.4.3.3 Irrotational harmonic waves (dilatation or pressure waves) 1.4.3.4 Shear waves (equivoluminal or rotational waves) 1.4.4 Phase and group velocities 1.4.5 Wave reflection at the boundary of a semi-infinite medium 1.4.5.1 Complex amplitude of harmonic and plane waves at oblique incidence 1.4.5.2 Reflection of (SH) waves on a free boundary 1.4.5.3 Reflection of (P) waves on a free boundary 1.4.6 Guided waves 1.4.6.1 Guided (SH) waves in a plane layer 1.4.6.2 Physical interpretation 1.4.6.3 Waves in an infinite elastic rod of circular cross-section 1.4.7 Standing waves and natural modes of vibration 1.4.7.1 Dilatation plane modes of vibration 1.4.7.2 Dilatation modes of vibration in three dimensions 1.4.7.3 Shear plane modes of vibration 1.5 From solids to structural elements 1.5.1 Saint-Venant’s principle 1.5.2 Shape criterion to reduce the dimension of a problem 1.5.2.1 Compression of a solid body shaped as a slender parallelepiped 1.5.2.2 Shearing of a slender parallelepiped 1.5.2.3 Validity of the simplification for a dynamic loading 1.5.2.4 Structural elements in engineering Chapter Straight beam models: Newtonian approach 2.1 Simplified representation of a 3D continuous medium by an equivalent 1D model 2.1.1 Beam geometry 2.1.2 Global and local displacements 2.1.3 Local and global strains 2.1.4 Local and global stresses 2.1.5 Elastic stresses 2.1.6 Equilibrium in terms of generalized stresses 2.1.6.1 Equilibrium of forces 2.1.6.2 Equilibrium of the moments 2.2 Small elastic motion 2.2.1 Longitudinal mode of deformation 2.2.1.1 Local equilibrium 32 33 33 33 38 38 40 41 43 44 48 48 51 53 53 54 55 58 59 59 61 61 62 63 64 66 67 67 67 70 72 74 75 75 77 78 78 78 Contents 2.2.1.2 General solution of the static equilibrium without external loading 2.2.1.3 Elastic boundary conditions 2.2.1.4 Concentrated loads 2.2.1.5 Intermediate supports 2.2.2 Shear mode of deformation 2.2.2.1 Local equilibrium 2.2.2.2 General solution without external loading 2.2.2.3 Elastic boundary conditions 2.2.2.4 Concentrated loads 2.2.2.5 Intermediate supports 2.2.3 Torsion mode of deformation 2.2.3.1 Torsion without warping 2.2.3.2 Local equilibrium 2.2.3.3 General solution without loading 2.2.3.4 Elastic boundary conditions 2.2.3.5 Concentrated loads 2.2.3.6 Intermediate supports 2.2.3.7 Torsion with warping: Saint Venant’s theory 2.2.4 Pure bending mode of deformation 2.2.4.1 Simplifying hypotheses of the Bernoulli–Euler model 2.2.4.2 Local equilibrium 2.2.4.3 Elastic boundary conditions 2.2.4.4 Intermediate supports 2.2.4.5 Concentrated loads 2.2.4.6 General solution of the static and homogeneous equation 2.2.4.7 Application to some problems of practical interest 2.2.5 Formulation of the boundary conditions 2.2.5.1 Elastic impedances 2.2.5.2 Generalized mechanical impedances 2.2.5.3 Homogeneous and inhomogeneous conditions 2.2.6 More about transverse shear stresses and straight beam models 2.2.6.1 Asymmetrical cross-sections and shear (or twist) centre 2.2.6.2 Slenderness ratio and lateral deflection 2.3 Thermoelastic behaviour of a straight beam 2.3.1 3D law of thermal expansion 2.3.2 Thermoelastic axial response 2.3.3 Thermoelastic bending of a beam 2.4 Elastic-plastic beam 2.4.1 Elastic-plastic behaviour under uniform traction 2.4.2 Elastic-plastic behaviour under bending 2.4.2.1 Skin stress vii 79 79 82 84 86 86 88 88 88 89 89 89 89 90 90 90 90 91 99 99 100 102 103 103 104 104 114 114 116 116 116 117 118 118 118 119 121 123 124 124 125 viii Contents 2.4.2.2 Moment-curvature law and failure load 126 2.4.2.3 Elastic-plastic bending: global constitutive law 127 2.4.2.4 Superposition of several modes of deformation 128 Chapter Straight beam models: Hamilton’s principle 3.1 Introduction 3.2 Variational formulation of the straight beam equations 3.2.1 Longitudinal motion 3.2.1.1 Model neglecting the Poisson effect 3.2.1.2 Model including the Poisson effect (Love–Rayleigh model) 3.2.2 Bending and transverse shear motion 3.2.2.1 Bending without shear: Bernoulli–Euler model 3.2.2.2 Bending including transverse shear: the Timoshenko model in statics 3.2.2.3 The Rayleigh–Timoshenko dynamic model 3.2.3 Bending of a beam prestressed by an axial force 3.2.3.1 Strain energy and Lagrangian 3.2.3.2 Vibration equation and boundary conditions 3.2.3.3 Static response to a transverse force and buckling instability 3.2.3.4 Follower loads 3.3 Weighted integral formulations 3.3.1 Introduction 3.3.2 Weighted equations of motion 3.3.3 Concentrated loads expressed in terms of distributions 3.3.3.1 External loads 3.3.3.2 Intermediate supports 3.3.3.3 A comment on the use of distributions in mechanics 3.3.4 Adjoint and self-adjoint operators 3.3.5 Generic properties of conservative operators 3.4 Finite element discretization 3.4.1 Introduction 3.4.2 Beam in traction-compression 3.4.2.1 Mesh 3.4.2.2 Shape functions 3.4.2.3 Element mass and stiffness matrices 3.4.2.4 Equivalent nodal external loading 3.4.2.5 Assembling the finite element model 3.4.2.6 Boundary conditions 3.4.2.7 Elastic supports and penalty method 130 131 132 132 132 133 135 135 136 139 141 142 143 145 148 149 149 151 151 152 155 156 156 162 163 163 167 168 169 169 171 171 172 173 Contents ix 3.4.3 Assembling non-coaxial beams 3.4.3.1 The stiffness and mass matrices of a beam element for bending 3.4.3.2 Stiffness matrix combining bending and axial modes of deformation 3.4.3.3 Assembling the finite element model of the whole structure 3.4.3.4 Transverse load resisted by string and bending stresses in a roof truss 3.4.4 Saving DOF when modelling deformable solids 174 Chapter Vibration modes of straight beams and modal analysis methods 4.1 Introduction 4.2 Natural modes of vibration of straight beams 4.2.1 Travelling waves of simplified models 4.2.1.1 Longitudinal waves 4.2.1.2 Flexure waves 4.2.2 Standing waves, or natural modes of vibration 4.2.2.1 Longitudinal modes 4.2.2.2 Torsion modes 4.2.2.3 Flexure (or bending) modes 4.2.2.4 Bending coupled with shear modes 4.2.3 Rayleigh’s quotient 4.2.3.1 Bending of a beam with an attached concentrated mass 4.2.3.2 Beam on elastic foundation 4.2.4 Finite element approximation 4.2.4.1 Longitudinal modes 4.2.4.2 Bending modes 4.2.5 Bending modes of an axially preloaded beam 4.2.5.1 Natural modes of vibration 4.2.5.2 Static buckling analysis 4.3 Modal projection methods 4.3.1 Equations of motion projected onto a modal basis 4.3.2 Deterministic excitations 4.3.2.1 Separable space and time excitation 4.3.2.2 Non-separable space and time excitation 4.3.3 Truncation of the modal basis 4.3.3.1 Criterion based on the mode shapes 4.3.3.2 Spectral criterion 4.3.4 Stresses and convergence rate of modal series 4.4 Substructuring method 4.4.1 Additional stiffnesses 4.4.1.1 Beam in traction-compression with an end spring 174 177 177 180 186 188 189 190 190 190 193 196 196 200 200 205 207 207 209 210 210 211 213 213 214 217 218 220 220 221 222 222 224 229 231 231 232 Solid mechanics In Section 1.3, Hamilton’s principle of least action is used to extend Lagrange’s formalism to deformable bodies This analytical approach will be used abundantly (but not exclusively) in the subsequent chapters to model the basic structural components of common use in mechanical engineering as equivalent 2D and 1D continuous media In Section 1.4, a few notions needed to analyse the propagation of material waves in elastic solids are presented Here, interest is focused on wave reflection at the boundaries and on standing waves The latter are identified with the natural modes of vibration of the elastic solid, provided with conservative (elastic or inertial) boundary conditions Modal frequencies and wavelengths provide suitable scaling factors to validate the simplifying assumptions adopted in structural modelling to analyse dynamical problems Finally, in Section 1.5, Saint-Venant’s principle, which allows one to distinguish between local and global effects in the response of solids, is discussed in the context of the simplifying assumptions which allow one to model a 3D solid as an equivalent 2D or 1D solid 1.2 Equilibrium equations of a continuum 1.2.1 Displacements and strains When loaded, the solid body is deformed, but, in most cases of practical interest, very slightly in comparison with the deformations experienced by fluids So, in a solid, material points which are initially very close together remain close together during deformation and the Lagrangian description is well adapted to formulate the equations of mechanical equilibrium The motion is described by a displacement vector field which is referenced to the initial (non-deformed) configuration (Figure 1.1) If the body is deformed during motion, the distance between two material points is changed So, the deformation rate has to be related in some suitable Figure 1.1 Lagrangian displacement and strain fields of two closely spaced points Structural elements manner to the relative change of length of an infinitesimal = segment, giving rise to the concept of strain tensor, denoted in symbolic notation ε The tensor nature of = ε arises as a consequence of the fact that the change of length generally depends upon the direction, but not upon the coordinate system See Appendix A.1 for a brief presentation of vector and tensor calculus Let P0 and Q0 be two infinitely neighbouring material points of the initial configuration (time t = 0) Their position is defined in a Cartesian coordinate system of unit vectors i, j , k as: P0 : Q0 : r0 = x0 i + y0 j + z0 k ′ ′ ′ ′ r0 = x0 i + y0 j + z0 k [1.1] At a later time t, P0 and Q0 are mapped into the slightly displaced points P and Q respectively: P : r = x i + y j + zk Q: r ′ = x ′ i + y ′ j + z′ k [1.2] where the coordinates are referred to the initial configuration and described by functions of space and time of the type: x(xo , yo , zo ; t); ′ ′ ′ x ′ (xo , yo , zo ; t) y(xo , yo , zo ; t); ′ ′ ′ y ′ (xo , yo , zo ; t) z(xo , yo , zo ; t); ′ ′ ′ z′ (xo , yo , zo ; t) [1.3] The displacement vectors of P and Q are defined as: − → − → −→ − − X (r0 ; t) = OP − OP0 = P0 P = r − r0 −→ − → − → − − − ′ ′ X r0 ; t = OQ − OQ0 = Q0 Q = r ′ − r0 [1.4] −− −→ Let d r0 denote the infinitesimal vector P0 Q0 of Cartesian components dx0 , dy0 , − → dz0 , and d r the infinitesimal vector P Q, of Cartesian components dx, dy, dz It is easily shown that: −→ − → − − ′ X r0 ; t = OQ − OQ0 = r + d r − (r0 + d r0 ) = X (r0 ; t) + d r − d r0 [1.5] As the coordinates [1.3] are continuous functions of space which can be differentiated at least to the first order, the chain derivation rule can be used to Solid mechanics obtain: ∂x dx0 + ∂x0 ∂y dx0 + dy = ∂x0 ∂z dx0 + dz = ∂x0 dx = ∂x dy0 + ∂y0 ∂y dy0 + ∂y0 ∂z dy0 + ∂y0 ∂x dz0 ∂z0 ∂y dz0 ∂z0 ∂z dz0 ∂z0 [1.6] [1.6] can be written in symbolic notation as: d r = grad r · d r0 [1.7] grad r is a tensor of the second rank called the gradient of the position vector r Its Cartesian components identify with the partial derivatives appearing in [1.6] A form like [1.7] is termed intrinsic as it makes no specific reference to a coordinate system In the same manner, the components of X (r0 ; t) can be differentiated to give: ∂X dx0 + ∂x0 ∂Y dx0 + dY = ∂x0 ∂Z dx0 + dZ = ∂x0 dX = ∂X dy0 + ∂y0 ∂Y dy0 + ∂y0 ∂Z dy0 + ∂y0 ∂X dz0 ∂z0 ∂Y dz0 ∂z0 ∂Z dz0 ∂z0 [1.8] written in intrinsic form as: d X = grad X · d r0 On the other hand, from [1.5] and [1.7] it follows that: = ′ X r0 ; t − X (r0 ; t) = d X = d r − d r0 = grad r− I · d r0 [1.9] = = where I is the identity tensor such that d r0 =I · d r0 Then, from [1.8] and [1.9], it follows that: = grad X = grad r− I [1.10] At this step, the mathematical manipulations which are necessary to proceed in the definition of the strain tensor are carried out more easily by shifting either Structural elements to a matrix or to an indicial notation, rather than by using directly the symbolic vector and tensor notation This is because matrix and indicial notations deal with the scalar components of vectors and tensors, as defined in a specific coordinate system In this way, the rules of algebra with scalars are immediately applicable It is important to get well trained in the symbolic, matrix and indicial notations because all of them are used with equal frequency in the literature on continuum and structural mechanics As we have to deal here with Cartesian tensors of the first rank (vectors) and of the second rank only, the matrix notation is particularly convenient for dealing with the present problem The results will be converted into the two other kinds of notation afterwards Accordingly, the linear system [1.6] is rewritten in matrix form as: ∂x/∂xo dx dy = ∂y/∂xo dz ∂z/∂xo ∂x/∂yo ∂y/∂yo ∂z/∂yo dxo ∂x/∂zo ∂y/∂zo dyo dzo ∂z/∂zo [1.11] that is, in concise form as: [dr] = [J ][dr0 ] [1.12] [dr] and [dr0 ] are the column vectors built with the Cartesian components of d r and d r0 , respectively [J ] is the gradient transformation matrix, also called the Jacobian matrix of transformation, which is built with the Cartesian components of grad r If the material is deformed when passing from the initial to the actual configuration, the length of [dr] differs from that of [dr0 ] Using [1.12], it follows that: dr = [dr]T [dr] = [dr0 ]T [J ]T [J ][dr0 ] = [dr0 ]T [C][dr0 ] where the upper script T [1.13] stands for a matrix transposition This quadratic form is independent of the coordinate system and is used to define the Cauchy kinematic tensor C, written as the symmetric matrix: [C] = [J ]T [J ] [1.14] In the particular case of a rigid body motion, d r = d r0 , hence [C] reduces necessarily to the identity matrix [I ] (diagonal elements equal to one, and nondiagonal elements equal to zero) Furthermore, using [1.10], it is possible to express C in terms of grad X First, [1.10] is rewritten in matrix notation as: [J ] = [I ] + [grad[X]] [1.15] Solid mechanics Then, substituting [1.15] into [1.14], and applying the rules of matrix product, we arrive at: [C] − [I ] = [grad[X]] + [grad[X]]T + [grad[X]][grad[X]]T [1.16] As in a rigid body motion [C] reduces to [I ], it is appropriate to use the righthand side of [1.16] to define the strain tensor It turns out that a suitable definition is the so called Green–Lagrange strain tensor represented by the matrix: [ε] = [grad[X]] + [grad[X]]T + [grad[X]][grad[X]]T ([C] − [I ]) = 2 [1.17] Obviously, [ε] is symmetric Physically, the Green–Lagrange strain tensor provides a means to measure the relative change of length of an infinitesimal segment by the relation: [dr0 ]T [ε][dr0 ] = [dr]T [dr] − [dr0 ]T [dr0 ] [1.18] Though [1.18] is suitably independent from the coordinate system, its pertinence to the measurement of deformations is not obvious at first glance The two following specific cases can be used to understand it better The so called engineering strain, denoted here by εE , measures the relative change of length of the straight segment AB (see Figure 1.2) by: εE = dx − dx0 dx0 Figure 1.2 Extension of a straight segment [1.19] Structural elements It is noticed that the definition [1.19] can also be transformed as follows: (dx − dx0 )(dx + dx0 ) (dx − dx0 ) ⇒ εE = dx0 (dx + dx0 ) dx0 (2 + εE ) dx = (εE + 1) dx0 εE = [1.20] If εE is sufficiently small, as will be always the case in this book, the relative change of length can also be expressed by the Green–Lagrange strain denoted here εG : ε E ≃ εG = dx − dx0 2dx0 = ∂X + ∂x ∂X ∂x [1.21] Motion of a rigid body, for instance a rotation, must induce no strain at all This can be checked by rotating a straight segment of length L through a finite angle θ , see Figure 1.3 The easiest way is to use the Cauchy kinematic tensor The initial configuration is defined by A(x0 , y0 ) and the actual one by A′ (x, y) The coordinates are transformed according to the formula: cos θ x = sin θ y − sin θ cos θ x0 y0 ⇒ cos θ − X = sin θ Y − sin θ cos θ − x0 y0 The Jacobian matrix identifies with the rotation matrix; consequently, [C] = [J ]T [J ] = [I ] and [ε] = [0], as suitable It is also of interest to express the components of the Green–Lagrange tensor directly; which emphasises that the Figure 1.3 Rotation of a straight segment Solid mechanics quadratic terms of ε must be taken into account to obtain the correct result: εxx = ∂X + ∂x0 = cos θ − + εyy = ∂X ∂x0 ∂Y + ∂y0 + ∂Y ∂x0 (cos θ − 1)2 + sin2 θ =0 2 ∂X ∂y0 + ∂Y ∂y0 (cos θ − 1)2 + sin2 θ =0 ∂Y ∂X ∂X ∂Y ∂Y ∂X + + + = ∂y0 ∂x0 ∂y0 ∂x0 ∂y0 ∂x0 = cos θ − + 2εxy = − sin θ + sin θ + − sin θ (cos θ − 1) + sin θ (cos θ − 1) = Nevertheless, if the displacements and the strains are small enough, the nonlinear terms of the Green–Lagrange strain tensor can be omitted, giving rise to the so called infinitesimal, or small strain tensor: [ε] = 1.2.2 [grad[X]] + [grad[X]]T [1.22] Indicial and symbolic notations As detailed in Appendix A.1, the indicial notation is a way to describe vectors and tensors in any orthogonal coordinate system by their generic component For instance, the Cartesian coordinates x, y, z of the position vector r are denoted collectively as xi (i = 1, 2, 3), where the index i takes on the values 1, 2, Those of grad r are denoted ∂xi /∂x0j , where the indices i and j take on the values 1, 2, independently from each other Besides the advantage of conciseness, the indicial notation allows one to deal with scalar variables only, avoiding thus the need to worry about the specific operation rules appropriate for scalar, vector and tensor quantities For instance, using the indicial notation, relation [1.7] is written as: dxi = ∂xi dx0j = Jij dx0j ∂x0j [1.23] where use is made of the convention of implicit summation on the repeated index j 10 Structural elements The index notation for the Green–Lagrange strain tensor [1.17] is: εij = ∂Xj ∂Xi ∂Xk ∂Xk + + ∂x0j ∂x0i ∂x0j ∂x0i [1.24] As an exercise, formula [1.24] can be derived directly, starting from: xi = x0i + Xi ⇒ ∂xi ∂Xi = δij + ∂x0j ∂x0j where δij = if i = j and zero otherwise From [1.14], it follows that Cij = Jki Jkj , further expressed as: Cij = ∂Xk ∂xk ∂xk = δki + ∂x0i ∂x0j ∂x0i δkj + ∂Xk ∂x0j then, Cij = δij + ∂Xk ∂Xk ∂Xk ∂Xk δkj + δki + ∂x0i ∂x0j ∂x0i ∂x0j and finally, in agreement with [1.17]: Cij − δij = 2εij = ∂Xj ∂Xi ∂Xk ∂Xk + + ∂x0i ∂x0j ∂x0i ∂x0j On the other hand, the small strain tensor is written as: εij = ∂Xj ∂Xi + ∂xj ∂xi [1.25] where the subscript (0 ) in the position coordinates is dropped, since no difference is made between the initial and the actual configurations in the case of infinitesimal motions Returning finally to the symbolic notation, the intrinsic expression of the Green– Lagrange strain tensor is written as: grad r + grad r ε= C−I = T + grad r T · grad r [1.26] where the dot product stands for the contracted product, marked in indicial notation by a repeated index Solid mechanics 1.2.3 11 Stresses The concept of stress extends the notion of internal restoring force in discrete systems and that of pressure, familiar in hydrostatics The basic idea is to consider the equilibrium of the solid at the local scale of an infinitesimal element Both the force and moment resultants acting on it must cancel out at static equilibrium When writing down such balances, it is appropriate to distinguish between the body forces, acting on elements of volume of the body, such as the weight or the inertia, and the surface forces exerted by the ‘exterior’ on the boundary (S) of the element, which are proportional to the area of (S) The latter are often termed ‘contact forces’ Thus, if the element is infinitesimal, body forces can be neglected in comparison with contact forces, since they are less by one order of magnitude Let consider a solid body notionally cut into two portions by a plane It can be arbitrarily decided that the cross-sectional surface (S) separating the two portions belongs to one of them, forming thus a facet of that portion, see Figure 1.4 in which (S) is assumed to belong to the portion (I ) Let (dS) be an infinitesimal surface element of the facet The contact force exerted by the portion (II ), through (dS) of area dS, is T = t dS, where t is the stress vector A priori, t depends both on the position and on the orientation of (dS) The latter is defined by the unit normal vector n, conventionally taken as positive when directed outward from the portion (I ), as indicated in Figure 1.4 Let us consider an infinitesimal tetrahedron with three facets parallel to the three Cartesian coordinate planes (unit vectors i, j , k) Static equilibrium implies that the force T exerted by the external medium on the oblique facet of the tetrahedron, defined by the area dS and the unit normal n (coordinates nx , ny , nz ), has to be balanced by all the forces Tx , Ty , Tz which act on the other tetrahedron facets and which are induced by the Figure 1.4 Section of a solid by a facet 12 Structural elements Figure 1.5 Equilibrium of an infinitesimal tetrahedron stress vectors: tx = txx i + txy j + txz k : facet with normal i and algebraic area: − dSnx ty = tyx i + tyy j + tyz k : facet with normal j and algebraic area: − dSny tz = tzx i + tzy j + tzz k : facet with normal k and algebraic area: − dSnz The minus sign arises as a consequence of the outward orientation of the normal vectors, see Figure 1.5 The force balance is thus written as: T + Tx + Ty + Tz = (t − (tx nx + ty ny + tz nz )) dS = It follows that the Cartesian components of t verify the matrix relation: tx txx tyx tzx nx ty = txy tyy tzy ny tz txz tyz tzz nz This matrix defines the Cauchy stress tensor, which is usually written σ (r): nx σxx σyx σzx tx t = σ · n ⇔ ty = σxy σyy σzy ny [1.27] nz σxz σyz σzz tz The tensor nature of σ is evidenced by noting that the virtual work δW = (δd)([n]T [σ ][n]) dS, produced by a virtual displacement (δd)n of the facet, does Solid mechanics 13 Figure 1.6 Stress Cartesian components on the facets of a cubical element not depend on the coordinate system Hence [n]T [σ ][n] is also invariant Using Cartesian coordinates and indicial notation, [1.27] can be written as: tj = σij ni i, j = 1, 2, [1.28] As a definition, σij is the j -th component of the stress force per unit area through the facet of normal unit vector ni The positive Cartesian components of the stress tensor are shown in Figure 1.6 (sketch on the left-hand side) The diagonal terms are called normal stresses, which may be either tensile or compressive, depending whether the sign is positive or negative The other components are termed tangential or shear stresses Of course, all these components are local quantities, which are defined at each position r within the solid and at the boundary So σ (r) is a tensor field The equilibrium of moments leads to the symmetry of σ This very important property can be understood referring to the sketch on the right-hand side of Figure 1.6 The resulting moment of stresses about the origin of the axes must be zero For (1) instance, the component dMx = −(σzy dx dy) dz is balanced by the component (2) dMx = +(σyz dx dz) dy, consequently σyz = σzy , etc Eigenvectors of the symmetric stress matrix are termed principal stress directions, which can be used to define an orthogonal coordinate system, and the eigenvalues are termed principal stresses 1.2.4 Equations of dynamical equilibrium Let us isolate mentally a finite portion of a solid medium, which occupies a volume (V(t)), bounded by a closed surface (S(t)) Such a portion may be viewed as a solid body Let X(r; t) be the Lagrangian displacement field of all the material particles within (V(t)) and ρ(r; t) be the mass per unit volume The portion of material remains in dynamical equilibrium at any time t, when subjected to inertia, 14 Structural elements stress and external forces The latter may comprise body forces f (e) (r; t) and contact forces t (e) (r; t) acting respectively in (V(t)) and on (S(t)); that is f (e) is assumed to vanish on (S(t)) and t (e) within (V(t)) The global force balance over the body is written as: (V (t)) ¨ −ρ X dV + (S (t)) σ · n dS + (V (t)) f (e) (r; t) dV = σ (r) · n (r) − t (r) (r; t) = t (e) (r; t) ∀r ∈ (S(t)) [1.29] [1.30] The contact forces t applied to (S(t)) can stand for external loads t (e) (r; t) and/or for reaction forces t (r) (r; t) induced by some support conditions, or prescribed displacements Equation [1.29] stands for the balance of the internal and external forces acting on the body itself and equation [1.30] for that of the internal and external forces acting on the boundary Further, in [1.29] the surface integral can be suitably transformed into a volume integral by using the divergence theorem (cf Appendix A.2, formula [A.2.5]) The internal terms are then collected on the lefthand side and the external forces on the right-hand side of the equations [1.29] is thus transformed into: (V (t)) ă ( X − div σ ) dV = f (e) dV [1.31] (V (t)) As (V(t)) can be chosen arbitrarily, the ‘global’ equilibrium is equivalent to the ‘local’ equilibrium defined by the two local equations: ă X div = f (e) (r; t); σ (r) · n (r) = t(r; t); ∀r ∈ (V(t)) [1.32] ∀r ∈ (S(t)) The first equation of this system asserts that any point of the volume (V(t)) is in dynamical equilibrium, and the second gives the boundary conditions which must be fulfilled at each point of (S(t)) As a general case, they may include support reactions, external contact forces t (e) and prescribed motions D(r; t), as sketched in Figure 1.7 The latter can be interpreted as a particular type of external contact forces, not given explicitly but expressible in terms of constraint reactions (cf [AXI04], Chapter 4) As a consequence, it is illegal to assign the values of a prescribed motion and of an external contact force at the same position; that is t (e) and D must be specified on two complementary parts of (S), denoted (S1 ) and (S2 ) = (S) ∩ (S1 ) respectively, as already mentioned in the introduction On the other hand, a boundary condition applied to a subdomain (S3 ) is said to be homogeneous if it does not prescribe any external loading (neither Solid mechanics 15 Figure 1.7 Solid with homogeneous and inhomogeneous boundary conditions prescribed forces nor motions); otherwise, the boundary condition is said to be inhomogeneous As the theory is restricted here to conservative systems, we will consider only elastic supports In linear elasticity, they are represented by a homogeneous relation of the type: σ (r) · n(r) − KS [X] = ∀ r ∈ (S3 ) [1.33] KS is a linear stiffness operator defined ∀r ⊂ (S), which can be a scalar (stiffness coefficient of springs), or a differential operator (stiffness operator of another solid used as a supporting device) Furthermore, the relation [1.33] can also describe the condition for a given degree of freedom to be free or, alternatively, to be fixed, by connecting it to a spring and letting the stiffness coefficient tend either to zero, or to infinity It is appropriate to conclude the present subsection by writing down the equations of dynamical equilibrium in the following general form which includes the particular cases discussed just above: ă ρ X − div σ + f (i) = f (e) (r; t); X(r; t) = D(r; t); σ (r) · n(r) − KS [X] = t (e) ∀r ∈ (V(t)) ∀r ⊂ (S1 ) (r; t); ∀r ⊂ (S2 ) [1.34] 16 Structural elements where (S2 ) = (S) ∩ (S1 ) In the system [1.34], an additional body force density f (i) is included, which depends on the problem studied For instance, f (i) can stand for a component such as a gravity force ρ g, where g is the acceleration vector of gravity, or for centrifugal and Coriolis forces On the other hand, if KS does not vanish on (S1 ) the constraint force induced by the prescribed displacement depends on KS Finally, the system of equations [1.34] is said to be mixed, because it is formulated partly in ă terms of kinematical variables X, X, and partly in terms of forces The formulation involves three scalar equations and nine unknowns, namely the six stress and the three displacement components The system is thus underdetermined, except if use can be made of six additional independent relationships, which are given by the material laws This is precisely the case if the material behaves elastically, as detailed in the next subsection 1.2.5 Stress–strain relationships for an isotropic elastic material This study is restricted here to materials in which the stresses depend on strain components σ = B(ε) solely The simplest law of this kind is the generalized Hooke’s law, which defines the so called linear elastic material law It is written both in symbolic and indicial notations as: h : ε = σ ⇐⇒ σij = hij kl εkl i, j , k, l = 1, 2, [1.35] where ε is the small strain tensor and h is the Hooke elasticity tensor The symbol (:) indicates that the product is contracted twice, as evidenced in the index notation by the presence of the repeated indices k and l In a 1D medium, Hooke’s law states that the stress is proportional to the strain, the coefficient of proportionality being the elasticity constant A priori, in a 3D medium a law of the type [1.35] would lead to the definition of 34 = 81 elastic constants for describing the elastic properties of the material However, since ε and σ are both symmetric, h is symmetric with respect to i, j and k, l respectively, which leads to at most × = 36 independent elasticity constants Furthermore, if the material is isotropic, the number of independent elasticity constants reduces to only two, which are defined either as the Lamé parameters, λ and µ (µ is the shear modulus, often denoted G in structural engineering), or as the Young’s modulus E and Poisson ratio ν The relations between these parameters are: λ= νE ; (1 + ν)(1 − 2ν) µ=G= E 2(1 + ν) [1.36] Solid mechanics 17 In Appendix A.6, numerical values are given for a few materials of common use Hooke’s law for an isotropic material can be written as: σ = λ Tr ε I + G ε = (λ div X) I + Gε [1.37] T r ε = T r[ε] = εii is the trace of the matrix [ε]; equal to the divergence of the displacement vector, it measures the relative variation of volume induced by the strains, see Appendix A.2, formula [A.2.3] Inversion of [1.37] results in: ε= 1+ν ν σ − Tr σ I E E [1.38] Relation [1.38] is derived by transforming first [1.37] into: ε= 3λ λ σ − Tr ε I ⇒ Tr ε = Tr σ − Tr ε 2G 2G 2G 2G ⇒ Tr ε = Tr σ 2G + 3λ then, ε= σ λ − Tr σ I 2G 2G(2G + 3λ) To write down the final result [1.38], use is made of the relations [1.36] between λ, G and E, ν 1.2.6 Equations of elastic vibrations (Navier’s equations) In linear elasticity, the equations of dynamical equilibrium are often called vibration equations since they describe small oscillations of the elastic material in the neighbourhood of a permanent and stable state of equilibrium, chosen as the configuration of reference Here, the latter is chosen as a static and stable state of equilibrium, in which the stresses and strains are identically zero These equations are expressed in terms of displacements Using Hooke’s law [1.37], and the small strain tensor [1.26], the equations of dynamical equilibrium [1.34] are ... 4.4 .1. 1 Beam in traction-compression with an end spring 17 4 17 7 17 7 18 0 18 6 18 8 18 9 19 0 19 0 19 0 19 3 19 6 19 6 20 0 20 0 20 5 20 7 20 7 20 9 21 0 21 0 21 1 21 3 21 3 21 4 21 7 21 8 22 0 22 0 22 1 22 2 22 2 22 4 22 9... 90 90 91 99 99 10 0 1 02 10 3 10 3 10 4 10 4 11 4 11 4 11 6 11 6 11 6 11 7 11 8 11 8 11 8 11 9 12 1 12 3 12 4 12 4 12 5 viii Contents 2. 4 .2. 2 Moment-curvature law and failure load 12 6 2. 4 .2. 3 Elastic-plastic... 23 5 23 7 23 8 24 0 24 0 24 3 24 5 24 7 24 8 25 4 25 6 25 9 26 0 26 0 26 0 26 2 26 2 26 2 26 3 26 3 26 3 26 5 26 5 26 5 26 6 26 7 27 0 27 0 27 2 27 3 27 5 27 5 27 7 27 8 27 8 Contents xi 5.3.4.5 In-plane