Springer mechanics and strength of materials

Vitor Dias da Silva Mechanics and Strength of Materials ABC Vitor Dias da Silva Department of Civil Engineering Faculty of Science & Technology University of Coimbra Polo II da Universidade - Pinhal de Marrocos 3030-290 Coimbra Portugal E-mail: vdsilva@dec.uc.pt Library of Congress Control Number: 2005932746 ISBN-10 3-540-25131-6 Springer Berlin Heidelberg New York ISBN-13 978-3-540-25131-6 Springer Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable for prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springeronline.com c Springer-Verlag Berlin Heidelberg 2006 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Typesetting: by the author and TechBooks using a Springer LATEX and TEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN: 10996904 89/TechBooks 543210 Preface To The English Edition The first English edition of this book corresponds to the third Portuguese edition Since the translation has been done by the author, a complete review of the text has been carried out simultaneously As a result, small improvements have been made, especially by explaining the introductory parts of some Chapters and sections in more detail The Portuguese academic environment has distinguished this book, since its first edition, with an excellent level of acceptance In fact, only a small fraction of the copies published has been absorbed by the school for which it was originally designed – the Department of Civil Engineering of the University of Coimbra This fact justifies the continuous effort made by the author to improve and complement its contents, and, indeed, requires it of him Thus, the 423 pages of the first Portuguese edition have now grown to 478 in the present version This increment is due to the inclusion of more solved and proposed exercises and also of additional subjects, such as an introduction to the fatigue failure of materials, an analysis of torsion of circular cross-sections in the elasto-plastic regime, an introduction to the study of the effect of the plastification of deformable elements of a structure on its post-critical behaviour, and a demonstration of the theorem of virtual forces The author would like to thank all the colleagues and students of Engineering who have used the first two Portuguese editions for their comments about the text and for their help in the detection of misprints This has greatly contributed to improving the quality and the precision of the explanations The author also thanks Springer-Verlag for agreeing to publish this book and also for their kind cooperation in the whole publishing process Coimbra March 2005 V Dias da Silva Preface to the First Portuguese Edition The motivation for writing this book came from an awareness of the lack of a treatise, written in European Portuguese, which contains the theoretical material taught in the disciplines of the Mechanics of Solid Materials and the Strength of Materials, and explained with a degree of depth appropriate to Engineering courses in Portuguese universities, with special reference to the University of Coimbra In fact, this book is the result of the theoretical texts and exercises prepared and improved on by the author between 1989-94, for the disciplines of Applied Mechanics II (Introduction to the Mechanics of Materials) and Strength of Materials, taught by the author in the Civil Engineering course and also in the Geological Engineering, Materials Engineering and Architecture courses at the University of Coimbra A physical approach has been favoured when explaining topics, sometimes rejecting the more elaborate mathematical formulations, since the physical understanding of the phenomena is of crucial importance for the student of Engineering In fact, in this way, we are able to develop in future Engineers the intuition which will allow them, in their professional activity, to recognize the difference between a bad and a good structural solution more readily and rapidly The book is divided into two parts In the first one the Mechanics of Materials is introduced on the basis of Continuum Mechanics, while the second one deals with basic concepts about the behaviour of materials and structures, as well as the Theory of Slender Members, in the form which is usually called Strength of Materials The introduction to the Mechanics of Materials is described in the first four chapters The first chapter has an introductory character and explains fundamental physical notions, such as continuity and rheological behaviour It also explains why the topics that compose Solid Continuum Mechanics are divided into three chapters: the stress theory, the strain theory and the constitutive law The second chapter contains the stress theory This theory is expounded almost exclusively by exploring the balance conditions inside the body, gradually introducing the mathematical notion of tensor As this notion VIII Preface to the First Portuguese Edition is also used in the theory of strain, which is dealt with in the third chapter, the explanation of this theory may be restricted to the essential physical aspects of the deformation, since the merely tensorial conclusions may be drawn by analogy with the stress tensor In this chapter, the physical approach adopted allows the introduction of notions whose mathematical description would be too complex and lengthy to be included in an elementary book The finite strains and the integral conditions of compatibility in multiplyconnected bodies are examples of such notions In the fourth chapter the basic phenomena which determine the relations between stresses and strains are explained with the help of physical models, and the constitutive laws in the simplest three-dimensional cases are deduced The most usual theories for predicting the yielding and rupture of isotropic materials complete the chapter on the constitutive law of materials In the remaining chapters, the topics traditionally included in the Strength of Materials discipline are expounded Chapter five describes the basic notions and general principles which are needed for the analysis and safety evaluation of structures Chapters six to eleven contain the theory of slender members The way this is explained is innovative in some aspects As an example, an alternative Lagrangian formulation for the computation of displacements caused by bending, and the analysis of the error introduced by the assumption of infinitesimal rotations when the usual methods are applied to problems where the rotations are not small, may be mentioned The comparison of the usual methods for computing the deflections caused by the shear force, clarifying some confusion in the traditional literature about the way as this deformation should be computed, is another example Chapter twelve contains theorems about the energy associated with the deformation of solid bodies with applications to framed structures This chapter includes a physical demonstration of the theorems of virtual displacements and virtual forces, based on considerations of energy conservation, instead of these theorems being presented without demonstration, as is usual in books on the Strength of Materials and Structural Analysis, or else with a lengthy mathematical demonstration Although this book is the result of the author working practically alone, including the typesetting and the pictures (which were drawn using a selfdeveloped computer program), the author must nevertheless acknowledge the important contribution of his former students of Strength of Materials for their help in identifying parts in the texts that preceded this treatise that were not as clear as they might be, allowing their gradual improvement The author must also thank Rui Cardoso for his meticulous work on the search for misprints and for the resolution of proposed exercises, and other colleagues, especially Rog´erio Martins of the University of Porto, for their comments on the preceding texts and for their encouragement for the laborious task of writing a technical book This book is also a belated tribute to the great Engineer and designer of large dams, Professor Joaquim Laginha Serafim, who the Civil Engineering Department of the University of Coimbra had the honour to have as Professor Preface to the First Portuguese Edition IX of Strength of Materials It is to him that the author owes the first and most determined encouragement for the preparation of a book on this subject Coimbra July 1995 V Dias da Silva Contents Part I Introduction to the Mechanics of Materials I Introduction I.1 General Considerations I.2 Fundamental Definitions I.3 Subdivisions of the Mechanics of Materials 3 II The Stress Tensor II.1 Introduction II.2 General Considerations II.3 Equilibrium Conditions II.3.a Equilibrium in the Interior of the Body II.3.b Equilibrium at the Boundary II.4 Stresses in an Inclined Facet II.5 Transposition of the Reference Axes II.6 Principal Stresses and Principal Directions II.6.a The Roots of the Characteristic Equation II.6.b Orthogonality of the Principal Directions II.6.c Lam´e’s Ellipsoid II.7 Isotropic and Deviatoric Components of the Stress Tensor II.8 Octahedral Stresses II.9 Two-Dimensional Analysis of the Stress Tensor II.9.a Introduction II.9.b Stresses on an Inclined Facet II.9.c Principal Stresses and Directions II.9.d Mohr’s Circle II.10 Three-Dimensional Mohr’s Circles II.11 Conclusions II.12 Examples and Exercises 9 12 12 15 16 17 19 21 22 22 24 25 27 27 28 29 31 33 36 37 XII Contents III The Strain Tensor III.1 Introduction III.2 General Considerations III.3 Components of the Strain Tensor III.4 Pure Deformation and Rigid Body Motion III.5 Equations of Compatibility III.6 Deformation in an Arbitrary Direction III.7 Volumetric Strain III.8 Two-Dimensional Analysis of the Strain Tensor III.8.a Introduction III.8.b Components of the Strain Tensor III.8.c Strain in an Arbitrary Direction III.9 Conclusions III.10 Examples and Exercises 41 41 41 44 49 51 54 58 59 59 60 60 63 64 IV Constitutive Law 67 IV.1 Introduction 67 IV.2 General Considerations 67 IV.3 Ideal Rheological Behaviour – Physical Models 69 IV.4 Generalized Hooke’s Law 75 IV.4.a Introduction 75 IV.4.b Isotropic Materials 75 IV.4.c Monotropic Materials 80 IV.4.d Orthotropic Materials 82 IV.4.e Isotropic Material with Linear Visco-Elastic Behaviour 83 IV.5 Newtonian Liquid 84 IV.6 Deformation Energy 86 IV.6.a General Considerations 86 IV.6.b Superposition of Deformation Energy in the Linear Elastic Case 89 IV.6.c Deformation Energy in Materials with Linear Elastic Behaviour 90 IV.7 Yielding and Rupture Laws 92 IV.7.a General Considerations 92 IV.7.b Yielding Criteria 93 IV.7.b.i Theory of Maximum Normal Stress 93 IV.7.b.ii Theory of Maximum Longitudinal Deformation 94 IV.7.b.iii Theory of Maximum Deformation Energy 94 IV.7.b.iv Theory of Maximum Shearing Stress 95 IV.7.b.v Theory of Maximum Distortion Energy 95 IV.7.b.vi Comparison of Yielding Criteria 96 IV.7.b.vii Conclusions About the Yielding Theories 100 IV.7.c Mohr’s Rupture Theory for Brittle Materials 101 IV.8 Concluding Remarks 105 Contents IV.9 XIII Examples and Exercises 106 Part II Strength of Materials V Fundamental Concepts of Strength of Materials 119 V.1 Introduction 119 V.2 Ductile and Brittle Material Behaviour 121 V.3 Stress and Strain 123 V.4 Work of Deformation Resilience and Tenacity 125 V.5 High-Strength Steel 127 V.6 Fatigue Failure 128 V.7 Saint-Venant’s Principle 130 V.8 Principle of Superposition 131 V.9 Structural Reliability and Safety 133 V.9.a Introduction 133 V.9.b Uncertainties Affecting the Verification of Structural Reliability 133 V.9.c Probabilistic Approach 134 V.9.d Semi-Probabilistic Approach 135 V.9.e Safety Stresses 136 V.10 Slender Members 137 V.10.a Introduction 137 V.10.b Definition of Slender Member 138 V.10.c Conservation of Plane Sections 138 VI Axially Loaded Members 141 VI.1 Introduction 141 VI.2 Dimensioning of Members Under Axial Loading 142 VI.3 Axial Deformations 142 VI.4 Statically Indeterminate Structures 143 VI.4.a Introduction 143 VI.4.b Computation of Internal Forces 144 VI.4.c Elasto-Plastic Analysis 145 VI.5 An Introduction to the Prestressing Technique 150 VI.6 Composite Members 153 VI.6.a Introduction 153 VI.6.b Position of the Stress Resultant 153 VI.6.c Stresses and Strains Caused by the Axial Force 154 VI.6.d Effects of Temperature Variations 155 VI.7 Non-Prismatic Members 157 VI.7.a Introduction 157 VI.7.b Slender Members with Curved Axis 157 VI.7.c Slender Members with Variable Cross-Section 159 VI.8 Non-Constant Axial Force – Self-Weight 160 XIV Contents VI.9 Stress Concentrations 161 VI.10 Examples and Exercises 163 VII Bending Moment 189 VII.1 Introduction 189 VII.2 General Considerations 190 VII.3 Pure Plane Bending 193 VII.4 Pure Inclined Bending 196 VII.5 Composed Circular Bending 200 VII.5.a The Core of a Cross-Section 202 VII.6 Deformation in the Cross-Section Plane 204 VII.7 Influence of a Non-Constant Shear Force 209 VII.8 Non-Prismatic Members 210 VII.8.a Introduction 210 VII.8.b Slender Members with Variable Cross-Section 210 VII.8.c Slender Members with Curved Axis 212 VII.9 Bending of Composite Members 213 VII.9.a Linear Analysis of Symmetrical Reinforced Concrete Cross-Sections 216 VII.10 Nonlinear bending 219 VII.10.a Introduction 219 VII.10.b Nonlinear Elastic Bending 220 VII.10.c Bending in Elasto-Plastic Regime 221 VII.10.d Ultimate Bending Strength of Reinforced Concrete Members 226 VII.11 Examples and Exercises 228 VIII Shear Force 251 VIII.1 General Considerations 251 VIII.2 The Longitudinal Shear Force 252 VIII.3 Shearing Stresses Caused by the Shear Force 258 VIII.3.a Rectangular Cross-Sections 258 VIII.3.b Symmetrical Cross-Sections 259 VIII.3.c Open Thin-Walled Cross-Sections 261 VIII.3.d Closed Thin-Walled Cross-Sections 265 VIII.3.e Composite Members 268 VIII.3.f Non-Principal Reference Axes 269 VIII.4 The Shear Centre 270 VIII.5 Non-Prismatic Members 273 VIII.5.a Introduction 273 VIII.5.b Slender Members with Curved Axis 273 VIII.5.c Slender Members with Variable Cross-Section 274 VIII.6 Influence of a Non-Constant Shear Force 275 VIII.7 Stress State in Slender Members 276 VIII.8 Examples and Exercises 278 XII.7 Examples and Exercises 513 P A B l C D 2l l E l F A 2l l Fig XII.25 C B D l l E 2l Fig XII.26 A B P C D l l Fig XII.27 XII.28 The beam represented in Fig XII.28 is made of a material with linear elastic behaviour with an elasticity modulus E Its cross-section has a moment of inertia I On this beam a weight P falls from a height h = 2l The impact may occur on any cross-section of the beam (a) Determine the point of the beam where a static load P must be applied, in order to get the maximum bending moment in crosssection B (b) Determine the bending moment induced in cross-section B by the fall of the weight on that cross-section (c) Does the fact that the deformation caused by the shear force is not considered introduce an advantageous or disadvantageous error for the safety of the structure ? XII.29 In which of the two beams represented in Fig XII.29 does the impact of the weight cause higher internal forces ? Justify the answer P l A B C 2I l l l Fig XII.28 h 2I I l l h Fig XII.29 XII.30 On point A of the beam represented in Fig XII.30 a weight P falls If an additional support is placed at point C, does the bending moment caused by the impact on cross-section B increase or decrease ? Justify the answer XII.31 The structure represented in Fig XII.31 is made of a material with linear elastic behaviour with an elasticity modulus E The cross-sections of bars AB and BC have the moments of inertia 2I and I, respectively 514 XII Energy Theorems A A B C 50r P l D l A P 10r B C B C 20r l Fig XII.30 Fig XII.31 Fig XII.32 (a) Determine the bending moments introduced into the structure by the impact of the weight P (b) If the moment of inertia of the cross-section of bar AB were to be reduced to the half, would those bending moments increase or decrease ? Justify the answer XII.32 The structure represented in Fig XII.32 is contained in a horizontal plane The two bars are perpendicular to each other and have a circular cross-section with a radius r The support at point B prevents the vertical displacement (a) Determine the internal forces caused in the structure by the impact of the weight P when it falls from the given height (consider G = 0.4E) (b) If the length of bar AB were to increase to double (100r), would the internal forces increase or decrease ? Justify the answer Answers to Proposed Exercises Chapter II II.3 (a) Substituting the given expressions into the differential equations of equilibrium (5) it is immediately seen that they are not satisfied (b) X = ηρ λ , Y = ηρ Chapter III III.1 εx = 10x + 3y εxy = 32 x + 7y + 32 z εxz = 32 y + 2z εy = 6x + 8y εyz = 2y + 2z + 32 εz = 4x + 12y III.2 εx = 12A + 72A + 72D2 + 312.5G2 III.3 (a) 4; (b) 4; (c) ωxy = − 32 x − y − 32 z ωxz = 32 y − 2z ωyz = 2y − 2z + 32 Chapter IV IV.7 Monotropic material: revolution ellipsoid with the revolution axis parallel to the monotropy direction; Orthotropic material: ellipsoid with principal axes parallel to the material orthotropy directions σ IV.8 (a) εv = Eσxx (1 − νxy − νxz ) + Eyy (1 − νyx − νyz ) + Eσzz (1 − νzx − νzy ) (x, y and z are the orthotropy directions) 516 XII Energy Theorems (b) εv = σx El (1 − 2νl ) + (σy + σz ) 1−νt Et − (x is the monotropy direction) IV.9 Ellipsoid with principal axes Rx = + σ R and Rz = + 0.775 E 1−νl El σ 30 E σ R, Ry = + 0.15 E R σ ∆t IV.10 (a) Ud = 0η (b) Ud = 38 Eε02 IV.11 Axisymmetric stress state where the symmetry axis has the direction of the material symmetry direction IV.12 Nine, since it is an orthotropic material with linear elastic behaviour: the material has symmetric rheological properties with respect to the plane of the layers and with respect to the two other planes that are perpendicular to it and are parallel to the fibres in the 1st , 5th , 9th , etc., and in the 2nd , 6th , 10th , etc., layers, respectively 1−νxy −νxz 1−νyx −νyz 1−νzx −νzy σm IV.13 (a) εv = + + Ex Ey Ez (b) εv = ⎡ l1 IV.14 [ l ] = ⎣ l2 l3 1−4νl El + 2−2νt Et σm (σm is the isotropic stress) ⎤ ⎡ σ1 0 m1 n1 [ σp ] = ⎣ σ2 ⎦ m2 n2 ⎦ m3 n3 0 ⎤σ3 ⎡ σx τyx τzx t (a) - [ σ ] = [ l ] [ σp ] [ l ] = ⎣ τxy σy τzy ⎦; τxz τyz σz ⎡ ⎤ εx εyx εzx - [ σ ] −→ [ ε ] = ⎣ εxy εy εzy ⎦ (85); εxz εyz εz - εv = εx + εy + εz (invariant) (b) Operations and equal to answer (a); ⎤ ⎡ εx εy x εz x t - [ ε ] = [ l ] [ ε ] [ l ] = ⎣ εx y εy εz y ⎦ εx z εy z εz (x ≡ 1, y ≡ 2, z ≡ – principal directions of the stress state) IV.15 The relaxation curve is similar to that corresponding to the Maxwell model (Fig 33), except in relation to the asymptote, since, after relaxation (t → ∞) a residual stress remains which is necessary to keep the additional spring deformed IV.16 (a) Monotropy direction and all directions of the isotropy plane (b) Orthotropy directions ⎤ σy2 +σz2 σx El + Et α σ0 Ue = 2α − 2e− 100 Ue = 12 E (ε∆t) ˙ ; ν ν IV.17 U = − Ell (σx σy + σx σz ) − Ett σy σz + IV.18 + e− 50 ; Ud = IV.19 α Ud = η ε˙2 ∆t σ0 α α 100 − 2 τxy +τxz 2Gt α + 1+νt Et τyz α + 2e− 100 − 12 e− 50 XII.8 Chapter VII 517 Chapter VI VI.19 NAB = 66 NBC = 29 P NCD = − 108 29 P 29 P 20 20 VI.20 σa = − Eα∆t and σb = Eα∆t ( 135) may be used to compute the stresses, although the cross-section does not have symmetry axes In fact, because of the way as the areas occupied by the two materials are distributed in the cross-section, the resulting moment of the stresses obtained by means of (135) vanishes This means that the equilibrium conditions are verified in the conditions for which (135) has been deduced: constant strain throughout the cross-section, that is, with the bar axis remaining straight during the deformation √ VI.21 e = 5−1 c ≈ 0.618c This value is computed by equating to zero the resulting moment of the stresses caused by a temperature variation, when the bar axis remains straight (constant strain in the cross-section) XII.8 Chapter VII VI.3 (a) I v = bh2 ; (b) I v = bh6 ; (c) I v = bh2 VII.10 (a) rhombus with height h3 and width 3b ; (b) circle of radius 4r ; (c) rectangle of base 6b and height h6 ; (d) equilateral triangle with side length a4 ; (e) ellipse with semi-axes lengths 4b and h4 max = 0.16 σc b3 , Mpmax = 0.264 σc b3 , ϕ = 1.65 VII.18 Mel VII.19 Width = a ⇒ D ≥ 3a ε1c − , Width = 3a ⇒ D ≥ a ε1c − D – diameter of the interior face of the winding (drum’s diameter) VII.20 (a) ∆l = 2.4 α∆t l; M (b) ρ1 = 34 c4 E ; (c) y = ⇒ σ = 1.2Eα∆T, M y = ±c ⇒ σb = ± 34 c3 + 1.2Eα∆T, m y = ±c ⇒ σa = ± 34 c3 − 1.2Eα∆T, 12 M y = ±2c ⇒ σa = ± 34 c3 − 1.2Eα∆T M ε0 VII.21 (a) ρel = 568 c4 σ ; (b) Material b, because the yielding strain of this material is smaller than that of material a and the distance of the farthest fibres from the neutral axis is greater in material b than in material a; M σb = 284 c3 518 XII Energy Theorems VII.22 The section modulus increases 19.2% The bending strength of the bar increases by the same amount, for the same reasons as explained in the resolution of example VII.6 In the case of a ductile material, removing the small rectangles increases the section modulus, but decreases the plastic modulus, that is, the plastic moment In fact, this modulus increases whenever area is added to the cross-section Removing the small rectangles causes a 3 c σc in the plastic moment of the cross-section decrease ∆Mp = 125 29 VII.23 (a) M = σc a ; y = ±2a ⇒ σ = ± 14 (b) y = ±a ⇒ σ = ∓ 15 44 σc , 44 σc ; 15 σc (c) ρre = 44 Ea ; M M max VII.24 σbmax = 192 = 576 185 c3 , σa 185 c3 VII.25 d = 32 a With this value of d the two principal moments of inertia are equal When this happens, any axis is a principal axis, that is bending is plane for any direction of the action axis of the bending moment Chapter VIII 12 pl VIII.18 dE dz max = 37 c dE VIII.19 (a) dz max = 9.7455p (b) The plane containing the loading should pass through the intersection point of the centre lines of the two rectangles XII.9 Chapter IX Ml IX.17 (a) θD = 6EI ; IX.18 (a) MA = M (b) δ = P l3 4EI ; (c) δ = P l3 EI ; l (d) δB = − M 3EI 2pl Pl (b) MB = P20l − 7pl 15 , MA = − − 15 Mp 2Mp IX.19 (a) Pcol = l ; (b) Pcol = 3l ; (c) Pcol = M (d) pcol = 6.9641 l2p Mp l ; XII.9 Chapter IX 519 Chapter X T X.15 (a) f = 2πr (shear flow in the cross-section) 25T 2.083T (b) θ = 12πr G ≈ πr G (result obtained by means of the expressions deduced in example X.6-a) X.16 (a) τa ≈ 4.081 aT3 τb ≈ 6.122 aT3 T (b) θ ≈ 4.122 Ga (this result and that of Sub-question X.16.a have been obtained by means of the expressions deduced in example X.6-b; note that A = (0.99a) ) 358 a X.17 (a) Open cross section: Tall = 179 a τall, Ja = a ; f Closed cross section: Tall = 800a τall, Jf = 32000 a ; f Tall a Tall J = 13.4078, Jfa = 89.3855 (b) f = τall a T f f X.18 Considering only the closed part, we get: τmax = 400a = 3, θ T t with the whole cross-section, we get: τmax ≈ 426.667a3 , T 3413.333Ga4 ; f τmax t τmax = θf θt T 3200Ga4 ; t θ ≈ ≈ 1.0667 T T X.19 (a) τa = 21546a τb = 10773a 3; −6 T (b) θ = 1.4023 × 10 Ga4 γ X.20 θ = 6a0 −0.01 ln 0.12 = 0.0106 ; 2r r 80000 P δV = δM = πEr ; X.21 (a) X.22 (b) M = 1.6τ0r π 20 P P 1500 πEr ; δT = 75000 πEr If the warping of the built-in cross-section is prevented, the last component (δT ) will be smaller, as explained in Footnote 75 Chapter XI XI.18 XI.19 XI.20 XI.21 XI.22 XI.23 e = 16 αb pcr = 5.12EL I I Ω > 125π 576 l2 ≈ 2.14184 l2 The spring is necessary; Pcr = 23 El EI pcr = 80.568 l3 The problem is solved in the same way as example XI.14 Considering the same reference system, the equation of the deflection curve is: Q Q P sin(kz) − 2P z, with k = EI y = 2P k cos ( kl2 ) Q tan kl The maximum value of the bending moment is : Mmax = 2k 520 XII Energy Theorems Chapter XII XII.20 The influence line of the displacement in cross-section C is the deflection line of the beam when a unit load is placed on point C In fact, Maxwell’s theorem leads to the conclusion that the displacement in cross-section C, caused by a unit load acting on any given crosssection, is equal to the displacement in that cross-section, caused by a unit load acting on point C XII.21 A unit load applied in coordinate causes in coordinate a generalized displacement (relative rotation of the end cross-sections) equal l2 ; a generalized unit force in in coordinate (a unit moment to 32 EI at each end of the beam) causes a displacement with equal value in coordinate XII.22 Drawing the approximate shape of the influence line, we conclude that the maximum negative ordinate occurs in beam segment BC, slightly to the left of the mid point of this segment, since cross-section A cannot rotate, while a small rotation of cross-section C takes place; thus, the deformation of segment BC is larger than the deformation of segment AB; the maximum positive ordinate occurs in segment CD at the distance √13 l ≈ 0.577l of support D (see example XII.10-a) XII.23 The load should be placed on the whole segment BC, since the influence line has positive ordinates in segment AB and negative ordinates in segment BC; the area of the negative part of the influence line is clearly larger than that of the positive part XII.24 The load should be placed slightly to the left of the middle of beam segment BC (at the distance 0.577l of support C, cf example XII.10a), since the ordinates of the influence line in this segment are greater than in segment AB, because the bending stiffness is smaller XII.25 Vmax in the right section of support C: segments BC, CD and EF ; Mmax in the middle of segment CD: segments AB, CD and EF XII.26 MAmax : load on point E; max : load on point E, since the slope of the influence line has the VB−left same absolute value in cross-sections B and D XII.27 Considering the structure without the cable, with a coordinate composed by a pair of horizontal forces applied to points A and B, positive when they cause an increase of the distance AB, and a vertical downwards coordinate in any point of the beam CD, we conclude easily that a positive force (pair of forces) in coordinate causes a negative displacement in coordinate Maxwell’s theorem leads to the conclusion that a positive force in coordinate causes a negative displacement in coordinate 1, that is the distance between points A and B decreases XII.28 (a) The load should fall on the point at the distance (1 − √13 )l ≈ 0.4226l of support B XII.9 Chapter IX (b) MB = 0.0962P l + 1+ 73.47EI P l2 521 (c) The error is advantageous to safety, since a stiffer beam is considered This increases the dynamic coefficient, and so higher internal forces than the actual ones are computed XII.29 In the left beam, since it is stiffer XII.30 The support increases the stiffness of the structure in the vertical displacement of point A which reduces the value of Us , increasing the value of the dynamic coefficient As segment AB is statically determinate, the bending moment in cross-section B depends only on the weight and the dynamic coefficient, so that that moment increases 6EI XII.31 (a) MB = P l + + 5P l2 (b) The moments would decrease, since this geometry change does not change the internal forces if the load is statically applied, but it does reduce the dynamic coefficient, as the rotation of crosssection B increases XII.32 (a) Twisting moment in AB: 20rP ψ; Maximum bending moment in BC: 20rP ψ; ψ = + + 15Eπr 79000P (b) The internal forces would be reduced (same justification as in answer XII.31.b) References A.J Durelli, E.A Phillips, C.H Tsao, Introduction to the Theoretical and Experimental Analysis of Stress and Strain, McGRAW-HILL BOOK COMPANY, INC., 1958 Y.C Fung, Foundations of Solid Mechanics, PRENTICE-HALL, INC., Englewood Cliffs, New Jersey, 1965 Ch Massonnet, S Cescotto, M´ecanique des Mat´eriaux, EYROLLES, Paris, 1980 S.P Timoshenko, J.N Goodier, Theory of Elasticity, third edition, McGraw-Hill Book Company, 1970 Bronstein, Semendjajew, Taschenbuch der Mathematik, 24th edition, Verlag Harri Deutsch Thun und Frankfurt/Main, 1989 D.R.J Owen, E Hinton, Finite Elements in Plasticity - Theory and Practice, Pineridge Press Limited, Swansea, U.K., 1980 Odone Belluzzi, Ciencia de la Construccion, Aguilar s a de ediciones, 1973 Charles Massonnet, R´esistance des Mat´eriaux, II volume, DUNOD, Paris, 1965 J.S Farinha, A Correia dos Reis, Tabelas T´ ecnicas, edi¸ca ˜o P.O.B., Set´ ubal, 1993 (in Portuguese) 10 prEN 1993-3: 20xx, Eurocode 3: Design of steel structures: Part 1-1: General structural rules, 2001 11 S.P Timoshenko, J.M Gere, Theory of Elastic Stability, second edition, McGraw-Hill Book Company, 1961 12 Curt F Kollbrunner, Konrad Basler, Torsion, Springer-Verlag, Berlin and Heidelberg, 1966 13 W.T Koiter, General theorems for elastic-plastic solids, in Progress in Soild Mechanics, Vol 1, I.N Snedon and R Hill (Eds.), Chapter 4, North- -Holland, Amesterdam, 1960 14 V Dias da Silva, Introdu¸c˜ ao a ` An´ alise N˜ ao-Linear de Estruturas, Departamento de Engenharia Civil, Faculdade de Ciˆencias e Tecnologia, Universidade de Coimbra, 2002 (in Portuguese) 15 Ted Belytschko, Wing Kam Liu, Brian Moran, Nonlinear Finite Elements for Continua and Structures, John Wiley & Sons Ltd., 2000 Index action action axis 192 of the bending moment of the shear force 193 analogy hydrodynamical 364 membrane 364 physical 364 anticlastic 206 axial force 141 axial stiffness 143 behaviour models 67 bending 189 composed 190, 202 in elasto-plastic regime 221 inclined 197, 199 non-uniform 189 nonlinear 219 of composite members 213 plane 192 pure or circular 189 bending moment 189 bending stiffness 195 Bernoulli’s hypothesis 139 Betti’s theorem 473 boundary balance equations 16 Bredt’s formulas 376 buckling modes 442, 445, 454, 457 bulk modulus 79 characteristic equation 19, 20 of the stress state 19 characteristic values 135 Clapeyron’s theorem 468 coefficient buckling 415 dynamic 490 homogenizing 154, 216 of thermal expansion 132 Poisson’s 76, 124 retardation 73 safety 137, 406 stiffness 452 collapse mechanism 320 compatibility of deformations 144 composite material conjugate beam method 302 conservation of energy 80, 309, 359, 468 of plane sections 138 constitutive law continuity conditions 299 Continuum Mechanics core of a cross section 201 creep 69 creep modulus 73 critical phase 390 curvature 189 curvature equation 298 Castigliano’s theorem 469 Cauchy equations 16 centroid 142 deflection curve deflection plane deformation 192 193 193 526 Index compatible 51 elastic 68 homogeneous 43 plastic 68 pure 49 visco-elastic 71 visco-plastic 69 viscous 68 deformation energy 86, 126 degree of connection 53 degree of indeterminacy kinematic 143, 153 deviatoric tensor 26 differential equations of equilibrium 14 dimensional tolerance 134 direction cosines 16 displacement method 144 displacement-strain relations distortion 46 Drucker-Prager’s criterion 104 effective length 404 elastic limit stress 129 elastic phase 93 elasto-plastic analysis 145, 223 bending 223 elasto-plastic phase 147 energy deformation 86 dissipated 88, 113, 127 elastic potential 80, 126 kinetic 511 potential 80, 390 total potential 485 equation of three moments 317 equation of two moments 317 equations of compatibility integral 54 local 54 of the strain 44 equilibrium conditions Euler’s hyperbola 408 Euler’s problem 410 Eulerian formulation 300 execution imperfections 134 external forces of mass of surface virtual 484 external friction 465 fatigue failure 128 fatigue limit stress 129 fibre 193 first area moment 192, 253 flow lines 364, 365 Fluid Mechanics 85 force method 144 force-stress relations framed structures 138 generalized displacements 469 generalized forces 469 generalized Maxwell model 74 geometrical stiffness 442 hardening 122 natural hardening 128 strain hardening 127 homogenization 215, 377, 378 Hooke’s law 67, 75, 105 hyperstatic unknowns 153 hypothesis of continuity imperfections (effect of) 396 inertial forces 5, 14 influence lines 475 instability 389 by divergence 433 by equilibrium bifurcation 398 in axial compression 414 in composed bending 411 interaction formula 415 internal forces 5, internal friction 88, 466 intrinsic strength curve 101 invariants 19 of the strain tensor 49 of the stress tensor 20 irradiation poles 31 isotropic tensor 24 Johnson’s parabola 407 Kelvin chain 74, 83 Kelvin’s solid 71 kinematic coordinates 454 kinematic method 321 kinematic unknowns 144 Index Lagrangian formulation 300, 413 Lam´e’s constant 79 Lam´e’s ellipsoid 22 level curves 365 limit states 133 of serviceability 133 ultimate 133 linear visco-elasticity 74, 83 liquid 69 load collapse 321 critical 414 elasticity limit 122 Euler 413 proportionality limit 131 yielding 131 longitudinal modulus of elasticity 76, 124 longitudinal shear flow 254 longitudinal shear force 252, 310 longitudinal strain 44 Mă uller-Breslaus principle 476 material brittle 69, 121 composite continuous ductile 69, 121 elastic 87 isotropic 68 monotropic 68 orthotropic 68 material stiffness 442 mathematical models 3, 67 Maxwell’s model 73 Maxwell’s theorem 477 mean rotation 50 Mechanics of Materials Menabrea’s theorem 473 method of integration of the curvature equation 298 minimum energy theorem 473 minimum loads 153 Mohr’s circle 30 Mohr’s criterion 104 Mohr’s representation 58 three-dimensional 34 moment of inertia 195, 218 moment-area method 304 multiply connected body 527 53 neutral axis 193 neutral equilibrium 390, 403 neutral surface 193, 206 Newtonian liquid 84 nominal values 135 normal stress 11 octahedral stress 24 partial factors 136 perfect liquid 84, 364 physical models 68, 150 plane of actions 193 plane strain 59 plane stress state 39 plastic hinge 225 plastic moment 223 plastic section modulus 224 polar decomposition theorem 50 post-critical behaviour 393 stable 395 symmetrical 395 unstable 398 pressure centre 201 prestressing technique 150 principal axis of inertia 198 principal directions of the stress state 19 principal strains 58 principal stress trajectories 276, 352 principal stresses 19 principle conservation 80, 309, 359, 468, 479 of energy 80 of Mă uller-Breslau 476 of Saint-Venant 130 of stationarity of the 485 of superposition 76, 131 potential energy 485 probabilistic approach 134 probabilistic density curve 134 product of inertia 198, 239 proportionality limit stress 131, 407 quantiles 136 reciprocity of shearing stresses 12 528 Index redistribution of internal forces redistribution of stresses 161 reduced area 310 reinforced concrete relaxation 69 relaxation modulus 74 resilience 126, 127 retardation time 73 rheological behaviour 3, elastic 143 elasto-visco-plastic 69 rigid body motion 152 safety stresses 136 Saint-Venant’s hypothesis 252 Saint-Venant’s principle 130 secant formula 411 second-order theories 391 section modulus 195 semi-normal of the facet 10 semi-probabilistic approach 135 shape factor 223 shear centre 270, 493 shear flow 265, 358 shear force 254 shear modulus 77 shearing strain 44, 77 simplifying hypotheses 120, 134, 257, 361 simply connected body 53 slender members 138 cross-section 138, 222, 274 non-prismatic 157, 209, 273 prismatic 138 with curved axis 157, 212, 273 slenderness ratio 403 solid 69 Solid Mechanics 7, 37, 120, 121, 466 stability 389 stable equilibrium 390, 395 state of deformation 43 around a point 49 isotropic 58 state of stress 17 around a point 17 axisymmetric 24 isotropic 22 three-dimensional 92 static method 321 statically determinate structures statically indeterminate structures 143, 315 statistical dispersion 134 stiffness 123 stiffness matrix 441 of a compressed bar 445 of a tensioned bar 451 strain strain tensor 6, 40 Strength of Materials 120 stress 5, 10 stress concentration 161, 364 stress tensor 6, 17, 20 support conditions 303 143 tangent elasticity modulus 124 tangential 11 tenacity 126, 134 tensor tensorial quantities Tetmeyer’s line 406, 407 theorem of virtual displacements 479 theorem of virtual forces 482 theory of elasticity 119 theory of strain theory of stress torque 347 torsion 346 of circular cross-sections 347 of thin-walled cross-sections 356, 360 torsion centre 271, 370 torsion modulus 351 torsional moment 347 torsional stiffness 351, 359, 368, 369 transversal modulus of elasticity 77 transversal strain 75 twisting moment 347 uncertainties 133 unstable equilibrium 390, 442 virtual displacements 479 virtual stress 482 viscosity modulus 85, 86 volumetric modulus of elasticity von Karman convention 36 yielding bending moment 223 79 Index yielding criteria 93, 96 Beltrami 98, 106 Rankine 106 Saint-Venant 106 Tresca 98 Von Mises 95 yielding stress 68, 92, 125 yielding surface 99, 100 yielding zone 121 Young’s modulus 124 529 ... theoretical texts and exercises prepared and improved on by the author between 1989-94, for the disciplines of Applied Mechanics II (Introduction to the Mechanics of Materials) and Strength of Materials, ... awareness of the lack of a treatise, written in European Portuguese, which contains the theoretical material taught in the disciplines of the Mechanics of Solid Materials and the Strength of Materials, ... Introduction to the Mechanics of Materials I Introduction I.1 General Considerations Materials are of a discrete nature, since they are made of atoms and molecules, in the case of liquids and gases, or,