Theory of Elasticity

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1 Theory of Elasticity

2

3 Aldo Maceri Theory of Elasticity 123

4 Prof. Dr.-Ing. Aldo Maceri Universitá Roma Tre Departimento di Ingegneria Meccanica e Industriale Via della Vasca Navale, Roma Italy aldo.maceri@fastwebnet.it ISBN e-isbn DOI / Springer Heidelberg Dordrecht London New York Library of Congress Control Number: Springer-Verlag Berlin Heidelberg 2010 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 to prosecution under the German Copyright Law. 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. Cover design: estudio Calamar S.L. Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 Preface The Theory of elasticity studies the behavior of those bodies that recover their initial state when the causes which produce deformations are removed. Its results constitute the foundations of the Theory of structures and then are of maximum importance for engineers. The Theory of elasticity moves freely within an unified mathematical framework that provides the analytical tools for calculating stresses and deformations in a strained elastic body. All the elastic problems can be exactly analyzed employing the classical Mathematical analysis, with the exception of the unilateral problems for which the employment of the Functional analysis is mandatory. The Theory of elasticity was founded by the famous mathematician Cauchy inthe eighteenth-century. During its historical development this scientific sector proposed to the mathematicians various problems that have contributed or entirely generated the development of complex mathematical theories, as the Variational calculus and the Finite element method. The matter analyzed in this book is three-dimensional problems (Chap. 1), and particularly the problem of Saint Venant (Chap. 1), two-dimensional problems, as panels, plates, shells (Chap. 3), one-dimensional problems, as ropes, beams, arches (Chap. 4), thermal stress problems (Chap. 5), stability problems (Chap. 6), anisotropic problems, that constitute the basic tool for the analysis of structures in composite material (Chap. 7), nonlinear elastic problems, as finite elasticity and unilateral problems (Chap. 8). In this book I have constantly kept in mind the practical application of the theoretical results. So I have always tried to give to engineers, in a simple form, a clear indication of the necessary fundamental knowledge of the Theory of elasticity. In the past some techniques of calculation were developed for particular elastic problems that cannot be organized in mathematical theories but are extremely simple to apply. Such technical theories have always furnished results experimentally verified v

6 vi Preface with good approximation and then among them I have presented those that are still useful tools of verification in the Structural design. Throughout the analysis of the elastic problems my constant focus has been to achieve the maximum clarity and because of this I have sacrificed various bright discussions. I have developed the treatment of the subjects in classical way, but to the light of the modern Mathematical theory of the elasticity and with more accented relief to the connections with the Thermodynamics. Just for this, to give a clear justification of the fundamental equation of the Thermoelasticity I have applied a technique of analysis proper of the Fluid dynamics. However in the discussion of the unilateral problems, where the Functional analysis is compulsory, I have related in details the mathematical aspects of the theoretical analysis. Roma, Italy October 2009 Aldo Maceri

7 Contents 1 The Three-Dimensional Problem AnalysisofStrain Components of Displacement InfinitesimalDeformation Elongation and Shearing Strain SmallDeformations Components of Strain PrincipalDirectionofStrain InvariantsofStrain PlaneStateofStrain Equations of Compatibility MeasurementofStrain AnalysisofStress StressVector Normal Stress Shearing Stress Components of Stress Symmetry of τ Differential Equations of Equilibrium Cauchy s Boundary Conditions SymmetryofStressVector Relations Between Normal or Shearing Stress and Components of Stress PrincipalDirectionofStress InvariantsofStress Mohr scircle Mohr sprincipalcircles Determination of the Maximum Normal Stress or Shearing Stress by the Mohr sprincipalcircles PlaneStateofStress UniaxialStateofStress MeasurementofStress PrincipleofVirtualWorks PrincipleofVirtualWorks vii

8 viii Contents 1.4 RelationsBetweenStressandStrain Tensile Breaking Test Homogeneous and Isotropic Materials Navier s Relations Bounds for the Elastic Modulus The Elastic Equilibrium Problem ClassicalFormulations Variational Formulations StrainEnergy Elements of Thermodynamics Thermodynamics of the Problem of the Elastic Equilibrium StrainWork TheElasticPotential Work Theorems Strength Criterions StructuralSafety The Maximum Shearing Stress Criterion The Octahedral Shearing Stress Criterion TheEnergeticCriterion TheIntrinsicCurveCriterion The Problem of Saint Venant GeometryofAreas Centroid Inertia Centroidal Ellipse Antipolarity InertiaCentroidalKernel The Problem of Saint Venant Introduction StateofStress Bending Right Bending of Axis x Right Bending of Axis y Deviated Bending AxialLoad CentroidalAxialLoad NonCentroidalAxialLoad MaterialNonResistanttoTraction Torsion The Exact Solution TheCircularCrossSection The Stress Concentration ClosedThinWalledCrossSection OpenThinWalledCrossSection NonUniformTorsion Shear The Exact Solution

9 Contents ix TheApproximateSolution TheCircularCrossSection OpenThinWalledCrossSection ClosedThinWalledCrossSection The Two-Dimensional Problems Panels The Problem of the Panel Rectangular Panels Circular Panels EffectofaHole Plates Small Deflections of Thin Plates Thin Plates on Elastic Foundation Shells Membranes ThinShells PlaneStrainProblems PlaneStrainProblems The One-Dimensional Problems Ropes The Funicular The Ropes Beams The Deflected Beam The Analogy of Mohr PrincipleofVirtualWorks StrainEnergy Deflected Beams on Elastic Foundation Arches Arches with Small Curvature Arches with Great Curvature Thermoelasticity Mechanics of Continuous Media Introduction Classical Thermodynamics The Equations of Balance Thermodynamics of the Irreversible Processes Fluid Dynamics The Mathematical Model TheCharacteristicNumbers NonDissipativeFlows DissipativeFlows Mechanics of Solids The Dynamic Thermoelastic Problem

10 x Contents TheThermoelasticDissipation The Uncoupled Thermoelastic Problem Thermoelasticity The One-Dimensional Problem Stability Stability of the Elastic Equilibrium TheBuckling The Ultimate Strength EnergyMethod Introduction SomeElementaryApplication StaticMethod The Beam Axially Loaded ApproximateAnalysis Exact Analysis EffectoftheImperfections Limit Slenderness OtherWaysofBuckling Second Type Instability The Snapping Anisotropy The Three-Dimensional Anisotropic Problem Introduction Constituent Links TheAnisotropicElasticBody Energetic Aspects The Anisotropic Saint Venant s Problem The Macroscopic Anisotropy CompositeMaterials StructuralAnisotropy Nonlinear Elasticity Nonlinear Problems The Nonlinearity Causes FiniteDeformations Three-Dimensional Problem Large Deflections of Thin Plates UnilateralProblems Introduction ContactProblems UnilateralConstraints Bibliography Author Index Subject Index

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