This book provides a lucid introduction to the theory of elasticity as applied to isotropic, specially orthotropic and laminated structures. With an application-oriented approach, the contents emphasize the need for rigorous analysis and illustrate its utility for a variety of problems. The simultaneous treatment of comparable isotropic and orthotropic problems enables one to easily visualize the changes in structural behaviour due to material orthotropy. Though intended as a textbook for graduate engineering study, this book is valuable as a self-study aid for practicing engineers as well.
Author(s): K. Bhaskar, T. K. Varadan
Publisher: Springer
Year: 2022
Language: English
Pages: 202
City: Cham
Preface
Contents
About the Authors
1 Introduction and Mathematical Framework
1.1 Theory of Elasticity versus Conventional Engineering Theories
1.2 Field Variables of the Theory of Elasticity
1.3 Two-Dimensional Elasticity
1.3.1 Plane Stress
1.3.2 Plane Strain
1.3.3 Field Variables of Two-Dimensional Elasticity
1.4 The Field Equations
1.4.1 Constitutive Relations—Generalised Hooke’s Law
1.4.2 Equations of Equilibrium
1.4.3 Strain–Displacement Relations
1.5 Stress Approach—Compatibility Equations
1.6 Displacement Approach—Navier Equations
1.7 Stress Transformation
1.8 Strain Transformation
1.9 Principal Stresses and Strains
1.10 Boundary Conditions
1.11 Summary
2 Plane Problems in Cartesian Coordinates
2.1 Airy Stress Function
2.2 Compatibility Equation in Terms of Stresses and Airy Stress Function
2.2.1 Plane Stress—Isotropic Case
2.2.2 Plane Strain—Isotropic Case
2.2.3 Plane Stress—Specially Orthotropic Case
2.2.4 Plane Strain—Specially Orthotropic Case
2.3 Use of Polynomial Stress Functions
2.3.1 The Simple Case of Uniaxial Tension
2.3.2 Pure Bending of a Beam
2.3.3 A Tip-Loaded Cantilever
2.3.4 A Simply Supported Beam Under Uniform Load
2.4 A Fourier Series Solution for the Simply Supported Beam
2.5 Justification of End Conditions Specified in Terms of Integrals
2.6 St.Venant’s Principle
2.6.1 Deduction from the Simply Supported Beam Solutions
2.6.2 Eigensolutions for Isotropic/Orthotropic Rectangular Strip
2.6.3 Implications of St.Venant’s Principle
2.7 Factors Governing Shear Deformation Effect
2.7.1 Clamped Edge Conditions
2.7.2 Localised Loading
2.8 What is a Long Beam?
2.9 Summary
3 Plane Problems in Polar Coordinates
3.1 Field Equations in Polar Coordinates
3.1.1 Equilibrium Equations
3.1.2 Constitutive Relations
3.1.3 Strain-Displacement Relations
3.1.4 Compatibility Equation
3.1.5 Equations for the Axisymmetric Problem
3.2 Circular Cylinder Under Internal and External Pressure (Lamé’s Problem)
3.3 Some Special Cases of Lamé’s Problem
3.3.1 Solid Cylinder
3.3.2 Externally Pressurised Cylinder with a Pin-Hole
3.3.3 Pressurised Hole in an Infinite Body
3.4 Isotropic Plate with a Circular Hole (Kirsch’s Problem)
3.5 Some Similar Stress Concentration Problems
3.5.1 Isotropic Plate with an Elliptic Hole (Inglis’ Problem)
3.5.2 Orthotropic Plate with a Circular Hole
3.6 Violation of Principle of Complementary Shear
3.7 Linear Crack in an Isotropic Plate (Williams’ Solution)
3.8 Stresses Under Concentrated Loads
3.8.1 Concentrated Normal Force on the Boundary of a Half-Plane (Flamant’s Problem)
3.8.2 Concentrated Load on an Isotropic Simply Supported Beam (Wilson-Stokes Method)
3.9 Frictionless Contact Between Isotropic Cylinders (Hertz Problem)
3.10 Bending of a Semicircular Beam
3.10.1 Load Case 1: End Moments
3.10.2 Load Case 2: End Shear
3.10.3 Calculation of the Transverse Normal Stress
3.11 Summary
4 Torsion of Non-circular Sections
4.1 Formulation for Isotropic Shafts
4.1.1 Displacement Approach (St. Venant’s Warping Function Formulation)
4.1.2 Stress Approach (Prandtl’s Stress Function Formulation)
4.2 Solutions for Isotropic Simply-Connected Domains
4.2.1 An Elliptical Section
4.2.2 The Special Case of a Circular Section
4.2.3 An Equilateral Triangle
4.2.4 A Rectangular Section
4.2.5 A Comparison of Various Shapes
4.3 Membrane Analogy
4.4 Approximate Analysis of Thin-Walled Open Sections
4.5 Multiply-Connected Domains
4.5.1 Formulation
4.5.2 Some Simple Solutions
4.5.3 Approximate Analysis of Thin-Walled Tubes (Bredt-Batho Theory)
4.5.4 Closed Tube versus Slit Tube
4.6 A Brief Discussion of Some Geometrically Complicated Sections
4.6.1 Protruding Sharp Corners
4.6.2 Re-Entrant Corners
4.6.3 Slots and Grooves
4.6.4 Eccentric Hole in a Circular Section
4.6.5 Effect of the Hole Shape
4.6.6 Optimum Shape of a Hollow Section
4.6.7 Irregular Sections
4.7 Orthotropic Shafts
4.7.1 Rectilinear Orthotropy—Simple Solutions
4.7.2 Rectlilinear Orthotropy—Laminated Rectangular Shaft
4.7.3 Rectilinear Orthotropy—Thin-Walled Open Sections
4.7.4 Shape-Intrinsic Orthotropy
4.8 Effect of Warping Restraints
4.9 Summary
5 Some Other Problems of Interest
5.1 Concentrated Normal Force on the Boundary of an Isotropic Half-Space (Boussinesq’s Problem)
5.2 Frictionless Contact Between Two Isotropic Spheres (Hertz Problem)
5.3 Free Edge Phenomenon in Composite Laminates
5.4 Functionally Graded Structures
Appendix
A1 Transformation of Stresses and Strains
A1.1 Stress Transformation Rules
A1.2 Strain Transformation Rules
A1.3 Principal Stresses and Strains
A2 The Isotropic Elastic Constants
A2.1 Derivation of G in Terms of E and μ
A2.2 Derivation of K in Terms of E and μ
A2.3 Limiting Values for μ
Index
