Solid Mechanics and Its Applications

Introduction.- 1.1 The Engineering design process;1.2 Design optimization;1.2.1 Predicting the behaviour of the component.- 1.2.2 Approximate solutions.- 1.3 Relative magnitude of different effects.- 1.4 Formulating and solving problems.- 1.4.1 Use of procedures.- 1.4.2 Inverse problems.- 1.4.3 Physical uniqueness and existence arguments.- 1.5 Review of elementary mechanics of materials.- 1.5.1 Definition of stress components.- 1.5.2 Transformation of stress components.- 1.5.3 Displacement and strain.- 1.5.4 Hooke's law.- 1.5.5 Bending of beams.- 1.5.6 Torsion of circular bars.- 1.6 Summary.- Problems.- 2 Material Behaviour and Failure.- 2.1 Transformation of stresses.- 2.1.1 Review of two-dimensional results.- 2.1.2 Principal stresses in three dimensions.- 2.2 Failure theories for isotropic materials.- 2.2.1 The failure surface.- 2.2.2 The shape of the failure envelope.- 2.2.3 Ductile failure (yielding).- 2.2.4 Brittle failure.- 2.3 Cyclic loading and fatigue.- 2.3.1 Experimental data.- 2.3.2 Statistics and the size effect.- 2.3.3 Factors influencing the design stress.- 2.3.4 Effect of combined stresses.- 2.3.5 Effect of a superposed mean stress.- 2.3.6 Summary of the design process.- 2.4 Summary.- Problems.- 3 Energy Methods.- 3.1 Work done on loading and unloading.- 3.2 Strain energy.- 3.3 Load-displacement relations.- 3.3.1 Beams with continuously varying bending moments.- 3.3.2 Axial loading and torsion.- 3.3.3 Combined loading.- 3.3.4 More general expressions for strain energy.- 3.3.5 Strain energy associated with shear forces in beams.- 3.4 Potential energy.- 3.5 The principle of stationary potential energy.- 3.5.1 Potential energy due to an external force.- 3.5.2 Problems with several degrees of freedom.- 3.5.3 Non-linear problems.- 3.6 The Rayleigh-Ritz method.- 3.6.1 Improving the accuracy.- 3.6.2 Improving the back of the envelope approximation.- 3.7 Castigliano's first theorem.- 3.8 Linear elastic systems.- 3.8.1 Strain energy.- 3.8.2 Bounds on the coefficients.- 3.8.3 Use of the reciprocal theorem.- 3.9 The stiffness matrix.- 3.9.1 Structures consisting of beams.- 3.9.2 Assembly of the stiffness matrix.- 3.10 Castigliano's second theorem.- 3.10.1 Use of the theorem.- 3.10.2 Dummy loads.- 3.10.3 Unit load method.- 3.10.4 Formal procedure for using Castigliano's second theorem.- 3.10.5 Statically indeterminate problems.- 3.10.6 Three-dimensional problems.- 3.11 Summary.- Problems.- 4 Unsymmetrical Bending.- 4.1 Stress distribution in bending.- 4.1.1 Bending about the x-axis only.- 4.1.2 Bending about the y-axis only.- 4.1.3 Generalized bending.- 4.1.4 Force resultants.- 4.1.5 Uncoupled problems.- 4.1.6 Coupled problems.- 4.2 Displacements of the beam.- 4.3 Second moments of area.- 4.3.1 Finding the centroid.- 4.3.2 The parallel axis theorem.- 4.3.3 Thin-walled sections.- 4.4 Further properties of second moments.- 4.4.1 Coordinate transformation.- 4.4.2 Mohr's circle of second moments.- 4.4.3 Solution of unsymmetrical bending problems in principal coordinates.- 4.4.4 Design estimates for the behaviour of unsymmetrical sections.- 4.4.5 Errors due to misalignment.- 4.5 Summary.- Problems.- 5 Non-linear and Elastic-Plastic Bending.- 5.1 Kinematics of bending.- 5.2 Elastic-plastic constitutive behaviour.- 5.2.1 Unloading and reloading.- 5.2.2 Yield during reversed loading.- 5.2.3 Elastic-perfectly plastic material.- 5.3 Stress fields in non-linear and inelastic bending.- 5.3.1 Force and moment resultants.- 5.4 Pure bending about an axis of symmetry.- 5.4.1 Symmetric problems for elastic-perfectly plastic materials.- 5.4.2 Fully plastic moment and shape factor.- 5.5 Bending of a symmetric section about an orthogonal axis.- 5.5.1 The fully plastic case.- 5.5.2 Non-zero axial force.- 5.5.3 The partially plastic solution.- 5.6 Unsymmetrical plastic bending.- 5.7 Unloading, springback and residual stress.- 5.7.1 Springback and residual curvature.- 5.7.2 Reloading and shakedown.- 5.8 Limit analysis in the design of beams.- 5.8.1 Plastic hinges.- 5.8.2 Indeterminate problems.- 5.9 Summary.- Problems.- 6 Shear and Torsion of Thin-walled Beams.- 6.1 Derivation of the shear stress formula.- 6.1.1 Choice of cut and direction of the shear stress.- 6.1.2 Location and magnitude of the maximum shear stress.- 6.1.3 Welds, rivets and bolts.- 6.1.4 Curved sections.- 6.2 Shear centre.- 6.2.1 Finding the shear centre.- 6.3 Unsymmetrical sections.- 6.3.1 Shear stress for an unsymmetrical section.- 6.3.2 Determining the shear centre.- 6.4 Closed sections.- 6.4.1 Determination of the shear stress distribution.- 6.5 Pure torsion of closed thin-walled sections.- 6.5.1 Torsional stiffness.- 6.5.2 Design considerations in torsion.- 6.6 Finding the shear centre for a closed section.- 6.6.1 Twist due to a shear force.- 6.6.2 Multicell sections.- 6.7 Torsion of thin-walled open sections.- 6.7.1 Loading of an open section away from its shear centre.- 6.8 Summary.- Problems.- 7 Beams on Elastic Foundations;7.1 The governing equation.- 7.1.1 Solution of the governing equation.- 7.2 The homogeneous solution.- 7.2.1 The semi-infinite beam.- 7.3 Localized nature of the solution.- 7.4 Concentrated force on an infinite beam.- 7.4.1 More general loading of the infinite beam.- 7.5 The particular solution.- 7.5.1 Uniform loading.- 7.5.2 Discontinuous loads.- 7.6 Finite beams.- 7.7 Short beams.- 7.8 Summary.- Problems.- 8 Membrane Stresses in Axisymmetric Shells.- 8.1 The meridional stress.- 8.1.1 Choice of cut.- 8.2 The circumferential stress.- 8.2.1 The radii of curvature.- 8.2.2 Sign conventions.- 8.3 Self-weight.- 8.4 Relative magnitudes of different loads.- 8.5 Strains and Displacements.- 8.5.1 Discontinuities.- 8.6 Summary.- Problems.- 9 Axisymmetric Bending of Cylindrical Shells.- 9.1 Bending stresses and moments.- 9.2 Deformation of the shell.- 9.3 Equilibrium of the shell element.- 9.4 The governing equation.- 9.4.1 Solution strategy.- 9.5 Localized loading of the shell.- 9.6 Shell transition regions.- 9.6.1 The cylinder/cone transition.- 9.6.2 Reinforcing rings.- 9.7 Thermal stresses.- 9.8 The ASME pressure vessel code.- 9.9 Summary.- Problems.- 10 Thick-walled Cylinders and Disks.- 10.1 Solution method.- 10.1.1 Stress components and the equilibrium condition.- 10.1.2 Strain, displacement and compatibility.- 10.1.3 The elastic constitutive law.- 10.2 The thin circular disk.- 10.3 Cylindrical pressure vessels.- 10.4 Composite cylinders, limits and fits.- 10.4.1 Solution procedure.- 10.4.2 Limits and fits.- 10.5 Plastic deformation of disks and cylinders.- 10.5.1 First yield.- 10.5.2 The fully-plastic solution.- 10.5.3 Elastic-plastic problems.- 10.5.4 Other failure modes.- 10.5.5 Unloading and residual stresses.- 10.6 Summary.- Problems.- 11 Curved Beams.- 11.1 The governing equation.- 11.1.1 Rectangular and circular cross sections.- 11.1.2 The bending moment.- 11.1.3 Composite cross sections.- 11.1.4 Axial loading.- 11.2 Radial stresses.- 11.3 Distortion of the cross section.- 11.4 Range of application of the theory.- 11.5 Summary.- Problems.- 12 Elastic Stability.- 12.1 Uniform beam in compression.- 12.2 Effect of initial perturbations.- 12.2.1 Eigenfunction expansions.- 12.3 Effect of lateral load (beam-columns).- 12.4 Indeterminate problems.- 12.5 Suppressing low-order modes ..- 12.6 Beams on elastic foundations.- 12.6.1 Axisymmetric buckling of cylindrical shells.- 12.6.2 Whirling of shafts.- 12.7 Energy methods.- 12.7.1 Energy methods in beam problems.- 12.7.2 The uniform beam in compression.- 12.7.3 Inhomogeneous problems.- 12.8 Quick estimates for the buckling force.- 12.9 Summary.- Problems.- A The Finite Element Method.- A.1 Approximation.- A.1.1 The 'best' approximation.- A.1.2 Choice of weight functions.- A.1.3 Piecewise approximations.- A.2 Axial loading ..- A.2.1 The structural mechanics approach.- A.2.2 Assembly of the global stiffness matrix.- A.2.3 The nodal forces.- A.2.4 The Rayleigh-Ritz approach.- A.2.5 Direct evaluation of the matrix equation.- A.3 Solution of differential equations.- A.4 Finite element solutions for the bending of beams.- A.4.1 Nodal forces and moments.- A.5 Two and three-dimensional problems.- A.6 Computational considerations.- A.6.1 Data storage considerations.- A.7 Use of the finite element method in design.- A.8 Summary.- Problems.- B Properties of Areas.- C Stress Concentration Factors.- D Answers to Even Numbered Problems.- Index.