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Preface | |

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Basic concepts | |

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Introductory concepts of solid mechanics | |

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Introduction | |

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Forces, loads and reactions - idealisations | |

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Types of loads | |

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Representation of forces and loads | |

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Reactions and constraints - idealisations | |

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Intensity of internal forces - average stresses | |

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Intensity of a normal force acting over an area - refinement of the concept: normal stress at a point | |

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Average stresses on an oblique plane | |

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Variation of internal forces and stresses with position | |

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Strain as a measure of intensity of deformation | |

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Mechanical behaviour of materials | |

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Summary | |

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Problems | |

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Internal forces and stress | |

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Introduction | |

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Internal force resultants | |

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State of stress at a point: traction | |

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Traction | |

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Sign convention | |

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The stress tensor | |

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Equality of the conjugate shear stresses | |

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Stress equations of motion and equilibrium | |

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Relations between stress components and internal force resultants | |

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Stress transformation laws for plane stress | |

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Derivation | |

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Remarks on the transformation laws (stress as a tensor; invariants of a tensor) | |

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Transformation law of a vector: the vector as a tensor | |

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Principal stresses and stationary shear stress values | |

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Principal stresses: stationary values of [sigma subscript n] | |

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Maximum and minimum shear stress components | |

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Summary of results | |

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Parametric representation of the state of stress: the Mohr circle | |

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Cartesian components of traction in terms of stress components: traction on the surface of a body | |

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Problems | |

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Deformation and strain | |

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Introduction | |

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Types of deformation | |

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Extensional or normal strain | |

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Shear strain | |

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Strain-displacement relations | |

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Some preliminary instructive examples | |

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Strain-displacement relations for infinitesimal strains and rotations | |

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State of strain | |

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Two-dimensional transformation law for infinitesimal strain components | |

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Geometric derivation | |

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Analytic derivation of the transformation laws | |

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The infinitesimal strain tensor - two-dimensional transformation laws | |

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Principal strains and principal directions of strain: the Mohr circle for strain | |

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The strain rosette | |

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Volumetric strain - dilatation | |

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Problems | |

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Behaviour of materials: constitutive equations | |

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Introduction | |

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Some general idealisations (definitions: 'micro' and 'macro' scales) | |

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Classification of materials: viscous, elastic, visco-elastic and plastic materials | |

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Elastic materials | |

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Constitutive equations for elastic materials: general elastic and linear elastic behaviour, Hooke's law | |

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Elastic strain energy | |

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Mechanical properties of engineering materials | |

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Behaviour of ductile materials | |

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Behaviour of brittle materials | |

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Behaviour of rubber-like materials | |

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Plastic behaviour: idealised models | |

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Problems | |

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Summary of basic results and further idealisations: solutions using the 'mechanics-of-materials' approach | |

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Introduction | |

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Superposition principles | |

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Superposition of infinitesimal strains | |

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Basic principle of superposition for linear elastic bodies | |

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The principle of de Saint Venant | |

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Applications to simple elements | |

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Axial loadings | |

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Introduction | |

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Elastic behaviour of prismatic rods: basic results | |

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Some general comments | |

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Extension of results: approximations for rods having varying cross-sections | |

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Statically indeterminate axially loaded members | |

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Temperature problems: thermal stresses | |

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Elastic-plastic behaviour: residual stresses | |

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Problems | |

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Torsion of circular cylindrical rods: Coulomb torsion | |

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Introduction | |

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Basic relations for elastic members under pure torsion | |

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Deformation analysis: conclusions based on axi-symmetry of the rod | |

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Basic relations | |

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Some comments on the derived expressions: extension of the results and approximations | |

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Comments on the solution | |

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An approximation for thin-wall circular tubular cross-sections | |

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Extension of the results: engineering approximations | |

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Some practical engineering design applications of the theory | |

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Circular members under combined loads | |

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Statically indeterminate systems under torsion | |

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Elastic-plastic torsion | |

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Problems | |

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Symmetric bending of beams - basic relations and stresses | |

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Introduction | |

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Resultant shear and bending moments - sign convention | |

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Some simple examples | |

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Sign convention | |

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Differential relations for beams | |

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Some further examples for resultant forces in beams | |

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Integral relations for beams | |

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Symmetrical bending of beams in a state of pure bending | |

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Some preliminary definitions and limitations - deformation analysis | |

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Moment-curvature relations and flexural stresses in an elastic beam under pure bending: Euler-Bernoulli relations | |

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Axial displacements of beams under pure bending | |

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Comments on the solution - exactness of the solution | |

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Methodology of solution - the methodology of 'mechanics of materials' | |

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Flexure of beams due to applied lateral loads - Navier's hypothesis | |

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Shear stresses in beams due to symmetric bending | |

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Derivation | |

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Limitations on the derived expression | |

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Shear effect on beams - warping of the cross-sections due to shear | |

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Re-examination of the expression for flexural stress [sigma subscript x] = My/I: further engineering approximations | |

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Examination of equilibrium state | |

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Flexural stress in a non-prismatic beam - an engineering approximation | |

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Engineering design applications for beams | |

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Bending of composite beams | |

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Combined loads | |

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Elastic-plastic behaviour | |

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Fully plastic moments - location of the neutral axis | |

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Moment-curvature relation for beams of rectangular cross-section in the plastic range | |

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Problems | |

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Symmetric bending of beams: deflections, fundamental solutions and superposition | |

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Introduction | |

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Linearised beam theory | |

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Accuracy of the linearised beam theory | |

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Elastic curve equations for some 'classical' cases | |

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Axial displacements due to flexure of a beam under lateral loads | |

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Deflections due to shear deformation | |

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Singularity functions and their application | |

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Definition of singularity functions | |

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Applications | |

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Solutions for statically indeterminate beams by integration of the differential equation | |

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Application of linear superposition in beam theory | |

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Analysis of statically indeterminate beams: the force method | |

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Development of the force method | |

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Comments on the force method | |

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Superposition - integral formulation: the fundamental solution and Green's functions | |

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Development and applications | |

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Generalisation: Green's functions for shears, moments, etc. in beams | |

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Some general comments | |

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The fourth-order differential equation for beams | |

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Development and applications | |

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The fourth-order differential equation for concentrated force and couple loadings | |

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Moment-area theorems | |

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Problems | |

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Thin-wall pressure vessels: thin shells under pressure | |

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Introduction | |

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Thin cylindrical shells | |

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Thin spherical shells | |

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Comments and closure | |

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Problems | |

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Stability and instability of rods under axial compression: beam-columns and tie-rods | |

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Introduction | |

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Stability and instability of mechanical systems | |

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Stability of rigid rods under compressive loads: the concept of bifurcation | |

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Stability of an elastic rod subjected to an axial compressive force-Euler buckling load | |

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Elastic buckling of rods under various boundary conditions | |

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Rods under eccentric axial loads-the 'secant formula' | |

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Rods under combined axial and lateral loads: preliminary remarks | |

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Differential equations of beams subjected to combined lateral loads and axial forces | |

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Stability analysis using the fourth-order differential equation | |

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Beam-column subjected to a single lateral force F and an axial compressive force P | |

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Some comments on the solution: use of linear superposition | |

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Tie-rods | |

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General comments and conclusions | |

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Problems | |

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Torsion of elastic members of arbitrary cross-section: de Saint Venant torsion | |

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Introduction | |

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Semi-inverse methods: uniqueness of solutions | |

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The general de Saint Venant torsion solution | |

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Torsion of a member of elliptic cross-section | |

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Torsion of a member of rectangular cross-section | |

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The membrane analogy | |

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Torsion of a member having a narrow rectangular cross-section | |

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Derivation of membrane analogy solution | |

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Comparison of exact solution with membrane analogy for narrow rectangular sections | |

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Torsion of thin-wall open-section members | |

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Shear stress at a re-entrant corner: approximate solution | |

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Torsion of closed-section members: thin-wall sections | |

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Torsion of multi-cell closed thin-wall sections | |

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Closure | |

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Problems | |

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General bending theory of beams | |

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Introduction | |

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Moment-curvature relation for elastic beams in flexure | |

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Sign convention and beam equations for bending about two axes | |

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Sign convention | |

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Differential beam equations | |

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General expression for stresses due to flexure | |

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Derivation: stresses in beams under pure bending | |

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Extension of expression for flexural stress in beams due to applied lateral loads | |

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Some particular cases | |

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General case | |

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Shear stresses due to bending of beams | |

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Derivation | |

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Comments on the expressions | |

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Distribution of shear stresses in a thin-wall section: shear centers | |

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Shear stress distribution | |

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The shear center | |

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Some remarks and comments | |

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Deflections and rotations of a beam under applied loads | |

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Shear stresses in closed thin-wall sections | |

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Problems | |

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Energy methods and virtual work | |

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Basic energy theorems, principles of virtual work and their applications to structural mechanics | |

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Introduction | |

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Elastic strain energy | |

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Review of results for the uniaxial state of stress | |

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General stress state | |

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Examples of strain energy for linear elastic bodies | |

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The principle of conservation of energy for linear elastic bodies | |

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Derivation of the principle | |

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Application of the principle | |

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Betti's law and Maxwell's reciprocal relation: flexibility coefficients | |

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Castigliano's second theorem | |

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Geometric representation (complementary strain energy and Castigliano's first theorem) | |

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The principle of virtual work | |

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Introduction | |

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Definitions of external and internal virtual work: virtual displacements | |

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Proof of the principle of virtual work: comments on the principle | |

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The principle of virtual work for flexure of beams | |

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Application of the principle of virtual work to evaluate reactions and internal stress resultants: the 'method of virtual displacements' | |

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Influence lines for reactions, shears and moments in beams by the principle of virtual work | |

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The principle of complementary virtual work | |

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Introduction | |

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Development and derivation of the principle | |

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Comparison and analogues between the two principles | |

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Expressions for internal complementary virtual work in terms of internal stress resultants: generalised forces and displacements | |

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Internal complementary virtual work in linear elastic rods and beams: explicit expressions (some generalisations) | |

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Application of the principle of complementary virtual work to evaluate displacements of linear elastic bodies: the 'method of virtual forces' | |

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The principle of stationary potential energy | |

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Derivation of the principle and some applications | |

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Approximate solutions-the Rayleigh-Ritz method | |

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Summary and conclusions | |

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Problems | |

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Stability of mechanical systems by energy considerations: approximate methods | |

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Introduction | |

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Classification of equilibrium states according to energy criteria | |

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Stability of a rigid rod subjected to a compressive axial force | |

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Determination of critical loads using a small deflection analysis-pseudo-neutral equilibrium | |

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The total potential for small displacements: reconsideration of the stability criteria | |

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Systems having several degrees-of-freedom-small displacement analysis | |

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Two-degree-of-freedom system | |

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n-Degree-of-freedom systems | |

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Stability of an elastic rod: the Rayleigh quotient | |

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The Rayleigh method for critical loads | |

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Development of the method | |

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Proof of the upper boundedness of the Rayleigh load (restricted proof) | |

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The Rayleigh-Ritz method for critical loads | |

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Problems | |

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Properties of areas | |

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General properties: centroids, first and second moments of areas | |

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Properties of selected areas | |

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Some mathematical relations | |

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Curvature of a line y = y(x) | |

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Green's theorem | |

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The divergence theorem (Gauss' theorem) | |

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The membrane equation | |

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Material properties | |

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Table of structural properties | |

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Reactions, deflections and slopes of selected beams | |

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Answers to selected problems | |

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Index | |