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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 | |