Book contents
- Frontmatter
- Contents
- Preface
- List of symbols
- 1 Introduction
- 2 Generalised Hooke's law for an element of a shell
- 3 Cylindrical shells under symmetric loading
- 4 Purely ‘equilibrium’ solutions for shells: the membrane hypothesis
- 5 The geometry of curved surfaces
- 6 Geometry of distortion of curved surfaces
- 7 Displacements of elastic shells stressed according to the membrane hypothesis
- 8 Stretching and bending in cylindrical and nearly-cylindrical shells
- 9 Problems in the behaviour of cylindrical and nearly-cylindrical shells subjected to non-symmetric loading
- 10 Cylindrical shell roofs
- 11 Bending stresses in symmetrically-loaded shells of revolution
- 12 Flexibility of axisymmetric bellows under axial loading
- 13 Curved tubes and pipe-bends
- 14 Buckling of shells: classical analysis
- 15 Buckling of shells: non-classical analysis
- 16 The Brazier effect in the buckling of bent tubes
- 17 Vibration of cylindrical shells
- 18 Shell structures and the theory of plasticity
- Appendices
- 1 Theorems of structural mechanics
- 2 ‘Corresponding’ load and deflection variables
- 3 Rayleigh's principle
- 4 Orthogonal functions
- 5 Force-like and stress-like loads
- 6 The ‘static-geometric analogy’
- 7 The area of a spherical polygon
- 8 The ‘sagitta’ of an arc
- 9 Rigidity of polyhedral frames
- 10 Fourier series
- 11 Suggestions for further reading
- Answers to selected problems
- References
- Index
2 - ‘Corresponding’ load and deflection variables
Published online by Cambridge University Press: 02 February 2010
- Frontmatter
- Contents
- Preface
- List of symbols
- 1 Introduction
- 2 Generalised Hooke's law for an element of a shell
- 3 Cylindrical shells under symmetric loading
- 4 Purely ‘equilibrium’ solutions for shells: the membrane hypothesis
- 5 The geometry of curved surfaces
- 6 Geometry of distortion of curved surfaces
- 7 Displacements of elastic shells stressed according to the membrane hypothesis
- 8 Stretching and bending in cylindrical and nearly-cylindrical shells
- 9 Problems in the behaviour of cylindrical and nearly-cylindrical shells subjected to non-symmetric loading
- 10 Cylindrical shell roofs
- 11 Bending stresses in symmetrically-loaded shells of revolution
- 12 Flexibility of axisymmetric bellows under axial loading
- 13 Curved tubes and pipe-bends
- 14 Buckling of shells: classical analysis
- 15 Buckling of shells: non-classical analysis
- 16 The Brazier effect in the buckling of bent tubes
- 17 Vibration of cylindrical shells
- 18 Shell structures and the theory of plasticity
- Appendices
- 1 Theorems of structural mechanics
- 2 ‘Corresponding’ load and deflection variables
- 3 Rayleigh's principle
- 4 Orthogonal functions
- 5 Force-like and stress-like loads
- 6 The ‘static-geometric analogy’
- 7 The area of a spherical polygon
- 8 The ‘sagitta’ of an arc
- 9 Rigidity of polyhedral frames
- 10 Fourier series
- 11 Suggestions for further reading
- Answers to selected problems
- References
- Index
Summary
A central notion in the concept of virtual work (appendix 1) is that both the external force and displacement quantities and the internal tension and elongation quantities are related to each other in the sense that the product of corresponding variables represents a quantity of work. If a single force P acts at a joint, the ‘corresponding’ measure of displacement of the joint is the component of the displacement in the (positive) direction of the line of action of the force. More generally, if the components of a force are specified, say U, V, W, in mutually perpendicular directions, the ‘corresponding’ displacements are the components of displacement u, v, w in the same directions; and the appropriate (scalar) work product is simply Uu + Vv + Ww.
We are not, however, limited to discussion of loads on structures in terms of force as such. A structure may be loaded by a couple, for which the corresponding displacement is an angle of rotation (measured in radians); or a pressure, for which the corresponding displacement is a ‘swept volume’; or a uniform line load, for which the corresponding displacement is a ‘swept area’.
In relation to internal variables we saw in appendix 1 that we must multiply the tension in a bar by the elongation in order to obtain the appropriate work quantity. For a uniform bar of length L and cross-sectional area A, precisely the same quantity would be obtained by evaluating σ∈V, where σ = T/A is the tensile stress, ∈ = e/L is the tensile strain and V = AL is the volume of the bar.
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- Theory of Shell Structures , pp. 721 - 722Publisher: Cambridge University PressPrint publication year: 1983