Chapter
3.5 The ring-loaded long cylindrical shell
3.6 The band-loaded cylindrical shell
3.7 Cylindrical shell loaded sinusoidally along its length
3.7.1 A 'two-surface * interpretation of the behaviour of the shell
3.7.2 The use of Fourier series
3.7.3 The nonlinear effect of axial tension
3.8 Edge-loading of finite elastic cylindrical shells
4. Purely 'equilibrium' solutions for shells: the membrane hypothesis
4.2 A simple problem: the plane 'string'
4.3 Equilibrium equations for a doubly-curved shell
4.4 Equilibrium equations for cylindrical shells
4.4.1 Some simple examples
4.5 Axisymmetric loading of shells of revolution
4.5.1 Equations of equilibrium
4.6 Equilibrium equations for nearly-cylindrical shells
4.7 Boundary conditions in membrane analysis
4.7.1 A 'framework' analogy
5. The geometry of curved surfaces
5.2 The idea of a surface
5.3 General properties of plane curves
5.4 Curvature of a surface in terms of the geometry of crosssections: principal curvatures, etc.
5.5 Gaussian curvature: an intrinsic view of surfaces
5.5.1 The solid angle subtended by the vertex of a 'roof
5.5.2 Th e curva ture of a polygonalised surface
5.6 Inextensional deformation of surfaces
5.7 Nontriangular polygonalisation of surfaces
6. Geometry of distortion of curved surfaces
6.2 'Change of Gaussian curvature' in terms of surface strain
6.3 Connection between the two aspects of Gaussian curvature
6.4 Strain-displacement relations for a cylindrical shell
6.5 Inextensional deformation of a cylindrical surface
6.5.2 Remarks on boundary conditions
6.6 Strain-displacement relations for nearly-cylindrical surfaces
6.7 General remarks on the inextensional deformation of shells
6.8 Compatibility between surface strain and change of curvature in curvilinear coordinates
6.8.1 Gaussian curvature of original surface in terms of Lame parameters
6.8.2 Change of Gaussian curvature due to surface strain
6.9 Symmetrical deformation of shells of revolution
6.9.1 Strain-displacemen t relations
7. Displacements of elastic shells stressed according to the membrane hypothesis
7.2 Testing the validity of the membrane hypothesis
7.3 Cylindrical shell with doubly-periodic pressure-loading
7.3.1 Panel flexibility factor
7.3.2 Criteria for validity of membrane hypothesis
7.3.3 Periodic tangen tial loading
7.3.4 A different set of boundary conditions
7.4 Use of the Airy stress function in the calculation of deflections in a cylindrical shell
7.5 A 'beam analogy' for 'quasi-inextensional' deformation of cylindrical shells
8. Stretching and bending in cylindrical and nearly-cylindrical shells
8.2 The 'two-surface' idealisation: equilibrium equations
8.3 Response of the 'bending surface' to pressure-loading
8.4 Cylindrical shell subjected to a doubly-periodic pressureloading
8.5 DonnelPs equations for nearly-cylindrical shells
8.5.1 Roots of DonnelVs equation
8.5.2 Physical interpretation of the roots
8.6 The improvement of Donnell's equations
8. 7.1 Separation of boundary conditions for the long- and short-wave solutions
8. 7.2 Twisting moments Mxy applied at an edge
8.7.3 Displacement boundary conditions
8.7.4 'Flat-plate' region
8.8 Summary and discussion
9. Problems in the behaviour of cylindrical and nearlycylindrical shells subjected to non-symmetric loading
9.2 A preliminary example: a cylindrical shell acting as a beam
9.3 Beam on elastic foundation
9.4 Long-wave solution: the formal 'beam' analogy
9.5 Cylindrical shell with one edge free and the other edge clamped
9.5.1 Other simple boundary conditions
9.5.2 Shell with thick flat-plate closure
9.6 Cylindrical shell loaded by radial point forces
9.7 Concentrated load on a spherical shell
10. Cylindrical shell roofs
10.2 A simple cylindrical shell roof
10.2.1 A preliminary analysis, treating the shell as a beam
10.2.2 A more refined calculation
10.2.3 Another mode of displacement
10.2.4 Continuity with adjacent shells
10.3 The effect of edge-beams: a simple example
10.3.1 The effect of edge-beams: more general analysis
10.3.2 Maximum tensile stress as a function ofFx and F2
10.3.3 Sharing of applied load between shell and edge-beams
11. Bending stresses in symmetrically-loaded shells of revolution
11.2 Equations of the problem
11.2.1 Equilibrium equations
11.2.2 Equations of kinematics
11.2.3 Generalised Hooke 's law
11.2.4 Governing equations
11.2.5 Geckeler's simp lifica tion
11.3 The effects of an 'imperfect' meridian
11.3.1 A 'change of slope'imperfection
11.3.2 Periodic imperfections
11.3.3 A 'change of curvature' imperfection
11.3.4 Clusters of imperfections
11.3.5 Moderation of a 'change of slope' imperfection
11.4 Some pressure-vessel junction problems
11.4.1 Spherical closure of a cylindrical vessel
11.4.2 Torispherical pressure-vessel heads
11.4.3 Stress-concentration factors
11.4.4 Cylindrical branches in spherical pressure-vessels
11.4.5 The analysis of branches in shallow shells
11.5 Reconciliation of the present scheme with the 'two-surface' approach
12. Flexibility of axisymmetric bellows under axial loading
12.2 Analysis of flexibility by an energy method
12.2.1 Geometry of deformation
12.2.2 Expressions for strain energy and flexibility
12.3 Comparison with previous work
12.4 Approximate analysis of strain in bellows
13. Curved tubes and pipe-bends
13.2 A curved tube subjected to internal pressure
13.3 Pure bending of a curved two-flange beam
13.3.1 Alternative derivation by means of complementary energy
13.4 Pure bending of a curved tube
13.4.1 A more complete treatment
13.4.2 Peak valu es of s tress
13.5 The effect of internal pressure
13.6 End-effects in the bending of curved tubes
13.6.1 Examples of join ts with flanges
14. Buckling of shells: classical analysis
14.1 Buckling of structures
14.2 Eigenvalue calculations according to the 'two-surface' model
14.2:1 A simple pin-ended column
14.2.2 Eigenvalue calculations for flat plates
14.2.3 Eigenvalue calculations for cylindrical shells
14.3 Cylindrical shell under uniform axial compression
14.3.2 Doubly-periodic modes
14.3.3 Boundary conditions
14.4 Axial compression and 'side' pressure combined
14.5 Necessary corrections for small values of circumferential wavenumber n
14.6 The effect of clamped and other boundary conditions
14.6.1 Buckling of oil-storage tanks
14.7 Buckling of cylindrical shells in torsion
14.7.1 Buckling of a 'long' cylindrical shell
14.7.2 Torsional buckling of shells of finite length: an energy method
14.8 Experimental observations
14.9 A simple design problem
14.10 Unidirectionally reinforced shells
14.11 A special boundary condition
15. Buckling of shells: non-classical analysis
15.2 A simple model for the study of buckling
15.2.1 The to tal po ten tial energy function
15.2.2 The introduction of an imperfection
15.3 A re-examination of the 'classical' calculation
15.4 A nonlinear analysis of buckling
15.4.1 The two-mode calculation
15.4.2 Equilibrium paths for imperfect shells
15.5 Distribution of tangential-stress resultants during buckling
15.6 Nonlinear behaviour of the S-surface
15.8 Axial buckling of pressurised cylindrical shells
15.9 Nonlinear effects in the buckling of cylindrical shells under pure torsion
16. The Brazier effect in the buckling of bent tubes
16.2 The Brazier effect in a simple beam
16.3 The Brazier effect in a tube of circular cross-section
16.5 The effect of interior pressure
16.6 The effect of finite length
16.7 An improvement on Brazier's analysis
17. Vibration of cylindrical shells
17.2 Vibrations of a simple ring
17.2.1 Extensional vibrations
17.2.2 Inextensional (bending) vibrations
17.2.3 A finite-thickness effect
17.2.4 Standing and travelling waves
17.3 Vibration of a cylindrical shell in 'shallow' modes
17.4 Low-frequency approximations
17.5.1 Free vibration of a beam on an elastic foundation
17.6 Natural frequencies for cylindrical shells having different boundary conditions
18. Shell structures and the theory of plasticity
18.1.1 Plastic theory of structures
18.1.2 Elastic and plastic philosophies of structural design
18.1.3 The general equations of plastic theory
18.2 Cylindrical shell subjected to axisymmetric loading applied at an edge
18.3 Upper-and lower-bound theorems
18.4 Cylindrical shell subjected to axisymmetric band-loading
18.5 Lower-bound analysis of axisymmetric pressure-vessels
18.5.1 A shallow-shell example
18.6 Use of lower-bound theorem to design reinforcement for pressure-vessels
18.6.1 A physical argumen t
18.7 An upper-bound method
18.7.1 Application to a boss-loaded shell
18.7.2 A calculation for a 'sandwich' shell
18.7.3 Calculation for changing geometry
1. Theorems of structural mechanics
I The principle of virtual work
II The theorem of minimum complementary energy
III The theorem of minimum strain energy
IV The theorem of minimum total potential energy
2. 'Corresponding'load and deflection variables
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
11. Suggestions for further reading
Answers to selected problems
Theory of Shell Structures
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