Small Elastic Deformations of Thin Shells

In the last decade or so the theory of shells has undergone a tremendous increase in development. Formerly a subject of interest only to a few special- ists and for which the literature was relatively smalI, the needs of structures for aerospace missions instigated a torrent of papers on all facets of the theory which also found application in the less glamorous earthbound shell struc- tures important in everyday life. Some idea of the rapidity of the development can be gained from the fact that a bibliography* completed in 1953 listed some 1455 books and papers as the sum total ofthe literature on shell theory to that date. Three years later, however, a supplementt added another 884 papers to the list, an increase of 60 per cent in that short period of time. The number of papers published since these listings has increased to an extent that does not bear contemplation. Obviously no single volume could contain all that constitutes the theory of shells and so this book is restricted to that portion of the theory associated with small deformations of elastic shells. Plastic deformations of shells, which is hardly developed, and nonlinear deformations and stability, which would require at least aseparate volume, are thus excluded. Even with this restriction, however, the present volume represents a long overdue compro- mise between completeness and finiteness. In making this compromise I have undoubtedly omitted discussions of many topics and references to many excellent papers which should have been included.

I The Linear Theory of Shells.- 1 Geometry of Surfaces.- 1.1 Geometric Relations for Surfaces.- 1.2 Lines of Curvature.- 1.3 Surfaces of Revolution.- 1.4 Parallel Surfaces.- 1.5 Small Deformations of a Surface.- 2 The Linear Theory of Transversely Rigid Shells.- 2.1 Fundamental Definitions.- 2.2 Equations of Three-Dimensional Elasticity.- 2.3 Assumptions of Shell Theory.- 2.4 Displacements and Strains.- 2.5 Stress Resultants and Equilibrium Equations.- 2.6 Statically Equivalent Force Systems.- 2.7 Overall Force-Strain Relations.- 2.8 The Principle of Virtual Work and Boundary Conditions.- 2.9 Strain Energy and the Principle of Minimum Potential Energy.- 2.10 Stress Functions.- 2.11 Transverse Shear and Normal Stresses.- Appendix Boundary Conditions at Edges which are Not Lines of Curvature.- 3 Simplifications of Shell Theory.- 3.1 Modification of Overall Force-Strain Relationships.- 3.2 Internal Stress Distribution.- 3.3 Approximate Reduction of Shell Theory: Vlasov’s Equations.- 3.4 Shallow Shells.- 3.5 Exact Reduction of Shell Theory for Zero Poisson’s Ratio (I) Complex Stress Resultants.- 3.6 Exact Reduction of Shell Theory for Zero Poisson’s Ratio (II) Complex Displacements.- 3.7 Approximate Reduction of Shell Theory for Nonzero Poisson’s Ratio.- 3.8 The Membrane Theory of Shells.- 3.9 Transformation of the Equations of Membrane Theory.- II Axisymmetric Deformations of Shells of Revolution.- 4 The Cylindrical Shell.- 4.1 General Relations for Cylindrical Shells.- 4.2 Edge-Loaded Cylindrical Shells.- (A) Semi-Infinite Cylinder.- (B) Symmetrically Loaded Finite Cylinder.- (C) Antisymmetrically Loaded Finite Cylinder.- (D) Finite Cylinder Loaded at One Edge.- 4.3 Particular Integrals for Surface-Loaded Cylindrical Shells.- 4.4 Some Solutions of Cylindrical Shell Problems.- (A) Compressed Cylinder with Rigid End Rings.- (B) Pressurized Cylinder with Rigid Movable End Plates.- (C) Effect of Restraint of Longitudinal Movement.- (D) Effect of End Plate Flexibility.- 4.5 Pressurized Cylinder with an Abrupt Thickness Change.- 4.6 Cylinder in a Rigid Collar.- 5 The Conical Shell.- 5.1 General Relations for Conical Shells.- 5.2 Solution of the Homogeneous Deflection Equation.- 5.3 Edge-Loaded Conical Shells.- (A) Semi-Infinite Cones.- (B) Complete Cones.- (C) Finite Conical Frustums.- 5.4 Particular Solutions of the Conical Shell Equations.- 5.5 Complete Conical Shell Subjected to External Pressure.- 6 The Spherical Shell.- 6.1 General Relations for Spherical Shells.- 6.2 A Particular Solution of the Bending Equations.- 6.3 Some Relations for the Complementary Solution.- 6.4 Solutions by Means of Legendre Functions.- 6.5 Shallow Spherical Shells.- 6.6 Closed Spherical Cap under Edge Loading.- 6.7 Tangent Cone Approximation.- 6.8 Particular Solutions for Spherical Shells.- 6.9 Complete Sphere Subjected to Internal or External Pressure.- 6.10 Spherical Shell with Concentrated Loads at the Poles.- 6.11 Spherical Cap Loaded at the Apex.- 6.12 Effect of Method of Load Application.- 7 Shells of Arbitrary Meridian.- 7.1 General Equations for Arbitrary Shells of Revolution.- 7.2 An Approximate Complementary Solution of the Bending Equations.- 7.3 The Variable ?.- 7.4 Approximate Complementary Solutions Valid at Points with Horizontal Tangents.- 7.4.1 The Ellipsoidal Shell.- 7.4.2 The Circular Toroidal Shell.- 7.5 Membrane Theory as a Particular Solution of the Bending Equations.- 7.6 Some Particular Solutions for Circular Toroidal Shells.- 8 Torsion and Circumferential Bending of Shells of Revolution.- 8.1 Torsion of Shells of Revolution.- 8.2 Circumferential Bending of Shells of Revolution.- III Asymmetrically Loaded Shells.- 9 Asymmetric Deformations of Spherical Shells.- 9.1 General Relations for Asymmetrically Loaded Spherical Shells.- 9.2 ‘Wind’ Loading of Spherical Shells.- 9.3 Edge-Loaded Spherical Shells: Inextensional Deformations and Membrane Stress States.- 9.4 Complex Variable Representation of Membrane Stress States.- 9.5 Edge-Loaded Spherical Shells: Mixed Bending-Stretching Solutions.- 9.6 Bending of a Spherical Shell by Moments at the Poles: Exact Solution.- 9.7 Bending of a Spherical Shell by Moments at the Poles: Shallow Shell Theory.- 9.8 Bending of a Spherical Shell by Tangential Loads.- 9.9 Some Other Problems of Edge-Loaded Spherical Shells.- 9.10 Surface-Loaded Spherical Shells.- 10 Asymmetric Deformations of Circular Cylindrical Shells.- 10.1 General Relations for Cylindrical Shells.- 10.2 Simplifications of the Equations for Radial and Edge Loading.- 10.3 Cylinders Loaded Along Circular Edges.- 10.4 End-Loaded Cantilevered Cylinder with a Rigid End Ring.- 10.5 Infinite Cylindrical Shell Loaded By Diametrically Opposed Concentrated Loads.- 10.6 Results for Some Other Concentrated Loads.- 10.7 Particular Solutions for Complete Cylindrical Shells.- 10.8 Edge-Loaded Cylindrical Strips.- 10.9 Infinite Cylindrical Strips.- 10.10 Stress Concentration Around Holes.- 10.10.1 Effect of a Hole on a Circular Cylinder under Uniform Tension.- 11 Results for Other Shells.- 11.1 ‘Wind’ Loading of Arbitrary Shells of Revolution.- 11.2 Conical Frustums Subjected to End Loads.- 11.3 Numerical Solution of the Bending Equations for Shells Of Revolution.- 11.4 Shallow Translational Shells.- 11.5 Finite Element Analysis of Shells of Arbitrary Middle Surface.- 12 Nonuniform Anisotropic Shells.- 12.1 Necessity for Modifications of the Theory of Shells.- 12.2 Effect of Thermal Loading.- 12.3 Effect of Nonuniform Wall Thickness.- 12.3.1 Derivation of Equations for Shells of Nonuniform Wall Thickness.- 12.3.2 Axisymmetric Deformations of Nonuniformly Thick Shells of Revolution.- 12.3.3 Cylindrical Shell of Linearly Varying Thickness.- 12.3.4 Conical Shell of Linearly Varying Thickness.- 12.4 Effects of Anisotropy.- 12.5 Layered Shells.- 12.6 Stiffened Shells.- IV Dynamics of Shells.- 13 Free Vibrations of Shells.- 13.1 Equations of Motion.- 13.2 Free Harmonic Vibrations.- 13.3 Orthogonality Conditions for Vibration Mode Shapes.- 13.4 Free Vibrations of Cylindrical Shells.- 13.4.1 Simply Supported Cylindrical Shells.- 13.4.2 Minimum Frequency of Vibration.- 13.4.3 Effects of Other Edge Conditions.- 13.5 Free Vibrations of Spherical Shells.- 13.5.1 Vibrations of Spherical Shell Segments.- 13.5.2 Torsional Vibrations of Spherical Caps.- 13.5.3 Approximate Solutions for Shallow Spherical Caps.- 13.5.4 Vibrations of Complete Spherical Shells.- 13.6 Vibration Characteristics of Conical Shells.- 14 Response of Shells to Dynamic Loading.- 14.1 Solution of Initial Value Problems.- 14.2 Solution of Problems Involving Time-Dependent Surface or Edge Loading.- 14.3 Periodic Surface and Edge Forces.- 14.4 Solution of Problems Involving Time-Dependent Edge Displacements.- 14.5 Some Examples of Modal Analysis.- 14.6 Other Methods of Solution of Shell Response Problems.- Author Index.