Local loads in plates and shells

Thin walled structures so extensively used nowadays in industry and civil engineering are usually loaded by very complex systems of forces acting on their edges or over their surfaces. In calculating the strength of a structure we replace real loads by certain idealized loads distinguishing between typical surface loads distributed over a great area of the structure and loads acting over a small area. The latter are called concentrated loads. When the area under the load is very small in comparison with the dimensions of the surface of the structure, for example, when the diameter of the loaded area is smaller than the wall thickness, the load can be considered as a single force or a mo­ ment acting on the structure at one point only. The real loads which are met in practice can always be replaced by a combination of components such as forces normal and tangential to the wall as well as bending and twisting moments. Knowing the distribution of the stresses in the structure produced by each component, we can find it under any arbitrary load using the principle of superposition. There are two main reasons for the appearance of the concentration of stresses in the structure. It can be produced by notches, rapid changes of the cross-section, holes, cutouts, etc. on one hand and by concentrated loads resulting from the interaction of the elements of the structure on the other.

1 Basic equations of the theory of plates and shells.- 1.1 A short introduction to the tensor calculus.- 1.2 Strains and displacements of a three-dimensional body.- 1.3 Stresses and equilibrium equations of a three-dimensional body.- 1.4 Fundamental assumptions in the theory of plates and shells.- 1.5 Some fundamental results from the theory of surfaces.- 1.5.1 First metric tensor.- 1.5.2 The physical components.- 1.5.3 The metric tensor of the curvature.- 1.6 The geometry of the shell.- 1.7 Deformation of the shell.- 1.8 Constitutive equations, equations of equilibrium.- 1.9 Effect of transverse shear and normal stresses.- 1.10 Compatibility equations.- 1.11 Simplified shell equations.- 1.11.1 Shells of slowly varying curvatures.- 1.11.2 Relations in rectangular coordinates.- 1.11.3 Polar coordinates.- 1.12 Classical theory.- 1.12.1 Shells of slowly varying curvature.- 1.12.2 Shallow shells.- 1.13 Loads tangential to middle surface.- 1.14 The strain energy.- 1.15 Thermal stresses.- 1.16 Orthotropic shells.- 1.16.1 Cylindrical shells. Equations of the classical theory.- 1.16.2 Shallow shells.- 1.16.3 Equations in polar coordinates.- 1.17 The boundary conditions.- 1.18 Sandwich shells.- References 1.- 2 Fundamental equations of plates.- 2.1 Differential equations of isotropic plates.- 2.2 Linear equations.- 2.3 The strain energy of the plate.- 2.4 Variational equations.- 2.5 Boundary conditions for plates.- 2.6 Classical theory of plates under lateral loads.- 2.7 Sandwich plates.- References 2.- 3 Plates under lateral loads at an interior point.- 3.1 Methods of solution.- 3.2 Isotropic circular plate loaded by a concentrated force at its centre.- 3.3 Circular plate with cylindrical orthotropy loaded by a concentrated force.- 3.4 Solutions by means of Fourier series and Fourier integrals.- 3.4.1 Introduction to the theory of distributions.- 3.4.2 Sequence approach.- 3.4.3 Functional approach.- 3.4.4 Methods of integration.- 3.5 Navier solution for the rectangular plate subjected to a concentrated load.- 3.6 Particular solution accomplished by means of Fourier integral.- 3.6.1 Infinite plate.- 3.6.2 Semi-infinite plate and wedge plate.- 3.7 Thick circular plate.- 3.7.1 Equations of the theory of elasticity of the symmetrical body.- 3.7.2 The stresses in a semi-infinite elastic body.- 3.7.3 Axi-symmetrically loaded thick plate.- 3.8 Plate on an elastic foundation.- 3.9 Infinite plate loaded by a concentrated bending moment.- 3.9.1 Moment of two normal forces.- 3.9.2 Concentrated bending moment.- 3.10 Bending moment introduced through a rigid insert.- 3.11 Thermal singularities.- 3.12 Influence surfaces and the Green functions.- 3.12.1 Influence surfaces for the deflection.- 3.12.2 Influence surfaces for the internal moments.- 3.12.3 Singularity and the influence surface for the edge moment of a clamped plate.- 3.13 Plates of various shapes under lateral loads.- 3.13.1 Method of images.- 3.13.2 Singular solutions for semi-infinite and wedge plates.- 3.14 Singularities in the orthotropic plate.- 3.14.1 Solution by means of Fourier integral.- 3.14.2 Solution in a double Fourier series.- 3.14.3 Infinitely long strip plate. Solution by means of single trigonometric series.- 3.15 Singular solutions for sandwich plates.- 3.15.1 Normal force.- 3.15.2 Concentrated bending moment.- References 3.- 4 Concentrated lateral loads at the edge of a plate.- 4.1 Cantilever plate loaded at its free edge by a lateral concentrated force.- 4.2 Plate loaded by a concentrated moment at its free edge.- 4.3 Plate on an elastic foundation loaded at its free edge by a concentrated force.- References 4.- 5 Plates loaded only in their middle plane.- 5.1 General equations.- 5.2 Infinite plate loaded at an interior point by a concentrated force.- 5.3 Circular plate loaded by a tangential load.- 5.4 Infinite orthotropic plate loaded at an interior point.- 5.5 Plates loaded at their edges.- 5.5.1 Concentrated force normal to edge.- 5.5.2 Force tangential to edge.- 5.5.3 Wedge loaded by concentrated force at its apex.- 5.5.4 Wedge loaded by concentrated moment at its apex.- 5.6 Orthotropic plate loaded at free edge.- 5.7 Plates of various shapes under loads in their plane.- References 5.- 6 Plates under lateral loads and loads in the middle plane.- 6.1 Large elastic deflections of a plate.- 6.2 Circular plate.- 6.3 Stability of plates.- 6.3.1 Method of differential equations.- 6.3.2 Energy method.- 6.4 Rectangular plate loaded by compressive forces.- 6.5 Stability of circular plate.- References 6.- 7 Membrane shells under concentrated loads.- 7.1 Shells of revolution.- 7.2 Spherical shell.- 7.3 Conical shell.- References 7.- 8 Spherical shell. Solutions of the general theory.- 8.1 Spherical shell loaded by a concentrated normal force.- 8.2 Spherical shell loaded by a concentrated moment.- 8.3 Spherical shell loaded by a concentrated tangential force.- 8.4 Spherical shell subjected to a twisting moment.- 8.5 The effect of boundary conditions. General solution.- 8.5.1 Shallow shell.- 8.5.2 Non-shallow spherical shell.- References 8.- 9 An arbitrary shell loaded by a normal concentrated force.- 9.1 Deflection and stress function.- 9.2 Stresses in the shell.- 9.3 Solution in polar coordinates.- 9.3.1 Deflection and stress function.- 9.3.2 Internal forces and moments.- 9.3.3 Nearly spherical shell.- 9.4 Load distributed over small surface.- 9.5 Experimental investigation.- 9.6 Simplified solution. Division of loads.- 9.7 Solutions for local load. Shell loaded by a normal concentrated force.- 9.8 The effect of variability of the shell curvature.- 9.9 Method of singular solutions.- 9.10 Load distributed along a small length.- 9.11 Experimental investigation.- 9.12 Conclusions.- References 9.- 10 Shells of cylindrical and nearly cylindrical shape subjected to normal force.- 10.1 The nearly cylindrical shell loaded by two oppositely directed radial forces.- 10.2 Stresses in nearly cylindrical shells.- 10.3 Cylindrical shells.- 10.3.1 General equations of the cylindrical shell.- 10.3.2 Infinitely long shell loaded by two oppositely directed radial forces.- 10.3.3 The cylindrical segment of infinite length.- 10.4 Shallow cylindrical shells.- 10.4.1 Singularities of shallow cylindrical shells.- 10.4.2 Shallow cylindrical shell subjected to hot spot.- 10.5 Effect of boundary conditions.- 10.5.1 General solution of homogeneous equations.- 10.5.2 Shell with simply supported edges.- 10.5.3 The other variant of the solution.- 10.5.4 Solution in double trigonometric series.- 10.5.5 Cylindrical shell with free edges.- 10.6 Cylindrical orthotropic shell.- 10.6.1 Shell with simply supported edges.- 10.6.2 Infinitely long shell.- References 10.- 11 Shells under various concentrated loads.- 11.1 Shell loaded by a concentrated bending moment.- 11.2 The cylindrical shell loaded by a moment distributed over a small surface.- 11.3 Shell loaded by a force tangential to its surface.- 11.3.1 Local loads.- 11.3.2 Solution in polar coordinates.- 11.3.3 The cylindrical shell loaded by a tangential force.- 11.4 Asymptotic relations.- References 11.- 12 Edge loads.- 12.1 Shell loaded at free edge by a lateral force.- 12.1.1 Solution in Fourier integrals.- 12.1.2 Local load.- 12.1.3 Solution by means of Fourier series.- 12.2 Shell loaded by a force normal to the edge and tangential to the middle surface.- 12.3 The shell loaded at its edge by a concentrated bending moment.- References 12.- 13 Large deflections of shells.- 13.1 Spherical shell.- 13.2 Arbitrary shell of positive double curvature.- References 13.- 14 Design of plates and shells under concentrated loads.- 14.1 Infinitely large plate jointed to a long bar and loaded at an interior point.- 14.2 Strip plate jointed with a stringer of finite length.- 14.3 The bar of finite length jointed with an infinite plate.- 14.4 Optimum design of elements introducing the load.- 14.4.1 Bar of variable cross-section.- 14.5 Plate strengthened at the edge by a bar with bending rigidity.- 14.6 An infinite rigid body.- 14.7 A bar of variable rigidity.- 14.8 A plate strengthened by a semi-infinite bar.- 14.9 Optimum design of plates under concentrated forces acting in its middle plane.- 14.9.1 Elastic design.- 14.9.2 Plastic design of plates.- 14.9.3 Design for minimum weight.- 14.9.4 Support for a single force.- 14.10 Design of circular plates under lateral loads.- 14.10.1 Plastic design.- 14.10.2 Circular plate of uniform strength loaded by a lateral concentrated force.- 14.11 Optimum design of shells.- 14.12 Basic equations of the problem.- 14.13 Shells of constant thickness.- 14.14 Design of a shell of variable thickness.- 14.14.1 Shell loaded by a concentrated force and by a pressure.- 14.14.2 Shell carrying its own weight.- 14.15 Experimental design of the shell.- 14.15.1 Designing with liquid analogy.- 14.15.2 Designing in the plastic state.- References 14.- Appendix. Certain remarks on the problem of a concentrated force acting at the edge of the plate.- Tables.- Author index.