I. Yu. Khoma

Stress-strain state of non-thin plates and shells. Generalized theory (survey)

ConclusionsThus, this part of our survey has presented the main approaches that have been taken to the construction of two-dimensional (in terms of the space coordinates) equations of a generalized theory of plates and shells. The solutions of these equations represent a certain approximation of the solution of the initial three-dimensional problem. They are based on expansion of the sought functions into Fourier series in Legendre polynomials of the thickness coordinate. Studies completed on the basis of the given variants of plate and shell theory were systematized and analyzed. In terms of the method of its construction, the theory involves a regular process of replacing the solution of the three-dimensional problem by the solution (or sequence of solutions) of two-dimensional boundary-value problems or initial-boundary-value problems. Numerical results illustrating the convergence of the successive approximation were presented. It should be noted that to make comparison with the results of classical or applied theories, several of the studies cited here presented solutions of problems for thin plates and shells with allowance only for the initial terms of expansions of the stress and displacement components into base functions (Legendre polynomials).

On One Technique of Constructing the General Solution to Equilibrium Equations for Nonthin Plates

The mathematical theory of plates based on the expansion of functions into Fourier series in terms of Legendre polynomials is used to state a method for determining the general solution to a system of equilibrium equations describing the stress–strain state of nonthin transversally isotropic plates. The state is assumed symmetric about the median plane

Stress-deformation state of thick shells and plates. Three-dimensional theory (review)

ConclusionsIn the first part of the present review we surveyed systematically published results of investigations of the stress-deformation state of thick-walled spheres, ellipsoids, cones, circular cylinders, as well as thick slabs, obtained by exact analytic solutions of spatial problems of elasticity theory. Several quantitative results were given of the variation of displacements and stresses with shell or plate thickness, and their comparative analysis was provided, making it possible to establish the validity limits of the corresponding applied theories. We also surveyed systematically published specific results of the spatial stress-deformation state of nearly canonical thick-walled shells, as well as non-thin plates of varying thickness, obtained by effective approximate analytic methods and known exact solutions for the corresponding canonical regions. Especially noted were characteristic mechanical (including boundary) effects on the stress-deformation state of the bodies under consideration. These effects are generated, in particular, by variations in the radius of curvature of the surface, the thickness parameter, the amplitude and frequency of the corrugated surface, material, inhomogeneity, conditions of mechanical contact between layers, the nature of self-balancing loads, and other factors.However, the possibilities of exact and effective approximate analytic solution of boundary value problems of this class in the three dimensional statement are restricted. In the case of shells and plates of mean thickness these results can be substantially supplemented by qualitative and quantitative data, obtained on the basis of analytic solutions in the generalized theory of shells and plates, based on expansions of components of the stress-deformation state in Legendre polynomial series, and making it possible, in principle, to approximate the three-dimensional solution with any required accuracy. This is one of the basic features distinguishing it from the classical and applied theories of shells and plates. The second part of this review will be devoted to systematic and comparative analysis of the results of investigations carried out within the generalized theory of non-thin shells and plates.

Integration of the Equilibrium Equations for Inhomogeneous, Transversely Isotropic Plates

A system of differential equilibrium equations for inhomogeneous transversely isotropic plates is derived based on the Fourier series in terms of Legendre polynomials. It is assumed that Poissons ratios are constant and the elastic moduli are linear functions of the transverse coordinate. A method of finding the general solution to the system of equations derived is set forth

Bending of nonthin transversally isotropic plates with curvilinear openings

The problem on the spatial stress–strain state of a nonthin transversally isotropic plate with a curvilinear nearly circular opening is considered. The plate is assumed to be subject to bending moments on the opening contour and at infinity. To solve the problem, it is proposed to make mutual use of two well-know methods — expansion of the displacement and stress components into Fourier series in Legendre polynomials of the thickness coordinate and boundary-shape perturbation. The equations and recurrent relations necessary for solution of the problem in an arbitrary approximation are presented

Determination of the stressed-strained state of nonthin transversally isotropic spherical shells with elastically stiffened curvilinear holes

An approximate analytical method has been developed for determining the stressed-strained state of nonthin transversally isotropic shallow spherical shells with noncircular elastic inclusions or stiffened holes, under both ideal and nonideal contact at the interface. The method is based on two tried and tested analytical methods—the method of Fourier expansion of the required functions using Legendre polynomials and the second version of the perturbation method for the boundary shape.

Some Nonlinear Relations of the Mathematical Theory of Nonthin Shells

The expansion of functions into Fourier series in terms of Legendre polynomials is used to state some relations of the geometrically nonlinear theory of nonthin anisotropic shells with a variable thickness. A system of equilibrium equations and corresponding boundary conditions are constructed

On the representation of the solutions of the equilibrium equations of a piezoelectric transversally-isotropic spherical shell

A method is proposed for constructing equilibrium equations for thickness-polarized transversally isotropic piezoceramic shells. The method is based on Fourier expanding the required functions in Legendre polynomials. The appropriate system of differential equations is formed for the expansion coefficients as functions of two independent variables. The equilibrium equations are given in particular for transversally isotropic spherical shells. A method is given for constructing the general solution in the first approximation.

Generalized thermoelastic equations for an anisotropic shell

Generalized thermoelastic equations are derived for an anisotropic shell. In deriving the equations, it is assumed that the heat flux g{sup i} (i = 1,2,3) in the body does not build up instantaneously, as in the classical theory, but is characterized by a finite relaxation time. Then the law of heat conduction take the form lg{sup i}=-{lambda}{sup ij}{delta}{sub j}{theta}, where {delta}{sub j} is the symbol for the covariant derivatives of the space, {lambda}{sup ij} is the heat conduction tensor, {theta} = T-T{sub 0}, where T is the absolute temperature and T{sub 0} is the temperature of the natural state of the body, l is an operator in which 5{sub r} is the relaxation time of the heat flux. The shell is treated as a three-dimensional body and the orthogonal curvilinear coordinate system x{sup i}(i=1,2,3) fixed to a reference surface G located inside the body is defined. 12 refs.

Representation of Solutions of the Deflection Equilibrium Equations for Thick Transversely Isotropic Plates

A system of equilibrium equations for the deflection of thick transversely isotropic plates with constant thicknesses is derived by expanding the stress and displacement components as Fourier series in Legendre polynomials of the thickness coordinate. A method for constructing a general solution of this system is described. It is made up of three types of solutions: biharmonic, potential, and rotational.