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The bending problem for a cylindrical shell with oblique cuts is solved using the equations of a model based on the straight-line hypothesis. The two-dimensional boundary-value problem obtained is reduced by the spline-approximation method to one-dimensional one solved by the method of discrete orthogonalization. The results calculated are compared with those obtained using the classical theory. Shells with different values of the cut angles are analyzed.

An approach to solving static problems for ring plates with parameters varying in two coordinate directions is proposed. The system of equations and boundary conditions are formulated for displacements, forces, and moments. The two-dimensional boundary-value problem is reduced to one-dimensional one using the spline-collocation method. This problem is solved with the stable numerical method of discrete orthogonalization. The numerical results presented as a table are analyzed.

An approach to the solution of problems of the statics of shallow orthotropic shells is proposed. It is based on reducing a two-dimensional boundary value problem to a one-dimensional one using the spline-collocation method and solution of the problem by the stable numerical method of discrete orthogonalization. Solutions are presented for problems on the stress state of orthotropic shells of double curvature for several values of the elastic constants of the material.

Since solving problems of the stressed-strained state of elastic bodies in an exact three-dimensional poses considerable mathematical and computational difficulties, only for a small number of problems have the results of the solution been put in numerical form [8, 10, 15, 16, 18]. The stressed-strained state of three-dimensional bodies must be known in order to evaluate the strength capabilities of structural elements, both as a reference for models of applied theories of shells and as solutions obtained by approximate methods. In this article we consider the class of problems of the stressed-strained state of transversely isotropic hollow cylinders of finite length under asymmetric loads applied to the lateral surfaces and with different boundary conditions at the ends. The solution of that class of problems is constructed on the basis of separation of variables in the circumferential coordinate and solution of the two-dimensional boundary-value problem by the spline-collocation and discrete orthogonalization methods [4, 5, 6, 12, 13]. This approach was used earlier [1, 2, 7, 11] to solve two-dimensional problems of the theory of shells in different formulations. The influence of boundary conditions on the displacement and stress distribution in cylinders is analyzed on the basis of the results obtained. We consider transversely isotropic hollow cylinders in which the plane of isotropy coincides with the tangent to the cylindrical surface and the mechanical characteristics vary only along the normal to it. Asymmetrically distributed loads are applied to the lateral surfaces of the cylinder and boundary conditions are given at the ends. The problem is described on the basis of the three-dimensional theory of elasticity in a cylindrical coordinate system

The study of the stress-strain state of plates and shells subjected to various types of loads with different support conditions entails the formulation of boundary-value problems that generally involve systems of partial differential equations with variable coefficients. The complexity of solving these problems stems not only from the high order of the system and the variability of the coefficients, but also from the need to exactly satisfy prescribed boundary conditions. The use of a given method to obtains a solution with a satisfactorily high degree of accuracy depends to a significant extent on the geometric and mechanical parameters characterizing certain aspects of the problem and the type of boundary conditions. These factors sometimes limit the possibilities of solving problems in the important (in a practical sense) cases in which the stiffness of the shell or plate supports is also variable. In addition, problems of the shell theory entail local and edge effects, which imposes certain stiffness conditions on boundary-value problems related to the phenomenon of instability in the computation. Spline functions have recently come into wide use to solve such problems in the areas of computational mathematics, mathematical physics, and mechanics. The popularity of this approach stems from the advantagesmore » offered by spline functions compared to other methods. Among these advantages: the stability of splines in relation to local perturbations, i.e. the behavior of the spline in the neighborhood of a point has no effect on the behavior of the spline as a whole (which is not the case in a polynomial approximation); good convergence of a spline-interpolation, in contrast to a polynomial interpolation; simplicity and ease of use of algorithms that construct and calculate splines on computers. 65 refs., 5 figs., 7 tabs.« less

Spline functions have come into increasingly wide use recently in the solution of boundary-value problems of the theory of elasticity of plates and shells. This development stems from the advantages offered by spline approximations compared to other methods. Among the most important advantages are the following: (1) the behavior of the spline in the neighborhood of a point has no effect on the behavior of the spline as a whole; (2) spline interpolation converges well compared to polynomial interpolation; (3) algorithms for spline construction are simple and convenient to use. The use of spline functions to solve linear two-dimensional problems on the stress-strain state of shallow shells and plates that are rectangular in plan has proven their efficiency and made it possible to expand the range of problems that can be solved. The approach proposed in these investigations is based on reducing a linear two-dimensional problem to a unidimensional problem by the spline unidimensional problem by the method of discrete orthogonalization in the other coordinate direction. Such an approach makes it possible to account for local and edge effects in the stress state of plates and shells and obtain reliable solutions with complex boundary conditions. In the present study, wemore » take the above approach, employing spline functions to solve linear problems, and use it to also solve geometrically nonlinear problems of the statics of shallow shells and plates with variable parameters.« less

Geometrically nonlinear relationships of the theory of thin layered shells are applied to analyze axisymmetric strain of bimetallic shells of revolution in a temperature field. One-dimensional nonlinear boundary-value problems are solved by a combination of the linearization method and the discrete orthogonalization method. A numerical approach is proposed to solve the boundary-value problems in the supercritical strain region.

An approach is proposed to numerical analysis of the stress state of shells of revolution in the geometrically nonlinear setting. It is based on a combination of linearization and discrete orthogonalization methods. Stress concentration is analyzed for a flexible spherical shell with a central hole, depending on load intensity, hole size, and the boundary conditions on the contours in the precritical strain region.

The paper presents a numerical analysis of the deformation of an infinitely long flexible cylindrical panel of variable thickness immersed in a liquid with a concentrated force applied along the panel generator in the symmetry plane. The nonlinear boundary-value problem is solved by the linearization method combined with the discrete orthogonalization method. We examine the effect of variable thickness on the deformation of the panel for various initial values of the radius of curvature.

The bending problem for laminated orthotropic trapezoidal plates of variable thickness is solved by parametrizing the domain of interest and reducing the two-dimensional boundary-value problem to one-dimensional using the spline-collocation method. The resulting problem is solved by the stable discrete-orthogonalization method. The influence of orthotropy and the variability of the thickness on the displacement and stress fields is analyzed