Abstract. The present report proposes an efficient approach to solving within the framework of the classic and refined models the stress-strain problems of shallow shells as well as the problems on free vibrations. In accordance with the approach the initial system of partial differential equations is reduced to one-dimensional problems by using approximation of the solution in terms of basic splines in one coordinate. The boundary-value problems obtained and eigenvalue boundary-value problems for systems of ordinary differential high-order equations are solved by the stable numerical method of discrete ortogonalization.
The stress-strain state of thick rectangular plates is determined using a numerical-analytic approach based on the three-dimensional theory of elasticity. The problem is made one-dimensional using the method of spline collocation in two coordinate directions. The boundary-value problem for a system of ordinary differential equations of high order is solved with the stable numerical method of discrete orthogonalization. The stress-strain state of plates with different geometrical parameters and hinged or clamped edges is analyzed Introduction. Thick rectangular plates are widely used in many fields of modern engineering. To ensure the strength and reliability of such structural elements, it is important to be able to determine their stress-strain state. Using the three-dimensional theory of elasticity involves severe computational difficulties. Because of this there are only few publications on the subject [1, 2]. Theoretical and experimental displacement and stress fields in rectangular thick plates with different boundary conditions were examined in [5, 6] based on solutions to boundary-value problems of linear elasticity. Partially embedded finite-and infinite-length plates were addressed in [5].Here we propose an efficient numerical-analytic approach to analyzing the stress-strain state of rectangular thick isotropic plates based on the theory of elasticity. The approach employs spline collocation in two directions to reduce the original three-dimensional boundary-value problem for a system of partial differential equations to a problem for a system of ordinary differential equations of high order. This system is solved with the stable numerical method of discrete orthogonalization. Some two-dimensional static and dynamic problems for plates and shells were solved in [4, 7-12] using the spline-collocation method.1. Starting Equations. Consider a rectangular plate of constant thickness in an orthogonal coordinate system X, Y, Z. We start with the following equations of linear elasticity [3]:
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