Issue 49

A. Baryakh et alii, Frattura ed Integrità Strutturale, 49 (2019) 257-266; DOI: 10.3221/IGF-ESIS.49.25 261 The constructed finite element analog (9) of variational Eqn. (1), owing to the use of nonlinear shape functions (as analytical solution of a stress problem for a single layer) in the FEM (finite element method) semi-analytical scheme, allows us to considerably decrease the number of the unknown. When using this approach, the total number of the unknown in the elastic equilibrium problem for a system of plane-parallel layers is equal to 4 ( 1) K N  , where K is the number of terms in a Fourier series retained in expansion (2), and N is the number of layers. Symmetry about the y- axis, makes it possible to reduce the number of the unknowns by half. T EST EXAMPLE s a test example, we consider a massless plane, which is under the action of a distributed load. The computational scheme for the problem is presented in Fig. 2. Figure 2 : The computational scheme of single layer test system The specified static boundary conditions are the following:     0, ; 0 0, ; 0 0; 0. xy y xy y x y h q x a y h q x a x x L U                    An analytical solution of this problem can be constructed with the use of the Fourier series. According to [10], an equation for the Airy stress function [11]: 4 4 4 4 2 2 4 2 0 x x y y              (10) will be identically satisfied if it is presented as follows:   sin m x f y l    (11) where m is any integer number, and function   f y is a solution of following equation:       4 2 2 0 IV f y f y f y       (12) y h -h x 0 a b L q=-1 A