The method of equivalent strength conditions in calculating composite structures with a regular structure using multigrid finite elements

Plates, beams and shells with non-uniform and micro-inhomogeneities regular structure are widely used in aviation and rocket and space technology. At the preliminary design stage, it is initially important to know whether the design safety factor meets the specified strength conditions. To determine the margin factor, it is necessary to solve the elasticity problem for the designed structure by the finite element method (FEM), taking into account its inhomogeneous structure, which requires large computer resources. In this paper, we propose a method of equivalent strength conditions (MESC) for calculating the static strength of elastic structures with a inhomogeneous regular structure. The proposed method is reduced to the calculation of the strength of isotropic homogeneous bodies using equivalent strength conditions. The MESC is based on the following statement. For any composite body V 0 , there exists such an isotropic homo b geneous body V b and such a number p (equivalence coefficient) that if the body V b stock coefficient satisfies n 0 the equivalent strength conditions pn 1 £ n 0 £ pn , then the body V stock coefficient satisfies n the given strength conditions n 1 £ n 0 £ n 2 , and Vice versa, n 1 , n 2 - given, the coefficients n 0 , n 0 , meet the exact solutions of elasticity problems constructed for bodies V 0 , V b . The method under consideration is reduced to FEM strength calculation of isotropic homogeneous bodies, which is the easiest to implement and requires less computer memory than a similar calculation of composite bodies taking into account their inhomogeneous structure. The procedure for determining the equivalence coefficients for a number of composite plates, beams and shells of rotation is described. High-precision multigrid finite elements generating discrete models of small dimension and solutions with small error are used in the construction of elastic solutions according to FEM for isotropic homogeneous bodies. The adjusted equivalent strength conditions are of the form pn 1 (1 + e1 ) £ nb £ pn 2 (1 - e2 ) , where nb is the body V b reserve coefficient and the values e1 , e2 correspond to the approximate solution constructed for the body V b . Implementation of FEM for multigrid discrete models requires several 103 ¸106 times less computer memory than for basic models. The calculation of the strength of a beam with a micro-homogeneous regular structure with the help of MESC is given.