MOMENT FINITE ELEMENT FOR SOLVING 3D ELASTICITY PROBLEMS

Abstract

The implementation of a new 8-node finite element for solving 3D elasticity static problems is considered. The idea of this finite element is based on the projection of a rare mesh FEM scheme into a space of lower dimension. The resulting scheme of the 8-node hexahedral finite element has a number of features compared to the scheme of the classical 8-node polyline element. This is one integration point in the element compared to eight in the classical element, as well as the presence of four parameters that allow you to adjust the convergence of the computational process. In a given finite element, the stresses as well as their moments (three bending and one torsional) are assumed to be constant within the element. Also, the continuity of displacement fields is preserved only in the nodes of finite elements. This circumstance is not a disadvantage and is characteristic of many new numerical FEM schemes (suffice it to mention the recently actively developed direction of the discontinuous Galerkin method). The issues of using this finite element in solving the well-known problem of increased shear stiffness (shear locking) of a number of well-known FEM schemes are discussed. The problem of hourglass instability, which is typical for FEM schemes with reduced integration and a number of other numerical schemes, in particular, the Wilkins scheme, which is popular in solving dynamic elasticity and plasticity problems, is also considered. The implementation of a technique for the numerical solution of three-dimensional static problems of the theory of elasticity on the basis of a given finite element within the framework of the traditional FEM technique using a vector-matrix notation is described. The results of solving test static problems of elasticity theory are presented.