Strength calculation algorithm for shell structures based on a four-node discretization element

Abstract

The article describes a method for calculating shell structures taking into account transverse shear deformations during elastoplastic deformation. The geometric relationships between the increments of deformations in an arbitrary layer of the shell and the increments of the components of the displacement vector, the components of the vector of inclination angles of the normal are obtained on the basis of the hypothesis of S.P. Tymoshenko. The slope of the normal and the increment of the slope are performed from the original position of the normal. The physical relationships between the increments of the stress tensor components and the increments of the components of the deformation tensor at the loading step were obtained on the basis of the hypothesis that they were proportional, based on the deformation theory of plasticity. In the finite element implementation of the developed algorithm, a four-node fragment of the middle surface of the shell structure with nodes located at the vertices of the quadrangular fragment was used. The increments of the displacement vector components, their first derivatives with respect to curvilinear coordinates, as well as the increments of the components of the normal rotation angle vector were selected as the nodal variable parameters of the four-node finite element at the loading step. The size of the stiffness matrix of the four-node finite element was 44x44. The developed algorithm was tested on test problems for calculating a cylindrical shell using a duralumin alloy with a deformation diagram in the form of a two-link broken line as a material.