CALCULATION OF SHELLS BASED ON THE MIXED FEM VARIANT WITH TENSOR-VECTOR APPROXIMATION OF THE DESIRED VALUES

Abstract

An algorithm for calculating thin shells based on the mixed finite element method (FEM) in a two-dimensional formulation is proposed. To correctly take into account possible displacements of the shell as a rigid body, the algorithm implements the developed tensor-vector form of the interpolation procedure of the desired values, which were chosen as the tensors of deformations and curvature at the point of the middle surface and the displacement vectors of this point. The discretization element was a quadrangular fragment of the middle surface with unknown nodes in the form of location and their first derivatives, as well as components of strain and curvature tensors among the inner surface at the nodes of the finite element. After minimizing the modified mixed functional, a 36?36 finite element stiffness matrix is formed. In order to verify the developed algorithm, a number of test problems were solved for calculating a fragment of an elliptical cylinder with an analytical solution, as well as for calculating a shell with spring supports that allow the ellipsoidal shell to move as an absolutely rigid body. Analysis of the obtained results showed that when calculating a fragment of an elliptical cylinder, the calculation scheme of which makes it possible to obtain an analytical solution, the numerical values of bending moments and normal stresses completely coincide with the values calculated from the condition of static equilibrium. An analysis of the calculation results for an ellipsoidal shell with spring supports led to the conclusion that the developed tensor-vector interpolation procedure in the mixed FEM version allows one to correctly take into account the displacements of shells as solids and obtain an adequate assessment of their stress-strain state.