Stress-strain state of elastic shell based on mixed finite element

Abstract

In a mixed formulation, a four-node finite element was developed, which is a fragment of the middle surface of the elastic shell. Longitudinal forces and bending moments, as well as displacements and their first derivatives with respect to curvilinear coordinates, were taken as nodal unknowns. To obtain the compliance matrix, the Reissner functional was used, in which the stresses, when using the direct normal hypothesis, are represented by dependences on the forces and bending moments of the middle surface, the approximation of which was carried out by bilinear functions. In the interpolating expressions for the kinematic sought quantities, Hermite polynomials of the third degree were used. As a result of minimizing the transformed functional with respect to the force and kinematic nodal unknowns, the compliance matrix of the accepted discrete element was formed. Verification of the developed discrete element in a mixed formulation was carried out on the examples of calculations of cylindrical shells with circular and elliptical cross sections. The values of the force parameters found using the developed algorithm adequately satisfied the conditions of static equilibrium (the calculation error was less than 0.5 %). An analysis of the obtained finite element solutions showed the effectiveness of the developed algorithm and made it possible to note the possibility of its use in calculations of thin-walled structures made of incompressible materials.

Keywords

• thin-walled shell-type structure
• modified mixed functional
• compliance matrix
• four-node sampling element
• force and kinematic unknowns