Two-Field Prismatic Finite Element Under Elasto-Plastic Deformation

For elasto-plastic analysis of structures at a particular load step, a mixed finite element in the form of a prism with triangular bases was obtained. Displacement increments and stress increments were taken as nodal unknowns. The target quantities were approximated using linear functions. Two versions of physical equations were used to describe elasto-plastic deformation. The first version used the constitutive equations of the theory of plastic flow. In the second version, the physical equations were obtained based on the hypothesis of proportionality of the components of the deviators of deformation increments to the components of the deviators of stress increments. To obtain the stiffness matrix of the prismatic finite element, a nonlinear mixed functional was used, as a result of the minimization of which two systems of algebraic equations with respect to nodal unknowns were obtained. As a result of solving these systems, the stiffness matrix of the finite element was determined, using which the stiffness matrix of the analysed structure was formed. After determining the displacements at a load step, the values of the nodal stress increments were determined. A specific example shows the agreement of the calculation results using the two versions of the constitutive equations of elasto-plastic deformation.