Mixed and Compatible Finite Elements in the Analysis Problem of Elastoplastic Structures

A problem of ideal elastoplastic structures stress-strain field determination is considered The dual general and discrete mathematical models of analysis problem are made on the basis of the extremal energy principles and finite element method. In these models the possible discontinuity of displacements and the dissipation of energy in the place of those discontinuities, also the different external effects (load, initial strains and support settlements) are estimated. At first, on the basis of the mixed functional and mixed finite elements the discrete expressions of fundamental relationships (geometric equations, yield conditions) and the discrete mathematical model of mixed formulation of the problem are made. This mathematical model corresponds to the minimum total energy principle for a kinematically admissible displacements. The dual static formulation of the problem is obtained by Lagrange's multipliers method; this corresponds to the minimum complementary energy principle. The kinematic formulation of the problem is obtained in the case of linear yield conditions. These mathematical models permit to determine the lower values of the stress and displacements of structures. It has shown that the approximation of geometric equations and yield conditions by Bubnov-Galiorkin's collocation method gives the more accurate results.

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