On 3D and 1D mathematical modeling of physically nonlinear beams

In this work, mathematical models of physically nonlinear beams (and plates) are constructed in a three-dimensional and one-dimensional formulation based on the kinematic models of Euler-Bernoulli and Timoshenko. The modeling includes achievements of the deformation theory of plasticity, the von Mises plasticity criterion and the method of variable parameters of the Birger theory of elasticity. The theory is built for arbitrary boundary conditions, transverse loads, and stress-strain diagrams. The issue of solving perforated structures is also addressed. The numerical investigations are based on the finite element method and the method of variable elasticity parameters. Convergence of the method is also investigated.