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.

在这项工作中，基于Euler–Bernoulli和Timoshenko的运动学模型，以三维和一维公式构建了物理非线性梁（和板）的数学模型。该模型包括可塑性变形理论，冯·米塞斯（von Mises）可塑性准则以及比尔格（Birger）弹性理论的可变参数方法的成果。该理论是为任意边界条件，横向载荷和应力-应变图构建的。还解决了解决穿孔结构的问题。数值研究基于有限元方法和可变弹性参数方法。还研究了该方法的收敛性。