Construction of the bending model of micropolar elastic thin beams with a circular axis and its implementation using finite element method

Abstract

This paper considers the problem of transition from the system of two-dimensional equations of the micropolar (moment) theory of elasticity in a thin curved area to the one-dimensional system of equations describing deformation of the micropolar elastic thin beam with a circular axis. During the transition process, Timoshenko's hypotheses generalized to the micropolar case are applied. As a result, the applied model (with independent fields of displacements and rotation) of a micropolar elastic thin beam with a circular axis has been constructed. It is shown that the model includes the law of conservation of energy, energy theorems and variation principles. All main functionals for the model of the micropolar elastic thin beam with a circular axis are obtained from the functional of the two-dimensional micropolar theory of elasticity, containing only the first derivatives of displacements and rotations. The finite element method (FEM) is taken to study the boundary problems of statics and dynamics of applied model of the micropolar elastic thin beam with a circular axis. The basic concepts and stages of the FEM are formulated: the discretization, the selection of basic nodal unknowns, the approximation of the solution, and the construction of the basic FEM equations. The finite-element solutions of some problems of statics and problems on natural vibrations of beams with a circular axis are considered according to the micropolar theory of elasticity. A comparative analysis with similar problems of beams with a circular axis according to the classical theory of elasticity is carried out. Based on the results, some effective properties of beams with a circular axis have been established when considering their deformations in the context of the micropolar theory of elasticity.