Mikhail A. Dudaev

Irkutsk State Transport University

There is physico-mathematics dependences conclusion of a finite element method for the Timoshenko beam finite element at the article, which is working by the shear bend at static and dynamic load conditions and allows more accurately approximate displacement field in comparison with the classical bending theory. It's shown, the softening of a finite element in this case is resultant of two components compose: “bending” caused by the bending moment action and “shear”, caused by the shear force action, and cross section angle is different from the beam elastic axis tangent angle; also the shear softening is appeared at short beams, where length is commensurable with height of the beam cross section. The mathematical model is built by the base of variational energy principle of the finite element method. For the indicated type of finite element, polynomial shape functions are obtained, which serve to approximate the displacement field, using differential dependencies known from the courses of resistance of materials and the theory of elasticity. There was gotten the gradient matrix, which allows to determine the deformation vector by the values of the node translation pole and the stiffness matrix, which is had the key meaning at the solving of the system of linear algebraic equations of the finite element method, value of the stress vector and internal force factors at the shear bending, which allows to determine it by the nodes translations. For the finite element model was made convergence analysis of the numerical finite element method by relative to the analytical calculation and are shown results deviation when the model of pure bending and Timoshenko beam model were used.

finite element, finite element method, Timoshenko beam, shear bend, shear, form functions, gradient matrix, stiffness matrix