Finite element model of Reisner’s plates in stresses

Structural mechanics

A method for calculating bending plates by the finite element method based on Reisner's theory is proposed. The method is based on the fundamental principles of minimum of additional energy and possible displacements. For discretization of the subject area, arbitrary quadrangular finite elements are used. Over the area of the finite element, the moment fields and shear forces are approximated by constant functions that satisfy the differential equilibrium equations in the area of the finite element in the absence of a distributed load. Using the principle of possible displacements, algebraic equilibrium equations of the nodes of the finite element grid are compiled. In accordance with Reisner's theory, vertical displacements and angles of rotation of the middle surface of the plate are taken as nodal possible displacements as independent. The proposed method of calculation allows you to calculate both thick and thin plates. There is no effect of «locking» of the solution for thin plates, which is confirmed by calculations of rectangular plates with different support conditions of side and different ratios of thickness to plate sizes. The solutions obtained by the proposed method for plates of various shapes are compared with analytical solutions. Sufficiently fast convergence and accuracy of the proposed calculation method for both thick and thin plates is shown.