Bending of orthotropic scalene triangle plates: finite difference modeling

Abstract

The object of the study is a transversely bent triangular plate made of an orthotropic material, fixed along the edges of the plate, under the action of a uniformly distributed load. The fourth-order differential equilibrium equations with variable orthotropy parameters were used. The equations were approximated by finite differences for a grid of scalene triangles. Such a grid describes well the boundary contour of triangular plates. The boundary conditions for the grid were written taking into account the orthotropy of the plate material. Seven typical finite difference equations were developed taking into account the boundary conditions along three edges of the plate and the presence of three angles of an irregular triangle. A finite difference matrix was obtained. The matrix structure allows calculating a triangular plate at different angles at the base. It is possible to vary the boundary conditions in the form of rigid or hinged support of the triangular plates. The calculation method takes into account the parameters of the orthotropy of the material in two mutually perpendicular planes. The adaptation of the numerical method to the calculation of orthotropic plates of arbitrary shape was described. The relationships for determining the rigidity characteristics of orthotropic materials were given. An algorithm for simple engineering calculation of triangular orthotropic plates was proposed that allowed performing accurate calculations in variant design. The scientific and applied results of the proposed article will find wide application in mechanics of deformable solids in the field of studying two-dimensional thin-walled structures, as well as in calculating plates of complex geometry with non-uniform mechanical characteristics of their materials.