Calculations of 3d Anisotropic Membrane Structures Under Various Conditions of Fixing

Abstract

A unified approach in solving equilibrium problems of standard cells of membrane structures made of various absolutely flexible materials, including anisotropic (orthotropic) materials is presented in this paper. The objects of study are rectangular membranes under various conditions of fixing and/or supporting. The problems were considered in a geometrically nonlinear formulation, with the deformations and the squares of the rotation angles thereunder being considered to be comparable with each other, but small compared to unity. A resolving system of differential equations in partial derivatives written in displacements is obtained therewith. These equations combining with the presented boundary conditions are numerical models of a number of fragments of real membrane structures. The closed nonlinear system of equations was integrated using the discrete braking method. Therewith both longitudinal and transverse vibrations of anisotropic masses were analyzed using a conditional dynamic model to select the initial values of the displacements. The problem of equilibrium of a square isotropic membrane with a free boundary under a uniformly distributed load is presented as an example. The resulting graphs and tables show the distribution of forces and displacements. They may be used for calculating membrane structures. The developed technique may be applied to those values of the initial parameters under which the calculations have not yet been made.