Critical Loads of Uniformly Compressed Orthotropic Rectangular Plate on an Elastic Base

Abstract

Introduction. The problem of critical loads of a compressed orthotropic rectangular plate on an elastic base was considered. The following orthotropy parameters were set for the plate: Poisson coefficients, Young's modules for the main directions, and the shear modulus of the plate material. The components of the compressive load were uniformly distributed along two opposite edges of the plate and acted parallel to the coordinate axes. The edges of the plate were loosely pinched or pivotally supported. Cases were also considered when two parallel edges of the plate were free from loads, and the other two were freely pinched or pivotally supported.Materials and Methods. The problem was studied on the basis of a system of nonlinear Kármán-type equilibrium equations. The critical values of the load parameter were determined from a linearized problem based on a trivial solution. At the same time, the variational method in combination with the finite difference method was used to solve the boundary eigenvalue problem.Results. The problem was reduced to solving a parametric linear boundary eigenvalue problem. In case of boundary conditions of a movable hinge support, exact formulas of eigenvalues and eigenfunctions were given. While in case of free edge pinching, a variational method was used in combination with a finite-difference method, and a computer program for solving the problem was built. It was established that one or two eigenfunctions expressing the deflection of the plate could correspond to the critical value of the compressive load parameter at which the stability of the compressed plate was lost. The results of numerical calculations of the critical values of the compressive load at different values of the orthotropy parameters were presented, and graphs of the corresponding equilibrium forms were constructed. For the case of a long orthotropic plate on an elastic base, it was established that the main term of the asymptotic expansion of the solution to the linear eigenvalue problem was determined from the problem of critical loads of a compressed beam on an elastic base with an elastic modulus that coincides with the elastic modulus of the plate in the longitudinal direction.Discussion and Conclusions. The problem of critical loads of an orthotropic plate compressed in two directions lying on an elastic base was investigated. As the compressive load component increased along one direction, the critical value of the load compressing the plate along the other direction decreased. If an orthotropic plate was compressed by a load along a direction that corresponded to a greater bending stiffness, then the critical value of the loss of stability was greater than the critical value of the compressive load acting along the direction of a lesser bending stiffness. The presence of an elastic foundation increased the bearing capacity of the compressed plate.

Keywords

• critical load
• elastic orthotropic plate
• equilibrium stability
• elastic base
• parametric eigenvalue boundary problem