Investigation of Stress-Strain State of Transversely Isotropic Plates Under Bending Using Equation of Statics {1,2} –Approximation

Abstract

The study examined the construction of the fundamental solution for the equations of statics {1,2}-approximation for transversely isotropic plates under bending with the action of concentrated force. Equations {1,2}-approximation were obtained by the decomposition method in the thickness coordinate using the Legendre polynomials. These equations take into account all the components of the stress tensor, including the transverse shear and normal stresses. Since the classical theory of Kirchhoff-Love doesn't take account of these stresses, the study on the basis of refined theories of stress-strain state of transversely isotropic plates under the action of concentrated force effects is an important scientific and technical problem. The fundamental solution of obtained equations results using a two-dimensional Fourier integral transform and inverse treatment techniques, built with the help of a special G-function. This method allows reducing the system of resolving differential equations for statics of flat plates and shells to a system of algebraic equations. After that, the inverse Fourier transform restores the fundamental solution. The work was carried out numerical studies that demonstrate patterns of behavior of components of the stress-strain state, depending on the elastic constants of transversely isotropic material. The results play a decisive role in the study of boundary value problems in the mechanics of thin-walled elements of constructions, including under the influence of concentrated and local diverse forces.

Analysis of fundamental solutions to the equations of statics constructed for transversal-isotropic plates

“…Among the publications, which apply a generalized theory of the {m,n}-approximation and the procedure of mapping, built with the help of specialized G-function, those articles should be noted that deal with the construction of fundamental solutions of the equations of statics for the transversal-isotropic plates based on the {1,0}-approximation for the state of bending [8] and based on the {1,2}-approximation for the zero spin stressed state and the state of bending [9,10]. Worth noting are also the publications, in which authors obtained fundamental solutions of differential equations on thermoelasticity of the {1,0}-and {1,2}-approximation for the transversal-isotropic plates [11,12].…”