Investigation of the behavior of generalized displacements obtained using the theory of {m,n}-approximation

Abstract

A concentrated force action on a transversaly-isotropic plate is considered. Three-dimensional equations of the elasticity theory are reduced to the two-dimensional ones by expanding unknown functions into Fourier series of Legendre polynomials. The chosen {m,n}-approximation theory is the most suitable for obtaining two-dimensional equations of the elasticity theory, because it is not based on any hypotheses. Also, this approach allows us to consider not only thin plates, but plates of medium and large thickness. The accuracy of the solutions obtained depends on the number of terms that are retained in the expansions of the given and unknown functions. Obtained equations using this approach 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 behavior of displacements based on the refined {m,n}-approximation theory, depending on the elastic constants of transversely isotropic material. Further analysis of the stress-strain state of plates on the basis of the generalized theory of {m,n} -approximation is analyzed. 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.