# The study of the stress-strain state of symmetrical rectangular plates of arbitrary geometry based on the refined theory

### Аuthors

Firsanov V. V.*, Doan Q. H.**

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: k906@mai.ru
**e-mail: dqhieu57@gmail.com

### Abstract

The article presents a refined theory of the stress-strain state calculating of rectangular plates symmetric in respect to the plane and arbitrary geometry in the longitudinal direction. Equations of the plate state are described by the relationships in the three-dimensional theory of elasticity. The required displacements of the plate are decomposed into two polynomials in a coordinate, normal to the plate surface, two degrees higher than that in the classical theory of the Kirghoff-Love type.

The system of basic equations of the refined theory and the corresponding boundary conditions were obtained by the variation Lagrange principle.

One of the distinguishing features of the proposed refined theory consists in employing direct integration of the equilibrium equations of the three-dimensional elasticity theory to determine the transverse normal and tangential stresses.

For isotropic rectangular plate of various thicknesses, a system of differential equations of equilibrium displacements with various coefficients, containing additional terms, accounting for the effect of a thickness change on the stress-strain state of the plate was obtained by the Levy method.

To solve this boundary value problem, the finite different method was employed. An example of the stress state computing of a rectangular plate with a thickness varying according to linear and parabolic laws was investigated. A comparison between the results obtained by the refined and classical theories was performed. It was established, that when studying the stress state in the zones of its distortion (joints, local stress zones, etc.), a refined theory should be used, since the additional corresponding stresses were of the same order as the values of the ground stress state.

In this article, the basic equations of the refined theory for the plate were obtained using the Lagrange variation principle and expansion of the desired displacements in thickness. This technique allows account for both the thin plates, and the plates with average thickness. As an example, the calculation of stress-strain state for a rectangular isotropic plate with variable thickness under the action of a distributed load is considered. The comparison of the strain-stress state of the plate computing according to the refined and classic theories is presented.

### Keywords:

rectangular plate, arbitrary geometry, Lagrange variational principle, finite difference method, “boundary layer” layer stress-strain state

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