Edge stress state of variable thickness rectangular plate based on refined theory

DOI: 10.34759/trd-2020-110-10

Аuthors

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

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

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

Abstract

Recently, a growing interest in developing refined theories of plates and shells is observed. This interest is aroused by the necessity of studying stressed-deformed state (SDS) while developing elements of modern structures, including aerospace engineering products. The article considers the edge stress state of isotropic rectangular plate of variable thickness under action of local load based on the refined theory. A mathematical model of additional stress state of “boundary layer” type, occurring near the clamped edge of the plate was developed. Three-dimensional equations of elasticity theory are applied while this mathematical model development. The displacements are approximated by the polynomials by the coordinate normal to the median plane two powers higher relative to the classical theory of Kirchhoff-Love type.

A system of basic equations of the refined theory and corresponding boundary conditions were obtained by the variation Lagrange principle. The solution of the formulated boundary value problem is accomplished the trigonometric Fourier series methods, finite differences, and matrix sweeps. One of distinguishing features of the proposed refined theory consists in the fact that direct integration of the equilibrium equations of the three-dimensional elasticity theory is employed while transversal and normal tangential stresses determining.

A refined mathematical SDS model of the rectangular plate with variable thickness, symmetric relative to the median plane in the longitudinal direction, was developed in this work. The article presents a comparison of the results obtained by the refined theory with the classical theory data. This technique allows consider not only the thin plates but also the plates of medium thickness. It was established that this refined theory should be used when studying the stress state in the zones of its distortion (joints, local loading zones, etc.). Additional, in relation to the classical theory, transverse normal stresses, are appeared to be of the same order with maximum values of the main bending stress. This result is important as it allows obtain more reliable evaluation of the strength and crack resistance of aircraft structural elements, as well as other machine building objects at the design stages.

Keywords:

rectangular plate, tightly clamped edge, three-dimensional equations of the theory of elasticity, Lagrange variation principle, decomposition into trigonometric series, finite difference method, matrix sweep method, stress-strain state “boundary layer”, transverse normal stresses

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