Modified Equations of FiniteSize Layered Plates Made of Orthotropic Material. Comparison of the Results of Numerical Calculations with Analytical Solutions
Yu. M. Volchkov^{1,2}
^{1}Lavrent’ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090 Russia ^{2}Novosibirsk State University, Novosibirsk, 630090 Russia
Keywords: уравнения изгиба слоистых пластин, ортотропный материал, полиномы Лежандра, bending equations for laminated plates, orthotropic material, Legendre polynomial
Abstract
This paper describes the modified bending equations of layered orthotropic plates in the first approximation. The approximation of the solution of the equation of the threedimensional theory of elasticity by the Legendre polynomial segments is used to obtain differential equations of the elastic layer. For the approximation of equilibrium equations and boundary conditions of threedimensional theory of elasticity, several approximations of each desired function (stresses and displacements). The stresses at the internal points of the plate are determined from the defining equations for the orthotropic material, averaged with respect to the plate thickness. The construction of the bending equations of laminated plates for each layer is carried out with the help of the elastic layer equations and the conjugation conditions on the boundaries between layers, which are conditions for the continuity of normal stresses and displacements. The numerical solution of the problem of bending of the rectangular laminated plate obtained with the help of modified equations is compared with an analytical solution. It is determined that the maximum error in determining the stresses does not exceed 3%.
