Numerical stress-strain analysis of a nonthin elastic plate

Abstract

A numerical solution to elastic-equilibrium problems for nonthin plates is proposed. The solution is obtained by using the curvilinear-mesh method in combination with Vekua's method. The efficiency (rapid convergence and accuracy) of this approach is demonstrated by solving test problems for thick plates that can also be solved exactly or approximately by other methods. A numerical solution is obtained to the bending problem for orthotropic nonthin plates of constant and varying thickness Keywords: nonthin plate, constant and varying thickness, bending, numerical solutions, comparison of resultsIn searching for numerical solutions describing the elastic deformation and stability of thin plates and shells, finite-difference schemes are used to approximate the governing equations and relations of the theory of thin shells [3][4]. The discrete-orthogonalization method is successfully used to solve spatial problems for thick shells [8][9][10][11]. Note that to determine the stress-strain state (SSS) in stress concentration zones (holes or abrupt changes in the geometry of the boundary surfaces) with adequate accuracy, it is necessary to use algebraic systems of equations or stiffness matrices with a great number of variables. Such systems as a rule converge weakly when solved by iteration. The reason is that finite-difference or finite-element schemes of approximating the differential relations of the theory of shells disregard rigid-body displacements of the displacement vector.The approach proposed in [4] made it possible to efficiently solve a number of problems for thin shells in view of rigid-body displacements. It was also used to find a numerical solution to the stress-strain problem for shells and to compare it with the exact solutions of test problems.Here we set forth an efficient numerical algorithm, based on the approach from [4], for solving the elastic problem for nonthin plates.

Three-dimensional elastic deformation of a functionally graded coating/substrate system

“…It was shown by Kashtalyan (2004) that as the inhomogeneity parameter becomes smaller, the three-dimensional solution for an FGM rapidly converges to that for a homogeneous material. For example, for sufficiently small values of the inhomogeneity parameter (smaller than 10 À5 by the absolute value), the difference between the computed out-of-plane displacements in functionally graded and homogeneous plates constituted less than 0.03% (Kashtalyan, 2004;Chernopiskii, 2006).…”

Stress distribution around a circular cylindrical cavity in a prestressed plate

“…A method to derive equilibrium equations for nonthin anisotropic shells with prestresses is outlined in [6,9]. It involves the expansion of unknown functions into Fourier-Legendre series in the thickness coordinate [8,10,11]. The present paper deals with the system of equations of elastic equilibrium for nonthin transversely isotropic plates with a uniform prestress field.…”

Refined theory of orthotropic plates subject to tangential surface loads

“…Numerical examples are given. The results are compared with those produced by other theories [2,4,8,[15][16][17][18].…”