The simply supported rectangular plate: An exact, three dimensional, linear elasticity solution

Abstract

An exact three dimensional solution for the problem of a transversely loaded, simply supported rectangular plate of arbitrary thickness is presented within the linear theory of elastostatics. The solution, obtained in a semi-inverse fashion, satisfies all the boundary conditions of the problem in a pointwise manner and is in the form of a double Fourier sine series. The classical Navier solution for the problem is shown to be the limit of the present solution as the plate thickness aspect ratio approaches zero. It is noted that the solution presented provides a benchmark against which approximate theories of transversely loaded plates may be measured.The new elasticity solution also provides a heuristic basis for a novel theory of thick plates of arbitrary planform and edge support recently given by the author.

Three-dimensional elastic solutions for functionally graded circular plates

“…In this way, we obtain the coefficients A j , B j , C j , D j by using the boundary conditions on the stress (2.2); the condition w(b, z) = 0 is automatically verified while, the radial displacement u A (b, z) (where the superscript A refers to the problem A) is obtained from (3.19). The solution of problem A gives so rise to an infinite sum of terms, each of them generating displacement fields w(r, z) and u(r, z) satisfying the so called semi-inverse Levinson assumptions defined in Levinson (1985) and generalized, for cross section of arbitrary shape, in Nicotra et al (1999). Hence, we define the problem B with the following boundary conditions: free loading ends,…”

Shells with thickness distension

“…• A scant supply of exact solutions to equilibrium problems for linearly elastic, three-dimensional plate-like bodies is available in the literature (see, e.g., Levinson (1985), Pan (1991), Rogers et al (1992)). Conceivably, these solutions can serve as benchmarks to test and compare the approximations offered by one or another plate theory.…”

On edge effects for the semi-infinite plate

“…An exact linear elastostatic solution for the transversely loaded simplysupported plate has recently been presented by Levinson [1] based upon the kinematic assumptions…”

“…It is shown in [ 1] that taking W(x, y) in the form of the double Fourier sine series W(x, y) = W,,~ Sin rmrx/a sin mry/b leads to an exact solution to the Navier equations of linear elasticity for the transverse loading of a rectangular plate simply-supported on the four edges x = 0, x = a , y = O , y = b . The assumptions (1) have been applied by Levinson to two other related problems: in [2] they provide an exact solution, within the spirit of linear elastodynamics, to the free vibration of the simply-supported rectangular plate; aside, obviously, from the time dependency, the displacements in [2] differ from the static case by the re-definition of the plane z = 0 as the mid-surface of the plate, when the faces of the plate are at z = +_hi2.…”