Single Finite Fourier Sine Integral Transform Method for the Flexural Analysis of Rectangular Kirchhoff Plate with Opposite Edges Simply Supported, Other Edges Clamped for the Case of Triangular Load Distribution

Abstract

This paper presents the single finite Fourier sine integral transform method for the flexural analysis of rectangular Kirchhoff plate with opposite edges simply supported, and the other edges clamped for the case of triangular load distribution on the plate domain. The finite sine transform method is adopted because the sinusoidal integral kernel function satisfies all the Dirichlet boundary conditions along the simply supported edges. The governing domain equation which is a nonhomogeneous biharmonic equation is converted by the transformation to an integral equation over the solution domain. The integral equation is reduced by the linearity properties of the transformation, Leibnitz rule and boundary conditions along the simply supported edges to ordinary differential equations (ODEs). Classical methods of solving ODEs in closed-form were used to obtain the solutions that satisfy the boundary conditions along the clamped edges. Thus exact solutions to the boundary value problem that satisfy the governing equation at all points in the solution domain as well as at all points along the four edges are obtained. The resulting expression for the deflection is a single infinite series with demonstrated rapidly convergent properties. The bending moments Mxx, Myy expressions were obtained using the bending moment-deflection relations as infinite series with demonstrated rapid convergence. The converged values of the maximum deflection and bending moments Mxx, Myy evaluated at the plate centre are in excellent agreement with the previous exact results to the problem obtained by researchers who used the Kantorovich-Vlasov method; Levy's method; and the superposition method.

Bending Analysis of Isotropic Rectangular Kirchhoff Plates Subjected to a Thermal Gradient Using the Fourier Transform Method<strong> </strong>

“…The governing equation of an isotropic Kirchhoff plate is given by (6) where q(x,y) denotes the applied transverse load per unit area. Substituting Equations (5a-c) into Equation (6) yields the governing equation of the isotropic plate subjected to a thermal gradient (7) The Kirchhoff shear forces per unit length are introduced to satisfy the boundary and continuity conditions; they combine shear forces and twisting moments, and can be expressed for an isotropic plate as follows:…”

“…Kamel [6] described the operational properties of the finite Fourier transform method, with the purpose of solving boundary value problems of partial differential equations, which has some applications on potential and steady-state temperature. Mama et al [7] presented the single finite Fourier sine integral transform method for the flexural analysis of rectangular Kirchhoff plate with opposite edges simply supported, and the other edges clamped for the case of triangular load distribution on the plate domain. Sayyad et al [8] assessed a trigonometric plate theory for the static bending analysis of plates resting on Winkler elastic foundation; the theory considered the effects of transverse shear and normal strains.…”

Bending Analysis of Isotropic Rectangular Kirchhoff Plates Subjected to Non-Uniform Heating Using the Fourier Transform Method<strong> </strong>

“…The governing equation of an isotropic Kirchhoff plate is given by (6) where q(x,y) denotes the applied transverse load per unit area. Substituting Equations (5a-c) into Equation ( 6) yields the governing equation of the isotropic plate subjected to a NUH (7) The Kirchhoff shear forces per unit length are introduced to satisfy the boundary and continuity conditions; they combine shear forces and twisting moments, and can be expressed for an isotropic plate as follows:…”

Analysis of Arbitrarily Loaded Rectangular Thin Plates with Two Opposite Edges Supported, One or Both of them Clamped, Using the Flexibility Method and a Modified Lévy Solution

“…Many exact solutions for isotropic linear elastic thin plates have been developed by Timoshenko [5]; the simple trigonometric series of Lévy was mostly considered. Mama et al [6] presented the single finite Fourier sine integral transform method for the flexural analysis of rectangular Kirchhoff plate with opposite edges simply supported, and the other edges clamped for the case of triangular load distribution on the plate domain. Sayyad et al [7] assessed a trigonometric plate theory for the static bending analysis of plates resting on Winkler elastic foundation; the theory considered the effects of transverse shear and normal strains.…”