The Kantorovich-Vlasov method was used, in this study, for the flexural analysis of rectangular Kirchhoff plates with opposite edges (x = 0, and x = a) simply supported and the other opposite edges (y = 0, and y = b) clamped (CSCS plates). The plate was subjected to a linear distribution of load over the entire plate domain. Vlasov method was used in finding the coordinate function in the x-direction, and Kantorovich method was used to consider the displacement function for the plate. The total potential energy functional, and the corresponding Euler-Lagrange differential equations were then obtained for the plate problem. This was solved subject to the boundary conditions in the y direction to obtain the displacement function which minimized the total potential energy functional. Bending moment distributions were obtained using the bending moment-displacement equations. The solutions obtained for deflection and bending moment distributions were found to be rapidly convergent single series. Deflections and bending moment computed at the center of the plate were also rapidly convergent series. The solutions obtained for deflections and bending moments (M xx and M yy) were exactly identical with solutions presented by Timoshenko and Woinowsky-Krieger who used the method of superposition.
Abstract-In this work, the mathematical theory of elasticity has been used to formulate and derive from fundamental principles, the first order shear deformation theory originally presented by Mindlin using variational calculus. A relaxation of the Kirchhoff's normality hypothesis was used to account for the effect of the transverse shear strains in the behaviour of the plate. This made the resulting theory appropriate for use for moderately thick plates. A simultaneous use of the strain-displacement relations for small-deformation elasticity, stress-strain laws and the stress differential equations of equilibrium was used to obtain the differential equations of static flexure for Mindlin plates in terms of three unknown generalised displacements. The equations were found to be coupled in the unknown displacements; but reducible to the Kirchhoff plate equations when the removed Kirchhoff normality hypothesis was introduced. This showed the Kirchhoff plate theory to be a specialization of the Mindlin plate theory. The theory of elasticity foundations of the Mindlin and Kirchhoff plate theories are thus highlighted.
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