Exact Analytical Solutions to Bending Problems of SFrSFr Thin Plates Using Variational Kantorovich-Vlasov Method

This article applies the variational Kantorovich-Vlasov method to obtain exact mathematical solutions to the bending problem of thin plate with two opposite simply supported edges and two free edges. Vlasov method was adopted simultaneously in the variational Kantorovich method, and the deflection function w(x, y) is expressed in variable-separable form as single infinite series in terms of the unknown function g(y) and known sinusoidal functions of x coordinate variable f(x) where f(x) satisfies Dirichlet boundary conditions at the simple supports. The total potential energy functional , expressed in terms of g(y) and the derivatives g(y), g(y) is then minimized with respect to g(y) using the Euler-Lagrange differential equations. The resulting equation of equilibrium is a fourth order inhomogeneous ordinary differential equation (ODE) in g(y). The general solution is found and boundary conditions are enforced to find the integration constants. The expression found for w(x, y) satisfies the governing equations on the domain and the boundaries and is thus exact within the scope of thin plate theory adopted to idealize the plate. Moment-deflection equations are used to obtain exact analytical expressions for the bending moments Mxx, Myy. Deflection and bending moments are computed at the plate center; as well as at the middle of the free edges. Comparison of the plate center deflections and bending moments for various aspect ratios illustrate that the exact solutions by the present work are in the agreement with Levy solutions presented by Timoshenko and Woinnowsky-Krieger and symplectic elasticity solutions presented by Cui Shuang. The present results for bending moments at the free edges for various aspect ratios agree with the Levy results presented by Timoshenko and Woinowsky-Krieger and symplectic elasticity results presented by Cui Shuang.