Series solution of the biharmonic equation in the rectangular domain with some applications to mechanics

Abstract

SummaryA double series solution of the biharmonic equation is derived in the rectangular domain. The solution utilizes the finite double Fourier sine transform and the analysis shows how the transform may be evaluated for quite arbitrary boundary conditions, the solution then being found from the corresponding inversion formula.The method is applied to some classical boundary value problems which have been investigated by other means by earlier workers. It is felt that some of the series solutions given here are more amenable to computation than some of the earlier methods.In some of the problems considered, notably the elastostatics of the rectangular plate and the bending of clamped plates, the method reduces to solving an infinite set of linear equations. In such cases, a finite leading matrix of the coefficients has to be considered. In the examples considered, this technique produced results which converged to known solutions obtained by other methods. However, the mathematical criteria which permit the infinite system to be curtailed, have not been established.