Solving the Problem of the Bending of a Plate with Fixed Edges by Reduction to an Infinite Systems of Equations

Abstract

We consider the known problem of the bending of a plate with fixed edges and a homogeneously distributed load. The solution can be found by solving infinite systems of equations. Based on the construction of a solution of biharmonic equations with boundary conditions, it is proven that these infinite systems have a finite unique solution. We use the existence of a special particular (“strictly particular”) solution (SPS) of the systems to which the solution obtained by the simple reduction method converges. Such a solution always exists, if the general system is consistent and it has the following particular properties: 1) it is a unique particular solution, which is expressed by Cramer’s formula; 2) it does not contain the nontrivial solution of the corresponding homogeneous system as an additive term; 3) the well-known principal solution of the infinite system coincides with the SPS. The SPS allows calculating the values of the plate deflection, bending moments and pressures at its contour. It is shown that the construction of the SPS (in fact, the exact solution) of the obtained infinite system does not depend on its regularity or irregularity.