The biharmonic problem and progress in the development of analytical methods for the solution of boundary-value problems

A comparative analysis of harmonic and biharmonic boundary-value problems for 2D problems on a rectangle is given. Some common features of two types of problems are emphasized and special attention is given to the basic distinction between them. This distinction was thoroughly studied for the first time by L. N. G. Filon with respect to some plane problems in the theory of elasticity. The analysis permits to introduce an important aspect of the general solution of boundary-value problems. The procedure for solving the biharmonic problem involves both the method of homogenous solutions and the method of superposition. For some cases involving self-equilibrated loadings on one pair of sides of the rectangle, the complete solution, including calculation of the quantitative characteristics of the displacements and stresses, is given. The efficiency of the numerical implementation of the general solutions is shown. The analysis of the quantitative data allows to elucidate some main points of the Saint-Venant principle.