Application of numerical Green's functions in the design of plates using symbolic computation

Abstract

In order to solve even simple plate problems, one may need to set up quite an amount of computer codes for the implementation of general numerical methods such as finite element analysis. Using the semi-analytical approach introduced in this paper, the amount of time-consuming work may be reduced significantly. It would be even more convenient if the solutions were given in the forms of algebraic expressions instead of numerical values. A primary concern in the design of plates would be the deflections of plates due to loadings at various points inside the plates. The deflections can be obtained from an influence function known as the Green's function of the plates. However, except a few known simple cases, it is difficult to obtain Green's function of the plate for various shapes and boundary conditions analytically. To circumvent such difficulties in finding this function, a numerical Green's function is derived in this paper for the first time. Once the numerical Green's function is obtained the solution, a lateral deflection in this paper, may be calculated by carrying out simple integration. Also, since the numerical Green's function obtained in this paper is given in terms of algebraic polynomials, it is suitable for differentiation or integration without introducing numerical errors using symbolic computation. Utilizing the numerical Green's functions of plates, a simple and accurate yet flexible approach to the subject is addressed.