On some aspects of implementation of boundary elements method in plate theory

Abstract

The article deals with the problem arising in the construction of a numerical scheme of the first-order boundary element method for plate theory. During construction of such a scheme, the initially smooth boundary of the plate is replaced by a polygonal chain. Due to this replacement the deviation of the numerical results from the actual distribution of deflections and other characteristics is arisen. The reason for this deviation lies in the so-called Sapondzyan's paradox. According to it, the deflection of a plate bounded by a regular polygon does not converge to the deflection of a circular plate with increasing of the polygon sides number. In the paper, on the basis of an analytical consideration of Sapondzyan's problem, the components of the numerical scheme of the boundary element method, which are responsible for the mentioned deviation, are pointed out. The modification of the boundary element method scheme that allows to eliminate given problem is presented. This approach is tested on the example of solving two pairs of problems for elliptical and rectangular plates. The results of numerical solution of those problems confirmed the adequacy of the proposed modification.

Analytical description of adjustment of rolls for manufacturing parts from elastic sheet material

“…In contrast to [6], according to the developed method, it is possible to adjust conical rolls for the production of conical parts from sheet material with a known spring coefficient. However, the question arises how to make a conical part for which the radii of both bases are given.…”

“…During bending of a completely elastic sheet, its original shape is completely restored. On the other hand, when an elastic sheet is bent, the shape is partially restored [6]. It is important to take this into account during preliminary calculations in order to obtain a cylindrical part with the required radius after partial expansion of the sheet [7].…”