In the direct boundary element method (DBEM) for thin plate bending analysis. the integration of the kernelshape-function products can lead to complications due to a combination of factors. The first is the existence of a singularity at the source point, leading to the singular kernel, which increases in strength as the integral equation is differentiated once for rotation. again for moment and yet again lor shear. The second factor arises from the use of piecewise interpolation polynomials.
In the first part ofthe DBEM analysis where the unspecified boundary values are computed, the numerical procedures are fairly straightforward. It is found in this paper that special care must be taken in calculating internal actions at domain points. The problems which can be encountered are demonstrated, and a modified boundary interpolation procedure is described. Examples are given which show the superiority of the recommended approach.
BACKGROUND
In an earlier paper,' the authors dealt with a procedure for carrying out the integrals around the plate boundary as the first step in the direct boundary element method (DBEM), where the specific concern was with the calculation of integrals having singular kernels. For these integrals the source point locations were on the boundary itself, with the integration path going through these sources. Serious problems arise when Gauss quadrature is employed for the integration of the functions which contain, or arc close to, singularities, but these can be avoided if the integrations arc carried out analytically and the results coded directly. This technique was developed in Reference 1.
In the present paper, the focus is on the calculation of interior values. Again, integrals around the entire boundary of the plate are involved, and although the point loads are not on the integration path, there are special problems since the kernels are more strongly singular than those encountered in the first part of the analysis.
This paper is a continuation of a longer term study which, it is hoped, will eventually provide practicable procedures for modelling and analysing the floors of building frames. The previous work of others must be acknowledged. Stern and his associates'. have provided a general framework which the present authors are attempting to extend. 'The concept of analytic integration in boundary element methods has been previously studied by Banerjee and Cathie4 and Vable.5 The authors have done some further work which is reported in Reference 1, and supplemented herein. Finally, for the authors, this has been a continuing exercise in finding ways to solve difficult integrals. It would not have been possible to have come this far without the background given in Hartmann's work (for example References 6 and 7).