Boundary Integral Formulations for Some Special Elastostatic B.V.Ps

Abstract

The study of the static and dynamic behaviour of structures made up of bars and beams, i.e. in trusses and frames, is one of the main tasks in applied mechanics and engineering. Although, in this field, the F.E.M is already very well developed and almost exclusively sovereighing, there are certain problems, e.g. shape optimization, where the boundary indeed plays the predominant role and, because of that, boundary methodologies had better been prefered. In this chapter we deal with certain boundary integral formulations for some of the basic beam, bar and plate problems. To elucidate the derivation possibilities of Ch.2, all the previously introduced methods are applied in this Chapter in order to obtain the integral equations. Let us consider first the bending of beams. Bending is assumed to be independent of the stretching in the framework of a geometrically linear theory (see 3.1>). The longitudinal displacement u(x) = u ₁(x) and the deflection w(x) = u ₃(x) have to satisfy the two independent basic differential equations

$$ \frac{{d^2 }}{{dx^2 }}\left[ {EI\frac{{d^2 }}{{dx^2 }}w(x)} \right] = q(x)on0 \le x \le 1, $$

(3.1)

$$ - \frac{d}{{dx}}\left[ {EA\frac{d}{{dx}}u(x)} \right] = p(x)on0 \le x \le 1. $$

(3.2)