The Solution of the Generalized Boussinesq's Problem for Elastic Foundation. I
Proceedings of the Japan Academy
Y. TANAKA, M.J.A., Oct. 12, 1955) Since the time of Boussinesq, theoretical basis for the problem of the safety of elastic foundation has been discussed and developed by various investigators, of which the works by Prof. Terazawa' and Love2' are noteworthy. The former discussed it in detail when the loaded area is of circular form, and the latter gave the integration of Boussinesq's potentials when a uniform pressure extends over a rectangular area. But as is well known, the Boussinesq's
... al method cannot be compatible with shearing forces at all, which would be of considerable importance especially in the case of soft foundation. In addition the Boussinesq's potentials are difficult to perform their integrations, and only the simplest case cited has been treated by Love. Numerical process of their integration is also almost impossible by ordinary methods of numerical integration, since integrands involved have an infinite number of singular points. The boundary-value problem here treated is that, within a rectangular form of loaded area, any distributions of two kinds of shearing forces as well as of normal pressure are given on the semi-infinite elastic solid, provided these three kinds of external forces are expressible in terms of Fourier's integral in two dimensions. The procedure of the calculation is for convenience due to a new set of functions which has been proposed by me and might be called stress-functions in three dimensions.3' The resulting solution is obtained in the forms of Fourier's integral, and the evaluation of the integrals was relied on the method of mechanical cubature4'5 because of the difficulty in its analytical performance. As regards stresses there is no singularity in the integrands, and to secure first two or three significant figures in the numerical result is not so laborious. Displacements can also be integrated, in spite that each of integrands involved has one singularity at the origin of parametric coordinates. Applications of the general solution will be given to several cases: (1) uniform pressure, (2) uniformly varying pressure, (3) uniform shearing force, (4) uniformly varying shearing force, etc.