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The analysis of the bending of thin deep rectangular beams by concentrated loads has already been treated by L.N.G. Filon and Th. v. Karman. These authors consider an infinitely long beam and express the load as a Fourier integral, obtaining integral solutions for the stresses and displacements. In their integral fonm, these results are rather difficult to interpret, although F. Seewald, using Karman's analysis, has calculated numerical values for the case of a single concentrated load. In this paper, a totally different approach to the problem is made. The loaded area is conformally transformed to a circle, and, using Muschelisvili's transformation of the boundary conditions, the problem is solved in the plane of the circle. The solution is then obtained in the form of a complex power series. In this manner, a direct solution, to any required degree of accuracy, is readily obtained. 1. Theory. Consider a long deep rectangular beam resting on two supports placed at the same level, and loaded by a weight placed midway between them. The weight of the beam will be neglected, and the beam will be assumed to be in a state of generalised plane stress, the plane of the mean stress being that of the length and depth. To determine the local effects of the concentrated loads on the stresses and deflexion it is convenient to move the supports to infinity and treat the beam as an infinitely long strip of depth 2h as is shown in Fig. 1. This procedure is that adopted by L.N.G. Filon, H. Lamb,

An iterative solution to the effects of concentrated loads applied to long rectangular beams