IV. On the strains in the Interior of beams

Abstract

I have long desired to possess a theory which should enable me to express and to compute numerically the actual strain or strains upon every point in the interior of a beam or girder, under circumstances analogous to those which occur in ordinary engineering applications,—partly for information on the amount of force actually sustained by the different particles of the cast or wrought iron in a solid beam, partly as a guide in the construction of lattice-bridges. The memoirs and treatises on the theories of elasticity and strains, to which I have referred, have given me no assistance. I have therefore constructed a theory, in a form which (I believe) is new, which solves completely the problems that I had proposed to myself, and which, as I think, may, with due attention to details, be applied to all the cases that are likely to present themselves as interesting. This theory, with some of its first applications, I ask leave to place before the Royal Society. 1. It is supposed, in the following investigations, that the beam consists of one lamina in a vertical plane,—the idea of a solid beam being supplied by the conception of a multitude of such laminae side by side, all subject to similar strains, and therefore exerting no force one upon another. It is also supposed that the thickness of the lamina is uniform, and that its form is rectangular, the depth of the beam being equal throughout: these suppositions are made only for the sake of simplicity, as there does not appear to be any difficulty of principle in applying the theory to cases not restricted by these conditions, although the complexity would be much increased. It also appears necessary to suppose that the material of the beam yields equally, with equal forces, in different directions. Another physical supposition, which appears to be necessary for complete solution of the problem, will be stated when we reach the discussion of the first instance.