I propose, in the first place, to give a brief account of the principal theories of the vibrations and flexure of a thin elastic plate hitherto put forward, and afterwards to apply the method of one of them to the case when the plate in its natural state has finite curvature. Passing over the early attempts of Mdlle. Sophie Germain, the first mathematician who succeeded in obtaining a theory of the flexure of a thin plane plate was Poisson. In his memoir he obtains the differential equation for the deflection of the plate, which is generally admitted, and certain boundary-conditions, which have met with less general acceptance. The idea of Poisson's method may be simply stated. The plate being very thin, we may expand all the functions which occur in the equations of equilibrium and boundary-conditions in powers of the variable expressing the distance of a particle from the middle-surface in the natural state, then, taking only the terms up to the third order, we obtain the differential equations for the determination of the displacements which are generally admitted. The meaning of Poisson’s boundary-conditions is as follows:—Suppose the plate to form part of an infinite plate, and to be held in its actual position, partly by the forces directly applied to its mass, and partly by the action of the remainder of the plate exerted across the boundary; if the plate be now cut out, it will be necessary, in order to hold it in the same configuration, to apply at every point of its edge a distribution of force and couple identical with that exerted by the remainder before the plate was cut out. Now, it has been shown by Kirchhoff that these equations express too much, and that it is not generally possible to satisfy them; but the method proposed by Thomson and Tait gives a rational explanation of Kirchhoff’s union of two of Poisson’s boundary-conditions in one, and renders his theory complete. However, the objection raised by de St. Venant to the fundamental assumption that the stresses and strains in an element can be expanded in integral powers of the distance from the middle-surface, seems to require a different theory.
XVI. The small free vibrations and deformation of a thin elastic shell
