Eigenfunctions of plane elastostatics I. The strip

Abstract

By the use of an analytic continuation technique, the problem of the strip in plane elastostatics is reduced to the solution of a differential-difference equation by Fourier transforms. The resulting integrals are evaluated by residue theory to give eigenfunction expansions. Previous workers have used either Fourier integral methods, or eigenfunction expansions, and it is shown how these two distinct approaches can be unified in a single theory. The completeness of the eigenfunction expansions is established and it is shown how the singularities at the origin of the integrands of the Fourier integrals give rise to polynomial solutions, corresponding to the eigenvalues zero. These polynomials represent the classical beam bending and stretching theories, while the other terms in the eigenfunction expansions illustrate St Venant’s principle for the strip. Particular examples of isolated loads on the boundaries are used to illustrate the theory. In the latter part of the paper, the problem of the semi-infinite strip is discussed. A closed solution is obtained for the particular case of plate flexture in which the short edge is simply supported and the two long edges are clamped. A closed solution in the general case having not yet been found, a technique for obtaining a good approximate solution is suggested.