Asymptotic solutions for shells with general boundary curves

Abstract

Abstract.The influence of arbitrary edge loads on the stresses and deformations of thin, elastic shells with general boundaries is studied by means of asymptotic expansions of a general tensor equation. Expansions are made in terms of an exponential or an Airy function and a series in powers of a small-thickness parameter. Most of the steps in the procedure are effected by using the dyadic form of the tensors. Solutions are obtained that are valid in the large, with no restrictions on the loading or on the boundary geometry.Results indicate that the behavior of shells with arbitrary boundary geometry can be quite different from that in the ordinary case. Specific results show the presence of an interior caustic which is the envelope of the characteristics of an eiconal equation. The exponential expansion becomes singular at the caustic, which would generally be expected to be a local region of stress concentration.The results have a close identity with asymptotic solutions obtained in geometric optics. Following some new techniques used for solution of the reduced wave equation, a solution of the shell equation is obtained using an asymptotic expansion in terms of Airy functions which provides a solution that is uniformly valid in the neighborhood of the caustic. Introduction.The primary purpose of this study is to investigate the influence of general edge loads on the stresses and deformations of thin elastic shells. The shell equations were cast into a single, compact dyadic form by Steele [1] that will be used here without development. Some details of that derivation and additional algebraic details of equations developed in this paper are given in [2], The type of solution sought is a generalization of the well-known solution, with a decaying behavior from the shell boundary into the shell interior, commonly referred to as an edge-effect solution.Analytic solutions available in the literature are applicable only to shells with restricted boundaries and loading conditions, or are valid only close to the boundary. Certainly the most widely used edge-effect solution is that for the axisymmetricallyloaded circular cylinder with a boundary on a circumferential line of curvature, and that result may be obtained as a special case of our general solution.Perhaps the best known solution for a shell with an arbitrary boundary is that of Goldenveizer [3]. However, that solution involves a basic assumption that restricts the region of applicability of the solution to a small boundary layer near the shell edge.

Bending waves in shells

“…In this case, the dependence on d is not entirely eliminated in the time-averaged Lagrangian, so the Euler equation (3.13) would appear to gain additional terms. However, by the straightforward expansion procedure used by Steele [8], Prat [5] and Logan [3], which is valid for complex k and for the static case of u -> 0, the identical relations (3.23) -(3.26) are obtained. This indicates that, instead of using the time-averaged Lagrangian, a more fundamental approach could be taken, in which the possibility of an evanescent wave would be admitted, but which produces the same relations (3.23)-(3.26).…”