An Eigenvalue Problem for a Semi - infinite Pretwisted Strip

‘For an isotropic, homogeneous and linearly elastic sheet subjected to selfequilibrating edge stresses over a portion of its boundary and free of surface and edge load otherwise, an explicit rate of decay of the stresses in the sheet was given in [1 ]. In particular, the stresses in a semi-infinite rectangular strip subjected to self-equilibrating in-plane normal and shear stress resultants along its short edge become exponentially small at a distance large compared to the strip width away from the loaded edge. For a strip with a small amount of pretwist, one would expect the rate of stress decay to be qualitatively similar to that of a flat strip. However, it was noted in [2] that a Saint-Venant type solution for a pretwisted rectangular strip subjected to equal and opposite sheet end bending moments, first obtained in [3], becomes unrealistic whenever a dimensionless pretwist parameter h is large compared to unity. With p proportional to the amount of pretwist and inversely proportional to the sheet thickness, the pretwist parameter may be large even for a slightly pretwisted strip, if the sheet is sufficiently thin. Therefore, an analysis of the (self-equilibrating) edge load problem for the semi-infinite pretwisted strip should be instructive. For a strip with a relatively small amount of pretwist so that pal, it is apparent that a straightforward perturbation solution [4] would give four families of eigenfunctions for the relevant eigenvalue problem, all with a decay length which is essentially that of a flat strip with a small correction term of order p. As we shall see, the situation for ,~>l is qualitatively different from that of the flat strip. Of the four possible families of solutions in this case, three have a decay length short (of order ,u-‘i3) compared to the strip width. The decay length of the remaining family is much longer, by a factor p, than the strip width. Therefore, a Saint-Venant type solution for the sheet bending problem is of dubious value beyond the range of p considered in [3]. In particular, the detailed distribution of applied edge loads is likely to be significant for a pretwisted strip sufficiently thin so that @+ 1.