Decay Rates for the Hollow Circular Cylinder

The self-equilibrated end load problem for a hollow circular cylinder is considered using the Papkovitch-Neuber solution to the elastostatic displacement equations of equilibrium; both axi- and nonaxisymmetric solutions are derived. The requirement of zero traction on the surface generators of the cylinder leads to an eigenequation whose roots determine the rate of decay with axial coordinate. The locus of the smaller roots is plotted for circumferential harmonic loadings n = 0, 1, 2, and 3, for different wall thicknesses, and supplement previously known decay rates for the solid section and the circular cylindrical shell which are the extremes of diameter ratio. The loci are of considerable intricacy, and for small wall thickness, simple shell theory and two modes of decay for the semi-infinite plate are employed to identify the various modes of decay. Whereas for the solid cylinder the characteristic decay length of Saint-Venant’sprinciple is the radius (or diameter), for the hollow cylinder it becomes possible to discriminate between “wall thickness” and “$rmt$” modes of decay according to the limiting behavior as the cylinder assumes shell-like proportions; the one exception is “membrane bending” for which self-equilibrating end loading does not decay as thickness tends to zero.