The Prismatic Bar

Abstract

We have seen in Chapter 25 that three-dimensional solutions can be obtained to the problem of the solid or hollow cylindrical bar loaded on its curved surfaces, using the Papkovich-Neuber solution with spherical harmonics and related potentials. Here we shall show that similar solutions can be obtained for bars of more general cross-section, using the complex-variable form of the Papkovich-Neuber solution from §21.6.

We use a coördinate system in which the axis of the bar is aligned with the z-direction, one end being the plane z = 0. The constant cross-section of the bar then comprises a domain Ω in the xy-plane, which may be the interior of a closed curve Г, or that part of the region interior to a closed curve Г₀ that is also exterior to one or more closed curves Г₁, Г₂, etc. The following derivations and examples will be restricted to the former, simply connected case, but it will be clear from the methods used that the additional complications asssociated with multiply connected cross-sections arise only in the solution of two-dimensional boundary-value problems, for which classical methods exist. As in Chapter 25, we shall apply weak boundary conditions at the ends of the bar, which implies that the solutions are appropriate only for relatively long bars in regions that are not too near the ends.