Elastic flexural-torsional instability of structural arches under hydrostatic pressure

This paper addresses the mechanics of the flexural-torsional buckling instability of pin-ended elastic circular arches, which are acted upon by a hydrostatic loading. This loading arrangement differs from the gravity-based loading usually considered in the literature, in that the load changes its direction with the deformation of the elastic arch during its flexural-torsional buckling, always remaining normal to the contour profile of the arch. The previous treatments of the mechanics of the problem, that assume the load direction remains invariant during flexural-torsional buckling, have been motivated by applications in structural engineering in which this loading regime is valid, but there are a number of applications in more general mechanics where this assumption cannot be made and a solution is needed. Both a mathematically based virtual work principle and a mechanical visualisation of the mechanics of the deformation are considered separately, and they are shown to arrive at the same formulation of the linear differential equations of equilibrium of the buckled arch when the buckling deformations are considered infinitesimal. The differential equations for buckling under radial loading that is distributed uniformly around the circumference of the arch are shown to be solvable in analytic form, resulting in a closed form solution for the elastic buckling load of the arch that hitherto has not been formulated. The buckling equation demonstrates that an arch is stiffer under hydrostatic loading than under gravity loading in its resistance to elastic flexural-torsional buckling.