Arresting the collapse of a catenary arch

Abstract: It is well known that viable architectural structures can be identified by locating the critical points of the gravitational potential energy congruent with some fixed surface metric. This is because, if the walls are thin, the lowest energy modes of deformation are strain-free, and thus described by surface isometries. If it is to stand, however, an arch had better possess some minimum rigidity. The bending energy consistent with this construction protocol, we will show, can only depend on curvature deviations away from the reference equilibrium form. The question of stability, like the determination of equilibrium, turns on the geometry. We show how to construct the self-adjoint operator controlling the response to deformations consistent with isometry. As illustration, we reassess the stability of a simple catenary arch in terms of the behavior of the ground state of this operator. The energy of this state increases monotonically with the bending rigidity and it is possible to identify the critical rigidity above which the arch is rendered stable. While this dependence may be monotonic, it exhibits a number of subcritical kinks indicating significant qualitative changes in the ground state associated with eigenvalue crossovers among the unstable modes in the spectrum; the number of such competing modes increasing rapidly as the rigidity is lowered. The initial collapse of a subcritical arch is controlled by the ground state; on the critical threshold, there are two unstable modes of equal energy, one raising the arch at its center, the other lowering it. The latter dominates as the instability grows. The qualitative behavior of the ground state changes as the rigidity is lowered---its nodal pattern as well as its parity undergoing abrupt changes as the intervals between crossovers converge---complicating the prediction of the dominant initial mode of collapse.