Stability of masonry piers and arches

A system of finite dimensional rigid bodies, such as a masonry arch, can be interpreted as a nonholonomic system in which there are constraints on the generalized coordinates. The potential energy function for a system of rigid blocks can be written as a mathematical programming problem: Minimize the potential energy subject to kinematic constraints on the degrees of freedom. A solution to this problem is a stable equilibrium state. Well-known results from the theory of optimization are applied to the solution. This formulation of the problem leads to a useful interpretation of the Lagrangian multipliers, from which the lower-bound condition of plastic analysis is directly obtained as a sufficient condition for the stability of the system. The upper-bound condition, which is also recovered from this formulation of the problem, is not a sufficient condition for instability of all systems. However, it is shown that for most systems of practical significance, the upper-bound condition is a sufficient condition for instability, and the lower-bound condition is a necessary condition for stability.