Limit behaviour of masonry arches in the presence of finite displacements

A procedure is proposed for the calculation of the equilibrium paths of a masonry arch with elastic abutments. The elasticity of the abutments implies the existence of a non-linear, non-trivial equilibrium path, with the development of three hinges from the beginning of the load history. The shape of the equilibrium path suggests that shallow masonry arches should be classified in three different ways according to the nature of their failure load. In the first class a fourth hinge can develop as the live load is increased, and a classical failure load is reached when a four-hinge mechanism occurs. In the second case the equilibrium path shows a limit point, and finally it is possible to have a monotonically increasing equilibrium path, with no limit point or bifurcation point. If the arch is very shallow, only these two latter possibilities must be examined, and the presence of non-rigid abutments can cause instability of the arch. This instability phenomenon could be particularly important for flying buttresses of cathedrals, if they were built along rivers, or more generally on elastic soil.