Finite-friction least-thickness self-standing domains of symmetric circular masonry arches. Part III: Complementary representations, extremal conditions and collapse multiplicity

This work completes an analytical investigation on the role of finite (Coulomb) friction, in the classical “Couplet-Heyman problem” of least-thickness optimization of self-standing symmetric circular masonry arches. Following a source (Part I) companion work, considering classical, approximate Heyman-like uniform self-weight distribution along geometrical centreline, and a Part II addendum, dedicated to the generalization to the true Milankovitch-like uniform self-weight distribution, accounting for the real centres of gravity of the ideal wedge-shaped infinitesimal chunks of the continuous arch, it highlights, for the latter real distribution, diverse final key aspects, for a comprehensive mathematical and physical interpretation. Namely: (a) the complete two- and three-dimensional representation maps of all characteristic solution variables as a function of geometrical and friction parameters are outlined, to show the rich, multi-faceted mechanical features of the solution in terms of physical states; (b) the peculiar extremal conditions for intrinsic non-dimensional horizontal thrust and inner failure joint angular positions are newly derived, at variable inherent friction coefficients, to inspect how friction may affect such self-supporting abilities, with respect to typical underlying purely-rotational collapse modes with no sliding failure at high or infinite friction; (c) the mathematical multiplicity of the collapse states at the interfaces of the failure domains is analyzed, to unveil distinctive kinematical characteristics of the feasible collapse modes. The analytical achievements, corroborated by a separate numerical experimentation, validation and illustration, by a home-made Complementarity Problem/Mathematical Programming (CP/MP) implementation, allow for a full perception of the problem at hand, in the Mechanics (statics) of masonry arches, and specific related role of finite friction, in providing crucial self-bearing structural capacity.