Vector Analysis and the Stationary Potential Energy for Assessing Equilibrium of Curved Masonry Structures

Curved structures have enabled masons, engineers and architects to carry heavy loads and cover large spans with the use of low-tensile strength materials for centuries, while creating the marvels of the world’s architectural heritage. Despite the long history of these practices, finding optimal structural forms and assessing the stability and safety of curved structures remains as topical as ever. This is due to an increasing interest to preserve heritage structures and reduce material use in construction, while replacing steel and concrete with low-carbon natural materials.
The analogy between inverted hanging chains and the optimal shape of masonry arches is a concept deeply rooted in our structural analysis practices. The paper revisits the equilibrium of the hanging chain, following the transition from Newtonian to Lagrangian Mechanics. Understanding these ideas reveals that hanging chains and arches are two incompatible structural systems. The paper discusses the limitations of describing the equilibrium of two-dimensional objects with finite thickness (e.g. arches) by using one-dimensional objects of infinitesimal thickness (e.g. hanging chains, funicular lines) on the geometry of the curved structure and the loading conditions that can be assumed in practice.
These limitations manifest themselves by applying force equilibrium that carefully considers the stereotomy exercised, which becomes particularly critical when studying the stability of vertical sections or considering seismic loads. The paper shows that by taking the logical progression towards Lagrangian Mechanics, one may obtain rigorous solutions for the limit equilibrium state of curved structures by applying the principle of stationary action. This approach liberates the analyst from the need to consider equilibrium of each individual block or describing geometrically the load path of thrust forces.