Vaulted Egyptian toy house frem approximately 2400 BC.
Photo: K. Hertz
Arches, and vaults have been applied for buildings for at least 7000 years.
A compressive force in a curved arch can be in equilibrium with a distributed load.
In practice most arches must take some moment load from eccentric live load. This can often be done by unloading compression in the one side and increasing compression in the other. In addition many arches and vaults have been made as compromise between what is ideal from a load-bearing point of view and what is ideal for application of the space beneath the arch.
Formulas for parabolas, catenaries, super-ellipses, and super-lines.