Editing Arch (section)

=== Funicular shapes ===
{{main|Funicular curve}}
When evaluated from the perspective of an amount of material required to support a given load, the best solid structures are compression-only; with the flexible materials, the same is true for tension-only designs. There is a fundamental symmetry in nature between solid compression-only and flexible tension-only arrangements, noticed by [[Robert Hooke]] in 1676: "As hangs the flexible line, so but inverted will stand the rigid arch", thus the study (and terminology) of arch shapes is inextricably linked to the study of hanging chains, the corresponding curves or polygons are called ''funicular''. Just like the shape of a hanging chain will vary depending on the weights attached to it, the shape of an ideal (compression-only) arch will depend on the distribution of the load.{{sfn|Allen|Ochsendorf|West|2016|p=40}}
<gallery>
File:Analogy between an arch and a hanging chain and comparison to the dome of St Peter's Cathedral in Rome.png|Analogy between an arch and a hanging chain and comparison to the dome of [[Saint Peter's Basilica]] in Rome ([[Giovanni Poleni]], 1748)
File:Maqueta funicular.jpg|A complex funicular model ([[Church of Colònia Güell]] by [[Gaudi]], 19th century)
</gallery>

[[File:Pressure polygon.png|thumb|Arch diagram with pressure polygons drawn. A defect at the R-S portion of the intrados makes the arch susceptible to extra force along the line M-N, where the polygon curve can be pushed out of the envelope of the arch causing a collapse]]
While building masonry arches in the not very tall buildings of the past, a practical assumption was that the stones can withstand virtually unlimited amount of pressure (up to 100 [[Newton (unit)|N]] per mm<sup>2</sup>), while the tensile strength was very low, even with the mortar added between the stones, and can be effectively assumed to be zero. Under these assumptions the calculations for the arch design are greatly simplified: either a reduced-scale model can be built and tested, or a funicular curve ([[pressure polygon]]) can be calculated or modeled, and as long as this curve stays within the confines of the voussoirs, the construction will be stable{{sfn|Calvo-López|2020|pp=8-9}} (a so called "[[safe theorem]]").