Catenary & Funicular Arch

Self-supporting arch following an inverted hanging-chain curve so gravity loads resolve purely into axial compression.

Specifications & Metrics

type: Long-Span Bridge & Arch System
load mechanism: Eliminates bending moments and shear by aligning the arch profile with the funicular line of thrust
governing relation: y = a · cosh(x / a) = (a / 2) · (e^(x/a) + e^(−x/a)) (uniform catenary curve)
classic configurations: Uniform inverted catenary vault, weighted (flattened) catenary arch, spun catenary dome
exemplars: Taq Kasra (Arch of Ctesiphon), St. Louis Gateway Arch, Antoni Gaudí's Casa Milà & Sagrada Família

Stated by Robert Hooke in 1675 as the Latin anagram 'Ut pendet continuum flexile, sic stabit contiguum rigidum inversum' ('As hangs a flexible cable so, inverted, stand the touching pieces of an arch'), an inverted catenary y = a·cosh(x/a) directs its own weight entirely along its centerline in pure compression with zero bending moment.
Built without centering falsework in the 3rd to 6th centuries CE, the Sasanian brick vault of Taq Kasra at Ctesiphon (Iraq) — still the world's largest single-span unreinforced brick vault at 37 meters high and 25 meters wide — follows a catenary profile.
Antoni Gaudí engineered the complex vaults of the Church of Colònia Güell and Sagrada Família using inverted hanging-chain-and-shotbag models, while Eero Saarinen's 192-meter St. Louis Gateway Arch follows a 'weighted catenary' (y = A(cosh(Cx/L) − 1)) tailored to its tapered stainless-steel cross-section.