catenary: one construct, many systems

In mathematics, the catenary is the curve a uniform flexible chain or cable settles into when hung from two points and loaded only by its own weight, described by the hyperbolic cosine. Lockwood's A book of curves treats its construction and properties, including that it is not the parabola Galileo first guessed but a distinct curve whose shape depends only on the balance of tension and weight along it. The referent is a specific plane curve defined by that equilibrium, and it is the same underlying object that engineers hang an overhead wire to follow and that masons invert into an arch.[1]

Unit

The hyperbolic-cosine shape of a self-weighted hanging cable

Recovery means

Not applicable

Measured by

Described analytically by the hyperbolic-cosine equation and its parameter, fitting sag, span, and tension of a hanging cable

In the structural analysis of masonry, the inverted catenary is the shape a pure arch would take to carry its own weight in compression alone, the principle behind Hooke's maxim that as a flexible cable hangs, so, inverted, an arch will stand. Heyman's The stone skeleton develops this into the thrust-line analysis by which a masonry arch is safe as long as a line of compression close to the inverted hanging curve can be fitted within the stonework. The referent is the same hanging-chain curve as in mathematics, now turned upside down and used to judge whether an arch or vault stands, measured by how the thrust line sits within the masonry.[2]

Unit

The inverted hanging-curve thrust line fitted within an arch

Recovery means

Not applicable

Measured by

Assessed by fitting a line of thrust (the inverted catenary of the loads) inside the masonry section to test stability