A weighted catenary (also flattened catenary, was defined by William Rankine as transformed catenary [1] and thus sometimes called Rankine curve [2] ) is a catenary curve, but of a special form: if a catenary is the curve formed by a chain under its own weight, a weighted catenary is the curve formed if the chain's weight is not consistent along its length. Formally, a "regular" catenary has the equation
for a given value of a. A weighted catenary has the equation
and now two constants enter: a and b.
A freestanding catenary arch has a uniform thickness. However, if
it becomes more complex. A weighted catenary is needed.
The aspect ratio of a weighted catenary (or other curve) describes a rectangular frame containing the selected fragment of the curve theoretically continuing to the infinity. [6] [7]
The Gateway Arch in the American city of St. Louis (Missouri) is the most famous example of a weighted catenary.[ citation needed ]
Simple suspension bridges use weighted catenaries. [7]