Weighted catenary

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

${\displaystyle y=a\,\cosh \left({\frac {x}{a}}\right)={\frac {a\left(e^{\frac {x}{a}}+e^{-{\frac {x}{a}}}\right)}{2}}}$

The Gateway Arch is a weighted catenary: thick at the bottom, thin at the top.

for a given value of a. A weighted catenary has the equation

${\displaystyle y=b\,\cosh \left({\frac {x}{a}}\right)={\frac {b\left(e^{\frac {x}{a}}+e^{-{\frac {x}{a}}}\right)}{2}}}$

and now two constants enter: a and b.

Significance

A hanging chain is a regular catenary — and is not weighted.

A freestanding catenary arch has a uniform thickness. However, if

1. the arch is not of uniform thickness,^[3]
2. the arch supports more than its own weight,^[4]
3. or if gravity varies,^[5]

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]

Examples

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]