Weighted catenary

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

A weighted catenary (also flattened catenary, was defined by William Rankine as transformed catenary and thus sometimes called Rankine curve) is a catenary curve, but of a special form. A "regular" catenary has the equation

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

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

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

and now two constants enter: a and b.

Significance

A catenary arch has a uniform thickness. However, if

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

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.

The St. Louis arch: thick at the bottom, thin at the top.

Examples

The Gateway Arch in the American city of St. Louis (Missouri) is the most famous example of a weighted catenary.

Simple suspension bridges use weighted catenaries.

External links and references

General links

• One general-interest link

On the Gateway arch

• Mathematics of the Gateway Arch
• On the Gateway Arch
• A weighted catenary graphed

Commons

• Category:Catenary
• Category:Arches