# Weighted catenary

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

A weighted catenary is a catenary curve, but of a special form. A "regular" catenary has the equation

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

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 catenary arch has a uniform thickness. However, if

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

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. [4][5]

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.[5]

## References

1. ^ Robert Osserman (February 2010). "Mathematics of the Gateway Arch". Notices of the AMS. {{cite web}}: Missing or empty |url= (help)
2. ^ Re-review: Catenary and Parabola: Re-review: Catenary and Parabola, accessdate: April 13, 2017
3. ^ MathOverflow: classical mechanics - Catenary curve under non-uniform gravitational field - MathOverflow, accessdate: April 13, 2017
4. ^ Definition from WhatIs.com: What is aspect ratio? - Definition from WhatIs.com, accessdate: April 13, 2017
5. ^ a b Robert Osserman (2010). "How the Gateway Arch Got its Shape" (PDF). Nexus Network Journal. Retrieved 13 April 2017.