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

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

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.

Why they are important

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.

Note that "aspect ratio" is important, which see, [4], [5].

The Saint Louis arch: fat at the bottom, skinny at the top.

Examples

The Gateway Arch in the American city of Saint Louis is the most famous example of a weighted catenary.

Simple suspension bridges use weighted catenaries, [6], or parabolas, [7], [8].

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

Category:Catenary

Category:Arches