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, ,
2. the arch supports more than its own weight, ,
3. or if gravity varies, ,

it becomes more complex. A weighted catenary is needed.

Note that "aspect ratio" is important, which see, , .

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, , or parabolas, , .

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