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[]

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