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

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.

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, , .

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