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Weighted catenary

A weighted catenary is a catenary curve, of a special form, with two constants

Aweighted catenary (also flattened catenary, was defined by William Rankineastransformed catenary^[1] and thus sometimes called Rankine curve^[2]) 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[edit]

Acatenary arch has a uniform thickness. However, if

1. the arch is not of uniform thickness,^[3]
2. the arch supports more than its own weight,^[4]
3. or if gravity varies,^[5]

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. ^[6][7]

Examples[edit]

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.^[7]

References[edit]

External links and references[edit]

General links[edit]

• One general-interest link

On the Gateway arch[edit]

• Mathematics of the Gateway Arch
• On the Gateway Arch
• A weighted catenary graphed

Commons[edit]

• Category:Catenary
• Category:Arches