A weighted catenary (also flattened catenary, was defined by William Rankine as transformed 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

A catenary 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

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

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