A weighted catenary (also flattened catenary, was defined by William Rankine as transformed catenary and thus sometimes called Rankine curve) is a catenary curve, but of a special form: while a catenary is the curve formed by a chain under its own weight, a weighted catenary is the curve formed if the chain's weight is not consistent along its length. Formally, a "regular" catenary has the equation
for a given value of a. A weighted catenary has the equation
and now two constants enter: a and b.
Significance
A freestanding catenary arch has a uniform thickness. However, if
the arch is not of uniform thickness, the arch supports more than its own weight,Re-review: Catenary and Parabola: Re-review: Catenary and Parabola, accessdate: April 13, 2017 or if gravity varies,MathOverflow: classical mechanics - Catenary curve under non-uniform gravitational field - MathOverflow, accessdate: April 13, 2017
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. Definition from WhatIs.com: What is aspect ratio? - Definition from WhatIs.com, accessdate: April 13, 2017
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.
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