A weighted catenary (also flattened catenary, was defined by

William Rankine William John Macquorn Rankine (; 5 July 1820 – 24 December 1872) was a Scottish mathematician and physicist. He was a founding contributor, with Rudolf Clausius and William Thomson (Lord Kelvin), to the science of thermodynamics, particularly ...

as transformed catenary and thus sometimes called Rankine curve) is a

catenary In physics and geometry, a catenary ( , ) is the curve that an idealized hanging chain or wire rope, cable assumes under its own weight when supported only at its ends in a uniform gravitational field. The catenary curve has a U-like shape, ...

curve, but of a special form: if 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 :

y = a \, \cosh \left(\frac\right) = \frac

for a given value of ''a''. A ''weighted catenary'' has the equation :

y = b \, \cosh \left(\frac\right) = \frac

and now ''two'' constants enter: ''a'' and ''b''.

Significance

A freestanding

catenary arch A catenary arch is a type of architectural arch that follows an inverted catenary curve. The catenary curve has been employed in buildings since ancient times. It forms an underlying principle to the overall system of vaults and buttresses i ...

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 The aspect ratio of a geometry, geometric shape is the ratio of its sizes in different dimensions. For example, the aspect ratio of a rectangle is the ratio of its longer side to its shorter side—the ratio of width to height, when the rectangl ...

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 The Gateway Arch is a monument in St. Louis, Missouri, United States. Clad in stainless steel and built in the form of a weighted catenary Catenary arch, arch, it is the world's tallest arch and List of tallest buildings in Missouri, Missouri's ...

in the American city of

St. Louis St. Louis ( , sometimes referred to as St. Louis City, Saint Louis or STL) is an independent city in the U.S. state of Missouri. It lies near the confluence of the Mississippi and the Missouri rivers. In 2020, the city proper had a populatio ...

(

Missouri Missouri (''see #Etymology and pronunciation, pronunciation'') is a U.S. state, state in the Midwestern United States, Midwestern region of the United States. Ranking List of U.S. states and territories by area, 21st in land area, it border ...

) is the most famous example of a weighted catenary.{{Citation needed, date=November 2024

Simple suspension bridge A simple suspension bridge (also rope bridge, swing bridge (in New Zealand), suspended bridge, hanging bridge and catenary bridge) is a primitive type of bridge in which the deck of the bridge lies on two parallel load-bearing cables that ar ...

s 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

* Catenary * Arches Plane curves Arches and vaults Architectural history