A weighted catenary is a

catenary In physics and geometry, a catenary (, ) is the curve that an idealized hanging chain or 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, superfici ...

curve, but of a special form. 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

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 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{{cite web , url=http://www.msri.org/people/staff/osserman/papers/NNJ_v12n2_Osserman_pp167-189.pdf , title=How the Gateway Arch Got its Shape , publisher=Nexus Network Journal , date=2010 , accessdate=13 April 2017 , author=Robert Osserman St Louis night expblend cropped

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 arch, it is the world's tallest arch and Missouri's tallest accessible building. Some sources consider ...

in the American city of

St. Louis St. Louis () is the second-largest city in Missouri, United States. It sits near the confluence of the Mississippi and the Missouri Rivers. In 2020, the city proper had a population of 301,578, while the bi-state metropolitan area, which e ...

(

Missouri Missouri is a state in the Midwestern region of the United States. Ranking 21st in land area, it is bordered by eight states (tied for the most with Tennessee): Iowa to the north, Illinois, Kentucky and Tennessee to the east, Arkansas t ...

) is the most famous example of a weighted catenary.

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