Weighted catenary
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
- [math]\displaystyle{ y = a \, \cosh \left(\frac{x}{a}\right) = \frac{a\left(e^\frac{x}{a} + e^{-\frac{x}{a}}\right)}{2} }[/math]
for a given value of a. A weighted catenary has the equation
- [math]\displaystyle{ y = b \, \cosh \left(\frac{x}{a}\right) = \frac{b\left(e^\frac{x}{a} + e^{-\frac{x}{a}}\right)}{2} }[/math]
and now two constants enter: a and b.
Significance
A catenary arch has a uniform thickness. However, if
- the arch is not of uniform thickness,[3]
- the arch supports more than its own weight,[4]
- 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
- ↑ Osserman, Robert (February 2010). "Mathematics of the Gateway Arch". Notices of the American Mathematical Society 57 (2): 220–229. ISSN 0002-9920. https://www.ams.org/notices/201002/rtx100200220p.pdf.
- ↑ Andrue, Mario (2020). "The arches of the facade of the Palau Güell. Hyphotesis about its conformation". Antonio Gaudi Foundation. https://fundacionantoniogaudi.org/Publicaciones/the-arches-of-the-facade-of-the-palau-guell.pdf.
- ↑ Robert Osserman (February 2010). "Mathematics of the Gateway Arch". Notices of the AMS. https://www.ams.org/notices/201002/rtx100200220p.pdf.
- ↑ Re-review: Catenary and Parabola: Re-review: Catenary and Parabola, accessdate: April 13, 2017
- ↑ MathOverflow: classical mechanics - Catenary curve under non-uniform gravitational field - MathOverflow, accessdate: April 13, 2017
- ↑ Definition from WhatIs.com: What is aspect ratio? - Definition from WhatIs.com, accessdate: April 13, 2017
- ↑ 7.0 7.1 Robert Osserman (2010). "How the Gateway Arch Got its Shape". Nexus Network Journal. http://www.msri.org/people/staff/osserman/papers/NNJ_v12n2_Osserman_pp167-189.pdf. Retrieved 13 April 2017.
External links and references
General links
On the Gateway arch
Commons
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