Weighted catenary

A weighted catenary is a catenary curve, of a special form, with two constants
A hanging chain is a regular catenary — and is not weighted.

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

y = a cosh ⁡ ( x a ) = a ( e x a + e − x a ) 2 {\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

y = b cosh ⁡ ( x a ) = b ( e x a + e − x a ) 2 {\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]

The St. Louis arch: thick at the bottom, thin at the top.

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

  1. ^ Osserman, Robert (February 2010). "Mathematics of the Gateway Arch" (PDF). Notices of the American Mathematical Society. 57 (2): 220–229. ISSN 0002-9920.
  2. ^ Andrue, Mario (2020). "The arches of the facade of the Palau Güell. Hyphotesis about its conformation" (PDF). fundacionantoniogaudi.org. Antonio Gaudi Foundation. Retrieved 5 January 2024.
  3. ^ Robert Osserman (February 2010). "Mathematics of the Gateway Arch" (PDF). Notices of the AMS.
  4. ^ Re-review: Catenary and Parabola: Re-review: Catenary and Parabola, accessdate: April 13, 2017
  5. ^ MathOverflow: classical mechanics - Catenary curve under non-uniform gravitational field - MathOverflow, accessdate: April 13, 2017
  6. ^ Definition from WhatIs.com: What is aspect ratio? - Definition from WhatIs.com, accessdate: April 13, 2017
  7. ^ a b Robert Osserman (2010). "How the Gateway Arch Got its Shape" (PDF). Nexus Network Journal. Retrieved 13 April 2017.

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