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Weighted catenary

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(Redirected from Rankine curve) A weighted catenary is a catenary curve, of a special form, with two constants
The Gateway Arch is a weighted catenary: thick at the bottom, thin at the top.

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: 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 ⁡ ( 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 hanging chain is a regular catenary — and is not weighted.

A catenary arch has a uniform thickness. However, if

  1. the arch is not of uniform thickness,
  2. the arch supports more than its own weight,
  3. or if gravity varies,

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.

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

  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. ^ Robert Osserman (2010). "How the Gateway Arch Got its Shape" (PDF). Nexus Network Journal. Retrieved 13 April 2017.

External links and references

General links

On the Gateway arch

Commons

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