Weighted catenary

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The Gateway Arch is a weighted catenary: thick at the bottom, thin at the top. St Louis night expblend cropped.jpg
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 [1] and thus sometimes called Rankine curve [2] ) is a catenary curve, but of a special form: while 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

Contents

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

and now two constants enter: a and b.

Significance

A hanging chain is a regular catenary -- and is not weighted. Kette Kettenkurve Catenary 2008 PD.JPG
A hanging chain is a regular catenary — and is not weighted.

A freestanding 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]

Examples

The Gateway Arch in the American city of St. Louis (Missouri) is the most famous example of a weighted catenary.[ citation needed ]

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. 1 2 Robert Osserman (2010). "How the Gateway Arch Got its Shape" (PDF). Nexus Network Journal. Retrieved 13 April 2017.

On the Gateway arch

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