Weighted catenary

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. A "regular" catenary has the equation

${\displaystyle y=a\,\cosh \left({\frac {x}{a}}\right)={\frac {a\left(e^{\frac {x}{a}}+e^{-{\frac {x}{a}}}\right)}{2}}}$

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

${\displaystyle y=b\,\cosh \left({\frac {x}{a}}\right)={\frac {b\left(e^{\frac {x}{a}}+e^{-{\frac {x}{a}}}\right)}{2}}}$

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,
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

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

1. Catenary
2. Gateway Arch
3. Catenary arch
4. Arch
5. William Rankine
6. History of St. Louis
7. List of National Historic Landmarks in Missouri
8. Railway electrification in the Soviet Union
9. Parametric design
10. Electric locomotive
11. Submarine pipeline
12. Anchor
13. Glossary of algebraic geometry
14. Downline (diving)
15. FS Class E.636
16. Two New Sciences
17. Longline bycatch in Hawaii
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