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Weighted catenary

Aweighted catenary 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.

Why they are important

Acatenary arch has a uniform thickness. However, if

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

it becomes more complex. A weighted catenary is needed.

Note that "aspect ratio" is important, which see,^[4][5]

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.^[5]

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

Category:Catenary

Category:Arches