information:classification, structural:expansion
Catenary of equal strength

In a catenary of equal strength, the cable is strengthened according to the magnitude of the tension at each point, so its resistance to breaking is constant along its length. Assuming that the strength of the cable is proportional to its density per unit length, the weight, w, per unit length of the chain can be written T / c, where c is constant, and the analysis for nonuniform chains can be applied.

In this case the equations for tension are

Combining gives

and by differentiation

where ρ is the radius of curvature.

The solution to this is

In this case, the curve has vertical asymptotes and this limits the span to π c. Other relations are

The curve was studied 1826 by Davies Gilbert and, apparently independently, by Gaspard-Gustave Coriolis in 1836.

Recently, it was shown that this type of catenary could act as a building block of electromagnetic metasurface and was known as "catenary of equal phase gradient".