How To Calculate Catenary

A catenary is the shape that a cable assumes when it's supported at its ends and only acted on by its own weight. It is used extensively in construction, especially for suspension bridges, and an upside-down catenary has been used since antiquity to build arches. The curve of the catenary is the hyperbolic cosine function which has a U shape similar to that of a parabola. The specific shape of a catenary may be determined by its scaling factor.

Step 1

Calculate the standard catenary function:

\(y=a\cosh (x/a)\)

where ​y​ is the ​y​ Cartesian coordinate, ​x​ is the ​x​ Cartesian coordinate, cosh is the hyperbolic cosine function and ​a​ is the scaling factor.

Step 2

Observe the effect of the scaling factor on the catenary's shape. The scaling factor may be thought of as the ratio between the horizontal tension on the cable and the weight of the cable per unit length. A low scaling factor will therefore result in a deeper curve.

Step 3

Calculate the catenary function with an alternate equation. The previous equation can be shown to be mathematically equivalent to:

\(y=\frac{a}{2}(e^{x/a}+e^{-x/a})\)

where e is the base of the natural logarithm and is approximately 2.71828.

Step 4

Calculate the function for an elastic catenary as:

\(y=\frac{y_0}{1+kt}\)

where ​_y₀​ is the initial mass per unit length, ​k​ is the spring constant and ​t_​ is time. This equation describes a bouncing spring instead of a hanging cable.

Step 5

Calculate a real-world example of a catenary. The function

\(y = -127.7 \cosh \bigg(\frac{x}{127.7}\bigg) + 757.7\)

describes the St. Louis Arch where the measurements are in units of feet.