How to Calculate Catenary

How to Calculate Catenary
••• THSON/iStock/GettyImages

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.

    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.

    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.

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


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

    Calculate the function for an elastic catenary as:


    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.

    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.

Related Articles

How to Calculate Pendulum Force
How to Figure a Delta Angle
How to Calculate Lifting Capacity
To Calculate Arcsine, What Buttons Do You Press on...
How to Calculate the Cubic Feet of a Hole
How to Find Mass in Weight
What Is a Delta Angle?
How to Calculate Taper Ratio
How to Calculate Architectural Scales
How to Find a Z Score
How to Calculate Superelevation
How to Calculate Abutment & Wingwall Length
How to Calculate Perimeter and Area Ratio
How to Work Out a Gradient of a Curve
How to Find the Slope of a Nonlinear Line
How to Calculate Ellipse Eccentricity
How to Calculate Axial Stress
How to Find Volume in Meters Cubed
How to Find a Tangent Line to a Curve

Dont Go!

We Have More Great Sciencing Articles!