Catenary Calculator (Equal Support Heights)
Use this cable sag calculator for chains, overhead lines, and suspended cables where both supports are at the same elevation.
What Is a Catenary?
A catenary is the natural curve formed by a flexible cable or chain hanging under its own weight. Unlike a parabola, a true catenary follows a hyperbolic cosine function. You see this shape in power lines, suspension systems, hanging chains, and even structural arches designed as inverted catenaries.
How This Catenary Calculator Works
This calculator uses the standard symmetric catenary model where both supports are at the same height. You provide span and sag, and the tool solves for the catenary parameter a numerically. Then it computes geometry and tension values from that parameter.
Core equations used
- Catenary equation (origin at lowest point): y = a cosh(x/a) - a
- Sag relation for equal supports: f = a(cosh(L/(2a)) - 1)
- Total cable length: S = 2a sinh(L/(2a))
- Horizontal tension: H = w a
- Support tension magnitude: Tsupport = H cosh(L/(2a))
Input Definitions
1) Horizontal span (L)
The straight horizontal distance between supports. This is not cable length.
2) Midspan sag (f)
The vertical drop from the support level down to the cable at midspan. Bigger sag means a looser cable; smaller sag generally means higher tension.
3) Weight per length (w)
Enter the cable self-weight per unit actual cable length. If you leave this as zero, the calculator still returns geometric values but tension outputs become zero.
4) Position x
A point along the horizontal span measured from the left support. The calculator reports the cable elevation above the lowest point and drop below support level at that x location.
Practical Uses
- Estimating overhead conductor sag and tension ranges
- Design checks for hanging decorative lighting or banner cables
- Preliminary sizing for suspension elements in architectural concepts
- Educational demonstrations of catenary vs parabola behavior
Important Notes and Limitations
This is a static, idealized model. Real installations can involve temperature effects, wind loads, ice loading, creep, elastic stretch, and unequal support elevations. Use this calculator for conceptual estimates and quick checks, not as a substitute for stamped engineering design.