catenary curve calculator

What Is a Catenary Curve?

A catenary curve is the natural shape a perfectly flexible, uniform cable takes when suspended under its own weight and supported at both ends. It looks similar to a parabola, but it is mathematically different. In engineering and architecture, catenaries show up in power lines, suspension systems, mooring lines, chain drapes, and even arch design.

How This Catenary Calculator Works

This calculator uses the equal-height support case. You enter:

• Span (L): horizontal distance between supports.
• Sag (f): vertical drop from support level to the lowest point.
• Unit weight (w): weight per meter of cable length (optional, used for tension outputs).
• Position x: a point along the span where you want the cable height relative to the lowest point.

From those values, the tool solves for the catenary parameter a numerically, then computes the cable arc length, support angle, and force components.

Core Equations

For a symmetric catenary with lowest point at x = 0:

• Curve equation: y(x) = a(cosh(x/a) - 1)
• Sag condition: f = a(cosh(L/(2a)) - 1)
• Arc length: S = 2a sinh(L/(2a))
• Horizontal tension: H = wa
• Support tension: T = H cosh(L/(2a))

Why Engineers Care About Catenary Calculations

Getting the geometry and loads right matters for both safety and cost. A cable with too little sag can experience very high tension; too much sag can cause clearance and serviceability problems. Catenary math helps balance:

• Structural capacity and safety margins
• Material selection and installation constraints
• Clearance requirements above roads, terrain, or water
• Long-term performance under temperature and loading changes

Quick Example

Suppose your span is 100 m and sag is 10 m. The calculator will find a value of a that satisfies the sag equation, then report the true cable length (which is always longer than the horizontal span). If you also provide unit weight, it estimates horizontal and end tensions for that geometry.

Important Assumptions and Limits

• Supports are at the same elevation.
• The cable is perfectly flexible and carries only axial tension.
• Unit weight is constant along cable length.
• No wind, ice, dynamic effects, or thermal strain are included.

For detailed design, use this tool as a fast preliminary estimate and confirm with project-specific analysis standards and load combinations.

FAQ

Is a catenary the same as a parabola?

No. They are close for small sag-to-span ratios, but mathematically distinct. The catenary comes from self-weight distributed along cable length.

What if my supports are at different heights?

That is an asymmetric catenary problem. This page handles the symmetric equal-height case for simplicity and speed.

What units should I use?

Use any consistent unit system. If span and sag are in meters, use weight in N/m to obtain forces in newtons.