Why a hanging cable pulls so hard
A cable, rope or wire strung between two supports always sags, and the less it sags the harder it pulls on its anchors. Clotheslines, zip lines, guy wires, catenary lighting, fence wires and overhead cables all follow the same simple relationship between span, sag, weight and tension.
The formulas
For a cable with weight w per unit length, span L and sag s at mid-span, with supports at the same height, the parabolic approximation gives:
- Horizontal tension H = w × L² ÷ (8 × s), the same at every point of the cable
- Vertical load at each support V = w × L ÷ 2
- Maximum tension T = √(H² + V²), at the supports
- Cable length ≈ L + 8s² ÷ (3L)
A point load P hung at mid-span adds P × L ÷ (4s) to the horizontal tension and P ÷ 2 to each vertical reaction. The approximation is accurate as long as the sag stays below about 10% of the span.
Example
A cable weighing 0.5 lb/ft over a 100 ft span with 2 ft of sag:
- Horizontal tension: 0.5 × 100² ÷ (8 × 2) = 312.5 lbf
- Vertical load on each support: 25 lbf
- Maximum tension: 313.5 lbf, at an angle of 4.6° to the horizontal
- Cable length: 100.11 ft
A 50 lb cable pulls more than 300 lb on each anchor. Halve the sag to 1 ft and the tension doubles to 625 lbf, while the cable gets only about an inch shorter.
Practical points
- Allow enough sag: a sag of 2 to 5% of the span keeps the tension reasonable.
- Ice, wind and people pulling on a line add load, so use a safety factor of at least 3 to 5 on the working load of the cable and fittings.
- Anchors must resist the full tension along the cable direction, not just the weight.
- Temperature changes the length of metal cables and therefore the sag; a cable tightened in summer sags less and pulls harder in winter.