Sag and tension of a cable

A cable carrying a uniform load along its span hangs as a parabola. From the span and the sag: the horizontal tension, the vertical reactions, the maximum tension at the supports, the length of the cable and the angle at the ends, with its shape plotted.

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Cable sag and tension calculation with the cable shape plotted.

The calculation

w := 1.2 kN/m
s := 40 m
d := 2 m
H := w*s^2/(8*d) = kN
V := w*s/2 = kN
T_max := sqrt(H^2 + V^2) = kN
S := s + 8*d^2/(3*s) = m
theta := atan(4*d/s)*180/pi =
y(x) := 4*d*x*(s - x)/s^2
plot(-y(x)/(1 m), x, 0 m, s)

Results for these inputs

Horizontal tension 120 kN, vertical reaction 24 kN at each support.
Maximum tension 122.4 kN.
Cable length 40.27 m for a 40 m span.
Angle at the supports 11.3°.

How it works

For a uniform load w over the span s with sag d, the horizontal tension is H = ws²/(8d); the vertical reaction is ws/2; the length of a shallow parabola is s + 8d²/(3s). Halving the sag doubles the tension.

These calculations use classical textbook mechanics, not a design code. For design, use the method, factors and checks that apply to your work and have the result checked by a competent person.

Change the sag

Make the sag a variable and plot the tension against it to find a sensible proportion, or solve for the sag that gives a chosen tension with solve.

Questions

How is cable tension calculated from sag?

For a uniformly loaded cable, the horizontal tension is H = w·s²/(8·d); the maximum tension at the supports is √(H² + (w·s/2)²).

How long is a sagging cable?

For a shallow parabola, L ≈ s + 8d²/(3s), where s is the span and d the sag.

Also used for

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