Maximum Cable Tension from the Sag Ratio (Parabolic)

The same support tension, rewritten so that the horizontal tension is out in front and everything else is a pure number: $T_{\max} = H\sqrt{1 + 16(d/L)^{2}}$. Nothing about the load or the span survives except through the sag ratio, which is what makes this the form worth memorizing.

Three values are worth carrying in your head. At a sag ratio of 1 in 20 the multiplier is 1.005 — half a percent, invisible. At 1 in 10, the traditional sag of a suspension-bridge main cable, it is 1.020. At 1 in 5 it is 1.077. Those three numbers answer the question "does it matter which tension I used?" before you touch a calculator, and the answer is no, sometimes, and very much yes.

The 16 is not arbitrary. It is $4^{2}$, and the 4 is the same 4 as in the slope at the support: the parabola's tangent there is $4d/L$, so the vertical leg of the tension triangle is $H$ times $4d/L$, and squaring it gives $16d^{2}/L^{2}$. Every appearance of a 4, a 16 or an 8 in this shard traces back to the same piece of parabola geometry, and once you see that, the formulas stop needing to be memorized separately.

This form and the load-based form are the same equation, not two approximations of one. Substitute $H = wL^{2}/(8d)$ into either and you get the other exactly. The anchors check that they agree to the last digit of a sixteen-figure expectation, which they do, because there is no approximation anywhere between them. Which one you reach for is purely a question of what you happen to know: the load and span, or the sag ratio.

The design direction is the useful one. You know the allowable tension — that is a property of the cable and its fittings and the factor of safety, and it is fixed before anything else is. Divide by the multiplier and you have the horizontal tension you may string at. Getting this order backwards, picking $H$ first and hoping the support tension lands under the allowable, is how a deep-sag design ends up 8% over on the very component that fails first.

One field use worth knowing: two tension readings — one taken at the low point and one at the dead-end — give the sag without any knowledge of what the cable weighs, because the ratio alone determines it. If the sag that comes back does not match the sag you can see with a rod and a level, one of the two instruments is lying, and finding out which is a better afternoon than discovering it later.