Cable Arc Length (Parabolic Series)

A cable is always longer than the span it crosses, and the amount by which it is longer is the number this page exists to give. Integrating $\sqrt{1 + (y')^{2}}$ along a parabola has no elementary closed form, so the standard route is to expand the square root in a binomial series and integrate term by term. Two terms are kept: $s = L\left(1 + \frac{8}{3}(d/L)^{2} - \frac{32}{5}(d/L)^{4}\right)$.

It is a truncation, and the page says so rather than presenting it as exact. The full expansion runs on forever in powers of $(d/L)^{2}$. At a sag ratio of 1 in 40 the second term contributes about 0.17% and the third contributes a few parts per million; at 1 in 10 the second contributes 2.7% and the third about 0.06%; and by 1 in 3 the series is visibly struggling. Past a sag ratio near 1 in 2.2 the fourth-order term overtakes the second and the truncated expression starts predicting a cable that gets shorter as it sags further, which is the algebra's way of announcing that it has left its own domain. The catenary arc length, $2a\sinh(L/2a)$, is closed form and has no such limit — it is the exact answer for the self-weight load case, and worth running alongside this one whenever the sag is deep.

The excess length is what everything practical depends on. It is the slack, and slack is what has to absorb a summer afternoon, an ice storm, and thirty years of creep. A conductor strung at 15 °C is measurably longer at 50 °C — aluminium expands about 23 microstrain per degree — and that extra length has nowhere to go but into sag, which comes straight off the clearance to whatever is underneath. This is why a stringing chart is a family of curves rather than a single number, and why the sag you are told to pull to depends on the temperature of the wire at the moment you pull it.

Notice how insensitive the relationship is in the other direction, because it is a trap. A cable only 0.1% longer than its span sags close to 2% of the span. The square root flattens everything: a small error in a measured length becomes a large error in the sag it predicts. Measuring cable and inferring sag works only if the length is good to millimetres, which is why nobody does it that way and everybody measures the sag directly.

And this is not the length to cut. This is the length of the cable as it hangs, loaded, stretched. The unstressed length is shorter, because the cable has elongated elastically under its own tension and will go on creeping for years afterward — a stranded aluminium conductor's permanent creep can be an appreciable fraction of its elastic strain, which is why conductors are often over-tensioned briefly at stringing and then let back. Recovering the cut length needs the axial stiffness $EA$, the creep model, and the erection sequence, and none of those is modelled anywhere on this site. What is here is geometry; what a stringing contract needs is geometry plus a materials model plus a construction plan.