Cable Sag Ratio

Sag divided by span. It looks too simple to deserve a page, and it is the single most informative number about a suspended cable — because everything else on this shard depends on it and on nothing else. The support tension is $\sqrt{1 + 16r^{2}}$ times the horizontal tension. The excess length over the span is $\frac{8}{3}r^{2}$ to leading order. The angle at the support has tangent $4r$. Not one of those depends on the span, the load, or what the cable is made of.

Engineers usually speak it as "one in something" rather than as a decimal, and the vocabulary is worth having. 1 in 40 to 1 in 30 is transmission-line territory: a conductor strung tight because clearance to ground is the binding constraint and the wire is strong for its weight. 1 in 10 is the traditional suspension-bridge main cable, and it is not a calculation but a century of practice, sitting at the compromise where the cable tension is reasonable and the towers are not absurdly tall. 1 in 5 and deeper is a chain, a slack rope, a decorative festoon, or a conveyor's return-side catenary.

The ratio is also the switch between the two models on this shard, and that is why it matters here. Below about 1 in 10, the parabola and the catenary agree within about 1% in sag, length and tension, and every practical calculation is done with the parabola because it is algebraic and it inverts. Above it, they part company quickly — by 1 in 4 the parabola understates a self-weight cable's sag by roughly 8% and its support tension by a similar margin. Neither model is a correction to the other; they are exact answers to different load cases, and the sag ratio simply tells you how much the difference between those cases has grown.

Two more things follow from the ratio's dominance. First, everything scales: a 1:10 model of a cable and the full-size cable have identical tension multipliers, identical slack fractions, identical angles at the supports. Sag ratio is the similarity parameter of the whole subject. Second, choosing the ratio is the real design decision, and it is made early. Clearance, structure height, tension, cable size and the number of towers are all downstream of it, and moving it after the fact moves all of them.

One caution on the sign. This whole shard assumes the cable hangs — sags downward, in tension. Invert the geometry and you have an arch, where the same shape carries the same load in pure compression; that is why an inverted catenary is the ideal masonry arch and why Gaudí famously designed with hanging chain models and photographed them upside down. The mathematics carries across intact. The failure mode does not: a cable fails by breaking, an arch by buckling, and buckling arrives long before the material's strength is reached.