Cable Horizontal Tension (Parabolic)

Cut a hanging cable anywhere and look at the piece from the low point out to your cut. Three forces act on it: the horizontal pull $H$ at the low point, where the cable is level and has no vertical component at all; the weight of the piece; and the tension at the cut. Take moments about the cut and the sag falls straight out. For a load spread evenly along the horizontal — $w$ per metre of span — the result is $H = wL^{2}/(8d)$, and the curve is a parabola.

The single most useful thing in that equation is the $d$ in the denominator. Sag and tension trade against each other, hyperbolically, and you cannot have both small. Halve the sag and you double the pull. Drive the sag toward zero and the tension runs away to infinity, which is not a mathematical curiosity — it is the reason no cable anywhere is ever strung flat, and the reason a clothesline you tighten past a certain point snaps rather than straightens. It is also why a guy wire is never pulled dead taut: a wire with almost no sag has almost no capacity left for the load it was put there to resist.

Notice what $w$ means, because getting it wrong is the most common error on this page. It is the load per metre of span, measured along the ground, not per metre of cable measured along the curve. For a suspension bridge those are effectively the same thing: the deck is what weighs anything, the hangers are light, and the deck is laid out horizontally. For a bare conductor or a chain they are not the same thing, because the cable's mass follows the cable. That distinction is the entire boundary between this page and the catenary pages, and it is a difference of load case rather than of accuracy.

The classic field mistake is measuring the sag from the wrong datum. Sag is the vertical drop from the chord — the straight line joining the two support points — down to the lowest point of the cable. Not from the ground. Not from a level line drawn through one support. On a level span the chord is horizontal and nobody ever notices the distinction; the first inclined span is where it bites. On a slope the low point is no longer at midspan, it slides toward the lower support, and on a steep enough span it leaves the span entirely — the cable is then rising the whole way across and the "low point" is a virtual one, out beyond the bottom tower.

The other classic mistake is treating $H$ as the tension. It is not; it is the horizontal component of the tension, and it happens to be the same number at every point along the cable, which is exactly why it earns a symbol of its own. Nothing pushes sideways on a hanging cable between its supports, so the horizontal pull cannot change along it. The total tension is smallest at the low point, where it equals $H$ exactly because there is no vertical component there, and largest at the supports. Size an anchorage on $H$ and you are under by the sag-ratio multiplier — a fifth of a percent on a taut conductor, and eight percent on a deep one.

Because sag goes as $L^{2}$, doubling a span at constant tension quadruples the sag. That single fact governs the layout of a transmission line: a long river crossing does not just need stronger conductors, it needs taller structures, and it is usually cheaper to add a tower than to raise two.