Article — Cable Sag Calculator
Cable sag calculator: parabolic sag, tension, and cable length
Cable sag is the vertical distance from the midpoint of a suspended cable to the straight line connecting its supports. The parabolic approximation gives d = s² × w / (8T), where s is span length, w is weight per unit length, and T is horizontal tension. The formula is accurate within 3% as long as the sag-to-span ratio stays below 1:8; above that, use the catenary equation.
Every suspended wire sags — from utility distribution lines to suspension bridges to backyard clotheslines. The math is the same; only the dominant load changes.
What is cable sag?
A cable suspended between two supports cannot stay straight. Its own weight pulls every point downward, and the only way to balance that gravity load is for the cable to curve. The amount of curvature depends on the horizontal tension at the supports. High tension keeps the cable flat; low tension lets it sag more. Sag is the vertical drop at midspan, where the curve reaches its lowest point.
Engineers care about sag for three reasons. First, ground clearance — an overhead line that sags too far becomes a safety hazard. Second, tension — sag and tension trade off, and tension determines what the supports must withstand. Third, cable length — the sagging cable is longer than the span, and that extra length must be ordered when stringing the line.
Galileo argued in 1638 that a suspended chain takes the shape of a parabola. He was wrong. The true shape is the catenary curve, derived independently by Huygens, Leibniz, and Bernoulli in 1691. For small sags the two are almost identical — under 12.5% sag-to-span ratio the difference stays below 3%. For deep sags like the cables of a suspension bridge, the catenary is the real shape.
The parabolic sag formula
Standard form: d = s² × w / (8T). Sag d is in the same units as span s (meters or feet). Weight per unit length w must be in compatible units (kg/m gives sag in meters once you convert mass to force by multiplying by g = 9.81 m/s²). T is horizontal tension at the supports.
The formula derives from moment balance on half the cable. Taking moments about a support point: T × d (at midspan) equals the distributed load weight times its lever arm, integrated from support to midspan. The integral gives ws²/8, and rearranging produces the sag formula. The same balance, rearranged, gives T = s²w / (8d) when you know the sag and want the tension.
d = s² × w / (8T)T = s² × w / (8d)L = s + 8d² / (3s)ratio = d/s (use 1:N format)valid if d/s < 1/8Parabolic sag vs catenary
The parabola assumes weight is distributed uniformly along the horizontal axis. The catenary assumes weight is distributed uniformly along the cable itself. For a real cable suspended in gravity, the catenary assumption is exactly right. The parabola is a simplification that works well only when the cable is nearly horizontal — when sag is small relative to span.
The boundary is 1:8 sag-to-span ratio. Below that, parabolic error stays under 3%. At 1:5 (deeper sag), error grows to about 8%. At 1:3, parabola underestimates sag by more than 20%. Suspension bridge designers always use catenary math; transmission line engineers stick with parabolic; both groups produce safe designs for their applications.
- Parabola = simple closed form, works for d/s < 1/8
- Catenary = y = a cosh(x/a), exact for all geometries
- 1:30 sag-to-span = transmission line typical
- 1:10 = decorative festoon, visible sag
- 1:5 = suspension bridge main cable
- 1:3 = deep loop, catenary required
Sag-to-span ratio explained
Engineers describe cable installations by sag-to-span ratio, usually written as 1:N. A 1:30 ratio means sag is 1/30th of span: a 30 m span with 1 m sag. The ratio is dimensionless, so it transfers across unit systems — 1:30 is the same whether measured in meters or feet.
Power transmission lines target 1:30 to 1:60 to maintain ground clearance under heavy loading conditions. Distribution lines run 1:20 to 1:30. Service drops from pole to house tighten to 1:15. Suspension bridges deliberately go deep at 1:8 to 1:12 because the main cable shape needs to carry deck weight via vertical hangers. Decorative outdoor lights aim for visible 1:5 to 1:10 sag because the visual matters more than the engineering.
How temperature affects cable sag
Cables expand when heated and contract when cold. The thermal expansion coefficient α determines how much: steel runs about 12 × 10⁻⁶ per °C, aluminum 23 × 10⁻⁶/°C, ACSR (aluminum conductor steel reinforced) sits between the two. A 100 m steel cable warmed 30 °C grows 36 mm; the same aluminum cable grows 69 mm.
The extra length goes into sag. A 100 m span with steel cable might sag 1 m in winter and 1.3 m on the hottest summer afternoon — a 30% sag increase from the same span and tension. Utility line designers calculate sag at three temperatures: minimum (highest tension), normal (operational), and maximum (greatest sag). The maximum sag must clear regulated ground clearances even at the hottest expected temperature plus design load.
Sag is the limiting condition for overhead conductor clearance. A line that meets ground clearance at 20 °C may dip below the safety threshold at 50 °C on a still summer afternoon. The NESC requires sag calculations at 49 °C (120 °F) operating temperature in most US zones, higher in southwestern desert regions. Build in margin or face violations the first heat wave.
Ice and wind on cable sag
Ice deposition during freezing rain or fog can transform cable loading. A quarter-inch radial ice coating on a half-inch cable triples its effective diameter and adds 1-2 lb/ft of weight. The cable now sags more (more weight per unit length) and tension goes up at the supports, sometimes catastrophically.
The NESC defines three loading districts in the US: light (no ice), medium (1/4 inch radial ice), heavy (1/2 inch radial ice). The medium district covers most of the central and northeastern US. Cables in heavy ice zones must be designed to carry full ice load plus a 40 mph wind on the iced cable — an enormous load multiplier that drives both line tension and support design. Failed ice loading on Quebec lines in 1998 collapsed thousands of transmission towers and cut power to 4 million people for weeks.
Cable sag in design
Engineers iterate cable sag design until tension stays below 25-50% of the cable's rated breaking strength under maximum loading (worst-case temperature, ice, and wind combined). Higher tension means flatter cable but more stress; lower tension means more sag but smaller safety margin and more required ground clearance.
The design process: start with the required ground clearance at maximum operating temperature, compute the allowable sag, work backward through d = s²w/(8T) to find the tension, then verify the tension stays below the rated working load. Iterate spans, tensions, and conductor sizes until everything balances. Modern utility design uses tension software (PLS-CADD, LineWorks) that handles thousands of spans simultaneously.
Common cable sag mistakes
The first mistake is ignoring temperature. A line sized for current weather works until the season changes. The second is forgetting ice and wind — bare cable tension calculations look great until a January freezing rain doubles the effective weight. The third is using parabolic math beyond its valid range; once sag-to-span exceeds 1:8, the parabola underestimates both sag and peak tension at the supports.
The fourth, and most common in DIY installations, is sizing the support hardware for the static tension only. The supports must resist tension under design loading conditions, not the as-built tension. A guy wire installed at 1,000 lb tension on a calm 70 °F day might pull 2,500 lb on a winter night with an inch of ice. Hardware rated 1,500 lb fails; the wire snaps; whatever the guy was holding falls down.
For DIY clothesline, fence wire, or decorative cable installations, target a 1:20 sag-to-span ratio. This keeps tension reasonable (about 5% of breaking strength for typical wire), gives moderate sag that handles temperature swings, and looks natural rather than overtight or droopy.