Calculation
f = q·L²/(8·H)
A freely hanging cable forms a catenary (the name of the cable curve, whatever the component). For small sag relative to span, it is very closely approximated by a parabola.
Dubbel Mechanik B 1.9.2 Seil unter konstanter Streckenlast: FH = q·x2²/(8f)
Cable Sag Calculator
A freely hanging cable forms a catenary (the name of the cable curve, whatever the component). For small sag relative to span, it is very closely approximated by a parabola.
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Result
Select a target and calculate.
Sag f versus horizontal tension H
f = q·L²/(8·H): halving the sag needs twice the horizontal tension, and a perfectly taut rope (f → 0) would need infinite force. The parabolic approximation applies to shallow ropes with small sag relative to the span.
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Understand the inputs
- Sag f — Largest vertical distance of the cable from the line joining the two support points.
- Distributed load q — Uniformly distributed self-weight or additional load per unit cable length.
- Span L — Horizontal distance between the two equal-height support points.
- Horizontal tension H — Horizontal component of cable tension, equal to the full cable tension at the lowest point.
Example
q=10 N/m, L=50 m and H=5,000 N give f=10·2,500/(8·5,000)=0.625 m.
Assumptions and limits
Slack, inextensible cable with uniform load and equal-height supports, small sag (parabolic approximation); cable stretch, temperature effects, ice and wind load are excluded.
Understand Cable Sag Calculator
This calculator determines the sag of a slack cable, overhead line or drive belt under uniform distributed load using the classic parabolic approximation.
What does this quantity describe?
A freely hanging cable takes the shape of a catenary under its own weight. The name refers only to this mathematical curve (historically studied on hanging chains) and applies equally to ropes, lines and chains. For small sag relative to span, the catenary is very closely approximated by a parabola: f=q·L²/(8·H), with q the distributed load, L the span and H the horizontal cable tension.
Why does an inextensible cable sag?
“Inextensible” only means the cable length does not change under load (Dubbel B 1.9: longitudinal strain is neglected). The cable still sags because it is slightly longer than the distance between the supports; this excess length, together with the load, sets its shape. For shallow cables between equal-height supports Dubbel gives the approximation cable length ≈ L·(1 + 8·f²/(3·L²)). In the example with L = 50 m and f = 0.625 m, the cable is only about 2.1 cm longer than the span. Doubling the horizontal tension halves the sag to 0.31 m and reduces the excess length to about 0.5 cm, so re-tensioning does not stretch the cable; it shortens the freely hanging length by just over 1.5 cm. A cable never becomes perfectly taut (f = 0): its weight can only be carried by the vertical component of the cable force, which exists only where the cable is inclined. With no sag, infinite tension would be required.
Formula and variables
f = q·L²/(8·H)
f = q·L²/(8·H)H = q·L²/(8·f)
| Symbol / input | Meaning |
|---|---|
| Sag f | Largest vertical distance of the cable from the line joining the two support points. |
| Distributed load q | Uniformly distributed self-weight or additional load per unit cable length. |
| Span L | Horizontal distance between the two equal-height support points. |
| Horizontal tension H | Horizontal component of cable tension, equal to the full cable tension at the lowest point. |
Choose the inputs correctly
q is the uniformly distributed load per unit length, L the span between the two equal-height supports, H the horizontal component of cable tension.
How to use the calculator
Determine q from the cable's or line's self-weight (plus ice or wind load where applicable), L from support spacing, H from the set cable tension.
Worked example
q=10 N/m, L=50 m and H=5,000 N give f=10·2,500/(8·5,000)=0.625 m.
Understand the result and units
Greater horizontal tension H reduces sag; a larger span L increases it quadratically.
q is a distributed load, L and f are lengths, H is a force.
Typical applications
Rough sizing of overhead lines, cableways, clotheslines and similar freely hanging cables under uniform distributed load.
Assumptions, limits and common mistakes
Slack, inextensible cable with uniform load, equal-height supports and small sag relative to span (parabolic approximation rather than exact catenary); cable stretch, temperature effects, and additional ice or wind load are excluded unless already included in q.
Common mistake: Do not confuse the tension at the lowest point (H, horizontal) with the maximum cable tension at the supports; the latter is somewhat larger there due to the slope.
Frequently asked questions
What is “Sag of a cable between two equal-height supports” used for?
Rough sizing of overhead lines, cableways, clotheslines and similar freely hanging cables under uniform distributed load.
Where do the input values come from?
q is the uniformly distributed load per unit length, L the span between the two equal-height supports, H the horizontal component of cable tension.
What does the result not cover?
Slack, inextensible cable with uniform load, equal-height supports and small sag relative to span (parabolic approximation rather than exact catenary); cable stretch, temperature effects, and additional ice or wind load are excluded unless already included in q.