BIZOMAT MAKEOS · Mechanics
Catenary Cable Sag, Tension and Length
What tension and length does a cable or chain of this weight have when it sags this much over the span?
Exact catenary for a uniform cable or chain hanging between two supports at the same level: catenary parameter, horizontal and maximum tension, cable length and support angle, compared with the parabolic approximation.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Span between supports | m | Horizontal distance between the supports (at the same level). |
| Sag at mid-span | m | Vertical sag at mid-span. |
| Cable weight per metre | N/m | Weight per unit length (mass per metre × 9.81, plus any uniform ice or attached load). |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Span between supports
- What it is
- The horizontal distance between the two supports, which must be at the same level.
- Units and format
- Enter a number in m: metres. The form accepts 0.000001 m to 100000 m. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100 m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use the horizontal span between the supports, not the length of the cable.
What this means: Sag at mid-span
- What it is
- The vertical drop from the level of the supports to the lowest point of the cable, at mid-span. It must be smaller than the span.
- Units and format
- Enter a number in m: metres. The form accepts 0.000001 m to 10000 m. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5 m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Less sag means much higher tension: the parabolic estimate of horizontal tension is inversely proportional to sag.
What this means: Cable weight per metre
- What it is
- The weight of the cable or chain per metre of its length, plus any load spread evenly along it such as ice; mass per metre times gravity gives weight per metre.
- Units and format
- Enter a number in N/m: N/m. The form accepts 0.000001 N/m to 1000000 N/m. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 N/m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- All the tensions rise in proportion; the shape and cable length do not change.
- Get it right
- It is a force per metre, not a mass: multiply kg/m by gravity to get N/m.
Method and formulas
- y = a cosh(x ÷ a): the parameter a solves d = a [cosh(S ÷ 2a) − 1].
- Horizontal tension H = w a; maximum tension at the supports = w (a + d).
- Length = 2a sinh(S ÷ 2a); parabolic estimate H ≈ w S² ÷ 8 d.
- Example: 10 N/m over 100 m with 5 m sag carries 2,558.3 N at the supports.
Limitations
- Uniform weight along the cable, supports at equal level, inextensible cable.
- Concentrated loads and elastic stretch change the shape.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check the tension in a hanging cable, hose or chain
- Work out the cable length to order for a span and sag
- Set sag for festoon and catenary supply systems
Who it is for
- Structural engineers
- Electrical engineers
- Rigging supervisors
What you get
- Horizontal and maximum tension
- Cable length
- Support angle
Interpreting the result
- Halving the sag roughly doubles the tension; tight cables carry very high loads.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2999. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Tune guy ropes and cables away from the machine
In Vibrating Machine Elements by the Numbers
What it covers: Find a guy rope's modes from its tension, see why a rope tuned to the fan speed matches every harmonic, screen wind-driven vibration and retension to a target.
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