Inputs
Example Data
| Scenario | Span (m) | Height Diff (m) | Unit Weight | Horizontal Tension | Typical Use |
|---|---|---|---|---|---|
| Temporary line | 30 | 0 | 12 N/m | 25 kN | Overhead guidance or light utility |
| Unequal supports | 45 | -2 | 18 N/m | 35 kN | Slope crossing between two elevations |
| Length-fit | 20 | 0 | 10 N/m (optional) | — | Known rope length, unknown tension |
Formula Used
This tool models a hanging cable under uniform distributed load using the catenary curve. The general form is: y(x) = a·cosh((x − x0)/a) + C
- a = H / w (tension mode), where H is horizontal tension and w is unit weight (N/m).
- Boundary conditions: y(0)=0 and y(L)=Δh determine x0 and C.
- Cable length between supports: S = a·sinh((L−x0)/a) + a·sinh(x0/a)
- Support tension magnitude: T(x)=H·cosh((x−x0)/a)
Parabolic quick estimate (equal support elevations, small sag): f ≈ w·L² / (8·H).
How to Use This Calculator
- Choose a mode: tension mode if w and H are known, length mode if cable length is known.
- Enter span and any elevation difference between supports.
- Provide unit weight in N/m (or select another unit to convert).
- Click Calculate. Results appear above the form and include sag, length, and (when possible) support tensions.
- Use CSV/PDF export for sharing and record keeping.
Where Catenary Sag Matters on Site
Catenary sag planning is critical for temporary utility crossings, hoisting tag lines, façade access ropes, and suspended lighting runs. Small errors can reduce headroom, violate exclusion zones, or overload anchors. A quick sag estimate supports lift planning, scaffold ties, and routing around plant, traffic, and pedestrian paths.
Inputs That Control Sag and Clearance
Span length increases sag rapidly, so confirm support spacing with a tape, total station, or as-built coordinates. Unit weight should include cable self-weight plus any evenly distributed attachments, such as festooned hoses or bundled conductors. Horizontal tension reflects rigging hardware, winch settings, and allowable pull on terminations, so record the method used. Use the length-fit mode when the rope is pre-cut and you need the resulting sag before tensioning. Enter the measured end-to-end length between attachment points, then verify that the length exceeds the straight-line distance; otherwise no stable hanging shape exists for the chosen geometry.
Interpreting Midspan Sag and Lowest Point
For equal-height supports, the lowest point occurs near midspan, but unequal elevations shift it toward the higher support. The midspan sag below the straight chord is a practical clearance check because crews often sight along the supports. Compare both the lowest-point sag and chord sag when obstructions are near one end.
Tension, Reactions, and Safety Margins
Support tension is highest at the ends, not at the lowest point. If unit weight is provided, the calculator reports end tensions and vertical reactions to help size anchors, clamps, and temporary frames. Apply your project safety factor to allowable capacity, and confirm that fittings, bolts, and substrate capacity match the governing load case.
Common Field Checks and Documentation
Before use, inspect connectors, saddles, and abrasion points, and confirm that the cable is free to align without kinks. Recheck sag after temperature swings, wetting, or loading changes because length and tension can drift. Export the calculation summary, attach photos of support heights, and keep the report with the lift or temporary works permit.
FAQs
1) What does “midspan sag below chord” represent?
It is the vertical gap between the cable at midspan and the straight line connecting both supports. It is a practical clearance metric when obstacles sit near the middle of the span.
2) Why does the lowest point move when supports are uneven?
A height difference changes the boundary conditions of the curve. The lowest point shifts toward the higher support, and the two end sags become different even with the same span.
3) When should I use the cable-length mode?
Use it when the end-to-end cable length is known but tension is not. The calculator fits the catenary shape to match span, elevation difference, and length.
4) Does unit weight include attached items like hoses or festoons?
Yes. Use the combined distributed load per meter along the span. If loads are not uniform or include point loads, results become approximate and should be checked with a more detailed model.
5) Why is end tension higher than tension at the lowest point?
At the supports, the cable carries both horizontal tension and a vertical component from the suspended weight. The vector sum increases the total tension compared with the lowest point.
6) How should I treat temperature effects shown here?
The thermal output estimates length change using a simple expansion relationship. Real sag change depends on how the system restrains movement and how tension is rebalanced, so treat it as a planning indicator.