How Sag and Tension Interact
Designers of power transmission lines, suspension footbridges, and lifting slings must understand the relationship between the sag of a cable and the forces that develop at its supports. The hanging shape of an ideal flexible cable under uniform gravity loading is a catenary, yet for modest sag-to-span ratios typical in construction a simple parabolic approximation is sufficiently accurate. This calculator accepts span length, distributed weight, and an observed or desired midspan sag, then evaluates the horizontal and total support tensions using the fundamental relation where is the weight per unit length, is the span, is sag, and represents the horizontal component of tension. The total tension at each support arises from combining this horizontal component with half the vertical load: . Understanding these relationships helps engineers maintain safe clearances, limit structural loads, and provide aesthetically pleasing profiles.
Parabolic Approximation to the Catenary
The exact catenary curve is defined by , where the parameter relates to horizontal tension through . For spans with small sag, expanding the hyperbolic cosine term produces the parabolic form which is far easier to manipulate. Although the approximation slightly underestimates sag for very long spans, its simplicity makes it common in line design manuals and crane rigging charts. When precision is critical, designers iterate with the full catenary equation, but the parabolic assumption remains a useful first step for feasibility studies and quick checks.
Typical Cable Weights
Weight per unit length influences sag more than any other parameter. Heavier conductors require greater tension to achieve the same clearance, which may necessitate stronger poles or towers. The table below lists representative weights of several common conductor and rope types per meter of length. These values are approximate and can vary with manufacturer and coating, yet they provide a starting point for preliminary calculations.
| Cable Type | Weight (kN/m) |
|---|---|
| Steel Wire Rope, 25 mm | 0.27 |
| Aluminum Conductor Steel Reinforced (ACSR) "Drake" | 0.11 |
| Fiber Rope, High Strength | 0.02 |
| Suspension Bridge Main Cable | 0.45 |
Checking Clearances
For power lines and ropeways, regulations specify minimum clearances above ground, roads, and waterways. After choosing an initial sag, engineers calculate the resulting support tension and ensure it does not exceed the safe working limit of the cable or the strength of the supporting structures. If the calculated tension is too high, they can either allow more sag or select a lighter cable. The horizontal tension also influences the anchor design; deadmen, rock bolts, or foundations must resist this load without creeping or pulling out. The vertical component informs the design of towers and poles, which must support the conductor weight without buckling.
Temperature and Creep
Real-world cables expand and contract with temperature, changing sag throughout the day. For metallic conductors, thermal expansion can add several centimeters of sag per ten degrees Celsius increase. Creep, the gradual elongation of material under constant load, also increases sag over years. Designers therefore analyze multiple cases: an initial unloaded stringing temperature, an extreme cold condition with high tension, and a hot, low-tension condition where clearances are critical. The parabolic formulas adapt to these scenarios by updating the weight per length and sag for each temperature case.
Worked Example
Suppose a 60 m span of steel wire rope weighs 0.27 kN per meter and exhibits 1.5 m of sag at midspan. The horizontal tension is kN. Half the vertical load is kN. Combining components yields a support tension of roughly kN. With this value in hand, the engineer can select hardware rated for at least four times the calculated tension to maintain an adequate safety margin.
Beyond Uniform Loads
This tool assumes uniform gravity loading. In practice, ice accumulation, wind pressure, and concentrated weights from warning spheres or spacers alter sag and tension. A heavy layer of ice increases weight per length while wind induces horizontal forces, effectively reducing sag. For long spans crossing valleys or rivers, engineers may account for curvature of the Earth and wind-induced oscillations. Dynamic effects such as galloping and vortex shedding require damping devices or special conductors. While those advanced topics lie beyond this calculator, the parabolic relations remain a foundation for preliminary sizing and serve as a stepping stone to more sophisticated finite-element analyses.
Summary
Understanding the interplay between sag and tension is essential for the safe and efficient design of overhead cables. The parabolic approximation provides a quick means to estimate horizontal and total support loads, guiding choices in cable size, support spacing, and anchorage details. By iterating sag values and reviewing the resulting tensions, practitioners can balance clearance requirements with structural limitations, ensuring reliable performance throughout the life of the installation. Always consult detailed design standards and engage qualified professionals for final engineering, but let this calculator serve as a versatile starting point in your design workflow.