A cable hung between two supports takes up a shape set by its own weight and the tension in it. Pull harder and it flattens; let it off and it drops. The cable sag calculator above works that relationship in either direction on a level span, using the parabolic approximation that overhead line and rigging practice has used for a century, and reports the tension at the supports and the length of cable inside the span alongside it.
This is a preliminary and teaching tool published by Arb Digital. It is not a design, it is not stamped, and it does not replace a qualified engineer. It is also, on its own, an incomplete picture of any real span: sag and tension on a real cable change with temperature, with creep over the life of the installation, and with ice and wind load, and none of those effects are modelled here. A rigging engineer or a structural engineer sizes, specifies and inspects any load-bearing cable, and no number on this page should be treated as evidence that a cable, a fitting or an anchorage is adequate.
What This Cable Sag Calculator Does
It solves the level-span parabolic relationship between four quantities: span, weight per unit length, horizontal tension and midspan sag. Fix any three and the fourth follows. From those it also reports the tension at the support, which is always larger than the horizontal tension, the arc length of cable inside the span, and the sag expressed as a fraction of the span, which is the ratio engineers actually talk in.
The added-load field lets you rerun the same span with ice, fittings or a wind resultant included as extra weight per unit length, so you can see how much the sag and tension move. What figure to use is not something this page will tell you. Design ice thickness and wind pressure are jurisdictional quantities set by the governing standard, and they are an input here for the same reason a mapped ground snow load is an input on our snow load calculator.
What it does not do is model the physics that governs a real installation. There is no temperature term, no elastic stretch, no creep, no ruling span, no wind angle, no ice shedding and no dynamic response. Those omissions are the subject of a whole section below, and they are the reason this is a teaching tool.
Which Standard Governs
It depends entirely on what the cable is doing. Overhead electrical lines in the United States are governed by the National Electrical Safety Code, IEEE C2, which sets the required clearances and the loading districts, adopted and amended state by state through utility regulators. The structural loading side is covered by ASCE Manual of Practice 74, Guidelines for Electrical Transmission Line Structural Loading, published by the American Society of Civil Engineers, which also publishes ASCE/SEI 7, Minimum Design Loads and Associated Criteria for Buildings and Other Structures. Rigging and lifting applications fall instead under the ASME B30 series of safety standards for cableways, cranes, slings and rigging hardware, alongside the workplace safety regulations in force where the work is done.
Those are different worlds with different rules, different design factors and different inspection regimes, and a number that is routine in one may be entirely unacceptable in the other. Local code and amendments differ by state, province and country. This page implements none of those documents. It computes one relationship from mechanics and leaves the standards to the people qualified to apply them.
No allowable-load, breaking-strength or safe-working-load figure appears anywhere on this page, and none should be inferred. Rated capacities belong to specific products, in specific conditions, with specific end fittings and specific design factors, and they come from the manufacturer and the applicable standard.
How to Use It
- Work in one consistent system. Metres with newtons per metre, or feet with pounds per foot. The arithmetic does not know which you meant.
- Use the manufacturer's weight per unit length for the actual product, including any messenger, jacket or lashing.
- Decide which direction you are solving. Tension to sag for stringing, sag to tension for a geometry or clearance question.
- Rerun with the added load your engineer specifies to see how far the sag and tension move under ice or wind.
- Take the result as a first look, not an answer. The temperature, creep and load effects that dominate a real span are not in this calculation.
The Formula / How It's Calculated
For a cable of uniform weight w per unit length hung between supports at the same level a span L apart, with horizontal tension H, the parabolic approximation gives the midspan sag as d = w L² / (8H). Rearranged, H = w L² / (8d).
The vertical reaction at each support is half the weight of the cable, V = wL / 2, and the tension at the support is the resultant of the two components, T = √(H² + V²). The horizontal component is constant everywhere along the span; the total tension is smallest at the lowest point and largest at the supports.
The arc length of the cable inside the span, to the usual second-order approximation, is S = L + 8d² / (3L). Notice how small that is: on a well-tensioned span the cable is only a fraction of a per cent longer than the straight line between supports, which is precisely why a tiny change in length — from a few degrees of temperature — produces a large change in sag.
Worked example with the loaded values. A 100 span carrying 10 per unit length at a horizontal tension of 5,000 sags 10 × 100² / (8 × 5,000) = 100,000 / 40,000 = 2.5. The vertical reaction is 10 × 100 / 2 = 500, so the tension at the support is √(5,000² + 500²) = 5,024.94, about half a per cent above the horizontal component. The cable length in the span is 100 + 8 × 2.5² / (3 × 100) = 100.1667, so 0.167 more than the span. The sag is 2.5 per cent of the span.
Parabola or Catenary?
A cable hanging under its own weight is strictly a catenary, the curve described by a hyperbolic cosine, not a parabola — a distinction first settled in the seventeenth century and set out at Wolfram MathWorld's Catenary entry. The parabola is what you get if the load is uniform along the horizontal projection rather than along the cable itself — which is exactly true of a suspension bridge deck and only approximately true of a bare cable.
The approximation is very good while the sag is a small fraction of the span. At a sag ratio of a few per cent the two curves differ by an amount smaller than the measurement error in the field, which is why utility and rigging practice uses the parabolic form. As the sag ratio climbs the two diverge, and past roughly a tenth of the span the parabola starts to understate both the length and the tension noticeably. For the exact curve, the hyperbolic parameter and a direct comparison against this approximation, use our catenary curve calculator. That is the boundary between the two pages: this one is the engineering sag-and-tension case in the form the trades work in, and that one is the mathematical curve itself.
One further limitation of the form used here: it assumes the two supports are at the same level. An inclined span behaves differently — the low point may move outside the span entirely, and the tensions at the two ends differ. That case needs the full catenary treatment and is not covered by this page.
What Actually Changes Sag on a Real Span
This is the section that matters most, because the arithmetic above is the easy part of the problem.
Temperature. Cable expands when warm and contracts when cold. Because the length change needed to alter sag substantially is tiny, a swing of a few tens of degrees can change midspan sag by a large fraction of its value. Sag is therefore always quoted at a stated temperature, stringing charts give a target sag for the temperature on the day, and a span strung on a hot afternoon and measured on a cold morning will not match.
Creep. Metallic conductors and synthetic ropes both elongate permanently under sustained load over months and years. Overhead line design accounts for this explicitly, because a span that met its clearance on the day of installation may not a decade later. No calculation from geometry alone can tell you about it.
Ice. Radial ice on a cable adds weight, and it adds it as a large multiple of the bare cable's own weight rather than a small percentage. It also increases the projected area presented to wind. Ice loading is a governing case in many climates and is set by the standard, not estimated.
Wind. Wind acts horizontally, so the resultant load on the cable is inclined and the cable blows out of the vertical plane. Sag increases, tension increases, and the swing itself may govern clearance to a structure alongside. Wind can also drive aeolian vibration and, on iced conductors, galloping, which are dynamic problems with their own mitigations.
Everything else. Terminations, insulator strings, splices, dampers, lashed services and clamps all add concentrated weight that the uniform assumption does not describe, and connection details govern far more failures than cable strength does. Our factor of safety calculator is useful for understanding margins in general, and the material weight calculator for the mass of the cable itself, but neither substitutes for a specific design.
The One Relationship Worth Remembering
Sag and tension are inversely proportional. Halve the sag and you double the horizontal tension; take a quarter of the sag and the tension quadruples. That single fact explains most cable problems in the field.
It explains why a span that has been tensioned by eye to look tidy can be carrying several times the tension the designer intended, with all of that load passing into the end anchorages and the structures they are fixed to. It explains why the anchorage is nearly always the critical element rather than the cable, since anchorage loads scale with the same factor. And it explains why the temptation to take out a bit more sag on a job that already looks a little slack is one of the most consistently dangerous instincts in rigging.
The corollary is that a cable can never be pulled straight. As sag approaches zero, tension approaches infinity, so there is always some sag no matter how hard the pull. Any specification that appears to call for a dead-straight horizontal cable has been misread. For related structural work, our beam load calculator and arch calculator carry the same framing and the same limits.
Arb Digital builds free calculators and interactive tools that earn organic search traffic for engineering, utility and construction businesses. Browse the library, or tell us what your customers keep asking you to work out.
Browse Free Tools Talk To Arb DigitalCommon Mistakes to Avoid
- Quoting a sag without a temperature — sag is meaningless on its own, because a swing of a few tens of degrees moves it substantially.
- Confusing horizontal tension with tension at the support — the horizontal component is constant along the span, and the support tension is always the larger of the two.
- Using the parabola at large sag ratios — past roughly a tenth of the span it understates both length and tension, and the exact catenary is needed.
- Applying a level-span formula to an inclined span — the two end tensions differ and the low point can fall outside the span entirely.
- Treating a computed tension as proof of adequacy — rated capacities belong to specific products with specific fittings and design factors, and none appear on this page.
Related Free Tools From Arb Digital
For the exact hanging curve and a comparison against this approximation, use the catenary curve calculator. For related structural work, the beam load calculator, the snow load calculator and the factor of safety calculator carry the same limits, and the material weight calculator covers cable mass. The bolt torque calculator deals with the fixings at the ends. Every calculator we publish is on the free online tools hub.
Frequently Asked Questions
They are inversely proportional. Midspan sag equals the weight per unit length times the span squared, divided by eight times the horizontal tension. Halving the sag therefore doubles the tension, and a cable can never be pulled perfectly straight.
Because cable expands and contracts, and the length change needed to alter sag is very small. A swing of a few tens of degrees moves midspan sag by a large fraction of its value, which is why stringing charts give a target sag for the temperature on the day.
No. The horizontal component is constant along the whole span, and the support tension is the resultant of that component and the vertical reaction, so it is always the larger figure. On a shallow sag the difference is small, but it grows with sag.
A cable under its own weight is strictly a catenary. The parabola is an excellent approximation while the sag is a small fraction of the span and is what utility and rigging practice uses. Past roughly a tenth of the span the exact catenary is needed.
Only insofar as you enter an added load per unit length. The design ice thickness and wind pressure are jurisdictional quantities set by the governing standard and the engineer of record, and this page publishes none of them.
No. The relationship implemented here assumes both supports are at the same level. On an inclined span the tensions at the two ends differ and the low point can fall outside the span, which needs the full catenary treatment.
No, and it publishes no strength, breaking load or safe working load. Rated capacities belong to specific products with specific end fittings and design factors, and a rigging or structural engineer sizes and inspects any load-bearing cable.
Because anchorage loads scale with tension, and tension rises sharply as sag is reduced. A span tensioned tighter than intended puts that increase straight into the end fittings and the structures they are fixed to.
This tool applies a published approximation to values you supply, for education and preliminary work only. It is not a design, it is not stamped, it models no temperature, creep, ice or wind effect, it publishes no capacity of any kind, and any load-bearing cable must be designed, specified and inspected by a qualified engineer under the applicable standard.
