Enter span, sag, and weight per unit length to compute cable shape, horizontal tension, max tension, cable length, stress, and elastic elongation — catenary (exact) or parabolic (approximate).
Parameters
Span L
m
Sag d
m
Sag ratio d/L = 10.0%
Weight/length w
N/m
Cross-section A
mm²
Elastic modulus E
GPa
Steel:200 · CFRP:150 · Al:70
Results
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Horiz. Tension H [kN]
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Max Tension T_max [kN]
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Cable Length S [m]
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Sag Ratio d/L [%]
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Stress σ [MPa]
—
Elastic Elong. ΔL [mm]
Cable
Theory & Key Formulas
Catenary (exact):
$$y = a\cosh\!\left(\frac{x}{a}\right) - a, \quad a = \frac{H}{w}$$
Cable length: $S = 2a\sinh\!\left(\dfrac{L}{2a}\right)$, Max tension: $T_{max}= H + w\,d$
Parabolic approximation (small d/L):
$$y = \frac{4d}{L^2}x(L-x), \quad H = \frac{wL^2}{8d}$$
$$T_{max}= H\sqrt{1+\!\left(\frac{wL}{2H}\right)^{\!2}}, \quad S \approx L\!\left(1+\frac{8}{3}\!\left(\frac{d}{L}\right)^{\!2}\right)$$
Stress: $\sigma = T_{max}/A$, Elastic elongation: $\Delta L = T_{max}\cdot S / (A\cdot E)$
What is a Catenary Cable?
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What exactly is a "catenary" cable? Is it just a hanging cable?
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Basically, yes—but it's the specific, natural shape a perfectly flexible cable or chain forms under its own weight. It's not a simple parabola. In practice, the exact shape depends on the horizontal tension and weight. Try moving the "Sag" slider in the simulator above; you'll see the curve change from tight to very loose.
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Wait, really? So when engineers design a suspension bridge cable, they have to use this complex catenary math?
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They often start with it! For instance, the main cable of a bridge is a classic catenary under its dead weight. But for many cases—like when the cable is much tighter than its span—a simpler parabolic approximation works well. In the simulator, you can switch between "Catenary" and "Parabolic" models to see how the results compare when you change the Span and Weight per length.
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So the "Maximum Tension" shown is the most important result? Where does that happen?
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Exactly. The maximum tension is always at the highest support points, never at the lowest sag point. A common mistake is to think the middle is under the most stress. For the catenary, it's $T_{max}= H + w \cdot y_{support}$. Try increasing the "Sag" parameter: you'll see the max tension decrease for the parabolic model, but the relationship is more nuanced for the exact catenary.
Physical Model & Key Equations
The exact catenary shape is defined by the hyperbolic cosine function, where the key parameter a is the ratio of horizontal tension to cable weight.
$$y(x) = a \cosh\!\left(\frac{x}{a}\right) - a, \quad a = \frac{H}{w}$$
Here, $H$ is the horizontal tension (N), $w$ is the weight per unit length (N/m), and $a$ (m) is the catenary parameter. The sag $d$ is not directly in the equation; $a$ must be found by solving $a\left[\cosh(L/(2a)) - 1\right] = d$ iteratively.
The parabolic approximation is valid for small sag-to-span ratios ($d/L < ~1/8$). It provides a direct, closed-form solution for horizontal tension.
$$y(x) \approx \frac{4d}{L^2}x(L-x), \quad H \approx \frac{w L^2}{8d}$$
Here, $L$ is the span (m) and $d$ is the sag (m). This simplification is powerful because it avoids iterative solving. The maximum total tension at the support is $T_{max} \approx H + w \cdot d$ for both models, but the exact catenary uses the actual support height.
Frequently Asked Questions
If the sag is small relative to the span (guideline: sag ratio of 1/8 or less), the parabolic approximation provides sufficient accuracy and is easier to calculate. For large sags or when high precision is required, use the catenary curve (exact solution). Since this tool calculates both simultaneously, you can compare the differences and make your selection.
Basically, you should input the cable's own weight per unit length (N/m), but in actual design, please input the combined load including ice and snow loads, wind loads, etc. Although this tool uses a physical model of pure self-weight only, by inputting the combined load as an equivalent unit weight, practical tension calculations are possible.
Horizontal tension is the tension at the lowest point of the cable and is used for designing the horizontal reaction force at the supports. Maximum tension occurs at the supports (ends) and is used for the strength design of the cable itself and selection of end fittings. When considering safety factors, always base them on the maximum tension.
In that case, the input sag may be physically impossible. In a catenary curve, as the sag approaches zero, the cable length approaches the span, but if the sag is extremely small, numerical calculation errors may cause a reversal. Please increase the sag slightly and recalculate.
Real-World Applications
Suspension Bridge Main Cables: The massive cables supporting the deck are analyzed as catenaries under their own weight. Engineers use the calculated initial shape and tension to design the towers and anchorages. The parameter $a$ is often used directly to define the initial geometry in Finite Element Analysis (FEA) software like ABAQUS.
Overhead Power Transmission Lines: Sag and tension calculations are critical for safety and clearance. Engineers must ensure lines don't sag too low in summer heat (increased length) or become too tight and snap in winter cold. The parabolic model is often sufficient for these long, relatively taut spans.
Tensile Fabric Roofs & Edge Cables: The perimeter cables of membrane structures often follow a catenary shape. Accurate tension prediction is essential for structural stability and for determining the reaction forces on supporting masts and foundations.
Marine & Mooring Systems: Heavy chains and cables used to moor ships, buoys, and offshore platforms form underwater catenaries. The analysis determines holding capacity and ensures the cable doesn't fully lift off the seabed, which would cause a shock load.
Common Misconceptions and Points to Note
When starting to use this tool, there are several pitfalls that engineers, especially those with less field experience, often fall into. First is the idea that "if the sag is small, a parabolic approximation is sufficient". It's true that if the sag ratio (d/L) is 0.1 or less, the error is small, within 1%, but this is purely about the shape. For tension calculations, results can differ between the catenary and parabola even under the same conditions. For example, for a span of 100m, a sag of 10m, and a unit weight of 10 N/m, the parabolic approximation formula $H = wL^2/(8d)$ gives a horizontal tension of 1250N, while the exact catenary solution gives about 1235N—a difference of about 1%. For large-scale structures, this difference cannot be ignored.
Next, consistent units for input parameters. This is basic but often a source of confusion. Mixing SI units and practical units—span [m], weight [N/m], cross-sectional area [mm²], Young's modulus [GPa]—can ruin your calculation. Especially for Young's modulus, even if you know "210 GPa for steel", if the input field expects Pa, you need to enter 210,000,000,000, which is prone to digit errors. Before using the tool, make it a habit to convert all values to SI units based on [N] and [m].
Finally, regarding the limitations of "elastic elongation" calculation. This tool's elastic elongation calculation is based solely on linear elongation from small deformation theory. Actual cables, especially wire ropes, experience initial "bedding-in" elongation and creep phenomena, which can result in greater elongation than the calculated value. In design, you must use this calculated value as a base and apply a safety factor based on manufacturer data or measured values.