Explore cable sag and tension by changing span, weight, reference tension and temperature. The model assumes equal-height supports and a perfectly flexible cable.
Cable Parameters
m
N/m
kN
°C
×10⁻⁶
Results
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Max Sag (m)
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Horizontal Tension H (kN)
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Max Tension at Support (kN)
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Cable Length (m)
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Parameter a (m)
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Sag / Span
Hanging Cable (Real-Time)
Catenary y=a·cosh(x/a) Parabolic approx. Sag Tension vector
Lower the tension and the cable loses to its weight, sagging deeper while the support tension vectors swing steeper. At the lowest point tension equals the horizontal tension H (horizontal); it grows larger and steeper toward the supports.
Maximum tension occurs at the supports. At the lowest point \(T_{\min}=H\) (horizontal). H₀ and w₀ are reference-temperature inputs (ΔT=0). Set S=S₀(1+αΔT), then solve the catenary arc-length condition at the same span. Total weight is conserved: w=w₀/(1+αΔT), H=wa. Supports are fixed at equal height and the cable is perfectly flexible. Axial elastic extension, creep, wind, ice and point loads are excluded.
What is a Catenary Cable?
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What exactly is a "catenary" curve? I've heard it's the shape of a hanging cable, but why is it special?
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Basically, it's the exact mathematical shape a perfectly flexible, uniform cable or chain forms when hanging under its own weight. It's special because it's the shape of pure tension equilibrium. In practice, it looks like a parabola but is mathematically different. You can see it in action here—try increasing the "Unit Weight" slider. The cable sags more dramatically because its own weight is pulling it down harder.
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Wait, really? So if it's not a parabola, what controls its shape? And what's that "a" parameter in the equation?
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Great question. The shape is controlled by the balance between the cable's weight and the horizontal tension pulling it taut. The key parameter is $a = H / w$, where $H$ is the horizontal tension and $w$ is the weight per unit length. A larger $a$ means a flatter curve. For instance, in a power line, you want high tension ($H$) to minimize sag. Try it: slide the "Horizontal Tension" control up and watch the sag decrease.
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That makes sense! But what about temperature? I see there's a "Temperature Change" parameter. Why does that matter for a cable?
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In practice, temperature is a huge design factor. Metals expand when hot and contract when cold. This changes the cable's length, which directly affects its sag and tension. A common case is a power line on a hot summer day—it sags much lower, which is a safety hazard if it gets too close to the ground. Play with the ΔT and α (thermal expansion) sliders to see how much the sag increases for a given temperature rise.
Physical Model & Key Equations
The fundamental catenary curve is derived from a force balance on a cable segment. The shape is described by the hyperbolic cosine function.
Where: $y(x)$ = vertical height at horizontal position $x$ (m). $a$ = catenary parameter = $H/w$ (m). $H$ = horizontal tension in the cable (N). $w$ = weight per unit length (N/m). $\cosh$ = hyperbolic cosine function.
Two other critical engineering values are the cable's total length (arc length) and the maximum tension, which occurs at the highest support point.
$$ S = 2a \cdot \sinh\left(\frac{L}{2a}\right) $$
$$ T_{max}= H + w \cdot y_{max}= H + w \cdot (y(L/2) - y(0)) $$
Where: $S$ = total arc length of the cable (m). $L$ = horizontal span between supports (m). $T_{max}$ = maximum tension, at the support (N). $\sinh$ = hyperbolic sine function.
The thermal elongation is calculated from the reference length $S_0$ as $\Delta S = \alpha \cdot S_0 \cdot \Delta T$.
Frequently Asked Questions
Enter reference-temperature weight w₀ in N/m. Convert mass in kg/m by multiplying by 9.80665; multiply kgf/m by 9.80665 as well. Enter H₀ in kN at ΔT=0. The results show the horizontal tension H after the temperature change.
H₀ and w₀ are reference-temperature inputs (ΔT=0). Set S=S₀(1+αΔT), then solve the catenary arc-length condition at the same span. Total weight is conserved: w=w₀/(1+αΔT), H=wa. Supports are fixed at equal height and the cable is perfectly flexible. Axial elastic extension, creep, wind, ice and point loads are excluded. Outside this model: if cooling makes the prescribed cable length no longer than the span, axial elastic extension must be modeled. Reduce reference tension or review the temperature change.
Maximum tension typically occurs at the support points (span endpoints) at both ends of the cable. In a catenary shape, tension is minimal at the lowest point (horizontal tension H) and increases toward the support points. In design, ensure that this maximum value is below the allowable tension of the cable.
This tool models static vertical self-weight only; it does not model wind or ice loads. Wind can act horizontally and ice changes the cable load. Use a separate analysis that accounts for their directions, combinations and applicable design requirements.
Real-World Applications
Overhead Power Lines: This is the most common application. Engineers must calculate sag precisely to ensure safe clearance from the ground under all conditions (wind, ice load, high temperature). The simulator's thermal expansion feature directly models the dangerous summer sag increase.
Suspension Bridge Cables: The main cables of bridges like the Golden Gate Bridge follow a catenary curve under their own weight. The accurate calculation of cable length and tension is critical for constructing the bridge deck and ensuring structural integrity.
Telecommunication & Ropeway Cables: For aerial cable cars or gondolas, the catenary calculation determines the tower height needed and the tension in the haul cable, which directly impacts the motor power required and passenger comfort.
Marine & Mooring Systems: Heavy chains and ropes used to moor ships, buoys, or offshore platforms form catenaries. The shape absorbs energy and provides a restoring force, acting as a natural shock absorber against waves and currents.
Common Misunderstandings and Points of Caution
First, a common misconception is unconsciously assuming that the support points are at the same height. In actual field conditions, it's almost always the case that support heights differ due to ground level variations where towers are erected or differences in building attachment points. This simulator assumes equal heights, so if there is a height difference, the calculation results cannot be applied directly. For example, for transmission lines in mountainous areas, a different calculation formula that accounts for this height difference is required.
Next, a pitfall in parameter setting is mistakes in unit consistency. Particular care is needed for the "unit weight w". For instance, if you input a value without checking whether the cable manufacturer's catalog specifies "N/m" or "kgf/m", the calculation result will be completely different. In practice, it's an ironclad rule to unify to the SI unit system (N, m, Pa). For example, the unit weight of a 20mm diameter steel wire is approximately 24.5 N/m. If you input just "24.5" and forget the unit, you're inviting a major disaster.
Another crucial point of caution is overlooking the initial condition setting. The tension H and sag calculated here represent an "equilibrium state" under specific temperature and load conditions. However, the temperature when installing the cable (applying the initial tension) is not necessarily the design reference temperature (e.g., 15°C). If you install it too taut on a hot summer day, excessive tension may occur in winter, which is dangerous. The key in practice is to treat the temperature change ΔT as the change from the installation temperature.
