M024 · Cable shape and friction

Catenary & Suspension Cable

Is every hanging cable a parabola? No. A cable that carries only its own weight hangs in a catenary. It becomes a parabola when the load is spread evenly across the horizontal span, as when a heavy deck hangs from it.

Hang a cable between two supports at the same height and load it with its own weight, with a bridge deck, or with both. Change the span and sag, compare the shape with a separately computed parabola, check the forces at the anchors and measure the tension along the cable.

Probe coordinate $0$

Physical probe tension $0\,\mathrm N$

Required cable length $0\,\mathrm m$

Model boundary $\text{Static}$

Acquired tension stations $0$

Fitted horizontal tension $\text{Need 3 stations}$

Tension fit RMS residual $\text{Need 3 stations}$

Height above the cable low point $0\,\mathrm m$

Local cable slope $0$

Constant horizontal tension $0\,\mathrm N$

Vertical reaction at each anchor $0\,\mathrm N$

Rope plus deck weight $0\,\mathrm N$

Integrated vertical force residual $0\,\mathrm N$

Maximum same-sag parabolic height difference $0\,\mathrm{mm}$

Measure the tension along a cable

Is every hanging cable a parabola?

Tension vectorsFixed scale per trialSupport span

L

12.00\,\mathrm m

Sag / span

d/L

0.18\,

Rope mass per arc length

\lambda

1.50\,\mathrm{kg/m}

Deck load per horizontal metre

q

0.00\,\mathrm{N/m}

Probe fraction along the span

\xi

0.50\,

Bounded relative tension noise

\varepsilon

0.00\,\%

Both supports are at the same height. The rope's weight is spread evenly along its length, and the deck's weight is spread evenly per horizontal metre. The span and sag of each trial fix its own cable length, so changing them designs a new cable instead of stretching the same rope. The hangers drawn on screen stand for a continuous load, not separate point loads.

Record or scan simulated tension readings. The fit for the horizontal force takes the cable geometry from the model; the geometry is not measured separately. The gray dashed curve is a parabola with the same span and sag, drawn for comparison. Changing a physical parameter or the noise clears the record; moving the probe keeps it.

The cable is flexible and static. It hangs in one plane from rigid supports at the same height, under vertical loads spread continuously along it. The chosen sag sets how long the cable must be. Elastic stretch, wind, bending stiffness, structural strength and vibration are left out. Hanger spacing and rope thickness are drawn only for illustration; the deck carries the chosen effective line load.

Start with the essentials

Focus question: When is the parabola a good model?
One-sentence intuition: Whether the weight is spread along the cable or along the horizontal decides the equation and the shape of the cable.

Core mathematical model

Two ways to distribute weight

H y^{\prime\prime}=\lambda g\sqrt{1+(y^\prime)^2}+q

Rope weight is spread per metre of cable; deck load is spread per horizontal metre. The horizontal part of the tension is the same everywhere.

Self weight gives a catenary

y=a[\cosh(x/a)-1],\qquad a=\frac H{\lambda g}

The origin is at the lowest point. The two anchors sit half a span to either side, both raised by the chosen sag above the origin.

Horizontal load gives a parabola

y=\frac{q x^2}{2H},\qquad H=\frac{qL^2}{8d}

This exact limit assumes zero rope weight. A light rope under a large deck load approaches it.

Tension along the cable and the anchor forces

T=H\sqrt{1+(y^\prime)^2},\qquad 2V=\lambda g\ell+qL

Together, the vertical forces at the two anchors hold up the rope weight and the deck load. The model integrates the cable length numerically, so the drawing does not affect it.

Fitting the tension with the model's slopes

T_i^{\mathrm{obs}}\approx\widehat H\sqrt{1+p_i^2}

The model supplies the cable slopes. The fit estimates the horizontal force from simulated noisy tension readings; it does not measure the shape.

Common difficulties

The shape depends on how the load is spread

Typical misconceptionEvery hanging cable is exactly a parabola.

Better mental modelWeight spread per metre of cable gives a catenary; weight spread per horizontal metre gives a parabola.

Changing the sag means a new cable

Typical misconceptionWhen I change the sag, the same inextensible cable stretches.

Better mental modelEach span and sag needs its own cable length, and the model recomputes it.

The dashed parabola is a reference, not a measurement

Typical misconceptionThe dashed parabola was reconstructed from sensor readings.

Better mental modelIt is calculated separately, as a parabola with the same span and sag.

Run the experiment

01
Start with self weight
Choose Self weight and move the probe from the centre to an anchor.
What to observe: The horizontal part of the tension stays the same, and the total tension grows toward the anchor.
02
Add a bridge deck
Choose Deck dominated and compare the cable with the dashed parabola.
What to observe: With a light rope, the cable comes close to the parabola.
03
Change the sag
Drag the orange anchor handle or use its keyboard slider.
What to observe: A shallower cable needs a larger horizontal force and a different length.
04
Measure the tension along the cable
Add bounded noise, press Scan 25 stations and export the observations.
What to observe: The fitted horizontal force uses the model's slopes; the residual shows how far the readings sit from the fit.