Presets
Playback
Bridge
Span, how deep the cable hangs, the weight of the deck and the cable size.
Loads
Traffic on a stretch of deck and a single heavy truck. You can also drag along the deck in the picture.
Weather and drawing
Temperature change and how much the deflection is exaggerated in the drawing.
Layers
Model checks
The science
A cable carries load as pure tension
The main cable is perfectly flexible, so it can only pull along itself. Between the hanger attachment points it is straight; at every hanger it is pulled down by the hanger force fi. Horizontal equilibrium makes the horizontal tension H the same everywhere. Taking moments about a point on the cable gives the funicular shape y(x) = M0(x) / H, where M0(x) is the bending moment the same loads would cause in a simple beam of the same span, and y is the sag below the line joining the towers.
Uniform load: the parabola
For a uniform deck load w over span L, M0 = w x (L - x) / 2, so the cable is a parabola with midspan sag d = w L2 / (8 H), i.e. H = w L2 / (8 d). The tension at a tower is T = sqrt(H2 + (w L / 2)2) and the cable length is close to L (1 + 8 d2 / (3 L2)).
What changes when the load is not uniform
The cable cannot change length freely. Live load, a heavy vehicle or a temperature change would like to move the cable to a different shape, but its material length is fixed. The page therefore solves for the new H by bisection so that the stretched geometric length of the cable equals the unstrained length: Σ dsi(H) = Σ s0,i (1 + α ΔT)(1 + Ti / (E A)), where Ti = H dsi/dx is the tension in segment i. The unstrained lengths s0,i come from the dead-load reference state, so every extra load, warm day or cold day is a genuine change from that state.
Stress and safety
The largest tension is at the tower, Tmax = sqrt(H2 + V2) with V the larger vertical reaction. Cable stress is σ = Tmax / A and the safety factor is σyield / σ, using 1,600 MPa for bridge wire. The deck deflection is the change in cable height at each hanger relative to the dead-load shape.
Model checks on this page
(1) With uniform load the bisection solver must return the closed-form H = w L2/(8 d). (2) The hanger forces must add up to the total load, and the cable shape must satisfy moment equilibrium about the right tower. (3) For a temperature rise, the computed extra sag must agree with the value from differentiating the parabola length series, Lc = L (1 + 8n2/3 - 32n4/5) with n = d/L, i.e. Δd = α ΔT Lc / [(16n/3)(1 - 24n2/5)], when the cable is made very stiff.
Honest simplifications
There is no stiffening girder: the deck simply follows the cable, so deflections are an upper bound for a real bridge, where the girder and the cable together carry moving loads. Hangers are vertical and inextensible, the towers do not move, the cable is one lumped section carrying all loads (two cables are combined), and wind and dynamic effects are ignored. Numbers are illustrative of the physics, not a design check.
What to try
Pick Hot summer day and watch the whole cable sink and the stress fall; pick Cold winter day for the opposite. Switch on Moving truck and look at the third graph: the deflection under a point load is largest near quarter span, not at the middle. Use the Traffic jam preset (a half-span of live load) and see the cable rise on the unloaded side. Reduce the cable area until the safety factor falls below 2 and the cable colour turns red.