A balanced cantilever bridge supports projecting arms from its main piers. Anchor arms provide balance, while a suspended middle span rests between the inward-projecting tips. Explore the load contributions on one arm below.
Cantilever Bridge Interactive Calculator
Vary the selected arm’s distributed load, length, suspended-span tip reaction, added live-load moment and root depth. Calculate its root bending moment and approximate chord couple.
Equation Used
- One selected cantilever arm carries uniform downward dead load and a downward tip force.
- Live-load moment is supplied by the user and added directly; no vehicle weight, travel position or load pattern is solved.
- Root depth is the effective separation of the upper and lower chord resultants. M/d is a simple bending-couple approximation.
- Overview is schematic; the selected-arm enlargement scales both L and root depth with the same drawing scale. Tip depth is illustrative.
- No deflection, buckling, shear capacity, fatigue, foundation or construction-stage analysis.
Root moment describes the selected cantilever arm at its pier section. Opposite-arm loading must also be included to find the net moment on the pier. The chord force is an approximate bending couple and does not establish member capacity.
Anchor arms, projecting arms and a suspended span
The overview shows two main piers carrying opposed arms. The outer arms connect back toward the abutments; the inner arms support a suspended span between their tips. The blue arm is the part evaluated by this calculator.
A downward load on that arm produces bending at its root near the pier. The whole bridge must also carry and balance loads from the other arms and its supports. Calculating one arm’s root moment is therefore different from calculating the net overturning moment at a pier foundation.
The bridge stays stationary in the animation. The changing highlight identifies each contribution to the calculation, followed by the chord couple that approximately resists the total bending moment.
Understanding a cantilever load path
Hiscox’s illustrated cantilever bridge shows anchored ends, piers and projecting portions supporting the crossing. The Forth Bridge provides a well-known built example of balanced cantilevers connected by suspended spans.
The simplified calculator is useful for comparing how arm length, load and structural depth affect a single root section. It is not a complete bridge model or a basis for choosing a safe span, bearing size or foundation.
Sum the moment contributions
For a uniform load w over an arm of length L, the resultant wL acts at L/2 from the root. Its moment magnitude is wL²/2. A downward tip reaction P contributes PL. Add the separately supplied live-load moment to obtain the selected arm’s total root moment.
Here w is in kN/m, L in metres and P in kN. Divide the first two moment terms by 1000 to express them in MN·m; the live-load input already uses MN·m.
For an approximate root chord couple, divide the moment in MN·m by the effective chord spacing d in metres. The result is in MN. Under the illustrated downward load, bending puts the upper chord in tension and the lower chord in compression. Actual member forces depend on the complete truss geometry and other actions.
Check the default calculation
At w = 450 kN/m and L = 200 m, the distributed-load contribution is 450 × 200² / 2000 = 9000 MN·m. A 12000 kN tip reaction adds 12000 × 200 / 1000 = 2400 MN·m.
Adding 3000 MN·m of live-load moment gives a root moment of 14400 MN·m. With a 40 m effective chord spacing, the approximate couple force is 14400 / 40 = 360 MN.
These are illustrative input values, not a design for a specific bridge. Doubling L quadruples the distributed-load contribution and doubles the tip-force contribution. Increasing depth reduces the simple chord-couple estimate without changing the applied moment.
What this estimate leaves out
The opposite arm and anchorage can counterbalance the selected arm, but their loads are not entered here. The displayed root moment cannot therefore be interpreted as a net foundation overturning moment.
The model also omits shear, axial effects, material strength, member buckling, bearing reactions, wind, fatigue and construction sequence. Neither the load-path animation nor a low chord-force estimate establishes an acceptable physical design.
Questions about the displayed results
What does arm root moment describe?
The formula gives the bending moment in the selected arm at its root. The net loading of the pier also depends on the opposite arm and the rest of the structure.
Why does increasing depth reduce chord force?
A wider separation lets a smaller force couple resist the same moment: F = M/d. This does not check whether the chords or connections can carry that force.
Does the animation show bridge deflection?
No. It highlights load contributions on a stationary structure. Stiffness and deflection are not calculated.
Where does the tip reaction come from?
It is the downward force transferred to this arm by the suspended span. Enter it from a separate analysis of that span.
Primary structural references
Gardner D. Hiscox, Mechanical Movements, Powers, Devices and Appliances (1901), figure 1505, page 359: anchored cantilever bridge layout.
The Forth Bridge World Heritage nomination: balanced cantilevers and suspended spans in a built bridge. The calculator’s dimensions and loads are illustrative and are not attributed to that structure.
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