What if a bridge were held up at one end only?

A beam bridge is like a long plank. It rests on something at each end.

What if one end had nothing under it? Would it still hold a truck?

A simple beam bridge is a long, strong beam with a support at each end. Both supports push up, sharing the weight of the beam and anything driving over it.

Now imagine one support is taken away, and the beam is held only where it is fixed at the other end, sticking out like a diving board. A beam held like this is called a cantilever. Does it still carry a truck, and how much more does it bend?

A simply supported beam rests on two supports, which share the load between them. A cantilever is fixed at one end only and carries its load out along the free part, so the fixed end must both hold up the weight and stop the beam from rotating.

This lab uses a beam-bending model to compare the two arrangements for the same 10 m steel beam and the same 20 t truck. Which one sags more, how much harder does the steel work, and where along the beam is the stress greatest?

In the lab, the steel beam rests on both banks of a river.

A big truck drives to the middle. The beam bends just a tiny bit.

In the model, a steel beam 10 meters long, 1 meter deep and half a meter wide rests on a support at each end. A 20-tonne truck stops in the middle.

The beam sags only about 1 millimeter (the readout shows 1.09 mm). The steel is using only a small part of its strength: the model's safety factor is 21.4, which means the steel could take about 21 times as much bending stress before it gave way.

Simply supported, with the 20 t truck at midspan and the beam's own weight included, the model gives a biggest sag of 1.09 mm and a biggest bending stress of 11.7 MPa. Against a yield strength of 250 MPa for common structural steel, that is a safety factor of 21.4.

The bending moment from the truck is greatest under it, P·L ÷ 4 at midspan, and falls to zero at both supports, which can only push up and cannot resist rotation.

Now Olly takes away the support at one end. The beam is held only at the other end.

It bends much more! But it still holds the truck.

Switch the supports to 'one end only'. The beam is now fixed at the left bank and sticks out over the river with nothing under its far end.

With the truck still in the middle, the sag jumps from 1.09 to 8.23 millimeters, more than seven times as much. The safety factor drops from 21.4 to 7.2. Drive the truck out to the free tip and it gets worse: 13.6 millimeters of sag and a safety factor of 5.4. The model still calls it safe, because this beam is deep and strong.

As a cantilever with the truck at midspan, the model gives 8.23 mm of sag, 34.9 MPa and a safety factor of 7.2. With the truck at the tip the moment at the fixed end is P·L plus the beam's own weight term w·L² ÷ 2: the stress reaches 46.6 MPa, exactly four times the simply supported midspan case, and the sag 13.6 mm, about 12.5 times.

That margin depends on the beam. Make it 0.5 m deep and the cantilever with the truck at the tip falls to a safety factor of 1.8 ('risky'), while the same shallow beam held at both ends stays at 7.1.

With two supports, both ends help hold the weight.

With one support, the fixed end does all the work. That is where the beam bends most.

When a beam rests on two supports, each one pushes up and takes a share of the load. Held at one end only, the fixed end must hold up all the weight and also stop the beam from tipping down like a seesaw. The farther out the truck is, the harder it twists the beam at that fixed end.

That is why, in the model, the beam works hardest right where it is fixed. Encyclopaedia Britannica explains another twist: in a cantilever the top of the beam is stretched and the bottom is squeezed, the opposite of a beam held at both ends.

For a cantilever, the bending moment at a distance x from the fixed end is M(x) = P(a − x) for a point load P at a, plus w(L − x)² ÷ 2 from the beam's weight. Both terms are largest at x = 0, so the peak stress, σ = M ÷ S with S = b·h² ÷ 6, always sits at the fixed end. Britannica notes that the upper half of a cantilever is in tension and the lower half in compression.

The model's limits: one solid rectangle of one material, small elastic bends, the truck as a point load. It does not check the fixed support itself, buckling, twisting, shear, fatigue from many trucks, wind or vibration, and real fixings are never perfectly rigid.

The Forth Bridge in Scotland has huge arms that stick out from its towers.

Trains have crossed it since 1890.

Cantilevers are everywhere. Britannica notes that any beam built into a wall with its free end sticking out is a cantilever, and buildings use them to hold up galleries, roofs and canopies over open space.

The Forth Bridge in Scotland is a famous cantilever bridge. Its giant steel arms reach out from towers on both sides, like very strong diving boards, and the arms pointing toward each other hold up the spans hung between them. When it opened in 1890 its main spans, 541 meters each, were the longest in the world, and trains still cross it today.

Britannica describes cantilever construction as a choice for large spans with heavy loads, and calls the Forth Bridge the classic example: three cantilevers joined by two suspended spans. PBS's Building Big explains that rigid arms project from both sides of each pier, so the inward arms are supported on one side only and carry the central span.

UNESCO records that the bridge opened in 1890 with the world's longest spans, 541 m, and is 2,529 m long overall. About 54,000 tons of mild steel, then a fairly new material, were rolled and riveted into tubes 4 m across that work in compression, with lighter steel spans in tension.