What if a bridge were twice as long?

A beam bridge is like a long plank across a river. Each end sits on a support.

When a truck drives on, the plank bends a tiny bit. What if the plank were twice as long?

The most common kind of bridge is the beam bridge: a strong beam resting on a support at each end. When a truck drives on, the beam bends a little, squeezed along its top and stretched along its bottom. The distance between the two supports is called the span.

So what happens if we double the span and keep everything else the same? Does the beam bend twice as much? In this lab you can stretch the bridge and measure the sag.

Britannica describes the beam as the most common bridge form: it carries vertical loads by bending, with horizontal compression along its top and tension along its bottom, while the supports pass the load down to the foundations.

This lab keeps a solid steel beam 1 m deep and 0.5 m wide, with a 20 t truck parked at midspan, and changes only one thing: the span, from 20 m to 40 m. Does the deflection double, or does it scale more steeply?

Olly's steel bridge is 20 meters long. A big truck stops in the middle.

The bridge bends down just a tiny bit. It is very safe.

In the model, the steel beam is 20 meters long, 1 meter tall and half a meter wide. A 20-tonne truck stops in the middle. The beam sags about 14 millimeters, and the model calls it safe.

The beam is also heavy: about 78.5 tonnes of steel, almost four times the weight of the truck. Part of that sag comes from the beam holding up its own weight.

With the truck at midspan and the beam's self-weight included, the model gives a maximum sag of 13.6 mm and a peak bending stress of 34.9 MPa. Structural steel in the model yields at 250 MPa, so the safety factor is 7.2.

The 78.5 t beam carries more of its own weight than truck weight. Switch self-weight off and the truck alone causes only 3.92 mm of sag; the beam's weight with no truck causes 9.63 mm.

Now Olly makes the bridge twice as long. Same steel, same truck.

The long bridge bends much, much more! It still holds the truck, but it sags a lot.

Now stretch the span to 40 meters and keep everything else the same. The sag jumps from about 14 millimeters to about 185 millimeters, nearly 14 times as much, not twice.

The beam also works much harder. The safety factor falls from 7.2 to 2.2, only just above the model's line of 2 for safe. Push the span past 41.5 meters and the model says the bridge is too risky.

At a 40 m span the model gives 185 mm of sag (about 13.7 times the 20 m value), 116 MPa of bending stress and a safety factor of 2.2. The beam's mass doubles to 157 t.

The model's verdict changes from safe to risky between 41.5 m (safety factor 2.0) and 42 m (1.97). At its 60 m limit the beam sags 886 mm with a safety factor of 1.0, barely holding. Bends that large stretch the model's simple beam theory, so treat that number as rough.

A long beam has more room to bend. The truck also has a longer lever to bend it.

And a longer beam is heavier. So it has to hold up more of its own weight.

Two things gang up on a long beam. First, the truck bends it more: with the supports twice as far apart, the truck in the middle has more leverage, and the beam has more length to curve. In the model, the truck's sag alone grows 8 times when the span doubles.

Second, a longer beam is heavier, and its own weight is spread along a longer, bendier span. The sag from the beam's own weight grows 16 times. Together, the total sag grows almost 14 times.

The USDA Wood Handbook gives a straight beam's bending deflection as proportional to W·L³ ÷ (E·I), where W is the load and I = b·h³ ÷ 12. For a fixed truck load that means L³: doubling the span multiplies the truck's sag by 8 (3.92 to 31.4 mm in the model). Self-weight scales with length too (W = w·L), so its sag goes as L⁴ and grows 16 times (9.63 to 154 mm).

Model limits: one solid rectangular steel beam, a single point load, no buckling, twisting, shear, wind, vibration or fatigue, and the supports themselves are not checked.

Engineers know long beams bend a lot. So for very long bridges, they use other shapes.

Some bridges have big arches. Some hang from strong cables on tall towers.

Engineers keep spans as short as they sensibly can. PBS's Building Big series explains that the farther apart a beam bridge's piers are, the weaker the beam becomes, and gives a rough rule of thumb: simple beam bridges rarely span more than about 76 meters (250 feet).

To go farther, engineers change the design. Trusses are frameworks of triangles. Arch bridges, mostly steel or concrete today, have great natural strength. Suspension bridges hang the road from cables and, by the same rough guide, can span about 600 to 2,100 meters (2,000 to 7,000 feet).

Britannica notes that spans are generally made as short as possible, with long spans justified where good foundations are hard to find, such as over estuaries with deep water. PBS's Building Big, an older teaching site, gives rounded rules of thumb rather than records: beam bridges rarely beyond 250 ft (about 76 m), suspension bridges 2,000 to 7,000 ft (about 600 to 2,100 m).

A suspension bridge carries loads through curved cables in tension, passing them to towers in compression and to anchorages (Britannica). Because its deck hangs in the air, the deck must be heavy, stiff or both to limit movement. PBS adds that most suspension bridges have a truss, a framework of triangles, under the roadway to resist bending and twisting.