Math Detours · Grade 6 · Connections · 13

Why bridges have gaps

A steel bridge 100 m long grows about 1.2 mm for every degree it warms. From a winter night to a summer afternoon that adds up to several centimetres.

Name ____________________Date ____________

Tonight's idea

Almost every solid grows a tiny bit when it warms and shrinks back when it cools. In a spoon you cannot see it. In a bridge you can.

For ordinary steel, each metre grows by about 0.012 mm for every degree Celsius it warms. That is 12 millionths of its length. (Structural steels vary from about 11 to 13 millionths. We use 12.)

For steel, roughly: growth in mm = length in m × 0.012 × temperature change in °C.

Twice the length gives twice the growth. Twice the temperature change also gives twice the growth.

A bridge held tight at both ends would push on its supports with a very large force. So one simple design holds one end still and leaves a gap at the other, called an expansion joint.

Cold day: the joint is open Hot day: the joint is nearly closed
Looking down on one kind of expansion joint, a finger joint. Cars cross from left to right. (A sketch, not to scale.)

The rule is a model: it assumes the steel is all at one temperature and free to move.

Watch one: a winter night to a summer afternoon

A steel bridge is 100 m long. On a winter night the steel is at \(-10\) °C. On a summer afternoon it is at 35 °C. How much longer is the bridge in summer?

From \(-10\) up to 0 is 10 degrees, and from 0 up to 35 is 35 more, so the temperature change is \(35 - (-10) = 45\) degrees. Each metre grows 0.012 mm, so 100 metres grow \(100 \times 0.012 = 1.2\) mm for each degree. Now all 45 degrees:

\[ \text{growth} = 100 \times 0.012 \times 45 = 1.2 \times 45 = 54 \text{ mm} \]

That is 5.4 cm. Steel does not squash easily, so the 54 mm has to go somewhere.

Your turn

  1. A steel footbridge is 20 m long. How much does it grow when the steel warms from \(-10\) °C to 35 °C? Compare your answer with the 54 mm of the 100 m bridge. What do you notice?
  2. Older railway track was laid as separate rails with a small gap at each join. Take 1 km of track made of steel rails 25 m long, each free to grow. The steel warms by 30 degrees. How much does one rail grow? How many rails are there, and how much do they grow altogether?
  3. A 30 cm ruler is made of the same kind of steel as the bridge. On a day 20 degrees colder, how much shorter is it? Would you notice? (Careful: the rule wants metres.)
  4. A steel bridge 200 m long is built on a day when the steel is at 15 °C. One end is held still. The steel may get as hot as 40 °C and as cold as \(-10\) °C. How wide must the gap at the free end be on building day so that it only just closes at 40 °C? How wide is that gap at \(-10\) °C?
  5. Look at the finger joint in the picture. A plain gap 120 mm wide across the road would give every car wheel a hard jolt. How do the teeth solve the problem?

Stretch: steel inside concrete

Many bridges are made of concrete with steel bars cast inside it. A typical concrete grows about 0.010 mm per metre per degree (it depends on the stone in the mix). Aluminium grows about 0.023 mm per metre per degree. Take a 10 m beam and warm it by 45 degrees. How much would steel, concrete and aluminium each grow if free to move? Why is it lucky that steel and concrete are so close?

Answers

  1. The change is 45 degrees, as in the worked example. \(20 \times 0.012 \times 45 = 10.8\) mm. The footbridge is one fifth as long as the 100 m bridge and it grows one fifth as much: \(54 \div 5 = 10.8\). Growth goes in step with length.
  2. One rail: \(25 \times 0.012 \times 30 = 9\) mm. There are \(1000 \div 25 = 40\) rails, so altogether \(40 \times 9 = 360\) mm, which is 36 cm. The rule agrees: \(1000 \times 0.012 \times 30 = 360\) mm. The gaps share it out, 9 mm at each join. (Much modern track is welded into long lengths and clamped down hard, so it cannot grow freely. The steel is squeezed instead, and our rule does not describe that.)
  3. 30 cm is 0.3 m. \(0.3 \times 0.012 \times 20 = 0.072\) mm shorter. That is less than a tenth of a millimetre, far too small to see, and for everyday measuring it does not matter. Small things change by small amounts. That is why long things like bridges and rails need gaps and rulers do not.
  4. From 15 °C up to 40 °C is 25 degrees: \(200 \times 0.012 \times 25 = 60\) mm, so the gap must be 60 mm on building day. From 15 °C down to \(-10\) °C is another 25 degrees, so the bridge shrinks by 60 mm and the gap opens to \(60 + 60 =\) 120 mm. Check: the whole range is 50 degrees, and \(200 \times 0.012 \times 50 = 120\) mm. A real designer would add a safety margin, so that the gap never quite closes.
  5. The teeth of one comb reach into the spaces of the other. When the bridge shrinks, the combs slide apart, but the teeth still overlap, so there is always steel under a wheel as it crosses. The open spaces left over are long and thin and run along the road, and each is only about as wide as one tooth, far narrower than the whole gap. The joint can open and close by many centimetres and the road stays nearly whole.

Stretch. Steel: \(10 \times 0.012 \times 45 = 5.4\) mm. Concrete: \(10 \times 0.010 \times 45 = 4.5\) mm. Aluminium: \(10 \times 0.023 \times 45 = 10.35\) mm. Steel and concrete differ by only \(5.4 - 4.5 = 0.9\) mm over 10 m. Aluminium and concrete would differ by \(10.35 - 4.5 = 5.85\) mm, more than six times as much. Bars cast inside concrete are not free to move: the two materials grip each other. If one tried to grow much more than the other, every warm day and cold night would strain that grip and could crack the concrete. Because steel and concrete grow by nearly the same amount, they warm and cool together almost as one material. Nobody designed that. It is a lucky match, and reinforced concrete depends on it.

← PreviousNext: Model a cup of tea →