The BLOODHOUND Supersonic Car is being built in the BLOODHOUND Technical Centre in Bristol. A large flat bed (the yellow surface table) supported by integral girders is used to ensure the weight of the car is supported and to provide a level reference point for metrology. The expansion and contraction changes of this plate are known by the Engineers. If we had built this plate into the floor and concreted it in there would have been problems.

Try this example to find out what possible problems there could be.

Note. Young’s Modulus for steel is 2.0 x 10^{11} N m^{-2}.

**A horizontal steel girder 3m long, with a cross sectional area of 25 cm ^{2}, has its ends embedded in concrete. If the girder was free to expand, its length would increase by 3.6 x 10^{-5} m for every 1°C rise in temperature.**

**1] Find the thrust in the girder when the temperature rises by 20°C.**

**A slightly harder extension:**

**2] If the elastic limit of the steel is 2.4 x 10 ^{8} N m^{-2}, find out by how much the temperature can rise before the elastic limit is reached and the girder distorts.**

problem set by Gerry Heather

### Solution

At a 20°C rise, the natural length of the girder is:-

3 + 7.2 x 10^{-4} = 3.00072 m

As the girder is held by force by the concrete at the original length of 3m it is compressed at 20°C by 7.2 x 10^{-4} m. The strain in the girder is therefore

7.2 x 10^{-4} ÷ 3.00072 or 2.4 x 10^{-4}.

Hookes Law is used here:-

Stress = Young’s Modulus (steel is 2.0 x 10^{11} N m^{-2}) x strain

Stress = 2 x 10^{11} x 2.4 x 10^{-4} N m^{-2}

Stress is acting on the cross-section area of 25cm^{2}

Therefore the thrust is:-

2 x 10^{11} x 2.4 x 10^{-4} x 25 x 10^{-4}

= 1.2 x 10^{5} N

**Thrust = 120 kN**

### Extension Solution

When the strain in the girder reaches the elastic limit :-

Strain = stress ÷ Young’s Modulus

= (2.4 x 10^{8}) / (2.0 x 10^{11})

= 1.2 x 10^{-3}

The natural length of the girder is 3 + 3.6 x 10^{-5} t m (where t is the rise in temperature in °C)

The length of the girder is 3m.

So, compression = 1.2 x 10^{-3} x 3

= 3.6 x 10^{-3} m

The girdeer reaches its elastic limit when:

3.6 x 10^{-5} x t = 3.6 x 10^{-3}

or t = 100°C

__ This means that the girder will fail when the temperature rises by 100°C__.

However,it will have buckled enough to make measurement accuracy impossible at lower temperatures. If you look at the photo of the real platform above it has no restrictions, so will not deform with temperature changes just freely expand and contract to known figures. This is why it must never be concreted into the floor.

Gerry Heather