Chapter 4: Materials

The chapter problem: the crane cable

A crane lifts a steel beam on a building site. The beam hangs from one vertical steel cable.

These are the facts to use all the way through the chapter.

• The beam has a mass of 2500 kg.
• The cable is 20.0 m long when no load hangs on it.
• The cable has a cross-sectional area of 1.5 × 10⁻⁴ m².
• The steel has a Young modulus of 2.0 × 10¹¹ Pa.
• The cable has a breaking stress of 1.5 × 10⁹ Pa.
• Lifting rules need a safety factor of 5. The working stress must not exceed one fifth of the breaking stress, which is 3.0 × 10⁸ Pa.

By the end of the chapter you will find the stress and the strain in the cable. You will work out how far the cable stretches and the energy stored in it. You will then decide whether the cable is safe for the beam, and find the heaviest beam it can lift under the safety rule.

First contact: no working yet. Estimate. The cable is 20 m long. Under the load it stretches a little. Does it stretch by about a millimetre, about a centimetre, or about a metre? Write down your estimate and one sentence of reasoning.

By the end of this chapter you will be able to

• explain why engineers use stress and strain instead of force and extension;
• calculate the tensile stress in a wire or cable;
• calculate the tensile strain, and state that it has no unit;
• define the Young modulus, and use it to find a stress, a strain or an extension;
• find the spring constant from the gradient of a force-extension graph;
• describe the limit of proportionality, the elastic limit and plastic deformation;
• find the energy stored in a stretched material from the area under its force-extension graph;
• use $E = \tfrac{1}{2}F\Delta L$ when the graph is a straight line through the origin;
• check whether a cable is safe for a load.

Where it is used

A structural engineer chooses the cables of a bridge, a lift or a crane. The choice depends on the stress the cable will carry, not on the load alone. A climbing rope is made to stretch, so it can absorb the energy of a fall. A dentist chooses a wire for a brace that bends by the right amount. The same ideas explain why a thin wire breaks under a load that a thick bar of the same steel holds safely.

Your turn to connect it

Think of something that changes shape when a load acts on it. It could be a rubber band, a bungee cord, a trampoline mat or the handle of a plastic bag. Write two examples. For one of them, say whether it returns to its original shape when the load is removed.

Roadmap

1. 4.1 Stress, strain and the Young modulus
2. 4.2 Force-extension graphs and stored energy
3. Review