Applications of elastic behaviour

How thick should a crane rope be?

Narrated video. Transcript:

Elastic behaviour matters in everyday engineering. Watch a crane at a building site lift a load of 10 tonnes, that is 10⁴ kilograms, with a steel rope.

The rope must never be stretched past its elastic limit. If it yields, it is permanently deformed and can fail. So the stress in it must stay below the yield strength of steel, about 3 × 10⁸ newtons per square metre.

The stress is the weight M g divided by the area π r squared. Setting M g over π r squared less than or equal to σ y gives r at least the square root of M g over π σ y.

Put in the numbers: the square root of 10⁴ times 9.8, divided by π times 3 × 10⁸. That gives a radius of about 1 centimetre.

But a real crane must also survive jerks, wear and rust. Engineers build in a safety factor of about ten: the rope is designed for ten times the load.

Ten times the load needs ten times the area, so the radius grows by the square root of ten, to about 3 centimetres.

A single solid steel rod 3 centimetres thick would be stiff and hard to wind on a drum. So the rope is made of many thin wires braided together, like a plait. It is strong, and still flexible.

Bridges are designed the same way. A bridge must carry the weight of traffic, the force of winds and its own weight, without bending too much or breaking.

Bending beams and I-girders

Narrated video. Transcript:

Lay a bar of length l, breadth b and depth d across two supports near its ends, and hang a load W at its centre. The bar bends, and its middle sags by a distance δ.

The sag is δ equals W l cubed, over 4 b d cubed Y, where Y is Young’s modulus of the material.

To reduce bending, first choose a material with a large Young’s modulus. A steel beam sags far less than a wooden one of the same size.

Then look at the shape. The sag depends on b to the power one, but on d to the power three. Doubling the depth cuts the sag by a factor of eight.

Here is a wooden plank across a drain. Laid flat, it sags a lot. Turn it on its edge, so the depth is large, and it hardly bends at all.

But a very deep, thin beam has a problem. Under a heavy load it can twist and buckle sideways, just like a ruler standing on its edge.

The answer is the I-shaped cross-section used in railway tracks and building girders. The wide top and bottom flanges carry the load and resist buckling, while the thin web gives the depth.

An I-beam keeps most of the stiffness of a solid beam of the same outer size, with much less steel. That means less weight and less cost.

Pillars and columns work the same way. A pillar with rounded ends supports less load than one with a distributed, broadened shape at its ends.

How tall can a mountain be?

Narrated video. Transcript:

Elasticity even sets a limit on the height of mountains. Look at the rock at the base of a mountain of height h.

The weight of all the rock above presses down on it. The force per unit area at the base is h ρ g, where ρ is the density of the rock.

The sides of the mountain are free, so this is not the same as squeezing from all sides. The base rock feels a shearing stress of roughly h ρ g as well.

If this stress goes past the elastic limit of the rock, the rock at the bottom starts to flow, like very thick toothpaste, and the mountain sinks. So h ρ g must stay below the elastic limit.

The elastic limit of typical rock is about 30 × 10⁷ newtons per square metre. Its density is about 3 × 10³ kilograms per cubic metre. Take g as 10 metres per second squared.

Then h equals 30 × 10⁷, divided by 3 × 10³ times 10. That is 10⁴ metres, or 10 kilometres.

Mount Everest, the highest peak on Earth, is about 8.8 kilometres tall, just under this limit. No mountain on Earth can be much taller.

On Mars, gravity is only about 3.7 metres per second squared, so the limit is nearly three times higher. Sure enough, Olympus Mons on Mars is more than 20 kilometres tall.

Try it: design lab

Interactive simulation. Things to try:

• Crane rope: find the smallest radius that stays green for 10 t with a safety factor of 10.
• Beam: press “Turn on its side” to swap b and d, and watch the sag change.
• Beam: switch to the I-beam. It uses half the material but sags only about 1.3 times as much.
• Mountain: raise the height past the red limit line, then switch the planet to Mars.

Plank on its edge

A wooden plank 20 cm wide and 5 cm thick is laid flat across a drain, with a load at its centre. It is then turned on its edge, so it is 5 cm wide and 20 cm deep. The load and span are unchanged. What happens to the sag?

1. It stays the same, because the plank and load are unchanged
2. It becomes 4 times smaller
3. It becomes 16 times smaller
4. It becomes 64 times smaller

Rope for a smaller crane

A crane must lift 5 tonnes (5000 kg) with a steel rope of yield strength 3 × 10⁸ N/m², using a safety factor of 6. Take g = 9.8 m/s². What is the minimum radius of the rope?

1. About 0.7 cm
2. About 1.8 cm
3. About 4.3 cm
4. About 5.6 mm

Summary

• Crane rope: r ≥ √(Mg / πσy)
• Sag of a beam (load at centre): δ = Wl³ / (4bd³Y)
• Highest mountain: hρg ≤ elastic limit ⇒ h ≈ 10 km

• Structures are designed so that stresses stay well inside the elastic limit, with a safety factor.
• Crane ropes are braided from many thin wires for strength with flexibility.
• Beam sag falls as d³: deep beams bend less. I-beams give depth and resist buckling with less material.
• The elastic limit of rock limits mountains on Earth to about 10 km; Everest is 8.8 km.