Elasticity in design: ropes, beams and mountains

The safety factor

No engineer loads a cable up to its breaking stress. Metals have flaws, loads jerk, and wear builds up over the years. So a safety factor is used:

For a lift or a crane it is often $5$ to $10$. A rope is then chosen so that it could hold the largest load times the safety factor before breaking.

Beams

A beam resting on two supports with a load $W$ at its middle sags by

where $l$ is the length between the supports, $b$ the breadth and $d$ the depth.

You should find that doubling the depth cuts the sag to an eighth, but doubling the breadth only halves it.

So a plank stands on its narrow edge, and steel girders are I-shaped. Most of the metal sits at the top and bottom, which are squeezed and stretched the most. Little is wasted in the middle, which hardly changes length. Long beams can also buckle: they bend sideways under a push along their length. A deep, thin beam can do this, and the wide top and bottom of the I-shape help to stop it.

Mountains

The rock at the base of a mountain carries the weight of all the rock above it: a stress $\rho gh$. Above a certain stress, rock flows. With rock that flows at $2.5 \times 10^{8}\ \text{Pa}$ and a density of $2700\ \text{kg/m}^3$,

This is close to the height of the tallest mountains on the Earth.

Watch out. A safety factor divides the breaking stress, never the load: the working load is the smaller one.

A worked example

A crane must lift loads up to $2500\ \text{kg}$ with a safety factor of $6$. Its steel rope breaks at $4.5 \times 10^{8}\ \text{Pa}$. What is the smallest radius of rope?

1. Force to design for. $6 \times 2500\ \text{kg} \times 9.8\ \text{m/s}^2 = 1.47 \times 10^{5}\ \text{N}$.
2. Area. $\dfrac{1.47 \times 10^{5}\ \text{N}}{4.5 \times 10^{8}\ \text{Pa}} = 3.27 \times 10^{-4}\ \text{m}^2$.
3. Radius. $r = \sqrt{\dfrac{3.27 \times 10^{-4}\ \text{m}^2}{\pi}} = 1.02 \times 10^{-2}\ \text{m}$, about $1.0\ \text{cm}$.

Think first. Real crane ropes are made of many thin strands twisted together, which makes them flexible; the total area is what counts.

Practise with new numbers

Try it in Python

The program sizes the crane's rope and finds the tallest mountain.

Remember these

• Safety factor $=$ breaking stress $\div$ working stress.
• Beam sag $\delta = \dfrac{Wl^3}{4bd^3Y}$: depth cubed, length cubed.
• Tallest mountain: $h = \dfrac{\sigma}{\rho g}$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A cable breaks at $6.0 \times 10^{8}\ \text{Pa}$ and is used at up to $1.2 \times 10^{8}\ \text{Pa}$. What is its safety factor?

2. A beam's length between its supports is halved. By how many times does its sag fall?

3. Why does a plank sag less on its edge than lying flat?

4. Why can mountains be taller on a planet with weaker gravity?