Elastic Energy and Applications: Quick Revision

A brief note on Elastic Energy and Applications. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Elastic energy: $U = \tfrac12 F\,\Delta L = \tfrac12 k(\Delta L)^2$, with $k = \dfrac{YA}{L}$. Per unit volume, $u = \tfrac12\,\text{stress} \times \text{strain} = \dfrac{\text{stress}^2}{2Y}$.
• Rubber's loop: loading and unloading follow different curves; the area between them becomes heat.
• Series and parallel: end to end, the same force and the stretches add; side by side, the same stretch and the forces add.
• Thermal stress: a clamped rod heated by $\Delta T$ carries $Y\alpha\,\Delta T$.
• Design: safety factor $=$ breaking stress $\div$ working stress; a beam sags $\dfrac{Wl^3}{4bd^3Y}$; mountains are limited to $\dfrac{\sigma}{\rho g}$.

The facts at a glance

A worked example

A steel wire $2.5\ \text{m}$ long with a section of $1.0\ \text{mm}^2$ is stretched by $1.25\ \text{mm}$. Steel breaks at $4.0 \times 10^{8}\ \text{Pa}$ and a safety factor of $4$ is required. ($Y = 2.0 \times 10^{11}\ \text{Pa}$.)

1. Force. $F = YA \times \dfrac{\Delta L}{L}$, with $YA = 2.0 \times 10^{5}\ \text{N}$ and $\dfrac{\Delta L}{L} = 5.0 \times 10^{-4}$, so $F = 100\ \text{N}$.
2. Energy. $U = \tfrac12 \times 100\ \text{N} \times 1.25 \times 10^{-3}\ \text{m} = 0.0625\ \text{J}$.
3. Safe? The stress is $\dfrac{100\ \text{N}}{10^{-6}\ \text{m}^2} = 1.0 \times 10^{8}\ \text{Pa}$, exactly the allowed $\dfrac{4.0 \times 10^{8}\ \text{Pa}}{4}$: at the limit, so no more stretch.

Practice with new numbers

A program to try

The program checks the wire of the worked example.

Mistakes to avoid

• $F\,\Delta L$ for the energy. The force grew from zero: use $\tfrac12 F\,\Delta L$.
• Sharing the load in series. End to end, each wire carries the whole load.
• Equal shares in parallel. Stiffer wires take more.
• Length in thermal stress. It does not appear; only $Y$, $\alpha$ and $\Delta T$ do.
• Dividing the load by the safety factor. Design for the load times the factor.

Check yourself

1. A wire stores $0.05\ \text{J}$ when stretched by $1\ \text{mm}$. How much at $3\ \text{mm}$?

2. Why are bridges built with expansion joints?

3. A cable breaks at $5.4 \times 10^{8}\ \text{Pa}$ and works at up to $9.0 \times 10^{7}\ \text{Pa}$. What is its safety factor?

4. Why are girders I-shaped?

5. Two identical wires side by side share a load. Compared with one wire carrying it all, how much energy is stored in the pair?