Linear Expansion
From a real bridge to a law: why bridges need a joint
Physics · delta (3A) · 18 September 2026
Before we start
- What is the difference between temperature and heat?
- What does it mean for two objects to be in thermal equilibrium?
- Why can temperature never be negative in Kelvin?
The Golden Gate Bridge, San Francisco
Main span: 1275 m of steel. Temperature between −15 °C and 40 °C.
How much does it expand between the coldest and hottest day?
Click a button for each raised hand (Shift+click to undo). Source: OpenStax University Physics, thermal expansion example.
The law of linear expansion
Definition
The change in length of a solid is proportional to its initial length and to the temperature change.
\htmlClass{c-dL}{\Delta L}\;=\;\htmlClass{c-alpha}{\alpha}\cdot\htmlClass{c-L0}{L_0}\cdot\htmlClass{c-dT}{\Delta T}
\Delta Lchange in length (m)
\alphalinear expansion coefficient (°C⁻¹): depends on the material
L_0initial length (m)
\Delta Tchange in temperature, T − T₀ (same value in °C and K)
Try the law
Change the data, watch ΔL
Material (α)
Initial length L₀20 m
Temperature change ΔT+45 °C
Bar enlarged ×200: on a real bar the change is far too small to see.
Linear expansion coefficients
| Material | α (×10⁻⁶ °C⁻¹) | |
|---|---|---|
| Steel / iron | 12 | |
| Copper | 17 | |
| Brass | 19 | |
| Aluminium | 23 | |
| Glass | 9 |
Typical textbook values near room temperature.
Worked example
An iron rail 20 m long goes from −10 °C to 35 °C. How much does it expand?
1
\Delta T = 35-(-10)=45\ ^{\circ}\mathrm{C}
first, the temperature change
2
\Delta L = 12\times10^{-6}\times 20\times 45
then apply the law
3
\Delta L = 0.0108\ \mathrm{m}=1.08\ \mathrm{cm}
result with consistent units
A real fact: when steel can't expand
- Modern rails are welded into one continuous ribbon: no more old joints with gaps
- In summer the steel heats up far more than the air and wants to expand, but it's constrained by the sleepers: huge stress builds up
- If the stress exceeds the limit, the track bends sideways: it's called a "sun kink"
- July 2002, Maryland: an Amtrak train derailed near Washington DC, injuring over 100 people; the derailment was linked to a sun kink
Source: AP and CNN reports on the July 2002 Amtrak derailment at Kensington, Md.; engineering literature on "sun kink" in welded rail.
Now it's your turn
Set up first: data, unknown, principle, justification, equation.
- Copper cable, 50 m, from 5 °C to 40 °C
- Rail, 15 m, from −5 °C to 40 °C: just like the fact you just saw
- Copper bar, 3 m, from 20 °C to 150 °C
- (Harder) Aluminium bar: find α knowing ΔL
When the simple model isn't enough: the Eiffel Tower
- 300 m of puddled iron. If it were a single bar, α·L₀·ΔT with ΔT = 60 °C predicts over 20 cm of seasonal variation
- Engineers instead measure 12–15 cm: a third less than predicted
- Why? It's a lattice of 18,000 pieces, not all vertical, and the sun heats one side more than the others
- The model ΔL = αL₀ΔT is still correct: it works well for a single bar, less well for a complex structure
Source: engineering monitoring of the Eiffel Tower, reported by The Conversation and Interesting Engineering (2025).
Back to the bridge
Golden Gate Bridge: 1275 m of steel, ΔT = 55 °C (from −15 °C to 40 °C).
1
\Delta L = 12\times10^{-6}\times 1275\times 55
same law, real numbers
2
\Delta L \approx 0.84\ \mathrm{m}
almost a metre, spread across many expansion joints along the bridge
+
Area and volume expansion
(if there's time)
From linear expansion to the other two
By analogy
\beta\approx 2\alpha\qquad\qquad\gamma\approx 3\alpha
\Delta A=\beta\,A_0\,\Delta T\qquad\qquad \Delta V=\gamma\,V_0\,\Delta T
A=L^2:\quad A_0(1+\alpha\Delta T)^2=A_0\left(1+2\alpha\Delta T+\alpha^2\Delta T^2\right)
\alpha\Delta T\sim10^{-3}\ \Rightarrow\ (\alpha\Delta T)^2\sim10^{-6}\text{ is negligible; likewise for the cube}
A counterintuitive fact
Does the hole get smaller or bigger?
A metal plate has a hole. If I heat it, does the hole get smaller or bigger?
Bigger.
The hole expands as if it were made of the same material as the plate.
Temperature change ΔT0 °C
Shrink fitting
- Heat a ring or a bearing: its hole expands
- Slide it onto the shaft
- Let it cool: it shrinks and grips tightly
- A real technique used to mount wheels, gears and bearings in mechanical engineering
Standard mechanical engineering principle (shrink fitting).
Worked example: area expansion
A steel plate has a hole of area 5.00 cm² at 20 °C. Heated to 220 °C, what is the new area?
1
\Delta T = 200\ ^{\circ}\mathrm{C},\quad \beta = 2\times12\times10^{-6}=24\times10^{-6}\ ^{\circ}\mathrm{C}^{-1}
double α, don't start from scratch
2
\Delta A = 24\times10^{-6}\times 5.00\times 200 = 0.024\ \mathrm{cm}^2
the hole's area increases
3
A = 5.00+0.024=5.024\ \mathrm{cm}^2
the hole gets bigger, consistent with the fact just seen
Why cars have an "expansion tank"
- The engine coolant heats up a lot and increases in volume
- Without a tank to absorb the increase, the pressure would burst the pipes
- Same physics on a planetary scale: the ocean warms and increases in volume. According to NASA, about a third of the sea level rise observed by satellites since 2004 comes from this, not from melting ice
Source: NASA Sea Level Change, "Thermal expansion" (sealevel.nasa.gov). The share varies by year: in 2024 it was about two thirds.
Worked example: volume expansion
A brass pipe fitting has a volume of 500 cm³ at 20 °C, and is heated to 80 °C.
1
\Delta T = 60\ ^{\circ}\mathrm{C},\quad \gamma = 3\times19\times10^{-6}=57\times10^{-6}\ ^{\circ}\mathrm{C}^{-1}
triple α
2
\Delta V = 57\times10^{-6}\times 500\times 60 = 1.71\ \mathrm{cm}^3
small, but measurable
In summary
- \Delta L=\alpha L_0\Delta T: expansion depends on the material, the initial length and the temperature change
- The Golden Gate Bridge expands by 84 cm: that's why bridges need expansion joints
- Area and volume expansion: same law, with \beta\approx2\alpha and \gamma\approx3\alpha: from the hole that gets bigger to the expansion tank
- Next lesson: 22/9, Arduino Lab 1: measurement error becomes central again