Thermal Stresses & Composite Bars Notes

Last reviewed 16 Sept 2026 · 7 min read

Thermal strain

A bar of length $L$ heated by $Δ T$ tries to expand by

δ_{T} = α L Δ T

where $α$ is the coefficient of linear thermal expansion (per °C). The corresponding thermal strain is $ε_{T} = α Δ T$ .

If the bar is free to expand, it simply grows longer — no stress develops. Stress appears only when the expansion is prevented, fully or partly.

Material	$α$ (per °C), typical
Steel	about $12 \times 1 0^{- 6}$
Concrete	$10 \times 1 0^{- 6}$ to $12 \times 1 0^{- 6}$
Copper	about $17 \times 1 0^{- 6}$
Brass	about $19 \times 1 0^{- 6}$
Aluminium	about $23 \times 1 0^{- 6}$

Exam TipWhy RCC works

Steel and concrete have almost the same coefficient of thermal expansion, so temperature changes do not set up large stresses between the bars and the surrounding concrete. This is one of the reasons reinforced concrete is a sound composite material.

Fully restrained bar

If both ends are held rigidly, the bar cannot change length. The prevented expansion $α L Δ T$ must be cancelled by an equal compression:

\frac{σ L}{E} = α L Δ T \Rightarrow σ = α Δ T E

FormulaThermal stress (rigid supports)

σ_{T} = α E Δ T, P = α E Δ T A

Rise in temperature with ends fixed → compressive stress.
Fall in temperature with ends fixed → tensile stress.
The stress is independent of the length and area of the bar.

Partially restrained bar (support yields or gap exists)

If the supports yield by $Δ$ (or a gap $Δ$ is left), only the expansion in excess of $Δ$ is prevented:

σ = \frac{( α L Δ T - Δ ) E}{L}

If $α L Δ T \leq Δ$ the bar closes the gap without touching both supports — no stress.

Tapering bar held between rigid supports

For a bar tapering from $d_{1}$ to $d_{2}$ , the compressive force is found from $δ_{l o a d} = δ_{t h er ma l}$ :

\frac{4 P L}{π E d _{1} d _{2}} = α L Δ T \Rightarrow P = \frac{π E d _{1} d _{2} α Δ T}{4}

The maximum stress occurs at the smaller end: $σ_{ma x} = P / (π d_{2}^{2} /4)$ .

Composite bars under axial load

A composite bar is made of two or more materials joined so that they deform together — a steel tube filled with concrete, a copper rod inside a steel tube, an RCC column.

For a load $P$ shared by materials 1 and 2 of the same length:

Equilibrium: $P = σ_{1} A_{1} + σ_{2} A_{2}$
Compatibility: both shorten equally → $ε_{1} = ε_{2}$ → $\frac{σ _{1}}{E _{1}} = \frac{σ _{2}}{E _{2}}$

So $σ_{1} = \frac{E _{1}}{E _{2}} σ_{2} = m σ_{2}$ where $m = E_{1} / E_{2}$ is the modular ratio.

FormulaComposite bar — shared load

σ_{2} = \frac{P}{A _{2} + m A _{1}}, σ_{1} = m σ_{2}, m = \frac{E _{1}}{E _{2}}

The quantity $A_{2} + m A_{1}$ is the equivalent (transformed) area in terms of material 2 — the same idea used for RCC columns in the working stress method.

The stiffer material carries the higher stress. If the two parts have different lengths, compatibility is $δ_{1} = δ_{2}$ , i.e. $σ_{1} L_{1} / E_{1} = σ_{2} L_{2} / E_{2}$ .

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Thermal Stresses & Composite Bars