When two materials with different coefficients of thermal expansion (α₁ and α₂) are rigidly connected to form a compound bar, temperature changes induce internal stresses because each material wants to expand or contract differently.

The thermal stress developed in one of the bars is given by:

σ₁ = ( (α₁ - α₂) E₁ E₂ A₂ T ) / ( A₁ E₁ + A₂ E₂ )


Where:

α₁, α₂ = coefficients of thermal expansion

E₁, E₂ = Young’s moduli of the two materials

A₁, A₂ = cross-sectional areas

T = temperature change