Derived for uniform bars:

Herein, we have assumed that the two parts of the bar are made of two different materials. On the other hands, if the entire bar is of a single material of Young's Modulus E and

coefficient of linear expansion α,

E1 = E2 = E and α1 = α2 = α

Then σ₁ and σ₂ reduce to

σ₁ = ΔT E A₂ α L / A₁ L₂ + A₂ L₁

and

σ₂  =    ΔT E A₁ α L / A₁ L₂ + A₂ L₁           

Further, if the cross-section is uniform throughout (A₁ = A₂ = A), both the above expressions reduce to

σ₁  = σ₂  =σ = E α ΔT

a result that we have already derived for uniform bars.