Partially Restrained Stepped Bar:

Let us now assume the case of a stepped bar as in Figure, but in that is free to extend through an amount ΔL´ (ΔL´ < ΔL) but restrained thereafter. In that case, during the free expansion of ΔL´ the bar remains unstressed but thereafter develops the stresses σ₁ and σ₂ whose values will now be different. Through a same argument as in the previous section, we get

σ₂ = σ₁ A₁/A₂

σ₁ L₁/E₁  = σ₂ L₂/E₂ = (ΔL - ΔL′) = [ΔT (L₁α₁  - L₂α₂) - ΔL′]

 From the above two expressions, the thermal stresses σ1 and σ2 can now be written as:

σ₁ = E₁ E₂ A₂ [ΔT (L₁ α₁ + L₂ α₂ ) - ΔL′]/( A₁ E₁ L₂  + A₂ E₂ L₁ )

σ₂ = E₁ E₂ A₁ [ΔT (L₁ α₁ + L₂ α₂ ) - ΔL′]/( A₁ E₁ L₂  + A₂ E₂ L₁ )

In the case of a single material bar

σ₁ = EA₂ [ΔT α L - ΔL′]/ A₁ L₂ + A₂ L₁,

and

σ₂ =   EA₁  [ΔT α L - ΔL′]/  A₁ L₂  + A₂ L₁

For a uniform bar the stress expression decrease to

σ₁ = σ₂  = σ = E ( α ΔT - ΔL′/L)

a result already observed.