Partially Restrained Stepped Bar:

Now, let us assume the case of a stepped bar as in Figure , but which is free to extend by an amount ΔL´ (ΔL´ < ΔL) but restrained after that. In this case, through the free expansion of ΔL´ the bar remains unstressed but after that develops the stresses σ₁ and σ₂ whose values shall now be different than those of previous sec. By a similar argument as in the previous section, we get

σ₂ = σ₁ (A₁ /A₂)

 σ₁ L₁/E₁  = σ₂ L₂/E₂ = (ΔL - ΔL′) = [ΔT (L₁ α₁  - L ₂ α₂ ) - ΔL′]

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

σ ₁= 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

σ₂ = EA₂ [ΔT α L - ΔL′]/ A₁ L₂  + A₂ L₁

and

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

For a uniform bar the stress expression decrease to

σ₁ = σ ₂ = σ = E( α ΔT - ΔL′ /L)

a result already observed.