Partially Restrained Bar:

Now, let us assume that one of the supports of the bar in Figure (a) can move by a distance ΔL′ (ΔL′ < ΔL) while the bar is expanding. In this case, the bar is free to expand by an amount ΔL′ but is restrained in expanding further by rest of the amount (ΔL - ΔL′).

Therefore, the force of restraint, P, is now a force which would produce a change in length (ΔL - ΔL′) in the bar.

Thus,    P = (ΔL - ΔL′) AE /L= (αL ΔT - ΔL′) AE /L   

Strain = (αL ΔT - ΔL′)/L = α ΔT - ( ΔL′ /L)

Thermal Stress = Strain × E = E ( α ΔT - ΔL′ /L)