Partially Restrained Bar:

Now, let us suppose that one of the supports of the bar in Figure (a) may move by a distance ΔL′ (ΔL′ < ΔL) while the bar is expanding. In this particular 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 generate a change in length (ΔL - ΔL′) in the bar.

Therefore,        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 ))