A bar of length 'L' is held between rigid supports and is heated non-uniformly in such a manner that the temperature increases ΔT at distance 'x' from the support A is given by the expression -

\(Δ T = \frac{Δ T_Bx^3}{L^3} \), 

where ΔTB is the increase in temperature at end B of the bar.

The magnitude of stress in the bar is given by k(EαΔT_B). Here, E - modulus of elasticity of the material of bar and α - coefficient of thermal expansion. The value of k (if compressive stresses are taken as positive) -