Why do we apply the combine equation of thermal stresses in composite bars, when both bars are free to move?

Question

i ahve got a question in thermal stresses in composite bar,,, in this particular question two bars are just concentric neither shrink fit nor joined at ends,, just of same length and both are supporting a load axially,,,both are in comressive stress,,,problem is that the bars are now simultaneously cooled, we have to calculate the final stresses???? can we apply the composite bars thermal stresses equation for solving it

Answer 1

Basically what you have is two springs in parallel that are initially compressed.  Find the stiffness of each bar (A*E/L) and then the combined stiffness (springs in parallel).  Solve for the initial deflection in the system based on the initial stresses.  Then, calculate the deflection of each bar caused by cooling the system (unrestrained thermal deflection equation).  Subtract this value from the original deflection.  Now calculate the new stress based on the remaining stress in each bar.