Normal Stress in Rods Experiencing Temperature Changes

[Qube]

Homework Statement

The rods are experience different temperature changes. Find the normal stress in rod AB.

Relevant Equations

Change in length of a rod due to temperature = alpha*temperature change*original length

Change in length of a rod due to a normal force: FL/AE

Now here is the part where I'm sort of stumped myself:

Could someone let me know if my reasoning is valid? The professor explained it during office hours and all I got out of that was that something cancels out and the answer is 0.

 

[Chestermiller]

I'm having trouble understanding what you did. Let $\delta$ represent the displacement of point B to the right. Then the change in length of rod AB is $+\delta$ and the change in length of rod BC is $-\delta$. So the total strain in rod AB is $\epsilon=+\frac{\delta}{3L}$ and the total strain in rod BC is $\epsilon=-\frac{\delta}{2L}$. For a rod that experiences both thermal expansion and tensile stress $\sigma$, the relationship between tensile stress, total strain, and temperature change is $$\sigma=E(\epsilon-\alpha \Delta T)$$