Your first approach looks a lot like the guided cantilever method which was widely used before martix methods were computerized. Assuming that both ends are anchored, the elbow is a square corner that will not rotate and ignoring the axial load from restrained growth, the thermal load on each leg can be calculated solely on the lateral offset caused by the thermal strain of other leg. This deflection is used with the leg stiffness (the guided cantilever which is 4 times stiffer to transvers end load than a simple cantilever) to compute an end load and from that the bending moment and stress.
I do not understand your second approach. Don't feel too bad though. As I re-read my paragraph above I'm not sure, if you don't already know what I'm saying, that it will make sense!
Focus on the phrase "guided cantilever method", better yet, run it through CAESAR II for a more acceptable result.