To enable the general analytical method, presented in the companion paper, to be applied to beams with non-uniform temperature distributions, additional assumptions are made about the beam’s deflection profiles. In this paper, it is assumed that for a beam with non-uniform temperature distribution and under concentrated loads, its mechanical deflection profile should be the average of the deflection profile under uniformly distributed load and the free bending moment profile of the beam under concentrated load. The results of various validation examples show that this assumption and the general analytical method are able to give close approximation to numerical simulation results. From detailed investigations, it has been found that catenary forces in a beam are not sensitive to different non-uniform temperature distributions. Furthermore, for safe prediction of catenary forces, complete end axial restraint and zero end rotational restraint may be assumed.