Abstract: A network of multidirectional shallow cracks due to thermal fatigue was discovered in the mixing zone of hot and cold water in the pipe of residual heat removal systems in pressurized water reactors. The fatigue crack initiation direction can be predicted by the critical plane orientation. In this paper, the theoretical derivation was performed and the analytical solution of the critical plane orientation under biaxial fatigue loadings was obtained. A program based on C++ language was developed. The computational results of the critical plane orientation using both the C++ programmed analysis code and the finite element software Code_Aster were compared and validated by the analytical solution. The critical plane orientation is consistent with the observed thermal fatigue crack direction. It is concluded that the thermal fatigue crack direction is determined by the weld residual stress in the pipe of residual heat removal system.