The failure of graphite moderator bricks in an Advanced Gas cooled Reactor (AGR) is potentially a serious problem. This thesis describes the generation of self stress in the moderator brick during irradiation and the derivation of a simple analytical model to predict the magnitude of this stress. The magnitude of the self stress in the brick is affected by the variations in the material properties of the graphite used for the brick and this is also examined, developing a statistical approach to the analysis. Property variations between bricks are considered but no allowance has been made for material property variations within a brick. Finally, the thesis compares the self stress in one of the critical peak rated moderator bricks to the strength of the irradiated oxidised material on a statistical basis and predicts the failure probability of a brick due to self stress to be extremely low at 25.5 full power years (FPY). However, the failure probability rises steeply and for the peak rated bricks at 29 FPY it approaches 100%.