This paper provides engineering J-integral and crack opening displacement (COD) estimation equations for circumferential through-wall cracked pipes under internal pressure and under combined internal pressure and bending. Based on selected 3-D finite element calculations for the circumferential through-wall cracked pipes under internal pressure using the idealized power law materials, the elastic and plastic influence functions for fully plastic J-integral and COD solutions are found as a function of the normalized crack length and the mean radius-to-thickness ratio. These developed GE/EPRI-type solutions are then re-formulated based on the enhanced reference stress method. Such re-formulation not only provides simpler equations for J-integral and COD estimations, but also can be easily extended to combined internal pressure and bending. The proposed equations are compared with elastic-plastic finite element results using actual stress-strain data, which shows overall excellent agreement.