Reference stress based J and COD estimation of circumferential through-wall cracked elbows under in-plane bending

•Circumferential through-wall cracked elbows under in-plane bending is considered.•J and COD estimation equations are proposed based on the reference stress method.•The proposed equations are compared with detailed finite element results.•They are validated against eight published full-scale pipe test data. This paper proposes J-integral and crack opening displacement (COD) estimation equations based on the reference stress method for circumferential through-wall cracked elbows under in-plane bending. For best estimates of J and COD, an optimized reference load, used to define the reference stress, is given. The proposed equations are compared with detailed finite element results. They are further validated against eight published full-scale pipe test data. Comparison of crack initiation and maximum loads shows good agreement.