Int J Fract (2007) 146:249–264 DOI 10.1007/s10704-007-9166-2 ORIGINAL PAPER Limit loads and approximate J estimates for axial through-wall cracked pipe bends Yun-Jae Kim · Tae-Kwang Song · Jong-Sung Kim · Tae-Eun Jin Received: 4 September 2006 / Accepted: 2 November 2007 / Published online: 27 November 2007 © Springer Science+Business Media B.V. 2007 Abstract This paper presents plastic limit loads and approximate J estimates for axial through-wall cracked pipe bends under internal pressure and in-plane bend- ing. These loads and estimates are based on small strain finite element limit analyses using elastic-perfectly plastic materials. Geometric variables associated with the crack and pipe bend are systematically varied, and three possible crack locations (intrados, crown and extrados) are considered. Effects of the bend and crack geometries on plastic limit loads are quantified, and closed-form limit load solutions are given. Based on the proposed limit load solutions, a reference stress based the J estimation scheme for axial through-wall cracked pipe bends under internal pressure and in-plane bend- ing is proposed. Keywords Axial through-wall crack · Finite element analysis · J estimation · Plastic limit load · Reference stress Nomenclature c Half crack length Y.-J. Kim (B ) · T.-K. Song Department of Mechanical Engineering, Korea University, 1-5 Ka Anam-dong, Sungbuk-ku, Seoul 136-701, Korea e-mail: kimy0308@korea.ac.kr J.-S. Kim · T.-E. Jin Structural Integrity & Materials Department, Korea Power Engineering Company, Yongin-si, Gyeonggi-do 449-713, Korea E Young’s modulus E ′ =E/(1 - ν 2 ) for plane strain; =E for plane stress J , J e J -integral and its elastically calculated value K Linear elastic stress intensity factor M,M L In-plane moment and limit moment of a cracked pipe bend, respec- tively M o , M s o Limit in-plane moment of an un-cracked pipe bend and straight pipe, respectively n Strain hardening index (1 ≤ n< ∞) for the Ramberg–Osgood model, Eq. 18 P,P L Internal pressure and limit pressure of a cracked pipe bend P o ,P s o Limit pressure of an un-cracked pipe bend and straight pipe, respectively R Bend radius r Mean pipe radius t Thickness of a pipe α Coefficient of the Ramberg–Osgood model, Eq. 18 φ Normalised half crack angle, see Eq. 3 for the definition ρ Normalised half crack length, =c/ √ rt λ Bend characteristic, =Rt/r 2 ν Poisson’s ratio σ o Limiting stress of an elastic-perfectly plastic material; yield (0.2% proof) stress for hardening materials σ r ef , ε r ef Reference stress and strain 123