PSI - Issue 13

574 4 Yuebao Lei / Procedia Structural Integrity 13 (2018) 571–577 Y Lei/ Structural Integrity Procedia 00 (2018) 000–000 � � � � ���� � � for � � � � � � � �� � � for � � � (3) where n L and m L are normalised limit membrane stress and limit bending stress under combined loading, respectively, which can be calculated using the solutions given in Lei and Budden (2015), and � � �6 � � ⁄ , � � �� � ⁄ , � � �� � ⁄ (4) The reference stress of Eqn. (2) is then used to define L r (see Eqn. (6) below) in the J prediction or defect assessment. 3. J prediction using the reference stress method The reference stress J estimation scheme predicts the total J from the elastic J , J e , using the reference stress, ��� , and corresponding reference strain, ��� , via the following relationship based on the R6 Option 2 FAC [R6 (2015)] � � � � �� ��� � ��� � � �� � � ��� �� ��� (5) where the reference stress and strain follow the uniaxial stress-strain relationship of the material and L r may be defined by the limit load or the reference stress via � � ��� � ⁄ (6) where � is the yield stress or 0.2% plastic strain off-set stress of material. 4. Validation using the FE J solutions The developed local limit load model in Section 2 may be validated by comparing the J values predicted using the obtained reference stress via the reference stress J prediction method and the elastic-plastic FE results. A reference stress solution is conservative if the FE J values are overestimated when it is used in the reference stress J scheme. In this paper, well-documented solutions from the published literature are collected, reviewed and used to validate the local limit load/reference stress determined from the developed local limit load model. The FE data from the following publications are used in this validation: (i) Semi-elliptical surface cracks in plates under combined tension and through thickness bending due to Lei (2004(1), 2004(2), 2004(3)). (ii) Semi-elliptical surface cracks in plates under biaxial tension due to Wang (2006). (iii) Axial internal semi-elliptical surface cracks in cylinders under pressure due to Kim et al. (2004). (iv) Circumferential internal semi-elliptical surface cracks in cylinders under pressure/global bending due to Kim et al. (2002). (v) Circumferential external semi-elliptical surface cracks in cylinders under global bending due to Chiodo and Ruggieri (2010). (vi) Elbows with axial/circumferential internal/external semi-elliptical cracks under internal pressure and bending moment(s) [Kayser (2016)]. For each case with available FE J results, elastic uncracked-body stress analysis is carried out to determine the required model loads shown in Section 2 and the reference stress, ��� , is evaluated from Eqn. (2). For Ramberg Osgood type materials with strain hardening exponent n and stress normalisation � , the FE J / J e values, where J e may be from FE or handbook SIF solutions, are plotted against ��� � ⁄ (see Figs. (2) and (3)) and compared with the predicted results from Eqn. (5) (indicated by “Predicted (Opt-2 FAC)” in figures). Note that, for this case, Eqn. (5) is plotted in relevant figures against ��� � ⁄ with L r in Eqn. (5) being replaced by � ��� � ⁄ ��� � � ⁄ � . Totally 273 cases have been used in this validation including 74 cases for plates, 192 cases for cylinders and 7 cases for elbows. A selection of representative results is presented in Figs. (2) to (4) for plates, cylinders and elbows, respectively. � ��� � � � �� � �