Issue 50

M. Belhamiani et alii, Frattura ed Integrità Strutturale, 50 (2019) 623-637; DOI: 10.3221/IGF-ESIS.50.53 625 Second criterion: The limit load analysis The development of finite element methods and computational tools has stimulated limit stress analysis for complex structures. Koopman and Lance [11] analysed the plastic limit load using nonlinear mathematical programming firstly in 1965. after that they used this method to analyze the 2D plate and symmetry shell structure [12]. In 1989 Berak and Gerdeen [13] developed a P-norm method based on lower bound method of limit load, and then Liu et al in 1995. [14–15] analyzed the limit load for 3D structure, they used the penalty-duality algorithm and direct iteration method to analyze the pressure vessels with volume defects, taking into account the failure modes for different defects. Recently in 2002, Yun.J et al [16] quantified the effect of the crack shape, semi-elliptical or rectangular on the limit load pressure. Peng.F et al [17] proposed theoretical method to calculate a plastic collapse load for pressure vessel under internal pressure compared with ASME Boiler pressure vessel code.in 2015 X.-T. Mioa et al [18] analysed a limit loads for CT specimen based on the XFEM compared with EPRI method, twice slope method and J IC criterion, they concluded that the XFEM is useful method to estimate the limit load for cracked structure. G EOMETRICAL AND MATERIALS MODELS : ig. 1 illustrates the geometrical characteristics of the model. It should be noted that for the cracked pipe, R is the outer radius R o = 177.8mm, L P is the longitudinal length L P = 1000 mm, and t=12.7 mm is the wall's thickness. The cracks were repaired by using a Glass-epoxy composite patch bonded cylinder form with an adhesive cylinder (FM 73). The width of the repair patches is ep comp = 7mm and adhesive thickness is ep adh = 0.2 mm the overwrap has 200mm of length (L), the interaction surfaces are supposed to be perfect between pipe / adhesive and adhesive / composite. The end sections pipeline was subjected to no displacement in z direction (u 3 =0). The material properties of the pipeline, patch, and adhesive are summarized in Tab. 1. The pipe was made of X65 grade steel per API 5L X65 PSL2 [19] specifications. The stress–strain relationship was defined using the Ramberg–Osgood material model expression [ 20]:   n E E              (4) API 5L X65 properties [19] Young’s modulus (GPa) 205 Poisson’s ratio 0.3 Minimum yield stress (MPa) 415 Yield strain 0.5% Ramberg–Osgood’s model yield offset (α) 1.48 Strain hardening component (n) 18.99 Glass epoxy composite properties [21] Young’s modulus E 1 (GPa) 55 Young’s modulus E 2 , E 3 (GPa) 15.2 Poisson’s ratio ν 12 , ν 13 0.254 Poisson’s ratio ν 23 0.428 Shear modulus (GPa) 4.7 Adhesive FM73 properties [22] Young’s modulus (GPa) 3.28 Poisson’s ratio 0.45 Table 1 : Material’s properties. F