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Nuclear Engineering and Design 302 (2016) 2026 Contents lists available at ScienceDirect Nuclear Engineering and Design jou rn al hom epage : Technical Note A simplifi ed approach for assessing the leak-before-break for the fl awed pressure vessels P. Kannana, K.S. Amirthagadeswaranb, T. Christopherc, B. Nageswara Raod, aRamagundam Super Thermal Power Station, NTPC Ltd, Jyothinagar 505215, India bFaculty of Mechanical Engineering, Government College of Technology, Coimbatore 641013, India cFaculty of Mechanical Engineering, Government College of Engineering, Tirunelveli 627007, India dFaculty of Mechanical Engineering, School of Mechanical and Civil Sciences, K L University, Green Fields, Vaddeswaram, Guntur 522502, India a r t i c l e i n f o Article history: Received 26 October 2015 Received in revised form 3 February 2016 Accepted 4 February 2016 Available online 14 April 2016 a b s t r a c t Surface cracks or embedded cracks in pressure vessels under service may grow and form stable through- thickness cracks causing leak prior to failure. If this leak-before-break phenomenon takes place, then there is a possibility of preventing the vessel failure. This paper presents a simplifi ed approach for assessing the leak-before-break or failure of the fl awed pressure vessels. This approach is validated through comparison of existing test data. 2016 Elsevier B.V. All rights reserved. 1. Introduction Part-through cracks in pressure vessels under service loads may grow and form stable through-thickness cracks causing leak prior to failure known as the leak-before-break (LBB) phenomenon. If this phenomenon happens, then there is a possibility of preventing the vessels from failure. If the part-through cracks under service loading conditions grown to critical size, then the vessel may fail catastrophically prior to the formation of the through-thickness crack. The signifi cant parameters affecting the critical crack size in a pressure vessel are the applied stress levels, the location of the crack and its orientation, and the strength as well as the fracture toughness of the material. For safe design of pressure vessels, LBB is one of the important criteria (Pacholkova and Taylor, 2002). Designers apply LBB crite- rion to structural components (which are subjected to high or low fatigue loads) in nuclear power plants, liquid nitrogen tankers and chemical plants. The LBB concept is applied to high pressure ves- sels and related plant equipments (Nam and Abn, 2002). Kawaguchi et al. (2004) have examined the LBB behavior for axially notched X65 and X80 gas pipelines. Drubaya et al. (2003) have provided a guide for defect assessment at elevated temperature. Toughry (2002) has developed an acceptance/rejection criterion for high pressure steel and aluminum cylinders. Zhou and Shen (1996) have Corresponding author. Tel.: +91 8106762175. E-mail address: bnrao52 (B. Nageswara Rao). discussed on the LBB assessment methods. The concept of LBB was initially introduced by Irwin. An analogous method later on was developed by Irwin and Hood. These two methods are very simple and provide conservative estimates. Wilkowski (2000) states that Irwin has performed the linear elastic fracture mechanics (LEFM) analysis on pressure vessels specifying the axial crack length less than twice the shell thickness, and observed greater crack driv- ing force in radial direction than in the axial direction of the vessel. Experiments of Rana (1987) on gas cylinders containing a sur- face crack (whose length is four times the shell wall thickness) indicate the validity of LBB criterion. Sharples and Clayton (1990) have generated crack depth versus crack length curves for assessing leakage or break of the fl awed pressure vessels. Kim (2004) has per- formed LBB analysis on through-thickness cracked pipes. Kim et al. (2005) have proposed an elasticplastic J-integral approach to car- ryout LBB analysis for circumferential through-thickness cracked pipes. The plane strain fracture toughness (KIC) of the material can be evaluated from the Compact Tension (CT) specimens follow- ing the ASTM E399 standards (ASTM, 2013a), whereas the crack growth resistance curve (R-curve) of the material can be generated following the ASTM E561 standards (ASTM, 2013b). The fracture toughness (KC) where plane strain conditions are not fully met can be determined from the point of tangency between the R-curve and the crack driving force curve appropriate for the loading geometry. The crack growth observed in KICspecimens after failure is very small whereas it is appreciable in KCspecimens. Failure load esti- mates based on the lower bound KICvalues will be conservative and /10.1016/j.nucengdes.2016.02.013 0029-5493/ 2016 Elsevier B.V. All rights reserved. P. Kannan et al. / Nuclear Engineering and Design 302 (2016) 2026 21 Nomenclature a depth of a surface crack 2c length of a surface crack 2c* through-thickness crack KCplane-stress fracture toughness KIstress intensity factor KICplane-strain fracture toughness KF, m and p fracture toughness parameters in Kmax ?frela- tion (1) Kmaxstress intensity factor corresponding to the failure stress (?f) P internal pressure Pbbursting pressure of unfl awed cylindrical vessel Pffailure pressure of fl awed cylindrical vessel Riinner radius of the cylindrical shell t thickness C, ICdimensionless parameters (ratio of plastic zone size to thickness) for plane-stress and plane-strain situ- ations ?a incremental fl aw growth ?K stress intensity range ?b, ?mincrement in bending and membrane stresses ? crack shape parameter ?fhoop stress at failure pressure of fl awed vessel ?ysyield strength or 0.2% proof stress (RP0.2) ?uhoop stress at failure pressure of unfl awed pressure vessel ?ultultimate tensile strength (Rm) design based on KICrequires unreasonably thick panels in normally thin sectioned structural members as in aerospace industry. 1.1. Relationship between Kmaxand ?f Kannan et al. (2013) have examined the applicability of a mod- ifi ed two-parameter criterion (Christopher et al., 2004a, 2005a) while assessing the fracture strength of structural components. They utilized a relation between the stress intensity factor (Kmax) and the corresponding stress at failure (?f) as Kmax= KF ? 1 m ? f ?u ? (1 m) ? f ?u ?p? (1) Here, ?fis the hoop stress at the failure pressure of the fl awed vessel and ?uis the hoop stress at the failure pressure of the unfl awed vessel. For uniaxial tensile specimens, ?uis equal to the ultimate tensile strength (?ultor Rm) of the material. KF, m and p are fracture parameters to be determined from the test data of cracked confi gurations. The fracture parameter, KFhas the units of the frac- ture toughness (MPam), whereas the second parameter 0 m 1 and to account for plasticity the third parameter, p dependent on m is given by p = 1 ln?1 2(1 + ?)? ln ? 1 (1 m) ? 1 1 22 (1 + ?) ? 1 ? + ? 2 1?m ? (2) and ? in Eq. (2) is ? = 4 3 + 9 8m (3) If the fracture data corresponds to the plane strain fracture tough- ness (KIC) of the specimens, then the fracture parameters in Eq. (1) are: KF= KICand m = 0. When the stress intensity factor (KI) of the through-thickness cracked vessel under service loads is less than the plane strain fracture toughness (KIC), the vessel leaks initially, grow gradually to the critical size and fail. Detection of leaking at the initial stage will be helpful in preventing the failure of vessel. To assess the life of the fl awed vessel, it is essential to know the path of the part-through crack grown to the through thickness crack. If the stress intensity factor (KI) of the through-thickness cracked vessel for the stress levels falls below the failure assessment diagram, then the vessel leaks. For KI KIC, the crack growth will be slower. Crack propagation will result if KI KICof the material. 1.2. Relationship between KCand KIC The ratio of plastic zone size to thickness (C) is a convenient measure of the degree of shear-lip. The dimensionless parameters Cand ICfor plane-stress and plane-strain situations defi ned by Irwin are (Irwin, 1962; Irwin and de Wit, 1983): C= 1 t ? KC ?ys ?2 ; and IC= 1 t ? KIC ?ys ?2 . Here ?ysis the yield strength or 0.2% proof stress (RP0.2) and t is the thickness. An approximate empirical relationship between C and IC(valid for C c (13) P. Kannan et al. / Nuclear Engineering and Design 302 (2016) 2026 23 Fig. 1. Cylindrical pressure vessel with an axial surface crack under internal pres- sure. Me= M1+ ? ? ? c a M1 ? a t ?q (14) M1= 1.13 0.1 ?a c ? for a c, and M1= ? 1 + 0.03 c a ? c a for a c, (15) fs= (1 + 0.52?s+ 1.29?2 s 0.074?3 s) 1/2 for 0 ?s 10, (16) ?s= c ? Rit a t , (17) q = 2 + 8 ?a c ?3 (18) a is the depth and c is half the crack length of a surface crack, Riis the inner radius and t is the thickness of the cylindrical vessel. Substi- tuting the hoop stress (?) by the fracture stress (?f) the maximum stress intensity factor (Kmax) is obtained from Eq. (10). 4. Fracture strength equation For ? = ?f, Eq. (10) gives Kmaxfor the specifi ed crack size. Using this Kmaxin Eq. (1), one can fi nd the fracture strength equation in the form (1 m) ? f ?u ?p + ? m + ?u(?a)1/2M ?KF ? ?f ?u ? 1 = 0 (19) Here, the hoop stress of the unfl awed cylindrical pressure vessel at failure ?u= PbRi t (20) The bursting pressure of unfl awed cylindrical vessel from the Faupels formula, Pb= 2 3?ys ? 2 ?ys ?ult ? ln ? 1 + t Ri ? (21) ?ysis the yield strength or 0.2% proof stress (RP0.2) and ?ult is the ultimate tensile strengths of the material. Using the NewtonRaphson iterative method, the non-linear equation (19) is solved for ? f. The adequacy of the fracture strength equation (19) has been examined by considering fracture data of tensile speci- mens and cylindrical pressure vessels having surface cracks as well as through-thickness cracks. The fracture analysis results of the cracked confi gurations (viz. compact tension specimens, surface as well as through-thickness center crack tensile specimens, cylindri- cal pressure vessels having an axial or a through crack) are found to be in good agreement with test data (Nageswara Rao, 1992; Nageswara Rao and Acharya, 1989a,b, 1993, 1996, 1997, 1998; Govindan Potti et al., 2000; Rama Sarma et al., 2002; Christopher et al., 2002, 2004b, 2005b). 4.1. Failure pressure estimation The fracture strength (?f) of cylindrical vessel having an axial surface or through-thickness crack is obtained by solving the frac- ture strength equation (19) and the failure pressure is obtained from Pf= t?f/Ri. If the outer or inner surface crack in the cylin- drical vessel under internal pressure grows slowly in a stable manner, and the crack depth grows to the other inner or outer surface (i.e., the depth equals the thickness of the shell: a = t), the size of the through-thickness crack (2c*) can be estimated from the crack growth model. Leaking of the vessel can be expected after growing the surface crack to the through-thickness crack (2c*). It is obvious that the leaking pressure must be lower than the failure pressure of vessel having a through 2c = 2c, the max- imum stress intensity factor (Kmax) can be obtained from Eq. (10) for the through-thickness crack size (2c), which can be obtained from the fracture strength (?f). Substituting the hoop stress (?) by the fracture strength (?f), crack depth, a = t, and the crack length, 2c = 2c, the maximum stress intensity factor (Kmax) can be obtained from Eq. (10) for the through-thickness crack (2c). Crack growth data of materials for the specifi ed load spectrum (cyclic or sustained or combination of both cyclic and sustained loading) can be generated from the compact tension specimens as per the ASTM E647 standards. It is observed that crack extension takes place in the component prior to the initiation of failure. For negligibly small crack extension, the stress intensity factor evalu- ated from the critical load considering the initial crack size is called as the plane strain fracture toughness (KIC). The plane stress fracture toughness (KC) evaluated from the R-curve of the material consid- ers the signifi cant crack extension, which is thickness as well as geometry dependent. The lower bound fracture toughness KICis independent of both thickness and geometry of the specimen. In linear elastic fracture mechanics (LEFM), the plastic zone near the crack-tip is negligibly small when compared to the crack size, and KICbe the critical stress intensity factor, which is related to the crit- ical strain energy release rate (GIC), or to the critical J-integral value (JIC). In LEFM, GIC= JIC= (1 ?2)K2 IC)/E. Here E is the Youngs mod- ulus and ? is the Poissons ratio. Crack growth takes place prior to failure, if the stress intensity factor (KI) of the cracked body exceeds the KIC. This is applicable to all metallic structures whose critical stress intensity factor is equal to the plane stress fracture tough- ness (KC). It should be noted that if the fracture data corresponds to the plane strain fracture toughness (KIC) of the specimens, then the fracture parameters in Eq. (19) are: KF= KIC, m = 0, and p from Eq. (2). 4.2. Leak pressure estimation Since the relationship between Kmaxand ?fin Eq. (1) considers crack growth prior to failure the maximum stress intensity factor (Kmax) can be treated as the plane stress fracture toughness (KC). Using Eq. (4), one can estimate the plane strain fracture toughness (KIC). Setting KF= KIC, m = 0 and p from Eq. (2), one can estimate the pressure (or leak pressure prior to break) at the initiation of the through-thickness crack (2c) growth from Eq. (19). 5. Results and discussion In order to examine the adequacy of the present simplifi ed approach, test data of Rana (1987) from open literature is consid- ered. Rana (1987) has conducted pressure tests on the modifi ed AISI 4130 steel seamless high pressure gas cylinders having 7.6 mm 24 P. Kannan et al. / Nuclear Engineering and Design 302 (2016) 2026 Table 1 Fracture analysis on ruptured AISI 4130 steel cylindrical vessels having an axial surface crack. (Yield strength, ?ys= 1097 MPa; ultimate tensile strength ?ult= 1180 MPa; and fracture parameters KF= 210 MPam; m = 0.8136 and p = 35.4.) Diameter of the cylindrical vessel = 228.6 mm. Thickness, t (mm) Surface crack size (mm) a/t Failure pressure, Pf(MPa) Relative error (%) Length (2c) Depth (a) Test (Rana, 1987) Fracture analysis 7.5 25.4 5.7 0.760 47.3 45.6 3.6 7.2 25.4 6.1 0.847 39.4 41.3 4.8 7.876.26.00.769 26.5 26.5 0 7.4 76.2 6.3 0.851 22.3 22.0 1.3 nominal wall thickness, 229 mm diameter and 1.4 m height. Sharp, semi-elliptical shaped EDM fl aws (of varying lengths of 2576 mm and fl aw depth to thickness ratio of 0.50.9) were machined on the outer surface of the test cylinders. Four fl awed cylinders subjected to fatigue loading by pressurization, whereas six fl awed cylinders subjected to monotonically pressurization with water to failure. All ruptured cylinders exhibited fl at fracture with fairly large size shear lips, indicating plane stress fracture mode. Fracture analysis has been carried out on ruptured cylinders. Fracture parameters (KF, m and p) in Eq. (1) evaluated from the fracture data are KF= 210 MPa m; m = 0.8136; and p = 35.4 respectively. To examine the adequacy of these fracture parame- ters, failure pressure is estimated from the measured surface cracks on the cylindrical vessels. Table 1 gives comparison of estimated failure pressure of fl awed cylindrical vessels with test results (Rana, 1987). Fracture analysis results are found to be in good agreement with test results. Fig. 1 shows the failure assessment diagram gen- erated from the fracture parameters using Eq. (1). Fracture data of the vessels are also presented in Fig. 2. Test data (Rana, 1987) is close to the failure boundary. Two cylindrical vessels subjected to monotonically pressuriza- tion experience leaking. Failure pressure estimates of these vessels in Table 2 are found to be higher than the recorded leak pressure. From the crack growth analysis, the sizes of the through-thickness crack for these two vessels are 53.1 and 53.6 mm respectively. The expected failure pressure of vessels is 23.3 and 25.5 MPa respec- tively, which are less by 5 MPa to the observed leak pressure. Specifying the initial fl aw size of the vessels and using the fracture parameters, the estimates of failure pressure are 31.9 and 30.6 MPa respectively, which are in good agreement with the observed leak pressure values of 28.6 and 30.5 MPa. Matching of estimated failure pressure with leak pressure in this case is mainly due to the depth of the surface fl aw above 80% of the shell wall thickness. The stress intensity factor for such surface fl aws will be close to that for the case of through cracks having the same length of the surface fl aw. Fig. 2. Failure assessment diagram. Table 3 gives fracture analysis results on cylindrical vessels which were subjected to repeated cyclic pressure loading. Since these vessels are leaked, the sizes of through-thickness crack (2c) are estimated from the crack growth modeling and estimated the failure pressure by specifying 2c*. The estimated through crack size (2c) in Table 3 is found too close to the measured one. Failure pressure estimates are found to be lower than those recorded leak pressure values. For the measured through-thickness crack (2c) and the repeated cyclic pressure, the plane stress fracture tough- ness (KC) is evaluated. Using Eq. (4), the corresponding plane-strain fracture toughness (KIC) is worked out for obtaining the leak pres- sure. Fracture analysis results in Table 4 provide the leak pressure for each vessel, which is obtained by replacing the stress inten- sity factor in Eq. (10) with the plane-strain fracture toughness (KI