Assessment of the structural integrity of cracked cylindrical geometries applying the EVTUBAG program

Transcription

1 Rev. Téc. ng. Univ. Zuli. Vol. 32, Nº 3, , 2009 Assessment of the structurl integrity of crcked cylindricl geometries pplying the EVTUBAG progrm Luis Héctor Hernández Gómez 1, Guilllermo Urriolgoiti Clderón 1, Guillermo Urriolgoiti Sos 1, Jun Mnuel Sndovl Pined 2, Emmnuel Alejndro Merchán Cruz 2 y José Frncisco Gurddo Grcí 1 1 Unidd Profesionl Adolfo López Mteos Zctenco. Edificio 5, 2do. Piso, Col. Lindvist, C. P Unidd Azcpotzlco, Av. de ls Grnjs 682. Col. Snt Ctrin, Azcpotzlco, C.P Escuel Superior de ngenierí Mecánic y Eléctric, Sección de Estudios de Posgrdo e nvestigción, nstituto Politécnico Ncionl. México D.F., México Abstrct n this pper, the ssessment of the structurl integrity of pressurized cylindricl components contining defects like crcks ws performed, considering two types of filures. Brittle filure ws evluted considering Frcture Mechnics nd ductile filure ws clculted with plstic nlysis. For this purpose, three methods were used. Accordingly, the progrm EVTUBAG (evlution of crcked pipe by its initils in Spnish) ws written considering the methodologies proposed by Ruiz nd Corrn, ASME nd Rju-Newmn. This pper describes this progrm nd relevnt results re discussed. ey words: Longitudinl crcks, circumferentil crcks, stress intensity fctor, crcked cylindricl vessel nd limit nlysis. Evlución de l integridd estructurl de geometrís cilíndrics grietds plicndo el progrm EVTUBAG Resumen En este trbjo, se relizó un evlución de l integridd estructurl de componentes cilíndricos grietdos sujetos presión intern. Pr este efecto, se considerron dos tipos de fll. L frágil fue evlud bjo criterios de Mecánic de l Frctur y l dúctil se clculó con nálisis plástico. Pr este efecto, se desrrollron y plicron tres diferentes metodologís. Por lo que se implementó el progrm EVTUBAG (Evlución de Tuberí Agrietd), en el cul se considern tres procedimientos, propuestos por Ruiz y Corrn, ASME y Rju-Newmn. En este rtículo se describe l plicción de este progrm y se nlizn los resultdos relevntes obtenidos. Plbrs clve: Griets longitudinles, griets circunferenciles, fctor de intensidd de esfuerzos, recipiente cilíndrico grietdo, nálisis l límite. ntroduction The norml opertion nd geing process of mterils t industril instlltions produce crcks in pressurized cylindricl components. From the structurl integrity viewpoint, crcks in pressure vessels nd piping systems cn be grouped in the following bsic cses: (1) xil crcks subjected to internl pressure nd (2) circumferentil crcks subjected to opening moment nd xil lods. Crcks cn be through-wll or prt through-wll thickness. n

2 Assessment of the structurl integrity of crcked cylindricl geometries pplying EVTUBAG 191 first instnce, crck initition is evluted following the Frcture Mechnic principles due to frcture toughness cn be exceeded. While in the second cse, limit nlysis is required, when yield stress is reched t the uncrcked section. Relible results re in the open literture nd severl solutions hve been proposed by the use of the Finite Element Method (FEM) [1]. On the other hnd, there re some useful solutions which re obtined by the use of explicit expressions [2]. n industril ctivities, fst nd ccurte pproches re needed when there is crcked cylindricl component. n this cse, the opertor hs to tke the decision between: (1) the crck is too big, therefore repir work must be done s soon s possible nd (2) this crck is not too big, so the repiring work cn be done in the future. This pproch is very useful to void unnecessry unvilbility. Therefore, n evlution tool is required, in order to mke ccurte nd quick evlutions of these crcked configurtions. Accordingly, the progrm EVTUBAG (evlution of crcked pipe by its initils in Spnish) cn nlyze in simplified mnner the cses mentioned bove. Also, the regultory guidnce, which pplies to Nucler nstlltions, is considered. For this purpose, the methodology proposed by Ruiz nd Corrn [3], the ASME Code Section X [4] nd the numericl solutions of Rju-Newmn [5] were selected. The resons why these procedures were considered re the following: Simplified solutions re proposed in [3], by using Frcture Mechnics nd Limit Anlysis. t ws vlidted with some experimentl results reported in the open literture. Regrding the circumferentil crck cse, ductile filure is evluted in this pper only for through-wll thickness crck under opening moment nd xil lods. Although more loding cses of such methodologies re included in Tble 1. They re treted with more detil in [6]. Regrding the ASME Code Section X [4] procedure, it ws considered becuse this is prt of the regultory guidnce tht is pplied by some nucler instlltions. However, ASME Code procedures only evlute prt through-wll crcks. The Rju-Newmn [5] solutions re introduced s benchmrk. The proposed methodologies evlute the structurl integrity of cylindricl vessel under internl pressure. n the cse of longitudinl through-wll thickness crcks, geometry correction fctor proposed by Folis [7] is considered for the Frcture Mechnics Anlysis. n other words, the Stress ntensity Fctor (SF) is: ( 116. ) 1 2 l H 2 (1) l where 1, ( 2tD e ) 2 H is the hoop stress, l is the crck length, D e is the externl dimeter nd t is the cylinder wll thickness. n the cse of longitudinl prt through-wll crck, the following eqution is used (M is the shpe fctor): M ( l) 1 2 (2) H Tble 1 Scope of EVTUBAG progrm [6] 1. Longitudinl crcks (nternl pressure) 2. Circumferentil crcks (Opening moment nd xil lods) Through wll 1. Ruiz nd Corrn [3] (Brittle nd ductile filure). 2. Rju-Newmn (Brittle filure). 1. Ruiz nd Corrn [3] (Ductile filure). 2. Rju-Newmn (Brittle filure). Prt through-wll 3. Ruiz nd Corrn [3] (Brittle nd ductile filure). 4. ASME (Brittle, elsto-plstic nd ductile filure). 5. Rju-Newmn (Brittle filure). 3. Ruiz nd Corrn [3] (Ductile filure). 4. ASME (Brittle, elsto-plstic nd ductile filure). 5. Rju-Newmn (Brittle filure).

3 192 Hernández Gómez y col. Theoreticl Bsis of the EVTUBAG Progrm Filure nlysis proposed by Ruiz nd Corrn [3] The shpe fctor M (obtined in Figure 1) M is function of the crck depth nd length. f crck depth is greter thn 0.7 of the cylinder thickness, the crck my behve in one of the following two mnners. n first instnce, the crck hs infinite length nd depth. t is supposed tht crck propgtes through the cylinder thickness. The SF is clculted by: 112. (2) H n second instnce, the crck length increses nd the SF is: 071. l (2b) H Summrizing, the SF depends on the crcked geometry, the pplied stress nd crck dimensions. For this reson equtions (1), (2), (2) nd (2b) re similr. n the cse of the ductile filure, limit nlysis is done considering n dimensionl prmeter P l *, which reltes the required pressure for the genertion of generl yield of crcked pipe with the required pressure for generl yield of the sme pipe without crck. For longitudinl crck (eqution 3), where f is flow stress: * l P f t 2 1 (3) 05. n the cse of through wll thickness circumferentil crck, P c *, insted of P l *. Two cses re considered. The first one is when the crcked cylinder is under internl pressure, nd the following eqution is used: where: is the longitudinl stress. n the second cse, M* is the reltion between the ductile filure cused by bending moment (M b ) of crcked pipe nd the plstic filure bending moment (4 f tr 2 M ) of n uncrcked pipe M * b 4 2. f tr When this bending moment is prllel to the crck, the required reltion is: M * 1 cos sin 2 2 (5) Otherwise, when the bending moment is norml to the crck, the following clcultion hs to be done: M * ( cos ) 1 2 (5) n other words, M * is geometricl fctor nd it depends on the wy the bending moment is pplied. n the next step of this methodology, sfety fctor is obtined for brittle nd ductile filure. For brittle filure in xil crcks: SF Brittle Figure 1. Shpe fctor M [3]. C (6) * P c 2sin 2 f 1 sin (4) where C is criticl Stress ntensity Fctor of the mteril. For longitudinl crck, brittle filure occurs when the SF is equl or less thn one. n the cse of ductile filure of longitudinl crcks, similr sfety fctor is evluted:

4 Assessment of the structurl integrity of crcked cylindricl geometries pplying EVTUBAG 193 SF Ductile f Pl * (7) f the SF Ductile is equl or lower thn one, filure occurs. n this cse, f is the flow stress, is the design stress nd P * l is reltion between the fluency pressure of the uncrcked pipe nd the fluency pressure of the crcked pipe. n order to know the behviour of the crcked component, the sfety fctors re compred. n the cse of circumferentil crcks, the sfety fctor is clculted with the following reltion. n this cse longitudinl stress is generted nd it my be considered s the resultnt of n xil lod. When ductile filure occurs, the following evlution is crried on. SF Ductile t P * c (8) nd when there is bending moment nd ductile filure tkes plce: SF Ductile t M tr b 4 2 M * (9) Summrizing, eqution 7, 8 nd 9 evlute the sfety fctor relted with ductile filure. Their difference depends on crck geometry nd loding conditions. ASME code nlysis [4] Section X of the ASME Boiler nd Pressure Vessel Code estblishes the criteri nd the requirements for service nd pre-service inspection nd testing. For clss 1 components, the rticle WB 3000 of Division estblishes the cceptnce stndrds for crcked vessels, piping, bolting, pumps, etc. f the crck size is greter thn those llowed by the tble WB , then n nlyticl evlution cn be followed. n the cse of Ferritic steel pipe, the Code Cse N463 [8] cn be observed when the cceptnce stndrds re exceeded. Alterntively, crck evlutions in Austenitic pipe cn be done with the Code Cse N-436 [9]. n generl terms, the Code Cse N-463 hs two cceptnce criteri. The first one is function of crck size, while the second depends on the pplied stresses. Moreover, the filure cses re considered in the evlution phse, nmely (1) limit lod filure, (2) elstoplstic frcture mechnics, where ductile crck propgtion my occur before reching the limit lod nd (3) brittle filure, which is nlyzed with elstic Frcture Mechnics. Regrding cse N-436, only limit nlysis is considered, becuse ustenitic steels hve high frcture toughness. Numericl solutions proposed by RAJU-NEWMAN [5] Rju nd Newmn hve evluted numericlly the SF for wide rnge of ellipticl crcks loded under Mode. FEM ws used for longitudinlly crcked cylinder under internl pressure, when the defect is on the internl or externl surfce (Figure 2). Furthermore, four stress distributions through the thickness re considered, nmely (1) Uniform, (2) Linel, (3) Qudrtic, nd (4) Cubic. From ll these cses, the influence coefficients, G j, re obtined for other stress distributions like internl pressure or therml shock. According to Figure 2, the SF of n externl surfce crck, long its front is clculted with the following eqution: Q G t j,,, (10) c t R Where j = 0 to 3. Q is shpe fctor: Q c 165. (11) nternl surfce crcks in cylinders under internl pressure re evluted with the following reltion: pr t Q F R i,,, (12) c t t where F i is correction fctor, which in terms of G j nd Lmé eqution is: t 2 R Fi R R R G R G o R G R G (13) o

5 194 Hernández Gómez y col. Figure 2. Finite element model of semi-ellipticl superficil crck in cylinder. n the cse of externl longitudinl crcks, the fctor F i is chnged by F e, F e t 2 R R R R G R G o G R 2 o o 4 R o 3 G 3 (14) The vlues re obtined from [5] nd they re integrted in EVTUBAG. Summrizing, equtions 10 nd 12 evlute the SF. They depend on crck locliztion (internl or externl surfce). Progrm Description EVTUBAG ws written in Quick Bsic 4.5 nd runs in Pentium PC with 16 MB Rm memory nd 540 MB hrd disk minimum. This progrm hs four subroutines, Figure 4 shows its flow digrm. The min one is EVTUBAG, which hndles the input dt tht cn be used in the other three subroutines (RUZ, ASME, RAJU). The subroutine RUZ follows the procedure described in [3]. Figure 3. Mximum bending evlution of full scle tests. The cses which cn be nlyzed re prt through-wll nd through wll thickness longitudinl crcks under the scope of brittle nd ductile filure. Moreover, the ductile filure of circumferentil crcks is covered. For this purpose, xysimetric prt through-wll crcks under internl pressure nd through wll thickness crcks under internl pressure nd/or bending moment my be clculted. The ASME

6 Assessment of the structurl integrity of crcked cylindricl geometries pplying EVTUBAG 195 Figure 4. Generl flow digrm for EVTUBAG progrm [6]. subroutine follows the crck size cceptnce procedure proposed in WB-3514 section X for nucler power plnt components for two situtions: (1) longitudinl crcks under internl pressure nd (2) circumferentil crcks under internl pressure, bending moment nd/or xil loding. t hs to keep in mind tht the scope of ASME Code only considers the cse of prt through-wll crcks. When crck size is bigger thn the cceptble stndrds, ASME subroutine hs two clcultion procedures. The first one is for ustenitic piping (Code Cse N-436) [9]. Alterntively, when the nlyzed pipe is mde by ferritic steel, the Code Cse N463 [8] is followed. n the RAJU subroutine, the stress intensity fctors re evluted. Accordingly, prt through-wll nd through wll thickness longitudinl crcked pipe under internl pressure my be nlyzed. Besides, there re two lterntives for the nlysis of circumferentil crcks. The first one is relted with xysimmetric nd semi ellipticl prt through-wll crcks under internl pressure. On the other hnd, the second cse is relted with the through wll thickness crcks under internl pressure, bending moments nd/or tensile loding. Ech of the subroutine mentioned bove my be used independently or linked. This progrm is complemented with dtbse, contining tbulted nd grphicl dt of ASME Code or the influence coefficients of Rju-Newmn solutions. A detiled description of the complete progrm cn be found in [6]. Vlidtion of the Progrm The Mexicn Regultory Body required progrm, which evlutes in short period of time nd ccurtely the structurl integrity of crcked cylindricl geometries. Brittle nd ductile filure mechnisms hve to be considered in such evlutions. n this cse, the proposed pro-

7 196 Hernández Gómez y col. cedures by ASME, Ruiz nd Rju Newmn were included. The bove mentioned methodologies re widely ccepted nd they hve been vlidted elsewhere in the pst. For the cse of this pper, EVTUBAG is vlidted with the following cses: Longitudinl crck CASE (Longitudinl prt through-wll wll crck) The internl rdius nd wll thickness of the cylinder nlyzed is m nd m respectively. Crck length is m nd its depth is m. Mteril properties re; Young Modulus is GP, Yield Stress is MP nd Poison s Rtio is 0.3. Two loding cses were considered. n the first cse, the internl pressure ws 15.6 MP nd in the second cse, the internl pressure ws MP. The results re shown in Tble 2. n this cse, brittle frcture ws nlyzed. For the purpose of vlidtion, it ws considered the nlysis reported in [10], in which the J-integrl vlue ws obtined. n order to mke comprison, the J-vlues were trnsformed to the SF in plne strin conditions. As it cn be seen, ll the results mtch. CASE (Longitudinl through wll thickness crck) One brittle filure cse ws proposed, which ws previously solved by the uthors with the Finite Element Method using ANSYS 9.0 code. The externl dimeter nd wll thickness re m nd m respectively. Crck length is m. Regrding the mteril properties, Young s Modulus is 210 GP nd Poisson Rtio is 0.3. As frcture conditions re nlyzed, the elstic properties re only required. All the clcultions were performed with n internl pressure of 36.7 MP. n order to mke complete comprison, ws clculted with other equtions which hve the following generl form =M F T ( ). The min difference is its geometricl fctor, which it tkes in to ccount the crck length,, the vessel rdius R, nd the vessel wll thickness t. n ll cses, the hoop stress ws introduced. Results re shown in Tble 3. n the evlution of the results, it is importnt to keep in mind tht the Rju-Newmn procedure is bsed on FEM nlysis. Also, both solutions re in greement with the one obtined with the geometricl fctor reported in [11]. On the other hnd, the solution of Ruiz nd Corrn nd the one obtined with the correction fctor reported in [12] re similr. All these nlyses re pproprite for brittle behviour nd the geometry of the crcked body plys n importnt role. Therefore, ll the solutions my be s rnge of solutions. Circumferentil crck CASE (Circumferentil through wll thickness crck) The experimentl results obtined in the US Nucler Regultory Commission Degrded Piping Progrm Phse [13], were considered. n this cse, full scle test of stright pipe, under bending loding t 288 C, ws done. ts externl dimeter nd thickness re m nd m respectively. The pipe mteril ws steel A516 Gr 70. One of the objectives ws the determintion of the mximum llowble bending moment. Therefore, two conditions were nlyzed. n the first cse, short circumferentil crck ws introduced; its length ws 6% of the circulr perimeter. n second instnce, long circumferentil crck ws evluted. ts length ws 37% of the circulr perimeter of the pipe. The experimentl evidence hs shown tht circumferentil short nternl Pressure MP Tble 2 Comprison of SF vlues (Cse ) Vlues from Reference [9] Results obtined with EVTUBAG for (MP m 1/2 ) J (m) (MP m 1/2 ) / 0 Ruiz Rju ASME / 0 / 0 /

8 Assessment of the structurl integrity of crcked cylindricl geometries pplying EVTUBAG 197 Tble 3 Comprison of SF vlues obtined by diverse methods (Cse ). Eqution (GP m 1/2 ) / 0 Reference M F Rt [10] FEM ANSYS 9.0 M F Rt Rt [11] Ruiz EVTUBAG Rju-Newmn EVTUBAG ASME EVTUBAG crcks tend to hve ductile behviour, while circumferentil long crcks tend to hve brittle behviour (Figure 3). Also, in this figure, the results obtined with the Net Section Collpse (NSC) re plotted. This cn be considered s the upper limit of the rnge of solutions. On the other, the lower limit of the rnge of solutions is obtined with the WB 3650 procedure of ASME. The lst two evlutions were done within the full scle test progrmme mentioned bove. The Ruiz nd Corrn procedure ws run for different crck lengths. The results re close to those obtined experimentlly. Also, such results tend to be close to those obtined with the WB Alterntively, the modified Rju- Newmnn results overestimte the filure moment. t is importnt to keep in mind tht the limit lod solutions provided by Zhoor [14] were introduced s complement for the Rju- Newmn procedures. These results re close to the NSC solution. n generl terms, both procedures give results within the rnge of solutions described before. The results obtined with the Ruiz nd Rju-Newmn-Zhoor procedures re shown in Tble 4. They were compred ginst the experimentl results nd the evlutions performed with Net Section Collpse (NSC) nd WB 3650 of ASME procedures reported in [13, 15 nd 16]. n this cse n dimensionl comprison is mde, the rtio between the mximum moment pplied in the experimentl tests nd the mximum moment clculted with the methods considered. n fct, the solutions which re closed to the unity my be considered s the best. For the cse of the short crck (6% of the pipe perimeter), n elstoplstic behviour ws observed. t is importnt to keep in mind tht it is pproprite for the nlysis of brittle behviour. n the cse of the long crck, it ws lso observed n elsto plstic filure. The results obtined with the Ruiz- Corrn behviour were close to the experimentl result. The modified Rju-Newmnn nlysis lso gve result which is close to the experimentl filure moment. As it ws expected, the NSC nd the WB 3650 evlute the filure moment, which re in the limits of the rnge of solutions. n generl terms, the evlutions of EVTUBAG re in line with the experimentl evlution. Conclusions t ws shown in this pper tht the progrm EVTUBAG cn perform complete structurl integrity ssessments on crcked cylindricl components. n this cse, it is voided the use of numericl nlysis, such s the Finite Element method, which demnds lrge mount of computing resources. Besides, this sort of nlyses requires lot of time. All the cses tht re presented in this pper were clculted in short period nd they only required few mount of computing resources. Actully, this progrm is being used by the Mexicn nucler regultory body. Another dvntge of EVTUBAG progrm is tht diverse methodologies re concentrted in

9 198 Hernández Gómez y col. Tble 4 Mximum llowed bending moment (Cse ) Test conditions Mximum moment pplied intests Rtio Mximum moment clculted Test Crck length/pipe circulr perimeter l/(d) Mximum experimentl moment (kn-m) NSC WB 3650 Ruiz Rju Newmn Zhoor [14] (short crck) [12] 0.37 (long crck) one progrmme nd they cn be run together. Furthermore, ductile nd brittle filure cn be evluted. n the first cse, frcture mechnics concepts re tken into ccount, while in the second cse, the filure is evluted since the point of view of limit nlysis. Therefore, the nlyst hs scope of solutions which cn be compred mong them. The nlyst hs to keep in mind tht during opertion, the mteril properties could be degrded. n consequence, it is importnt to use in the evlutions the ctul properties of the mteril. The results presented in this pper nd other cses tht were used to evlute the performnce of the progrm showed good convergence with the solutions. Even if EVTUBAG uses simple equtions, this fst method hs cceptble greement with the known forml solutions. Acknowledgements The uthors grtefully cknowledge the finncil support grnted by de Consejo Ncionl de Cienci y Tecnologí nd the nstituto Politécnico Ncionl. Also, it is in high recognition the support given by Mr. Pblo Ruiz López of the Comisión Ncionl de Seguridd Nucler y Slvgurds. Nottion : Hlf width D e : Externl dimeter G j : nfluence coefficients : Stress ntensity Fctor (mode ) C : Criticl Stress ntensity Fctor (mode ) l : Crck length M : Shpe fctor M* : Adimensionl moment prmeter M b : Bending moment P c * : Adimensionl pressure prmeter, for pipes with circumferentil crcks P l * : Adimensionl pressure prmeter, for pipes with longitudinl crcks Q : Geometricl shpe fctor t : Cylinder wll thickness : Longitudinl stress f : Flow stress : Hoop stress H References 1. Dickson, T. L., Bss, B. R. nd Willims, P. T., A comprison of frcture mechnics methodologies for postulted flws embedded in the wll of nucler rector pressure vessel, Pressure Vessel nd Piping Design Anlysis, ASME Publictions PVP-Vol. 430, (2001), Wllin,., Comprison of the scientific bsis of Russin nd Europen pproches for evluting irrdition effects in pressure vessel steels, Europen Network on Ageing Mterils nd Evlution, VTT Mnufcturing Technology, Finlnd, (1994) Ruiz, C. nd Corrn, J. S., Prcticl ppliction of extreml elstic-idelly plstic solutions for the ssessment of the severity of crcks, nterntionl Journl of Pressure

10 Assessment of the structurl integrity of crcked cylindricl geometries pplying EVTUBAG 199 Vessels nd piping, Vol. 10, No. 5, (1982), Americn Society of Mechnicl Engineers Boiler nd Pressure Vessel Code, Section X, Rules for inservice inspection of nucler power plnt components, Appendix A, Anlysis of flws, Article A-3000, Method for determintion, Americn Society of Mechnicl Engineers, New York, (1998). 5. Rju,. S. nd Newmn, Jr. J. C., Stress intensity fctors for internl nd externl surfce crcks in cylindricl vessels, Journl of Pressure Vessel Technology, Vol. 104, No. 4, (1982) Gurddo-Grcí, J. F., Evlución de l integridd estructurl de recipientes cilíndricos grietdos sometidos presión intern, Tesis de Mestrí en Ciencis. nstituto Politécnico Ncionl, SEP-ESME, (1998). 7. Folis, E. A., n mechnics of frcture, Vol., Sih. G. C. (Ed), Noordhoff, ASME, Cses of ASME Boiler nd Pressure Vessel Code, Cse N-463: Evlution Procedures nd Acceptnce Criteri for Flws Clss 1 Ferritic Piping tht Exceed the Acceptnce Stndrds of WB , Section X, Division 1, ASME, New York, (1988). 9. ASME, Cses of ASME Boiler nd Pressure Vessels Code, Cse N-436-1: Alterntive Methods for Evlution of Flws in Austenitic Piping, Section X, Division 1, ASME, New York, December, (1987). 10. Bloom, J. M. nd Lee, D. R., Function solutions for pressurized cylindricl vessels hving semi-ellipticl surfce flws, Journl of Ftigue nd Frcture Mechnics in Pressure Vessels nd Piping, Vol. 304, pp 105, Ewlds, H, L. nd Wnhill, R. J. H., Frctures Mechnics, EAP, (1969) Folis, E. S., A finite line crck in pressurized sphericl shell, nterntionl Journl of Frcture, Vol. 1, ssue 1, (1965) Wilkowski, G. M. nd Snkusks, G., Degrded piping progrm-phse, Summry of technicl results nd their significnce to lek-before-brek nd in-service flw cceptnce criteri, Mrch 1984-Jnury 1989, Bttelle, NUREG/CR-4802, U.S. Nucler Regultory Commission, Vol. 8, (1989). 14. Zhoor, A., Ductile frcture hndbook; Circumferentil throughwll crcks, Vol. 1, EPR Report NP-6301-D, Electric Power Reserch nstitute, Plo Alto C. (1984). 15. Wilkowski, G. M. nd Scott, P. M., Short Crcks in Piping Welds, Semi-nnul Report, Bttelle, NUREG/CR-4599, U.S. Nucler Regultory Commission, Vol. 1, No. 1, April- September, (1992). 16. Corrn, R. S. J., Dvies, P H. nd Ruiz, C. Engineering of Frcture Mechnics, 16 (1982), pp 585. Recibido el 08 de Septiembre de 2008 En form revisd el 28 de Septiembre de 2009