Flow Induced Vibration Crack

8/4/2019 Flow Induced Vibration Crack

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F ractu re o f P ip elin es an d C ylin ders C on ta in in g aC ircumferen tia l C rackby F. Erdogan and H. Ezzat

CONTENTSAbstract 11 , it ..1. Introduction .

1.1 Fracture MechanicsApproach ....... 11.2 Brief Surveyof the Field . . .. .. .. .. . . .. 21.3 Circumferential Flaws ............... 32. Stress Intensity Factors .............. 3

2.1 Results for aSurface Crack ........... 32.2 Stress Intensity Factors for a ThroughCrack it'" it . i t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3. Crack Opening Displacement ......... 94. The Pipe Experiments.. ...... ...... 104.1 Stresses in a Pipeunder "Four Point"Bending.............................. 114.2 Mechanical Properties ofthe Material .. 124.3 The Fatigue Experiments. . . . . . . . . . . . . . . 134.4 Fracture Tests........... . . .. 144.5 Examination of Fracture Surfaces ..... 18

5. Conclusions ..II .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 196. References II .. .. .. .. .. .. .. .. .. 237. Acknowledgments ................ 23AbstractThis study is concerned with the problem of a pipe con-taining a part-through or a through circumferential crack.First, the stress intensity factorsfor an internal or anexternalcircumferentialsurfacecrack in a pipe are obtained.Themainobjectivehere isto givethe necessarytheoretical informationfor the treatment ofsubcriticalcrackgrowthprocess.Next theproblem of a through crack in the presence of large scaleplastic deformations is considered. The crack opening dis-placement (COD) is used as the main parameter to analyzethe fracture instability problem and to correlate the experi-mental results. In the analytical part ofthe study Reissner'sshell theory and an elasticor elastic-plasticline springmodel

are used to formulate the problem. The experiments wereperformed on 20-inchdiameter X60line pipes.AO.025-inchwide starter notch was introduced to the pipes whichwerethen subjectedto cyclicloadingunder fourpoint bending.Thelimited data obtained from the fatigue tests givethe expectedresult, namely that the crack propagation rate in pipes maybe predicted from the baseline data obtained from simplespecimens provided the stress intensity factors for pipes arecalculated with sufficient accuracy. The ductile fracture re-F. Erdogan isProfessor ofMechanics, Lehigh University, Bethlehem, Pa., H.Ezzat is Assistant Professor of Mechanical Engineering, University of Pitts-burgh,Pa.The research reported inthis paper was supported by the U.S. Departmentof Transportation, Office of University Research.Publication ofthis report was sponsored bythe Weldability Committee ofthe Welding Research Committee.

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suits show that the technique based on the asymptotic be-havior of COD may be quite useful in determining a conser-vative estimate ofthe fracture instability load.1. IntroductionThis report presents the theoretical and the experi-mental results of a four year study on the fracture ofcircumferentially cracked pipelines and relatively

thin-walled cylindrical containers. The research wassponsored by the U.S. Department of Transportation,Research and Special Programs Administration, Officeof University Research. It was part of a coordinatedDOT program in which the National Bureau of Stan-dards and Johns Hopkins University were the otherparticipants.The primary objective of the research program was(a) to identify the possible modes offracture failure inpipelines and in relatively thin-walled cylindrical con-tainers containing various types of initial circumfer-ential flaws, (b) to review and develop appropriatefracture criteria and to carry out the necessary analyt-ical investigations which may be applied to variousphases of fracture failure in circumferentially crackedpipes and containers, and (c ) to design and perform anexperimental research program in order to test the va-lidity of the related analytical models.1.1 Fracture MechanicsApproachDepending on the thermo-mechanical behavior of thematerial and the nature of applied loads and environ-

mental conditions, in the design of pipelines, tank cars,and a variety of other pressurized containers, it isoftennecessary to consider fatigue or corrosion crack propa-gation and fracture among the possible modes of failure.This requires, inaddition to the application of standardfailure theories specified by the existing design codes,the treatment of the problem of acceptance and safetyfrom the viewpoint of fracture mechanics. In using thisapproach the flaws or certain types of imperfectionswhich may initially exist in the material are treated as"cracks." These initial flaws which may have the po-tential of growing into a macroscopic fatigue or corro-sion fatigue crack are generally weld defects (such asslag inclusions, excessive or inadequate weld penetra-tion, incomplete fusion, gas pockets, arc burns, etc.),notches caused by possible initial misalignment during

Fracture of Pipelines and Cylinders 1


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welding, accidental dents, scratches, Orgouges put onthe pipe or the cylinder during manufacturing andtransportation, and accidental damages which occurduring the period of operation. Thus, in designingpipelines and pressurized containers used for trans-porting goods a safe design philosophy requires that theexistence of such defects be taken into consideration.Fracture mechanics approach has been highly suc-cessful in dealing with this so-called fracture controlproblem which involves subcritical propagation of fa-tigue and stress corrosion cracks, brittle or quasi-brittlefracture, and ductile fracture. In a broad sense the ob-jective of the pipeline and container fracture analysisis the determination of the maximum allowable initialflaw size (of certain specified shape, location and or-ientation) which would not grow into a crack goingthrough the entire thickness of the cylinder wall undera specified load history and environmental conditions.The general procedure for the fracture analysis may beoutlined as follows:(a) The strength characterization of the material

with regard to fatigue and corrosion fatigue crackpropagation, stress corrosion cracking, brittle orquasi-brittle fracture, and ductile or post-yieldfracture.(b) Determination of the time profile of all significantexternal loads and operating temperature and acomplete stress analysis (including the residualstresses) of the structure by ignoring the exis-tence of any flaws.(c) Obtaining the map of significant existing flawsby means of appropriate nondestructive flawdetection techniques, or making a realistic (andconservative) assumption regarding the location,size, and orientation of dominant flaws.(d) Performing the fracture analysis to estimate thelife ofthe structure. Generally, this step requiresthe calculation of the relevant stress intensityfactor and the use of a proper sub critical crackgrowth model to estimate the propagation rateof the part-through crack in the initial elasticrange, the application of a modified crack growthmodel to estimate the crack propagation rate inthe elastic-plastic range, and the use of an ap-propriate post-yield fracture criterion to estimatethe critical net ligament thickness capable ofsustaining the specified peak load.

1.2 Brief Survey of the FieldFor the purpose ofa brief review one may consider theresearch efforts regarding pipeline fracture analysis inthe following categories: (i) theoretical work aimed atthe calculation of quantities such as stress intensityfactor, crack opening displacement, or J-integral whichmay be used in an appropriate fracture or fatigue theoryas the representative of the external loads and flaw ge-ometry, (ii) the theoretical and experimental workaimed at the development of proper "models," "crite-ria," or "theories" for fatigue crack propagation, cor-rosion fatigue, stress corrosion cracking, and fracture

in pipeline materials, (iii) experimental work aimethe verification or demonstration of the related theoin pipes, including crack morphology studies, (iv)namics of crack propagation in pipelines. The technliterature in the field is quite extensive (see, for examRef. 1 for an extensive list of references and a critreview). Most of the theoretical work in categorywhich is relevant to pipeline studies has been onelastic solutions for part-through cracks in platesfor through cracks in shells. Some of the significsolutions and results regarding the part-through crin plates may be found in Ref. 2. A review of the mrecent studies and rather accurate finite element resmay be found in Ref. 3. The problem of a through cin cylindrical and spherical shells has been discussea review article, Ref. 4, where, in addition to a necomplete list of references, a summary of the exisresults has been included. The plasticity effects inlindrical shells with an axial crack have been considein Ref. 5. Reference 6 summarizes some of the recapproximate and finite element results on the thwalled cylinders with a part-through crack.The processes of brittle and quasi-brittle fractuand the subcritical crack propagation due to fatigcorrosion fatigue, and stress corrosion cracking appto be, at least from an empirical viewpoint, well-derstood and the models dealing with such phenomhave been adequately standardized (see, for examRef. 7). The stress intensity factor is almost universaccepted and used as the primary correlation paramein all these models. However, in the presence of rtively large scale plastic deformations the effectsspecimen and crack geometry, elastic-plastic strestrain behavior of the material, and the nature of

external loads on the fracture initiation and propagatis much too great to permit the treatment of the pnomenon by means of a single parameter (such as KGIG, or Jlc). Thus, the relatively successful ducfracture models contain generally more than oneterial constant. For example, the crack extensionsistance curve (R of KR - curvel' ' is a continuoudistributed parameter model and Newman's criteriis a two-parameter model. A discussion of theseother ductile fracture models and related referenmay be found in the review article.t? A gooddeal ofwon pipeline fracture has been done at Battelle-Colubus Laboratories. A partial summary of the resultsthe techniques used in these studies may be foundRef. 11 and an extensive review of fracture mechanapproaches dealing with girth weld discontinuitiesgiven in Refs. 12 and 13.The importance of dynamic problems lies in thethat in natural gas pipelines once the axial throucrack appears in the pipe wall it rapidly growsreaches a velocity which is generally greater thandecompression wave velocity of the gas in the pHence, the crack is subjected to a constant driving fand, unless it is arrested by some obstacle or is diverin the hoop direction, it may run rather long distancausing considerable damage to the surroundings. T


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papers in Reference 14 provide a good sampling of thework in this area (see also Ref. 15 for more recent workand references, and Ref. 16 for work in finite volumecontainers).1.3 Circumferential FlawsIn pressure vessels and piping generally the primaryload component is the internal pressure. Consequently,in the past considerable emphasis has been placed on

the studies of fracture problems arising from axial flawsin the cylinder (see, for example, Refs. 11 and 14 fortypical fracture studies and Refs. 17-21 for the elasticsolutions of through and part-through axial crackproblems in cylinders), On the other hand, particularlyin piping, the secondary loads are mostly axial andgenerally time-dependent. Relatively high frequencyand low amplitude flow-induced vibrations in heat ex-changer tubing and in line pipes near the pumpingstations may be cited as some examples. Severe bendingin pipes due to fit-up or that in offshore piping whilelaying off the barge, and variety of constraint stressescaused by thermal fluctuations, ground settlements, and ,earthquakes may be mentioned as examples of axialloading with very low frequency and relatively highamplitude (see, for example, Refs. 12, 13, 21, 22, and 23for sample studies).From the viewpoint of analytical modeling the fatiguecrack propagation and fracture problems in pressurevessels and piping may be considered in three loosegroupings. The first is the problem of fatigue crackpropagation and fracture in pipes or cylinders havinga dominant part-through or through crack in which theplastic deformations are confined to a small region alongthe crack front [Fig. l(a) and (b)]. In such problems the

stress intensity factor appears to be highly satisfactoryas the correlation parameter and hence an elastic solu-tion of the related crack problem is sufficient for thefracture or fatigue analysis. The second group relatesto the "ductile fracture" problems which involve ex-tensive plastic deformations around the crack frontthrough the net ligament [Fig. l(c)]. In this case a va-riety of parameters or criteria (such as COD, CTOD, Jintegral JR-curve, net ligament plastic strain instability,and plastic collapse load) are used to analyze the frac-ture initiation, stable crack growth, and unstable frac-ture. However, particularly in piping and other rela-tively thin-walled components this ductile fractureprocess seems to be very highly dependent on the grossmechanics of the structure. Therefore, no single pa-rameter criterion ignoring the relevant aspects of theoverall mechanics problem would be adequate to modelthe process. The third group is concerned with the dy-namic fracture propagation in pipelines which mayfollow the instability of an axial crack.In this paper, after a brief outline of the techniqueused to analyze the problem, first some calculated stressintensity factors for a circumferential part-throughcrack in cylinders are presented. With a proper sub-critical crack growth model and appropriate baselinedata, these results may be used to estimate fatigue or

Ad o I 20 I). hI

(a) (b)

'I p 1 20 -, - p -,

Plast ic Zone (c)

,(d) (e)

20 I I . p - I(f)

Fig, 1-Evolut ion of a through crack. {a l initial flaw and subcritical crackpropagation, (bl part-through crack with confined plastic zone andlargely elastic net ligament, (c) part-through crack with ful ly yielded netligament, (d ) progressive growth of part-through crack, (e) plasticnecking of th e net ligament, (I)through crack with relatively large plasticzones

corrosion fatigue crack propagation rate in the cylinder.Next, some sample results obtained from the elastic-plastic solution of pipes having a relatively large crackwith a fully yielded net ligament are given and theirpossible use in estimating the fracture initiation andinstability loads is discussed. Finally, the experimentalresults on the fatigue crack propagation and fracture ofa 20-inch diameter line pipe tested under four pointbending are presented and discussed.2. Stress Intensity Factors2.1 Results for a Surface CrackThe general formulation of the cylindrical shellcontaining an axial or a circumferential through crackwas given in Ref. 22. In this study Reissner's transverseshear theory was used to formulate the problem. Thecorresponding part-through crack problem was con-sidered in Ref. 21. In this section additional axial andcircumferential crack results for certain standard linepipes as well as some results for spirally oriented cracksare given.The basic geometry ofthe shell with a circumferentialcrack is given in Fig. 2. In most cases the stress state inthe neighborhood of the crack isapproximately uniform.Thus, if the loading is symmetric with respect to the


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X I

Fig. 2-The cylinder with a circumferential through crack

plane of the crack, the problem may be solved by as-sumingthatN ll(X 2) =N 00 = constant, MUX2) = 0, or (1)Mn(X2) =Moo = constant, NU(X2) = 0, (2)

where N11 and M11 are respectively the membrane andbending resultants. Aside from the uniform loadinggiven by Eqs. 1 and 2, in pipes the loading which is ofconsiderable practical interest is the gross bending withbending moment Moat the cracked section Xl = (Fig.2). In this case in the absence of crack the stress in thecylinder may be expressed as0' = Mozo = MoR ( 1 _ X22)U I I 2R 2

+ MoX3 ( 1 _ X22) 1 = Rh(4R2 + h2) (3)I 2R2 'where 20 is the distance from the neutral plane. Thus,by observing that local membrane and bending stressesare related to the stress and moment resultants by

and by letting 0'11 = 0'11m + 0'11b, from Eqs. 3 and 4 weobtain the resultants which are to be used as externalloads in the crack problem as follows:

Nll(X2) = Mo:h ( 1 - ~ : ) = N 00 ( 1 - : ; : ) , (5)Mll(X2) = Moh3 ( 1 - X22) = M", ( 1 _ X22) . (6)1~ 2R 2 ~2

Sample results giving the stress intensity factor at themaximum penetration point L = Lo of a semi ellipticinner or outer circumferential crack in standard linepipes are given in Tables 1-8 (see the insert in Fig. 21).Similar results for the semi-elliptic axial inner or outersurface crack may be found in Ref. 24. Additional gen-eral results for surface cracks in cylindrical shells interms of the dimensionless variables Lo/h, a/h and A =[12(1 - p2)]lf4a/v'Rh are given inRef. 21. Reference 21also contains the stress intensity factors for a surfacecrack in the cylinder subjected to local bending of theshell wall. The analysis used for the calculation of stressintensity factors is based on the Reissner's shell theoryand the elastic line spring model and is described inRefs. 21, 22 and 24 in detail. Reference 21 gives a spotcomparison of the analytical results obtained from the

Table 1. K/Ko in a line pipe with 00 - 48 in h : 0.625 In.I'ill 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.a/h~ Outer circumferent ial crack N . 0 M o . - a12:~ 1~:;6~ Ig :~~U~:g :~ : I g :~~0 .~ 47 1 0. [4 7 g : V ~ U0.340 0.209 0.3.0 0.976 0.911 0.817 0.695 0.554 0.405 0.257 0.132 0.4.0 0.980 0.928 0.848 0.739 0.605 0.455 0.296 0.155 0.5.0 0.983 0.939 0.868 0.770 0.643 0.494 0.329 0.175 06.0 0.985 0.946 0.882 0.792 0.672 0.525 0.357 0.193 0.7.0 0.987 0.951 0.893 0.809 0.694 0.551 0.381 0.209 0.8.0 0.988 0.954 0.900 0.821 0.712 0.572 0.401 0.223 0

Outer clrcumferenti aT crack N . - a v aI I . u I ~ : ; = I~:;~:~:m : ; ~ f ~ :m~:~:~g : r i ~~:~~6g.03.0 0.975 0.906 0.79B 0.658 0.492 0.31B 0.163 0.047 -0.4."0 0.980 0.923 0.832 0.708 0.55T 0.377 0.209 0.072 -0.5.0 0.983 0.934 0.855 0.742 0.595 0.422 0.246 0.094 -0.6.0 0.9B5 0.942 0.B70 0.767 0.628 0.485 0.278 0.114 07.0 0.986 0.947 0.882 0.785 0.653 0.488 0.304 0.131 0.8.0 0.987 0.951 0.890 0.800 0.673 0.511 0.327 0.147 0.Inner circumferent i a1 crack N . ~ . M o . - aDr I ~ : ; ~ ~:;~~: m t~::4~~6 : ; ~ I g : ~ 6 ~~:~~.03.0 0.974 0.905 o.sos 0.677 0.532 0.384 0.242 0.124 0.4.0 0.978 0.920 0.832 0.716 0.575 0.425 0.273 0.142 0.5.0 0.981 0.930 0.850 0.742 0.607 0.456 0.297 0.156 0.6.0 0.983 0.937 0.864 0.762 0.631 0.480 0.318 0.169 O7.0 0.984 0.941 0.873 0.776 0.650 0.500 0.335 0.180 08.0 0.985 0.945 0.880 0.787 0.664 0.517 0.350 0.T9O 0.Inner circumferentiQl crack N,,-O ~I t g I g : ; ~ g : ; ~ ~~ :~~6g : ; ~ ~~ : ~ ; ~ ~ : m I ~ : g ; ~- g : g { ~: g3.0 0.973 0.898 0.783 0.635 0.465 0.292 0.144 0.037 -0.4.0 0.978 0.915 0.814 0.679 0.5T6 0.339 0.179 0.056 -0.5.0 0.981 0.925 0.835 0.709 0.552 0.375 0.207 0.071 -0.6.0 0.982 0.932 0.849 0.731 0.579 0.403 0.230 0.085 -0.7.0 0.984 0.937 0.859 0.747 0.600 0.426 0.249 0.096 -0.8.0 0.985 0.941 0.867 0.759 0.616 0.444 0.265 0.107 0.

(4)Table 2. K/~ In a 11ne pipe with 00 - 48 1n h - 0.75 In.

L J h d 0.1 I 0.2 I 0.3 I 0.4 I 0.5 I 0.6 I 0.7 I 0.8 I 0a/h+ Outer circwnferent ial crack, N . I(), H.-OI~:g,g:; :~Ig:~ Ig :~: Ig :~: Ig : :~ I ~ : ~ ~ I g : 2 ~ ~ g:~~ g :3.0 0.976 0.911 0.8T6 0.695 0.553 0.405 0.258 0.133 0.04.0 0.980 0.927 0.847 0.738 0.604 0.454 0.297 0.156 0.05.0 0.983 0.938 0.B67 0.768 0.641 0.493 0.329 0.176 0.06.0 0.985 0.1I44 0.881 0.790 0.669 0.523 0.356 0.T93 0.07.0 0.986 0.949 0.890 0.805 0.690 0.548 0.379 0.209 0.08.0 0.987 0.952 0.897 0.8T7 0.707 0.567 0.398 0.222 O.

Outer c1rcumferentia 1 crack N " a Q . _ M . J I j )1 2 : ~ , g : ~ 6 :~:~:g : m g:;~g : : ~ ; I g : 2 4 ~g :~g-g :g2~g3.0 0.975 0.905 0.797 0.657 0.492 0.318 0.164 0.048 -0.04.0 0.980 0.922 0.831 0.706 0.550 0.376 0.209 0.073 -0.05.0 0.983 0.933 0.853 0.740 0.593 0.421 0.246 0.095 -0.06.0 0.985 0.941 0.868 0.764 0.624 0.456 0.276 0.114 0.07.0 0.986 0.946 0.879 0.781 0.648 0.483 0.302 0.131 0.08.0 0.987 0.949 0.886 0.794 0.666 0.505 0.323 0.146 0.0

Inner ci rcwnferent1 al crack N ,J O. K "to OI t g I g : ; : ; 6 : 8 1 : g : m 6 : 6 1 ' 4 g :~ Ig :~~Ig:m g:~~~g :3.0 0.973 0.903 0.802 0.674 0.529 0.382 0.240 0.124 0.04.0 0.978 0.9T9 0.829 0.712 0.571 0.421 0.270 0.141 0.05.0 0.981 0.928 0.847 0.737 0.602 0.451 0.294 0.155 0.06.0 0.982 0.934 0.860 0.756 0.625 0.475 0.314 0.168 0.07.0 0.984 0.939 0.869 0.770 0.642 0.493 0.330" 0.178 0.08.0 0.985 0.942 0.875 0.781 0.656 0.509 0.344 0.187 .0.0Inner c1rcumferentfal crack. N,,-O'uIt g Ig:; : : I g : : ~ ,g :~~~~:;;~: ~ I g : 2 2 ;~:~~~ : 6 1 ; :g3.0 0.973 0.897 0.780 0.632 0.461 0.289 0.142 0.036 -0.04.0 0.977 0.913 0.811 0.675 0.510 0.335 0.176 0.054 -0.05.0 0.980 0.923 0.831 0.704 0.545 0.369 0.203 0.070 -0.O6.0 0.982 0.930 0.844 0.7Z4 0.571 '0.396 0.224 0.082 -0.07.0 0.983 0.934 0.854 0.740 0.591 0.417 0.243 0.094 -0.08.0 0.984 0.938 0.861 0.751 0.606 0.434 0.258 0.104 0.0


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Table 3. K/Ko in a line pipe with 00 M 36 In h 0.5 'In.

Y"-!:IO.l I 0.2 I 0.3 I (1.4 I 0.5 I 0.6 I 0,7 I 0.8 I 0.9a/h+ Outer c1rclWferent1al crack N. o. 'M.-Otg ,O.!I4!) 1517 10664 10.505 10.J66 10,247 0.147 O'I}~~ g :6 :.967 0.882 0.766 0.628 0.481 0.340 0.210 0.1063.0 0.976 0.911 0.817 0.695 0.554 0.405 0.267 0,132 0.0604.0 0.980 0.928 i).848 0.739 0.605 0.455 0.297 0.155 0.0705.0 0.983 0.938 0.868 0.169 0.642 0.494 0.330 0.175 0.0796.0 0.985 0.945 0.882 0.791 0.671 0,525 0.357 0.193 0.0887.0 0.987 0.950 0.892 0.808 0.693 0,550 0.380 0.209 0.0968.0 0.988 0.954 0.900 0,820 0.710 0.570 0.400 0,223 0.104Outer cfrcumferentfal crack It.-a UItg 10944 10605 10627 10.~~ 10.273 10.133 ~ : m tg~A : g : g r a.966 0.874 0.741 0.581 0.407 0.2423.0 0.975 0.905 0.798 0;657 0.492 0.318 0.164 0,048 -0.0214.0 0.980 0.923 0.832 0.707 0.551 0.376 0.209 0.072 -0.0125.0 0.983 0.934 0.854 0.741 0.594 0.422 0.246 0.094 -0.0046.0 0.985 0.941 0.870 0.766 0.627 0.458 0,277 0,114 O . O O S7.0 0.986 0.947 0.881 0.784 0.652 0.486 0.304 0,131 0.013B.O 0.987 0.950 0.889 0.798 0.671 0.509 0.326 0;146 0.020Inner cfrclll lferentfal cr~ck N..l1O.M..-Otg Ig :~ rsg :~ }18:~~ 18:m 18:~} IQ243 10.145 U~~: g : :.328 0.2013.0 0.974 0.904 0,803 0.676 0.531 0.383 0.241 0.124 0.OS84.0 0.978 0.920 0.831 0.714 0.574 0.424 0.272 0.142 0.0665.0 0.981 0.929 0.850 0.741 0.605 0.454 0.296 O~156 0.0746.0 0.983 0.936 0.862 0.760 0.629 0.478 0.316 0.169 O . o e o7.0 0.984 0.940 0.872 0.774 0.647 0.498 0.333 0.179 0.0868;0 0.985 0.944 0.879 0.785 0.662 0.514 0.348 0.189 0.091Inner cf rcumferentta 1 craek, N...-D ~Its !g :~~~g::~ g :~~ g: :~~g:;~ ~ ~0:O37 ; ; z : m -~'ll~0.099 -0.0303.0 0.973 0.898 ' 0.782 0.634 0.464 0.291 0.143 0.037 -0.0244.0 0.978 0.914 0.813 0.678 0.514 0.338 0.178 0.055 -0.0185.0 0.980 0,924 0.833 0.707 0.549 0.373 0.205 o.on -0.0136.0 0.982 0.931 0.847 0.729 0.576 0.401 0.228 0.084 -0~0017.0 0.984 0.936 0.857 0.745 0.597 0.423 0.247 0.095 -0.0028.0 0.985 0.940 0.865 0.757 0.613 0.441 0.262 0.106 0.003

Table 4. K/Ko in a Tine pipe with 00 ~ 36 in h 0.625 in.

yn-!:IO.1 I 0.2 I 0.3 I 0.4 I 0,5 I 0.6 I 0.7 I 0.8 I 0.9a/h. Outer circumferential crack. N . . i 0 M. , sOt g 10945 10817 1~664 I~:~~~ !g :~ ~~~ :m i g : m 0.074 o.~~~0.967 0.882 0.766 0.107 0.0493.0 0.975 0.911 0.816 0.695 0.553 0.405 0.258 0.133 0.0604.0 0.980 0.927 0.846 0.737 0.603 0.454 0.297 0.156 0.0715.0 0.983 0.987 0.866 0.767 0.640 0.492 0.329 0.176 0.0816.0 0.985 0.944 0.879 0.788 0.667 0.522 0.356 0.193 0.0907.0 0.986 0.948 0.889 0.803 0,688 0.545 0.378 0.209 0.0988.0 0.987 0.951 0.895 0.814 0.704 0.564 0.396 0.222 0.105Outer cfrcumferential crack Nm"O Mo.,IO

I~:g I~ :~ : :~ :~~g : m I~:m g : m Q.T34 -0-;040 : t g 1 l : g : g ~ ~.242 0.1103.0 0.975 0.905 0.797 0.656 0.491 0.318 0.164 0.048 -0.0204.0 0.980 0.922 0.830 0.705 0.549 0.375 0.209 0.073, -0.0115.0 0.982 0.933 0.852 0.738 0.591 0.419 0,245 0.095 -0.0026.0 0.984 0.940 0.866 0.761 0.622 0.453 0.275 0.114 0.0067.0 0.986 0.944 0.877 0.778 0.645 0.480 0.300 0.130 0.0148.0 0.986 0.948 0.884 0.790 0.662 0.401 0.320 0.145 0.022

Inner cfrcumferentfal crack. N 0 fl\.sO1.0 0.9~ 0.~!4 0.658 0.502 0.360 0:Z4:f 0.T4S 0.0?3 0.0332.0 0.965 0.876 0.754 0.613 0.465 0.326 0.200 0.102 0.0483.0 0.973 0.903 0.800 0.672 0.527 0.380 0.239 0.123 0.0584.0 0.978 0.918 0.827 0.709 0.568 0.419 0.269 0.140 0.0675.0 o.sso 0.927 0.645 0.734 0.598 0.448 0.292 0.155 0.0746.0 0.982 0.933 0.857 0.752 0.620 0.471 0.311 0.167 0.Oa17.0 0.983 0.937 0.866 0.766 0.638 0.489 0.327 0.177 0.0868.0 0.984 0.941 0.872 0.760 0.651 0.504 0.341 0.186 0.092Inner circumferenti a1 crack Nm=OUOt~,0.942 0.601 0.620 0.435 0.266 0.128 0.037 -O.v~~ : g : 8 ~ ~ ~.964 0.867 0.727 0.562 0.387 0.224 0.097 0.0153.0 0.972 0.896 0.777 0.629 0.459 0.287 0.141 0.036 -0.0244.0 0.977 0.912 0.809 0.671 0.507 0.332 0.174 0.054 -0.0185.0 0.980 0.922 0.828 0.700 0.541 0.365 0,200 0.069 -0.0126.0 0.981 0.928 0.841 0.720 0.566 0.391 0.221 0.081 -0.007

7.0 0.983 0.933 0.651 0.735 0.585 0.412 0.239 0.092 -0.0018.0 0.984 0.936 0.858 0.746 0.600 0.429 0.254 0.102 0.004

Table 5., K/Ko in a line pipe with 00 a 30 in" h .. 0.375 in.

lJh~ 0.1 I 0.2 I 0.3 I 0.4 I 0.5 I 0.6 ! 0.7 I 0,8 I 0.9a/h+ Outer cfrctlllferential crack N. I{J. M..-OI t g ~.945 ~.517 ~.664 1~50B 10366 10.247 i~~147 g:~~ ~:g~.967 0.883 0.766 0.628 0.481 0.340 0.2093.0 0.976 0.912 0.817 0.696 0.554 0.405 0.257 0.132 0.0604.0 0.981 0.928 0.848 0.740 0.605 0.455 0.296 0.158 0.070'5.0 0.983 0.939 0.869 0.770


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Table 7. K/~ in a lfne pipe with 00 ~ 24 tn., h : 0.344 in.

YR-:tIO.l I 0.2 I 0.3 I 0.4 I 0.5 I 0.6 I 0.7 ! 0.8 I 0.9a/h+ Outer c1rcumferenti a1 crack N o a M",. .ot~I~ :~~~ : :~:~:;::~ :~~g: :~ g : ~ : h g : ~ ~ ~ g :~~ g : g ~ ~3.0 0.976 0.911 0.817 0.695' 0.553 0.405 0.257 0.132 0.0604.0 0.980 0.928 0.847 0.739 0.604 0.455 0.297 0.155 0.0705.0 0.983 0.938 0.868 0.769 0.642 0.493 0.329 0.175 0.0806.0 0.985 0.945 0.882 . 0.791 0.670 0.524 0.357 0.193 0.0887.0 0.987 0.950 0.892 0.807 0.692 0.549 0.380 0.209 0.0978.0 0.987 0.953 0.899 0.819 0.709 0.570 0.400 0.223 0.104

Outer circumferential crack N . . . . Q K . J I OI t g 1 9 : ; : : I g : ~ : g : m : g : m g:~~ g : m g : r o ~ - ~ : g ~ 5 : g : g ~ ~3.0 0.975 0.905 0.798 0.657 0.492 0.318 0.164 0.048 -0.0214.0 0.980 0.923 0.832 0.707 0.551 0.376 0.209 0.072 -0.0125.0 0.983 0.934 0.854 0.741 0.594 0.421 0.246 0.094 -0.0046.0 0.985 0.941 0.869 0.765 0.626 0.457 0.277 0.114 0.0057.0 0.986 0.946 0.880 0.783 0.651 0.485 0.303 0.131 0.0138.0 0.987 0.950 0.888 0.797 0.670 0.508 0.325 0.146 0.020Inner ci rcumferentt a1 crack N o ' 0 M " , . . Ql ~ : g 1 9 : ; ~ g : ~ j 7 g : ~ ~ ~g :m Ig: :~ g : m ~:~~~: ~ b ~ g : g ~3.0 0.974 0.904 0.803 0.675 0.530 0.383 0.241 0.124 0.0584.0 0.978 0.919 0.831 0.714 0.573 0.423 0.271 0.141 0.0665.0 0.981 0.929 0.849 0.740 0.604 0.453 0.296 0.156 0.0746.0 0.983 0.935 0.862 0.759 0.628 0,477 0.316 0.168 0.0807.0 0.984 0.940 0.871 0.773 0.646 0.497 0.332 0.179 0.08618.0 0.985 0.943 0.878 0.784 0.660 0.513, 0.347 0.189 0.091Inner circumferential crack N ..a IY 'OIU I g : ; 6 ~ ~ : : i B ~:~~:5:;g : ; b ~:~~~: g ~ ~ - g : g l ~ : g : g ~3.0 0.973 0.897 0.782 0.634 0.463 0.291 0.143 0.037 -0.0244.0 0.977 0.914 0.813 0.677 0.513 0.337 0.177 0.055 -0.0185.0 0.980 0.924 0.833 0.706 0.548 0.372 0.204 0.070 -0.0126.0 0.982 0.931 0.846 0.728 0.575 0.399 0.227 0.084 -0.0077.0 0.983 0.936 0.856 0.743 0.595 0.421 0.246 0.095 -0.0028.0 0.984 0.939 0864 0.755 0.611 0.439 0.261 0.105 0.003

Table 8. K/Ko in a line pipe with 00" 20 tn., h .. 0.344 in.y n ~ 10.1 I 0.2 I 0.3 10.4 I 0.5 I 0.6 I 0.7 I 0.8 I 0.9a /h+ Outer c1rcumferential crack N..; a M", -O1.0 g: ;6~0811 10.~~ o.~~ I g : 4 8 l I O : ~ 4 0 i g : 2 i b O : b 7 g : g : ~.0 0.882 0.766 0.6283.0 0.975 0.911 0.816 0.695 0.553 0.405 0.258 0.133 0.064.0 0.980 0.927 0.846 0.737 0.603 0.454 0.297 0.156 0.0715.0 0.983 0.937 0.866 0.767 0.640 0.492 0.329 0.176 0.0816.0 0.985 0.944 0.879 0.788 0.667 0.522 0.356 0.193 0.0907.0 0.986 0.948 0.889 0.803 0.688 0.546 0.378 0.209 0.0988.0 0.987 0.951 0.896 0.815 0.704 0.565 0.396 0.222 0.105

Outer circumferent ial crack N..O H.~O~ :gI g :: : I g :~ I g :~~ f~::gt~~g :m ,g:m - ~ : ~ ~ l : ~ : g ~ ~3.0 0.975 0'.905 0.797 0.657 0.491 0.318 0.164 0.048 ~0.0204.0 0.9SO 0.922 ' 0.830 0.705 0.549 0.376 0.209 0.073 -0.0115.0 0.982 0.933 0.852 0.738 0.591 0.419 0.245 0.095 -0.0026.0 0.984 0.940 0.867 0.762 0.622 0.454 0.276 0.114 0.0067.0 0.986 0.945 0.877 0.779 0.645 0.480 0.300 0.131 0.0148.0 0.986 0.948 0.884 0.791 0.663 0.501 0.320 0.145 0.022

IMer circlll lferential crack N . IQ , M.-OI t g I g : ; 6 ; I g : m I ~ : ~ ; ~ g :m g : ~ ~ g : ~ ~ :t~6 g : ~~~g :g~3.0 0.973 0.903 O.SOl 0.672 0.527 0.380 0.240 0.124 0.0584.0 0.978 0.918 0.828 0.709 0.569 0.419 0.269 0.141 0.0675.0, 0.980 0.927 0.845 0.735 0.599 0.448 0.292 0.155 0.0746.0 0.982 0.933 0.857 0.753 0.621 0.471 0.312 0.167 0.0807.0 0.983 0.938 0.866 0.766 0.638 0.490 0.327 0.177 0.08618.0 0.984 0.941 0.873 0.777 0.652 0.505 0.341 0.186 0.092Inner circulllf'erent1al cnek N ~..a H .J IO! t g :~:;~g : ~ j I g : ~ ~~g :;~ ~g :~ Ig :m g:~~ w g : m :g :g;~3.0 0.972 0.896 0.779 0.629 0.459 0.287 0.141 0.036 -0.0244.0 0.977 0.912 0.809 0.672 0.507 0.332 0.174 0.054 -0.0185.0 0.980 0.922 0.828 0.700 0.541 0.366 0.200 0.068 -0.0126.0 0.981 0.928 0.841 0.720 0.566 0.392 0.222 0.081 - 0 . 0 0 77.0 0.983 0.933 0.851 0 . 7 3 5 0.586 0.412 0.239 0.092 -0.0018.0 0.984 0.936 0.858 0.747 0.601 0.429 0.254 0.102 0.004

shell theory and three dimensional fmite element resgiven in Ref. 18. The agreement appears to be qsatisfactory.The normalizing stress intensity factor Ko whappears in the tables is the corresponding plane strvalue for an edge-cracked plate of thickness h and crdepth Lo and is given bytc, = x; = ~ '" .jhgt(Lo/h),

&W = v'1i1(1.1216 + 6.5200~2- 12.3877~4+ 89.05- 188.6080~8+ 207.3870~10- 32.0524~12),

~=Lo/h,for membrane loading, and

gb(~) = V1i1(1.1202 - L8S72~ + 18.0143~2- 87.3851~3+ 241.9124~2 -319.9402~5

+ 168.0105~6), ~ = Lo/hfor (local) bending.Tables 1 to S give the Mode I stress intensity facat the maximum penetration point L = Lo of a seelliptic surface crack defined by

L2 X22 -2+ -2 = 1;or L = Lo sin 1>,X2 a cos 1>.Lo aThe stress intensity factor at other locations alongcrack front may be obtained by using the followingproximate formula:K(1)) = K (i)[1 - (1 + cos 21>)(0.2323 - 0.0615 ~ owhere K(7r/2) is given by Tables I-S.Ifthe stress state in the crack region isnot "uniformit may be decomposed into appropriate membranebending resultants which are functions of X 2 Forample, in the case of gross bending Mo , the local restants are given by Eqs, 5 and 6. In this case, too, sigenerally X22 2R2, the stress intensity factors mbe approximated by the uniform loading results gi

in the tables by assuming that N"" = MoRh/ I and= Moh3/121. Some sample results showing the effeccurvature (i.e., of the term involving X 22/2R 2 in Eqand 6) in the case of gross bending are given in TablIn this table, too, the respective normalizing stresstensity factors are given by Eqs, 7 and 8. Note thatthe pipe dimensions under consideration the resultsthree to, four orders of magnitude smaller thancorresponding stress intensity factors under unifoloading.2.2 Stress Intensity Factors for a Through CrackIf the geometry of the cracked shell or the exter


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Table 9. The stress intensity factor ratio 104(1(/~) at the deepest pene-tration point of a semf-el1fptie outer circumferential crack ina pipe subjected to parabolic loadings. OD 20 In., h 0.344 tn.,v = 0.3.

~ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.91.0 0.024 0.062 0.0130 0.076 0.063 0.043 0.026 .012 0.0041.5 0.032 0.089 0.129 0.138 0.121 0.089 0.055 0.026 0.0102.0 0.083 0.247 0.379 0.432 0.400 0.311 0.200 0.096 0.0352.5 0.175 0.538 0,663 1.032 1.003 0.813 0.542 0.271 0.0973.0 0.320 1.006 0.673 2.077 2.098 1.764 1.217 0.625 0.2273.5 0.529 1.699 2.696 3.711 3.875 3.366 2.396 1.263 0.4674.0 0.S10 2.645 4.616 6.076 6.530 5.843 4.277 2.312 0.8754.5 1.169 3.882 6.904 9.302 1Q.261 9.429 7.088 3.928 1.5265.0 1.618 5.429 9.822 13.514 15.254 14 .366 11.064 6.281 2.512

~ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 O.S 0.91.0 0.025 0.069 0.097 0.098 0.082 0.057 0.032 0.013 .0021.5 0.033 0.100 0.154 0.173 0.157 0.119 0.072 0.311 0.0662.0 0.127 0.402 0.659 0.766 0.756 0.604 0.385 0.175 0.0432.5 0.184 0.602 1.032 1.290 1.300 I . D B Z 0.723 0.344 0.0933.0 0.336 1.129 1.997 2.591 2.715 2.358 1.628 0.805 0.2363.5 0.555 1.899 3.453 4.620 5.003 4.491 3.207 1.641 0.5184.0 0.850 2.956 5.495 7.547 8.410 7.7130 5.724 3.026 1.0154.5 1.229 4.333 8.205 11.531 13.182 12.521 9.468 5.154 1.8285.0 1.698 6.053 11.653 16.709 19.540 19.020 14.751 S.251 3.077

loads are not properly symmetric, then Mode II andMode III components of the stress intensity factorwould no longer be zero. Furthermore, in this case thecrack initiation (for example, under cyclic loading)would be in a plane other than meridional or circum-ferential. The problem may be important in spirallywelded line pipes having weld defects. The solution ofsuch problems requires the formulation of the mostgeneral crack problem in cylindrical shells, namely thatof an arbitrarily oriented crack under general non-symmetric loading conditions. The solution of thisproblem and extensive results for a through crack aregiven in Ref. 25 . Again,selecting the local coordinatesystem as in Fig. 21, for a spirally oriented crack whichoccupies Xl = 0, -a < X2 < a, -h/2 < Xs < h/2 theModes I, II, and II I stress intensity factors around thecrack tip X2 = a may be defined as follows:

KI(XS) = lim [21r(X2 - a )]l /2 0 '1 l( 0, X2,XS), (11)X2-a

Kn(X 3) = lim [21r(X2 - a)]l/20'dO,X2,Xa),X2-a

(12)Km(Xs) = lim [21r(X2 - a )jI /2 0 '1S (0 ,X2,XS ),

X2-a(13)

where O'ij , (iJ = 1,2,3) are the stress components re-ferredto XI, X 2, X a coordinates (see Figs. 2 and 21 forthe coordinate system).In the shell analysis the external loads are given interms of stress and bending resultants. Therefore, theproblem may be solved more conveniently by assumingonly one of the five resultants to be nonzero at a time.The general solution may then be obtained by a suitablesuperposition. As in the symmetric case, there is onestress intensity factor associated with each of the fiveresultants. The stress component corresponding to the

membrane, bending, and transverse shear resultantswhich may be applied to the cylinder along the crack aregiven byo-. = Nll/h,(Jb = 6M ll /h2,(Js = N12/h,

a, = 6M12/h2,O'v = (3/2)V1/h. (14)The nominal stresses defined by Eq. 14 will be desig-nated as "membrane," "bending," in-plane "shear,""twisting," and "transverse shear," and N11,M11,N12,M12, and VI are (a measure or amplitude of) the corre-sponding crack surface loading.Some sample results for a pipe which contains athrough crack along a 450 spiral and which is subjectedto torsion are given in Table 10. In this case the mem-brane resultant N11 is the only nonzero load component.However, because of the lack of geometric symmetrynone of the other stress intensity components vanishes.The normalized stress intensity factors k im, (i =m ,b ,8 ,t ,V ) given in the table are related to the Modes I,II, and III stress intensity factors at the crack front asfollows:

kmm = KI(O)/(Jm.J""if f i ,kb m = [K I(h /2) - K I(O )]I(Jm v;;:a,ksm = Kn(O)/O'm.J""if f i ,k tm = [ Kn (h /2 ) - K n(O )]! O'm .J ""iffi,kum =Km(O)/(Jm6a,O'm =Nu/h, (15)

where h, 2a, and R are again the thickness, the cracklength and the mean radius. Needless to say, in thisinplane membrane loading K I is the dominant stressintensity factor; the components k im, (i = bs ,t,v) rep-resent the coupling effects. Note that as R ......o theproblem reduces to that of a flat plate, kmm becomes 1and the remaining stress intensity componentsvanish.In the special case of a circumferential through crackshown in Fig. 2 , ifthe loading is symmetric with respectto the plane of the crack, then Ku and KIll would bezero. For this practical case the membrane and bendingcomponents of stress intensity factors for a cylinderunder uniform tension Nu (or gross bending Mo) aregiven in Figs. 3 and 4. In these figures the shell param-eter A 2 is defined by (see Fig. 2)

(16)Similarly, if the cylinder is subjected to local bendingso that in the corresponding perturbation problem thesymmetric bending momentM uapplied to the cracksurfaces is the only nonzero external load, then theprimary and coupling stress intensity factors may bedefined as

kb b = [K I(h /2 ) - K I(O )]/O 'b v:;;:a ,kmb = K I(O )/O 'b .J""iffi (17)

where O 'b is given by Eq, 14. For the crack geometryshown in Fig. 2 the results are given in Figs. 5 and 6.


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Table 10. Stress. intensity factor ratios in an isotropic 0.2cylindrical s~ell containfng an inclined crack underuni fom membrane 1oad1 ng Nll ' '1/. 0.3. II 45.

~ 1 Z 3 5 10R115 .L097. _l..._302 1-544. ~"II ~.411"1 10 1.049 1.167 1.321 2.516k m n 15 .033 1.116 1.230 ..2.l99Z~ .020 1.072 1.148 4 1 88650 1.010 1.037 1.079 ,56300 .005 1.019 1.041 a ,337200 1.002 1.010 1.021 1.056 1.192':' .084 0.122 0.100 - . 69 - 761'10 0.058 O. 08 0.126 0.079 -0.2991 15 0.OA6 0.093 0.121 IB - 125k b m 25 0.032 0.073 0.104 0.132 0.023'.~O. , u < : u U.U'l~ 0.016 0.117 0.120100 0.0 0.051 0.OB9 0.139'zoo 0.033 0.062 O. 201 5 - - -0,190 -0.333 _0.51'1 10 -0. - )61 -0, 13 -0.22: -0.424ks m 1 15 -0.01 - )4- -0,081 -0.73 -0.3651 25 -0.00 - .025 -D.052_ _n.11Q -n.2891 50 -0.00 - .013 -0.02P; _O.nAA _n.1Q?1 100 -0.002 - 007 -0.014 -0.037 -01171 200 -0.001 - 003 - 00 -0.019 -0.0671 5 0.012 - 029 - 19 -0.432 -4.2-321 10 0.010 -0.008 -0.053_ -_0~219 - 1 . B 5 3

k t m 1 15 0.008 -0.002 - .031 -n.144 -1.2441 25 0.006 ..0_._002 -0.015 - o s : -0,7571 50 0.004 0.003 -0.004 - 3 -0.379_100 0.003 0;003 .000 - -O.JJ ili1 200 0.002 .002 .00 - -0.0'1 5 -0.051 -0.139 -0.261 - -2 6 01 10 -0.026 -0.070 - .13- - - .1 7k v m 1 15 -O.DJ8 -D.O'I] -0.088_ .. e ,201 -0.7361 25 -0.011 -0.0:'9 __~053 -0 12' -0.44'

1 I 5 C -0.005 -0.015 .. ~O~.Q28_ -0.062 " o . .zzl1 100 -0.0.03 -O.OIlB -0.014 -0.032 -0.1 200 ' -0.001, -0.004 __O.OOB -0.0'7 -0.056

Fig, 3-Membrane component of the stress intensity ratio for a cir-cumferentiaJly cracked cylinder under uniform membrane loading (N'1;< ! 0, Mll = 0)

Fig, 4-Bending component of the stress intensit y rat io for a circuferent ially cracked cylinder under uni form membrane loading (N'10,Ml1 = 0)

1 0

o 5Fig. 5-Bending component of the st ress intensit y ratiO for a circuferent ially cracked cyl inder under uniform crack surface bending mment (Nl1 =0, Ml1 ~ 0)


9/24

. . "

Fig. 6-Membrane component of the stress intensity ratio for a cir-cumferentially cracked cylinder under uni form crack surface bendingmoment (N11 =0, M11 ~ 0)

3. Crack Opening DisplacementIfthe crack in the cylinder wall is relatively deep andlong, the material is not brittle and the external loadsare sufficiently high, then there may be extensive plasticdeformations around the crack region. In this case un-like the J-controlled predominantly plane strain typecontained plastic deformation problems, in the mod-eling and analysis of the related ductile fracture processthe crack-structure geometry and the global mechanicsof the phenomenon would be expected to playa majorrole. For the initiation of ductile fracture some localdeformation or strength parameter at the crack frontmust reach a critical condition. On the other hand, forthe subsequent stable or unstable crack growth theglobal energy balance condition must be satisfied.Clearly, the phenomenon in shells is far too complex fora single parameter characterization. However, for thepurpose of analyzing and correlating the experimentalresults an appropriate single fracture mechanics pa-rameter has obvious advantages. Among the required

features of such a parameter one may mention the fol-lowing: (a) The parameter must be sufficiently repre-sentative of the intensity of the local deformation state,(b ) its theoretical evaluation must be relatively insen-sitive to the accuracy of the continuum modeling of therelated elastic-plastic crack problem, (c ) the corre-sponding mechanics problem should be analyticallytractable and (d) it must be a relatively simple and ac-curately measurable quantity.In our view, for the shell problem the crack openingdisplacement (COD) seems to come closest fulfillingthese requirements. Whatever the actual mechanics ofthe fracture process, the intensity oflocal deformations

may be looked upon as an acceptable measure of thefracture resistance as well as the intensity of the appliedloads. COD is certainly a reasonably good representativeof the local deformation state and can be measured ac-curately. At the same time it is a global quantity in thesense that it represents the integrated effects of theinelastic deformations in the crack region. Thus, it maybe assumed that the calculated value of COD would notbe as sensitive as some other fracture parameters to thedetails of the elastic-plastic modeling of the part-through crack problem in shells.The continuum plasticity problem for the shells ap-pears to be analytically intractable. In this study amodified version of the standard plastic strip model willbe used to model the problem. This is nothing but afully-plastic line spring model. In the model it is as-sumed that the net ligament and a certain region aroundthe crack is fully yielded, with the size p of the yieldzones as well as the membrane and the bending resul-tants N and M in the yield zones being unknown. In theformulation of the shell problem these three additionalunknowns are accounted for by the conditions that themembranes and bending components of the stress in-tensity factor be zero at X 2 = 'f (a + p) and Nand Msatisfy a yield condition, namely

(18)(19)(20)

Km(a+p)=O,Kb(a + p) = 0,_E_+lML = 1hl1F h2(IF

where I1F is the "flow stress" of the material.In the particular formulation of the shell problemsuch as that described in Ref. 21, the calculated quan-tities are the crack surface displacement Ul(0,X2) andthe crack surface rotation fh (O,X2) at the neutral sur-face. Thus, in the symmetric problem under consider-ation, the crack opening displacement at any point(0,X2,XS) on the crack surface may easily be obtainedfromo(X2,XS) = 2Ul(+0,X2) + 2XS!31(+0,X2),( - a


10/24

00 =30Hh -O.S"a/h -i.s

5

o 1.0Fig. 7-COD VS. ( folaF for a 30-in. diameter pipe

elastic response of the cracked shell. For greater valuesof (J'olo'p as expected the relationship is severely non-linear and, for given crack and shell dimensions, at acertain value of the stress ratio COD behaves nearly inan asymptotic manner. From the viewpoint of the me-chanics of the problem this behavior may be interpretedas some kind of instability phenomenon.

00 =30"h.=0.5"a/h2.0

Fig. 8-COD vs. a of a F for a 30-in. diameter pipe

Fig. 9-COD VS. a01 aF for a 3D-in. diameterpipe

4. The Pipe ExperimentsThe experimental work on a 20-inch line pipe wundertaken primarily to verify the validity of somethe analytically obtained elastic and elastic-plastic rsults for a part-through circumferential crack andperform crack morphology studies. The pipe was testeunder four-point bending. The dimensions of the pipand the locations of the loading pads and the crack ashown in Fig. 11.

00 = 30"h =0.5"a/h4

Fig. 10-COD VS. ( fof( fF for a 30-in. diameter pipe


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p p-jLSI"1----871"1----87;"----1.'1" I--,

Fig. ii-The geometry and dimensions of the pipe specimens

4.1 Stresses in a Pipe under "Four-Point" BendingIn solving the crack problems in pipes it was indicatedthat for a given loading condition the stresses in the pipewithout the crack are known. The crack problem wasthen solved under self-equilibrating loads applied to the

crack surfaces only. In the problem under considerationthe pipe is under "four-point" bending (Fig. 1). Gen-erally in such problems it is assumed that the pipe issubjected to gross bending and the stresses may be ob-tained by treating it as a "beam." However, in mostcases, it is necessary to verify the results given by thebeam theory by carrying out a somewhat more realisticstress analysis of the pipe and by considering the detailsof the loading fixtures.At the four points shown in Fig. 11 the loads wereapplied to the pipe by 6-in. wide semicircular saddles.To prevent a possible collapse ofthe pipe wooden blocksof 4 in. X 4 in. cross-section were inserted into the pipeat the load locations (Fig. 12). Other relevant dimen-sions are shown in Fig. 11.A shell theory was used to calculate the stresses in thepipe. (KSHEL developed by Professor A. Kalnins atLehigh University.) This is a numerical technique inwhich all field quantities including the external loadsare expanded into Fourier series in 0 and a segmentalintegration is used in the axial direction. The shellequations are expressed in terms of a system of firstorder differential equations. The resulting "two-pointboundary value problem" is then solved by reducing itto an initial value type problem. In order to avoid ahighly complicated contact problem, the form of the"contact" stresses at the locations of the loads was as-sumed beforehand (see Fig. 12). Following were themain assumptions: (a ) the contact at all locations isfrictionless, (b ) the pressure distribution under thesaddles is independent of the axial coordinate and hasa cosine distribution in 0, and (c) the pressure distri-bution between the wooden blocks and the shell isuniform. Thus, the transverse load N(8) applied to thepipe would be of the form shown in Fig. 12, where N(O)= No cos 8,Nl is unknown, and () 1 = c/2R 0;;: 0.2 rad., c= 4 in. being the width of the wooden block. From theequilibrium condition No is found to be

; ,I : :: :L '~i~~~

Pressure Profile Under theSoddleN = (No cos e)

Wooden Column UnderLoad PointTransmitted by Column(No -Nt)e }

N

(b)Fig. 12-Assumed dist ribution of the appl ied load through the saddlesand the wooden blocks in the pipe specimens

5 ",/2 N(8) cos 0 R d() =P,No = 2PhrR.-",/2 (23)The load N 1 is determined from the following dis-placement compatibility condition:W(O) + Weir) = (Jw1w , (24 )Ew

where W() is the radial displacement in the shell(positive if outward), (Jw is the stress in and lw and Eware the length and the Young's modulus of the woodenblock.In the analysis the loading condition shown in Fig. 12is used only in the interior load locations x = =F15 in.(Fig. 11). For simplicity the pipe ends were assumed tobe "simply-supported," that is, at x = "fl02 in. it wasassumed that ,(N",q,,M,,,q,,N,,,o , W ) = 0, (25)where and 0 are respectively the axial and the cir-cumferential coordinates and Nij and Mij (iJ = ,() arethe membrane and bending resultants.Some calculated results for the dimensions shown inFig. 11 are given in Figs. 13-15. Fig. 13 shows the8-distribution of the circumferential stress (Joo and theaxil stresses (Jxi and (Jxo in the pipe at the plane of

Fracture of Pipelines and Cylinders 11


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STRESS. psi/lb - -X'"O

Fig. 13-Circumferential variation of the hoop (U8o) and the axialstresses (uxi , Uxo ) at x = 0 plane in the pipe under "four-pointbending"

symmetry x :::: where the subscripts 0 and i stand forpoints on the "outer" and the "inner" surface of theshell, respectively. O"xo ~ (fxi implies local bending of theshell wall; that is,

lI,in.

- - Beam--- Shell

,,III,IIIIIIIIIII~

Fig. 14-Variation of the maximum axial stress (0 " }r / h2 is approximately 3.7% of the membrastress (fm = N < I > " , / h ) . Similarly, the difference betwethe stresses at the crack location obtained from the sand the beam theories is approximately 3.1% ofnominal value (Fig. 14). Therefore, in the pipe unconsideration using the beam theory to calculatestresses would involve no substantial error.Fig. 15 gives some idea about the ovalization ofpipe at x = plane. In this figure W(O) is the radcomponent of the displacement (positive if outwarItshould be emphasized that the problem was solvfor P = 1 by using the actual dimensions of the pgiven in Fig. 11.Thus all quantities shown in Fig. 13are per unit applied load and are not dimensionless.4.2Mechanical Properites ofthe MaterialBefore testing, specimens were cut from the pipethe longitudinal direction to obtain simple mechanicproperties of the material. A sample result ofthe tentests is shown in Fig. 16. Even though the material wnominally designated as being X60 the yield and umate strengths were found to be UYS ~ 68 ksi, UUL82.9 ksi. Various views of the ruptured tensile specime


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. . . . . .S TRE S S ( X SI )60

60

40

20

00

~E" 30.000 KSlCTvs" 88 kSlerUL" 82.9 kSI%[LONG"26.~%CTF' CTyS+ O.S(CTUL - ern)

0.024STRAIN (;.1;0)

0.032.008 0.016

Fig. 16-MechanicaJ properties of the pipe material. nominally des-ignated as X60, obtained from a tensile specimen cut parallel to the axisof the pipe (i.e. the rolling direction)

and are shown in Fig. 17. The most striking feature ofthe fracture surfaces was the severe delamination of thematerial parallel to the pipe surfaces. There was alsosevere necking in both thickness and circumferentialdirections before fracture.Specimens were also cut from the pipe in the longi-tudinal direction for Charpy V-notch experiments. In

z

(a l

o

( b)

Fig. 17-Various views of the X60 tensile specimen cut from the pipe;r , ( },and z refer to the radial, c ircumferential, and axial directions, re-spectively

these specimens the notch was cut in the circumferential(or 8) direction (that is, the loads were applied in 8-zplane, see Fig. 17). The test results are shown in Fig. 18where the solid line represents an approximate fit to thedata. The results are characteristic of pipeline steels,namely they indicate relatively high toughness, and riodistinct shelf values.4.3 The Fatigue ExperimentsThe length and load point locations of the pipespecimens shown in Fig. 11were determined largely toaccommodate the capacity of the Amsler hydraulic jacksused for loading the specimens. The machine capacityhad to be sufficient to produce a bending moment in theunnotched pipe near the full yield or "hinge" value ofthe moment. Another consideration was the bucklingof the pipe on the compression side. Since prematureelastic instability could spoil the entire program, someelementary buckling calculations had to be made. First,the following empirical formula developed for the elasticbuckling of thin shells under bending was used to cal-culate an equivalent critical stress26:

Eh(Jcr = RV3(1 _ p2 ) [1- 0.731(1- e - < I ] ,

1r p = 16 v ' R l F i , (27)where (Jcr is the critical stress for elastic buckling initi-ation, R the mean radius, h the thickness, and E, parethe elastic constants. For the 20-in. diameter steel pipeunder consideration (Eq. 27) gives (Jcr ~ 500 ksi indi-cating no danger of buckling according to (Eq. 27).A second calculation was made by assuming that thepipe is under axial compression. Following Ref. 27 thecritical stress for this case is given by

ENERGY (ft. -lb.)

. Fig. 18-Results of the Charpy tests for the X6D pipe material


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(28)1 +0.004~

(fPLwhere (fPL is the proportionality limit of the material.Equation 28 takes into account initial imperfections(i.e., deviations of the shell surface from an ideal circularcylinder) and assumes that the deflections may not besmall. Taking (JPL = 58 ksi, Eq. 28 gives (fer ~ 205 ksi,It then appears that the elastic buckling should notpose a problem in the pipe tests. A circumferentialstarter notch was introduced to the pipe specimens byusing a l-in. diameter 0,025-in. thick abrasive disk. Twoor three overlapping initial cuts were made to have thedesired initial flaw size and particularly to create a"chevron" effect to shorten the crack initiation time.The pipe was then placed in the test frame with thecrack on the compression side and the notch wassubjected to a precompression stress of approximately75%of the yield strength of the material. The reason forthis was to further speed up the crack initiation process.A sketch of the pipe cross-section through the crackplane is shown in Fig. 19.The objective of the fatigue tests was twofold. Thefirst was to introduce a natural crack to the pipe wallprior to the fracture tests, The second was to collectsome fatigue crack propagation data in shells. Theprogrammed loading and crack front marking techniquewere used to collect the fatigue data, The same hy-draulic jacks were used for cyclic (at 250 cpm) and forstatic loading. The stress intensity factor calculated in

Machined Notch

Fatigue Crack

Net Ligament Tear

Circumferential Growth (delaminationsl

~ Braze

Fig. 19-A sketch of the fracture surface of pipe #6 showing the lo-cations of var ious specimens used in the scanning electron micro-scope

Section 2 of this paper was used to correlate the fatiguresults. The limited results obtained for the pipes ashown in Fig. 20 superimposed on the results obtainefor flat plates with a part-through surface crack and fsingle edge notched specimens ofX-70 steel. The solline in the figure represents an approximate fit to thair data obtained by Vosikovsky for X70 specimens.PAll test specimens shown in Fig. 20 had the same cracorientation with respect to the rolling direction of thmaterial. Assuming that the materials X60 and X7have similar fatigue crack growth characteristics, Fi20 shows that the fatigue crack propagation rates in thpipe can be predicted from the data obtained frosimple two-dimensional specimens provided the theretical stress intensity factors for the pipe are availablConversely, in applications the calculated results sucas those given by Tables 1-8 may be used in a simplifiefatigue crack growth model obtained from the fatigucharacterization ofthe material (e.g.,Fig. 20) to estimathe crack propagation rate.4.4 Fracture TestsFollowing each fatigue experiment the transversloads P were slowly increased in order to observe thdevelopment of ductile fracture in the pipe (Fig. 11The fracture tests were carried out by using the samhydraulic jacks and the same load frame as used in thfatigue experiments. Strain gages were mountedvarious locations on the specimen to monitor the d

+ SPECS.6. S PE C 1 0 SPEC 14Q PL, l .TEtil PIPE-VOSIKOVSJ


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. .formations in the pipe and the relative magnitudes ofthe loads applied by the two jacks. A precalibrated clipgage was used to measure the crack mouth openingdisplacement. The outputs of gage 1 and the clip gagewere connected to an x-y recorder for continuous re-cording of the transverse load P vs. the crack openingdisplacement COD. An eight channel oscilloscope wasused as a back-up to the x-y recorder and to store theinformation on a disk. A digital data acquisition systemwas used to record the outputs from the strain gages atcertain values of the load. In order to detect the loadlevel corresponding to the initiation of net ligamentrupture, a photo cell was installed inside the pipe op-posite the crack, the pipe was darkened by blocking theends and the light was directed at the crack from out-side.Except for the initial fatigue-sharpening the cracks,the experimental procedure followed and essentially theresults found in this study are quite similar to thosereported in Ref. 29 by Wilkowski and Eiber. As in Ref.29, the loading technique used in the present experi-ments was basically "displacement-controlled." Thismeans that the experimental P vs. COD curves gothrough a maximum and then P starts decreasing as theload point displacement and COD increases. In a"load-controlled" experiment the maximum P thusattained would have been the fracture instabilityload.Altogether six pipes were tested. In two of the pipesthe fatigue crack was permitted to propagate throughthe entire pipe wall. In the remaining four some effortwas made to have a part-through fatigue crack ofvari-ous specific dimensions. The experimentally obtainedP vs. COD curves are shown in Figs. 21-26. Unlike someof the results given in Ref. 29 and except for pipe #2,

LOAD, kips

20" 4.28 Lo/h" 1 .0-- Exp.--- Theor. COD. in.

a 0. 1 0.2Fig. 21- Transverse load P vs. COD for pipe # 1

. :1.:": . ' , ~ d::!~

LOAD. kips -----------,..-" "V60

II40

20 20 = 1.688" Lo/h = 0.545-- Exp.--- Theor. C OD ,in.

aFig. 22- Transverse load P vs, COD for pipe # 2

the curves are all "smooth." That is, there were no"kinks" in the curves which would have been an indi-cation of "fracture initiation" or "net ligament rupture."The reason for this is believed to be fatigue sharpeningof the crack prior to static loading. Pipe #2 has a rela-tively short and shallow fatigue crack (2 a = 1.688 in.,Lo/h = 0.545). In this case the pipe "failed" as a conse-quence of structural instability (i.e., buckling) rather

50

20 "1.77in.Lo/h" 1.0

- EXPERIMENTAL--- THEORETICAL

0.1 0.2Fig. 23- Transverse load P vs. COD for pipe # 3


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LOAD, kipsThrough Crock

40

20 20 = 2.0625" LolL = 0.727-Exp.-- - Theor. COD,in.

o 0.1 0.2Fig. 24-Transverse load Pvs. COD for pipe # 4

than fracture instability. There was no evidence of crackgrowth in the pipe wall during the loading process on thetest bench. Severe nonlinearity observed in Fig. 22 inthe P vs. COD curve prior to reaching the peak load isan indication ofplastic deformations in the crack region,particularly, in the net ligament. The load fell offsharply upon reaching the structural instability value(Fig. 22). The actual development of the buckling of thepipe wall may be seen in Fig. 27.As pointed out earlier, the elastic instability in the

40

20 20' 2.0625" Loth. 0.773-Exp--- Theor. C OD : In .

o 0. 1 0.2.Fig. 25- Transverse load P vs. COD for pipe # 516

z . , , , , 1.97 in.Lo/h 0.68

- EXPERIMENTAL--- THEORETICAL

0.1Fig. 26- Transverse load P vs. COD for pipe # 6

pipes was not a likely mode of failure, that is, the cculated instability loads were much too high for tmaterial strength to sustain them. However, as sefrom Fig. 27, the instability observed in the pipe # 2inelastic buckling. Even though the peak load in tpipe # 6was the same as that in # 2,there was no visibsign of buckling in # 6. This may be due to the compance change in the pipe #6 resulting from the propgation of the through crack on the tension side anperhaps more likely, to the highly imperfection senstivity of the buckling process. An important factor in tinelastic buckling of pipes in the present study is tnearly rigid saddles used to transmit the load from t

WRC Bulletin 288Fig. 27-Buckling of pipe #2 on the compression side


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. .hydraulic jacks to the specimens. There was indeed anindication of slight buckling initiation in all pipes tested.They were all on one side and very near the saddle. Thelocal "bending" in the pipe wall near and at the leadingedge of the saddle seems to be one of the main factorsfor the reduction in the observed instability load. Thedegree of buckling instability was also responsible forthe difference in behavior of the measured load vs. CODcurves obtained from pipes #4 and #5 which hadnearly identical initial part through fatigue cracks. Thebuckling in the pipe #5 started at a smaller COD valuethan in # 4 which consequently resulted in the reduc-tion of the load at a comparatively smaller COD value.To give an idea about the comparative behavior of themeasured P vs. COD curves obtained from various pipesthey are reproduced in Fig. 28 in superimposed form.Except for the pipe #1which had a relatively long ini-tial through crack (and to some extent #3 which hadan initial through crack), the elastic behaviors (that isthe initial parts of the curve) in all pipes seem to bequite similar, whereas buckling played a major role inthe inelastic range. Of the six pipes tested only in one(pipe #6) there was no evidence of any structuralbuckling on the compression side. In this pipe after thenet ligament rupture the through crack continued togrow in a slowstable fashion. At some point the clip gageran out of space and fell and the test was terminated.When the test was stopped the crack (which had anoriginal length of 1.97 in.) was approximately 7 in. long.In the remaining pipes there was very small stablegrowth of the through crack.For the pipes tested, Figs. 21-26 also show the loadvs. CODrelationship obtained from the elastic-plasticanalysis described in the Section 3 of this paper. Forpart-through cracks shown in Figs. 24-26 four calcu-lated curves are given: one for the part-through crackwith the profile as given by the fatigue experiment, thesecond for the corresponding through crack and thethird and fourth for intermediate net ligament thick-nesses. For a given COD the curve based on the fatigue

120

100

80PIPE 2a Loth PM.X (I(ip~)1 4.280 1.000 9960 2 : 1.688 0.545 1093 1.770 1.000 1064 2.063 0.727 10540 5 2.063 0.773 106.56 1.970 0.680 109

20COD l in. )

00 0.1 0.2

Fig. 28- Transverse load P vs, COD in the pipes tested as 'reproducedfrom the xy recorder .

,; - ~I ,Ij.;;:~~

crack has the highest and that based on the throughcrack the lowest values of the load P. The curves cor-responding to the intermediate crack depth fall betweenthese two limiting values. The "net ligament rupture"point (or the load corresponding to the "through crack"initiation) PR is marked on the experimental curve(which was obtained from the photocell). Clearly, forloads greater than PR , the crack should be treated as athrough crack. The initial linear portion of the curvescorrespond to elastic loading. Between the load P Ncorresponding to the plastic necking or tearing initiationof the net ligament and PR corresponding to total netligament rupture, intermediate values of net ligamentthickness must be used to obtain the theoretical P vs.COD curve.Again, it should be emphasized that the ductilefracture process involving relatively thin-walledstructures and large flaws is very highly geometry-dependent and cannot be characterized by a single pa-rameter. The empirical or semi-empirical models de-signed for this purpose would generally be satisfactoryonly for the geometry they were developed. The simpleidea underlying the current study is that if one can de-fine or designate a certain parameter which is an ade-quate measure of the intensity of the applied loads andof the severity ofthe flaw under conditions of large scaleplastic deformations, and the value of which may notbe highly sensitive to the details of the elastic-plasticmodel assumed for the purpose of calculating it, thenthe asymptotic behavior of this parameter may be usedto estimate a gross stability load for the flawed com-ponent. As argued before in this paper COD comesperhaps closest to fulfilling the conditions of such aparameter.The asymptotic behaviors of the experimental andthe theoretical P vs. COD curves are ex.pected to bedifferent. The experimental curve is obtained from adisplacement controlled test and hence exhibits amaximum for the load (Pma:J. The theoretical CODcurves on the other hand possess a true asymptote (P= P m f j , X ) . These asymptotic values of P are the theoret-i ca l estimates ofthe instability load ineach pipe. For thesix pipes tested, Table 11 shows the comparison ofmeasured and estimated instability loads.An alternative way of presenting the results may beseen, for example, in Fig. 29 where the normalized COD

Table 11. Experimentally measured and theoretically estimatedfra~ture instability values of Transverse load P in pipes.

Pipe # 2a L/h (PllRtx)exp (Pmax)Theor.(kips'}'1 4.280 1.0 99 91

(*) 2 1.688 0.545 109 11 23 1.770 1.0 106 1034 2,063 0.727 105 1025 2.063 0.773 106.5 1026 1.970 0.680 T0 9 105

(*) Inelastic buckling, no fracture.


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COO/auF /E

Fig. 29-Normalized CODVS. moment ratio for pipe #6

(with respect to a (JF/E, a, (JF and E being the half cracklength, the flow stress, and the Young's modulus) isplotted against the moment ratio M /M p Here M is themoment applied to the pipe and M is the fully plastic(or collapse) value ofM which is given byS o 7r J:R+hI2Mp = 2(JF r2 sin 8drd8o R-h12

In all pipes tested for ductile fracture, at peak valueof the load the region of the shell containing the crackwas fully yielded and after the tests were terminatedpermanent deformations in the form of gross bendingwas observable. However, the elastic plastic analysis ofthe crack problem in the pipe was still valid. Consideredas a beam, the pipe still had a very large "elastic core"after the extremities were plastically deformed. Thus,unlike the plate problem under similar situations, therewas no "net section collapse." Also, in the elastic-plasticshell analysis the end points of the plastic zone in theplane of the crack extended into the elastic region in thepipe.Originally, it was thought that the net ligament wouldsuddenly become unstable and one may have some dy-namic effects on the tearing of the resulting throughcrack. However, as seen from COD vs. P records (Fig.

28) in the type of problems under consideration thetearing or necking-tearing process in the net ligamentis gradual and, for the circumferential cracks, stable.

Therefore, it does not seem to be practical to talk aboa "net ligament instability" load. Since the developmeof the through crack and its initial growth are stable, tonly meaningful instability load is that of the througcrack.4.5 Examination of Fracture SurfacesAs in plates containing a surface crack,24 in fatigucracked pipes subjected to fracture the evidence

considerable necking was observable in the net liment, particularly from the inside surface of the piopposite to the crack. The examination ofthe two halof the fractured specimen indicated that after thevelopment of the stretch zone the crack started alothe front and slowly propagated in thickness directioThe stability of this phase of the crack propagation wevident from the fact that prior to and during the nligament rupture (initiation of which was detectedthe photo cell) the load was still on the rise. The diretion of the crack propagation was perpendicular to tpipe surface. This may easily be seen from the fact thin the net ligament region the two halves of the fratured pipe wall were perfectly symmetric with respeto the original plane of the crack. There is every incation that upon the initiation of ductile tear the craprofile near and at the leading edge maintained(symmetric) V shape as it propaged through the nligament. Fig. 19 shows a sketch and Fig. 30 a photgraph of the fracture surface.One of the basic microscopic fracture mechanismthat almost always presents itself in cases of ductfracture is microvoid coalescence. The stress inducfracture and, in some cases, complex dislocation inteactions lead to the formation of microcracks or porwithin the stressed component. As the stress levelcreases these voids grow larger and start coalescingform a broad crack front. There are, roughly speakinthree main processes for void formation and coalescenwhich depend on the stress state existing in the coponent. Under simple uniaxial loading conditions, tmicrovoids will tend to form in association with fratured particles and/or interfaces and grow out in a plagenerally normal to the direction of the applied streThe resulting "equiaxial dimples" are believed torelated in some fashion to the fracture energy. Howevwhen the failure is predominantly influenced by shestresses, the voids that nucleate in the manner citabove grow and subsequently coalesce along planesmaximum shear stress. Consequently, these voids teto be elongated and result in the formation of paraboldepressions on the fracture surface. Finally, if the staof stress is that of combined membrane and bendinstresses, again the voids would be elongated, pointinback at the origin of the crack.From Fig. 30 one may easily distinguish three dferent zones on the fracture surface, namely the fatigcrack, net ligament rupture, and the through-thicknesshear fracture of the pipe wall. Even though both the nligament and the pipe wall were undergone ductifracture, their appearance were quite different. The n


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. .

C a )

(b)

(c)

F ig . 3 0- V ar io us vie ws o f t he fr actu re s ur fa ce o f the p ip e # 6 , L a =0.68h in., 2a = 1.97 in.

ligament appeared to have the structure of a fine-textured dimple fracture and had no signs of delami-nation. On the other hand, the through-thicknessfracture beyond the crack tips had much coarser sur-faces and had the appearance of shear fracture of alaminated material with clearly observable delamina-tion cracks (Fig. 30, see, also, Fig. 17b).For a closer examination of various fracture surfacesphotomicrographs were taken at various locations inpipe #6 by scanning electron microscope (SEM). Fig.19 shows a sketch of the fracture surface of the pipe # 6which indicates the locations of the samples (601, 602,603,604) used in SEM. The symbols r , c, and c.g. shownin Figs. 31-33 refer to the (outward) radial, circumfer-ential, and macroscopic crack growth directions, re-spectively. The views of the fracture surface oppositeto that used in SEM study are shown in Fig. 30. Thebraze shown in Fig. 19 was part of an effort to use anacoustic emission device for detecting the crack initia-tion. This attempt did not prove to be very reliable.The series of photomicrographs shown in Fig. 31 aretaken from the sample 603. Fig. 31a shows fatigue sur-face and part of the stretch zone (marked by A). The

" Lf: : !l .w :~c

stretch zone and the beginning of the tear region areshown inFig. 31b. Views further into the tear region areshown in Figs. 31c and d. The orientation of the dimplesin these figures indicate that the direction of the crackpropagation was radial. On the fracture surface therewas no evidence of shear fracture in the net ligamentpropagating in circumferential direction.Fig. 32 shows various views of the sample 602. Figs.32a and b show lowmagnification photomicrographs of

the whole pipe wall and a portion of the peculiar bandwhich was developed during the fracture process. In thesection of the pipe wall shown in Fig. 32a the regions ofmachined surface, fatigue crack, and the ductile fracturesurface including the delaminations and the "band" areclearly visible. The band is also seen in Fig. 32b. Figure32c shows a photomicrograph of the fatigue surface. Thestretch zone adjacent to the fatigue surface and thetransition region (to ductile fracture) are shown in Fig.32d. The "band" observed in Figs. 32a and b is alsovisible (this time in light color) near the crack tips in Fig.30 and is believed to be due to the interruption of thetest momentarily for manual readjustment of theloading jacks. * This may have caused a crack closure,resulting in "smearing" or "flattening" of the dimples.Itis, nevertheless, clear that the bands seen in Fig. 30along the entire thickness of the pipe correspond to thecrack front at a particular time during the propagationof the through thickness fracture.The photomicrographs of the sample 601 are shownin Fig. 33. Fig. 33a shows the view from a "valley" be-tween delaminations where the voids tend to be moreequiaxial. As one climbs along the side of a delaminationthe shearing effect becomes more visible and the di-mples tend to be more elongated (Fig. 33b). The dif-ference between the through-thickness tear in the netligament and the circumferential tear in the pipe wallseems to be purely a matter of stress state and geometricconstraints. Circumferential tear region contains highlypronounced delaminations which may have initiatedfrom the impurities inthe steel whereas the net liga-ment is completely free of such delaminations.5. ConclusionsOne of the main conclusions of this study is that inshell structures containing a relatively large initial crackgenerally the fracture instability load is highly depen-dent on the overall mechanics of the problem (i.e., the

geometry and loading conditions) as well as on thefracture resistance characteristics of the material, anda properly selected and fairly accurately calculatedparameter such as COD may be used to estimate theinstability load. By examining the results given in Figs.21-26 and in Table 11 it may be observed that the es-timate which may be obtained from the current analysis The drop of the load P to zero and reloading is not shown in Fig. 26. Fig.26wasreproduced from the oscilloscope record in which unloading and loadingwas ignored. However, the corresponding trace in the ;I:'Y recorder shows thatthe unloading and loading were perfectly elastic, followed the same straight line(in P vs, COD plane), and there was no sign of any discontinuity or kink in Pvs, COD record.


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c.g.

(c)e . g .

A

AB-(b)

1e . g .

(d) i.g .Fig. 31-Scanning electron micrographs (SEM) of the fracture surface at the locat ion 603 (see Fig. 19). (a) fat igue surface and stretch zone (A(b) stretch zone (A) (overload) and tear region, (e ) tear region, (0') further into the tear region (1000X). (I: outward radial direction, c: Circumferentiadirection, e.g.: crack growth direction)

20 WRC Bulletin 288


21/24


22/24

(a)e . g . (b) e . g .

Fig. 33-SEMs of the fracture surface at the location 601 (Fig. 19). (a) In a valley between delaminations, (b) along the side of a delaminat i(1000X)

appears to be sufficiently close to the instability loadand furthermore seems to be consistently conservative.One should also remark that if the tests were performedunder "load-controlled" conditions, qualitatively theexperimental P vs. COD curves would have been verysimilar in behavior to the theoretical curves, in that theywould not have had a maximum and would have beenasymptotic to P = Pmax lines.As indicated before there was considerable neckingin the net ligament from inside the pipe wall, the crackgrowth through the net ligament was perpendicular tothe pipe surface and was stable, and after the initiationof ductile tear the crack profile near and at the leadingedge maintained its (symmetric) V shape as it propa-gated. On the other hand, after the net ligament rupturethe through crack propagated essentially in shear modebetween planes formed by delamination cracks. Eventhough Fig. 30 appears to indicate a finer texturedfracture surface for the net ligament, scanning electronmicrographs showed that both the net ligament and thewall were undergone basically the same dimple frac-ture.The particular X60 line pipes which were subjectedto four point bending and which contained a part-through circumferential fatigue crack on the tensionside provided to be very highly resistant to ductilefracture. Even though it was possible to rupture the netligament (i.e., the pipe wall under the crack) in all butone of the pipe specimens, the gross failure under

gradually increased static bending occurred mostlya result of inelastic buckling of the pipe wall on tcompression side rather than fracture instability (iunstable crack growth) on the tension side. The one pispecimen in which the net ligament did not rupture ain which there was severe buckling on the compressioside conained a relatively shallow fatigue crack. In ospecimen there was no sign of buckling and the strutural failure resulted from fracture instability. Incases the theoretical model gave conservative estimatfor the fracture instability load.The research program, which was partially supporteby NSF and NASA-Langley, led to the publicationthe following articles:"Transverse Shear Effect in Circumferentially CrackCylindrical Shells," Quarterly of Applied Mathmatics, Vol. 37, 1979, pp. 239-258 (F. Delale andErdogan)."Line-Spring Model for Surface Cracks in a ReissnPlate," Int. J.Engng. Science, Vol. 19, pp. 1331-41981 (F. Delale and F. Erdogan)."Ductile Fracture of Pipes and Cylindrical Containerwith a Circumferential Flaw," J. Pressure VessTechnology, Trans. ASME, Vol. 103, pp. 160-161981 (F. Erdogan and F. Delale)."Elastic-Plastic Problem for a Plate with a ParThrough Crack under Extension and Bending," InJ. Engng. Science, Vol. 20, 1982 (M. B. CivelekanF. Erdogan).


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. . . I"Application of the Line-Spring Model to a CylindricalShell Containing a Circumferential or an AxialPart-Through Crack," J. Applied Mechanics, Trans.ASME, VoL 49, 1982, pp. 97-102 (F. Delale and F.Erdogan),"Stress Intensity Factors in a Hollow Cylinder Con-. taining a Radial Crack," Int. J.Fracture Mechanics,Vol. 19, 1983 (F. Delale and F. Erdogan)."The Elasticity Problem for a Thick-Walled CylinderContaining a Circumferential Crack," Int. J. FractureMechanics, Vol. 19, 1983 (H. F. Nied and F. Er-dogan)."The Transient Thermal Stress Problem for a Cir-cumferentially Cracked Hollow Cylinder," Journalof Thermal Stresses, 1983 (H. F. Nied and F. Er-dogan)."Elastic-Plastic Fracture of Cylindrical Shells Con-taining a Part-Through Circumferential Crack,"ASME paper to be presented at 1982Winter AnnualMeeting (F. Erdogan and H. A. Ezzat).

6. References1. Reich. M, and Esztergar, E. P."Compilation ofReferences. Data Sources.and Analysis Methods for LMFBR Piping System Components," BrookhavenNational Laboratory. BNL-NUREG-50650, March 1977.2. The Surface Crack, Swedlow, J. L., ed., ASME. New York, 1972.3. Raju, I. S . and Newman, J. C., Jr., "Improved Stress Intensity Factorsfor Semi- Elliptic Surface Cracks in Finite Thickness Plates," NASA TMX-72825, 1977.4. Erdogan, F., "Crack Problems in Cylindrical and Spherical Shells, ' inPlates and Shells and Cracks, Sih, G. C., ed., Noordhoff Int. Publ., Leyden,1977.5. Erdogan, F. and Ratwani, M., "Plasticity and Crack Opening Displace-ment in Shells," Int. J.Fracture Mechanics, Vol. 8,1972, pp. 413-426.6. Kobayashi , A.S. , Polvanich, N., Emery, A. F. and Love, W. J. , "Innerand Outer Cracks in Internally Pressurized Cylinders," J. Pressure VesselTechnology, Trans. ASME, Vol. 99,1977, pp. 83-89.7. Rolfe, S. T. and Barsom, J. M., Fracture and Fatigue Control in Struc-tures, Prentice- Hall, 1977.8. Fracture Toughness Evaluation by RCurve Methods, ASTM-STP 527,1973.9. Newman, J. D., Jr. "Fracture Analysis ofSurface and Through-CrackedSheets and Plates," J. Engng. Fracture Mechanics, Vol. 5, 1973, pp. 667-689.10. Erdogan, F., "Ductile Fracture Theories for Pressurized Pipes andContainers," Int. J. Pressure Vessels and Piping, Vol. 4,1976, pp. 253-283.11. Kiefner,J. F., Maxey, W. A.,Biber, R J. and Duffy,A R, "Failure StressLevels of Flaws in Pressurized Cylinders," in ASTM,STP 536, 1973, pp.461-481.12. Wilkowski , G.M. and Eiber, R J. , "Review ofFracture Mechanics Ap-proaches to Defining Critical Size Girth Weld Discontinuities," WRC Bull. 239,July 1978.13. Reed, R P., McHenry, H. I. and Kasen, M. B., "A Fracture MechanicsEvaluation ofFlaws in Pipeline Girth Welds," WRC Bull. 245, Jan. 1979.14. Crack Propagation in Pipelines, Inst. Gas Eng., London, 1974.15. Popelar, C., Rosenfield, A.Rand Kanninen, M. F., "Steady-State CrackPropagation in Pressurized Pipelines," J. Pressure Vessel Technology, Trans.ASME, Vol. 99,1977, pp. 112-121.

16. Erdogan, F., Delale, F. and Owczarek, J. A., "Crack Propagat ion andArrest in Pressurized Containers," J. Pressure Vessel Technology, Trans.ASME, Vol. 99, 1977, pp. 90-99.17. Krenk, S. ," Influence ofTransverse Shear on an Axial Crack in a Cylin-drical Sheil," Int. J. Fracture, Vol. 14, 1978, pp. 123-143.18. Newman, J. C. and Raju, I.S., "Stress Intensi ty Factors for InternalSurface Cracks in Cylindrical Pressure Vessels," NASA Technical Memoran-dum 80073, July 1979.19. McGowan, J. J. and Raymund, M., "Stress Intensity Factor Solutionsfor Internal Longitudinal Semi-Elliptical Surface Flaws in a Cylinder underArbitrary Loadings," Fracture Mechanics, ASTM, STP 677, 1979.20. Heliot, J., Labbens, R. C.and Pellisier- Tanon, A" "Semi-Elliptic Cracksin a Cylinder Subjected to Suess Gradients," Fracture Mechanics, ASTM, STP677,1979. pp. 341-364.21. Delale, F. and Erdogan, F., "Application of the Line Spring Model to aCylindrical Shell Containing a Circumferential or Axial Part-Through Crack."J. Appl. Mech., Trans. ASME, Vol. 49, 1982, pp. 97-102.22. Delale, F. and Erdogan, F., "Transverse Shear Effect in a Circumferen-tially Cracked Cylindrical Shell," Quart. Appl. Mathematics, Vol. 37, 1979,pp. 239-258.23. Erdogan, F. and Ratwani, M., "Fatigue and Fracture ofCylindrical ShellsContaining a Circumferential Crack," Int. J. of Fracture Mechanics, Vol. 6,1970, pp. 379-389.24. Erdogan, F., "Theoretical and Experimental Study of Fracture inPipelines Containing Circumferential Flaws," Final Report, U.S. Departmentof Transportation Contract DOT-RC-.82007, Aug. 1982.25. Yahsi, O. S., "Effect ofTransverse Shear and Material Orthotropy in aCylindrical Shell Containing an Arbitrarily Oriented Crack," Ph.D. Dissertation,Lehigh University, 1981.26. Timoshenko, S.P. and Gere, J. M., Theory of Elastic Stability, 2nd Ed.,McGrawHill, New York, 1961.27. Vinson, J. , Structural Mechanics: Behavior of Plates and Shei ls, J.Wiley, 1974.28. Vosikovsky, 0., "Effects of Stress Ratio on Fatigue Crack Growth Ratesin X70 Pipel ine Steels in Airand Salt Water ," Journal of Testing and Evalu-ation, Vol. 8, 1980, pp. 68-73.29. Wilkowski, G.M. and Eiber, R. J., "Evaluation ofTensile Failure ofGirthWeld Repair Grooves in Pipe Subjected to Offshore Laying Stresses," Journalof Energy Resources Technology, Trans. ASME, Vol. 103, 1981, pp. 48-55.

7. AcknowledgmentsThe authors greatly appreciate the contributions madeto this study by the following: F. Delale, H. F. Nied, M.B. Civelek, A. C. Kaya, S. Baskan, and R. Roberts.Special thanks are also extended to Professor R.Slutter for providing the facilities of Lehigh University'sFritz Engineering Laboratory for the experimental partof this study and for his valuable advice in setting upand in carrying out the tests, Professor A.W. Pense andDr. B. R. Somers for their help in the fractographicstudies, Dr. Orringerof the Transportation SystemsCenter for his advice and encouragement during theperiod of this research program, Dr. M. Lauriente ofD.O.T. for his interest and encouragement, Dr. J. H.Gross of U.S. Steel for providing the X-70 line pipematerial for plate experiments, and Stupp Brothers, Inc.of Saint Louis, Missouri for providing the X-SO linepipes for the pipe experiments.


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