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Assessment of inspection criteria and techniques for recertification of natural gas vehicle (NGV) storage.. Ribarits, S. G. 1992-12-31

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ASSESSMENT OF INSPECTION CRITERIA AND TECHNIQUESFOR RECERTIFICATION OF NATURAL GAS VEHICLE (NGV)STORAGE CYLINDERSByS. G. RibaritsB. A. Sc. (Mechanical Engineering), The University of British Columbia, 1983A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTERS OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIES(DEPARTMENT OF MECHANICAL ENGINEERING)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAApril 1992© S. G. Ribarits, April 1992In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature) Department of ^A-4 6"...-e- i 1-, (.7"...-/ AThe University of British ColumbiaVancouver, CanadaDate ^/1-i2/2/ e.-^/ °1 °12,DE-6 (2/88)AbstractIn-service natural gas vehicle (NGV) storage cylinders are subject to various formsof damage which can degrade structural integrity. In recognition of this, the CanadianTransport Commission (CTC) requires that cylinders be recertified for service every fiveyears.Current standards for NGV cylinder recertification are based on cylinder behaviourduring a hydrostatic test (pressurization to 1.67 x service pressure) and specify that acylinder should be removed from service if either the measured plastic expansion exceeds10 % of the total cylinder expansion or the cylinder ruptures. These criteria have beenestablished, for the most part, to ensure a minimum cylinder wall thickness during service.A serious deficiency of the current assessment criteria is that they do not specificallyaddress the possibility of sub-critical crack growth that may occur between inspectionsdue to the combined effects of aggressive environmental conditions and cyclic loading.Evidence of sub-critical crack growth in sectioned in-service NGV cylinders however, hasraised concerns over the effectiveness of the current recertification criteria and points tothe need to determine whether these criteria are adequate to ensure against in-servicefailure.The objective of this research is to evaluate the current standards for NGV cylinderrecertification.In this investigation, various fracture mechanics based methodologies are investigatedin view of their applicability to predicting cylinder rupture. Elastic-plastic finite elementmodels of untracked and cracked NGV cylinders are utilized to predict cylinder behaviour(volumetric expansion/crack opening displacement) during a hydrostatic test. Theseiianalyses, in conjunction with small scale critical crack tip opening displacement (CTOD)test results and full scale burst test results indicate that a critical CTOD approach.modified by a plastic collapse analysis for small defects can be used to accurately predictcylinder rupture.Analysis of full scale numerical and experimental results indicates that there existsa range of defect sizes which, if present in a cylinder, would not violate the currenthydrostatic test acceptance criteria (i.e., volumetric expansion/rupture). This finding,in conjunction with available fatigue crack growth estimates indicates that it is possiblefor cylinders containing this range of defect sizes to fail in service during the subsequentfive year inspection interval. This result indicates that current hydrostatic test criteriaare non-conservative.In view of the inability of current retest procedures to reject cylinders containingdefects which may lead to in-service failure, the feasibility of utilizing acoustic emission(AE) techniques for NM,' cylinder inspection is also investigated. Full scale cylinder AEtests indicate that it is possible for cylinders with similar defect sizes to exhibit markedlydifferent AE characteristics (i.e, hit rate, amplitude distribution, etc.) due to other AEsources such as corrosion. However, the potential for using AE techniques for inspectionof cylinders is discussed.iiiTable of ContentsAbstract^ iiList of Tables^ ixList of FiguresNomenclature^ xixAcknowledgement xxiii1 INTRODUCTION 11.1 Background ^ 11.2 Statement of Problem ^ 31.3 Objectives ^ 41.4 Scope of Research ^ 52 TECHNICAL BACKGROUND 72.1 Cylinder Geometry and Material Properties ^ 72.2 Environment ^ 72.3 Service Loading 92.4 Current Standards for NGV Cylinder Recertification ^ 92.4.1^Minimum Wall Thickness Requirement ^ 92.4.2^Hydrostatic Testing ^ 112.4.3^Hydrostatic Test Failure Criteria ^ 12iv3 FRACTURE MECHANICS METHODS APPLICABLE TONGV CYLINDERS^ 143.1 General Concepts  ^143.2 Linear-Elastic. Fracture Mechanics  ^153.2.1 Stress Intensity Factors for Through-Wall and Surface Defectsin Cylinders  ^173.3 Plasticity Corrected LEFM ^  203.4 Elastic-Plastic Fracture Mechanics  ^213.4.1 The CTOD Approach ^  213.4.2 J-integral Methods  ^333.5 Plastic Collapse Methods ^  363.5.1 Limit Load Analysis  ^363.5.2 Analytical and Numerical Estimates of CTOD for Surface Flawsin the Large Scale Yielding Regime ^  383.5.3 Erdogan and Ratwani Formulation  403.5.4 Line-Spring Models  ^423.6 Design Methods ^  453.6.1 The CTOD Design Curve ^  453.6.2 The Central Electricity Generating Board R6 Method ^ 473.6.3 AGA Model (Battelle Empirical Analysis) ^ 484 INVESTIGATION OF SMALL SCALE BEHAVIOUR^514.1 Numerical Modelling of CTOD Specimens ^  524.1.1 Specimen Design  ^524.1.2 Mesh, Boundary Conditions, and Loading ^ 534.1.3 Numerical Results  ^544.2 Experimental Measurement of Critical CTOD ^  554.2.1 Specimen Preparation ^  554.2.2 Test Procedure  ^564.2.3 Small Scale Test Results  ^575 NUMERICAL INVESTIGATION OF FULLSCALE BEHAVIOUR^ 615.1 Model Description  625.1.1 Material Properties  ^625.1.2 Cylinder/Defect Geometry ^  625.1.3 Finite Element Mesh  635.1.4 Loading  ^685.1.5 Calculation of COD, CTOD, and Volumetric Expansion ^ 695.2 Numerical Results ^  725.2.1 Crack Opening Displacement (COD) ^  725.2.2 Development of the Plastic Zone ^735.2.3 Crack Tip Opening Displacement (CTOD) ^ 745.2.4 Volumetric Expansion  ^756 EXPERIMENTAL INVESTIGATION OF FULLSCALE BEHAVIOUR^ 776.1 Introduction to Acoustic Emission  ^776.2 Experimental Setup and Procedure ^  796.2.1 Hydrostatic Testing Equipment  ^796.2.2 Acoustic Emission Monitoring Equipment  ^796.2.3 Test Cylinder Preparation  ^816.2.4 Procedure  ^83vi6.2.5 AE Settings  ^886.3 Results  ^906.3.1 Hydrostatic Test Results  ^906.3.2 Acoustic Emission Results  ^916.3.3 Burst Test Results  ^987 CORRELATION BETWEEN EXPERIMENTAL ANDNUMERICAL/ANALYTICAL FAILURE PREDICTIONS^1017.1 Finite Element Failure Predictions ^  1017.2 PD 6493 Failure Predictions  1037.3 CEGB R6 Method Failure Predictions ^  1057.4 Plastic Collapse Failure Predictions  1067.5 Battelle Empirical Analysis ^  1067.6 Choice of an Acceptable Failure Criterion for NGV Cylinders ^ 1087.6.1 Comparison of Fracture Mechanics Based Failure Predictions ^1087.6.2 Limiting Defect Sizes for Rupture of NGV Cylinders During aHydrostatic Test ^  1108 ASSESSMENT OF THE CURRENT STANDARDS FORRECERTIFICATION OF NGV CYLINDERS^ 1128.1 Summary of Findings ^  1128.2 In-Service Failure  1148.2.1 Subcritical Crack Growth ^  1148.2.2 Expected Mode of In-Service Failure ^  1168.3 Limitations of Current Standards for NGV Cylinder Recertification^1168.4 Interior versus Exterior Defects ^  117vii9 CONCLUSIONS AND RECOMMENDATIONS^ 1199.1 Conclusions ^  1199.1.1 General conclusions ^  1199.1.2 Specific conclusions regarding current and potential standards forNGV cylinder recertification ^  1219.2 Recommendations ^  1239.2.1 Recommendations to ensure NGV cylinder integrity ^ 1239.2.2 Recommendations for further work ^  123Bibliography^ 125Appendices 280A ELEMENT DESCRIPTION^ 280A.1 Two-Dimensional Eight-Node Isoparametric Solid Element ^ 280A.2 Three-Dimensional 20-Node Isoparametric Solid Element  280A.3 Eight-Node Isoparametric Thin Shell Element ^  281A.4 Element Stiffness Matrices ^  281B ELEMENTARY ELASTIC-PLASTIC FINITEELEMENT THEORY^ 282B.1 Solution Methods  282B.2 Plasticity Theory ^  285B.3 Incremental Plastic Strain ^  286B.4 Specializations for Bilinear Kinematic Hardening Materials ^ 287B.5 Notation ^  289C PROGRAM LISTINGS^ 290viiiList of Tables2.1 Nominal weights and dimensions of steel NGV cylinders. ^ 1342.2 Material properties of NGV cylinder (AISI 4130X) steel. 1342.3 Chemical composition of NGV cylinder (AISI 4130X) steel ^ 1342.4 Maximum contractural and typical levels of natural gas contaminants. . 1344.1 Initiation load and COD for small scale test specimens ^ 1354.2 Initiation CTOD for small scale test specimens. 1354.3 Limit loads and ratios of maximum test load to limit load for small scaletest specimens. ^ 1355.1 Run designation and defect sizes analysed in finite element analysis of fullscale cylinder behaviour ^ 1365.2 Predicted failure pressures (finite element analysis of full scale cylinderbehaviour). ^ 1366.1 Test cylinder designation / defect sizes ^ 1376.2 Measured test cylinder elastic expansion at 34.48 MPa (5000 psi). 1376.3 Measured and rejection test cylinder plastic expansions. ^ 1377.1 Measured and predicted cylinder burst pressures. 1387.2 Number of cycles/years to failure for test cylinders. ^ 1388.1 Comparison of expected modes of cylinder failure (finite element analysisof full scale cylinder behaviour) ^ 139ixList of Figures2.1 60 liter NGV cylinder geometry^  1403.1 Regimes of crack tip behaviour [79]  1413.2 Diagramatic illustration of approach used to analyse defects in cylinders. 1423.3 Membrane and bending components of stress magnification factor forthrough-wall defects in cylinders [44] ^  1433.4 Elliptical surface defect in a wide plate  1443.5 Irwin plastic zone correction [53]. ^  1453.6 Dugdale plastic zone correction [42].  1453.7 Center cracked plate geometry. ^  1463.8 Crack tip opening displacement (CTOD) and crack openingdisplacement (COD) ^  1473.9 Single edge notch bend specimen (SENB) geometry ^  1483.10 The J-integral. ^  1493.11 Models for CTOD in large scale yielding ^  1503.12 Plastic zone correction from Erdogan and Ratwani formulation [45]. 1513.13 Crack tip open displacement from Erdogan and Ratwani formulation [45]. 1523.14 Line spring model geometry [59].   1533.15 Relationship between actual defect dimensions and the parameter ,Ttfor surface defects [24] ^  1543.16 Relationship between actual defect dimensions and the parameterfor embedded defects [24]. ^  1553.17 Reduction factor for long defects in curved shells containing pressure [24] ^ 1563.18 The CEGB R6 failure assessment diagram [27]. ^  1574.1 Small scale CTOD specimen geometry. ^  1584.2 Small scale CTOD specimen clip gauge and potential drop lead locations ^ 1594.3 Finite element mesh for small scale CTOD specimen. ^ 1604.4 Small scale CTOD specimen COD and point load displacement. ^ 1614.5 Non-dimensional stress intensity shape factor for small scaleCTOD specimen. ^  1624.6 Non-dimensional crack opening displacement shape factor for small scaleCTOD specimen. ^  1634.7 Non-dimensional point load displacement shape factor for small scaleCTOD specimen. ^  1644.8 Crack depth versus number of cycles for small scale CTOD specimen. ^1654.9 daldAr versus AK for for small scale CTOD specimen. ^ 1664.10 Types of load-COD records [23] ^  1674.11 Normalized load versus COD for Specimens 1 and 5. ^ 1684.12 Crack growth versus load for Specimens 3. ^  1694.13 Crack growth versus load for Specimens 5.  1705.1 Flowchart illustrating approach to assess criteria for NGVcylinder recertification ^  1715.2 Idealized stress-strain relationship for 4130X NGV cylinder steel. ^ 1725.3 NGV cylinder/defect geometry. ^  1735.4 Finite element mesh for cylinder.  1745.5 Finite element longitudinal and hoop membrane stress. ^ 1755.6 Finite element longitudinal and hoop bending stress.  1765.7 Finite element mesh for submodelled region of cylinder ^ 177xi5.8 Elliptical crack profile and crack tip (finite element meshfor submodelled region of cylinder) ^  1785.9 Submodelled region of cylinder.  1795.10 Elliptical transformation used in construction of submodel mesh ^1805.11 Non-dimensional stress intensity factors for the 5 x 15 mm defect. ^ 1815.12 Calculation of COD, CTOD and point of rotation. ^  1825.13 Incremental volume bounded by origin and three points on element face. 1835.14 Elastic and plastic expansion components of total expansion ^ 1845.15 Crack opening displacement versus pressure for interior cracks ^ 1855.16 Crack opening displacement versus pressure for exterior cracks. ^ 1865.17 Crack opening displacement as a function of crack sizeat 20.69 MPa (3000 psi) ^  1875.18 Crack opening displacement as a function of crack sizeat 34.48 MPa (5000 psi) ^  1885.19 Development of plastic zone for interior cracks. ^  1895.20 Development of plastic zone for exterior cracks.  1905.21 Crack tip opening displacement, versus pressure for interior cracks ^ 1915.22 Crack tip opening displacement versus pressure for exterior cracks. . . ^1925.23 Crack tip opening displacement as a function of crack sizeat 20.69 MPa (3000 psi) ^  1935.24 Crack tip opening displacement as a function of crack sizeat 34.48 MPa (5000 psi) ^  1945.25 Limiting defect dimensions for failure at 34.48, 37.92 and41.37 MPa (5000, 5500 and 6000 psi) ^  1955.26 Finite element expansion of a NGV cylinder ^  1965.27 Predicted and measured elastic expansion.  197xii5.28 Plastic expansion as a function of pressure for interior cracks. ^ 1985.29 Plastic expansion as a function of pressure for exterior cracks. ^ 1995.30 Plastic expansion as a function of crack size at34.48 MPa (5000 psi) (hydrostatic test pressure). ^  2006.1 Schematic illustration of hydrostatic testing facility  2016.2 Schematic illustration of cylinder instrumentation. ^  2026.3 Characteristics of an acoustic emission hit ^  2036.4 Measured elastic expansion (test and control cylinders) ^ 2046.5 Hits past previous pressure during first cycle of cylindertests (30 dB < Amp < 70 dB). ^  2056.6 Hits past previous pressure during second cycle of cylindertests (30 dB < Amp < 70 dB). ^  2066.7 Hits past previous pressure during third cycle of cylindertests (30 dB < Amp < 70 dB). ^  2076.8 Hits past previous pressure during first cycle of cylinderretests (30 dB < Amp < 70 dB). ^  2086.9 Hits past previous pressure during second cycle of cylinderretests (30 dB < Amp < 70 dB). ^  2096.10 Hits past previous pressure during third cycle of cylinderretests (30 dB < Amp < 70 dB). ^  2106.11 Hits past previous pressure during first cycle of cylindertests (40 dB < Amp < 70 dB). ^  2116.12 Hits past previous pressure during second cycle of cylindertests (40 dB < Amp < 70 dB). ^  2126.13 Hits past previous pressure during third cycle of cylindertests (40 dB < Amp < 70 dB). ^  2136.14 Hits past previous pressure during first cycle of cylinderretests (40 dB < Amp < 70 dB). ^  2146.15 Hits past previous pressure during second cycle of cylinderretests (40 dB < Amp < 70 dB). ^  2156.16 Hits past previous pressure during third cycle of cylinderretests (40 dB < Amp < 70 dB). ^  2166.17 Hold time hits during first cycle of cylindertests (30 dB < Amp < 70 dB). ^  2176.18 Hold time hits during second cycle of cylindertests (30 dB < Amp < 70 dB). ^  2186.19 Hold time hits during third cycle of cylindertests (30 dB < Amp < 70 dB). ^  2196.20 Hold time hits during first cycle of cylinderretests (30 dB < Amp < 70 dB). ^  2206.21 Hold time hits during second cycle of cylinderretests (30 dB < Amp < 70 dB). ^  2216.22 Hold time hits during third cycle of cylinderretests (30 dB < Amp < 70 dB). ^  2226.23 Hold time hits during first cycle of cylindertests (40 dB < Amp < 70 dB). ^  2236.24 Hold time hits during second cycle of cylindertests (40 dB < Amp < 70 dB). ^  2246.25 Hold time hits during third cycle of cylindertests (40 dB < Amp < 70 dB). ^  2256.26 Hold time hits during first cycle of cylinderretests (40 dB < Amp < 70 dB). ^  226xiv6.27 Hold time hits during second cycle of cylinderretests (40 dB < Amp < 70 dB). ^  2276.28 Hold time hits during third cycle of cylinderretests (40 dB < Amp < 70 dB). ^  2286.29 Hit rate (hits/MPa) during first cycle of cylindertests (Amp > 30 dB). ^  2296.30 Hit rate (hits/MPa) during second cycle of cylindertests (Amp > 30 dB). ^  2306.31 Hit rate (hits/MPa) during third cycle of cylindertests (Amp > 30 dB). ^  2316.32 Hite rate (hits/MPa) during first cycle of cylinderretests (Amp > 30 dB). ^  2326.33 Hit rate (hits/MPa) during second cycle of cylinderretests (Amp > 30 dB). ^  2336.34 Hit rate (hits/MPa.) during third cycle of cylinderretests (Amp > 30 dB). ^  2346.35 Hit rate (hits/MPa) during first cycle of cylindertests (Amp > 40 dB). ^  2356.36 Hit rate (hits/MPa) during second cycle of cylindertests (Amp > 40 dB). ^  2366.37 Hit rate (hits/MPa) during third cycle of cylindertests (Amp > 40 dB). ^  2376.38 Hite rate (hits/MPa) during first cycle of cylinderretests (Amp > 40 dB). ^  2386.39 Hit rate (hits/MPa) during second cycle of cylinderretests (Amp > 40 dB). ^  239XV6.40 Hit rate (hits/MPa) during third cycle of cylinderretests (Amp > 40 dB) . ^6.41 First cycle amplitude distributions (P = 20.69 MPa (30006.42 First cycle amplitude distributions (P = 22.75 MPa (33006.43 First cycle amplitude distributions (P = 24.82 MPa (36006.44 First cycle amplitude distributions (P = 26.89 MPa (39006.45 First cycle amplitude distributions (P = 28.96 MPa (42006.46 First cycle amplitude distributions (P = 31.03 MPa (45006.47 First cycle amplitude distributions (P = 33.10 MPa (48006.48 First cycle amplitude distributions (P = 34.48 MPa (50006.49 First cycle amplitude distributions (cylinder retests,240psi)) . ^ 241psi)) . ^ 242psi)) . ^ 243psi)) . ^ 244psi)). ^ 245psi)) . ^ 246psi)) . ^ 247psi)) . ^ 248P = 20.69 MPa (3000 psi)). ^  2496.50 First cycle amplitude distributions (cylinder retests,P = 22.75 MPa (3300 psi)). ^  2506.51 First cycle amplitude distributions (cylinder retests,P = 24.82 MPa (3600 psi)). ^  2516.52 First cycle amplitude distributions (cylinder retests,P = 26.89 MPa (3900 psi)). ^  2526.53 First cycle amplitude distributions (cylinder retests,P = 28.96 MPa (4200 psi)). ^  2536.54 First cycle amplitude distributions (cylinder retests,P = 31.03 MPa (4500 psi)). ^  2546.55 First cycle amplitude distributions (cylinder retests.P = 33.10 MPa (4800 psi)). ^  2556.56 First cycle amplitude distributions (cylinder retests.P = 34.48 MPa (5000 psi)). ^  256xvi6.57 Cumulative hits versus pressure for burst test of Tank A ^ 2576.58 Cumulative hits versus pressure for burst test of Tank B ^2586.59 Cumulative hits versus pressure for burst test of Tank C ^ 2596.60 Cumulative hits versus pressure for burst test of Tank D  2606.61 Cumulative hits versus pressure for burst tests. ^  2616.62 Amplitude distributions for cylinder burst tests  2626.63 Tanks A, B and C following burst tests. ^  2636.64 Fracture origin (Tank A). ^  2646.65 Fracture origin (Tank D).  2646.66 COD versus pressure for burst test of Tank A ^  2656.67 COD versus pressure for burst test of Tank B  2666.68 COD versus pressure for burst test, of Tank C ^  2676.69 COD versus pressure for burst test of Tank D  2687.1 Actual versus predicted failure pressures (finite element/CTOD approach). ^  2697.2 Actual versus predicted failure pressures (CTOD design curve). ^ 2707.3 Actual versus predicted failure pressures (plastic collapse). ^ 2717.4 Actual versus predicted failure pressures (CEGB R6 Method). ^ 2727.5 Actual versus predicted failure pressures (Battelle Empirical Analysis). ^2737.6 Comparison of LEFM and plastic collapse failure criteria ^ 2747.7 Comparison of numerical CTOD and plastic collapse limitingdefect dimensions. ^  2757.8 Limiting defect dimensions for failure during a hydrostatic test. ^ 2768.1 Number of fueling cycles to failure as a function of initialdefect depth [13]. ^  2778.2 Limiting defect dimensions for in-service failure [13]. ^ 278xvii8.3 Limiting defect sizes for in-service and hydrostatic test failure ^ 279xviiiNomenclaturea^crack depth2c^crack lengthd inner diameterz^distance to knife edgesD outer diameterCOD crack mouth opening displacementCTOD crack tip opening displacementE modulus of elasticityJ J-integralK stress intensity factor1^Dugdale plastic zone lengthill^stress magnification factorp^pressureP loadr^radial distance from the crack tiprr^Irwin plastic zone radiusR^mean radiusS spanthickness{u}^displacement vectorV^volume, clip gage displacementW^widthxixY^form factorxxGreek symbolsstrainA^shell parameterPoisson's ratio0^non-dimensional CTODa^stressangle from plane of crack{9}^rotation vectorSubscripts and superscriptsel^elasticpl^plasticF^flowcircumferentialI^mode IIc^mode I critical1^longitudinalL^limitLY^ligament yieldNSY^net section yieldultimate tensileS^surface flawT^through-wall defectY^yieldxxiMiscellaneousAE^Acoustic EmissionCCP^Center Cracked PlateCGA Compressed Gas AssociationCT^Compact TensionCTC^Canadian Transport ComissionSENB Single Edge Notch BendAcknowledgementAbove all, the author would like to express his sincere gratitude to Dr. D. P. Romillyfor his many invaluable suggestions, academic guidance and continued support through-out this project. Without his on-going efforts, completion of this thesis would not havebeen possible.The author would like to thank Mr. D. Chow, Ms. C. Taggart, and Mr. R. Baerg forassistance while these individuals contributed to the on-going investigation of AE basedrecertification techniques. The author would also like to thank Dr. A. Akhtar, P. Eng.and Dr. G. Bhuyan, P. Eng. of Powertech Labs, Inc. for many informative discussionsrelated to structural integrity of NGV cylinders.Further, the author would like to express his sincere appreciation to the followingresearch engineers/technicians for continued assistance and invaluable advice throughoutthe course of this investigation.Mr. G. Rohling, P. Eng. Research Engineer (Computing support)Mr. A. Steeves, P. Eng. Research Engineer (Computing support)Mr. D. Bysouth Technician (Instrumentation)Mr. A. Schreinders Machine Shop supervisorMr. A. Abel Technician (Machine shop)Mr. T. Besec Technician (Machine shop)The author would also to express his gratitude to the Canadian Gas Association andB. C. Science Council for funding this research.Chapter 1INTRODUCTION1.1 BackgroundIn the last decade there has been a significant increase in the use of natural gas as analternative fuel. This trend, which can be expected to increase has resulted in a significantresearch effort to assess and improve the technology of natural gas fuel systems. Oneimportant aspect of this technology is the on-board storage of the product.While the current trend is towards use of aluminum liner/composite wound cylinders,a great many existing and new natural gas fuel systems employ seamless steel cylinders foron-board storage of natural gas. Due to the susceptability of steels to corrosion, blisteringand stress corrosion cracking in a natural gas environment, there exists a significant threatto the structural integity of these cylinders.While there are no reported incidents involving NGV cylinders, there are reportedfailures of other classes of cylinders which were in natural gas service. Perhaps the mostprominent of these failures was that of a seamless Cr-Mo steel tube trailer in 1977, whichfailed only one week after beginning service [28]. The cause of failure was found due tostress corrosion cracking which had resulted from a combination of a particularly harshnatural gas environment, material susceptibility and marginal wall thickness.While a recent report has concluded that NGV cylinder material is relatively resistantto sulphide stress cracking (at 24.82 MPa (3600 psi) and current contractural limits ofnatural gas contaminants) [11], there is evidence to suggest that defects can develop in1Chapter 1. INTRODUCTION^ 2NGV cylinders during service. Microcracks have been discovered in certain Italian-madecylinders and further, a more recent study indicated that crack-like defects can even beintroduced into cylinders during manufacture [85].To ensure against in-service failures, the Canadian Transport Commission requiresthat steel NGV cylinders be inspected and recertified every five years. The CompressedGas Association (CGA) has set out standards for inspection and recertification of DOT-3HT seamless steel cylinders; these standards have been adopted by the Canadian Trans-port Commission (CTC) and are currently used to recertify NGV cylinders.A major component of the CGA/CTC standards is a criterion which limits volumet-ric expansion of a cylinder during a hydrostatic test, a test which involves pressurizinga cylinder to 34.48 MPa (5000 psi) (1.67x service pressure) 1 . The reason for use of thevolumetric expansion criterion is that it provides some measure of cylinder wall thicknessand hence, nominal cylinder wall stress. The volumetric expansion criterion reflects anassumption in the CGA/CTC standards that the predominant mode of cylinder degreda-tion is a uniform reduction in cylinder wall thickness due to internal corrosion. A majordrawback of this assumption is that it does not specifically address the significance ofvery localized reductions in wall thickness.Localized reductions in cylinder wall thickness, such as corrosion pits and stress cor-rosion cracks produce stress concentrations which can lead to failure at loads much lessthan would be calculated on the basis of the nominal stress acting over the remainingwall. In recognition of this, current standards for NGV cylinder recertification specifythat the region around a localized reduction in wall thickness be ground to a depth equalto the reduction. While this procedure, known as 'reconditioning', may have merit interms of eliminating stress concentrations, it relies on visual inspection to detect poten-tial defects. A further drawback is that defects on the interior surface of the cylinder'Volumetric expansion criterion and the hydrostatic test are discussed more fully in Sec. (2.4).Chapter 1. INTRODUCTION^ 3(where gas impurities accelerate corrosion and cracking) cannot be reconditioned.As a means of resolving any uncertainty associated with defects that may evade visualinspection or which cannot be reconditioned, current standards rely on the hydrostatictest. In addition to providing the means to measure volumetric expansion therefore, thehydrostatic test is intended as a proof test. It is assumed that any defect that does notlead to failure (rupture) of a cylinder during a hydrostatic test can be tolerated for thenext five years (following recertification) that a cylinder remains in service.1.2 Statement of ProblemWhile a proof test can be utilized to ensure the integrity of a wide range of structurescontaining defects, one can reasonably question the use of this test to recertify NGVcylinders. A proof test, for the most part, is effective only when it is possible to en-sure that defect size will remain less than critical during a given inspection interval.Because NGV cylinders are subject to an aggressive environment and alternating load(see Sec.s (2.2) and (2.3)), there exists the potential for accelerated sub-critical stresscorrosion and fatigue crack growth. Evidence of this potential (i.e., crack-like defects)has been found in sectioned, previously in-service cylinders.On the basis of these observations, and because any pre-existing defect is a potentialsite for the initiation of sub-critical crack growth, it becomes reasonable to questionwhether current standards are adequate to guard against in-service failure. The currentview is that these standards (the hydrostatic test/volumetric expansion criterion) are notsufficient to guard against in-service failure.In order to assess whether the currents standards for NGV cylinder recertification areadequate, a significant research effort has evolved to assess the significance of localizeddefects that can arise during NGV service. This effort has for the most part focused onChapter 1. INTRODUCTION^ 4the significance of defects from the viewpoint of sub-critical crack growth [12, 13, 15].A significant result of this research effort has been the recommendation of an alternaterecertification protocol for NGV cylinders. This recertification protocol would removecylinders from service based on the dimensions of defects detected during a scheduled5 year inspection.While the recommendation of an alternate recertification protocol for NGV cylindersis promising, it is important to note that there is only limited support for its implemen-tation. This is due to the fact that to date, there have been no quantitative studiesperformed which quantify the conservatism of current hydrostatic test criteria when ap-plied to cylinders containing very localized defects. Clearly such a study is required,as the expense of implementing any new recertification protocol cannot be fully justi-fied until existing standards (i.e., current hydrostatic test criteria) have been thoroughlyexamined.1.3 ObjectivesIn view of the need to examine and re-evaluate the current standards for NGV cylinderrecertification, and develop standards which will ensure on-going cylinder integrity, thefollowing research objectives were established.1. To develop and verify a fracture mechanics based failure criterion suitable for as-sessing the structural integrity of a steel NGV cylinder containing a defect.2. To evaluate the applicability, and degree of conservatism (if any) of the currenthydrostatic test recertification criteria through comparison with the developed frac-ture mechanics based approach.3. To investigate the feasibility of an acoustic emission based recertification protocol.Chapter 1. INTRODUCTION^ 51.4 Scope of ResearchBased on these objectives, the scope of this project encompassed the following:Review of fracture mechanics methods applicable to NGV cylindersThe various fracture mechanics methods which might be utilized to assess NGVcylinder integrity were reviewed. Based on this review, which covered methodsapplicable to the full range of crack tip behaviour, a CTOD (crack tip openingdisplacement) approach was adopted for further investigation.Small scale testing of steel cylinder materialA series of small scale tests were performed to determine critical CTOD of axialdefects in NGV cylinders. These tests were performed in general accordance witha recognized standard for critical CTOD testing (BS 5762).Development and verification of a computer-based model to predictvolumetric expansionA computer based finite element model of a NGV cylinder was constructed andadditional software developed to predict volumetric expansion of cracked/uncrackedNGV cylinders. Volumetric expansion results from this model were compared tocurrent hydrostatic test volumetric expansion criteria to identify defect dimensionswhich would be considered acceptable under current standards for NGV cylinderrecertification.Development of a computer-based model to predict failureutilizing CTOD conceptsThe finite element model was further developed and utilized to evaluate CTODfor various crack geometries and locations. CTOD estimates from the model werecompared with critical CTOD values obtained from the small scale testing to de-termine the failure pressure of cylinders containing defined defect dimensions, andto establish critical defect dimensions for failure at, a given cylinder pressure.Chapter I. INTRODUCTION^ 6Acoustic emission testing of cracked/uncracked NGV cylindersA series of acoustic emission tests were performed on cylinders containing repre-sentative size, artificially introduced external defects. Additional acoustic emissioncontrol tests were performed on cylinders containing no known defects. Informationover a range of pressures (from operating to recertification pressure) was obtained.Data from the tests (hits, hits past previous pressure, amplitude distribution, etc.)was analysed in a variety of ways in an attempt to correlate defect severity withacoustic emission activity. To gain as much information as possible from avail-able cylinders, an initial series of tests was followed by further fatigue cracking ofcylinders and retesting.Burst testing of cracked NGV cylindersCracked cylinders were pressurized to failure to obtain data to verify numerical andanalytical failure predictions and to gain information on acoustic emission activityup to failure.Correlation of analytical/numerical failure predictions withexperimental resultsNumerical and analytical predictions of failure pressure were compared with exper-imental results to determine an acceptable failure criterion for rupture type failureof 'NGV cylinders.Assessment of current standards for recertification of NGV cylindersNumerical estimates of volumetric expansion and failure pressure of cracked cylin-ders, and available fatigue crack growth data, were used to evaluate the conservatismof current standards for recertification of NGV cylinders.Chapter 2TECHNICAL BACKGROUNDKnowledge of design and service conditions is a prerequisite to an understanding ofthe possibility of in-service failure. For this reason, NGV cylinder geometry and materialproperties, environmental conditions and loading are discussed in this chapter.2.1 Cylinder Geometry and Material PropertiesSteel cylinders currently used in NGV service are available in 40, 60, and 70 liter capaci-ties. The nominal dimensions and weights of these designations are given in Tables (2.1).Fabrication of NGV cylinders involves deep drawing of AISI 4130X steel billets toform the bodies, followed by heating and spinning to form the end caps. The top endcaps (nozzle ends) are made hemispherical, to minimize stress and the bottom capstorospherical, to minimize overall cylinder length. Details of cylinder geometry of atypical Faber 60 liter cylinder are shown in Fig. (2.1).Mechanical properties of 4130X steel, determined from a series of tests performedby PowerTech Labs, are given in Table (2.2). The chemical composition of this steel,obtained from the manufacturer. is given in Table (2.3).2.2 EnvironmentNatural gas contains, in addition to the primary constituent methane (CH 4 ), a num-ber of impurities including hydrogen sulphide (H 2 S), carbon dioxide (CO 2 ), and water7Chapter 2. TECHNICAL BACKGROUND^ 8(H2 0). Experience in the oil and gas industry has shown that these impurities createan environment that promotes surface corrosion, blistering and cracking of steels. Theseenvironmentally induced forms of damage are all, in one way or another, due to thedissociation of hydrogen sulphide in water.The increased availability of atomic hydrogen (decrease in pH) that results when hy-drogen sulphide dissociates in water creates an acidic environment. This environmentpromotes surface corrosion through the formation of iron sulphide. Surplus atomic hy-drogen adsorbs to the metal surface and, possibly catalyzed by the sulphide ion, diffusesinto the matrix. Upon entering the metal matrix, the atomic hydrogen tends to migratetowards microvoids, dislocations and inclusions, or regions of high triaxial stress such asthe tips of sharp defects or cracks. At microvoids, atomic hydrogen recombines to formhydrogen gas. The partial pressure of the gas is often sufficient to produce blistering ofthe metal surface. Atomic hydrogen that has accumulated at regions of triaxial stressdecreases the cohesive strength between interatomic bonds. This reduction in strength,or `embrittlemenf, reduces apparent fracture resistance often leading to sustained, sub-critical crack growth. In addition to environmental variables the rate of crack growthdepends upon the magnitude of applied tensile stress. This form of sub-critical cracking,resulting from the dissociation of hydrogen sulphide and occurring due to an appliedstress, is commonly referred to as sulphide stress cracking.The susceptability of steels to sulphide stress cracking depends on a number of fac-tors such as material properties (composition, microstructure, strength and hardness),environmental conditions (concentration of impurities, pH, temperature, etc.) and nom-inally applied stress levels. In general, steels of higher strength and hardness are mostsusceptable. One approach to controlling sulphide stress cracking therefore, has been tolimit the strength and hardness of steels used in natural gas service.Although there appears to be a threshold strength (and applied stress level) belowChapter 2. TECHNICAL BACKGROUND^ 9which sulphide stress cracking will not occur, certain low strength steels can be suscept-able if the concentration of hydrogen sulphide is sufficiently high. Another approach tocontrolling sulphide stress cracking therefore, has been to regulate pipeline gas quality.The current Canadian contractural limits (and typical values) for the concentrations ofhydrogen sulphide, carbon dioxide, and water are given in Table (2.4).2.3 Service LoadingThe relatively low critical temperature for liquification of natural gas (-82.5 C at ambienttemperatures) makes on-board storage of liquid natural gas impractical. Acceptablevehicle range requires on-board storage at the relatively high pressure and thus, ratedmaximum service pressure for NGV cylinders is 20.69 MPa (3000 psi).Because NGV cylinders are drained through fuel consumption and refilled throughoutservice, they are subject to an alternating, or cyclic, loading. Estimates of the number ofcycles that a typical cylinder will experence are difficult to make, given the variablity invehicle operating conditions. An upper bound of 25 cycles per week however, has beenestimated based on taxi cab service. This figure represents approximately 6500 refillingcycles during the five year recertification period. NGV cylinders are typically refilled atapproximately 2.07 MPa (300 psi) resulting in a cyclic R ratio (ratio between maximumand minimum service pressures) equal to 0.1.2.4 Current Standards for NGV Cylinder Recertification2.4.1 Minimum Wall Thickness RequirementCurrent standards for certification of NGV cylinders (Section 73.34(e) of CTC regulationsfor the transport of dangerous goods [26]) are based on a minimum allowable service wallthickness approach. The standards limit the reduction in wall thickness that can occurChapter 2. TECHNICAL BACKGROUND^ 10during service, and specify that a cylinder should be removed from service when the wallthickness is reduced by^1 ^— tmin )^ (2. 1)where t is the 'as-received' wall thickness and t min is the 'minimum allowable' wall thick-ness. The minimum allowable wall thickness is determined from a condition of generalyield, i.e., it is the wall thickness at which the cylinder suffers complete plastic collapseat hydrostatic test pressure. Prediction of the pressure at which plastic collapse occursis made using the Bach Formula [32, 34]:D2 —d2^P = S 1.3D2 0.4d2where P is the test pressure (1.67x service pressure), S is the allowable wall stress, Dis the outer diameter, and d is the inner diameter. This equation, derived using the St.Venant maximum principal strain yield condition and the Lame solution for stresses ina long cylinder [67], can be used to obtain an expression for the minimum wall thicknesstmin,^D ^(S — 1.3P4P)^tmin = —2—^s 0 . (2.3)The minimum service wall criterion for cylinder recertification assumes that duringservice the wall thickness is decreased uniformly, i.e., the cylinder wall is reduced by thesame amount at all locations. This assumption is valid when the only mode of cylinderdegradation is surface corrosion (the formation of iron sulphide). The aggressive NGVenvironment, however, can produce more local forms of damage such as corrosion pits,lines and crevices, and sulphide stress cracks. Stress concentrations associated with suchdiscontinuities result in stresses greater than would be calculated on the basis of thenominal stress acting over the remaining ligament, or net section. The nominal stressesrequired for yielding of the net section are therefore usually much less that those required(2.2)Chapter 2. TECHNICAL BACKGROUND^ 11for yielding of a 'smooth' wall of equivalent thickness. While there is some recognitionof this in the current standards, no formulation is provided to assess the significance oflocalized defects; hydrostatic testing, as required by current regulations, is assumed tobe sufficient to remove the uncertainty associated with such defects.2.4.2 Hydrostatic TestingCylinder wall thickness can be determined using a variety of techniques. Direct mea-surement can be made using mechanical, electrical, ultrasonic, or radiographic methods.The time and expense involved in the use of these methods however, makes them imprac-tical for cylinder recertification. A practical alternative to direct measurement of wallthickness is an indirect measurement based on volumetric expansion during a hydrostatictest [31]. This test determines wall thickness indirectly, through a measurement of cylin-der compliance (the change in volume due to internal pressure). As noted previously, thehydrostatic test is also intended as a proof test, that is defects and cracks that do notbecome critical under test pressure are considered safe until the next inspection.Hydrostatic testing involves measurements of volumetric expansion while pressurizinga cylinder to 1.67 times the service pressure. Volumetric expansion can be measured eitherby enclosing a cylinder in a water jacket and measuring the volume of water displaced(the water jacket method) or by directly measuring the volume of water forced into acylinder (the direct method). The water jacket method, because of its relative simplicity,is most widely used.Regardless of the method employed, when the hydrostatic test is performed, twocomponents of expansion are typically observed: an elastic component and a plasticcomponent. Elastic expansion is due to cylinder compliance and is therefore directlyproportional to pressure. Plastic, or permanent expansion is a residual expansion due toplastic deformation during the hydrostatic test.Chapter 2. TECHNICAL BACKGROUND^ 122.4.3 Hydrostatic Test Failure CriteriaBoth the elastic and plastic component of volumetric expansion can be used to estimatethe reductions in wall thickness that have occurred during service. Volumetric expansionscorresponding to minimum wall thickness can therefore be used as the failure criteriaduring cylinder recertification.The elastic expansion, AV, / , of a cylinder can be related to pressure using an empiricalrelationship; this relationship, known as the Clavarino equation is given byAVei = KPV ^D2D2 — d2where K is a constant, P is the test pressure (1.67 X service pressure), V is the cylin-der volume, D is the outer diameter, and d is the inner diameter. The constant K,the `K-factor', represents the constraint imposed by the cylinder end caps and is deter-mined experimentally from 'as-manufactured' cylinders. The Bach Formula. given byequation (2.2), can be used in conjunction with the Clavarino equation to derive a rela-tionship giving elastic expansion at minimum allowable wall thickness. The result is theBach-Clavarino formula given byS OAPAVel = KV1.7 (2.5)This equation provides the failure criterion based on elastic expansion. According toCGA standards [32], cylinders with elastic expansion in excess of AV,/ no longer meetthe minimum wall thickness requirement, and should be condemned.Plastic expansion cannot be written directly in terms of variables such as allowablewall stress, test pressure, and cylinder geometry. Therefore, rejection plastic expansioncannot be determined absolutely. Experience with a large number of cylinders however,has shown that when permanent expansion exceeds 10 of the measured total expansion,(2.4)Chapter 2. TECHNICAL BACKGROUND^ 13i.e., when> 0.10AVtotatwhere AVtotai = AV,/ + Al/pi , wall thickness has been reduced to the minimum allowable.In theory, both elastic and plastic expansion failure criteria could be applied to NGVcylinder recertification. However, current Canadian regulations based on standards pre-pared by the Compressed Gas Association (CGA) specify that only the plastic componentof expansion be used as a criterion in NGV cylinder recertification. Under section 73.34(e)of the CTC regulations, a cylinder is considered unfit for further service when permanentexpansion exceeds 10 Vc of total (elastic plus permanent) expansion.Chapter 3FRACTURE MECHANICS METHODS APPLICABLE TO NGVCYLINDERSOver the years, a variety of fracture mechanics techniques have been developed whichcan be used to assess structural integrity of NGV cylinders. In this chapter, these tech-niques are reviewed as they apply to a range of material behaviours (i.e., linear-elastic,elastic-plastic, etc.). Also discussed are some design practices which have been estab-lished which may have potential as standards for NGV cylinder recertification.3.1 General ConceptsFracture mechanics assessments are based on the comparison of an applied parameterwhich quantifies the severity of a defect to an associated material property which quanti-fies resistance to crack extension. Failure is predicted when the applied fracture mechanicsparameter exceeds some critical material value. The applied fracture mechanics parame-ter is a function of defect/structure geometry and applied load: the critical value of thisparameter is a material property (which can often be determined in a small scale test of arepresentative section). To date, a variety of techniques have evolved for evaluating bothapplied and critical fracture mechanics parameters. Application of these techniques, inconjunction with the use of modern non-destructive testing methods to estimate defectsize, provide the means to assess the severity of defects in a wide range of structures andmaterials and make 'fitness for purpose' assessments.14Chapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^15Before examining these techniques it should be noted that, in general. fracture me-chanics methods fall into three categories as illustrated diagrammatically in Fig. (3.1).These categories, linear-elastic fracture mechanics (LEFM), elastic-plastic fracture me-chanics (EPFM) and plastic collapse methods apply to low, moderate and high toughnessmaterials respectively. Materials of low toughness require relatively low energy to initiatefailure. These materials typically fail in a 'brittle' manner due to cleavage along crys-tallographic planes. Materials of increased toughness require greater energy to initiatefailure. This energy is associated with plastic flow and consequently, moderate toughnessmaterials typically fail in a 'ductile' manner due to formation, growth and coalescence ofmicrovoids at the crack tip. Materials of very high toughness require the greatest energyto initiate failure. Failure of these materials does not occur due to crack extension, butrather, to rupture due to complete plastic collapse.3.2 Linear-Elastic Fracture MechanicsLinear-elastic fracture mechanics is based on the elasticity solution for stresses near acrack tip. This solution, due to Westergaard (1939) has the general formK1V27^r (3.1)where K1 is a stress intensity factor'. For an arbitrarily cracked body, the stress intensityfactor is a function of structure/defect geometry and applied loading. In the originalanalysis, it was shown, using dimensional analysis, that K 1 = o-10, where c is the cracklength and /3 is a factor which depends only on geometry. For an infinite plate with athrough thickness defect subject to remote tension, 3 =Since the time that the concept of a stress intensity factor was introduced, it has beencustomary in LEFM to utilize the same basic solution (i.e., that for an infinite plate)1 The subscript I is used here to distinguish mode I (opening) from modes II (sliding) and III (tearing).Chapter 3. FRACTURE MECHANICS METHODS APPLICABLE...^16modified by a 'form' factor, Y, to account for specific structure/defect geometry andloading. Hence, most solutions for stress intensity factor are given asK=a177-cY(Tc1 ) (3.2)where W is a generalized size parameter.Since all the terms in Equ. (3.1) except the stress intensity factor are independent ofthe specific geometry under consideration, the asymptotic behaviour in the near crack tipregion (i.e., strength of the singularity) for different cracked bodies differs in proportionto the ratio of the stress intensity factors. This implies that the stress intensity factoris a similitude parameter which can be used to relate the severity of defects in differentgeometries. Hence, if failure of one cracked body occurs at some critical value of K 1 (i.e.,at Kip), failure of another cracked body (of the same material) will occur whenKi > KIc (3.3)where Kip is a material property called fracture toughness. It should be noted that, forEqu. (3.3) to apply, there must be similitude between the state of stress (i.e., plane stressor plane strain) at the crack tip. This makes it necessary, in specifying fracture toughness,to distinguish between material plane strain and plane stress fracture toughness 2 . In mostpractical applications of LEFM, it is desirable to employ plane strain fracture toughnesssince this will yield the most conservative estimate of failure stress.From the preceeding discussion, it can be seen that to apply LEFM it is necessaryto have knowledge of both the material fracture toughness and applied stress intensityfactor for the particular structure/defect geometry and loading of interest. Most often,material fracture toughness is determined in a standardized test (e.g.. ASTM E399 [3]or BS 5447 [22]) which will require that certain validity requirements are to be met2 It is usual to denote plane strain fracture toughness with a roman I, i.e., lift, and plane stressfracture toughness with an arabic 1, i.e., KChapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^17to ensure that measured fracture toughness is in fact plane strain fracture toughness.Stress intensity factors for most geometries and loadings of interest can be found inthe literature. Because of the widespread use of pressure vessels, there exists a numberof solutions for stress intensity factors for through-wall and surface defects in cylinderswhich may be applicable to NGV cylinders. These solutions are discussed in the nextsection.3.2.1 Stress Intensity Factors for Through-Wall and Surface Defects inCylindersAs noted, it is customary in fracture mechanics to utilize the same basic form of thestress intensity factor (i.e., the flat plate result given by Equ. (3.2)) and relate this to aspecific geometry and loading through a geometry factor. Hence, most solutions for stressintensity factors in cylinders are given as the ratio of the stress intensity of a cylinder,Kcy1 , to that of a wide plate, Kp lateMT,S Kcyiplatewhere the subscripts T and S denote a through-wall or surface defect respectively (seeFig. (3.2)). In solutions of this form, the remote stress a in Equ. (3.2) is replacedby circumferential stress, ARP. Because induced bending tends to increase the stressintensity in cylinders. the ratio MT,S is always greater than one and is commonly referredto as a stress magnification factor.Analytical solutions for stress magnification factors for axial through-wall defects havebeen derived by several investigators [44. 47]. These factors are typically given as a linearcombination of a membrane component and a bending component. The membrane andbending components, in turn. are given in terms of a shell parameter A,A = ^RtChapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^18Erdogan and Kibler [44] give2zMT = Grn —t Gbwhere G, and Gb are the membrane and bending components respectively (see Fig. (3.3)),z is the distance from the cylinder midplane and t is the cylinder wall thickness. Folias [47]givesMT = [/2 (33 + 6v — 7v 2 )(1 v)  J213(9 — 7v)^ (3.5)where again, I and J are the membrane and bending components respectively. Foliasgives the components in graphic form for a range of A. For small values of A however,Equ. (3.5) can be written in the following approximate form.1C2 ] 2MT =. [1 + 1.61 (Rt)While stress magnification factors for through-wall defects in cylinders are given interms of the stress intensity factor for a through-wall defect in a wide plate, stress mag-nification factors for surface defects in cylinders are typically given in terms of the stressintensity factor for an elliptical defect in a wide plate; this solution, from Irwin [54] is,() (k) smVia (. 222EKI =^ cost (p) (3.6)where E(k) is the complete elliptical integral of the second kind, most often given ap-proximately by7r^a 2)E(k)^(3 -c-yand a, c, and^areare defined as shown in Fig. (3.4).The simplicity of Equ. (3.6) sometimes makes it desirable to apply this equationdirectly to cylinders. although strictly speaking it is valid only for plates. This is usuallyacceptable provided A is small. Some sources have also suggested simple modifications toEqu. (3.6) to increase its applicability to cylinders. These modifications include replacing(3.4)2Chapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^19the term pR/t by p(R/t + 1) for internal defects to account for pressure on the crackfaces [25] and multiplying by MT for external defects [48]) to account for outward bulging.A more accurate approach however, would be to multiply Equ. (3.6) by a stressmagnification factor derived specifically for surface defects in cylinders. Such stress mag-nifiation factors have been developed by Kobayashi (and co-workers) [60, 61] and havebeen presented in the formms M = mc(70)-uKs(0) (3.7)where Mc is a factor which accounts for various effects including the non-linear stressdistribution over the defect, induced bending, and internal pressure on the faces of in-ternal defects. The factor MKS is a front free surface correction that accounts for theproximity of the free surface ahead of the crack.In addition to analytical solutions for stress intensity factors for surface flaws, anumber of numerical solutions utilizing spring-line [40] and finite element methods [17, 75]have been developed. Perhaps, the most popular of these numerical solutions is that ofNewman and Raju [75], who give present results for internal and external defects for arange of a/c, a/t, t/R and (0.2 < a/c < 1.0. 0.2 < a/t < 0.8, 0.1 < t/R < 0.25 and0< < 7r/2).While LEFM solutions are relatively well developed and understood, they are oflimited applicability to NGV cylinders due to the relatively high toughness of NGVcylinder material. Use of LEFM relationships will, in general. provide over-conservativeestimates of burst pressures due to lack of consideration to the energy associated withplastic deformation at the crack tip. LEFM relationships do however provide a means ofpredicting fatigue crack growth and leak before break (LBB) behaviour.Chapter 3. FRACTURE MECHANICS METHODS APPLICABLE...^203.3 Plasticity Corrected LEFMPlasticity corrected LEFM evolved in an attempt to extend the applicability of LEFMsolutions to situations where small scale yielding occurs. The fundamental ideas behindthe approach were first proposed by Irwin and have been shown to be consistent with thesmall scale approximations of models dealing with larger scale plasticity due to Dugdaleand Barenblatt [25].Plasticity corrected LEFM solutions are based on some approximation of the plasticzone at the crack tip. Irwin [53] reasoned that the plastic zone could be approximatedby a circular disk of radius r p (see Fig. (3.5)) and derivedIf?Tp^(3.8)27r(Ai cry) 2where the factor A i was introduced to account for the constraint at the crack tip; Irwinsuggested that= 1V2V-2- Plane strainPlane stress(3.9)Dugdale [42] assumed that the plastic zone ahead of a crack could be approximated bya yielded strip of length 1 (see Fig. (3.6)) and derived7r al= a [sec (-2^I— 11 (3.10)Expanding the sec term in the LHS of Equ. (3.10) and noting that for small scale yielding(i.e., a/ay << 1) all but the first term can be neglected, gives7r ICl2I =^ (3.11)which is within 20 % of Irwin's plane stress plastic zone correction.In formulating plasticity corrected LEFM, Irwin reasoned that due to crack tip plas-ticity the crack effectively behaves as if it is slightly longer, i.e.,aeff = a + rp^(3.12)Chapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^21Using this crack length provides a plasticity corrected expression for the stress intensityfactor. For example, substituting Equ. (3.12) into Equ. (3.6) and rearranging, gives [36]a 2KI =^7ra sine +^cos 2 (t.)^(3.13)c2where Q, denoted the flaw shape parameter, isQ= E(k)2 — 0.2122:-CryMultiplying this result by a stress magnification factor (see Sec. 3.2.1) gives a plasticitycorrected solution which can be used to analyse surface defects in cylinders.Although they extend the applicability of LEFM relationships, plasticity correctedLEFM concepts are of limited value since there is no real justification for predictingelastic-plastic behaviour using relationships derived on the basis of linear elasticity. An-other difficulty with the approach is that one must typically solve for the stress intensityfactor iteratively. Despite these limitations, plasticity-corrected LEFM concepts haveproved of value since they have provided insight into development of another fracturemechanics parameter more suited to predicting failure in the elastic-plastic regime, thecrack tip opening displacement.3.4 Elastic-Plastic Fracture Mechanics3.4.1 The CTOD ApproachGeneral ConceptsThe crack tip opening displacement (CTOD) approach was first proposed by Wells [96]as a means of predicting failure in the elastic-plastic regime. The approach is based onthe concept that it is crack tip strain and hence, crack tip displacement. which controlsfailure. As with all fracture mechanics approaches, the CTOD approach predicts failurewhen the controlling parameter (i.e., crack tip displacement) exceeds some critical value.Chapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^22A simple argument to justify the CTOD approach can be made on the basis of thestate of stress and strain at the crack tip. Ignoring the effects of material hardening, oncea condition of general yield is reached, stress in the near crack tip region will increasevery little with increasing load. Strain on the other hand, will continue to increase untilultimately, sufficient plastic strain has accumulated to initiate failure. If crack tip strainis assumed equivalent to crack tip displacement, it follows that in general, failure occurswhen some critical value of crack tip displacement is exceeded.A relationship for CTOD can be derived using the LEFM solution for crack openingdisplacement (COD) in a wide plate (see Fig. (3.7)),COD =^^a2 — x 2^(3.14)and the Irwin plastic zone correction. Substituting Equ. (3.12) into Equ. (3.14) anddefining CTOD as the COD at x = a gives (assuming that rp << a),4o- ^CTOD = --E- Oarpwhich can be written (c.f. Equ. (3.2))CTOD = 4 K2Eay (3.15)A somewhat different relationship for CTOD is derived using the Dugdale plastic zonecorrection; this relationship is [21]8o-ya^7ro-CTOD = ^ In sec7rE^2ay (3.16)It should be noted that, in accordance with the assumed plastic zone corrections,Equ.s (3.15) and (3.16) each have limits of applicability in terms of the scale of yielding.While Equ. (3.15) is valid only for small scale yielding (i.e., r p << a), Equ. (3.16) isapparently valid for more extensive yielding 3. For small scale yielding however, it can be3 See Sec.s (3.6.2) and (3.6.3).Chapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^23shown by expanding the In sec term and neglecting all but the first term, that Equ. (3.16)reduces to7ro-2a^K 2CTOD = ^Eo ^EcryBecause this expression differs from Equ. (3.15) only by a constant factor (4/7r) whichreflects the assumed plastic zone approximation, in generalwhere2CTOD =  KDA20-1,(3.17)E ^Plane strain1 — v2E^Plane stressA survey of numerical and experimental studies which have attempted to quantify A2 [25]indicates that this factor is a function of the exact position where the CTOD is measured,constraint at the crack tip (i.e., plane stress or plane strain) and possibly, work hardeningcharacteristics. Depending on these factors, 1 < < 2.14.While the CTOD approach has proved of great value in understanding and predictingfailure in the elastic-plastic regime, it should be noted that practical application of theapproach is not without drawbacks. One drawback is that relatively few analytical modelsexist which predict CTOD when large scale yielding occurs, thus one cannot usually solvedirectly for a critical crack length. CTOD at failure however, does provide a measureof material toughness and therefore can used to compare fracture resistance of differentmaterials. Another drawback of the CTOD approach is disagreement in the literatureover how CTOD should be defined. This disagreement has arisen, in large part, dueto discrepancies between crack tip profiles predicted by analytical models and thoseobserved in experimental and numerical studies [37]. While attempts have been madeto define CTOD empirically, the most consistent definition remains the crack openingdisplacement at the original crack tip, as indicated in Fig. (3.7).Chapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^24Despite these drawbacks, the CTOD approach has gained considerable acceptance.With this acceptance has come continued refinement of analytical/numerical solutionsto predict CTOD and the development of experimental techniques to determine criticalCTOD. In addition, the CTOD approach has been incorporated into a design standard,BSI PD 6493 [24], which can be used to assess structural integrity of a wide range ofstructural components. These topics are discussed in the following sections.Experimental Measurement of CTODDirect experimental measurement of CTOD requires the use of relatively sophisticatedtechniques (e.g., silicone infiltration, radiography, etc.). Such techniques are difficult toimplement and for this reason, in practice, CTOD is measured indirectly. This is doneusing expressions which relate CTOD to COD, a quantity which can be measured withrelative ease somewhere along the crack faces (typically at the crack mouth as shown inFig. (3.8)).Critical CTOD for a given material is most often evaluated utilizing a small scale testspecimen. Of the various configurations which can be employed, the three point bendspecimen is most common. For this geometry, the relationship between CTOD and CODis (see Fig. (3.9))—CTOD =  r(W a) COD^ (3.18)r(W — a) + awhere r is a factor (rotational factor) which accounts for rotation of the crack face oncea plastic hinge has formed. -Using infiltration techniques, Robinson and Tetelman foundthat r = 0.0472 + 0.0939CTOD + 0.00931CTOD 2 0.00037CTOD 2 in' [25].Another (and to date the most accepted) expression relating CTOD to measuredCOD was proposed by Wells [96]. In this expression,K?^0.4(W — a)1/7,CTOD = ^ + (3.19)9o-y.E' 0.4W + 0.6a + zChapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^25CTOD is separated into elastic and plastic compontents (c.f. Equ.s (3.17) and (3.18))and a constant rotational factor r = 0.4 is assumed. While the assumption of a constantrotational factor may be justified, Anderson et al. [6] have pointed out that the valuer = 0.4 may not be appropriate for all materials. These investigators have suggestedthat a more accurate estimate of r can be obtained from1^1- 1/p Wr =   16W a +W — a 1. qpwhere qp is the plastic component of load line displacement.In addition to relationships for the three-point bend configuration, relationships havebeen derived which relate CTOD to measured COD in other geometries (e.g., the center-cracked plate). A review of these relationships can be found in Ref. [79].Finite Element Estimation of CTODModelling the Crack TipOf the various numerical techniques which can be used to solve problems in fracturemechanics, the finite element method is perhaps the most popular. This popularity isdue, in large part to the versatility of the method which has facilitated the developmentof specialized approaches to deal with the various regimes of crack tip behaviour. Earlyinvestigations were concerned mainly with the suitability of the finite element methodfor predicting LEFM parameters such as stress intensity factor and energy release rate.More recent investigations have been concerned with the suitability of the method forpredicting elastic-plastic parameters such as J-integral and CTOD. A common challengein the application of finite element method to both LEFM and EPFM has been modellingof the crack tip region.As noted, LEFM solutions predict a r - 1 singularity in stress and strain at the cracktip. Early attempts to deal with this singularity employed models that focused a largeChapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^26number of simple elements, e.g. constant strain triangles, at the crack tip. This approach,although justified by convergence theorems, has two main drawbacks. The first was cost;the second was that computational error increased with mesh density. These drawbacksultimately led to the development of other, more efficient means of modelling problemsin LEFM.One approach to more efficient modelling of problems in LEFM is the non-uniformmesh refinement method. This method is based on techniques that equalize the dis-cretization error in elements near a singularity. These techniques make element size afunction of the order of the singularity and of the element shape function. The resultsof numerical experiments performed by Johnson [55] indicate that non-uniform meshrefinement is a very efficient means of modelling problems in LEFM.Another more efficient approach is the use of specially constructed singularity or`crack tip' elements. This approach followed from reasoning that fewer elements wouldbe required overall if an inverse square root singularity could be incorporated into ele-ments at the crack tip. This reasoning led to the development, by a number of differentinvestigators (e.g., Blackburn [16]) of special elements which are constructed with shapefunctions that produce an inverse square root singularity at the crack tip. Althoughrelatively successful, these elements sometimes lacked the rigid body modes and constantstrain states that guarantee convergence. This drawback was overcome however, by thediscovery (by Henshell and Shaw [52], and Barsoum [8]) that an inverse square root sin-gularity could be produced in a standard eight-node isoparametric element by collapsingone face and moving the mid-side nodes on adjacent faces to the quarter-point position.Given the relative ease with which an inverse square root singularity can be obtained,finite element modelling of problems in LEFM has become an accepted approach. Morerecent studies have focused on extending finite element methods to problems in elastic-plastic fracture mechanics. This has lead a number of investigators to focus attentionChapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^27on development of crack tip elements which reproduce EPFM singularities. Formulationof these elements however, was more complicated due to the fact that in high tough-ness materials, the strain field near the crack tip is a function of material hardeningcharacteristics. For materials that obey a power law hardening rule.( Ty /^G )n(where T = ( 82.i j /2)1/2 ,^(26,3eij)1/2 and 3 , ez3 are, respectively, the deviatoriccomponents of stress and strain), the elastic-plastic crack tip singularities areC1cr = ^„ pr n+2 (0),^ez, = C2 qz.7(°)rn-flIt should be noted that for n = 1, (linear-elastic behaviour) the singularities reduce tothe inverse square root singularities predicted by LEFM.Despite the difficulties involved, several investigators (e.g., Tracey [93]) have success-fully developed a class of two- and three-dimensional crack tip elements which embodythe above singularities. An important feature of these elements is that they possess non-unique degrees of freedom at the crack tip. This facilitates accurate modelling of cracktip blunting, which as has been noted, occurs due to plasticity at the crack tip.A special case of power law material hardening is perfectly plastic behaviour. Forthis behaviour, n^0, and the singularities given by Equ.s (3.4.1) becomeC1(7i3 =P (0), E = i 2 q-(0)i3Levy et al. [65] have shown that these singularities can be reproduced in a standardfour node (linear) isoparametric element by collapsing one face and allowing the nodeson this face to move independently. Latter. Barsoum [9] demonstrated that these samesingularities could be reproduced in standard eight (two-dimensional) and 20 node (three-dimensional) isoparametric elements in a similar manner.Chapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^28To verify the accuracy of modified isoparametric elements in predicting perfectlyplastic behaviour, both Levey et al. and Barsoum compared their finite element resultsfor small scale yielding to the Prandtl slip line field. In these comparisons, the angularvariation of stress around the crack and the distribution of stress ahead of the crack wereinvestigated. Agreement between the element models and the slip line solution was good,and would appear to justify the use of collapsed isoparametric finite element methods inEPFM.While these results support the use of collapsed isoparametric elements in EPFM,it should be noted that there is evidence which suggests that in some cases, standardelements (such as the constant strain triangle) may be equally as accurate. In a studyof near crack tip displacement fields, Amstuiz and Seeger [2] investigated the effects ofelement type, mesh refinement, load incrementization and to a limited extent, materialhardening. Employing constant strain triangles (CSTs), these investigators constructed afinite element model of a through thickness cracked plate. An analysis was first performedassuming perfectly-plastic (n = 0) behaviour. In this analysis, the displacement fieldnear the crack tip was compared to that predicted by the Dugdale strip yield model(Equ. (3.16)). Differences between the finite element and analytical results were foundto depend strongly on the number of load steps, with better agreement occurring asthe number of load steps was increased. While a great number (on the order of 50)of load steps were required, good agreement with the analytical solution was ultimatelyobtained. Using the same model, Amstuiz and Seeger also performed an analysis assumedhardening (n = 3.33) behaviour. In this analysis, the displacement field near the cracktip was compared to that predicted by HHR (Hutchinson, Rosengren and Rice) fieldsolutions. Differences between the finite element results and these solutions were found tofollow the same trend (with respect to load incrementization) as was observed for elastic-plastic behaviour. Final agreement between the finite element results and the analyticalChapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^29solution was also found to be very good. In the same study, Amstuiz and Seeger went onto investigate the displacement field of a model constructed with isoparametric elements.In this model, collapsed isoparametric elements (HPEs) were employed at the crack tip.Assuming perfectly-plastic behaviour, and similarly comparing the displacement field ofthe finite element model with that predicted by the Dugdale solution, it was again foundthat differences between the finite element results and analytical solution decreased asthe number of load steps was increased. It should be noted however that when theHPEs were employed, much fewer (10) load steps were required and that final agreementbetween the finite element results and the Dugdale solution was much better (than thatobtained with CSTs). Assuming hardening behaviour, Amstuiz and Seeger went on, inthe final stage of the investigation, to compare the displacement field of the HPE modelwith that of the HRR solutions. In this case, it was found that agreement between thefinite element results and analytical solutions was poor.The results of Amistuiz and Seeger indicate that, although efficiency is sacrificed,it is possible to accurately model the crack tip with a fixed node element such as theCST when behaviour is perfectly plastic. These results also indicate that, while it ispossible to accurately model the crack tip with a fixed node element when hardeningoccurs, it is not possible to model the crack tip with a collapsed isoparametric elementwhen hardening occurs. While this finding is somewhat expected, given that there is notheoretical basis for modelling the crack tip with collapsed isoparametric elements whenhardening occurs, it should be noted that this is often what is done in practice.In closing the discussion on finite element modelling of the crack tip, it should benoted that as yielding becomes more extensive, there may be less need to accuratelymodel crack tip singularities. This follows from intuitive reasoning that the singularitieswill become lost in a large plastic zone and that, ultimately, plastic instability mechanismswill become more important. Some support for this follows from a study performed byChapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^30Worswick and Pick [98]. This study involved finite element analysis of a circumferentiallycracked pipe subjected to a moment loading. Three crack sizes (relatively short/shallow,short/deep and long/shallow) were analysed. Because material hardening was assumed,singularity elements were not employed. In the analysis, load was increased until theplastic zone extended well beyond the cracked region of the pipe. Asymptotic load-CODbehaviour was used to predict the failure moment. The failure moments predicted bythe finite element model were compared to failure moments predicted by an analyticalmodel 4 . Although poor for the long/shallow defect, agreement between the finite ele-ment and analytical model was found to be within several percent for the short/shallowand short/deep defects. Reviewing the results presented by Worswick and Pick indicatesthat relative to defect size, the plastic zone was larger for the small defects than for thelarge defect. This would appear to suggest that when the plastic zone is large comparedto defect size, sophisticated modelling of the crack tip region may not be necessary.Experimental Verification of Finite Element CTOD EstimatesSeveral investigators have also attempted to correlate finite element and experimentalresults. These correlations however, have been mainly limited to small scale fracturemechanics specimens and other relatively simple geometries (e.g., [81, 4, 19]. Thesestudies have typically involved comparisons of measured and finite element crack open-ing profiles, or measured and finite element load-COD records. For example, Schmittand Hollstien [81] performed both two- and three-dimensional finite element analysis ofcompact tension specimens and compared computed crack open profiles to measuredresults. Agreement between the finite element and measured results was, in general,good. It should be noted however, that experimental points were limited mainly to the4 Details of this analytical model are not given in Ref. [98]. The authors note however, that theanalytical model is supported by a considerable body of experimental evidence. On this basis. theanalytical model was considered an accurate benchmark in the comparison of results.Chapter 3. FRACTURE MECHANICS METHODS APPLICABLE...^31crack flanks, and that no detailed comparisons were made between profiles at the cracktip. Anderson [4] similarly compared finite element and measured crack profiles for agrooved bar. This comparison however, showed only qualitative agreement between thefinite element and measured results. Bleakley and Luxmoore [19] performed a roundrobin comparison of elastic-plastic finite element solutions with experimental data. Inthis investigation, a number of contributors were asked to perform finite element anal-ysis of compact tension (CT), single edge notch bend (SENB) and center cracked plate(CPP) geometries. Depending on the particular geometry, several sets of results i.e.,load versus COD, load versus J, crack opening profile, etc. from each of the analyseswere compared with existing experimental data. While some variability between thefinite element solutions and the experimental data was observed, there was in general,good agreement between the finite element and experimental results. Unfortunately, noexperimental results for the crack opening profile were available. Hence, while it waspossible to compare the various finite element crack opening profiles, it was not possibleto compare experimental and numerical crack opening profiles.Unfortunately, perhaps due the cost of carrying out a program of full scale numericaland experimental analysis, there are relatively few studies along these lines. One studywhich did compare full scale numerical finite element failure predictions with experimen-tal results was performed by Wellman et al. [95]. In this study, five cylinders containingaxial surface defects were pressurized to failure. Four of the cylinders (D = 378 mm,t = 12.4 mm, 1 = 122 m and ay 800MPa) contained extended axial notches and one(D = 686 mm, t = 152 mm, 1 = 1.27 m and 0y 487MPa) contained an ellipticalfatigue crack. In conjunction with the burst tests, a number of small scale CTOD testswere performed on the cylinder material to establish critical (initiation) CTOD. Thesesmall scale tests were performed according to BS 5762. Following the experimental fullChapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^32and small scale tests, a finite element analysis was performed. Two-dimensional analy-sis was performed for the cylinders containing the extended flaws and three-dimensionalanalysis was performed for the cylinder containing the elliptical flaw. The crack tip re-gion was modelled using collapsed isoparametric (perfectly plastic singularity) elementsdiscussed previously. To reduce the size of the analysis, the non-linear region near thecrack tip was sub-structured. The finite elements models were incrementally loaded untilthe analysis failed to converge. After the analysis was performed, pressure versus CTODcurves were constructed for each of the cylinders. The pressures corresponding the criti-cal CTOD (from the small scale tests) on these curves were compared to the actual burstpressures. Agreement between the predicted and actual failure pressures was found tobe very good; three of the cylinders failed within 1 % of the predicted failure pressureand two within 7 % of the predicted failure pressure. While these results are promising(in terms of finite element CTOD failure predictions), it should be noted that during theburst tests Wellman et al. failed to measure COD. Further comparison of finite elementand experimental results was therefore impossible.Correlation Between Analytical and Finite Element Estimates of CTOD There exists relatively few analytical solutions which predict CTOD when large scaleyielding occurs. For this reason, comparisons between analytical and finite element re-sults given here will be limited to the small scale yielding regime, that is, in the regimewhere Equ. (3.17) holds. A useful means of making comparisons between CTOD pre-dicted by Equ. (3.17) and finite element results is to rewrite this equationK 2A2 = ^IPayand compare A2 predicted by theory with that predicted by finite element analysis. Inmaking such comparisons however, it should be noted that variability exists even amongChapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^33the analytical estimates of )' 2 . The Irwin and Dugdale solutions discussed in Sec. 3.4.1give 4/7r and 1 respectively; J-integral computations by Rice [78] give 1.47.An initial comparison of analytical and finite element estimates of A2 can be madeusing the results of two-dimensional analysis. Examples of such analysis are the investi-gations (discussed previously) of Tracy [93], Levey et al. [65] and Barsoum [9]. In theseinvestigations A2 was found to be 1.85, 2.14, and 1.72 respectively. These values, whichvary considerably and are somewhat greater than the analytical estimates of A2, suggestthat finite element analysis will yield a conservative estimate of CTOD.Further comparison of analytical estimates and finite element estimates of A2 can bemade using the results of full three-dimensional finite element analysis. These compar-isons however, should be made with care since one must ascertain the state of stress (i.e.,plane strain or plane stress) in the structure which was modelled. In cylinders, one can bereasonably certain that a state of plane strain exists along edge cracks, or at the deepestpoint of a surface crack. Ranta-Maunus and Talja [76] performed an elastic-plastic finiteelement analysis of an edge cracked cylinder. In this investigation, J-integral and CTODwere analysed over the full elastic-plastic range and it was found that J = 1.06o CTOD.Comparing this result to the LEFM relationship between J and K1 [25],J E'2 (3.20)gives A2 = 1.06. Aurich et al. [7] and deLorenzi [41] performed a similar analysis ofsurface (elliptically) cracked cylinders. The results of these investigations indicate thatJ ti 1.0ay CTOD and that A2":: "' 1.0.3.4.2 J-integral MethodsAnother elastic-plastic fracture mechanics parameter which has been proposed is the J-integral. This parameter, although not utilized in this research, has received a great dealChapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^34of attention and therefore will be discussed here for completeness.The J-integral approach is based on the concept that the energy available for crackextension controls failure. Formally, the J-integral is defined by the following path inde-pendent integral [25].J = (Wdy — T au ds) (3.21)where W = lot az3 dE t3 . This integral represents a balance between the strain energy withinthe contour F and that crossing the contour F. It can be shown that if the contour F isclosed, J = 0; alternately, if the contour is not closed, i.e., if it is penetrated by a crackas shown in Fig. (3.10), J > 0.This property, and that of path-independence noted previously, make it possible toutilize J to predict failure in the elastic-plastic regime. Rice [78] has shown that theenergy difference between a closed contour and one penetrated by a crack is equivalentto the potential energy per unit of crack extensionSince J is path independent, it can be evaluated along a contour outside the plastic zoneat the crack tip where behaviour is elastic.When material behaviour is linear-elastic, J is equivalent to the energy release rateG. On the basis that elastic-plastic behaviour can be approximated by non-linear elasticbehaviour, it has been suggested that J can be used to predict failure in the elastic-plasticregime. This implies, in analogy with LEFM concepts discussed previously, that failurewill occur when J exceeds some critical value, i.e., whenJapplzed >^ (3.22)where JI, is a material property.Chapter 3. FRACTURE MECHANICS METHODS APPLICABLE...^35Practical application of the J-integral approach requires a solution to Equ. (3.21)for a particular structural/defect geometry and loading. While solutions for relativelysimple geometries (e.g., the SENB specimen) exist, the complexities of solving Equ. (3.21)often makes it necessary to employ some numerical scheme (e.g., finite element analysis,spring-line methods, etc.).Application of the J-integral approach also requires some means of estimating J1c.To this end, multispecimen and single specimen techniques have been developed whichcan be utilized to establish Jic for a given material. A review of these techniques can befound in Ref. [79]While a complete discussion cannot be given here, it should be noted that certainrestrictions apply to the use of the J-integral approach. One restriction, which followsfrom the use of non-linear elasticity to approximate elastic-plastic behaviour is thatno unloading should occur. The consequence of this restriction is that the J-integralapproach is valid only for stationary cracks since, typically, some unloading will occurat the flanks of advancing cracks. Other restrictions on the use of J-integral also exist.These restrictions are to ensure similarity between different structural configurations sothat Equ. (3.22) remains valid.In closing this section, it should be noted that relationships exist between J-integraland the fracture mechanics parameters (K and CTOD) discussed previously. Due to theequivalence of J and the energy release rate, G, the relationship between J and stressintensity factor K isJ = E' (3.23)From Equ.s (3.23) and (3.17), the relationship between J and CTOD isJ A 3 cryCTOD^ (3.24)Chapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^36where A3 is a constant, generally taken to be1^Plane stressA3 =1.6 Plane strainThe restrictions with apply to Equ.s (3.23) and (3.24) should be noted. Equation (3.23)is valid only when behaviour is linear-elastic; Equ. (3.24) is valid throughout the elasticplastic regime.3.5 Plastic Collapse Methods3.5.1 Limit Load AnalysisStructural components containing cracks may be so resistant to fracture that failure doesnot occur due to crack extension, but rather due to complete plastic collapse. While thissituation arises due to very high toughness, it can also occur if defect dimensions aresmall compared to section thickness. or alternately if defect dimensions are very largecompared to section thickness.To predict failure due to plastic collapse, a limit load solution must be employed.These solutions are. in general, derived from the upper and lower bound theorems ofplasticity. These theorems state that failure will occur when a mechanism can be foundin which either: 1. the rate of internal plastic work done becomes less than the rate ofwork done by external forces or 2, internal stresses cannot be redistributed such thatthey are less than yield.Due to the complexities involved in accounting for material hardening behaviour,most limit load solutions have been derived assuming elastic-perfectly plastic behaviourwhere the yield stress is normally replaced by flow stress. Although flow stress can be-PL = aF R,(t — a)PL  (R, + a)(t — a)(3.26)(3.27)Chapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^37determined empirically (see Sec. 3.6.3), it is most often taken to be [27](ay + au)^Plane stressF =1.5 ( cry + au ) Plane strainwhere the factor 1.15 in the plane strain definition reflects the apparent elevation of flowstress resulting from transverse (out of plane) constraint in this condition.A number of plastic collapse solutions of practical importance have been developed.For the single edge notch bend (SENB) specimen [6](W — a) 2 BPL = 0-F ^ (3.25)where the factor L is the notch constraint factor. This factor accounts for the geometricalconstraint produced by plastic strains near the crack tip. McClintock [69] assumed a slipline field for the SENB and calculated L = 1.543. Romilly et al. [80], in experimentalwork, used L = 1.543 for calculations when SENB specimens contained sharp cracks, buta somewhat lower value, L = 1.261 when specimens contained blunt notches.Due to the practical importance of cylindrical pressure vessels, there exists a numberof plastic collapse solutions for this geometry. For extended axial cracks, Chell [27] givesfor internal and external flaws respectively. Chell suggests however, that these relation-ships may be over-conservative for elliptical defects and recommends the term (t — a) bereplaced by(t — a)1 ^a^t(1+*) 2Chapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^38It should be noted that the effects of outward bulging have not been included in equa-tion (3.27). A relationship which accounts for these effects is [92]t^a/t —1  )PL— crF Ralt-11mwhere 1m = 1 + [0.263 (2c)2 2Rt(3.28)As can be seen, for small values of (2c) 2 /Rt this relationship predicts failure when thehoops stress reaches the material flow stress.3.5.2 Analytical and Numerical Estimates of CTOD for Surface Flaws in theLarge Scale Yielding RegimesA number of investigators have investigated the problem of developing solutions for theCTOD of surface flaws when large scale yielding occurs. While the approaches taken aresomewhat varied, a common theme in the solutions is the equivalence of the uncrackedligament of a surface flaw to certain closure forces (and possibly, moments). The ad-vantage of this approach is that the surface flaw becomes equivalent to a through-walldefect with the same closure force acting over some region of the crack faces. This equiv-alence makes it possible to develop solutions for surface flaws by modifying solutions forthrough-wall defects which are more developed.Critical CTOD ApproachIn the critical CTOD approach developed by Irwin [70]. it is assumed that the uncrackedligament of the surface flaw has fully yielded (see Fig. (3.11)a). This is assumed to occur5 The approaches discussed in Sections (3.5.2), (3.5.3) and (3.5.4) although not utilized in this researchare discussed for completeness.Chapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^39when the nominal stress acting over the ligament reaches the material flow stress; thecorresponding closure force is then2c(t — a)•FThe surface flaw thus becomes equivalent to a through-wall defect with closure stresscc = 2ct= (1 — 7) „acting over the crack faces. Assuming that this closure stress will effectively reduce theCTOD of the through-wall defect by an amountCTOD 4acThe CTOD for the surface flaw becomesCTOD 4(u — ac )cIrwin went on to refine the model by accounting for plasticity at the ends of thedefect. Replacing the actual crack length c by an effective crack length c eff c rp (c.f.Equ.s (3.8) and (3.12)), and substituting for a, givesCTOD 4(c E+ rP) [a^—at) aFJwhich is Irwin's final result for CTOD of a surface defect in the large scale yieldingregime.FModified Critical CTOD MethodRecognizing that the critical CTOD method neglects elastic contributions to the CTOD.Cheng et al. proposed the modified CTOD method [77] which separates the CTODChapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^40into elastic and plastic components. The elastic component is evaluated from LEFMsolutions and the plastic component is based on Irwin's CTOD solution. Based on thepremise that the LEFM solution remains valid to ligament yield, and that the criticalCTOD solution is valid between ligament yield and net section yield, the modified criticalCTOD approach givesforand,K? CTODEo-Faa< 1— —t crF(3.29)4(c-4- rp) [a — Cl—CTOD CTOD LY^cr.F]^(3.30)forwhere CTODLYac— -t ) crF < <^—^aFis the elastic component of CTOD (i.e., Equ. (3.29)) evaluated at 0-LY•3.5.3 Erdogan and Ratwani FormulationA limitation of the critical and modified critical CTOD approaches is that they do notaccount for out-of-plane bending due to the eccentrically loaded crack. While this limi-tation may not be severe when the models are applied to defects in flat plates, it may besignificant when the models are applied to defects in cylinders, where curvature furtherintensifies out-of-plane bending effects.A model which more accurately predicts the CTOD of surface defects in cylinders hasbeen developed by Erdogan and Ratwani [45, 46]. This model formulation is analagousto the critical CTOD approach in that the surface is modelled as a through-wall defectwith closure stress acting over the the crack faces. In contrast to the critical CTODmethod however, it is assumed that in addition to acting on the actual crack faces, theChapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^41closure stress acts beyond the ends of the defect (see Fig. (3.11b)). The surface defectis thus redefined as a through-wall defect with an effective length cp c p where pis a plastic zone correction which must be determined. To determine p, Erdogan andRatwani assume a yield condition based on a plastic strip at the ends of the defect. Thisyield condition (which can be derived on the basis of simple beam theory- 6 is6M = 1 (3.31)to-F 2 Fwhere N and M are defined as shown in Fig. (3.11b). To complete the formulation,two additional equations are derived on the basis of assumptions regarding finiteness ofstresses at c cp : these assumptions lead to two equations of the formKm (cp) 0 (3.32)Kb (cp ) = 0 (3.33)Examination of these equations (given in their full form in Ref. [45]) reveals that bothKm and Kb consist of terms representing the stress intensity due to internal pressure andthe closure force and moment, N and M, at the ends of the defect. These two equations,in addition to Equ. (3.31) provide three non-linear equations for p, N and M whichErdogan and Rawtani solve numerically.Some results of the solution, in terms of the non-dimensional parameter c/(c p)are plotted against the shell parameter A in Fig. (3.12). Utilizing p, N and M. and adisplacement solution which follows from their analysis, Erdogan and Rawtwani calculatethe CTOD. These results, plotted against A are shown in Fig. (3.13).Erdogan and Ratwani compared their analytical result with experimental results forburst tests of 2014-T6 aluminum and titanium (Ti-5A1-2.5Sn-ELI) cylinders. This com-parison generally verified the analytical model and the failure criterion CTOD CTOD,.'Maximum plastic load carrying capacity at the outmost fibers of a beam subject to a combinationof axial load and bending; a slightly different result is obtained if a plastic hinge condition is assumed.Chapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^423.5.4 Line-Spring ModelsThe line-spring method offers yet another means of estimating CTOD for a range ofelastic-plastic behaviour. The approach, originally utilized by Rice and Levy to determineLEFM stress intensity factors [59], has gained popularity due to the considerable accuracywhich can be obtained with minimal computational resources (i.e., in comparison tofull three-dimensional elastic-plastic finite element analysis). Against the economy ofthe approach however, is relatively greater effort and sophistication in formulating therequired equations. Due to the complexities inherent in the approach, only an overviewwill be given here.In analogy with the formulations discussed previously, the principal behind the line-spring approach is that a surface defect can be modelled as a through-wall defect withcertain closure forces and moments acting on its faces. These closure forces and momentsconstrain displacement and rotation of the faces of the through crack. To relate closureforces and moments to displacement and rotation, compliance expressions are derivedfrom the plane strain solution for an edge crack which is equal in depth (locally) to thesurface crack.In formulation of the line-spring model, mathematical complexities arise due to thefact that the displacement and rotation of every point on the crack faces is a functionof the displacement and rotation of every other point. This leads, in general, to a setof non-linear integral equations which must be solved numerically or alternately, using aspecially modified finite element scheme.A simplification which eliminates the need to solve such equations has however beenproposed by King [59]. In this formulation, which serves to illustrate line-spring concepts,the surface crack is assumed to have a constant depth a. This assumption simplifies theequations for displacement. A, and rotation. O. (see Fig. (3.14)) of the crack faces to aChapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^43set of linear algebraic equations,2(1 — v)tEA =  ^aign)^ (3.34)12(1 — v 2 )tE= ^ (ai2ac a22m) (3.35)where a-, = N/t, in = 6M/t 2 , v is Poisson's ratio and a i are compliance coefficientsderived from the solution for an edge crack.For a through crack with a, and in acting on the crack faces, the expressions for Aand 0 are==4c—8(1 + v) c m(3 + v) Et(3.36)(3.37)Equating these expressions to Equ.s (3.34) and (3.35) gives= evo"m =where ck and r3 are funtions of a/t, v and the compliance coefficients au . Having derivedexpressions for a, and rn, Equ.s (3.36) and (3.37) can be used to determine A and 0.From the geometry shown in Fig. (3.14) the CTOD of the surface defect is given byCTOD = A + 0(t — 2a)To complete the formulation, King adopted an approach similar to that employed inthe critical CTOD/modified critical CTOD models. That is, the CTOD remains elasticup to ligament yield, which is assumed to occur when the nominal stress reaches thematerial flow stress. CTOD beyond ligament yield is calculated assuming no furtherrotation of the crack faces once the ligament has yielded. King also employed an effectiveChapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^44crack length to account for plasticity at the ends of the defect. The final solution is4cre^+ 2(1 v)^2a^CTOD =   1E^(3 v) )for1 (1^)andCTOD = 4(c r ,E P) — ELY) + CTODLY (1 +for(1 a) < a aF (1 2ac)Wt )While King's formulation of the line-spring is admirable because of its simplicity, itshould be noted that due to out-of-plane bending the approach may not be accurate forsurface flaws in cylinders. King suggests that when applied to cylinders, use of the modelbe restricted to c/at > 1. Parks [74] however, has applied line-spring concepts to ageneral class of plates and shells. This approach however, is somewhat more sophisticatedin that for post yield behaviour, the equations relating diplacement and rotation to closureforces and moments are written in incremental form and solved based on an assumed slipline field (rather than average stress over the ligament). In addition, Parks suggest certainconstitutive refinements to account for material hardening.As noted, popularity of line-spring methods has arisen chiefly as a result of the con-siderable accuracy which can be obtained with minimal computational resources in com-parison to full three-dimensional elastic-plastic finite element analysis. While expenseremains an important factor in any analysis, it appears that with the decreased costs ofcomputing and increased availability of proprietary finite element packages, the trend inrecent years has been to more fully exploit finite element methods to analyse fractureproblems. This is particularly true for critical components where the cost of a detailedfinite element analysis can be justified.27rao-y— CTODE(I) (3.38)Chapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^453.6 Design Methods3.6.1 The CTOD Design CurveThe CTOD Design Curve was developed as a means of applying the CTOD approach topractical engineering problems and is based on a non-dimensional CTOD, (I), given byInitial motivation for the design curve was due to the the work of Burdekin andStone [21]. These investigators reasoned that by utilizing an equation of the form givenby Equ. (3.38), a maximum defect size could be established once material critical CTODis known. It remained however, to find an acceptable form for (1) which, to be of practicalvalue, would be conservative over the range of material yield stress. In the original formu-lation, Burdekin and Stone utilized the Dugdale strip yield CTOD solution (Equ. (3.16)).However, as experimental results became available the CTOD design curve underwentseveral empirical adjustments [39]. These adjustments, made to maintain conservatismand ease of use resulted in the following form of design curve, which is in use today.=^y ) 2 < 0 5Ey ^•_ _ 0.25 4- > 0.5(3.39)Equating this equation to Equ. (3.38) and rearranging, gives the following expressionsfor the maximum allowable defect.CTODc Ea.?, 27a2am axCTODcE 27- (a — 0.25a y )<0.5>0.5(3.40)The CTOD design curve has been incorporated into a document, BSI PD 6493 [24]which can be used to assess the severity in a wide range of structural components. Toextend the usefulness of the approach, i.e.. to make it possible to utilize a single curveChapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^46to analyse geometries other than through-wall defects. LEFM shape factors are utilizedto relate through-wall to surface and buried defects (see Fig.s (3.15) and (3.16)). Thisprovides an equivalent through-wall defect (72-) which can be compared to the maximumallowable given by Equ.s (3.40). Although not strictly valid, this procedure follows asuggestion by Dawes that under constrained yielding, use of LEFM relationships canbe justified [39]. For plates, Fig.s (3.15) and (3.16) can be used directly; for cylindershowever, due to outward bulging effects which have been noted previously, PD 6493requires that the equivalent crack length be further reduced by the factor D shown inFig. (3.17).One drawback with the CTOD design curve approach is that no explicit formulationis given for behaviour near plastic collapse. In recognition of this, PD 6493 requires thatsurface defects be recategorized as through-wall defects (of the equivalent length) whenthe net section stress reaches the material flow stress, i.e., whena t (1 — (3.41)An important point to note is that the CTOD design curve approach provides amaximum allowable and not a critical defect size. Due to the assumption of unconstrainedyielding and the nature of applied loading, maximum allowable defect size will possessan inherent factor of safety. Based on a study performed by Dawes [38], a factor of safetyof approximately 2.5 on crack sizes was estimated at a 95.4 confidence level.It should be noted that while a factor of safety of 2.5 is generally accepted, in certaininstances (i.e., at high net section stresses), a much more variable factor of safety canbe observed [5]. This observation, which relates to the fact that the upper region of thecurve is empirical, in addition to the one noted previously (i.e., that plastic collapse isnot an integral part of the formulation) means that the CTOD design curve cannot ingeneral be used to perform a critical engineering assessment. To resolve these difficulties.Chapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^47a three tier assessment (which utilizes a reference stress methodology to predict postyield behaviour) has been proposed. Details of this approach, which should provide amore universal means of assessing the significance of defects in structural components,can be found in Ref. [5].3.6.2 The Central Electricity Generating Board R6 MethodIn a study of the significance of defects in structural components, Dowling and Townleysuggested the 'two criteria approach' to failure assessment. The approach was subse-quently supported by other investigators (Harrison et al. and Milne et al.) and ulti-mately used by the Central Electricity Generating Board (CEGB) to develop the R6failure assessment diagram [84].The principal behind the two criteria approach is that failure will occur when theapplied load reaches either: I. the load to cause brittle fracture, PK, or 2, the loadto cause plastic collapse, P1 . The two criteria approach therefore assumes failure tobe either toughness controlled or flow stress controlled. This assumption is clearly notalways valid; it is however, a conservative one given that, actual failure load is usuallybounded by either Pi; or P1 .Dowling and Townley found that even in the transition from toughness controlledto flow stress controlled behavior, failure loads could be predicted to within 20 usingthe two criteria approach. Although this order of accuracy was considered reasonable inview of the simplicity of the approach. Dowling and Towley noted that increased accuracycould be achieved using the Dugdale strip yield model to interpolate between LEFM andplastic collapse failure criteria. This refinement of the two criteria approach ultimatelyled to the development of a more comprehensive means of predicting failure-the CEGBR6 method.The CEGB R6 method [27. 84] is based on the generalized failure assessment diagramChapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^48shown in Fig. (3.18). The ordinate and abscissa of this diagram, Sr , Kr , are given by thefollowing non-dimensional parameters.Sr^0^(3.42)T1Kr = K,^ (3.43)These parameters reflect the degree to which either toughness or flow stress controlfailure. It is assumed that the range of behaviour between fully toughness controlled,(Sr = 0, Kr = 1) and fully flow stress controlled (Sr = 1, Kr = 0) can be interpolatedusing the Dugdale strip yield model. This leads to_1Kr = I ^ In sec (-7r Sr )1 22which corresponds to the locus of points (Sr , Kr ) for which failure is expected to occur.Equations (3.42) and (3.43) can be used to determine an assessment point for a crackedbody. If this point lies on or outside the assessment curve, a prediction of failure is made.A feature of the R6 method is that a quantitative assessment of structural integritycan be made based on the proximity of a point ((Sr , Kr ) to the failure curve. This makesit possible to perform sensitivity analysis with respect to variables such as applied load,defect geometry, and material properties.3.6.3 AGA Model (Battelle Empirical Analysis)A number of investigators at the Battelle Memorial Institute have worked towards de-velopment of a model to predict failure of cylinders containing through wall and surfacedefects. This model, developed for the American Gas Association (AGA) was derived ontheoretical grounds and subsequently refined using the results of large scale tests. Theseempirical adjustments have been found to improve the accuracy of the model throughoutthe range of toughness controlled to flow stress controlled behaviour.Chapter 3. FRACTURE MECHANICS METHODS APPLICABLE... ^49In formulating the AGA model, Hahn et al. [50] reasoned that flat plate expressionscould be utilized to predict failure stress in a cylinder if a correction was made for outwardbulging. Combining Equ. (3.16) (the Dugdale strip yield model) with Equ. (3.17), theseinvestigators arrived atK1c 1  m-10 =cOPwhere cb is a plasticity correction, given by(rMa9 ) -2^Cr M cro 11 2In [sec2o-F^2o-Fand M is a stress magnification factor. Hahn et al. choseM= [1 + 1.612 1Rt)](3.44)(3.45)(3.46)which, as noted in Sec. 3.2.1, is due to Folias.To assess the validity of Equ. (3.44), Hahn et al. drew upon a number of exper-imental studies of pressure vessel failure. These studies included a range of materials(ductile and brittle steel, as well as alumimum and titanium alloys), and various defectgeometries (c2 /Rt ratios). Equation (3.44) was used to determine the fracture proper-.ties of the materials used in experimental studies. These properties were found to be inagreement with fracture properties determined from flat plates of the same material. Itwas concluded that Equ. (3.44) could be used to predict both toughness controlled andflow stress controlled failure.A similar conclusion was drawn by Maxey et al. [68] and Kiefner et al. [58] in studiesof intermediate to high toughness pipeline steels. These investigators however, recastEqu.s (3.44) and (3.45) into the following non-dimensional form.( ^ aOMT,P)^ = In sec8co-1^2o-F (3.47)Chapter 3. FRACTURE MECHANICS METHODS APPLICABLE...where50C2^c4 2MT^ 0.01351.1.255 (Tit-)=[1 +^+(R2t2)(3.48)(3.49)for through-wall defects and1 — a I tMP =1 — a/MTtfor surface flaws.Equation (3.47) relates a non-dimensional toughness parameter to a non-dimensionalflow parameter, and represents the locus of points along which failure can be expected tooccur. Maxey et al. and Kiefner et al. found, when comparing the experimental resultswith Equ. (3.47) that good agreement could be obtained if flow stress was taken to be= ay + 69 MP aMaxey et al. and Keifner et al. noted that the transition from toughness controlled toflow stress controlled behavior occurs atK 2 7r8co-1 -^st, 4 (3.50)and that beyond this transition, the failure criterion, Equ. (3.47) effectively becomes(76MT,P = CrFThe Battelle methodology discussed in this section and other elastic-plastic fracturemechanics approaches discussed previously in this chapter were considered in view of theirpotential to predict failure of NGV cylinders. Choice of any one methodology (or numberof methodologies) as a basis for assessing in-service integrity of NGV cylinders however.required greater understanding of the fracture resistance of NGV cylinder material. Forthis reason a series of small scale CTOD tests were performed. These tests are discussedin the following chapter.Chapter 4INVESTIGATION OF SMALL SCALE BEHAVIOURMaterial test results which were available when this investigation was begun [1, 13]indicated that NGV cylinder steel (AISI 4130 X) is a moderately tough material. Sincematerials of this nature typically exhibit some plasticity prior to failure, it was considerednecessary, in this investigation, to employ elastic-plastic fracture mechanics methods topredict failure. Consideration was therefore given to the relative merits of adoptingeither J-integral or crack tip opening displacement (CTOD) based methods. Althoughuse of the J-integral technique is applicable to this material behaviour, a fundamentaldrawback is that this parameter cannot in general be measured experimentally. Forthis reason, it was decided to adopt the CTOD approach. While the CTOD cannot bemeasured directly, it can be estimated from the crack opening displacement (COD), aquantity that can be measured experimentally (see Sec. 3.4.1). Another motivation foruse of the CTOD to predict failure was that a design methodology based on the CTODapproach is somewhat established and has been incorporated into a proposed standard,BSI PD 6493 [24].In order to apply the CTOD approach to predict failure of NGV cylinders, it was nec-essary to initially establish the critical value of CTOD for AISI 4130 X steel. One of thefirst stages of this investigation therefore, was small scale critical CTOD bend specimentesting. This testing was performed in general accordance with British Standard 5762(BS 5762) [23]. an accepted standard for critical CTOD testing.51Chapter 4. INVESTIGATION OF SMALL SCALE BEHAVIOUR^524.1 Numerical Modelling of CTOD Specimens4.1.1 Specimen DesignNGV cylinder steel was expected to exhibit considerable anisotropy due to the deepdrawing process used in the manufacturing process. Critical CTOD in turn, was expectedto be highly dependent on defect orientation. Since interest in this investigation was inaxial defects and because BS 5762 requires a full thickness specimen, it was necessary toemploy a curved CTOD specimen.The nominal stress field acting over an axially oriented defect in a pressurized NGVcylinder (a thin walled vessel) is biaxial tension. This stress field produces a high con-straint condition at the crack tip. To reproduce this condition in the small scale spec-imens, a three point bend loading configuration was employed. Guidelines for the finaldesign (see Fig. (4.1)) were found in the literature and British Standard BS 5762.While there are no inherent difficulties in employing curved fracture CTOD speci-mens, it does make a number of standardized relationships given in BS 5762 suspect,as they were derived assuming a straight fracture mechanics specimen. Since handbooksolutions for curved fracture mechanics specimens were unavailable, an alternate meansof determining the stress intensity and compliance of this unique CTOD specimen wasrequired. This led to the use of a finite element analysis to model the curved CTOD spec-imens. It should be noted that in addition to providing data necessary to calibrate thesmall scale test procedure. the finite element analysis of the curved fracture mechanicsspecimen provided experience in the modelling aspects of fracture mechanics problems.This experience was of value in the latter stages of this project, where a finite elementanalysis of full scale NGV cylinders containing defects was performed.Chapter 4. INVESTIGATION OF SMALL SCALE BEHAVIOUR ^534.1.2 Mesh, Boundary Conditions, and LoadingAs noted above, the purpose of finite element analysis in this stage of the investigationwas to determine stress intensity factors and specimen compliance for a range of crackdepth in the curved CTOD specimen. To facilitate this analysis, the finite element modelwas constructed to permit crack length a to be input as a parameter. This parameterwas varied so that the specimen a/W ratio ranged between 0.4 and 0.9.Figure (4.3) shows the finite element mesh for the CTOD specimen; this figure alsoshows the boundary conditions and applied load. The finite element mesh was con-structed using the eight-node (quadratic) isoparametric element of Zienkiewicz [99} 1 .Since specimen width was sufficient for a condition of plane strain, stiffness matrices wereformulated with a plane strain elasticity matrix. Note in Fig. (4.3) that due to symmetry,it was possible to perform the analysis with a model of one-half of the actual specimens.To model symmetry across x = 0, nodes were constrained against displacement in thex-direction. To model the roller supports of the bend fixture (see Fig. (4.2)), the nodeat x = 50.8 mm was constrained against displacement in the y-direction. Specimen loadwas modelled by a point force acting in the —y direction.As noted in Sec. 3.4.1, an inverse square root singularity is produced when the mid-side node of an isoparametric element is moved to the quarter-point position. Since thismodification more accurately represents the stress distribution in the crack tip region,the mid-side nodes of the four elements near the crack tip were moved to the quarterpoint position (see Fig. (4.3b)).Finite element modelling of the small scale CTOD specimens (and subsequent fullscale cylinders) during this investigation was performed using the commercial finite el-ement program ANSYS 2 . The ANSYS input file is listed in Appendix C. The analysis'This element is described in Appendix A.2 ANSYS was developed and is distributed by Swanson Analysis Systems, Inc..Chapter 4. INVESTIGATION OF SMALL SCALE BEHAVIOUR^54described in this phase was run on an Apollo DN 4000 workstation. Computer CPU timeper specimen analysis was approximately 2 minutes.4.1.3 Numerical ResultsAfter the finite element models were created and submitted, solution files were post-processed to extract stress intensity, crack opening displacement and specimen compli-ance values. This was accomplished by employing a user written subroutine which utilizedthe ANSYS command KCALC to compute stress intensity factor (K1 ) and the followingrelationships to compute crack opening displacement (A) and specimen compliance (C)A = 2uC = vwhere u is the x-displacement of the node at the crack mouth, P is the load, and v isthe y-displacement of the node to which the load was applied (see Fig. (4.4)).To aid in identifying trends in the numerical results, the following relationships wereemployeda\FraF (Tv-a ) (4.1)4Ea-a vi (4.2)v 5E'SV2 (l/V) (4.3)where a is the maximum bending stress (3PSOW2 t), E is the modulus of elasticity, Sis the load span and F, V1 and V2 are non-dimensional shape factors which are functionsof a/1/1/.To assess the differences in the compliance relationships, the results of the finiteelement analysis have been plotted in terms of the non-dimensional shape factors. F,Chapter 4. INVESTIGATION OF SMALL SCALE BEHAVIOUR^55V1 and V2 as shown in Fig.s (4.5) through (4.7). Superimposed on these figures areanalytical solutions from Ref. [91] calculated based on a straight specimen of dimensionsequivalent to those of the curved CTOD specimen for comparison. Overall, the resultsfrom the finite element analysis suggest that the curvature effects of the cylinder CTODspecimen are relatively insignificant. This is particularly evident in the figures thatcompare stress intensity and crack opening displacement results for the two specimengeometries (Fig.s (4.5) and (4.6)). For these results, the effect of curvature becomessignificant only when a/W > 0.8, with the maximum difference between the finite elementestimates and analytical solution being 9 % for the crack tip stress intensity and 5 %for the crack opening displacement. Increased differences between the finite elementestimates and analytical solution over the range of crack size can be seen in Fig. (4.7)which shows the results for point load displacement. For these results, a maximumdifference between the finite element estimates and analytical solution was 16 %.From the finite element analysis of the CTOD specimen, it was concluded that the ef-fects of curvature of the specimen up to a/W < 0.8 could be neglected thus validating theBS 5762 guidelines for CTOD testing. Hence, in subsequent small scale testing describedin the next section, the standardized relationships given in BS 5762 were employed.4.2 Experimental Measurement of Critical CTOD4.2.1 Specimen PreparationA total of 6 specimens were cut and prepared from a sectioned NGV cylinder. To facilitatefatigue precracking, a 0.15 mm slitting saw was then used to produce a notch. After thespecimens were notched, they were cycled in three point bending to grow a fatigue crackapproximately 0.125 mm (0.05 in) in length. To ensure uniformity in fatigue crack length,crack propagation gauges were mounted on the specimens to monitor crack growth. TheChapter 4. INVESTIGATION OF SMALL SCALE BEHAVIOUR^56location of these gauges are shown in Fig. (4.2). This figure also indicates the position ofthe potential drop probes used to measure crack growth in the subsequent CTOD testing.Fatigue crack growth data collected during fatigue precracking has been plotted inFig.s (4.8) and (4.9).4.2.2 Test ProcedureAfter each CTOD specimen was precracked, it was fitted with knife edges and a doublecantilever type clip gauge to monitor COD, and potential drop leads to monitor crackgrowth. The locations of the knife edges and potential drop leads are shown in Fig. (4.2).The specimen was then placed in the three point bend fixture of the test machine andloaded under displacement control at a ramp displacement rate of approximately 0.025in/min. Loading of each specimen proceeded well beyond the attainment of maximumload. This resulted in CTOD tests that lasted approximately three to four minutes. Dur-ing the CTOD tests, data from from the clip gauge and the load cell of the test machinewere collected on a DAS 20 data acquisition board installed in an IBM AT. Potentialdrop data was collected using an MDT DC potential drop system. Unfortunately, atthe time of testing, it was not possible collect load and COD, and potential drop datautilizing a single system. This made it necessary to design a system which would enablematching of load/COD data and potential drop data. A system was therefore designedto monitor potential drop power pulses to the specimen and send these pulses to theDAS 20 hoard. After the tests. pulses recorded on the DAS 20 board were matched topotential drop data recorded on the MDT board.To facilitate integration and analysis of load, COD and potential drop data files, aFORTRAN progam (CODREAD) was written. This program, listed in Appendix C,reads and merges binary DAS 20 and ASCII MDT files. applies calibration factors andoutputs the results for further examination. The potential drop data are converted toChapter 4. INVESTIGATION OF SMALL SCALE BEHAVIOUR^57crack length using Johnson's equation [81]:V^cosh' [cosh(y/2W)/ cos(a/2W)] Vo cosh -1 [cosh(y/2W)/ cos(a0/2W)]where V is the voltage drop corresponding to the crack length a, Vo is the voltage dropcorresponding to the original crack length a o , y is the length across which V and Vo aremeasured and W is the specimen width.After the CTOD tests, the specimens were immersed in liquid nitrogen and brokenopen to reveal the fracture surfaces. The fracture surfaces were then examined under anoptical microscope for evidence of cleavage and/or stable crack growth. With the aid ofa travelling table and a precision scale mounted on the base of the microscope, fatigueprecrack lengths, a0 , were measured.4.2.3 Small Scale Test ResultsDepending on the material crack growth resistance, several different types of materialbehaviour may be observed during a critical CTOD test. The various material responsebehaviour, i.e., cleavage, stable crack growth prior to failure, attainment of a maximumload plateau, etc., have been classified in BS 5762 according to characteristics of theload-COD as shown in Fig. (4.10). This figure also illustrates the point (or points) oneach load-COD record at which critical CTOD can be calculated. As has been noted,it was of interest in this investigation to determine CTOD at initiation (subscript i inFig. (4.10)).Determining critical CTOD at true crack initiation is, from a practical viewpoint, verydifficult. It can be seen from Fig. (4.10) that, while the points of instability or maximumload on the various load-COD curve (subscripts u and m) are well defined. the onsetof crack initiation is not easily determined. At the present time, there are no acceptedanalysis techniques for estimating the point of initiation. Where stable crack growth isChapter 4. INVESTIGATION OF SMALL SCALE BEHAVIOUR ^58suspected, BS 5762 outlines a multi-specimen method for developing a resistance curvethat can be used to determine crack opening displacement at initiation. This methodhowever, requires further testing and a minimum of four (but preferably six) additionalspecimens. An alternative to the use of the multi-specimen technique is to monitor crackgrowth directly as was done in the test described here.One approach to utilizing crack growth data to estimate CTOD at initiation is givenby Romilly [79] and was employed in this study. The approach makes use of a plot ofload, P, versus crack growth, Da. From this plot the load at the onset of stable crackgrowth, i.e., at Aa = 0 can be determined. This load is then transferred to the load-crack opening displacement curve to determine the plastic component of crack openingdisplacement, Vp. CTOD at initiation, CTOD, is then calculated using (ref. Sec. 3.4.1)CTOD K2(1 — v 2 )^0.4(W — a)Vp2ayE^0.4W + 0.6a z (4.4)where K is the stress intensity factor, v is the Poisson's ratio, E is the modulus ofelasticity, 1/17 is the specimen width, a is the crack length, and z is the distance betweenthe specimen and the point at which the COD is measured.Critical CTOD AnalysisReferring to Fig. (4.10) discussed previously, analysis of the load-crack opening displace-ment records of all of the CTOD specimens showed the behaviour of this material to besimilar to those of Type I and Type III in Fig. (4.10), indicating either cleavage or somestable crack growth prior to failure. Comparison of the load-crack opening displacementcurves suggested that appreciable stable growth prior to failure had occurred in only onespecimen. Specimen 5. This can be observed in Fig. (4.11) , where the load-crack openingdisplacement record of Specimen 1 is compared to that of Specimen 5. (The load-crackopening displacement records of the other specimens are similar to that of Specimen 1Chapter 4. INVESTIGATION OF SMALL SCALE BEHAVIOUR ^59and have therefore been omitted from Fig. (4.11) for clarity.) Visual records made duringtesting supported the assumption that, with the exception of Specimen 5, only limitedcrack growth occurred prior to failure.In order to verify the visual records, crack growth (Da) was plotted against load (P)as shown in Fig.s (4.12) and (4.13). These figures indicated that, with the exceptionof Specimen 5, initiation of crack growth occurred at maximum load. Following theprocedure outlined in the previous section, initiation CTOD was calculated using theload-COD curves. The loads and crack lengths used in these calculations are given inTable (4.1). Initiation CTOD was found to range from 0.027 to 0.049 mm, with a meanof 0.039 mm and a standard deviation of 0.008 mm (see Table (4.2)).To verify the small scale test results, a correlation between J-integral and CTOD wasemployed. This correlation (ref. Sec. 3.4.2) isJ = maFCTODwhere J is the J-integral corresponding to CTOD, m is a non-dimensional constantof proportionality (= 1.6 for plane strain) and a F is the material flow stress (0 F =(ay + au )/2 = 885 MPa). Substituting the value of flow stress and the average value forinitiation CTOD for the cylinder material into the above equation givesJ = (1.6)(885 MPa)(0.039 mm)= 55.2 N/mmwhich is nearly identical to J1, = 54 N/mm reported by Powertech Labs, Inc. [13]. Thisclose agreement would appear to support the small scale CTOD results of the presentinvestigation and further, suggests that the J-CTOD correlation is valid and useful forthis material.Chapter 4. INVESTIGATION OF SMALL SCALE BEHAVIOUR ^60Limit Load AnalysisThe limit load, PL, for three point bend specimens (ref. Sec. 3.5.1) isPL = 1.5430.F (W — a) 2 B4SThis expression was used to calculate the limit load for the five CTOD specimens. Thecalculated limit load and the ratio of maximum test load to limit load (P7-flax /PO foreach specimen has been entered in Table (4.3). As indicated in this table, the meanof the results for P / P- max, - L was 0.83 with a standard deviation of 0.02. These resultsindicate that all of the specimens failed significantly below the predicted limit load. Theresults of the limit load analysis confirm that unstable crack growth rather than plasticcollapse controlled failure of the small scale specimens. Although not conclusive, thissuggests that plastic collapse methods may not be well suited to predicting large scaleNGV cylinder behaviour in the presence of sharp defects.The critical CTOD results obtained from this small scale testing were subsequentlyutilized in FEM modelling to predict the full scale behaviour of NGV cylinders containingcracks. This is discussed in Chapters 5 and 6.Chapter 5NUMERICAL INVESTIGATION OF FULL SCALE BEHAVIOURAs noted in Chapter 1, the major goal of this investigation was to assess the currentstandards for NGV cylinder recertification. The approach adopted to achieve this goalwas to predict cylinder behaviour during recertification, i.e., during a hydrostatic test,and following recertification, during the five years a cylinder remains in service. Anoverview of this approach is shown in Fig. (5.1). As noted in Chapter 1, a significantportion of the study involved finite element analysis of NGV cylinders. This analysis isdiscussed in this chapter.The goal of the finite element analysis was to predict volumetric expansions andfailure (burst) pressures of cylinders containing a range of defect sizes. Finite elementanalysis was intended therefore, to predict cylinder behaviour during a hydrostatic test.The results of the analysis make it possible to determine whether or not a defectivecylinder will fail recertification, that is, whether or not a defective cylinder will exceedvolumetric expansion limits or rupture. As will be seen in Chapter 8, this information,together with available fatigue crack growth estimates (which predict whether or not acylinder will fail during service) provides the means to assess the current standards forNGV cylinder recertification.Before discussing the numerical analysis. it should be noted that a CTOD approachwas adopted to predict failure (burst) pressure. To this end, the critical CTOD materialdata from the small scale tests as discussed previously in Chapter 4 were employed.61Chapter 5. NUMERICAL INVESTIGATION OF FULL SCALE BEHAVIOUR 625.1 Model Description5.1.1 Material PropertiesTo perform the finite element analysis and subsequently analyse the results, it was neces-sary to know both the tensile and fracture properties of NGV (AISI 4130X) steel. At thetime this investigation was begun, a series of tensile tests had been performed at Pow-ertech Labs, Inc. These tests, as reported in Ref. [13], indicated that the yield strength,ultimate tensile strength and elongation of AISI 4130X steel are 822 MPa, 948 MPa,and 20 percent respectively. As discussed in Chapter 4, the fracture properties for thismaterial were determined by a series of small scale tests performed during present inves-tigation and indicated that the average critical CTOD for the initiation of crack growth(L-R direction) is 0.039 mm.For the purpose of finite element modelling, the stress-strain behaviour of the 4130Xsteel was idealized as linear strain hardening as shown in Fig. (5.2). Plasticity wasassumed to be governed by the von Mises yield criteria, associative (Prandtl-Reuss) flowrule and kinematic hardening rule'.5.1.2 Cylinder/Defect GeometryThe geometry of the 60 liter cylinder that was analysed is shown in Fig. (5.3). Nominaloutside diameter and wall thickness of this capacity cylinder are 316.6 mm and 7.5 mmrespectively: nominal length is 972.8 mm. The orientation of the defect considered isalso shown in Fig. (5.3). Note that an elliptical crack shape and axial orientation wereassumed. This geometry and orientation corresponds to a fatigue crack subjected to thehighest principal wall stress (i.e., the hoop stress), and hence represents the most severetype of defect encountered in service.1 A explanation of these laws, which come from plasticity theory. is given in Appendix B.Chapter 5. NUMERICAL INVESTIGATION OF FULL SCALE BEHAVIOUR 63To assess cylinder behaviour, i.e., fracture resistance, volumetric expansion, etc., overa range of defect size, defects of various a/t and a/c ratios were considered. Each defectsize was investigated as both an interior and exterior defect. The matrix of computeranalyses performed (run designation and crack size) is given in Table (5.1). Note fromthis table that the crack size was varied from short and shallow to relatively long anddeep.5.1.3 Finite Element MeshShell to Solid SubmodellingDue to a relatively small thickness to radius ratio (i.e., t/R), NGV cylinders can beclassified as thin walled, shell structures. The logical approach to finite element analysisof such structures is to utilize elements which have been derived from thin shell theory,i.e., shell elements. Shell elements are by nature, two-dimensional entities since onlyin-plane forces and moments have been assumed in their derivation.Because shell elements are well suited to analysis of NGV cylinders, it was desirableto use this type of element in the analysis. This, however, presented a difficulty asthe ultimate goal was to analyse surface defects in the cylinders. As surface flaws arethree-dimensional, some form of solid element was required. While it was possible toconstruct a mesh for a defective NGV cylinder exclusively with solid elements, this wasnot considered feasible due to the demands which would be placed on the computer CPUand memory. An alternate approach, known as shell to solid submodelling was thereforeemployed. In shell to solid submodelling, a technique developed specifically to analysestress concentrations in shell structures, two finite element models are constructed. Thefirst model, referred to as the coarse model, represents the structure of interest andis constructed with shell elements. The second model, referred to as the submodel,Chapter 5. NUMERICAL INVESTIGATION OF FULL SCALE BEHAVIOUR 64represents some region of the structure (i.e., a stress concentration) and is constructedwith solid elements.The basis for shell to solid submodelling is the specified boundary displacement ap-proach discussed by Kelley [57]. In this approach, an analysis is first performed withthe coarse model. This analysis yields a displacement solution which is used to specifydisplacements at the boundary of the submodel. Displacements of a node at the bound-ary of the submodel ({u} s ) are specified using the following result, which accounts fortranslation and rotation of elements in the coarse model [89].{u}s = {u} {9} x {r}where {u} is the translation vector of a point on the coarse model, {O} is the rotationvector of a point on the coarse model, and {r} is the position vector from the coarsemodel to a node on the boundary of the submodel.Because the displacement field of the submodel is made to match that of the coarsemodel, the stress field should also match. This will not be the case however, if thedimensions of the submodel have not been made sufficiently large for any local distur-bances within it to dissipate at the boundary. It is therefore critical in shell to solidsubmodelling that the dimensions of the submodel be sufficient for any local disturbancesproduced within this region to dissipate prior to reaching the boundaries.Coarse ModelFollowing the shell to solid submodelling approach, a coarse model representing an NGVcylinder plus a number of submodels representing defects in the cylinder were constructed.The coarse model mesh is shown in Fig. (5.4). This mesh was constructed using the eightnode isoparametric shell element of Cook [35] (see Appendix A). Note in Fig. (5.4), that2 Dissipation of any local effects is guaranteed by Saint-Venant's principal.Chapter 5. NUMERICAL INVESTIGATION OF FULL SCALE BEHAVIOUR 65due to symmetry, it was possible to employ a mesh of one-quarter of the actual cylinder.This was accomplished with the aid of symmetry boundary conditions which were appliedto nodes along the planes x = 0 and z = 0. These boundary conditions constrained nodaldisplacements perpendicular to the plane of symmetry and nodal rotations parallel andperpendicular to the element planes. In addition to the symmetry boundary conditions,an additional set of boundary conditions were applied to the nodes along the y = 0plane to constrain the model against rigid body motion in the y-direction. Althoughpossible to apply this last set of boundary conditions at any point in the y-direction,they were applied at the mid-section to permit use of symmetry boundary conditions inthe submodel (see Sec. 5.1.3).In a non-defective NGV cylinder, plasticity does not occur at pressures below 41.37 MPa(6000 psi) (the maximum pressure in the analysis-see Sec. 5.1.4). For this reason, finiteelement analysis of the coarse model was performed assuming linear-elastic materialproperties.Several refinements in the finite element mesh were performed prior to achievingthe desired order of accuracy. The final version, as shown in Fig. (5.4), contained 300elements and approximately 4500 active degrees of freedom. Run time on an ApolloDN 4000 workstation was approximately 15 minutes. The ANSYS input file for thecoarse model is listed in Appendix C.To gain some idea of the accuracy of the coarse model, membrane and bending stresses(in the circumferential and longitudinal directions) were compared with an analyticalsolution given by Coates [30]. This comparison is shown in Fig.s (5.5) and (5.6), weremembrane and bending stress from the finite element model and from Coates' solutionhave been plotted as functions of s//, where s is distance (measured from the nozzle end)along the cylinder wall and 1 is cylinder length. As can be seen, there is generally goodagreement between the finite element and analytical solution. Differences between theChapter 5. NUMERICAL INVESTIGATION OF FULL SCALE BEHAVIOUR 66two solutions appear mainly in the top end cap, near the nozzle, and in the bottom endcap. These differences however, should not necessarily be attributed to the finite elementmodel, as Coates' solution does not account for the constraint imposed by a nozzle andfurther, is based on an elliptical, rather than a torospherical end cap.SubmodelThe submodel mesh (for the 5 x 10 mm defect) is shown in Fig. (5.7). This meshwas constructed with the twenty-node isoparametric element of Zienkiewicz [99] (seeAppendix A). In the view of the submodel given in Fig. (5.7), the defect lies in the lowerright hand corner of the mesh on the plane z = 0. This region of the mesh has beenenlarged and is shown in Fig. (5.8(a)). The near crack tip mesh is shown in Fig (5.8(b)).To facilitate analysis of a range of defect sizes (ref. Sec. 5.1.2), the elliptical transfor-mationC2 - a21 + x 2 + y 2== zwas employed. This transformation, illustrated diagrammatically in Fig. (5.10), madeit possible to employ a single semi-circular mesh (representing a crack of depth a andlength 2a) for each crack length 2c. The advantage of using this transformation wasthat for each three crack lengths, it was necessary to construct only one submodel. Thesemi-circular crack submodels for each crack depth were constructed with the aid of thesolid modelling capabilities of ANSYS. It should be noted that while in general. solidmodelling greatly simplifies construction of a finite element mesh, initial attempts toconstruct an elliptical crack profile utilizing solid modelling exclusively were disappointingdue to current limitations of ANSYS solid modelling capabilities. Use of the aboveChapter 5. NUMERICAL INVESTIGATION OF FULL SCALE BEHAVIOUR 67transformation however, made it possible to take full advantage of solid modelling, asconstruction of a semicircular mesh is relatively straight forward.To further reduce the need to construct a large number of submodels, an additionaltransformation was employed to change the curvature of each submodel so that a singlesubmodel (i.e., for a given crack depth a and crack length 2c) could by utilized for bothinterior and exterior defects.To reduce the size of the submodel, symmetry boundary conditions were appliedto nodes along the planes y 0 and z 0. These boundary conditions constraineddisplacement normal to the planes of symmetry. To generate the defect on z = 0, thedisplacement constraints on the elliptical crack plane were released. Boundary conditionson the other two boundaries of the submodel consisted of specified displacements asdiscussed previously. Submodel boundary conditions, and the geometric relationshipbetween the submodel and the coarse model are illustrated diagramatically in Fig. (5.9).Development of a submodel that would accurately predict volumetric expansion andCTOD required preliminary studies to determine the dimensions of the submodel andthe number of elements to employ in the submodel. To determine submodel dimensions,membrane and bending stresses at the boundary were examined as the dimensions hand (see Fig. (5.7)) of the submodel with the largest defect were varied over the range75 < h < 150 mm. 40 < s < 80 mm. Based on the observation that membrane andbending stresses at the boundary were within several percent of those shown in Fig.s (5.5)and (5.6), the submodel dimensions h = 100 mm and s = 80 mm were chosen.Following the initial study to determine correct dimensions for the submodel, a studywas performed to determine the number of elements to employ in the submodel. Toillustrate the approach taken to determine the number of elements to focus at the cracktip. some results are plotted in Fig. (5.11). In this figure, which represents the final stagesof mesh refinement, non-dimensional LEFM stress intensity factors (c.f. Equ. (3.6)) areChapter 5. NUMERICAL INVESTIGATION OF FULL SCALE BEHAVIOUR 68plotted as functions of 2,,,V7 for three levels of refinement of the mesh near the crack tip(5 x 30 mm defect) 3 . As can be seen, there is little variation in the stress intensity factorwith increased degrees of freedom. This was taken to indicate that the near crack tipmesh was sufficiently refined and based on this observation, the mesh shown in Fig. (5.7)was chosen.The final version of the submodel mesh shown in Fig (5.7) contained approximately500 elements and 7000 active degrees of freedom. Run time for this model on an ApolloDN 10000 was approximately six hours. The ANSYS input file for the submodel is listedin Appendix C.5.1.4 LoadingTo load the finite element model in a manner consistent with actual service loading,uniform pressure was applied to the element faces on the interior of the model. Thispressure was applied to the element faces in both the coarse model and submodel; pressurewas also applied to the faces of the interior defects.The model was loaded to 41.37 MPa (6000 psi), a pressure 6.90 MPa (1000 psi)greater than the maximum pressure that a cylinder experiences during a hydrostatictest. Because the analysis was non-linear, it was necessary to step incrementally throughthe loading history. Since the greatest plasticity was expected for the largest defect(5x30 mm), the submodel for this defect was used to determine the correct loadingsequence. Based on the load at which yielding first occurred, the first load step for thisand all other submodels was set equal to 10.34 MPa (1500 psi). To determine the size ofsubsequent load steps and convergence within a load step, criteria based on the maximumplasticity ratio were employed. The plasticity ratio (at an integration point) is defined3 cp is defined in Fig. (3.4).chapter 5. NUMERICAL INVESTIGATION OF FULL SCALE BEHAVIOUR 69byAEPIEelwhere DEPT is the change in plastic strain (between load steps) and E el is the elastic strain(in the current load step). Based on a maximum plasticity ratio of 0.015, the load stepincrement was set equal to 1.72 MPa (250 psi). Within each load step, the solution wasconsidered converged when the maximum plasticity ratio was equal to 0.005.Several solution methods (the Newton-Raphson, modified Newton-Raphson and ini-tial stiffness methods) were investigated as a means of solving the non-linear problem. Itwas found that per load step, the Newton-Raphson and modified Newton-Raphson meth-ods required relatively few (three to four) iterations to converge, and that the initial stressmethod 4 required considerably greater number (up to 15) of iterations to converge. Thetime taken per iteration in the Newton-Raphson and modified Newton-Raphson methodshowever, actually made these methods slower than the initial stress method. To minimizerun time therefore, the initial stress method was employed throughout the analyses.5.1.5 Calculation of COD, CTOD, and Volumetric ExpansionTo predict failure of NGV cylinders and assess the current standards for cylinder recer-tification, it was necessary to extract COD, CTOD and volumetric expansion data fromthe finite element analysis. The methods used to calculate these quantities are describedin this section. It should be noted that, since there were no provisions for calculatingCOD. CTOD and volumetric expansion in the version of ANSYS (Ver. 4.3/4.4) usedfor the analysis. it was necessary to develop several ANSYS subroutines (macros) andFORTRAN programs for this purpose.In fracture mechanics literature, various definitions for CTOD have been suggested.4 A discussion of the initial stress method is given in Appendix B.Chapter 5. NUMERICAL INVESTIGATION OF FULL SCALE BEHAVIOUR 70These definitions include the displacement at the original crack tip, the crack tip radius,and the displacement at the elastic plastic boundary. [25]. The most often used, andperhaps the most consistent definition of CTOD is the displacement at the original cracktip. This definition of CTOD was adopted in this investigation.Employing this definition of CTOD, an approach was developed for extracting CTODfrom the finite element analysis. This approach is illustrated in Fig. (5.12). This figurerepresents the displacement of nodes perpendicular to the plane of the defect. Note thatdue to symmetry only one-half of the actual defect is represented. Indicated on Fig. (5.12)are the points at which COD and CTOD were measured. Note that the actual COD istwice the displacement of the node at the crack mouth, and that the CTOD is twice thedistance to a line projected from the crack faces.To extract the volumetric expansion from the finite element analysis, an approach wasdeveloped based on concepts typically employed in Computer Aided Design/Engineering(CAD/E). In CAD/E, it is often necessary to find the volume of an arbitrarily shaped,three-dimensional object. One approach often utilized is to subdivide the object intoa number of tetrahedra [71]. The volumes of the tetrahedron are then calculated andsummed to find the total object volume.Following this approach to find the cylinder volume, tetrahedra were defined by theorigin and three points on element faces as shown in Fig. (5.13). Note in this figure thattwo points of the tetrahedron (on the element face) correspond to nodal points while theremaining point (on the element face) corresponds to the center of the element face. Thecenter point was defined using element shape functions to map the element center fromlocal to global coordinates. The volume of the incremental tetrahedra were calculatedusing_ 1Vincr = 6[fril • ({rm} x {rc})]Chapter 5. NUMERICAL INVESTIGATION OF FULL SCALE BEHAVIOUR 71where {ri }, { rm } and { } are position vectors of points on the element face.To facilitate cylinder volume calculations, a FORTRAN program (see Appendix C)was written to perform the above vector operations and sum the incremental volumes. Ateach load step in this program, the nodal displacements are added to nodal coordinatesto update the position vectors.As noted in Chapter 2, there are typically two components which contribute to thevolumetric expansion. The first component is elastic and due to cylinder compliance;the second component is plastic and due to deformation of the cylinder walls. Sincethe current recertification criterion is based on plastic expansion, it was necessary toisolate this component of expansion from total cylinder expansion. This operation wasperformed as shown in Fig. (5.14). This figure illustrates diagrammatically the changein cylinder volume (i.e., from the initial unpressurized volume) as a function of pressure.In Fig. (5.14), it can be seen that with increasing pressure there is an increase in the rateof change in cylinder volume. This non-linear behaviour results in permanent cylinderexpansion since any unloading will occur along the elastic loading line. Hence, to calculateplastic expansion at each load, the elastic loading line was projected from the P-AV curveto P = O.Chapter 5. NUMERICAL INVESTIGATION OF FULL SCALE BEHAVIOUR 725.2 Numerical Results5.2.1 Crack Opening Displacement (COD)Figure (5.15) shows the COD versus pressure for the 5 x 30 mm (deep long), 3.5 x 20 mm(intermediate) and 2 x 10 mm (shallow short) interior cracks; Fig. (5.16) shows the CODversus pressure for the corresponding exterior cracks'. From these figures it can be seenthat significant differences exist in the growth of the COD over the range of crack sizesanalysed. COD versus pressure for the shortest cracks is nearly linear. This suggeststhat for these cracks, yielding is relatively confined. In contrast, COD versus pressurefor the largest cracks is approaching asymptotic with increasing pressure. This suggeststhat for these cracks, yielding is much more extensive.Figure (5.17) shows the COD as a function of crack size at 20.69 MPa (3000 psi)(NGV cylinder service pressure). This figure shows that, as crack depth increases, CODvaries more with crack length. Figure (5.17) illustrates the differences between CODfor the interior and exterior defects. It was found that the COD for smaller exteriorcracks is less than that for the corresponding interior cracks. The opposite of this trendis observed for larger cracks. These differences in COD between interior and exteriorcracks can be explained in terms of the differences in secondary loading. Induced bendingtends to close an internal crack and open an external crack. The faces of an internal crackhowever, are subject to internal pressure which tends to open the crack. From Fig. (5.17),it appears that when the crack size is small, loading due to internal pressure is moresignificant than induced bending. Hence. COD for small internal cracks is greater thanfor corresponding external cracks. When crack size is large however, induced bendingbecomes more significant. Hence, COD for large external cracks is greater than for large'These crack sizes are plotted in Fig.s (5.15) and (5.16) (and elsewhere) to illustrate behaviour overthe range of crack sizes analysed.Chapter 5. NUMERICAL INVESTIGATION OF FULL SCALE BEHAVIOUR 73internal cracks.Figure (5.18) shows the COD as a function of crack size at 34.48 MPa (5000 psi)(hydrostatic test pressure). This figure exhibits trends similar to those of Fig. (5.17).The secondary effects discussed previously are however, significantly more evident at34.48 MPa (5000 psi) than at 20.69 MPa (3000 psi).5.2.2 Development of the Plastic ZoneTo quantify the extent of yielding and determine whether a condition of instability mightbe expected for larger cracks, the development of the plastic zone was analysed. Fig-ures (5.19) and (5.20) show the percentage of the ligament that yields as a function ofthe COD for interior and exterior cracks. From these figures it can be seen that in severalcases the plastic zone had extended to either the inner or outer wall of the cylinder. Boththe size and location of the defect appear to influence the development of the plastic zone.Location of the defect, that is interior or exterior, appears to be particularly significant.This is most evident for the case of the intermediate size crack; the plastic zone extendingfrom the exterior crack penetrates through the wall while the plastic zone extending fromthe interior crack did not.It is of interest to note that the curves shown in Fig.s (5.19) and (5.20) are discon-tinuous. For each crack size, a point is reached in the development of the plastic zoneat which the rate of change of COD decreases. The reason for this is not entirely clear.Comparison of Fig.s (5.19) and (5.20) with Fig.s (5.15) and (5.16) indicates that discon-tinuities in COD versus rp /(t — a) correspond roughly to points at which COD versuspressure becomes nonlinear.Although of considerable interest, an in-depth study of the development of the plasticzone was beyond the scope of this investigation. This topic will be the subject of futurework.Chapter 5. NUMERICAL INVESTIGATION OF FULL SCALE BEHAVIOUR 745.2.3 Crack Tip Opening Displacement (CTOD)Figure (5.21) shows the CTOD versus pressure for the 5 x 30 mm, 3.5 x 20 mm, and2 x 10 mm interior cracks; Fig. (5.22) shows the CTOD versus pressure for the corre-sponding external cracks. These figures exhibit trends similar to those discussed for theCOD versus pressure; however the CTOD appears to increase somewhat more rapidlythan the COD. This behaviour is attributed to blunting and stretching of the crack tip.Figure (5.23) shows the CTOD as a function of crack size at 20.69 MPa (3000 psi)(NGV cylinder service pressure). Figure (5.24) shows corresponding results at 34.48 MPa(5000 psi) (hydrostatic test pressure). Note that as with the COD, the CTOD for cracks ofthe same depth varies more significantly as crack size increases. The effects of secondaryloading can also be observed.At 20.69 MPa (3000 psi) (Fig. (5.23)), the maximum CTOD for any of the cracksanalysed is less than approximately 0.033 mm, indicating that none of the cracks inves-tigated would be critical during service. (Recall that the critical CTOD for initiation ofcrack growth is estimated at 0.039 mm.) At 34.48 MPa (5000 psi) however, the CTODof five of the cracked cylinders does exceed 0.039 mm 6. Cylinders containing these crackswould therefore be expected to fail (rupture) during a hydrostatic test.For each defect size analysed, failure pressure was predicted using the CTOD versuspressure plots. These predictions, based on the pressure at which CTOD becomes equalto 0.039 mm are listed in Table (5.2).Further use was made of the CTOD versus pressure plots to predict the dimensions ofdefect which will lead to failure at 34.48, 37.92 and 41.37 MPa (5000, 5500 and 6000 psi).These defect dimensions are shown in Fig. (5.25). As will be seen in Chapter 8, thecurve corresponding to failure at 34.48 MPa (5000 psi) is of particular interest, as it6 CTOD of the 2 x 15 mm cracks is only slightly less that the critical value and should, given numericaland experimental error, be considered critical.Chapter 5. NUMERICAL INVESTIGATION OF FULL SCALE BEHAVIOUR 75predicts the defect dimensions which will lead to rupture type failure of a cylinder duringa hydrostatic test.5.2.4 Volumetric ExpansionFigure (5.27) shows the finite element modelling result for elastic expansion of a NGVcylinder. Also shown in Fig. (5.27) are the experimentally measured results. Theseresults were obtained from hydrostatic tests performed on two as received (uncracked)NGV cylinders. Agreement between the predicted and measured results is within 5 %indicating that both the finite element model and that computational method used tocalculate volumetric expansion were accurate.Figure (5.28) shows the finite element results for plastic expansion of cylinders con-taining 5 x 30 mm, 3.5 x 20 mm and 2 x 10 mm interior defects. Figure (5.29) showsthe results for cylinders containing corresponding exterior defects. Most significant isthe magnitude of the results shown in these figures. Comparison of Fig.s (5.28) and(5.29) with Fig. (5.27) indicates that the plastic expansion was, in all cases, several or-ders of magnitude less than the cylinder elastic expansion. Note however that the plasticexpansion for the larger defects was becoming asymptotic.Figure (5.30) shows the plastic expansion as a function of crack size at 34.48 MPa(5000 psi) (recertification pressure). From this figure, it can be seen that the plasticexpansion versus crack size follows a trend similar to the COD/CTOD versus crack sizerelationships. That is. the plastic expansion for a given crack depth varied increasinglyas crack length increased. Also similar to the COD/CTOD results was that, in general,plastic expansion of cylinders containing exterior defects was greater than for interiorcracks when defect depth size was large (i.e., a > 3.5mm, c > 10mm).As noted in Chapter 2, current standards for NGV cylinder recertification requirethat a cylinder should be removed from service if the plastic expansion measured duringChapter 5. NUMERICAL INVESTIGATION OF FULL SCALE BEHAVIOUR 76a hydrostatic test exceeds 10 % of the total (elastic plus plastic) expansion, i.e., whenVp/ > 0.100Vtotai^ (5.1)Noting that Otitotat = AVE./ AVpi, this criterion can be writtenAVpi > 0.110Vei^(5.2)to give the volumetric expansion criterion in terms of elastic expansion.Equation (5.2) can be used to predict which defects investigated would be consid-ered tolerable under current standards for recertification. As noted, elastic expansionsresults (Fig. (5.27) predict that elastic expansion is 360 cc at hydrostatic test pressure(34.48 MPa (5000 psi)). This means that to be considered unfit for service under cur-rent standards, cylinder plastic expansion should exceed 40 cc. Plastic expansion results(Fig. (5.30)) however, predict plastic expansions that are significantly less (by a factorof 100) than this.Thus, based the volumetric expansion criterion, finite element results indicate thatfor the crack size range investigated, all defects would be considered tolerable under thecurrent standards for cylinder recertification.Chapter 6EXPERIMENTAL INVESTIGATION OF FULL SCALE BEHAVIOURIn this stage of the investigation, a series of full scale hydrostatic tests were performed.These tests were conducted for several reasons. The primary reason was to gather datawhich could be used to verify numerical predictions of cylinder behaviour (failure pres-sures and volumetric expansions). The second reason for the large scale tests was toinvestigate the feasibility of an acoustic emission based recertification protocol for NGVcylinders.6.1 Introduction to Acoustic EmissionProcesses at the crack tip, such as sudden crack extension or plastic deformation releaseelastic energy. This energy is propagated in the form of longintudinal and transversewaves. As these waves approach and are reflected from a free surface, they cause finitedisplacement of the surface. This displacement can, with proper transducers and instru-mentation, be measured. Because P- and S-waves are effectively sound waves in a solidmedium, they are commonly referred to as acoustic emission.In a manner analagous to seismology, acoustic emission techniques are based on theanalysis/interpretation of the surface displacements which result due to remote processes(i.e., at the crack tip). The fundamental concept behind the approach is that acousticemission can be used to quantify the severity of the emitting source (i.e., defect). Tothis end, a great deal of theoretical (e.g. [72]) and empirical (e.g. [18, 10]) research hasbeen done in an attempt to correlate fracture mechanics parameters (i.e., stress intensity77Chapter 6. EXPERIMENTAL INVESTIGATION OF ...^ 78factor, plastic zone size, etc.) with AE signal characteristics. While progress in thisendeavor has been made, there is to date no unified approach to AE testing due tothe variables which influence AE (material properties, loading, etc.) and AE detection(frequency response of sensors, acoustic coupling of sensors to structure, backgroundnoise, etc.). For this reason, interpretation of AE is often based on empirical correlationsfor a given configuration and the experience of the AE technician.Despite noted drawbacks of AE testing, there are a number of advanges which makethe method a desirable non-destructive testing technique. The primary advantage is theability to passively monitor all regions of a structure remotely. This means, in the case ofpressurized components with must be periodically hydrotested, that recertification canbe done in-situ, eliminating the need for disassembly. This can mean a significant savingin cost due to the elimination of labour and component down-time.While a great deal of research on AE of pressurized components has been performed,perhaps the most significant from the point of view of the present investigation, is that ofBlackburn and Rana [18] and Barthelemy [10] who, independently, have studied the useof AE based recertification of seamless steel cylinders. Blackburn and Rana studied AEin DOT 3AAX and 3T cylinders. Employing a 40 dB threshold, these investigators wereable to isolate flaws which, based on available fatigue growth rates, would grow to criticalduring the following inspection interval. Based on the strength of these findings, the U.S.Department of Transport currently permits the use of AE testing for recertification inlieu of hydrostatic testing. Barthelemy has studied AE of defects in a similar class ofcylinders (in service in Europe) and noted the onset of AE from defects at pressuresabove approximately 1.5 x service pressure. While this investigator recommends use ofAE in addition to a standard hydrotest, he notes that on the basis of AE test results,the recertification interval could be doubled.Chapter 6. EXPERIMENTAL INVESTIGATION OF ... ^ 79Given the potential for application of AE techniques for recertification of NGV cylin-ders, it was decided to conduct a series of full scale AE tests prior to burst testing ofcylinders. These tests are described in the following sections.6.2 Experimental Setup and Procedure6.2.1 Hydrostatic Testing EquipmentA Galiso Hydroclose Model MAK-6B hydrostatic testing system with automatic controlswas used for pressurizing cylinders. This system, illustrated schematically in Fig. (6.1),consists of a test chamber and a control console, and utilizes an air driven two stagepump capable of delivering up to 68.95 MPa (10,000 psi) pressure. The test chamber isconstructed of heavy gauge steel, a feature considered essential to provide protection inthe event of cylinder failure during planned tests. Cylinders are suspended within the testchamber by means of a special high pressure spud attached to the lid of the test chamber.The lid itself is sealed in place by a pressurized rubber diaphram. The automatic controlsof the system permit choice of a variety of loading sequences, including extended hold ata given pressure and cyclic pressurization/depressurization.To facilitate planned AE tests, the test chamber of the Hydroclose unit was modifiedto permit passage of low noise signal cables from the cylinders to the Locan AcousticEmission (AE) system (described in Sec. 6.2.2). The control console was also modifiedto permit acquisition of a pressure signal from the unit's pressure transducer.6.2.2 Acoustic Emission Monitoring EquipmentA Physical Acoustics Corporation Locan AT based system was used to collect, and storeacoustic emission (AE) data. This system is capable of collecting AE data from up to sixchannels. and parametric (voltage) inputs from up to four channels. Acoustic emissionChapter 6. EXPERIMENTAL INVESTIGATION OF ... ^ 80data arrives at the Locan from sensors attached to the structure under investigation (seeinset, Fig. (6.2)). These sensors are essentially displacement transducers which utilize apiezoelectric crystal to sense motion beneath a wear-plate. Various types of AE sensors,with different resonant frequencies are available; in this investigation, 150 kHz sensors,with integral 40 dB preamplifiers were employed.Figure (6.3) shows the signal waveform of a typical acoustic emission event (or 'hit').Also indicated on this figure are a number of parameters which quantify a hit; theseparameters are:Time of hitThe time of the first threshold crossing; recorded with a resolution of 250 nanosec-onds.Energy countA measure of the area under the rectified signal envelope; recorded with a resolu-tion of 1 energy count.DurationThe time difference between the first and last threshold crossing; recorded with aresolution of 1 micosecond.CountsThe number of threshold crossings between the first and last threshold crossings.Rise TimeThe time difference between the first threshold crossing and the signal peak;recorded with a resolution of 1 microsecond.AmplitudeThe amplitude of the AE signal referenced to 1 microvolt at the sensor; recordedwith an accuracy of 1 dB.The Locan unit processes each hit to extract these quantities and writes this informa-tion to a disk file. Hence, it is the characteristics of a hit (i.e., time of hit, energy count,duration, etc.), not the actual signal waveform which is recorded. Due to reflection or ananomalous waveform, it is possible in some instances to record the same hit twice. Toreduce this possibility, the Locan provides several additional settings which can be usedto isolate incoming hits; these parameters are:Chapter 6. EXPERIMENTAL INVESTIGATION OF ...^ 81Peak Definition Time (PDT)The length of time that the system will continue to sample for a peak after de-tecting a local peak. Used to determine the true peak of the AE waveform; presetto within 1 microsecond.Hit Definition Time (HDT)The length of time that the system continues to sample for threshold crossings.Used to isolate one hit from the next; preset to within 1 microsecond.Hit Lockout Time (HLT)The length of time that the system 'locks out' after recording a hit. Used in con-juction with HDT to eliminate signal reflections; preset to within 2 microseconds.The Locan AT operates under MS-DOS, with test settings such as threshold ampli-tude, HDT, etc. software selected. During a test, incoming data is written to a diskfile, for later analysis, and to a CTR screen for real time analysis. The Locan softwarehowever, provides only limited support for post-processing. To overcome this limitation,specialized software was written in the C programming language to allow AE data tobe downloaded from a Locan AT data file to an APOLLO DN 3000 workstation. Thissoftware also makes it possible to filter the AE data based on hit characteristics such astime of hit, energy, etc. and allows for more versatile batch style processing of data files.6.2.3 Test Cylinder PreparationPrecrackingSix NGV cylinders in total were tested in the experimental investigation of large scalebehaviour. Four of the cylinders were defective and two were 'as received'. All of thecylinders were manufactured by Faber Industries of Italy. The four cracked test cylinders(manufactured in 1981) had previously been in NGV service; the two uncracked controlcylinders (manufactured in 1986) were new.Chapter 6. EXPERIMENTAL INVESTIGATION OF ... ^ 82The four cracked cylinders each contained an external, axially oriented elliptical sur-face crack near the cylinder midpoint. These cracks were artificially introduced' by firstelectrode discharge machining a starter notch, then pressurizing under cyclic hydrostaticpressure (ranging between 2.41 and 20.69 MPa (350 and 3000 psi)) to produce a sharpfatigue crack. The four cracked cylinders were originally prepared and utilized as part ofa parallel investigation 2 of acoustic emission sponsored by the Canadian Gas Association.Before defects were introduced, the four test cylinders were visually examined. Noevidence of external surface corrosion, denting, gouging or external cracks, other thanthose artificially introduced, was found. While it was not possible to perform a detailedexamination of the interior of cylinders, it was apparent that some corrosion of the interiorwalls had occurred. After the cylinders were examined, they were designated Tanks A-D;Table (6.1) lists the corresponding fatigue precrack sizes 3 .Acoustic Emission Sensor and Clip Gauge PlacementThe general arrangement of the experimental setup is illustrated diagrammatically inFig. (6.2). As indicated, four sensors mounted in a rectangular pattern were employedin each AE test. This arrangement placed the known defect in the center of the pattern.Two sensors where located approximately 32 cm above the defect, each 90 degrees aroundthe cylinder from the defect. The remaining two sensors were located approximately 32cm below the defect, again 90 degrees around the cylinder from the known defect.To ensure proper transmission of AE during the tests, the sensor mounting area wasfirst cleaned (using a combination of coarse (150 grit) and fine (400 grit) sandpaper) toremove rust and paint. After cleaning the sensor area, sensors were mounted on the cylin-ders using specially designed magnetic holders. These holders applied the force requiredI This operation was performed at PowerTech Labs, Inc.2 In which the author was involved.3 Measured during post burst test examination of the cylinders.Chapter 6. EXPERIMENTAL INVESTIGATION OF ... ^ 83to ensure good transmission of AE signal from the cylinder to the sensor (approximately2 lbf), and eliminated lateral movement of the sensors due to cylinder expansion. To fur-ther ensure good transmission of AE, an ultrasonic couplant was applied to each sensorwear-plate before the sensors were mounted. Prior to each test, an AE pulser unit wasused to check the response of each sensor and ensure proper acoustic coupling.Crack opening displacement during the AE tests was monitored using a double can-tilever type clip gauge. This gauge was fixed between two knife edges which were bondedto the cylinder with an epoxy adhesive. The distance between the knife edges (seeFig. (6.2)) was approximately 0.125 in. Output from the clip gauge was first run througha DC amplifier and then balanced using a bridge amplifying meter (BAM). After bal-ancing the clip gauge, output from the BAM was fed to one of the Locan AT parametricchannels. Before each test, the clip gauge was calibrated using a specially designed slidingblock/caliper fixture.Once the sensor and clip gauge placement and all calibrations were complete, thecylinder was hung inside the test chamber of the Hydroclose for testing. This chamber,as noted in Sec. 6.2.1, is constructed of heavy gauge steel. This feature, in addition tothat of a rubber diaphram (to secure the lid of the test chamber) and flexible tubing (tosupply pressure to cylinders), isolated the cylinders from extraneous noise which mayhave interfered with recording of the AE data.With all instrumentation ready and the cylinder in place, the AE test was begun.These tests were performed in accordance with a specially designed procedure, which isdiscussed in the following section.6.2.4 ProcedureThe testing procedure was designed to provide information to assess both the feasibility ofacoustic emission as an inspection technique, and the validity of hydrostatic test criteria.Chapter 6. EXPERIMENTAL INVESTIGATION OF ... ^ 84Since both acoustic emission, and volumetric expansion are load history dependent, itwas necessary to select suitable measurement techniques and correctly sequence the testprocedures to ensure valid data. This made necessary a relatively elaborate procedure,which would normally not be required in standard cylinder recertification.Acoustic Emission Test ProcedureA common observation in AE testing is relatively low levels of AE activity at loadsless than a previously attained maximum. This effect (known as the Kaiser effect), wasconsidered a constraint when designing the procedure, as it would degrade the quality ofresults which could be obtained from a cylinder subsequent to the application of givenpressure. Hence, to realize greatest economy from available cylinders, it was necessaryto design a procedure which would yield the maximum amount of data in a single test.In designing the procedure, it was also recognized (for safety reasons), that in anypotential AE recertification protocol, it would be desirable to utilize a pressure as close tooperating pressure as possible. That is, it would be desirable to utilize a recertificationpressure of 1.1 to 1.3 x operating pressure (22.75 to 26.89 MPa (3300 to 3900 psi)),rather that the current 1.67 x operating pressure (34.48 MPa (5000 psi)). Without aprior knowledge of the AE characteristics of NGV cylinders however, it was not possibleto ascertain a suitable test pressure in advance. Hence, it was decided to monitor AE upto the current hydrostatic test. pressure. To investigate the feasibility of utilizing a lowerrecertification pressure however, it was desirable to design a procedure which yieldedvalid AE data at a number of pressure increments which would be some fixed percentage(i.e., 110, 120, etc.) of normal service pressure.The procedure ultimately decided upon, with consideration to gaining the maximumdata from available cylinders and to assessing the feasibility of utilizing a lower recertifi-cation pressure. was to pressurize cylinders from 20.69 to 34.48 MPa (3000 to 5000 psi)Chapter 6. EXPERIMENTAL INVESTIGATION OF ... ^ 85in increments of 2.07 MPa (300 psi). To eliminate residual AE at each pressure incre-ment (i.e., Felicity ratio), cylinders were pressurized to each peak pressure a total ofthree times. Following this procedure resulted in an AE test which lasted approximately2 hours.Volumetric Expansion Test ProcedureTo determine whether test cylinders would be recertified under current standards, it wasnecessary to determine both total (elastic plus plastic), and plastic cylinder expansion at34.48 MPa (5000 psi). In practice (see Sec. 2.4.2), this would be done by noting the quan-tity of water displaced from the test chamber as a cylinder is pressurized to 34.48 MPa(5000 psi), and the volume of water displaced after the cylinder is depressurized. Sinceduring planned AE tests, cylinders would be pressurized to 34.48 MPa (5000 psi), it wasdesirable to measure volumetric expansion simultaneously as the AE tests were beingperformed. Difficulties in isolating the piezoelectric sensors from water however, madethis impossible.Since previous pressurization to a given pressure does not affect elastic expansionmeasured during a subsequent pressurization, it was possible to measure elastic expansionof the cylinders following the AE tests. Elastic expansion therefore, was measured duringa hydrostatic test performed after the AE tests. Plastic expansion of each cylinder wasmeasured indirectly, by calculating the difference in cylinder volume before and after theAE tests. Cylinder volumes were determined from the mass of water which cylindersheld. The following result was used to relate cylinder mass before and after the AE teststo plastic expansion.1— (e-± — 1) — — ni2)pl — 1P2^P2Chapter 6. EXPERIMENTAL INVESTIGATION OF ...^ 86where V1 is the inital cylinder volume, given by- M= Pland M is the mass of the empty cylinder, m 1 is the mass of the cylinder and water beforethe AE test, m 2 is the mass of the cylinder and water after the AE test, pl is the densityof water before the AE test, p 2 is the density of water after the AE test, and M is themass of the cylinder. The densities p1 and p2 correspond to the temperature of the waterwhen the cylinders were weighed.Summary of Test Procedure1. Visually inspect cylinder for damage (cuts, gouges, etc.) and corrosion.2. Fill cylinder with water and allow temperature to equalize. Weigh cylinder andrecord water temperature.3. Prepare cylinder for AE test (prepare sensor mounting area, mount knife edges,etc.).4. Mount AE sensors, verify sensor output using artificial source (pulser), calibrateand mount clip gauge. Insert cylinder into test chamber.5. Perform AE test according to the following schedule. Pressurize three times to eachpeak pressure and hold for two to five minutes at each peak pressure.Chapter 6. EXPERIMENTAL INVESTIGATION OF ...^ 87Pressure range % of OP0-20.69 MPa (0-3000 psi) 1000-22.75 MPa (0-3300 psi) 1100-24.82 MPa (0-3600 psi) 1200-26.89 MPa (0-3900 psi) 1300-28.96 MPa (0-4200 psi) 1400-31.03 MPa (0-4500 psi) 1500-33.10 MPa (0-4800 psi) 1600-34.48 MPa (0-5000 psi) 1676. Remove cylinder from test chamber. Remove sensors and clip gauge.7. Reweigh cylinder and record water temperature (to determine plastic expansion).8. Perform hydrostatic test (to determine elastic expansion).Retest/Burst Test ProcedureTo gather additional data which could be used to more clearly identify AE characteristicsof defective NGV cylinders, it was decided to retest the defective cylinders. Due tounexplainably high levels of AE observed during the initial test of control Tank Z, it wasalso decided to retest this cylinder.It was expected that, due to pressurization to 34.48 MPa (5000 psi) during the initialtests, some plasticity at the crack tip occurred. On the assumption that this plasticitywould contribute to the Kaiser effect and significantly degrade AE activity during plannedretests, it was decided to cycle cylinders between 2.41 and 20.69 MPa (350 and 3000 psi)to grow and sharpen the artificial defects. Hence, upon completion of the initial AE tests,each of the cracked cylinders was cycled between 2.41 and 20.69 MPa (350 and 3000 psi)pressure.Chapter 6. EXPERIMENTAL INVESTIGATION OF ...^ 88Fatigue cracking of cylinders was done in the Hydroclose unit described previously.After placing the cylinders in the test chamber. the automatic controls were set to pressur-ize to 20.69 MPa (3000 psi), hold for one second, and depressurize to 2.41 MPa (350 psi).The duration of each cycle was approximately one minute. During fatigue cracking, crackopening displacement was monitored via the clip gauge described previously. Pressuriza-tion/depressurization was continued until the clip gauge indicated an increase in crackopening displacement. This increase was taken as an indication that the crack had grownthrough the plastic zone, as due to crack tip plasticity, a latent period of constant crackopening displacement would be expected. An increase in crack opening diplacement wasobserved for all cylinders except Tank A, which was cycled 3000 times with no apparentcrack growth.When fatigue cracking of the cylinders was complete, the AE retests were performedaccording to the procedure described previously. At the end of the each AE retest,cylinders were monotonically pressurized to failure. During these burst tests, AE wasmonitored in the same manner as during cyclic AE tests. To perform the burst tests, theHydroclose pump was simply allowed to operate at maximum capacity until the pressurewas sufficient to rupture the cylinders.6.2.5 AE SettingsAs noted in Sec. 6.2.2, AE signals generated by an emission source are detected by meansof sensors which are constructed to take advantage of the mechanical resonance of thea piezoelectric crystal. To maximize sensitivity when testing steels, 150 KHz sensorsare typically employed. Hence, 150 kHz sensors with integral 40 dB preamplifiers wereemployed in the test performed in this investivation. To minimize noise. (i.e., to maximizethe signal to noise ratio), shielded cables and Locan band-pass filters were employed.Locan setting for the AE tests were selected on the basis of settings reported in theChapter 6. EXPERIMENTAL INVESTIGATION OF ... ^ 89literature (e.g. [18]) and through discussions with investigators at Powertech Labs, Inc. 4These settings, listed below were held constant for all tests to ensure consistency insubsequent comparison the AE results for the different cylinders.Amplitude threshold^30 dBGain^ 58 dBPDT 500 microsecondsHDT^ 1000 microsecondsHLT 1000 microsecondsTime driven data sampling^2 secondsA Locan AT threshold amplitude of 30 dB was selected to allow acquisition ofevents associated with crack tip plasticity, expected to have an amplitude of between30-40 dB [18]. Preliminary AE test performed on discarded cylinders indicated thathydraulic noise (due to pressurization) was significant at low pressures due to associatedhigh flow rates, but negligible above approximately 10.34 MPa (1500 psi). To eliminatethis noise, all AE data obtained and presented in this report has been filtered to excludeany hits which occurred at pressures less than 6.90 MPa (1000 psi). The AE data pre-sented has also been filtered to exclude any hits with a total count of one, since these hitswere believed to be noise (i.e., a signal which just exceeded the threshold amplitude).To allow the greatest flexibility in the data analysis, incoming AE data was not filteredon any AE characteristic or parametric input (e.g., pressure, COD) other than thosenoted above. In subsequent analysis, the C program (noted in Sec. 6.2.2) was employedto download and filter the AE results based on selected AE hit characteristics. As thedata was obtained over three pressure cycles, data from each cycle was first isolated.analysed and then compared with data from the other cycles.4 During this investigation, AE testing of steel NGV cylinders was simultaneously performed at Pow-ertech Labs, Inc. as part of the CGA sponsored project noted previously.Chapter 6. EXPERIMENTAL INVESTIGATION OF ... ^ 906.3 Results6.3.1 Hydrostatic Test ResultsElastic expansion results for the four test cylinders and the two control cylinders aregiven in Fig. (6.4). The figure shows results scattered about two straight lines; one linerepresenting the test cylinders, the other the control cylinders. Regression analysis onthe data givesAVei = (7.76 x 10 -2 )p — 2.93for the test cylinders, andAV,/ = (6.99 x 10 -2 )p — 2.28for the control cylinders. Comparison of the slopes indicates that the compliance of thetest cylinders is approximately 11 percent greater than that of the control cylinders. Max-imum elastic expansions (elastic expansions measured at 34.48 MPa (5000 psi), 1.67 xservice pressure) are given in Table (6.2); average values are 387.8 cc for the test cylin-ders and 348.5 cc for the control cylinders. The elastic expansion for which the cylinderswould be rejected was calculated and found to be 472.6 cc. The measured maximumsare all less than the rejection elastic expansion indicating that all test cylinders wouldbe recertified using the elastic expansion failure criteria.Plastic expansion results for three of the test cylinders and the two control cylindersare given in Table (6.3). (Temperature, used in the calculation of plastic expansion,was not measured for Tank A. therefore plastic expansion measurements for this tankhave been omitted.) Plastic expansion ranged from 21.7 to 27 cc for the test cylindersand from 0 to 0.9 cc for the control cylinders. The insignificant plastic expansion ofthe control cylinders is due to the previous hydrostatic testing to 34.48 MPa (5000 psi)during quality control, prior to delivery as part of factory inspection. The rejectionChapter 6. EXPERIMENTAL INVESTIGATION OF ... ^ 91plastic (10 % of total (elastic plus plastic) expansion) was calculated and is also listed inTable (6.3). Reviewing the results in this table indicates that all of the cylinders wouldbe recertified under current CTC regulations as all measured plastic expansions are lessthan the corresponding rejection values.6.3.2 Acoustic Emission ResultsThe AE data was analyzed and plotted in a number of ways in an attempt to identifya useful parameter or criteria for correlating the data to defect severity. The followingsections present and compare the results, and identify areas where correlation and possibleuse as inspection criteria may be possible.AE Hits Past Previous PressureAs hit activity is often considered a parameter which correlates with defect severity, AEdata was initially reviewed for trends in the number of hits up to peak pressure in eachcycle. This analysis suggested that the Kaiser effect had occurred between cycles. Thiseffect, as noted previously, tends to eliminate hit activity below a previously attained peakpressure. As the Kaiser effect had tended to eliminate hit activity below the previouslyattained pressure, it was deemed appropriate to plot and analyse hits past previouspressure. This approach was also considered more consistent, as it would effectively filtercycle to cycle variations of the Kaiser effect (i.e., cycle to cycle Felicity ratio).Hits past previous pressure for the full amplitude range of hits (30 to 70 dB) for theinitial test first, second and third cycles are plotted in Fig.s (6.5), (6.6) and (6.7) respec-tively: corresponding results for the retests are plotted in Fig.s (6.8). (6.9) and (6.10).(Note that the data from the initial test of cylinder A, the first test in the series. wastested at pressures based on an operating pressure of 24.13 rather that 20.69 MPa (3500rather than 3000 psi) and that the points are at intervals of 2.41 MPa (350 psi) ratherChapter 6. EXPERIMENTAL INVESTIGATION OF ...^ 92than 2.07 MPa (300 psi)). Hits past previous pressure for the filtered amplitude range(40 to 70 dB) of hits are plotted in Fig.s (6.11) through (6.16).Initial review of these figures does not yield any easily recognizable, generally increas-ing or decreasing trend. The results indicate a highly non-linear relationship betweenhits past previous pressure and peak pressure which is not reproduced from cycle to cycleor upon retesting the cylinders.First cycle results Focusing attention on intial test first cycle results for cylinders having approximately thesame size defect (Tanks B, C, and D) shown in Fig. (6.5), it can be seen that there isno clear correlation between the results for the different cylinders. Varied levels of AEhits emitted by Tank D below 28.96 MPa (4200 psi) tended to increase above 28.96 MPa(4200 psi). This contrasts the low levels of AE hits emitted by Tank B (which containeda slightly larger defect that Tank D), which tended to increase in the midrange pressuresand decrease again at higher pressures, and the relatively constant levels (20-30) of AEhits emitted by Tank C throughout the pressure range. Turning attention to the firstcycle retest results shown in Fig. (6.8), it can be seen that there is some weak correlationbetween the results for the different cylinders. High levels of AE hits emitted by all cylin-ders at low pressures (<;::-.,24.82 MPa (3600 psi)) tended to decrease to some minimumin the midrange pressures and increase again at higher pressures.Further review of the results in Fig.s (6.5) and (6.8) reveals that the levels of hitsemitted by the control cylinders varied significantly. Relatively constant and low levelsof AE hits emitted by Tank Y. contrast the varied and typically greater levels of AE hitsemitted by Tank Z. Hits emitted by Tank Z tended to be high at low pressures, somewhatless in the midrange and greater again in the high pressure range. Comparison with thedata for the cracked cylinders indicates that activity of Tank Z was for the most part,Chapter 6. EXPERIMENTAL INVESTIGATION OF ... ^ 93greater than that of cracked cylinders.At the particular time of testing Tank Z, mechanical problems with the hydrotestpump were experienced and it was suspected that the high levels of AE observed mightbe due to mechanical noise. After full reconditioning and testing of the pump, Tank Zwas retested 5 . Comparing figures for the inital and retest results indicates that during theretests, Tank Z reproduced the high levels and varied distribution of AE observed duringthe initial tests. In an attempt to resolve this clearly anomalous behaviour, Tank Z wassectioned and microscopically examined following the tests. This examination revealed alarge number of microcracks in the wall on the interior of the cylinder. These microcracksare believed responsible for the high levels of AE observed for Tank Z.Disregarding Tank Z results, it can be seen with reference to Fig.s (6.5) and (6.8) thatcontrol (Tank Y) cylinder AE hits emitted were generally less than the cracked cylinderAE hits emitted. This suggests, although far from conclusively, that some threshold levelof AE hits exists. Additionally, the general trend for cracked cylinder AE hits emittedto reach some minimum in the midrange pressures (at MPa (4200 psi)), andsubsequently increase suggests that cracked cylinder response is different from uncrackedcylinder response, as this trend was not observed in Tank Y results. While there is aneed for more substantiative results, these observations clearly enhance the potential forhits past previous pressure as an inspection criterion.An analysis of initial and retest first cycle amplitude filtered to retain only those hitsbetween 40 and 70 dB (Fig.s (6.11) through (6.14)) indicates reduced levels of AE hits,and thus suggests that the majority of hits emitted were of relatively low amplitude.These results however, still do not reveal a clear trend or threshold level of AE hits.While the relevence of filtering the data cannot be established, it clearly reduces and'As Tank Z was retested, retest results for this cylinder appear with retest results for the crackedcylinders in all the figures referred to in this chapter.Chapter 6. EXPERIMENTAL INVESTIGATION OF ...^ 94condenses the results and thus may in itself have merit.Second and third cycle resultsThese results. shown in Fig.s (6.6) and (6.7) (initial tests) and Fig.s (6.9) and (6.10)(retests), indicate a significant reduction in AE hits past previous pressure emitted byall cylinders throughout the pressure range. In comparison to the results for the crackedcylinders, relatively few hits were emitted by Tank Y. This would appear to supportearlier comments that a threshold in hits past previous pressure may exist. Althoughthere are several data points (e.g., Tank B (24.82 MPa (3600 psi)), Tank C (28.96-33.10 MPa (4200-4800 psi)), Tank D retest (22.75 and 33.10 MPa (3300 and 4800 psi)),etc.) which clearly do not support this observation, it would appear to enhance thepotential for utilizing AE hits past previous pressure as an inspection criterion.Unfortunately, amplitude filtering of second and third cycle results (Fig.s (6.12),(6.13), (6.15) and (6.16)) as described previously did not yield any further useful in-formation. As identification of a threshold level becomes even more uncertain, amplitudefiltering is for these results, not beneficial.AE Hit Rate/Pressure IncreaseFurther review of hit data suggested that hit rates (slopes of hits versus pressure curves)near peak pressures might have potential as an inspection criterion. This concept ap-peared justified on the grounds that hit activity which might occur due to crack extensionwould likely occur near peak pressure of a given cycle, whereas hit activity resulting fromother sources (i.e., corrosion) would likely occur throughout the pressure cycle.Adopting this approach, hit rates for the final 0.69 MPa (100 psi) of each cycle werecalculated. The results of these calculations for the first, second and third cycles of theinitial and retests (full amplitude range) are shown in Fig.s (6.29) through (6.34). ReviewChapter 6. EXPERIMENTAL INVESTIGATION OF ... ^ 95of these results indicates that, during the initial tests, hit rates of the cracked cylindersvaried with peak pressure and that hit rate generally decreased with each load cycle.During the retests, hit rates of the cracked cylinders (all cycles) tended to decrease in themidrange pressures, reach some minimum (between approximately 26.89 and 28.96 MPa(3900 and 4200 psi)) and then increase. Hit rates during these tests tended either toremain at the same level or to increase with each cycle. Hit rates of Tank Y (initialtests), were in general less than those of the cracked cylinders during either the initial orretests.To resolve discrepancies between the initial and retest results, hit rates were calculatedusing only amplitude filtered hits (i.e., hit amplitude greater than 40 dB). The resultsof these calculations (hit rates for the first, second and third cycles of the initial andretests) are shown in Fig.s (6.35) through (6.40). Reviewing these figures suggests thatamplitude filtering is not a valid approach, as it provides little in the way of isolatingtrends. Moreover, comparing Fig.s (6.29) through (6.34) which indicate a general decreaseand subsequent increase in hit rate, with Fig.s (6.35) through (6.40) suggests that filteringhit amplitude may actually obscure existing trends.Comparing initial test results for control Tank Y with those for the cracked cylindersindicates that a threshold level of hit rate might exist. While this observation enhancesthe potential for use of hit rate as an inspection criterion, inconsistencies between initialand retest results, and difficulties in identifying threshold hit rate, clearly diminish thepotential.AE Hold Time HitsHold time hits is a parameter which reflects time dependent energy release processes (i.e.,creep, plastic flow, incremental crack growth, strain hardening, etc.) at the crack tip.To assess the potential for use of this parameter as an inspection criterion, the numberChapter 6. EXPERIMENTAL INVESTIGATION OF ...^ 96of hits which occurred during the hold times were plotted as functions of peak cyclicpressure.Review of initial and retest results for the first, second and third cycles (Fig.s (6.17)through (6.22)) indicates that fewer hold time hits were emitted during the initial teststhan in the retests, and that hold time hits emitted by the cracked cylinders were, ingeneral, markedly greater in number than those emitted by the control cylinders. Thispromising observation however, is offset by the sporadic hold time hits emitted by thecontrol cylinders and the fact that there are no discernible trends in the number of holdtime hits emitted by either the cracked or control cylinders. Further, while hold time hitsemitted by the cracked cylinders are in general greater than those emitted by the controlcylinders, it is evident that the presence of a crack is in itself no guarantee that holdtime hits will occur (e.g. cylinder B containing a relatively large defect produced littleor no hold time hits) This fact, combined with the sporadic hold time hits emitted bythe control cylinders, clearly diminishes the potential for hold time hits as an inspectioncriterion.To further analyse the results, amplitude filtered (i.e., amplitude greater than 40 dB)hold time hits were plotted as shown in Fig.s (6.23) through (6.28). Reviewing thesefigures indicates that amplitude filtering reduced the results, but did not serve to isolateany new trend which might increase the potential for hold time hits as an inspectioncriterion.AE Amplitude DistributionHit amplitude is considered a parameter which correlates in some manner with defectseverity. On the basis that hit amplitude might serve as an inspection criterion, AE hitamplitude distributions were plotted for the three cycles of the initial and retests. Areview of these distributions indicated that the greatest frequency of hits was in the 30Chapter 6. EXPERIMENTAL INVESTIGATION OF ...^ 97to 45 dB amplitude range, and that no cycle to cycle trend existed. On the basis of theseobservations, only first cycle, 30 to 45 dB amplitude distributions will be presented.Amplitude distributions for the first cycle of initial and re-tests are plotted for eachpeak pressure in Fig.s (6.41) through (6.56). A review of these figures indicates thatpeak frequencies varied significantly between cylinders and from peak pressure to peakpressure. Generally, initial test distributions indicate that relatively low amplitude hitswere emitted. This contrasts retest amplitude distributions which indicate a significantincrease in higher (i.e. 40 dB or greater) amplitude hits. Although this behaviour mightbe attributed to slightly larger defects during the retests, it is not believed that cracklengths were increased enough during fatigue cracking to influence amplitude distribu-tions in this manner.While it was anticipated that as the pressure increased, the amplitude distributionsof the cracked cylinders would change to reflect some crack tip process such as plasticdeformation or crack growth, it is apparent upon examining the results that no clearlydefined change in fact occurred. Based on the observation that peak frequency generallyfell in the 32 to 40 dB range, it would appear that amplitude distribution is independentof both crack size and peak pressure. (As noted, there were also no significant changein amplitude distributions over the three cycles.) Unfortunately, comparison of the am-plitude distributions of the control cylinders with those of cracked cylinders does notserve to illuminate any trend which might be useful in further explaining results for thecracked cylinders. This clearly makes identification of specific amplitude characteristicsfor cracked cylinders difficult, and reduces the potential for amplitude distribution as aninspection criterion.Chapter 6. EXPERIMENTAL INVESTIGATION OF ... ^ 986.3.3 Burst Test ResultsBurst Test Cummulative AE HitsCumulative AE hits emitted during the burst tests of Tanks A, B, C and D are plottedagainst pressure Fig.s (6.57) through (6.61). The data in these figures indicates relativelylow levels of hits below 34.48 MPa (5000 psi), and an asymptotic increase in hits between34.48 MPa (5000 psi) and failure pressure. A reasonable explanation for the relativelylow levels of activity below 34.48 MPa (5000 psi) is the Kaiser effect. The asymptoticincrease in AE hits is believed due to plastic deformation at and beyond the defects, andthroughout the cylinder wall, as post-test examination of the cylinders indicated thatconsiderable yielding had occurred prior to failure. Although impossible to confirm atthis point, it appears that there is a significant correlation between the volume of yieldedmaterial and AE hit activity. If this correlation can be confirmed and quantified, it mayserve as a means of predicting impending failure during AE recertification.Burst Test Amplitude DistributionAmplitude distributions for the cylinder burst tests are shown in Fig. (6.62). This datashows the AE data obtained as the cylinder was pressurized to failure pressure, however,as the cylinder had been previously pressurized to a pressure of 34.48 MPa (5000 psi),the majority of AE hits were emitted at pressures above 34.48 MPa (5000 psi). Thus,the amplitude distribution for the burst tests are dominated by the characteristics ofthe AE which occurred near failure, rather than proportionally over the full pressurerange approaching failure. Reviewing this data indicates that the amplitude frequencypeak occurs at approximately 32 dB, notably lower than previously observed in the lowerpressure level testing. However, unlike the previous results, this burst data indicatesa definite and consistent amplitude distribution and peak amplitude frequency. If, asChapter 6. EXPERIMENTAL INVESTIGATION OF ... ^ 99suggested in the previous section, AE hits emitted near failure were due to plastic defor-mation, it would appear from these observations that a definite amplitude threshold forplastic deformation exists.Burst Test Crack Opening DisplacementAs has been noted, the primary reason for adopting the CTOD approach in this in-vestigation was that CTOD can be measured experimentally. This meant that actualfailure pressures during planned burst tests could be compared to analytical and numer-ical predictions of failure pressure. Comparison of experimental results with analyticaland numerical predictions was considered important, as this would make it possible toassess the accuracy and/or conservatism of these predictions.Burst tests were performed by monotonically pressurizing each of the four test cylin-ders containing defects to failure. Failure pressures and initial defect sizes for the cylindersare given in Table (7.1). Failure of all of the cylinders during burst tests was due to prop-agation of cracks that initiated at the artificial defects (see Figs (6.63) through (6.65)).Post-test examination of fracture surfaces revealed that in all cases, some yielding hadoccurred in the ligament, and in the wall beyond the ends of defects. The fracture sur-face of the ligament and slightly beyond the ends of defects (2 to 3 mm) was parallel tothe plane of defects, indicating that growth of defects through-wall occurred under planestrain conditions. Beyond this plane strain region, a 45 degree shear type fracture sur-face was observed indicating that growth of defects parallel to the cylinder axis occurredunder plane stress conditions.To assess the suitability of the CTOD approach in predicting failure of NGV cylinders,burst test COD was plotted against pressure as shown in Fig.s (6.66) through (6.69). Thetrend in these figures. as can be seen, is a distinct change in slope (an increase in the rateof change in COD with pressure) prior to failure. This indicates that yielding beyond theChapter 6. EXPERIMENTAL INVESTIGATION OF ...^ 100ends of defects and/or some stable crack growth (tearing) was occurring prior to failure.As noted above, post-test examination of the fracture surfaces indicated that yieldinghad occurred.Chapter 7CORRELATION BETWEEN EXPERIMENTAL ANDNUMERICAL/ANALYTICAL FAILURE PREDICTIONSIn this chapter. failure pressures measured during the large scale tests are comparedwith failure predictions based both on the numerical results discussed in Chapter 5 andwith failure predictions made using several of the analytical methods discussed in Chap-ter 3 (i.e., the CTOD design curve, the CEGB R6 method, the Battelle empirical analysisand a plastic collapse approach). The intent of this comparison is to illustrate the relativeaccuracy/conservatism of the various approaches to predict burst type failure of NGVcylinders. Following the comparison of experimental and numerical/analytical results, afailure criterion is identified which defines limiting defects dimensions for rupture duringa hydrostatic test (i.e., at 34.48 MPa (5000 psi)). In the chapter which follows, it willbe shown that this criterion relates directly to limitations of current standards for NGVrecertifi cation.7.1 Finite Element Failure PredictionsTo verify and assess the accuracy of the numerical approach adopted in this investigation,finite element results presented in Chapter 5 were used to predict COD versus pressurebehaviour and failure pressures for the four test cylinders. Because the defect sizes inthe test cylinders differed from those considered in the finite element analysis, it wasnecessary to interpolate (and in the case of Tank A. extrapolate) the numerical results.To facilitate this interpolation (/extrapolation), all surface defect dimensions were first101Chapter 7. CORRELATION BETWEEN EXPERIMENTAL AND... ^102converted to equivalent through-wall defect dimensions using Fig. (3.15).The predicted COD versus pressure have been plotted against observed COD versuspressure in Fig.s (6.66) through (6.69). Considering various sources of error, agreementbetween the numerical and measured results in the initial portion of the COD versuspressure relationships is good. The difference between these results (slopes of COD versuspressure) is less than 5 % for Tanks A and D (Fig.s (6.66) and (6.69)) and approximately25 % for Tanks B and C (Fig.s (6.67 and (6.68)). One factor that may have contributedto differences between numerical and observed results is internal surface corrosion of thetest cylinders; because corrosion reduces wall thickness, the -e-/t ratio of the test cylinderswas likely somewhat greater than that assumed in the finite element model. This greaterZ'/t ratio would explain the tendency in Fig.s (6.66) through (6.69) for observed COD tobe greater than predicted COD.Although initially (i.e., at lower pressures) in good agreement, differences betweenthe predicted and observed results for all of the cylinders tends to increase as measuredCOD begins to increase rapidly prior to failure. It is believed that this disagreementmay be due to stable crack growth occurring prior to failure of the test cylinders. Sinceno mechanism for stable crack growth was incorporated into the finite element, model,this behaviour would not be predicted. It should be noted that differences between themeasured and predicted results in both the initial and later portions of the COD versuspressure were most significant for the tanks containing the two largest defects (Tanks Band C).To determine the accuracy and estimate the conservatism (if any) of the finite elementfailure predictions made in Chapter 5, numerical CTOD results were again interpolated(/extrapolated) to determine failure pressures based on the critical CTOD (0.039 mm)from the small scale testing. These predicted failure pressures (indicated on Fig.s (6.66)Chapter 7. CORRELATION BETWEEN EXPERIMENTAL AND...^103through (6.69) and listed in Table (7.1)) have been plotted against actual failure pres-sures in Fig. (7.1) 1 . Based on the observed burst pressures, three of the four finite ele-ment/CTOD failure predictions are conservative. Predicted failure pressure for Tank Ahowever, was 73.09 MPa (10,600 psi); although plotted on Fig.(7.1) for completeness,the point representing Tank A should in fact lie to the right of the position indicated.This cylinder however, contained a relatively small (2.2 x 5.6 mm) defect. Subsequentanalyses utilizing different approaches (to be discussed) suggest that for this cylinder,plastic collapse mechanisms may have played a more significant role.7.2 PD 6493 Failure PredictionsIn Sec. (3.6.1), it was noted that the CTOD design curve has been incorporated intoa. design standard, PD 6493, which can be used to assess structural integrity of a widerange of structural components. To assess the applicability of PD 6493 to NM' cylinders,this standard was to predict failure pressures of the four test cylinders.It should be noted that while it is possible to calculate failure pressure using PD 6493,the intent of the standard is to provide a means of calculating allowable defect dimensionsbased on given loading, structural geometry, etc. Because of a number of assumptionsmade in the development of the CTOD design curve, allowable defect dimensions willtypically be much smaller (by up to a factor of 2.5) than critical defect dimensions. Forthis reason, failure pressure calculated using PD 6493 are typically very conservative.In accordance with the PD 6493 (the CTOD design curve approach), a non-dimensionalCTOD, (13., was calculated for each defect size using(I) = CTODc E2R- ay-e'During the burst test of Tank D. AE data saturation caused collection of pressure data to cease atapproximately 41.37 MPa (6000 psi). Actual failure pressure for this cylinder was visually noted to be43.78 MPa (6350 psi).Chapter 7. CORRELATION BETWEEN EXPERIMENTAL AND...^104where T. is the length of an equivalent through-wall defect determined using Fig. (3.15).This non-dimensional CTOD was equated to the CTOD design curve, defined by(Ey)k / Ey —E  0.5- 0.25 4- > 0.5 (7.1)where, from elementary theory of elasticity, the relationship between E I Ey and pressure,p, is given bye pR ( 1 1 v)—Ey ayi 9In Sec. (3.6.1) it was noted that if load is sufficient to cause yielding of the remainingligament, PD 6493 requires that a surface defect by recategorized as through-wall defect.Accordingly, ifa > aF (1 — —a ) (7.2)a surface defect of length 2c must be reclassified as a through-wall defect of length 2c.In making failure predictions as was done here, a defect cannot be recategorized onthe basis of Equ. (7.2) since it is a (i.e., p) which is unknown. Hence, for each cylinder,failure pressure was calculated based on both an equivalent and a recategorized cracklength. The results of these calculations are shown in Fig. (7.2). The open symbols inthis figure represent failure pressures calculated using a crack length 2c (i.e., recategorizeddefects) and the solid symbols represent failure pressures calculated using a crack lengthT. (i.e., equivalent through-wall defects determined from Fig. (3.15)).The results presented in Fig. (7.2) indicate that, with the exception of one point.the CTOD design curve provides conservative estimates of failure pressure. As can beseen, recategorization of defects significantly effects the conservatism of these predictions;failure pressures based on the recategorized defect length are on average approximately50 % lower than those based on an equivalent defect length. It is of note that the TankA prediction based on an equivalent crack length is non-conservative by a considerableChapter 7. CORRELATION BETWEEN EXPERIMENTAL AND... ^105margin while the prediction based on an equivalent crack length is conservative. Thiswould appear to support earlier comments that plastic collapse played an important rolein the failure of this cylinder.7.3 CEGB R6 Method Failure PredictionsTo assess the applicability of the R6 method (discussed in Sec. 3.6.2) to NGV cylinders,the approach was used to predict failure pressures of the four test cylinders. It shouldbe noted (as with PD 6493) that while possible to predict failure pressures using the R6method, the intent of the approach is to provide a means of assessing safe operating loadsfor structures containing defects.In accordance with the R6 method, the non-dimensional parameters (assessmentpoints)Kr =SrKp Pcollapse(7.3)(7.4)were calculated for each test cylinder based on measured failure pressure and defectdimensions. Kr was calculated using the finite element results for stress intensity factorsof Newman and Raju [75] and KID = 105 MPaVmm (from Ref. [13]). Sr was calculatedusing the plastic collapse solution given in Sec. 7.4.The results of the calculations are shown in Fig. (7.4). Recalling that failure ispredicted for any point that lies outside the assessment line, it can be seen that the R6method correctly predicts failure of all the cylinders. Because both Kr and Sr are linearfunctions of load. the ratio OP/OP' (see construction for Tank B in Fig. (7.4)) is theratio of actual to predicted failure pressure. Using constructions similar to that shownfor Tank B. predictions of failure pressure were made for the four test cylinders. TheseChapter 7. CORRELATION BETWEEN EXPERIMENTAL AND... ^106predictions are listed in Table (7.1). Comparing the predictions for the different cylinderssuggests that the R6 method is most conservative for larger defects.7.4 Plastic Collapse Failure PredictionsTo investigate the accuracy of plastic collapse failure predictions, the collapse pressurefor each cylinder was calculated using the solution given by Turner [92]. This solution(see Sec. (3.5.1) ist t /a —1Pcollapse = R aF t /a — 1/mwhere[0.263(2c) 2 1m = 1 +Rtand of is the material flow stress assumed here to be0-F = -2- (GrY + al-1)= 885 MPaThe results of the calculations, shown in Fig. (7.3), indicate that for the range of defectsizes investigated, a plastic collapse approach provides a relatively accurate and con-servative method for predicting burst failure of these cylinders. Noting that the pointrepresenting Tank A is within several percent of the actual failure pressure, it wouldappear that for small defects in NGV cylinders. plastic collapse methods are most ap-propriate.(7.5)7.5 Battelle Empirical AnalysisThe Battelle empirical analysis (see Sec. (3.6.3)), has successfully been applied to theanalysis of failure of seamless compressed gas cylinders containing cracks [28, 29]. ToChapter 7. CORRELATION BETWEEN EXPERIMENTAL AND...^107assess the accuracy of this approach, it was used to predict failure pressures of the fourtest cylinders.The failure criterion for the Battelle analysis is7r Mpae KL7In sec ^2 aF^secl (7.6)where Mp is1 — a/tMp = ^1 — a/MTtand MTCf2^C14  )MT = (1 + 1.255Rt 0.0135 R2t2 —The term Kl , is the material plane stress fracture toughness and the term c' is an equiv-alent crack length (for surface defects) This crack length is defined in terms of the actualsurface area of a part-through defect. For an elliptical defect2c' = l ire2In the original formulation of the Battelle empirical analysis described in Ref. [68],plane stress fracture toughness K1 , was determined using a correlation with Charpyimpact energy C v . Since in the present study neither K 1 , nor Cv data was available, acorrelation between K1, and plane strain fracture toughness (KID ) was employed. Thiscorrelation, from Ref. [18], is=^+ L.4 KIc 41 112t2and gives, for Kk = 105.5 MPa\/m. ay = 822 MPa and t = 7.8 mm, the value Klc =205 MPaVi--n.Using the above relationships, the non-dimensional parameters Mpao /aF and Ki2,7r/8cialwere calculated for each test cylinder at the observed failure pressure. The results of theChapter 7. CORRELATION BETWEEN EXPERIMENTAL AND... ^108calculations have been plotted in Fig. (7.5) along with the failure criterion expressed byEqu. (7.6). As can be seen, there is very good agreement (less that 2.5 % difference)between the experimental results and the curve which predicts failure.It can also be seen in Fig. (7.5) that for all the experimental points, K?c 7r /8c1 > 4.Maxey [68] has suggested that when this is the case, failure is predominantly flow stresscontrolled. For this behaviour, the failure criterion expressed by Equ. (7.6) can be writtenMP Cri9 = CIF^ ( 7. 7)Based on the limited experimental results of this investigation, it appears that this simpleequation may provide an acceptable failure criterion for NGV cylinders.7.6 Choice of an Acceptable Failure Criterion for NGV Cylinders7.6.1 Comparison of Fracture Mechanics Based Failure PredictionsThe fracture mechanics based failure predictions discussed in the last section have beensummarized in Table (7.1). A review of this table indicates that, with the exceptionof Tank A. numerical CTOD and CTOD design curve predictions of failure pressure areconservative (i.e., less than the observed failure pressures). Plastic collapse and R6 failurepredictions are also conservative. Battelle empirical analysis predictions are slightly non-conservative.Focusing attention on the conservative failure predictions, it can be seen that CTODdesign curve and R6 predictions are somewhat more conservative than the finite elementCTOD (ignoring Tank A) and plastic collapse failure predictions. Comparing the finiteelement CTOD and plastic collapse predictions indicates that, with the exception ofTank A. the finite element CTOD predictions are more accurate.If the Tank A prediction is ignored, the overall accuracy and conservatism of the finiteelement CTOD failure predictions for Tanks B, C. and D supports use of the CTODChapter 7. CORRELATION BETWEEN EXPERIMENTAL AND... ^109approach to predict failure of NGV cylinders. As has been noted. Tank A contained arelatively small defect. The relative accuracy of the plastic collapse failure prediction forTank A supports the view that this cylinder failed due to plastic collapse. This suggeststhat for small defects in NGV cylinders, plastic collapse analysis is appropriate.Given the experimental support for use of the finite element CTOD approach forlarger defects and a plastic collapse approach for smaller defects, it is interesting tofurther analyse the failure predictions made using these two approaches. Comparingthe failure predictions made using the two approaches indicates that. with the exceptionof Tank A, plastic collapse predictions are slightly more conservative than the finiteelement CTOD predictions. This observation can be explained with the aid of a simpleexample. For a wide plate with a through thickness defect subject to remote tension, theexpressions for failure stress areCTOD,EaFa, =for elastic-plastic behaviour and= (1 — w aFfor fully plastic behaviour. If these expressions are plotted as functions of c/14 7 as shownin Fig. (7.6), it can be seen that they cross and that to the right of the intersection (thelargest range of c/I47 ), the CTOD failure stress is most conservative. To the left of theintersection however, because materials cannot support infinite stress. plastic collapsefailure stress is most conservative.Based on the conservatism of plastic collapse failure predictions with respect to thenumerical CTOD failure predictions. it would appear that the defect sizes in the testcylinders correspond roughly to the intersection of the two curves shown in Fig. (7.6).Recalling that Tank A contained a relatively small defect (see Table (6.1)), the pointN./77—re (7.8)(7.9)Chapter 7. CORRELATION BETWEEN EXPERIMENTAL AND... ^110representing this cylinder would (notionally) lie to the left of those representing theother cylinders. This would explain, given the asymptotic behaviour of the CTOD failurestress, why predicted failure pressure of Tank A is highly non-conservative.7.6.2 Limiting Defect Sizes for Rupture of NGV Cylinders During a Hydro-static TestIn the chapter to follow, an assessment is made of the ability of current standards forNGV cylinder recertification to ensure in-service cylinder integrity. For reasons whichwill become apparent, it was necessary, in making this assessment, to predict the defectdimensions which will lead to cylinder failure during a hydrostatic test.In Sec. (5.2.3) numerical CTOD estimates of critical defect dimensions were employedto develop a set of curves (see Fig. (5.25)) which define critical defect dimensions (i.e.,a x 2c) at 34.48, 37.92 and 41.37 MPa (5000, 5500, and 6000 psi). Based on experimentalsupport for the finite element. CTOD approach, it was deemed appropriate to utilize thecurve corresponding to 34.48 MPa (5000 psi) to predict the dimensions of defects whichwill lead to failure during a hydrostatic test. The observation that the CTOD approachmay be non-conservative for small defects however, suggested that the lower region ofthe curve be modified. Accordingly, a curve which defines critical defect dimensions at34.48 MPa (5000 psi) was also developed using the plastic collapse solution given byEqu. (7.5). This curve, and the CTOD based curve are plotted Fig. (7.7).Noting that in this figure points which lie on or to the right of a respective curverepresent defect dimensions which will lead to failure at 34.48 MPa (5000 psi), it canbe seen that although the curves overlap, there are ranges of defects sizes for whichone approach predicts failure and the other does not. As expected, plastic collapsepredicts failure for relatively deep and short defects. This approach does not predictfailure however, for a considerable range of shallower, longer defects (i.e., a 5 mm,Chapter 7. CORRELATION BETWEEN EXPERIMENTAL AND...^1112c^15 mm). For this range of defect size, the numerical CTOD approach provides amore conservative estimate of critical defect size.Since it is considered desirable to utilize a single curve to define critical defect di-mensions at 34.48 MPa (5000 psi), a curve bounding both the critical CTOD and plasticcollapse curves was constructed. This curve is plotted in Fig. (7.8) along with the dimen-sions of the defects analysed in the numerical and experimental investigations. As canbe seen, the failure curve predicts failure of cylinders containing several of the numericaldefect dimensions. Further, it can be seen that the modified curve correctly predicts nofailure for the experimental cylinders. These observations, in conjuction with numeri-cal and experimental plastic expansion results are of considerable importance from thepoint of view of of assessing current standards for NGV cylinder recertification. Thisassessment is discussed in the following chapter.Chapter 8ASSESSMENT OF THE CURRENT STANDARDS FORRECERTIFICATION OF NGV CYLINDERS8.1 Summary of FindingsIn Chapter 1 it was noted that steel NGV cylinders must be recertified for service everyfive years. It was also noted in Chapter 1 (and further in Chapter 2) that the standardscurrently used to assess cylinder integrity are based on cylinder performance during ahydrostatic test. According to these standards. a cylinder is considered unfit for furtherservice if (during a hydrostatic test) either:1. plastic (permanent) volumetric expansion exceeds 10 % of the total (elastic plusplastic) cylinder expansion at 34.48 MPa (5000 psi) (1.67x service pressure), or,2. the cylinder ruptures (bursts or leaks).These hydrostatic test criteria, it will be recalled, are designed to ensure that reductionsin wall thickness which may have occurred during service (due to environment) havenot significantly degraded cylinder integrity. The second criterion, cylinder rupture,recognizes the existence of localized forms of damage such as cracks, and is intended toremove from service any cylinder containing a defect which may lead to failure duringservice. This criterion however, is not based on any quantitative fracture mechanicsassessment of the various forms of sub-critical crack growth which can occur duringNGV cylinder service. Several studies (discussed in the following section) which haveexamined sub-critical crack growth in NGV cylinders indicate that it is possible for some112Chapter 8. ASSESSMENT OF THE CURRENT STANDARDS... ^113part-through defects to grow through-wall within a five year period. The results of thesestudies further make it possible to predict the dimensions of a defect which will growthrough-wall within a five year period (the length of time a cylinder remains in servicefollowing recertification).Given that current standards for cylinder recertification are not based on any acceptedfracture mechanics approach, it is reasonable to question whether the hydrostatic testcriteria (i.e., 1 or 2 above) are adequate to ensure on-going cylinder integrity. Clearly,any recerification procedure must be capable of removing from service any cylinder whichcontains a defect which will grow to critical within five years. The goal of this investiga-tion has been to address this question.To this point in the discussion, the behaviour of defective (cracked) NGV cylindersduring a hydrostatic test has been analysed and quantified. Based on numerical results(which are supported by experimental results), this behaviour can be summarized asfollows:1. Permanent ExpansionPermanent (plastic) volumetric expansion of cylinders containing no other form ofdamage other than a crack is significantly less than 10 % of total volumetric ex-pansion. Numerical results indicate that plastic expansion of a cylinder containinga defect as large as as 5 x 30 mm, is of the order of 1 % of total expansion.2. Cylinder RuptureBurst failure of cylinders containing defects can occur during a hydrostatic test.Numerical CTOD results (in conjunction with small scale critical CTOD results)(modified by a plastic collapse approach for smaller defects) indicate that a rangeof defect dimensions exists (see Fig. (7.8)) which will lead to failure at 34.48 MPa(5000 psi).Chapter 8. ASSESSMENT OF THE CURRENT STANDARDS...^114To illustrate the significance of these findings, it is useful to consider a hypotheticalhydrostatic test to recertify cylinders containing defects of the dimensions analysed in thenumerical portion of the study. Several cylinders containing defects of these dimensionswould rupture during such a test. The defect sizes of these cylinders are identified inFig. (7.8), where the dimensions of defects analysed have be overlaid on the modifiedCTOD/plastic collapse failure curve discussed in Sec. 7.6.2. Cylinders which would notrupture, would be considered fit for service. This is because, as noted above, permanentvolumetric expansion would be less than 10 % of total expansion. These cylinders wouldbe recertified and returned to service. It should he noted that, based on observed vol-umetric expansion and failure pressures, all of the experimental cylinders would also beconsidered fit for service based on current standards.Defects in cylinders returned to service would however, be subject to environmentaland loading conditions which promote various forms of sub-critical crack growth. Studieswhich have investigated this form of degredation are discussed in the following section.8.2 In-Service Failure8.2.1 Subcritical Crack GrowthThe potential for subcritical crack growth in NGV cylinders has been the subject ofseveral investigations. An initial study considered the effects of environment [11]. Inthis study, the susceptability of NGV cylinder steel to sulphide stress cracking was in-vestigated using the NACE 1 test method. The study concluded that, based on currentcontractual limits for natural gas contaminants and nominal service loads, NGV cylindersteel is not particularly susceptable to sulphide stress crackingLater studies have investigated the effects of alternating load. In on-going research'National Association of Corrosion EngineersChapter 8. ASSESSMENT OF THE CURRENT STANDARDS...^115at Powertech Labs, Inc. [1, 13, 14] which has involved both small and large scale test-ing. fatigue crack growth rates have been determined for NGV cylinder material. Ina study which involved large scale testing, artificial axial defects were introduced intofour cylinders. Of the four cylinders, one was monotonically loaded to failure. Three ofthe cylinders were cyclically loaded between pressures of 2.41 and 24.13 MPa (350 and3500 psi) (the typical range of NGV service pressures) until fatigue cracks originatingfrom either the artificial or natural defects grew through the cylinder wall, causing leak-age of the pressurizing medium (water). During these large scale tests, crack growth inboth the depth and length direction was monitored. The data obtained was fitted to re-lationships to predict crack growth rate and flaw shape development. These relationshipswere then used to construct curves to predict crack growth as a function of initial defectsize (Fig. (8.1)) and the size of defect that will grow through the cylinder wall within thefive years (an estimated 6500 refueling cycles) that a cylinder remains in service beforethe next scheduled recertification (Fig. (8.2)).It should be noted that Fig.s (8.1) and (8.2) were constructed based on fatigue crackgrowth rates measured in an inert (air or water) environment. In more recent studies [12,15] of sub-critical crack growth in NGV cylinder material, the combined effects of thenatural gas environment and alternating load (including frequency and overload effects)have been investigated. In these studies, which involved measurement of fatigue crackgrowth rates in small scale specimens in a simulated natural gas environment, it wasconcluded that fatigue growth rates in a natural gas environment can be up to 60 timesgreater than those in air. This finding suggests that actual in-service fatigue crack growthrates in NGV cylinder may be significantly greater than those predicted by Fig.s (8.1)and (8.2). To date however, this has not been confirmed by any large scale studies offatigue crack growth in a simulated natural gas environment.Chapter 8. ASSESSMENT OF THE CURRENT STANDARDS... ^1168.2.2 Expected Mode of In-Service FailureSub-critical crack growth of a surface defect in a pressurized component can, in general,lead to one of two conditions. The growing surface crack can reach a critical size, resultingin instability and fast fracture (rupture) at the nominal service load. Alternately, thesurface crack can grow through the cylinder wall before becoming critical. This lattercondition, known as 'leak before break' (LLB) is most desirable as it makes detection ofthe defect possible. Removal of the component from service, or repair can be initiatedbefore instability occurs.Numerical CTOD results discussed in Sec. 5.2.3 (see Fig. (5.24)) indicate that, evenfor relatively large cracks (i.e., 5 x 30 mm, CTOD at maximum NGV service pressure(20.69 MPa (3000 psi)) is less than the critical CTOD for initiation of crack growth.These results suggest that in-service rupture of cylinders should not occur.In the fatigue studies cited previously (i.e., Ref. [13]) it was found during the largescale tests that cylinders failed in LLB mode. Based on this observation, it was concludedthat NGV cylinders fail in LLB mode. This conclusion was subsequently supported inRef. [14] on the basis of a correlation between plane stress and plane strain fracturetoughness.8.3 Limitations of Current Standards for NGV Cylinder RecertificationTo illustrate the limitations of the current standards for NGV cylinder recertification,the defects sizes (numerical and experimental) have been plotted in Fig. (8.2). From thisfigure, which defines defect dimensions which will grow through-wall within a five yearperiod, it can be seen that all but two (Run 12 and Tank A defects) would lead to failuresometime during service.The eleven cylinders containing defects which would grow through-wall should beChapter 8. ASSESSMENT OF THE CURRENT STANDARDS... ^117considered unfit for further service at time of recertification. It was noted in Sec. (8.1)however, that under current hydrostatic test criteria (i.e., permanent volumetric expan-sion/cylinder rupture) only four of these eleven cylinders (Run 01, 02, and 05) would beconsidered unfit for further service. A summary of the the two modes of cylinder failure(i.e., rupture in hydrostatic test versus in-service failure) is given in Table (8.1).In view of the possibility that cylinders containing some defects can pass hydrostaticinspection yet still fail in service, it must be concluded that current standards for cylinderrecertification are inadequate to ensure on-going cylinder integrity. To quantify the rangeof defects for which hydrostatic test criteria are inadequate and summarize the majorfindings of this investigation, the curves in Fig.s (7.8) and (8.2) have been overlaid asshown in Fig. (8.3). Recalling that defect dimensions to the right of Curve A lead torupture of cylinders during a hydrostatic test, defect dimensions to the left of this curvewill be considered tolerable under current standards for NGV cylinder recertification.Since defect dimensions to the right of curve B lead to in-service failure within five years,the defect dimensions in the hatched region represent those defects for which currentstandards provide no assurance of NGV cylinder integrity.8.4 Interior versus Exterior DefectsIn closing this chapter it should be noted that Fig. (8.3) was derived from results forexterior defects in NM' cylinders. A similar, and in fact stronger, conclusion as drawnin the previous section can however be made for interior defects. This can be shown ifit is recalled that Curve A of Fig. (8.3) bounds the finite element CTOD solution forshallow, long defects and the plastic collapse solution for deep. short defects.From Fig. (5.25) (discussed in Chap (5)), it can be seen that there are only severalpercent difference between CTOD failure curves for interior and exterior defects. ForChapter 8. ASSESSMENT OF THE CURRENT STANDARDS... ^118practical purposes, these differences can be ignored. With regard to the plastic collapsesolution used to construct Curve B, a comparison of plastic collapse solutions for interiorand exterior defects given in Ref. [27] indicates that, for a given failure pressure, thedefect dimensions of an interior crack required for failure will vary only slightly (severalpercent) from those of an exterior crack required for failure. Hence, given that boththe CTOD failure curve and plastic collaspe failure curve vary only slightly for interiorversus exterior defects, Curve A in Fig. (8.3) can, for practical purposes, be consideredapplicable to both interior and exterior defects.With regard to Curve B, it should be noted that interior defects in NGV cylindersare in direct contact with the natural gas environment. This implies, based on thestudies (cited in Sec. 8.2.1) of the combined effects of environment/loading, that Curve B(derived from fatigue crack growth rates measured in an inert environment), may in factbe conservative. Due to the effects of environment, it is reasonable to expect that forinterior defects, Curve B will lie closer to the lower left-hand region of Fig. (8.3). If thisis the case, a greater range of defect dimensions will lead to in-service failure than ispredicted by the existing Curve B. This would mean that the current hydrostatic testcriteria provide even less assurance against in-service cylinder failure than concluded onthe basis of exterior defects.Chapter 9CONCLUSIONS AND RECOMMENDATIONS9.1 ConclusionsThis thesis has investigated current standards for NGV cylinder recertification in viewof fracture mechanics methodologies which address the severity of defects that may de-velop during NGV cylinder service. Based on the results of this investigation, a numberof general conclusions (i.e., related to numerical and experimental results) and morespecific conclusions (i.e., related to current and potential standards for NGV cylinderrecertification) can be drawn. These conclusions are:9.1.1 General conclusions1. A finite element model was developed in this investigation to predict volumetricexpansion of uncracked NGV cylinders. Comparison of volumetric expansion resultsobtained from this model with experimental results has shown that the model isaccurate to within +5 %.2. A finite element model was developed to model the behaviour of cracked NGVcylinders and was utilized to predict COD and CTOD for a range of defect sizes.This model provides a powerful tool for assessing the significance of defects thatcan develop during NGV cylinder service and indicates that:(a) Length of defects in NGV cylinders strongly influences COD and CTOD. De-fects as shallow as 2 mm can lead to failure (at hydrostatic test pressure) if119Chapter 9. CONCLUSIONS AND RECOMMENDATIONS^ 120length is in excess of 30 mm. The dependence of CTOD on defect lengthincreases non-linearly as defect depth increases.(b) For equivalent defect sizes, maximum crack severity shifts from an internalsurface crack location for small crack dimensions to an external surface cracklocation for relatively larger defect dimensions as a result of induced bend-ing effects. This trend was also reflected in the extent of plastic volumetricexpansion of interior and exterior cracked cylinders.(c) The plastic expansion of a cylinder containing defects that will grow to criticalwithin five years is negligible; this expansion is, even for relatively large defects,of the order of accuracy that can be attained during an actual hydrostatic test,i.e. ±0.5 cc.(d) Hydrostatic test pressure (1.67x service pressure) is sufficient to cause ruptureof some cylinders containing defects that will grow to critical during a five yearinspection interval (based on available fatigue crack growth rates). There is,however a significant range of defect sizes that will not lead to rupture duringa hydrostatic test, but will lead to failure during a five year inspection inter-val. Cylinders containing these defects will be considered fit for service undercurrent standards for cylinder recertification since, as noted above, plasticexpansion will be less than 10 % of total expansion.3. Burst test results for cracked NGV cylinders indicate that, over the range of defectsizes investigated. the CTOD approach provides accurate and conservative predic-tions of failure pressure for large defects and non-conservative predictions of failurepressure for relatively small defects. For small defects, a plastic collapse analy-sis provides conservative predictions of failure pressure. Over the range of defectsizes analysed, the CTOD design curve (i.e.. PD 6493) and R6 failure assessmentChapter 9. CONCLUSIONS AND RECOMMENDATIONS^ 121diagram provide somewhat more conservative predictions of failure pressure thanthe CTOD approach; in contrast, the Battelle empirical analysis provides slightlynon-conservative predictions of failure pressure.4. Acoustic emission results for cracked, previously in-service NGV cylinders (i.e., hitspast previous pressure, hold time hits, hit rates and amplitude distributions) donot, in general, follow any readily recognizable trends which can be be correlatedto defect severity. Trends in this data do indicate however, potential criteria whichmay be used in NGV cylinder recertification and hence justify further research inthe application of acoustic emission techniques to NGV cylinder recertification (seeSec. 9.2).5. Acoustic emission hit activity results indicate that event rate increases non-linearlyas failure is approached.9.1.2 Specific conclusions regarding current and potential standards for NGVcylinder recertification1. The potential exists for localized defects such as cracks to develop during NGVcylinder service. Further, there exists a range of defect size which will, throughmechanisms of sub-critical crack growth (fatigue and stress corrosion cracking)grow through-wall within a five year inspection interval. Cylinders containing suchdefects should be considered unacceptable and be removed from service. Currentstandards for cylinder recertification which are based on hydrostatic test (volumet-ric expansion) failure criteria however, do not specifically address the significanceof such defects. Given that internal corrosion which may occur during service morestrongly influences volumetric expansion, it should be concluded that:Chapter 9. CONCLUSIONS AND RECOMMENDATIONS ^ 122(a) Plastic volumetric expansion of a cylinder containing a defect (as measuredduring a hydrostatic test) will provide no useful information regarding thepresence, or the severity of a defect.(b) The current hydrostatic test rejection criterion (plastic expansion less than10 % of total expansion) is not an adequate safeguard against in-service failuredue to subcritical crack growth.2. The CTOD method has been shown to accurately predict fracture behaviour inpressurized NGV cylinders and should therefore be considered a viable techniquefor defining critical and allowable defect sizes for NGV cylinder structural integrityassessment.3. A combined CTOD/plastic collapse failure assessment curve has been developedwhich accurately predicts the defect sizes for which the current hydrostatic testcriteria are insufficient to guard against in-service failure.Chapter 9. CONCLUSIONS AND RECOMMENDATIONS^ 1239.2 RecommendationsBased on the the results of this investigation and the above conclusions, there follows anumber of recommendations to ensure NGV cylinder integrity, and for further work.9.2.1 Recommendations to ensure NGV cylinder integrity1. Current standards for NGV cylinder recertification, which are designed to ensureminimum wall thickness, should be revised in a manner which incorporates a frac-ture mechanics assessment of localized defects; specifically, the standards should berevised to ensure that at time of recertification:(a) Localized defects in cylinders are located and quantified.(b) Cylinders containing defects equal to or greater than the dimensions indicatedin Fig. (8.2) should be removed from services.9.2.2 Recommendations for further work1. Work should be continued to specifically define an AE inspection technique andcriteria for in-situ inspection. Due to the complexity of the technique, fundamentalstudies to isolate AE characteristics of various types of cylinder degredation mustbe conducted prior to this work, specifically:(a) Further large scale acoustic emission tests should be performed utilizing cylin-ders in 'as-received' condition. Such tests would hopefully eliminate variables(such as internal corrosion or possible unknown defects) which are believedto have contributed to difficulties in identifying trends in AE data from testsperformed in this investigation.the current five year inspection interval is to be retained.Chapter 9. CONCLUSIONS AND RECOMMENDATIONS^ 124(b) Acoustic emission tests utilizing small scale specimens should be performed.Such tests would make it possible to more closely control conditions and elim-inate extraneous sources of AE leading to a better correlation between cracktip processes and AE activity (i.e. hold time hit dependence on fatigue crackgrowth rate).2. In accordance with (la) above, further research should be undertaken to developthe acoustic emission technique as a means to locate and quantify potential defects,specifically:(a) Acoustic emission wave speed characteristics should be investigated and quan-tified.(b) Existing location software (which relies on wave speed data as input) should beutilized, and if necessary modified (i.e., for wave speed dependence) to exploitand develop its potential as a means of locating defects in NGV cylinders.3. Experimental and numerical work should be continued to more precisely definethe role of stable crack growth in failure behaviour of NGV cylinder material,specifically:(a) The numerical models should be further developed to incorporate a mechanismfor stable crack growth. This would allow predictions which could be used toto further refine the definition of allowable crack size for inspection of NGVcylinders.(b) Crack growth should be monitored in future small scale and/or large scaletesting of NGV cylinders utilizing a technique such as D.C. potential drop.Bibliography[1] Akhtar, A. and Heenan, J., "Advanced On-Board Storage of Natural Gas," pre-sented at the ASME Energy Sources Technology Conference and Exhibit held inHouston, TX, 1989.[2] Amatuiz, H. and Seeger, T., "Problems of Numerical CTOD Analyses," in TheCrack Tip Opening Displacement in Elastic-Plastic Fracture Mechanics, K. H.Schwalbe (ed.), Springer-Verlag, 1985, pp. 21-44.[3] American Society for Testing and Materials, ASTM E399 (Standard Test Methodfor Plane-Strain Fracture Toughness of Metallic Materials), American Society forTesting and Materials, 1978.[4] Anderson, D. 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R., "Crack-Extension Force for a Part-Through Crack in a Plate," ASMEJournal of Applied Mechanics, 1962, pp. 651-654.[55] Johnson, A. R., "On the Accuracy of Polynomial Finite Elements for Crack Prob-lems," Int. J. Num. Meth. Eng., Vol. 17, 1981, pp. 1835-1842.[56] Kane, R. D., "The Roles of H 2 S in the Behaviour of Engineering Alloys," Symp.on the Effects of Hydrogen Sulphide on. Steels, Edmonton, Alta, The CanadianInstitute of Mining and Metallurgy, 1983, pp 2-6.[57] Kelley, F. S., "Mesh Requirements for the Analysis of a Stress Concentration bythe Specified Boundary Displacement Method," Proceedings of the Second Inter-national Computer Engineering Conference held Aug. 15-19, 1982 in San Diego,CA. (Computer Engineering Division, ASME.)[58] Kiefner, J. F.. Maxey, W. A., Eiber, R. J. and Duffy, A. R., "Failure Stress Lev-els of Flaws in Pressurized Cylinders," in Progress in Flaw Growth and FractureToughness Testing, ASTM STP 536, American Society for Testing and Materials,1973, pp. 461-481.[59] King, R. B., "Elastic-Plastic Analysis of Surface Flaws Using a Simplified Line-Spring Model,.' ^Fracture Mechanics, Vol. 18, No. 1, 1983, pp. 217-231.BIBLIOGRAPHY^ 130[60] Kobayashi, A. S., "A Simple Procedure for Estimating Stress Intensity Factor inRegion of High Stress Gradient," Significance of Defects in Welded Structures, T.Kanazawa and A. S. Kobayashi (ed.^), University of Tokyo Press, 1974, pp. 127-143.[61] Kobayashi, A. S., Polvanich. N., Emery, A. F. and Love, A. J., "Inner and OuterCracks in Internally Pressurized Cylinders," ASME Journal of Pressure VesselTechnology, 1977, pp. 83-89.[62] Kobayashi, A. S., Ziv, M. and Hall, L. R., "Approximate Stress Intensity Factorfor an Embedded Elliptical Crack Near Two Parallel Free Surfaces," Int. Journ. ofFracture Mech., Vol. 1, 1965, pp. 81-95.[63] Kumar, V. and German, M. D., "Studies of the Line-Spring Model for NonlinearCrack Problems," ASME Journal of Pressure Vessel Technology, Vol 107, 1985,pp. 412-420.[64] Kumar, V., German, M. D. and Schumacher, B. I., "Analysis of Elastic Sur-face Cracks in Cylinders Using the Line-Spring Model and Shell Finite ElementMethod," ASME Journal of Pressure Vessel Technology, Vol. 107, 1985, pp. 403-411.[65] Levy, N., Marcal, P. V., Ostergren, W. J. and Rice, J. R., "Small Scale YieldingNear a Crack in Plane Strain: A Finite Element. Analysis, " Int. Journ. of FractureMech., Vol. 7, 1971, pp. 143-156.[66] Maddox, S. J., "An Analysis of Fatigue Cracks in Fillet Welded Joints," Int. Journ.of Fracture, Vol. 11, 1975, pp. 221-243.[67] Martin, A. R., "A Review of Current Design Formulae Applied to High PressureAluminium Alloy Gas Containers," I. Mech. E. Publication, 1980.[68] Maxey, W. A. Kiefner, J. F. Eiber, R. J. and Duffy, A. R. "Ductile Fracture Ini-tiation, Propagation, and Arrest in Cylindrical Vessels," Fracture Toughness: Pro-ceedings of the 1971 National Symposium on Fracture Mechanics, Part II, ASTMSTP 514, American Society for Testing and Materials, 1972, pp. 70-81.[69] McClintock, F. A., "Plasticity Aspects of Fracture," in Fracture—An Advanced Trea-tise, Vol. III (Engineering Fundamentals and Environmental Effects), H. Liebowitz(ed.), Academic Press, New York, 1971, pp. 47-225.[70] McHenry, H. I., Read, D. T. and Begley, J. A., "Fracture Mechanics Analysis ofPipeline Girthwelds," Elastic-Plastic Fracture, ASTM STP 668, J. D. Landes. J.A. Begley and G. A. Clarke (ed.^), American Society for Testing and Materials,1979, pp. 632-642.BIBLIOGRAPHY^ 131[71] Mullineux, G., CAD: Computational Concepts and Methods, pp. 186-188.[72] Nabil, M., "Macroscopic Origins of Acoustic Emission," Nondestructive TestingHandbook, Vol. 5-Acoustic Emission Testing, American Society for NondestructiveTesting, 1987, pp. 45-61.[73] Pandey, R. K., Pratap, C. R. and Chinadurai, R., "Significance of Rotational Factorr in CTOD Determination and the Effect of Material and Loading Geometry onr," Eng. Frac. Mech., Vol. 31, No. 1, 1988, pp. 105-118.[74] Parks, D. M., "The Inelastic Line-Spring: Estimates of Elastic-Plastic FractureMechanics Parameters for Surface-Cracked Plates and Shells," ASME Journal ofPressure Vessel Technology, Vol. 103, 1981, pp. 246-254.[75] Raju, I. S. and Newman, J. C., "Stress-Intensity Factors for Internal and ExternalSurface Cracks Cylindrical Vessels," ASME Journal of Pressure Vessel Technology,Vol. 104, 1982, pp. 293-298.[76] Ranta-Maunus, A. and Talja, H., "Elasto-Plastic Analysis of a Cracked DuctileCylinderical Pressure Vessel," Int. J. Pres. Ves. and Piping, Vol. 13, 1983, pp. 169-182.[77] Read, D. T., McHenry, H. I. and Petrovski, B., "Elastic-Plastic Models of SurfaceCracks in Tensile Panels," Experimental Mechanics, Vol. 29, No. 2, 1989, pp. 226-230.[78] Rice, J. R., "A Path Independent Integral and the Approximate Analysis of StrainConcentration by Notches and Cracks," J. Appl. Mech., 1968, pp. 379-386.[79] Romilly, D., "Failure Initiation From Circumferential Defects In Large DiameterPipelines," PhD Thesis, University of Waterloo, 1984.[80] Romilly, D., Pick, R. J., Burns, D. J. and Coote, R. I., "Ductile Failure of HighToughness Line Pipe Containing Circumferential Defects," in Modelling Problemsin Crack Tip Mechanics, University of Waterloo, 1983, pp. 297-306.[81] Schmitt, W. and Hollstein, T., "Numerical Evaluation of Crack Tip Opening Dis-placements: 2D and 3D Applications," in The Crack Tip Opening Displacement inElastic-Plastic Fracture Mechanics, K. H. Schwalbe (ed.), Springer-Verlag, 1985,pp. 3-20.[82] Schwalbe, K. H. and Hellmann, D., "Application of the Electrical Potential Methodto Crack Length Measurements Using Johnson's Formula," Journal of Testing andEvaluation, 1981, pp. 218-220.BIBLIOGRAPHY^ 132[83] Shah, R. C. and Kobayashi, A. S., "Stress Intensity Factors for an Elliptical CrackApproaching the Surface of a Semi-Infinite Solid," Int. Journ. of Fracture, Vol. 9,1973, pp. 133-146.[84] Simpson, L. A., Hosbons, R. R., Davies, P. H. and Chow, C. K., "Fracture Con-trol Using Elastic-Plastic Fracture Mechanics," in Proceedings of the InternationalSymposium on Fracture Mechanics held Aug. 23-26 in Winnipeg, Canada, W. R.Tyson and B. 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E., "Design Methods," in Post-yield Fracture Mechanics, 2d ed., ElsevierApplied Science Publishers, London and New York, 1984, pp. 319-431.[93] Tracey. D. M., "On the Fracture Mechanics Analysis of Elastic-Plastic Materialsusing the Finite Element Method," PhD Thesis, Brown University, 1973.[94] Wellman, G. W. and Rolfe, S. T., "Engineering Aspects of Crack-Tip OpeningDisplacement Fracture Toughness Testing," in Elastic-Plastic Fracture Test Meth-ods: The Users Experience, ASTM STP 856, E. T. Wessel and F. J. Loss (eds.),American Society for Testing and Materials, 1985, pp 230-262.[95] Wellman, G. W., Rolfe, S. T. and Dodds, R. H., "Failure Prediction of NotchedPressure Vessels using the CTOD Approach," Welding Research Council Bulletin229, Nov. 1984, pp. 26-35.BIBLIOGRAPHY^ 133[96] Wells, A. A., "Unstable Crack Propagation in Metals—Cleavage and Fast Frac-ture." Proc. Crack Propagation Symposium, Cranfield, 1961, pp. 210-230.[97] Willougby, A. A. and Garwood, S. J., "Application of Maximum Load Toughnessto Defect Assessment in a Ductile Pipeline Steel," Fracture Mechanics: SixteenthSymposium, ASTM STP 868, M. Kanninen and A. T. Hopper (eds.), AmericanSociety for Testing and Materials, 1985, pp. 632-655.[98] Worswick, M. J. and Pick, R. J., "Investigation of Plastic Instability Criteria forFracture of Pipeline Girth Welds Containing Defects," in Proceedings of the Inter-national Symposium on Fracture Mechanics held Aug. 23-26 in Winnipeg, Canada,W. R. Tyson and B. Mukherjee (eds.), Pergamon Press, Toronto, 1988, pp. 215-226.[99] Zienkiewicz, 0. C., The Finite Element Method, McGraw-Hill, London, 1977.[100] Zienkiewicz, 0. C. and Cheung, The Finite Element Method in Structural andContinuum Mechanics, McGraw-Hill, London, 1967.TABLES^ 134Capacity(1)Weight(kg)Outer Dia., D(mm)Wall Thickness, t(mm)Length(mm)50 54.0 268.0 7.5 889.160 62.6 316.6 7.8 972.870 70.8 317.5 7.8 972.8Table 2.1: Nominal weights and dimensions of steel NGV cylinders.Property ValueYield strengthTensile strengthElongationHardnessKIDdic822 MPa948 MPa20 %Rc 36105.5 MPa N/mm54 N/mmTable 2.2: Material properties of NGV cylinder (AISI 4130X) steel.Element Percent by weightC 0.33Mn 0.61P 0.026S 0.018Si 0.4Cr 1.1Mo 0.28Table 2.3: Chemical composition of NGV cylinder (AISI 4130X) steel.Impurity Maximum TypicalHydrogen Sulphide (H 2 S)Carbon Dioxide (CO 2 )Water (H 2 O)1 grain/100 SCF3 % by volume7 lbs/MMCF0.05 grain/100 SCF0.4 Vc by volume2 to 4 lbs/MMCFTable 2.4: Maximum contractural and typical levels of natural gas contaminants.TABLES^ 135Specimen a, (mm) P (N) COD, (mm)1 4.4 2000 0.0392 4.4 2100 0.0343 4.4 2100 0.0424 4.2 2300 0.0345 4.4 2100 0.026Table 4.1: Initiation load and COD for small scale test specimens.Specimen CTOD, (mm)1 0.0432 0.0403 0.0504 0.0385 0.0270.039 (Mean)0.008 (Stnd. Dev.)Table 4.2: Initiation CTOD for small scale test specimens.Specimen PL (N) Pm,, (N) P../PL1 2506 2000 0.802 2506 2100 0.843 2506 2100 0.844 2809 2300 0.825 2506 2250 0.840.83 (Mean)0.02 (Stnd. Dev.)Table 4.3: Limit loads and ratios of maximum test load to limit load for small scale testspecimens.TABLES^ 136Run Depth, a (mm) Length, 2c (mm)01 5 3002 5 2004 5 1005 3.5 3006 3.5 2008 3.5 1009 2 3010 2 2012 2 10Table 5.1: Run designation and defect sizes analysed in finite element analysis of fullscale cylinder behaviour.Depth, a (mm) Length, 2c (mm) Failure Pressure Predicted (MPa)Interior Defect Exterior Defect5 30 24.13 23.515 20 28.75 28.825 10 39.37 40.133.5 30 29.17 29.373.5 20 33.16 33.923.5 10 41.37 >41.372 30 35.30 36.822 20 38.13 39.722 10 >41.37 >41.37Table 5.2: Predicted failure pressures (finite element analysis of full scale cylinder be-haviour).TABLES^ 137Tank Defect Size (mm)A 2.2x5.6B 4.2x12.4C 4.1x10.5D 3.5x9.9Table 6.1: Test cylinder designation / defect sizes.Tank Elastic Expansion(at 34.48 MPa)A 381.5B 394.4C 390.6D 384.5Y 349.4Z 347.0Table 6.2: Measured test cylinder elastic expansion at 34.48 MPa (5000 psi).Tank Plastic Expansion (cc)Measured RejectionA 38.2B 21.8 41.6C 27.0 41.8D 21.7 40.6Y 1.8 35.2Z 0.0 34.8Table 6.3: Measured and rejection test cylinder plastic expansions.TABLES^ 138Tank ObservedBurst Press. (MPa)PredictedBurst Press. (MPa)FEM CTOD Curve P1. Coll. R6 Meth. Battelle Empir.A 44.13 73.09 25.37 (68.81) 43.30 42.82 44.68B 43.09 38.27 18.13 (33.23) 33.78 35.30 45.23C 43.44 41.09 19.24 (36.75) 38.82 37.16 45.02D 43.78 43.78 19.65 (39.51) 40.40 38.41 44.89Table 7.1: Measured and predicted cylinder burst pressures.Tank Cycles to Failure (x103 ) Years to FailureA 11 8.5B 5 3.8C 5 3.8D 7 >5.0Table 7.2: Number of cycles/years to failure for test cylinders.TABLES^ 139Depth, a (mm) Length, 2c (mm) Failure DuringRecertificationFailure DuringService5 30 Yes Yes5 20 Yes Yes5 10 No Yes3.5 30 Yes Yes3.5 20 Yes Yes3.5 10 No Yes2 30 Yes Yes2 20 No Yes2 10 No NoTable 8.1: Comparison of expected modes of cylinder failure (finite element analysis offull scale cylinder behaviour).7.8 Ref.11; 4 A9756684A4A4 /Al r/O rrAISI 4130 XFIGURES^ 140Dimensions in mmFigure 2.1: 60 liter NGV cylinder geometry.FIGURES^ 141LINEARELASTICINCREASINGMATERIALTOUGHNESSELASTIC-PLASTICPLASTICCOLLAPSE— QUI.1 ISM OMNI._ oOPENING STRETCHINITIATION^STABLE(DUCTILE) GROWTHSMALL PLASTICZONE SIZE(NEGLECT)INCREASEDPLASTIC ZONESIZEEFFECTIVECRACK SIZE (sur)see *FtLARGE PLASTICZONEINSTABILITY(DUCTILE 1INITIATION S FRACTURE(BRITTLE)Figure 3.1: Regimes of crack tip behaviour [79].FIGURES^ 142_ p R— -F MT,S Kpiate^Kcyl2c L-1^1^1Plate^ CylinderMI=12 t2c 2cThrough wall defect^Surface DefectFigure 3.2: Diagramatic illustration of approach used to analyse defects in cylinders.C Gm GbFigure 3.3: Membrane and bending components of stress magnification factor for through-wall defects in cylinders [44].FIGURES^ 144 2cFigure 3.4: Elliptical surface defect in a wide plate.FIGURES 145FOR CRACKTIP AT Up ,o)FOR CRACKTIP AT (o,o)crFigure 3.5: Irwin plastic zone correction [53].t^t a20^r ayFigure 3.6: Dugdale plastic zone correction [42].(IDEALIZEDPLASTIC ZONEFIGURES^ 146Figure 3.7: Center cracked plate geometry.FIGURES^ 147COD USUALLYMEASURED ONSURFACE CTOD DEFINED ASDISPLACEMENT ATF ORIGINAL CRACK^ TIPDEFECT SIZEDURING LOADINGIITIAL CRACKDEPTH a oFigure 3.8: Crack tip opening displacement (CTOD) and crack opening displacement(COD).COD CTODP^1.4 a -or(w-o) [..-0 01r(W-0FIGURES^ 148wPLASTICHINGEFigure 3.9: Single edge notch bend specimen (SENB) geometry.FIGURES 149duJ = f(cd dY- T. —dx ds)rJ=0Figure 3.10: The J-integral.FIG URES 150 41111111W A a2cIrwin^P CP^Erdogan and RatwaniFigure 3.11: Models for CTOD in large scale yielding.FIGURES^ 1511.00.90.80.70.60.50.40.30.20.10 ^0 2=52=42=32=22=12=00.1^0.2^0.3^0.4 0.5^0.6 0.7 0.8^0.9 10Nat ayCc pFigure 3.12: Plastic zone correction from Erdogan and Ratwani formulation [451.EbC CTyFIGURES^ 1520.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9^1 0N,tFigure 3.13: Crack tip open displacement from Erdogan and Ratwani formulation [45].atCMOD A^CTODt/2FIGURES^ 153Actual crack^ ModelFigure 3.14: Line spring model geometry [59].atatFigure 3.15: Relationship between actual defect dimensions and the parameter a for surfacedefects [24].a+ a)a ^or2(p + a)atFigure 3.16: Relationship between actual defect dimensions and the parameter T ./ for em-bedded defects [24].1 . 00 0.800 0.6C00.4a)Cr0.200.01^0.1 1 .0^10.0^100.0VRt^or VRtFigure 3.17: Reduction factor for long defects in curved shells containing pressure [24].FIGURES^ 1571.2I .00.8Kr0.60.40.20.00.0^0.2^0.4^0.6^0.8^I.0^1.2S rFigure 3.18: The CEGB R6 failure assessment diagram [27].FIGURES^ 15836.0± 0.1 36.0± 0.1 -1 8.00± 0.068.000.15 --    ■4 2.8±0.11.8± 0.1^10.5Note: Dimensions in mmFigure 4.1: Small scale CTOD specimen geometry.^ Span = 50.8 mm\ P+a = 4.1 ± 0.3 mm2y, = 2.0 ± 0.2 mm2y2 = 2.5 ± 0.2 mmFIGURES^ 159Figure 4.2: Small scale CTOD specimen clip gauge and potential drop lead locations.FIGURES^ 160(a) CTOD specimen finite element meshyCrack tipSingularityElementsx(b) Near crack tip meshFigure 4.3: Finite element mesh for small scale CTOD specimen.FIGURES^ 161PP / 2^ P / 2Figure 4.4: Small scale CTOD specimen COD and point load displacement.FIGURES^ 1621.11.00.90.80.7L0.60 0.50.40.30.20.10.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9^1.0a/WFigure 4.5: Non-dimensional stress intensity shape factor for small scale CTOD specimen.FIG URES^ 1631 .61.51.41.31 .21.11.0a5- 0.9(NI0.80.70.60.50.40.30.20.10.00.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9^1.0a/WFigure 4.6: Non-dimensional crack opening displacement shape factor for small scaleCTOD specimen.FIGURES^ 1641 .61 .51 .41 .31 .21.11.00>.N^0.90.80.70.60.50.40.30.20.10.00.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9^1 .0a/WFigure 4.7: Non-dimensional point load displacement shape factor for small scale CTODspecimen.0 OnL10^20^30^40^50^602.8000.100.110.120.130.150.180.170.16700.1404.554.304.053.800 3.553.303.050CO coo Specimen 2^ Specimen 3A Specimen 4^ Specimen 50 Specimen 6FIGURES^ 165N (X 10 3 cycles)Figure 4.8: Crack depth versus number of cycles for small scale CTOD specimen.0 0 0A A0 63 VAo Specimen 2^ Specimen 3A Specimen 4^ Specimen 50 Specimen 6002520(1)-5EE15OXz 100-0500 200 400 600 8000.010000.30.5 X0.81.0z0CDCFIGURES^ 166AK (X 10 3 Iblin 3/2 )0^5^10^15^20^25AK (N/m-n 3/2 )Figure 4.9: da/dN versus AK for for small scale CTOD specimen.FIGURES 167-J0 OM 0 0010000110001000011000-0000000 101000010001111EU 0■0 OMmumI1113111111trsth7(:t ^F.sl op!^to° ST T_-^_1liner9" Vv."r """ TO S00000000._0000000000MMOMUdown;oro f0000110111001103EMILindicatecompliance" thebehaviour.^-point!ofmethodafter1^,and^of^instrumentOfpop-ininitiation"Cendilofis^aAlmMEIEnmum^No. 1. P v, 'n curves (ii4 to lvi:^1A":r cortoh ce, !elt:oldhs a:efde::edsit7 d;.Note 7. In curves PO and^Wel the:::toinosne of,I . ,0MONmum011000000011-01110, .i- ..wrominamanamminammonworms10 — OUPM aiummurnhemnaMM NUMBiMEIVE . MU NEWOUMAINUMNIMU U0 irgarAMMUCOMPO UMUM rii WAR immummin 80- EMU.WM IAKM110M.■ wMUOAmMUMMioffid0601,04111100.M0NOMM100 1110 1.■MU■WM' 0 Ed PP ^UN. ini:72FA1 WOE -FAWN WE II^;u/II I A MUOM.OAAWAMMUMU 8 OM'J1 .VUO MUM^MOVAIMUVeVe010 goups efrICIKFT•At V,Figure 4.10: Types of load-COD records: i, cleavage, ii, crack arrest (pop-in), iii , stablecrack growth, iv, stable crack growth followed by crack arrest, v, attainment of maximumload plateau [23].FIGURES^ 168 1 .201.10xE00COD (mm)Figure 4.11: Normalized load versus COD for Specimens 1 and 5.Specimen 3AFIGURES^ 169Aa (in)0.01^0.02^0.03^0.04^0.05250020000.003000400500600-0 150000100050000.000300 320010000.20^0.40^0.60^0.80^1.00^1.20Aa (mm)Figure 4.12: Crack growth versus load for Specimens 3.Specimen 5—0FIGURES^ 170Aa (in)0.01^0.02^0.03^0.04^0.050.00300060025002000-0 150000100050000.0050040020010000300^00.20^0.40^0.60^0.80^1.00^1.20Aa (mm)Figure 4.13: Crack growth versus load for Specimens 5.GenerateFE ModelFatigue CrockGrowth DataFE ModelParametersMaterial Prop.,Cylinder Geom.,Defect Geom.,Load Step Incr.IncrementLoadFailure DuringHydrostatic TestL _ _ _Criteria for CylinderRecertification Not MetCriteria for CylinderRecertification Met—1 Service Failure( Stop )FIGURES^ 171StartFigure 5.1: Flowchart illustrating approach to assess criteria for NGV cylinder recertifi-cation.FIGURES^ 172948 MPa822 MPaAssume linearstrain hardeningE = 20 x 10 3 MPa^1■,^0.41^ 20STRAIN (%)Figure 5.2: Idealized stress-strain relationship for 4130X NGV cylinder steel.FIGURES^ 173Defect GeometryFigure 5.3: NGV cylinder/defect geometry.FIGURES^ 174Figure 5.4: Finite element mesh for cylinder.D D t> D D D D r> r> D D » r>DDCrhCoates^r>0>DDD crhaa aaa 0-, FEMFIGURES^ 1752520015alp100.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9^1.0S/IFigure 5.5:^Finite element longitudinal and hoop membrane stress(s// = non-dimensional distance from nozzle end of cylinder). oil) 0 41 0 0 0 tl 0 0 \4CoatesDDDDD gh FEM<1<a« 0-110FIGURES^ 176105—0.0 1.00.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.93/IFigure^5.6:^Finite^element^longitudinal^and^hoop^bending^stress(s// = non-dimensional distance from nozzle end of cylinder).FIGURES^ 177YFigure 5.7: Finite element mesh for submodelled region of cylinder.FIGURES^ 178(a) Crack plane (Z = 0)(b) Crack tip (Y = 0)Figure 5.8: Elliptical crack profile and crack tip (finite element mesh for submodelledregion of cylinder).Shell Model=0luts =^+^xu y = 0Solid SubmodelFIGURES^ 179Figure 5.9: Submodelled region of cylinder.FIGURES^ 180 c2 — 'x' = x 1 +x 2 4- y 2= Yz' = zFigure 5.10: Elliptical transformation used in construction of submodel mesh.1.6FIGURES^ 1812.0N000CS-Nc_cn01.2 —1.00.0^0.1^0.2^0.3^0.4^0.5^0.6^0.7^0.8^0.9^1.02p/iv1.4 —1.8 0—00000 486 DOF00000 927 DOF»r:.» 1344 DOFFigure 5.11: Non-dimensional stress intensity factors for the 5 x 15 mm defect.CTODMoving centerof rotationVxCODFIGURES^ 182Figure 5.12: Calculation of COD, CTOD and point of rotation.FIGURES^ 183Figure 5.13: Incremental volume bounded by origin and three points on element face..16^ Aye'AV Pi --•'"FIGURES^ 184PAVFigure 5.14: Elastic and plastic expansion components of total expansion.ICU0.000^1000^2000^3000^4000Pressure (psi)5000 6000FIGURES^ 185Pressure (MPa)10.0^20.0^30.0^40.0Figure 5.15: Crack opening displacement versus pressure for interior cracks.E00C)0.000^1000^2000^3000^4000 5000 6000FIGURES^ 186Pressure (MPa)10.0^20.0^30.0^40.0Pressure (psi)Figure 5.16: Crack opening displacement versus pressure for exterior cracks.FIGURES^ 187 0.130.120.110.100.090.080.070.060.050.040.030.020.010.00E0O1.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.0^5.5^6.0a (mm)Figure 5.17: Crack opening displacement as a function of crack size at 20.69 MPa(3000 psi).FIGURES^ 1880.260.240.220.200.180.160.140.120CD0.100.080.060.040.020.001.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.0^5.5^6.0a (mm)Figure 5.18: Crack opening displacement as a function of crack size at 34.48 MPa(5000 psi).(mm) c ( mm)—e— 5.0 15.0 —e— 3. 5 10. 02. 0 5. 00FIGURES^ 1890.400.350.300.25I0.2000.150. 1 00.050.000^20^40^60^80^100rp At — a) (x 100%)Figure 5.19: Development of plastic zone for interior cracks.FIGURES^ 1900.400.350.300.25Ea 0.2000.150.100.05a ( mm) c ( mm)—e-- 5. 0^15. 0— e— 3. 5 10. 0— — 2. 0^5. 0 000.00 e01^1 20^40^60^80rp^— a) (x 100%)100Figure 5.20: Development of plastic zone for exterior cracks.0.000^1000^2000^3000Pressure (psi)4000 5000 6000FIGURES^ 191Pressure (MPa)10.0^20.0^30.0^40.0Figure 5.21: Crack tip opening displacement versus pressure for interior cracks.0.000^1000^2000^.3000^4000Pressure (psi)5000 6000FIG URES^ 192Pressure (MPa)10.0^20.0^.30.0^40.0Figure 5.22: Crack tip opening displacement versus pressure for exterior cracks.FIGURES^ 193I0OU0.0400.0350.0300.0250.0200.0150.0100.0050.0001.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.0^5.5^6.0a ( m m )Figure 5.23: Crack tip opening displacement as a function of crack size at 20.69 MPa(3000 psi).0.1000.0900.0800.0700.0600.0500.0400.0300.0200.0100.000E00FIGURES^ 1941.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.0^5.5^6.0o (mm)Figure 5.24: Crack tip opening displacement as a function of crack size at 34.48 MPa(5000 psi).FIGURES^ 195302520EE^15C\J1050 0^1^2^3^4^5^6^7^8( m m )Figure 5.25: Limiting defect dimensions for failure at 34.48, 37.92 and 41.37 MPa (5000,5500 and 6000 psi).FIGURES^ 196Figure 5.26: Finite element expansion of a NGV cylinder.s"--- 200.00150.00FIGURES^ 197Pressure (MPa)10.0^20.0^30.02000^3000^4000^5000Pressure (psi)Figure 5.27: Predicted and measured elastic expansion.FIGURES^ 198Pressure (MPa)0.550.500.450.400.350.300.250.200.150.100.050.000^1000^2000^3000^4000Pressure (psi)40.05000^60000.0^10.0^20.0^30.00.60Figure 5.28: Plastic expansion as a function of pressure for interior cracks.FIGURES^ 199Pressure (MPa)0.0^10.0^20.0^30.00.6040.00.550.500.450.400.350.300.250.200.150.100.05(mm) c (mm)5.0 15.03.5 10.02.0 5.00.000^1000^2000^3000^4000^5000^6000Pressure (psi)Figure 5.29: Plastic expansion as a function of pressure for exterior cracks.FIGURES^ 2000.150.140.130.120.110.100.090.08a 0.070.060.050.040.030.020.010.001.0^1.5^2.0^2.5^3.0^3.5^4.0^4.5^5.0^5.5^6.0C (mm)Figure 5.30: Plastic expansion as a function of crack size at 34.48 MPa (5000 psi) (hdrostatic test pressure).FIGURES^ 201Figure 6.1: Schematic illustration of hydrostatic testing facility.FIGURES^ 202 Housing--- ConnectorCrystal^ — Wear—PlateSensor DetailS4S2340 mmLocan ATBAMClip Gouge340 mmPressureS is3Figure 6.2: Schematic illustration of cylinder instrumentation.PEAK AMPLITUDE(45 is 1 AV Ai ^RISE !MEP.^PEAKDISCAHAIHAT ONTIMESIGNAL WAVEFORMFOR ONE ACOUSTIC EMISSION EVENT ENVELOPE DECAYRATE OR SLOPE(01 pol..80)COUNTS START ATFIRST THRESHOLD CROSSINGTHRE S HOL D (.S) --EN0 OFPROCE531110a_a a Av  - I 0DURATION ( see)FIRSTTHRESHOLDCROSSINGLASTTHRESHOLDCROSSINGFIGURES^ 203Figure 6.3: Characteristics of an acoustic emission hit.FIGURES^ 204Pressure (MPG)0.0^10.0^20.0^30.0^40.00^1000^2000^3000^4000^5000^6000Pressure (psi)Figure 6.4: Measured elastic expansion (test and control cylinders).Pressure (MPa)20.7^25.0^30.010090 —80 —•70 —60 —c50^ PI40 —35.0^40.000000 Tank A0000 ^ Tank Br>> 1>r>> Tank C" 4 « Tank D••••• Tank Y■■■■■ Tank Za O•30 —^ t>^LI^ 0DE>oD'o10 —^ ^ o PI• •^••a^o 6 •0^ 1^I^I I^•I^I^I3000 3300 3600 3900 4200Pressure (psi)20 —4500 4800 5100 5400 5700 6000Figure 6.5: Hits past previous pressure during first cycle of cylinder tests (30 dB < Amp < 70 dB).1008090I^I^I^I00000 Tank A^^^^^ Tank BDD 'L' Tank C<1 «<I'l Tank D••••• Tank Y■■■■■ Tank Z7060(r),_.•O.— 5I40 •30a20100P! 1^. t■ a•aL>aD^■0fp■a001■o0 E>•J.^• 1^0^1^0 1^•Pressure (MPo)20.7^25.0 30.0 35.0 40.03000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.6: Hits past previous pressure during second cycle of cylinder tests (30 dB < Amp < 70 dB).00000 Tank A^^^^^ Tank Bt>1>t>» Tank C«<« Tank D••••• Tank Y■■■■■ Tank Z  aN0 g^t■o^• 0EP^  1^0^1 Pressure (MPa)20.7^25.0^30.0^35.0^40.010090807060co504030201003000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.7: Hits past previous pressure during third cycle of cylinder tests (30 dB < Amp < 70 dB).■43-0-••OaO•^^■I^L^IPressure (MPa)20.7^25.0^30.0^35.0^40.010090807060ti)504030201003000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure00000 Tank A^ 0000 Tank Bt> E> t> » Tank C44 444 Tank D■■■■■ Tank ZFigure 6.8: Hits past previous pressure during first cycle of cylinder retests (30 dB < Amp < 70 dB).1008090•00000 Tank A^^^^^ Tank Br>t>t>» Tank C1<11< Tank D■■■■■ Tank Z70604030■^20100.0^ 0•I>■ ^■Pressure (MPa)20.7^25.0^30.0^35.0^40.03000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.9: Hits past previous pressure during second cycle of cylinder retests (30 dB < Amp < 70 dB).Pressure (MPa)EnI■20.7^25.0^30.010090 —80 —70 —60 —50 —0^ 040 Q—35.0^40.000000 Tank AO000^ Tank Br>r>>>> Tank C< 4414 Tank D■■■■■ Tank Z^30 —^r>^M^ l>0^20 —^ 4^•^Ei■ ^^10 —^4^8^ 49 0 r>^00a0^9)^I^I^il^a)^I^•^1^1^I3000 3300 3600 3900 4200 4500 4800Pressure (psi)5100 5400 5700 6000Figure 6.10: Hits past previous pressure during third cycle of cylinder retests (30 dB < Amp < 70 dB).Pressure (MPo)20.7^25.0^30.010090 —80 —70 —60 —35.0^40.000000 Tank A^now] Tank Br>t>t>» Tank C4444 '1 Tank D••••• Tank Y■■■■■ Tonk Zcn-,-,50 —i40 —30 —20 —^ a■ • •a1 0 —^().^a^a^■• o c>• o^® o0 ^id^o !^ili^:^?o^:^11 1^o^I3000 3300 3600 3900 4200 4500 4800 5100 540001 5700 6000Pressure (psi)Figure 6.11: Hits past previous pressure during first cycle of cylinder tests (40 dB < Amp < 70 dB).I^I^I^I00000 Tank A^ 0000 Tank B»t>t>> Tank C44 '144 Tank D••••• Tank Y■■■■■ Tank Z4.-IF-II^ii^9^s■^• o A^;Pressure (MPo)20.7^25.0 30.0 35.0 40.0100908070600-,— — 50I4030201003000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.12: Hits past previous pressure during second cycle of cylinder tests (40 dB < Amp < 70 dB).Pressure (MPo)20.7^25.0^30.010090 —80 —70 —60 —U)50 —40 —30 —20 a-35.0^40.000000 Tank A^^^^^ Tank Bt>c>i>» Tank C4444.1 Tank D••••• Tank Y■■■■■ Tank Z••^10 —^■a^a•a^I^■I_____i_______:L__112_1:__ j_e_± __e_i__^0 •^0^o3000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.13: Hits past previous pressure during third cycle of cylinder tests (40 dB < Amp < 70 dB).Pressure (MPa)20.7^25.0^30.010090 —80 —70 —60 —(r)50  40 —35.0^40.000000 Tank AO000^ Tank B>>>>> Tank C444 <<1 Tank D■■■■■ Tank Z •30 —20 d—^ r>t>^ I>D^ >■ a•10 —^^ t>• ^'Fcioo^g^a0^Et)^43^I^#3^I^63000 3300 3600 3900 4200 4500 4800Pressure (psi)^ I^15100 5400 5700 6000Figure 6.14: Hits past previous pressure during first cycle of cylinder retests (40 dB < Amp < 70 dB).00000 Tank A^ 000^ Tank Bl>t>t>E>r> Tank C<4444 Tank D■■■■■ Tank Z•>El 1:. D g43 I^1^110090807060403020100Pressure (MPo)20.7^25.0^30.0^35.0^40.03000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.15: Hits past previous pressure during second cycle of cylinder retests (40 dB < Amp < 70 dB).Pressure (MPa)20.7^25.0^30.010090 —80 —70 —60 —cn5040 — •30 —35.0 40.000000 Tank A^^^^^ Tank BE>r>t>» Tank C«44.1 Tank D■■■■■ Tank Z20 — D•B^ ■1 0 ^—^ t>^a ■ El^^a ■0^ <0^9'^(I)^g^11^0 1^l^I3000 3300 3600 3900 4200 4500 4800Pressure (psi)5100 5400 5700 6000Figure 6.16: Hits past previous pressure during third cycle of cylinder retests (40 dB < Amp < 70 dB).2000000 Tank A0^ 0000^ Tank Br>E>r>r>r> Tank C4444 '1 Tank Da^ ••••• Tank Y15 —^4^ ■■■■■ Tank Z4w+-‘ 10 a-D.4o^ 05 — •^ ■•D ■0 ^ L> 0Pressure (MPa)20.7^25.0^30.0^35.0^40.00 •^1^El^Ili^III^10^•^el^o^I^o 1 3000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.17: Hold time hits during first cycle of cylinder tests (30 dB < Amp < 70 dB).Pressure (MPa)20.7^25.0^30.0^35.0^40.02015U)÷-,•—^10i50I I I I00000 Tank AO 00130 Tank B1>t>c•r>> Tank C«444 Tank D••••• Tank Y■■■■■ Tank Za^a^aC>‹^  0 m^■I^I^ill^1^l0^1 0^1^1^13000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.18: Hold time hits during second cycle of cylinder tests (30 d13 < Amp < 70 dB).00000 Tank Annoop Tank Bc.t>>>> Tank C< 44 <l‹ Tank D••••• Tank Y■■■■■ Tank Z ■• Ili^0^1110^111 0 • I  -Pressure (MPa)20.7^25.0^30.0^35.0^40.02015503000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.19: Hold time hits during third cycle of cylinder tests (30 dB < Amp < 70 dB).Pressure (MPa)20.7^25.0^30.0^35.0^40.02015(r)- —^1 0i50N00000 Tank AD^ poop° Tank Bt>E>D» Tank C«4'14 Tank DD.^■■■■■ Tank Z<aO^i>C!^a^ a40^0^0^4^0^■.^I^I^I^1^ill^(6 i^I^I3000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.20: Hold time hits during first cycle of cylinder retests (30 dB < Amp < 70 dB).00000 Tank A^^olio Tank Bt>r>t>» Tank C«<4`1 Tank D■■■■■ Tank ZDa0a^a^r>sa^ a^DI^I^IPressure (MPa)20.7^25.0^30.0^35.0^40.02015cn-4--, O.^11503000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.21: Hold time hits during second cycle of cylinder retests (30 dB < Amp < 70 dB).Pressure (MPa)20.7^25.0^30.0^35.0^40.0201550N^ 00000 Tank A^^^^o Tank Br>E>E>» Tank Caaaaa Tank D■■■■■ Tank ZDDaa^O^^ ^ <^ 0O ‹^‹ III^I^1^b^ ^I^0 I^I^I3000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.22: Hold time hits during third cycle of cylinder retests (30 dB < Amp < 70 dB).00000 Tank A0000 ^ Tank BE> t> E> » Tank C'14444 Tank D      Tank Y■■■■■ Tonk Zaaa^ a• a0• I^0 ii^i^i^410^ill r II^1. ^ .4DPressure (MPa)20.7^25.0^30.0^35.0^40.02015cf)-±- 10503000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.23: Hold time hits during first cycle of cylinder tests (40 dB < Amp < 70 dB).201500000 Tank A00000 Tank BD.D.E>>> Tank C44444 Tank D••••• Tank Y■■■■■ Tank Z5 C3-• 4^ •0 4^4^0^0• I 0 I^0Pressure (MPa)20.7^25.0^30.0^35.0^40.03000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.24: Hold time hits during second cycle of cylinder tests (40 dB < Amp < 70 dB).a^■•a■ aI I00000 Tank A0000 ^ Tank Bi>> 1 » Tank C<14 '14 <1 Tank D••••• Tank Y■■■■■ Tank Z1^ill^1^io^1 0• I^o^[^0 1Pressure (MPo)20.7^25.0^30.0^35.0^40.02015cn._.=^10503000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.25: Hold time hits during third cycle of cylinder tests (40 dB < Amp < 70 dB).Pressure (MPa)20.7^25.0^30.0^35.0^40.020155000000 Tank A0000^ Tank Bt>ND.D.r> Tank C4'1444 Tank D■■■■■ Tonk Z0I^ill^ I^ill3000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.26: Hold time hits during first cycle of cylinder retests (40 dB < Amp < 70 dB).Pressure (MPa)20.7^25.0^30.0^35.0^40.020151 05000000 Tank A^ 0000 Tank B>>1>i>r> Tank C1'14 4 '1 Tank D■■■■■ Tank Za-r:.• 1 3000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.27: Hold time hits during second cycle of cylinder retests (40 dB < Amp < 70 dB).00000 Tank A0000^ Tank Bt>>>» Tank C«aaa Tank D■■■■■ Tank ZDDD■ toD.^ 0• 0^^ ‹ 1^1^i^ii^iii^1^0 1^1^1DDPressure (MPa)20.7^25.0^30.0^35.0^40.02015503000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.28: Hold time hits during third cycle of cylinder retests (40 dB < Amp < 70 d13).1.000000 Tank A0.9 — 0000^ Tank B›»1>c> Tank C44«4 Tank D0.8 —••••• Tank Yo_^ ■■■■■ Tank Z0.7 —a)Lci) 0.6 —cr),a)a- 0.54)0.4 —Cla)  0. 3cn4-,1 0.2 — •0^t>^a^ o0.1—^^0^0 oI^>I>^ ^ 0^>0^00.0 0^i^0 t^411^It :^T^  1^I^I•Pressure (MPa)20.7^25.0^30.0^35.0^40.03000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.29: Hit rate (hits/MPa) during first cycle of cylinder tests (Amp > 30 dB).a Pressure (MPa)20.7^25.0^30.0^35.0^40.01.0 I I I I00000 Tank A0.9 — 0000^ Tank Bi>r>c>» Tank C44444 Tank D- 0.8 -CL^ ::::: Tank YZcn-„,`—' 0.7a)(r) 0.6 —cna)ca- _cD 0.4a)0- 0.3cn± 0.20.1 - ^a • ■D.^•■ 4^D.^a^4^^  ^▪ 0 r>0.0 IN^I^° a^ ^i^jilo^  I^0^I^0 I 3000 3300 3600 3900 4200 4500 4800 5100 5400 5700Pressure (psi)0.56000Figure 6.30: Hit rate (hits/MPa) during second cycle of cylinder tests (Amp > 30 dB).0.60.5■•0 8^°1.00.9-C 0.4a)a 0.3U)± 0.20.10.000000 Tank A0000^ Tank Br>r>r>» Tank C'14444 Tank D••••• Tank Y■■■■■ Tank Z■4o1 0^ 1Pressure (MPa)20.7^25.0^30.0^35.0^40.03000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.31: Hit rate (hits/MPa) during third cycle of cylinder tests (Amp > 30 dB).Pressure (MPa)20.7^25.0^30.0^35.0^40.01.0 0.9 — 00000 Tank A00000 Tank BD.D.D.i>r> Tank C 0.8 —^ 4444'1 Tank Do_^ ■■■■■ Tank Z'---' 0.7 —v)• 0.6 —(1)Q)6- 0.5 —' C 0.4   ^ ± 0.2 —^■ ^ 6^0.1 —Z^ D.D.D.^ocii 8 o^c.0.0^Ja^II^1 1'^$^0 1^1^I3000 3300 3600 3900 4200 4500 4800 5100Pressure (psi)5400 5700 6000Figure 6.32: Hite rate (hits/MPa) during first cycle of cylinder retests (Amp > 30 dB).Pressure (MPa)20.7^25.0^30.0^35.0^40.01.0 I I I I0.9 —^ 00000 Tank A0000^ Tank Br>4.1r> r>44r> : TTaa n kk C• 0.8 —^ D(---' CL^ ••••• Tank Z'---' 0.7 —4)L.v) 0.6 —(f)^t>2a 0.5C' 0.4 —• •Q)a 0.3 0-u)^± 0.2 —^•^ r>^•8^0.1 — 10.0^ 1^?^I^/^d3^o 1^I^I^to^8 o^•D^^t>r>^ii5400 5700 60003000 3300 3600 3900 4200 4500 4800 5100Pressure (psi)Figure 6.33: Hit rate (hits/MPa) during second cycle of cylinder retests (Amp > 30 dB).0.8cn0_0.7a)cf.) 0.6U)0- 0.5D 0.4a)0- 0.3U)•0.200— 0Pressure (MPa)20.7^25.0^30.01.035.0^40.000000 Tank Aompo^ Tank BL>»t>> Tank C4444 Tank D■■■■■ Tank Z0.9 — ■ .1P0.1 —^ 4^•^1?)^4• ^0a0.0^i^el (i)^(i)^0 03000 3300 3600 3900 4200 4500Pressure (psi)6i3^0 1 4800 5100 5400 5700 6000Figure 6.34: Hit rate (hits/MPa) during third cycle of cylinder retests (Amp > 30 dB).Pressure (MPa)20.70.25 ^00000 Tank A^^^^^ Tank Bc>r>r>» Tank C4 '11 '4 '1 Tank D(7T0.20 —^••••• Tank Ya^ ■■■■■ Tank Z0-_,_cD0.10 —DCD^•aU)_,.,.10.05 — a a aa a^0isO^■^o  0^•^•0 00.00^Z^o (!)^9'^iii^i^ill^I^o^I^I 3000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)25.0^30.0^35.0^40.0Figure 6.35: Hit rate (hits/MPa) during first cycle of cylinder tests (Amp > 40 dB).Pressure (MPa)20.70.25 ^__.CX0.10 II-a)CL(r),_.± 0.0525.0^30.0• sa^a^■^^■ •o •35.0^40.000000 Tank A^^^^0 Tank B>>>>> Tank C<<44.1 Tank D••••• Tank Y■■■■■ Tank Z^0 r> 4 ■ 0 ■ D^ 400.00^111^I^♦^:^:^gl^• 1^o^1^o 1 3000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.36: Hit rate (hits/MPa) during second cycle of cylinder tests (Amp > 40 dB).Pressure (MPa)20.70.25 ^25.0^30.0^35.0^40.0. (7"-;0.2000000 Tank A0000^ Tank B>>>>> Tank C4 '1 "i Tank D••••• Tank Y■^■■■■■ Tank ZaD0.1047)a(f) •=0.05 • ■a^ ■0.00^o^c!}3000 3300 3600 3900 4200 45000 ■ 0^I^01 4800 5100 5400 5700 6000a-Pressure (psi)Figure 6.37: Hit rate (hits/MPa) during third cycle of cylinder tests (Amp > 40 dB).Pressure (MPG)20.7^25.0^30.0^35.0^40.00.25 00000 Tank A^o^^^ Tank Bt>t>i>>> Tank C44444 Tank D■■■■■ Tank Z•DD■ ■—.10.05 — D■ ^00 a000^ 00.00 Ell^1'^I^g^i^g^1^I^I 3000 3300 3600 3900 4200 4500 4800 5100 5400 5700 6000Pressure (psi)Figure 6.38: Hite rate (hits/MPa) during first cycle of cylinder retests (Amp > 40 dB).•■DPressure (MPa)25.0^30.0^35.0^40.000000 Tank A0000 ^ Tonk Br>» I>> Tank C 0.20 D^<4144<1 Tank DCl^ ■•■■■ Tank Za)(n 0.15 D0.10 —a)a..10.05 —D.^aO ^ ^ ■^ ^a  O i>^a ■^■U o^ I:10.00^11 I^I^I^%^03000 3300 3600 3900 4200 4500 4800Pressure (psi)I^1^I 5100 5400 5700 600020.70.25 ^Figure 6.39: Hit rate (hits/MPa) during second cycle of cylinder retests (Amp > 40 dB).Pressure (MPa)•00000 Tank Aoo^0 Tank BD.r>t>» Tank C14 14 4 Tank D■■■■■ Tank Z20.70.25 ^ •0X 0.1025.0^30.0 35.0^40.0a5^is^■^ ■ >a•cr) t>^ ■....—10.05 —^>a ^t>^ ^a a^■D aa^ ■Oo o^o^o^o0.00^(!) I T1^ti)^I^0 I^I^I3000 3300 3600 3900 4200 4500 4800 5100Pressure (psi)5400 5700 6000Figure 6.40: Hit rate (hits/MPa) during third cycle of cylinder retests (Amp > 40 dB).1000100U)96 100z30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49Amplitude (dB)50zFIGURES^ 241100330 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)Figure 6.41: First cycle amplitude distributions (P = 20.69 MPa (3000 psi)).FIGURES^ 2421003U) 1001E0zlaoo100Iff3100zI^ilt^III 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)CDCr)00z100co0 100z1000111111111111111111111 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)zFigure 6.42: First cycle amplitude distributions (P = 22.75 MPa (3300 psi)).1000 10001Eo todzC^111111111111111^ 111111111111111111111 ^30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB) Amplitude (dB)100(f)1.E'15^to0z1003^ 1000B100100z100YFIGURES^ 243Figure 6.43: First cycle amplitude distributions (P^24.82 MPa (3600 psi)).1000C10345 to0zI F30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)FIGURES^ 24430 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)Figure 6.44: First cycle amplitude distributions (P = 26.89 MPa (3900 psi)).dz30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)10301E0 100z1000z1000)'8 10 -0zFIGURES^ 245A1003100Cl)30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)Figure 6.45: First cycle amplitude distributions (P = 28.96 MPa (4200 psi)).FIGURES^ 2461000 FB'_EO lo30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)dI^I^I30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)Figure 6.46: First cycle amplitude distributions (P = 31.03 MPa (4500 psi)).zFIGURES^ 24710001000D100031EO 100z1111^11^11111 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)100cn1EOz(5,^10100030 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)Figure 6.47: First cycle amplitude distributions (P = 33.10 MPa (4800 psi)).1000!.-100U)100z1000 v1000 100zC/3FIGURES^ 248zI^I^I 30 3132 3334353637 38 39 4041 42 43 44 45 46 47 48 49 50Amplitude (dB)DII^t 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)Figure 6.48: First cycle amplitude distributions (P = 34.48 MPa (5000 psi)).2491000 z0z30 31 32 13 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)1000D30 31 32 33 34 15 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)Figure 6.49: First cycle amplitude distributions (cylinder retests, P = 20.69 MPa(3000 psi)).30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)FIGURES^ 2501000D1031111111111111111^1^1^f^t 30 31 32 33 34 35 36 37 3839 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)Figure 6.50: First cycle amplitude distributions (cylinder retests, P = 22.75 MPa(3300 psi)).Y1^I^I100030 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)10001001E48,^to0zFIGURES^ 2511003D1^1^130 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)100(1)O 10OzFigure 6.51: First cycle amplitude distributions (cylinder retests, P = 24.82 MPa(3600 psi)).1 000OzFIGURES^ 2521000A1001E48 toOz1000B10048; io6zz30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)10001Cll1E0 106zD1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^1^130 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)Figure 6.52: First cycle amplitude distributions (cylinder retests, P = 26.89 MPa(3900 psi)).FIGURES^ 2531000 t-100fnB46 ioO 100coE'13^10O100330 31 32 33 34 35 36 37 38 39 40 41 42 43 4.4 45 46 47 48 49 50Amplitude (dB)DO loO30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)Figure 6.53: First cycle amplitude distributions (cylinder retests, P = 28.96 MPa(4200 psi)).FIGURES^ 254A100010015 to0z1000 rC1001E-6 100zJ1000Dtoo0 t0OzI^II^l^I^l^l^l^l^l^t 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)30 3132 33 3435363738 39 40 41 42 43 44 45 4647 48 4950Amplitude (dB)Figure 6.54: First cycle amplitude distributions (cylinder retests, P = 31.03 MPa(4500 psi)).30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)FIGURES^ 25530 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)Figure 6.55: First cycle amplitude distributions (cylinder retests, P = 33.10 MPa(4800 psi)).FIGURES^ 2561000too1E450z01000 zB1000D100Cl)io0z1^1^1^1^11^1^111111111^1^1111 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)30Figure 6.56: First cycle amplitude distributions (cylinder retests, P = 34.48 MPa(5000 psi)).FIGURES^ 257Pressure (MPa)Pressure (psi)Figure 6.57: Cumulative hits versus pressure for burst test of Tank A.a)*-47:^20000E1500FIG URES^ 258Pressure (MPa)5^10^15^20^25^30^35^40^450^1000^2000^3000^4000^5000^6000^7000Pressure (psi)Figure 6.58: Cumulative hits versus pressure for burst test of Tank B.FIGURES^ 259Pressure (MPa)5^10^15^20^25^30^35^40^45Pressure (psi)tFigure 6.59: Cumulative hits versus pressure for burst test of Tank C.IFIGURES^ 260Pressure (MPa)0^5^10^15^20^25^30^35^40^45Pressure (psi)Figure 6.60: Cumulative hits versus pressure for burst test of Tank D.5 10 15 20 25 30 35 40 452500cn.4-,^2000EFIGURES^ 261Pressure (MPa)Pressure (psi)Figure 6.61: Cumulative hits versus pressure for burst tests.10301001EO 10O1000100U)FIGURES^ 262CA1000cn 100O1000U) 1001E10OB11111111111^1,1111111 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50D1111111^1^ 1 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50Amplitude (dB)^ Amplitude (dB)Figure 6.62: Amplitude distributions for cylinder burst tests.FIGURES^ 263Figure 6.63: Tanks A, B and C following burst tests.FIGURES^ 264Figure 6.64: Fracture origin (Tank A).Figure 6.65: Fracture origin (Tank D).FIGURES^ 265Pressure (MPa)0^5^1 0^15^20^25^30^35^40^45 0.50.40.30.20.10.0ED0Pressure (psi)Figure 6.66: COD versus pressure for burst test of Tank A.^^^^ o 00.092FIGURES^ 266Pressure (MPa)0^5^10^15^20^25^30^35^40^450.5Tank B^ Measured^ Finite Element— 0.01500.30.4ED0U ^0.200.10.00^5550I^I^I ^0.0001000^2000^3000^4000^5000^6000^7000Pressure (psi)0.010O— 0.005DO^^Figure 6.67: COD versus pressure for burst test of Tank B.FIGURES^ 267Pressure (MPa)0^5^10^15^20^25^30^35^40^450.5Tank C6. Measured^ Finite Element0.0150.40.3E0O0.010000.20.0050.1  -- 0.088 ^AAAAAA LIC1114'AAAAAA59600.00 0.00070001000^2000^3000^4000^5000^6000Pressure (psi)Figure 6.68: COD versus pressure for burst test of Tank C.^FIGURES^ 268Pressure (MPa)0^5^10^15^20^25^30^35^40^450.5 ———0. 10.0150.01000C.)0.0050.0000.40.3000.2Tank Dv Measured—Finite Element 1000^2000^3000 4000^5000 6000^7000Pressure (psi)Figure 6.69: COD versus pressure for burst test of Tank D.FIG URES^ 269Predicted Failure Pressure (MPa)19 29 34 39147000650024 44U)0_C)U)(/)a-0L_ Conservative3000 —^/2500 — /^A V 0->o Tank A^ Tank BA Tank C^ Tank D4400390DU)co34C)029^V_24196000550050004500o 4000j.CC)EC) 35000_XLJNon—conservative2000 / I^ 142000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000Predicted Failure Pressure (psi)Figure 7.1: Actual versus predicted failure pressures (finite element/CTOD approach)./V^0^ •^ ■o Tank A^ Tank BA Tank Cv Tank D(Solid symbols —defects notrecategorized)ConservativeNon—conservative•—e•—■7000650060005500U)U)50000 45006Lof 4000CE'E 3500X30002500200014/^I^I^I^I^I^I^I^I^I ^1419 24 29 34 39 442419293444Q_39c.)L_(/)(/)C)00LL0EC)cuaxFIGURES^ 270Predicted Failure Pressure (MPa)2000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000Predicted Failure Pressure (psi)Figure 7.2: Actual versus predicted failure pressures (CTOD design curve).FIGURES^ 271Predicted Failure Pressure (MPa)147000443424 2919 396500A 44Conservative60005500U)U)5000D 45006u-0 4000E'E 3500cox3000Non—conservativeo Tank A^ Tank BA Tank Cv Tank D00-39U)34 0_a)029^Li-0C.)E24xco192500/2000 /^I^I^I^I^I^I^I^I ^142000 2500 3000 3500 4000 4500 5000 5500 6000 6500 7000Predicted 7 a . lure Pressure (ps:)oFigure 7.3: Actual versus predicted failure pressures (plastic collapse).FIGURES^ 272K r1 .21.11 .00.90.80.70.60.50.40.30.20.10.00.0 0.1^0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9^1.0 1.1^1.2SrFigure 7.4: Actual versus predicted failure pressures (CEGB R6 Method).19e7e5M P g0/gF4320.12^3^4^5^6 7 8 9^2^3^4^5^6 7 8 9^20.1^ 1 10K,27/(8co-F2)Figure 7.5: Actual versus predicted failure pressures (Battelle Empirical Analysis).Plastic collapse (Equ. (7.9))LEFM (Equ. (7.8))1acc-Fc/WFIGURES^ 274Figure 7.6: Comparison of LEFM and plastic collapse failure criteria.FailureFHTte ElementCTOD( —Int. ---Ext.)Plastic Collapse^-(Turner)No FailureFIGURES^ 27535302520E(NJ^1510500^1^2^3^4^5^6^7^8a (mm)Figure 7.7: Comparison of numerical CTOD and plastic collapse limiting defect dimen-sions.FIGURES^ 27635302520E0CN1^1510500^1^2^3^4^5^6^7^8a (mm)Figure 7.8: Limiting defect dimensions for failure during a hydrostatic test (numericaland experimental defect dimensions plotted).2E0304.6005104FIGURES^ 2770^10^20^30^40^50^60Number of Filling Cycles to Failure (x 10 3)Figure 8.1: Number of fueling cycles to failure as a function of initial defect depth [13].FIGURES^ 27835302520ECN^151 0500^1^2^3^4^5^6^7^8a ( m m )Figure 8.2: Limiting defect dimensions for in-service failure [13] (numerical and experi-mental defect dimensions plotted).FIGURES^ 27935302520ECN^1510500^2^3^4^5^6^7^8(mm)Figure 8.3: Limiting defect sizes for in-service and hydrostatic test failure.Appendix AELEMENT DESCRIPTIONThis appendix describes the various finite elements used in this study. (Notation isgiven at the end of Appendix B.)A.1 Two-Dimensional Eight-Node Isoparametric Solid ElementDisplacements within this element vary quadratically according to8{u} =where177) = —4 ( 1 +^71772)(-1^7770 (A.1)for corner nodes^= +1,77 = +1) and1= 2 (1 — e)(1-77712 )2( 1^77 2 )( 1 -^)(A.2)(A.3)for midside nodes (" = 0 and 77 = 0 respectively). This element in discussed further inRef. [100].A.2 Three- Dimensional 20-Node Isoparametric Solid ElementDisplacements within this element vary quadratically according to20{u} = E280Appendix A. ELEMENT DESCRIPTION^ 281whereNg, 7i, C) = —8 (1 + az )( 1 + 77 70( 1 + (Cz )( -2^+ 7777z + CC )1^ (A.4)for corner nodes (4" = +1,77 = +1,C = +1) and-Arz(,C) =^4-2)(i^7/7/z)( 1 + C()^(A.5)Ni( , 71,C) = 7/ ( 1 + 77 2 )( 1 + (Cz)( 1 + az) (A.6)Ni( , 71,C) = 71 ( 1— ( 2 )( 1 + az)( 1 +71711)^(A.7)for midside nodes^= 0, i = 0 and ( = 0 respectively). The pressure load vector iscomputed using{P} p fA {N}dA^ (A.8)This element in discussed further in Ref. [99].A.3 Eight-Node Isoparametric Thin Shell ElementDisplacements within this element vary quadratically according to{u} = Ng. 77)C1-1 ({ai }{exi }^{b,}Ovi )2z=i^i=iThe shape functions are given by equ.s (A.1) to (A.3). The element pressure load vectoris determined from equ. (A.8). This element is discussed further in Ref. [35].A.4 Element Stiffness MatricesThe stiffness matrices for all elements were computed in the standard way using[lie ] = fv [B] T [D][B]dV (A.9)The integral was evaluated using a Gaussian quadrature scheme; 2 x 2 integration pointsare used for the two-dimensional isoparametric element and 3 x 3 integration points forthe three-dimensional isoparametric element and the shell element.Appendix BELEMENTARY ELASTIC-PLASTIC FINITE ELEMENT THEORYThis appendix provides a brief overview of elastic-plastic finite element theory relatedto the analysis discussed in Chapter 5. Solution methods are discussed first. Generalplasticity theory and some specializations of this theory for bilinear kinematically hard-ening materials follows.B.1 Solution MethodsIn finite element analysis, the equation of static equilibrium isfv [13] T {a}dV = {F"d al }^ (B.1)where [B] is the strain-displacement matrix, {a} is the stress vector and {Fnc'cial } is thenodal force vector. This equation follows directly from the principal of virtual work withno assumptions regarding material response. When behaviour is linear-elastic, stressesand strains are related by{a} = [D]{E}^ (B.2)where [D] is the elasticity matrix. Substituting this equation into Equ. (B.1), and notingthat{E} = [13]{u}gives the familiar result[K]fu} = {F n°d a 1}^ (B.3)282Appendix B. ELEMENTARY ELASTIC-PLASTIC... ^ 283where the matrix [K] is the stiffness matrix given by[K] = fv [B] T [D][B]dV^ (B.4)Equ. (B.3) is linear and therefore readily solved for u. Once u is known, the otherquantities of interest, i.e., stresses and strains can be determined using the stress-strainand strain-displacement relationships.The relative simplicity with which solutions can be obtained in linear-elastic analysisis a consequence of the linear stress-strain relationship. In elastic-plastic analysis thesituation is more complicated, as the stress-strain relationship is of the form{a} = ({E})^ (B.5)where Ø is some non-linear function. Non-linearity of this relationship means that theequilibrium equation (Equ. (B.1) cannot be written explicitly in terms of nodal displace-ments. Solution of the equilibrium equation therefore, requires some iterative scheme.Two such schemes are the Newton-Raphson and initial stress methods.In a manner analagous to that employed to find the roots of transcendental func-tions, the Newton-Raphson method is based on a Taylor series expansion of Equ. (B.5).Performing this expansion about E n_ i , substituting into Equ. (B.1) and rearranging givesiK Ti { Au} { Fnodal} — fv[B]T n 1}dV^(B.6)where [KT], the tangent stiffness matrix, is given by[KT] = I [B] T [ dl[B]dVv^dEand{Au} = {u n } — {u n _ 1 }Equation (B.6) is the equation of equilibrium in the Newton-Raphson method. Becausethe tangent stiffness matrix is a function of the current state of stress, it is reformed atAppendix B. ELEMENTARY ELASTIC-PLASTIC...^ 284each iteration. While this process leads to rapid convergence, it is time consuming andhence, expensive. This drawback is overcome in the initial stress method.In the initial stress method, the matrix [dO/dE] is taken to remain unchanged fromiteration to iteration. This corresponds to taking[dOdE = [D]in Equ. (B.6) and in turn, replacing the tangent stiffness matrix with the elastic stiffnessmatrix. Noting that total strain is the sum of elastic and plastic strains, i.e.,E = E el + EP1the term {an _ i } becomes [D]({E ri _ i } — {EP,1 1 }). Making these modifications, and rear-ranging gives[K] j Un} {F"dal} + [B]T[D]{41-1}dV^(B.7)which is the equation of equilibrium in the initial stress method. Hence, the equation ofequilibrium in the initial stress method is one in which non-linearity is accounted for byan unbalanced load vector term. Because terms (other than EP1 ) are not updated fromiteration to iteration, convergence in the initial stress method is typically slower (i.e.,requires more iterations) than in the the Newton-Raphson method. However, becausethe stiffness matrix is formed only once, a solution can often be obtained in less time.In elastic-plastic analysis, external loads are applied incrementally since there is thepossibility of redistribution of internal loads. Typically, an initial load is applied that isjust sufficient to cause yielding at one or more integration points. Successive load stepsare set to some fraction of the initial load step. Iterative solution of the equilibriumequations (Equ. (B.6) or (B.7)) at the first load step begins with the elastic solution asan initial guess for {u}; successive load steps use the previous solution for {u} as anAppendix B. ELEMENTARY ELASTIC-PLASTIC...^ 285initial guess. Iteration within each load step continues until the change in plastic strainis some fraction of elastic strain, that is untilAe7:1 <Eel^R^ (B.8)where R is some constant. Ref. [88] recommends that R < 0.05.B.2 Plasticity TheoryTo proceed with an elastic-plastic analysis, an expression must be developed which quan-tifies plastic strain at each point in the load history. This is done utilizing the yieldcriteria, flow rules and hardening rules from the theory of plasticity.Yield criteria define the state of stress which will cause plastic flow. These criteriaare expressed in general terms by expressions of the form[88]f({a})—a y = 0^ (B.9)where {o- } is the stress vector and o-y is the yield stress in uniaxial tension. An equationsuch as this describes a surface in the principal stress space. If the stress state is suchthat principal stresses lie on this surface, plastic flow will occur. The directions of theresulting plastic strains are determined by a flow rule; a flow has the form{dEP1 } = A {--190Qa^(B.10)where {dEP1 } is the incremental plastic strain vector, A is a plastic multiplier, and Q is aplastic potential.Material hardening rules are incorporated into yield criteria since. effectively, materialhardening causes the yield criterion to change with increasing plastic strain. One materialhardening rule is kinematic hardening. This rule states that the yield surface translatesAppendix B. ELEMENTARY ELASTIC-PLASTIC...^ 286in the principal stress space with progressive plastic straining. To account for this effect,the yield condition is writtenF({a}, {a}) ,--- 0^ (B.11)where {a} is the location of the center of the yield surface, given by{a} = f CIdEPI I^ (B.12)and C is a constant which depends on the material stress-strain relationship.B.3 Incremental Plastic StrainA quantity of fundamental importance in elastic-plastic finite element analysis is theincremental plastic strain. An expression for this quantity can be obtained from thedifferential of Equ. (B.11),and noting that{aF}T^oF-DT; {do-}^(T;} {da} =0{do-} = [D]IdEe l l= [g({dE) — {dEP1 }){da} = COEP1 1(B.13)(B.14)(B.15)(B.16)Substituting Equ.s (B.15) and (B.16) into Equ. (B.13), and substituting Equ. (B.10)gives, after rearrangingT{Ur [D]{de}{Va lT [D] {v} _ c{g } T {V}A =Appendix B. ELEMENTARY ELASTIC-PLASTIC...Substituting this equation in Equ. (B.10) gives{g} T{g {V}ala IT [D]^vc,}T taa J{dEP1 } =287{dE}^(B.17)From inspection, it can be seen that this equation gives incremental plastic strain interms of total strain.B.4 Specializations for Bilinear Kinematic Hardening MaterialsEquation (B.17) is general in that in its derivation, no assumptions regarding the spe-cific form of yield criterion F, flow rule Q or hardening rule are made. In the analysisperformed in this study, the von Mises yield condition and associated flow rule, andkinematic hardening were assumed.The von Mises yield criterion with kinematic hardening is written (c.f. Equ. (B.11))F = [Us} lan T ({s} {ct})]^(TY = 0^(B.18)where {s} is the deviatoric stress vector given by{s} = {a} —with a, =^+ ay + az ). a is the yield surface translation vector (Equ. (B.12)).An associated flow rule is one in which the plastic potential takes the same form asthe yield condition (i.e., Q = F). This gives, for the terms {(9Q/acr} and {3F/ao- } inAppendix B. ELEMENTARY ELASTIC-PLASTIC...^ 288Equ. (B.17) I aQ^aF^3ao-^1Tc7^)-", ({ s } — {a})where a, is an equivalent stress given by{({s} - {Q17 Us} - {c24From Equ. (B.18) the terms {OF/Oa} and faF/arcl in Equ. (B.17) are{aaFa }^32a, [{8}{a} = oWhen bilinear kinematic hardening is assumed, the constant C in Equ. (B.12) is2 EETC = 3 E — ETwhere E is the elastic modulus and ET is the tangent modulus (i.e., the slope of thehardening portion of the uniaxial stress-strain curve). The term coFiaamacoo-}is thereforeC ^tact} 1 °Q.1^EETtact j to- E — ETAppendix B. ELEMENTARY ELASTIC-PLASTIC... ^ 289B.5 Notation{u} = displacement vectorN = shape functionelement coordinateelement coordinateC = element coordinateti^element thicknessO ^rotation about local x-axisOy^rotation about local y-axis{a} = unit vector in the -direction{b} = unit vector normal in plane of element normal to {a}[B] = strain-displacement matrix[D] = elasticity matrix[Ke ] = element stiffness matrixA^element face areaV^element volumeSubscriptsi = nodal quantityAppendix CPROGRAM LISTINGS290Appendix C. PROGRAM LISTINGS^ 291CC***C*** ANSYS input file for CTOD specimen.C***^Written by S.G. Ribarits.C***C***^a = crack lengthC*** s = span/prep?/title,COD SPECIMEN (a = 3.9 mm, s = 50.8 mm)C*** Set model parametersa=3.9^*crack lengths=25.4 *spanx1=2.1x2=(a-2.8)*(x1/ (a-1.8))y1=7.5y2=((y1-a)*(x2/x1))y3=((yl-a)*(.25/x1))C*** Specify element typeet,1,82„,2C*** Specify material propertiesex,1,2e5C*** Define local coordinate systemloca1,11,0„-158.6C*** Place keypointsk, 1,(a+y3)k, 2, , (a+y2)k, 3, ,^ylk, 4, .25 , (a+y3)k, 5, x2 , (a+y2)k, 6, xl ,^y1k, 7, s ,^y1k, 9,^s ,^y1k,10, ,(a-.25)k,11, ,^2.8k,12, ,^1.8k,13, .075 ja-.25)k,14, .075 ,^2.8k,15, .075 ,^1.8k,16, .25 ,(a- .25)k,17, .25^,^2.8k,18, .25^,^1.8k,19, .25 ,(a-.25)k,20, x2 ,^2.8k,21, xl^,^1.8k,22, s 1.8k,24, s^1.8k,33, 7.8k,36, xl ,^7.8k,39, s^,^7.8k,42,k,45, .075k,48, .25k,51, xlAppendix C. PROGRAM LISTINGS^ 292k,54,^skmove,36,0, x1 ,999„1, 150.8,999kmove,51,0, z1 ,999„1, 158.6,999kmove,48,0, .25 ,999„1 158.6,999kmove, 7,0, s ,999„1,(158.6-y1),999kmove, 9,0,kmove,39,0,ss,999„1,(158.6-y1),999,999„1,^150.8,999kmove,22,0, s ,999„1, 156.8,999kmove,24,0, s ,999„1, 156.8,999kmove,54,0, s ,999„1, 158.6,999kmove,45,0, .075 ,999„1, 158.6,999C*** Draw linescsys,01,^1,^2,1,10,11,1,^2,^3,1,11,12,1,^1,^4,1,10,13,1,13,16,1,17,20,1,^4,19,1,^7,22,1,^3,33,1,12,42,1,22,54,csys,11,33,36,1,42,451,48,51,1, 6, 9,21124211229, 5$rp2, 3, 3$rp4, 3, 3$rp2, 3, 3$rp4, 3, 3$rp3, 1,^1$rp3,^1,^1$rp3,^1,^1$rp2, 1,^1$rp3, 1,^1$rp3, 3, 3$rp4, 3, 3$rp2, 3, 3$rp4,15,15C*** Merge coincident keypointsnummrg,kpoiC*** Define areasa, 1, 2, 5, 4a, 4, 5,20,16a,16,20,17,17a,13,16,17,14a,10,13,14,11a, 2, 3, 6, 5a, 5, 6,21,20a,17,20,21,18a,14,17,18,15a, 3,33,36, 6a, 6,36,39, 7a, 6, 7,22,21a,21,22,54,51a,18,21,51,48a,11,14,15,12a,12,15,45,42a,15,18,48,45C*** Mesh areasAppendix C. PROGRAM LISTINGS^ 293numstr,node,44numstr,elem,9elsize,1„1amesh,1,14C*** Generate crack tip elementscsys,11*create,nplacenode=arglkpoi=arg2*get,x,kx,kpoi*get,y,ky,kpoin,node,x,y*end*use,nplace,27,1*use,nplace,31,4*use,nplace,39,16*use,nplace,41,13*use,nplace,43,10n,1 „a^$rp17,1fi11,27,31,3fil1,31,39,7fil1,39,41,1fi11,41,43,1en,^1,27,29, 3, 1,28,19, 2,18rp8,^1,^2,^2, 2, 2,^2,^1, 2,^1C*** Reorder modelcsysvsortorC*** Modify crack tip elements*create,nmovenodl=arglnod2=arg2nmid=arg3csys*get,x1,nx,nod1*get,x2,nx,nod2*get,y1,ny,nodl*get,y2,ny,nod2xmid=((.75*x1)+(.25*x2))ymid=((.75*y1)+(.25*y2))nmodif,nmid,xmid,ymid*end*use,nmove, 1,27,18*use,nmove, 3,29,19*use,nmove, 5,31,20*use,nmove, 7,33,21*use,nmove, 9,35,22*use,nmove,11,37,23*use,nmove,13,39,24*use,nmove,15,41,25*use,nmove,17,43,26C*** Merge coincident nodesnummrg,nodesC*** Apply boundary conditionsAppendix C. PROGRAM LISTINGS^ 294lsrsel„ 1lsasel„ 7lsase1„28nline,1nasel„ 1nase1„18symbc„1,allnaillsallkd,54,nyC*** Apply loadkf,33,fy,(-100)afwritefinishC*** Ann model/exe/input 0 27finishC*** Postprocess/postl/ontput,cod,09set/noprcsysnrsel,x,.249,.251csys,1nrsel,x,158.59,158.61csys/goprprdisp/noprnailnrsel,x,-.001,.001csys,1nrsel,x,150.79,150.81csys/goprprdisp/noprnailcs,12„1,27,311path,1,26,43/goprkcalc,„,1finish/output,6Appendix C. PROGRAM LISTINGS^ 295PROGRAM CODREADc***** ****** ********* ****** *** ***** ***** ******** ** ***** *********** ******CC^PROGRAM TO ANALYSE RESULTS FROM COD TESTS. A BINARY DAS20 FILE ISC^READ AND PULSES RECORDED ON CHANNEL PCHAN ARE COUNTED AND MATCHEDC^TO POTENTIAL DROP DATA FROM Al ASCII MDT FILE. DATA FROM ALLC^CHANNELS IS THEN CONVERTED TO PHYSICAL QUANTITIES AND REWRITTEN TOC^AN ASCII FILE FOR FURTHER ANALYSIS.C^WRITTEN BY S.G. RIBARITS.CC^VARIABLESC A^= CRACK LENGTHC^AO^= INITIAL CRACK LENGTHC Al, B1 = CALIBRATION CONSTANTS FOR COD DATAC^A2, B2 = CALIBRATION CONSTANTS FOR LOAD DATAC A3, B3, C3C = CONSTANTS IN JOHNSONS EQUATIONC^COD^= CRACK OPENING DISPLACEMENTC CODMN = OFFSET IN COD DATAC^COUNT = NUMBER OF PULSES COUNTED ON CHANNEL PCHANC FSIZE = NUMBER OF RECORDS TO WRITE TO OUTPUT FILEC^G1^= GAIN FOR COD DATAC G2^= GAIN FOR LOAD DATAC^G3^= GAIN FOR PD DATAC LOAD = LOADC^LOADMI = OFFSET II LOAD DATAC MINI = MINIMUM COD DATAC^MIN2 = MINUMUM LOAD DATAC NCHAN = NUMBER OF CHANNELSC^IPOINT = NUMBER OF DATA POINTS PER CHANNELC NWRITE = NUMBER OF RECORDS IN ORIGINAL DATA FILE TO SKIPC^PCHAN = CHANNEL CONTAINING PULSESC UNIT1 = UNIT CONNECTED TO DAS20 BINARY DATA FILEC^UNIT2 = UNIT CONNECTED TO MDT ASCII FILEC U1IT3 = UNIT CONNECTED TO GAIN INPUT FILEC^UNIT4 = UNIT CONNECTED TO LOAD AID COD CALIBRATION INPUT FILEC UNIT7 = UNIT CONNECTED TO PD CALIBRATION INPUT FILEC^UNITS = UNIT CONTECTED TO OUTPUT FILEC RANGE = RANGE USED TO SCALE PD DATAC^THRHLD = TRESHOLD SET TO DETECT PULSES ON PCHANC UO^= VOLTAGE CORRESPONDING AO= SPECIMEN WIDTHC^Y^= HALF DISTANCE BETWEEN PD PROBESCC^ARRAYS= DATACCINTEGER*1 UNIT1, UNIT2, UNIT3, UNIT4, U1IT7, UNITS,PCHAN, NCHANINTEGER*2 1(35000,3), THRHLD, MINI, MIN2INTEGER COUNT, FSIZEREAL LOADPARAMETER(PI=3.141592654)CC^INITIALIZE VARIABLESCICHAN = 3IPOIIT = 12000THRHLD = 1024PCHAN = 3RANGE = 5.0CC^READ BINARY DAS20 FILECUNIT1 = 1OPENCUIIT=UNIT1,FILE='TEST1DAS.DAT,,Appendix C. PROGRAM LISTINGS^ 296 STATUS='OLD',ACCESS='DIRECT',FORM='UNFORMATTED',RECL=2)CALL READER(UNIT1,1CHAN,IPOINT,I)CLOSE(UNIT=UIIT1)CC^COUNT PULSES ON CHANNEL PCHANCCALL PULSE(PCHAN,X,IPOINT,THRHLD,COUNT)CC^READ POTENTAL DROP DATA CORRESPONDING TO PCHAN PULSE NUMBERCUNIT2 2OPEN(UNIT=UNIT2,FILE='TEST1PD.DAT', STATUS='OLD',ACCESS='SEQUENTIAL',FORM='FORMATTED , , BLANK='ZER0')CALL PDREAD(UNIT2,X,PCHAN,COUNT,IPOINT,RANGE)CLOSE(UNIT=UNIT2)CC^READ GAINS, CALIBRATION FACTORS, AND POTENTIAL DROP CONSTANTSCUNIT3 = 3UNIT4 = 4UIIT7 = 7OPEN(UNIT=UNIT3,FILE='GAINS1.DAT", STATUS=0OLD',ACCESS='SEQUENTIAL',FORM='FORMATTED', BLANK='ZER0')OPEN(UNIT=UNIT4,FILE='CFACTS1.DAT', STATUS='OLD',ACCESS='SEQUENTIAL',FORM="FORMITTED', BLANK='ZER0')OPEN(UNIT=UNIT7,FILE='PDCONST1.DAT', STATUS='OLD',ACCESS='SEQUENTIAL',FORM='FORMATTED', BLANK='ZER0')CREAD(3,*) Gl, G2, G3READ(4,*) Al, B1, A2, B2READ(7,*) W, Y, A0, UOCCLOSE(UNIT=UIIT3)CLOSE(UNIT=UIIT4)CLOSE(UNIT=UNIT7)CC^CALCULATE CONSTANTS IN JOHNSON'S EQUATIONCA3 = (2*V)/PIB3 = COSH((PI*Y)/(2*W))C3 = ACOSH(B3/COSUPI*A0)/(2*W)))CC^CONVERT INTEGER VALUES TO PHYSICAL QUANTITIES AND WRITE EVERYC^'WRITE DATA SET TO OUTPUT FILECFSIZE = 1000NWRITE = INT(NPOINT/FSIZE)UNITS = 8OPEN(UNIT=UNIT8,FILE='TEST1OUT.PRI', STATUS='UNKNOWN',ACCESS='SEQUENTIAL',FORM='FORMITTED', BLANK=,ZERD')MINI = 4096MIN2 = 4096DO 4 I = 1, IPOINT, NWRITEIF(I(I,1).LT.MIN1) MINI = X(I,1)IF(X(I,2).LT.MI12) MIN2 = X(I,2)4 CONTINUECCODMI = (MI11/4096.)*G1CODM1 = Al*CODMI + B1LOADMI = (MIN2/4096.)*G2LOADMI = A2*LOADMI + B2CAppendix C. PROGRAM LISTINGS^ 297PRINT*,,CODMI =',CODMN,'LOADMI =',LOADMICDO 2 I = 1, IPOINT, NWRITECOD = (I(I,1)/4096.)*G1COD = Al*COD + B1COD = COD - CODMNLOAD = (I(I,2)/4096.)*G2LOAD = A2*LOAD + B2LOAD = LOAD - LOADMIA = (X(I,3)/4096.)*G3A = A3*ACOS(B3/COSH(CA/U0)*C3))WRITE(UNIT8,1) COD, LOAD, A1^FORMIT(3(G12.5,1X))2 CONTINUECC^CLOSE FILES AND TERMINATE PROGRAMCCLOSE(UNIT=UNIT8)CSTOPENDCc**** ******** **************** ****** *** ***** ***** ******** * ******* ** ******CSUBROUTINE READERCIUNIT,ICHAN,NPOINT,X)CC^SUBROUTINE TO READ BINARY DAS20 DATA FILES.C^WRITTEN BY S.G. RIBARITSCC^DATA = DATAC ICHAN = CHANNELC^IUNIT = UNIT CONNECTED TO DAS20 BINARY DATA FILEC IRON = ROW II ARRAY X CORRESPONDING TO NREC AND ICHANC^NCHAN = NUMBER OF CHANNELSC 'POINT = NUMBER OF DATA POINTS PER CHANNELC^NUM^= 16 BIT VALUE ENCODED WITH DATA AND CHANNELC NREC = NUMBER OF RECORDS TO READC^VAL^= TEMPORARY VARIABLEC X^= ARRAY CONTAINING DATACINTEGER*1 'UNIT, ICHAN, NCHANINTEGER*2 NUM, DATAINTEGER*2 1(35000,3)CIREC = NPOINT*NCHANCDO 1000 IREC = 1, IRECREAD(IUTIT,REC=IREC) NUMIF (IUM.GE.0) THENVAL = NUM/16.0DATA = IFIX(VAL)ELSEVAL = (65536.0 + NUM)/16.0DATA = IFII(VAL)END IFICHAN = INT((VAL - DATA)*16.0)ICHAN = ICHAN + 1IRON = (IREC - ICHAN)/NCHAN + 1ICIROW,ICHAN) = DATA1000 CONTINUECRETURNENDCc*** ************ *********** ******** ********** ********* * ****** * **********CSUBROUTINE PULSE(PCHAN,X,IPOINT,THRHLD,COUNT)Appendix C. PROGRAM LISTINGS^ 298CC^SUBROUTINE TO COUNT CHANNEL PCHAN PULSES.C^WRITTEN BY S.G. RIBARITSCC^COUNT = NUMBER OF PULSES COUNTEDC IPOINT = NUMBER OF DATA POINTS PER CHANNELC^OFF,ON = LOGICAL VARIABLES USED TO FLAG PULSESC PCHAN = CHANNEL CONTAINING PULSESC^TIMID = THRESHOLD SET TO DETECT PULSES= ARRAY CONTAINING DATACINTEGERS1 PCHANINTEGER*2 1(35000,3),THRHLDINTEGER COUNTLOGICAL ON,OFFCOFF= .FALSE.ON= .FALSE.COUNT = 1CDO 1000 I = 1, IPOINTIF (I(I,PCHAN).GT.THRHLD) THENON = .TRUE.END IFIF (I(I,PCHAN).LT.THRELD.AND.ON) THENOFF^.TRUE.Of = .FALSE.END IFIF (OFF.AND..NOT.ON) THENCOUNT = COUNT + 1OFF = .FALSE.ON = .FALSE.END IF1(I,PCHAN) = COUNT1000 CONTINUERETURNENDCc*:************ ***** ******************* ******** ****** ******** ******* ***CSUBROUTINE PDREAD(IUNIT,I,PCHAN,COUNT,NPOIIT,RANGE)CC^SUBROUTINE TO READ MDT ACSII FILE AID MATCH DATA WITHC^PULSES COUNTED ON PCHAN.C^WRITTEN BY S.G. RIBARITSCC^COUNT = NUMBER OF PULSES COUNTEDC DATA = SCALED POTENTIAL DROP DATAC^IUNIT = UNIT CONNECTED TO MDT ASCII FILEC IPOINT = NUMBER OF DATA POINTS PER CHANNELC^PCHAN = CRANIAL CONTAINING PULSESC PDCHAI = POTENTIAL DROP CHANNELC^PDDATA = POTENTIAL DROP DATAC RANGE = RANGE USED TO SCALE POTENTIAL DROP DATA= ARRAY CONTAINING DATACINTEGERS1 'UNIT, PCHAN, PDCHAIINTEGER*2 1(35000,3), DATAINTEGER COUNTCICOUIT = 1'POINT = 11 IF (.10T.(ICOUNT.LE.COUNT.AND.IPOINT.LE.IPOINT)) GOTO 4READ(IUNIT,*) PDCHAI, PDDATADATA = NINT((PDDATA/RANGE)*4096.)2^IF (.NOT.(I(IPOINT,PCHAN).EQ.ICOUIT)) GOTO 3ICIPOIIT,PCHAN) = DATAAppendix C. PROGRAM LISTINGS^ 299IPOIIT = IPOINT + 1IF (.10T.CIPOINT.LE.NPOINT)) GOTO 3GOTO 23^CONTINUEICOUNT = ICOUIT + 1GOTO 14 CONTINUECRETURNENDCc************ ******* ********* ****** *** *************************** * ******CREAL FUNCTION ACOSH(X)CC^CALCULATES THE INVERSE HYPERBOLIC COSINE OF ARGUMENT ICIF (I.LT.1) THENWRITE(*,1)1^FORMAT('*ERROR* AN ARGUMENT THAT WAS LESS THAN 1',' WAS PASSED TO FUNCTION ACOSH')RETURNEND IFCACOSH = LOG(I + SQRT(X*I - 1))CRETURNENDAppendix C. PROGRAM LISTINGS^ 3000**** ******** *********** ****** ******* ******** ****** ******** *********C***C*** AISYS input file for NGV cylinder coarse model.C*** Written by S.G. Ribarits.C***C***^p = pressureC***/prep?/title, YGV Cylinder, Coarse model (P = 1500 psi)C*** Set model parameters*set,p,1500C*** Specify element typeset,1,93et,2,93C*** Define material properties/real constantsez,1,2e5r,1,7.8C*** Define local coordinate systemsloca1,11,1„334loca1,12,1,49.8,-334loca1,13,1„-266.354loca1,14,1„„„-90C*** Generate areask,1,9.771„488.391k,2,154.7„334k,3,154.7„-334k,4,111.991„-418.477k,5,19.9421„-454.198k,6,„-455.254csys,111,1,2,14csys,141,2,3,20csys,121,3,4,9csys,131,4,5,51,5,6,1csys,14k,100,„-100k,200,,,100arotat,1,2,3,4,5„100,200,90lsrse1„6,10ldvs,a11„6lsallC*** Mesh areastype,1elsize,35„2amesh,1,4elsize,35„1amesh,5C*** Specify elements for print controlsAppendix C. PROGRAM LISTINGS^ 301nrsel,y,-1,61nrsel,z,-1,201enode,1type,2emodif,all,0eallnailC*** Reorder elementsnsrse1„6nline,1lsallwstart,allnailwavesC*** Apply boundary conditionssymbc„1„.5symbc„3„.5nrsel,z,-.5, ^ 5d,a11,uynaillsrse1„6nline,1lsallddele,allnrotat,alld,a11,ux ^uy,rotx,roty,rotznailC*** Apply loadsp = 1500ep,a11,2,.006895*arglpodisp,-1postr,-1,1postr„2porf,-1afwritfinishC*** Assign output to FILE12.CRS and run model/get,12,file12,crs/input,27finishAppendix C. PROGRAM LISTINGS ^ 302cm*** ***** ********************** ****** ************** ********** ******C***C*** AISYS input file for IGV cylinder defect submodel.C***^Written by S.G. Ribarits.C***C***^a^= crack depthC*** arc = submodel arc lengthC***^c^= half crack lengthC*** ht = submodel heightC***^fact = ratio of pressure in current load stepC*** to pressure in previous load stepC***^ksqu = c**2 - a**2C*** loc = flag set to 1 for interior defect,C*** 2 for exterior defectC***^p0 = initial pressureC*** pl^pressure in previous load stepC***^p2 = pressure in current load stepC*** pinc = incremental pressureC***^post = flag set to 1 to print results for current loadC*** step, 1 suppress to results for current load stepC***^rsqu = x**2 + y**2C*** s^= geometry parameterC***^x,y = nodal coordinates in csys 12C***/prep7/title, Interior Crack Submodel, (a = 5, c = 15)C*** Set model parameters*set,ht,100*set,arc,30*set,loc,1*set,a,5.0*set,c,15.0*set,s,(180.0*((2.0/154.7)/3.14159))*set,post,1*set,p0,1500*set,pinc,250C*** Define material propertiesez,1,2e5knl,1n1,1,13, 10n1,1,19, 20,^25n1,1,25, 822, 822n1,1,31,5050,5050C*** Specify element typeset,1,45et,2,45et,3,45C*** Define local coordinate systems*if,loc,eq,2,:lb3dist = 150.8ang = 0*go,:lb4:1b3dist = 158.6ang = 180:1b4loca1,21 ^ -90Appendix C. PROGRAM LISTINGS^ 303loca1,11,1 ^ -90loca1,12,0,dist„,90„angloca1,13,1,dist„,90„angC*** Generate solid modelcsys,13k, 1, 0.05k, 8,a-2.0k,15,a+2.0kgen,3,1,15,7„17.5„1kgen,5,3,17,7„13.75„1csys,12k,22,11.140k,23,11.140,3.512k,24,11.140,7.8kgen,2,22,24,1,(ht-11.140),,,7k,24,11.140,7.8k,25,6.840,7.8k,26,4.060,7.8k,27,1.909,7.8k,28„7.8csys,13kgen,2,1,31,1,„s,31kgen,2,32,62,1,„(arc-s),311,1,8,21,32,39,21,63,70,21,39,46,21,70,77,21,15,22,21,46,53,21,77,84,2csys,121,22,23,11,53,54,11,84,85,11,22,29,5,101,53,60,5,101,84,91,5,101,32,63,9,10csys,13v,1,2,9,8,32,33,40,39v,15,16,23,22,46,47,54,53v,32,33,40,39,63,64,71,70v,39,40,47,46,70,71,78,77v,46,47,54,53,77,78,85,84csys,12v,22,23,30,29,53,54,61,60v,53,54,61,60,84,85,92,91$rp7,1,1$rp7,1,1$rp7,1,1$rp7,1,1$rp7,1,1$rp7,1,1$rp7,1,1$rp7,1,1$rp6,1,1$rp6,1,1$rp6,1,1$rp3,1,1$rp3,1,1$rp3,1,1$rp31,1,1$rp6,1,1,1,1,1,1,1,1$rp6,1,1,1,1,1,1,1,1$rp6,1,1,1,1,1,1,1,1$rp6,1,1,1,1,1,1,1,1$rp6,1,1,1,1,1,1,1,1$rp2,1,1,1,1,1,1,1,1$rp2,1,1,1,1,1,1,1,1C*** Place nodes at keypoints*create,nplace*set,kpoi,arg1Appendix C. PROGRAM LISTINGS^ 304*set,node,arg2*get,nz,kx,kpoi*get,ny,ky,kpoi*get,nz,kz,kpoin,node,nx,ny,nz*end*use,nplace,8,16rp7„1,20*use,nplace,39,17rp7„1,20*use,nplace,46,19rp7„1,20*use,nplace,15,20rp7„1,20fil1,16,20,1,1„7,20^$rp5,,,,1fil1,17,19,1,18,7,20fil1,1,16,2,6,5,5,1^Srp7,20,20„20C*** Generate crack tip elementstype,1e,6,7,2,1,26,27,22,21egen,4,1,-1egen,3,5,-4egen,6,20,-12C*** Mesh volumeselsize,35„2vmesh,allC*** Connect crack tip elements to solid modelnrse1„1,5,1nase1„21,25,1ninvmergenailnumcmp,nodeC*** Modify element typesvirse1„7,12,1vlase1„31,32,1evoluease1„4,12,4type,2emodif,a11,0eallvlallvirse1„1,6,1evoluease1„1,9,4type,3emodif,a11,0eallvlallSrp6„20,20Srp6„12,12$rp6„12,12C*** Detach elements from solid modelmodmsh,nocheckmodmsh,detachC*** Transform nodal coordinates to ellipticalcsys,12sset,ksqu,(c**2)-(a**2)Appendix C. PROGRAM LISTINGS^ 305screate,modfy1*get,x,nx,argl*get,y,ny,argl*set,rsqu,(x**2)+(y**2)*set,x,x*sqrt(1+(ksqu/rsqu))nmodif,arg1,x*end*use,modfy1,1rp802„1C*** Fix badly formed elementsnrsel,y,-.01,.06nmodif,all„le-10nail*create,efix*get,n1,enl,argl*get,n2,en2,argl*get,n3,en3,arg1*get,n4,en4,arg1*get,n5,en5,arg1*get,n6,en6,argl*get,n7,en7,arg1*get,n8,en8,arg1emodif,arg1,1,n2,n4,n3,n3,n6,n8,n7,n7ndele,n1ndele,n5*endtype,3*use,efix,73type,1*use,efix,97rp9„1numcmp,nodesC*** Change element type to 20 node isoparametricet,1,95et,2,95et,3,95emidC*** Transform nodal coordinates to cylindricalcsys,21trans,11„allC*** Reorder elementscsys,0vsort,znrsel,z,-.01,.01wstart,allnallwavesmergenumcmp,nodesC*** Write boundary nodes to file26.datcsys,11nfile„26nrsel,z,(ht-1.0),(ht+5.0)nasel,y,(arc-1.0),(arc+1.0)nwriteAppendix C. PROGRAM LISTINGS^ 306nailfinishCs** Perform cut boundary displacement conversion/get,12,file12,crs/auz1cbdsp,„,1finishC*** Reassign output to filel2.dat/get,12,file12,dat/prep?resumeC*** Apply boundary conditionsersel,type,2nelemcsys,11nrsel,y,-.001,.001nusel,z,(ht-1.0),(ht+5.0)symbc,11,2„.001naileallnrsel,z,-.001,.001nwel,y,(arc-1.0),(arc+1.0)symbc,11,3„.001nailC*** Set solution option and convergence criteriakay,9,2iter,-25„25kuse,0cnvr,.05C*** Apply 1500 psi pressure*set,p0,(p0*.006895)*set,pinc,(pinc*.006895)podisppostr,„5porf ,-1/input,24psf,11,1,150.8,p0„.001ersel,type,3nelemnrsel,y,-.001,.001psf,11,2„p0„.001nailealllwriteC*** Apply 6000 psi pressure in 250 psi incrementspost =0*create, loads*set,p2,(p0+(pinc*argel))*set,p1,(p0+(pinc*(arg1-1)))*set ,fact ,p2/p1lscale,fact„fact*if,post,eq,0,:1b3podisppostr,„5porf,-1Appendix C. PROGRAM LISTINGS^ 307post=0*go,:1b4:1b3podisp,-1postr,-1porf,-1post=1:1b4lwrite*end*use,loads,1rp18„1afwritfinishC*** Assign output and run submodel\get,12,file12,01i\input,27finishAppendix C. PROGRAM LISTINGS^ 308CC***C*** Routine to calculate and plot membrane and bending stressesCs** at boundary of IGV cylinder submodel.Cs** Written by S.G. RibaritsC***Cs**/postlCs** Define local coordinate systemloca1,11,„„-90csys,11dsys,11C*** Create macro STRPLT/shov,file33,dat,1screate,strpltset ,argl1path,604,674pdef,intr,szi,szpdef,save1path,619,719pdef,intr,szo,szpcalc,add,szb,szi,szo,+.5,-.5pcalc,add,szm,szi,szo,+.5,+.5pcalc,div,sigz,szb,szm/graph,labx,Y/graph,laby,SIGframe,-.95,.+95pviev,plot,sigz1path,674,320pdef,intr,syi,sypdef,save1path,719,581pdef,intr,syosycalc,add,syb,syi,syo,+.5,-.5calc,add,syb,syi,syo,+.5,+.5pcalc,div,sigy,syb,sbm/graph,labx,2/graph,laby,SIGframe,-.95,+.95pviev,plot,sigy*endC*** Execute STRPLT for 1500 psi, 2000 psi, 2500 psi...6000psi*use,strplt,1rp10„2finishAppendix C. PROGRAM LISTINGS^ 309CC***C*** Routine to calculate stress intensity factor asC*** a function of angle around an elliptical defect.C*** Written by S.G. RibaritsC***C***/postlset,1C*** Create macro CSCALC*create,cscalccsys*get,x1,nx,argl*get,x2,nx,arg2*get,y1,ny,arg1*get,y2,ny,arg2*get,z1,nz,argl*set,delx,x1-x2*set,dely,y1-y2*if,delx,eq,0,:lb1*set,ang,atan(dely/delx)*set,ang,(180*ang)/3.14159*go,:lb2:lbl*set,ang,90:1b2loca1,21,0,x1,y1,z1,ang„90csys*set,delx,x1-150.8*set,dely,y1*if,dely,le,0,:1b3*set,phi,atan(delx/dely)*set,phi,(180*phi)/3.14159*go,:lb4:1b3*set,phi,90:1b4csys,21*endC*** Calculate stress intensity factors*use,cscalc,1,21path,1,689,2kcalc,1*set,phil,phi*get,kil,kcalc,ki*use,cscalc,17,181path,17,739,18kcalc*set,phi2,phi*get,ki2,kcalc,ki*use,cscalc,113,1141path,113,1039,114kcalc*set,phi8,phi*get,ki8,kcalc,kiC*** Print resultsAppendix C. PROGRAM LISTINGS^ 310/output,kfactors,01i*stat/output,6finishc******************* ****** A,* ********* *********** ****** *** ******* *****C***Cs** Macro to calculate COD, CTOD and point of rotation r.Cs** Written by S.G. RibaritsC***Cs**/postlC*** Set parameters*set,a,5.0*set,loc,1C*** Define local coordinate system*if,loc,eq,2,:1b3dist = 150.8ang = 0sgo,:lb4:1b3dist = 158.6ang = 180:1b4loca1,22„dist,„ang„-90C*** Create macro to calculate COD, CTOD and r*create, ctodp = (250*(arg1-1)) + 1500setgcsys,22nrsel,z,-.001,.001nrsel,y,-.001,.001nrsel,x, - .001,(.65*a)ninvndele,allninvset , arglsuml = 0sum2 = 0sum3 = 0sum4 = 0n = 0*get,max,ndmx:1b1*get,node,ndmn*get,x1oc,x,node*get,disp,uy,nodexloc = xloc/asuml = xloc + sumlsum2 = disp + sum2sum3 = (xloc*disp) + sum3sum4 = (xloc**2) + sum4n = n + 1nusel„node*if,node,eq,max,:1b2*go,:lb1:1b2vall = (suml*sum2)/nAppendix C. PROGRAM LISTINGS^ 311val2 = (suml**2)/nc1 = (suln3 - vall)/(sum4 - val2)c2 = (sum2 - (cl*sum1))/ncod = c2ctod = c1 + c2r = -c2/c1nail*stet*endC*** Execute CTOD for 1500 psi, 2000 psi, 2500 psi...6000psi/output,CTOD,Oli*use,ctod,1rp10„2/output,6finishC******************* ***** ***** ******* * ****** ************ ************ **C***C*** Macro to read displacements of nodes on exteriorC*** of submodel.C***^Written by S.G. RibaritsC***C***/postlC*** Define local coordinate system and select nodesloca1,11,1„„„-90C*** Create macro to read displacementsnlines,1000/nopr*create dspreadset,arg1csys,11nrsel,x,158.59,158.61csys,0prdisp,allsendC*** Execute DSPREAD for 1500 psi, 2000 psi, 2500 psi .6000psi/output,disp,01i*use,dspread,1rp10„2/output,6finishAppendix C. PROGRAM LISTINGS^ 312PROGRAM ARRANGEC********************************* ********* ***** ********* ******** ******CC^PROGRAM TO ORDER NODE NUMBERS ON FACES OF ELEMENTS FOR VOLUMEC CALCULATIONS. NODE NUMBERS OF NODES ON THE EXTERIOR OF THE MODELC^(FILE26.RUI) ARE COMPARED WITH THE LODE NUMBERS DEFINING EACHC ELEMENT (FILE14.RU1) TO DETERMINE WHICH ELEMENT FACES LIE Of THEC^EXTERIOR OF THE MODEL. THE NODE NUMBERS DEFINING THESE FACES AREC THEN PLACED IN ARRAY IFACE IN AN ORDER WHICH DEPENDS ON ELEMENTC^CONNECTIVITY.C WRITTEN BY S.G. RIBARITSCC^VARIABLESC NELEM = NO. OF ELEMENTSC^!NODES = NO. OF NODESC RUN^= RUN CODECC^ARRAYSC IEL^= ANSYS ELEMENT NUMBERC^IELEM = NODES OH ELEMENTC IFACE = NODES OH ELEMENT FACEC^IREAL = ELEMENT REAL CONSTANT (ANSYS VARIABLE)C ITYPE = ELEMENT TYPE (AISYS VARIABLE)C^MAT^= ELEMENT MATERIAL CODE (AISYS VARIABLE)C INUM = NODE NUMBERS OF NODES ON EXTERIOR OF MODELC^X,Y,Z = GLOBAL COORDINATES OF NODESCCINTEGER COUNTIMPLICIT DOUBLE PRECISION (A-H 2O-Z)CHARACTER RUI*3DIMENSION NIUM(1000),X(1000),Y(1000),Z(1000),IELEM(500,20),IFACE(500,8),MAT(500),ITYPE(500),IREAL(500),IEL(500)CC^INITIALIZE RUNCRUN = '01I ,CC^OPEN INPUT/OUTPUT FILESCOPEN(UNIT=1,FILE='FILE26.'//RUI,STATUS=,OLD,, ACCESS= , SEQUENTIAL',FORM= , FORMATTED , ,BLAIX='ZERO')OPEN(UHIT=2,FILE='FILE14.V/RUN,STATUS= , OLD , , ACCESS='SEQUEITIAL , ,FORM= , FORMATTED',BLAIK='ZERO')OPEN(UNIT=3,FILE='FIODES.V/RUI,STATUS='UNKNOWN', ACCESS= , SEQUENTIAL',FORM='FORMATTED',BLAIA='ZERO')COUNT = 110 READ(1,20,END=30) NIUM(COUNT), X(COUIT), Y(COUIT), Z(COUNT)20 FORMAT(I5,3G16.9)COUNT = COUNT + 1GOTO 1030 REWIND(UlIT=1)MODES = COUNT - 1COUNT = 140 READ(2,50,END=60) (IELEM(COUIT,I),I=1,8), MAT(COUIT),ITYPE(COUIT), IREAL(COUIT), IEL(COUNT)READ(2,50) (IELEM(COUIT,I),I=9,20)50 FORMAT(12I6,8I)COUNT = COUNT + 1GOTO 4060 REWIND(UlIT=2)IELEM = COUNT - 1CWRITE(3,70) !NODES, IELEMCCAppendix C. PROGRAM LISTINGS^ 31370 FORMAT(2I6)CDO 140 I = 1, IELEMDO 80 J = 1, 8IFACE(I,J) = 0^80^CONTINUEFLAG = 0K = 1J = 090^IF (FLAG.EQ.1) GOTO 110DO 100 L = 1, 8IF(NIUM(K).EQ.IELEM(I,L)) THENIFACE(I,L) = 1J = J + 1END IF100^CONTINUEIF (K.EQ.NIODES.OR.J.EQ.4) FLAG = 1K = K + 1GOTO 90110^CONTINUEIF (J.EQ.4) THENIF (CIFACE(I,1).EQ.1).AID.(IFACE(I,2).EQ.1).AND.(IFACE(I,3).EQ.1).AND.(IFACE(I,4).EQ.1)) THEYIFACE(I,1) = IELEM(I,1)IFACE(I,2) = IELEM(I,9)IFACE(I,3) = IELEM(I,2)IFACE(I,4) = IELEM(I,10)IFACE(I,5) = IELEM(I,3)IFACE(I,6) = IELEM(I,11)IFACE(I,7) = IELEM(I,4)IFACE(I,8) = IELEM(I,12)END IFIF ((IFACE(I,5).EQ.1).AID.(IFA(E(I,6).EQ.1).AID.(IFACE(I,7).EQ.1).AND.(IFACE(I,8).EQ.1)) THENIFACE(I,1) = IELEM(I,5)IFACE(I,2) = IELEM(I,16)IFACE(I,3) = IELEM(I,8)IFACE(I,4) = IELEM(I,15)IFACE(I,5) = IELEM(I,7)IFACE(I,6) = IELEM(I,14)IFACE(I,7) = IELEM(I,6)IFACE(I,8) = IELEM(I,13)END IFIF ((IFACE(I,3).EQ.1).AND.(IFACE(I,4).EQ.1).AID.(IFACE(I,7).EQ.1).AND.(IFACE(I,8).EQ.1)) THENIFACE(I,1) = IELEM(I,3)IFACE(I,2) = IELEM(I,19)IFACE(I,3) = IELEM(I,7)IFACE(I,4) = IELEM(I,15)IFACE(I,5) = IELEM(I,8)IFACE(I,6) = IELEM(I,20)IFACE(I,7) = IELEM(I,4)IFACE(I,8) = IELEM(I,11)END IFIF (CIFICE(I,1).EQ.1).AND.(IFACE(I,2).EQ.1).AND.(IFACE(I,5).EQ.1).AID.(IFACE(I,6).EQ.1)) THENIFACE(I,1) = IELEM(I,1)IFACE(I,2) = IELEM(I,17)IFACE(I,3) = IELEM(I,5)IFACE(I,4) = IELEM(I,13)IFACE(I,5) = IELEM(I,6)IFACE(I,6) = IELEM(I,18)IFACE(I,7) = IELEM(I,2)IFACE(I,8) = IELEM(I,9)END IFIF ((IFACE(I,2).EQ.1).AND.(IFICE(I,3).EQ.1).AND.(IFACE(I,6).EQ.1).AID.(IFACE(I,7).EQ.1)) THENAppendix C. PROGRAM LISTINGS^ 314IFACE(I,1) = IELEM(I,2)IFACE(I,2) = IELEM(I,18)IFACE(I,3) = IELEM(I,6)IFACE(I,4) = IELEM(I,14)IFACE(I,5) = IELEM(I,7)IFACE(I,6) = IELEM(I,19)IFACE(I,7) = IELEM(I,3)IFACE(I,8) IELEM(I,10)END IFIF ((IFACE(I,1).EQ.1).AID.(IFACE(I,4).EQ.1).AND.(IFACE(I,5).EQ.1).AND.(IFACE(I,8).EQ.1)) THENIFACE(I,1) = IELEM(I,1)IFACE(I,2) = IELEM(I,12)IFACE(I,3) = IELEM(I,4)IFACE(I,4) = IELEM(I,20)IFACE(I,5) = IELEM(I,8)IFACE(I,6) = IELEM(I,16)IFACE(I,7) = IELEM(I,5)IFACE(I,8) = IELEM(I,17)END IFWRITE(3,120) (IFACE(I,M),M=1,8)120^FORMAT(8(3X,I4))END IFIF (J.IE.4) THENWRITE(6,130) IEL(I)130^FORMAT('*WARIING* LESS THAN 4 CORNER NODES WERE FOUND 01',ELEMEIT',I5)END IF140 CONTINUECCLOSE(UNIT=1)CLOSE(UIIT=2)CLOSE(UNIT=3)CSTOPENDAppendix C. PROGRAM LISTINGS^ 315PROGRAM VOLCALCC*********************** ***** ************************** ********* * *******CC^PROGRAM TO CALCULATE THE VOLUME OF A CIG CYLINDER FROM FINITEC ELEMENT RESULTS. THE INCREMENTAL VOLUME BOUNDED BY AI ELEMENTC^FACE AID THE ORIGIN IS CALCULATED AND SUMMED.C WRITTEN BY S.G. RIBARITSCC^VARIABLESC ILOAD^= COUNTERC^INODE^= NODE NUMBERC LSTEP^= LOAD STEPC^NELEM^= NO. OF ELEMENTSC IFACE^= NO. OF NODES ON ELEMENT FACEC^MODES = NO. OF NODESC PRESS^= PRESSUREC^V^= TOTAL VOLUMEC VINC^= INCREMENTAL VOLUMECC^ARRAYSC IFACE^= NODES ON FACE OF ELEMENTC^LOAD^= PRESSURE CORRESPONDING TO ILOADC UI,UY,UZ = NODAL DISPLACEMENTSC^X,Y,Z^= GLOBAL COORDINATES OF NODES= LOCAL COORDINATES OF NODESCCINTEGER PRESSIMPLICIT DOUBLE PRECISION (A-H 2 O-Z)CHARACTER RUI*3DIMENSION 1(5000),Y(5000),Z(5000),IFACE(500,8),UX(5000),UY(5000),UZ(5000),II(8),YY(8),ZZ(8),LOAD(6)DATA LOAD/1500,2000,3000,4000,5000,6000/CC^INITIALIZE RUICRUN = '01I'CC^OPEN INPUT/OUTPUT FILESCOPEN(UNIT=1,FILE='FILE26.'//RUN,STATUS='OLD', ACCESS="SEQUENTIAL',FORM='FORMATTED , ,BLANK='ZERO')OPEN(UNIT=2,FILE='FNODES.'//RUN,STATUS='OLD', ACCESS='SEQUENTIAL',FORM='FORMATTED',BLAHK='ZERO')OPEN(UNIT=3,FILE='DISP.'//RUN,STATUS='OLD', ACCESS='SEQUENTIAL',FORM='FORMATTED',BLANX='ZERO')OPECUIIT=4,FILE='VOLU'//RUN//'.PRI',STATUS='UNKIOWI', ACCESS='SEQUENTIAL',FORM='FORMATTED',BLANK= , ZER0')CC^READ NODES AND ELEMENTSCREAD(2,5) INODES,NELEM5 FORMAT(2I6)CDO 20 I = 1, NNODESREAD(1,10) INODE, Y(INODE), Y(INODE), Z(INODE)10^FORMAT(I5,3G16.9)20 CONTINUEDO 40 IELEM = 1, NELEMREAD(2,30) (IFACE(IELEM,I),I =1,8)30^FORMAT(8(31(,I4))40 CONTINUECC^WRITE HEADER TO OUTPUT FILECWRITE(4,45) RUNAppendix C. PROGRAM LISTINGS^ 31645 FORMIT('RUN',1X,A3//'LOAD STEP',3I,'PRESSURE (psi)',3I,'VOLUME (cc)')CC^LOOP THROUGH LOAD STEPS; READ LOAD STEP DATA AID DISPLACEMENTSCDO 130 ILOAD = 0, 6IF (ILOAD.GT.0) THENREAD(3,50) LSTEP50^FoRmAT(/////////14x,I3/////)ASSIGN 60 TODO 80 J = 1, IIODESIF (J.EQ.INODES.AND.ILOAD.LT .6) ASSIGN 70 TO NNREAD(3,NN) INODE, UX(INODE), UY(INODE), UZ(IIODE)60^FORMIT(2X,I5,11,3(1X,G15.8))70 FoRMIT(2x,i5,11,3(1x,G15.8)////)80^CONTINUEEND IFCC^CALCULATE INCREMENTAL VOLUMES AID ADD TO TOTAL VOLUMECV = 0.D0DO 110 IELEM = 1, HELENDO 90 INODE = 1, 8XI(INODE) = X(IFACE(IELEM,INODE))YY(IIODE) = Y(IFICE(IELEM,INODE))ZZ(INODE) = Z(IFACE(IELEM,INODE))IF (ILOAD.GT.0) THENII(INODE) = IX(INODE) + UX(IFACE(IELEM,INODE))YY(IIODE) = YY(IIODE) + UY(IFACE(IELEM,INODE))ZZ(IIODE) = ZZ(IIODE) + UZ(IFACE(IELEM,INODE))EID IF90^CONTINUE= -(11(1) + XX(3) + XX(5) + XX(7))/4.D0+(IX(2) + IX(4) + IX(6) + IX(8))/2.D0YK = -(YY(1) + YY(3) + YY(5) + YY(7))/4.D0+(YY(2) + YY(4) + YY(6) + YY(8))/2.D0ZK = -(ZZ(1) + ZZ(3) + ZZ(5) + ZZ(7))/4.D0+(ZZ(2) + ZZ(4) + ZZ(6) + ZZ(8))/2.D0DO 100 INODE = 1, 8II = INODENJ = INODE + 1IF (1J.EQ.9) NJ = 1II = II(NI)YI YUNI)ZI = ZUNI)KJ = KE(IJ)YJ = YY(IJ)ZJ = ZUNJ)VINC = DABSUII*(YJ*ZA - ZJ*YK) - YI*(XJ*ZK - ZJ*IK)+ KI*(KJ*YK - usx10)/6.1:0)V = V + VINC100^CONTINUE110^CONTINUECC^PRINT RESULTS FOR LOAD STEPCIF (ILDAD.EQ.0) THENLSTEP = 0PRESS = 0ELSEPRESS = LOAD(ILOAD)EID IFV = (4.D0*V)/1000.D0VRITE(4,120) LSTEP, PRESS, V120^FORMAT(3X,I3,10X,I5,10X,F8.3)130 CONTINUECAppendix C. PROGRAM LISTINGS^ 317C^CLOSE FILES AID TERMIIATE PROGRAMCCLOSE(UNIT=1)CLOSE(UIIT=2)CLOSE(UIIT=3)CLOSE(UIIT=4)CSTOPEND

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