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1 Copyright © 2000 by ASME
IGTI 2000ASME TURBO EXPO 2000: LAND, SEA, AND AIR
May 8-11, 2000 – Munich Germany
2000-GT-0421
A PROBABILISTICALLY-BASED DAMAGE TOLERANCE ANALYSIS COMPUTERPROGRAM FOR HARD ALPHA ANOMALIES IN TITANIUM ROTORS*
Harry R. Millwater Simeon H. K. Fitch
Y.-T. (Justin) Wu David S. Riha Michael P. Enright
Gerry R. Leverant R. Craig McClung
Chris J. Kuhlman G. Graham Chell Yi-Der Lee
Southwest Research Institute6220 Culebra Road
San Antonio, TX 78238-5166
ABSTRACTA probabilistically-based damage tolerance analysis
computer program for engine rotors has been developed underFederal Aviation Administration (FAA) funding to augment thetraditional safe-life approach. The computer program, in itscurrent form, is designed to quantify the risk of rotor failure dueto fatigue cracks initiated at hard alpha anomalies in titanium.The software, DARWINTM (Design Assessment of ReliabilityWith Inspection), integrates a graphical user interface, finiteelement stress analysis results, fracture-mechanics-based lifeassessment for low-cycle fatigue, material anomaly data,probability of anomaly detection, and inspection schedules todetermine the probability-of-fracture of a rotor disk as afunction of operating cycles with and without inspections. Theprogram also indicates the relative likelihood of failure of thedisk regions. Work is underway to enhance the software tohandle anomalies in cast/wrought and powder nickel disks, andmanufacturing and maintenance-induced surface anomalies inall disk materials.
*Funded under FAA Grant 95-G-041
INTRODUCTIONGas turbine industry experience has shown that the
occurrence of material and manufacturing anomalies canpotentially degrade the structural integrity of high energy rotors.Conventional rotor life management methodology does notexplicitly address the occurrence of these types of anomalies.The conventional methodology is founded on the assumption ofnominal material and manufacturing conditions. Undetectable
material and manufacturing anomalies represent a departurefrom the assumed nominal conditions.
As a result of the accident at Sioux City, Iowa in 1989,the FAA requested in 1991 that industry, through the AerospaceIndustries Association, review available techniques todetermine whether a damage tolerance approach could beintroduced to reduce the rate of uncontained rotor events. Theindustry working group concluded that additional enhancementsto the conventional rotor life management methodology couldbe established which explicitly address anomalous conditions.As a result, a research program was initiated in 1995 with teammembers consisting of Southwest Research Institute, AlliedSignal, Rolls-Royce Allison, General Electric, and Pratt &Whitney. These organizations formed a steering committee thatprovided guidance on program content, priorities and direction.
The objective of this program was to develop aprobabilistically-based, damage tolerance design code toaugment the current safe-life approach for life management ofcommercial aircraft gas turbine rotors/disks[1,2]. The designcode is not intended to replace existing design methods, but toprovide an additional tool that the engine manufacturers can usefor reliability assessment. Initial application of the codefocused on melt-related anomalies (hard alpha) in titanium.Additional work performed during this program but notdiscussed in this paper include: supplemental tests performed todetermine the fatigue crack growth and mechanical propertiesof hard alpha and titanium disk alloys, and development of a
2 Copyright © 2000 by ASME
forging code to predict the shape and orientation of hard alphaanomalies during processing.
Capability Overview
The DARWIN code is designed to provide an easy-to-use vehicle for engineers to compute the probability-of-fractureof a rotor disk with and without inspection. The datarequirements for DARWIN are summarized in Table 1 below.A library of industry-developed inputs are provided for anomalydistributions, probability of detection curves and fatigue crackgrowth material properties. The data requirements are discussedin detail in subsequent sections in this paper.
Table 1. Data Requirements for DARWIN
• Finite element stress results• Anomaly distribution+
• Inspection schedule• Probability of detection+
• Fatigue crack growth material properties+
• Stress-strain properties• Zone definitions + library provided
The input data are combined using the technologyareas of: a graphical user interface (GUI), probabilistic analysis,stress processing, and fracture mechanics to determine theprobability-of-fracture of a disk and the risk contribution factorswhich indicate the relative importance of the individual diskregions to the probability-of-fracture of the disk. Thisprocedure is shown schematically in Figure 1.
Probabilistic Fracture Mechanics
Probability of DetectionAnomaly Distribution
Finite Element Stress Analysis
Material Crack Growth Data
NDE Inspection Schedule
Pf vs. Cycles
Risk Contribution Factors
Figure 1. Schematic of DARWIN Analysis Procedure
Probabilistic Methodology
A “zone-based” system reliability methodology is used tocompute the probability-of-fracture of a disk as a function offlight cycles[3]. This methodology accounts for:
• the probability of having an anomaly in the disk,• the possibility that a hard alpha anomaly developed
during the titanium melt process could be in anylocation of the disk,
• the initial size distribution of the anomaly,• randomness in the time of inspection time, probability
of detection, finite element stresses and fracturemechanics analysis,
• the probability-of-fracture if an anomaly exists,• the probability of detecting an anomaly and removing
a disk before the disk has fractured.
The disk structure is discretized into a number of zones. Azone is a grouping of material such that all sub-regions in thezone have a generally uniform stress state, and the same fatiguecrack growth properties, inspection schedules, probability ofdetection curves, and anomaly distribution. In other words, therisk computed for any sub-region of material of the zone will bethe same; thus, the subregions are grouped into a zone. A finiteelement mesh and stress results are used as the framework forthe zone discretization.
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The methodology assumes, at most, only one anomalyexists in the disk (the probability of two or more anomalies in adisk is negligible). Based on this assumption, the probability-of-fracture of the zones can be assumed as independent events.The probability-of-fracture of the disk can be obtained from theequation
∑∏==
≈−−=∪∪∪
==N
ii
N
iiN FPFPFFFP
n any zonefracture iPdiskP
1121 ][])[1(1][
][][
! (1)
where iF indicates failure in zone i, ][ iFP indicates the
probability-of-fracture of zone i, and N is the total number ofzones.
The probability-of-fracture of a zone, i.e., ][ iFP requires an
anomaly be present and grow to failure. This can berepresented as
]|[*][][ iiii ABPAPFP = (2)
where ][ iAP is the probability of having an anomaly in zone i
and ]|[ ii ABP is the probability-of-fracture given an anomaly
in zone i. ][ iAP is the total number of anomalies per the
reference weight (e.g., one million pounds) times the weight ofthe zone.
The conditional probability ]|[ ii ABP is computed
using probabilistic fracture-mechanics-based life assessment forlow-cycle fatigue. The probabilistic solution methods MonteCarlo sampling or Importance sampling are used to quantify theprobability-of-fracture of a zone assuming an anomaly exists.The probability of detecting and removing anomalies is alsocalculated and, therefore, the probability-of-fracture with andwithout inspection is computed. Monte Carlo sampling is veryrobust and can be made as accurate as required by increasingthe number of samples. However, Monte Carlo sampling isoften very time consuming requiring millions of samples. TheImportance sampling methodology is a sampling methodtailored to the probabilistic fracture mechanics problemrequired for the disk risk assessment, see Figure 2. Thefundamental premise of the Importance sampling method is of“selective” sampling where only realizations of the randomvariables are sampled which will cause failure before theservice life is reached. Studies so far indicate that Importancesampling is one to two orders of magnitude more efficient insolution time than Monte Carlo sampling for the same error andconfidence.
1. Calculate the risk without inspection, f
p , by numerical
integration.
2. Generate realizations in the failure domain.
• Randomly generate a life scatter value (B).• Randomly generate a stress multiplier (S), given B.• Randomly generate a defect, given S and B.
3. Using the above samples, perform crack growth andinspection simulations to compute the number of disksremoved by inspection.
4. The probability-of-failure with inspection is calculatedas:
(with inspection) (without inspection) *
Number of failures with inspection
Total number of samples
f fp p=
Figure 2. DARWIN Importance Sampling Procedure:
The random variables considered by DARWIN areshown in Table 2. The anomaly distribution defines theprobability of having an anomaly and the size distribution of theanomalies. Variations in time of inspection, in terms of cycles,is used to model real world uncertainties in the inspection of afleet of engines. The probability of detection defines theexpected probability of detecting an anomaly as a function ofanomaly size. Variations in finite element stress results aresimulated using a multiplier that can be considered random, i.e.,σ = σ(FE)*S, where σ(FE) are the stresses obtained from thefinite element analysis and S is a random variable modeled witha lognormal distribution. A life scatter factor is implemented toconsider variations in predicted cycles-to-failure, i.e., N =N(FM) * B, where N(FM) is the predicted cycles-to-failure fromfracture mechanics analysis and B is a random variable modeledwith a lognormal distribution.
Table 2. Random Variables Considered in DARWIN
Anomaly distributionTime of inspectionProbability of detectionFinite element stresses (multiplier)Life scatter (fracture mechanics)
Graphical User Interface
A Java-based graphical user interface (GUI) has beendeveloped to assist the user in using DARWIN. The GUIoperates in pre- and post-processing modes with the riskassessment code. The output of the pre-processing mode is anASCII input file which contains the data needed for a riskassessment. The risk assessment analysis creates an output filewhich is read by the GUI for post-processing.
4 Copyright © 2000 by ASME
The GUI is displayed throughout this paper in the formof screen images.
Stress Processing
DARWIN does not contain a stress analysis solver butintegrates with finite element (FE) results through a neutral fileASCII format. The required stresses for each zone areextracted from the neutral file and processed as shown in Table3 and Figure 3.
Table 3. Summary of Stress Processing Features
For Each Zone:1 Extract FE stresses along gradient through crack,
parallel to edge of plate for all load cases2 Perform shakedown analysis to determine residual
stresses3 Perform rainflow analysis to determine contributing
cycles and max-min load pairs4 Synthesize all required stress information and output to a
reduced stress file
FE stresses and zone definition
stress gradient
Stress gradient extraction
����������������
3 4 5 6 7 0 1 2 3
Load Step
01020304050607080
Hoo
p S
tres
s (k
si)
Rainflow stress pairing
FE Analysis
0.0 0.2 0.4 0.6 0.8 1.0Normalized distance from the notch tip, x/r
-0.8
-0.4
0.0
0.4
0.8
1.2
1.6
2.0
σ/σo
(σz)relax
(σz)residual
(σz)elastic
Shakedown module
Computed relaxed stress
σelastic - σresidual
σ0/σ
Residual stress determination
Figure 3. Stress Processing Capabilities
Fracture Mechanics
DARWIN contains a default fatigue crack growth(FCG) module called FLIGHT_LIFE, which was specificallydesigned with the capabilities and computational speed for theprobabilistic disk analysis problem. DARWIN also has optionsfor the user to supply their own FCG code if desired.FLIGHT_LIFE currently contains stress intensity factorsolutions for corner, surface (semi-elliptical), embedded(elliptical), and through cracks that typically accept univariant6th order polynomial stress gradients, see Table 4FLIGHT_LIFE permits crack transitioning, e.g., breakthroughof an embedded crack to a surface crack, surface to corner, see
Figure 4. FCG material properties will also changeautomatically from vacuum to air at embedded-to-surfacetransitions. FLIGHT_LIFE provides different options forcommon FCG equations and models for stress ratio andtemperature effects, along with tabular FCG data capability.A summary of the stress intensity factor solutions andtransitions available in FLIGHT_LIFE is given in Table 4 andFigure 4. More detail on the fracture mechanics capabilitiescan be found in [1,4,5].
Table 4. Crack Geometries available in DARWIN
CC01 Corner CrackSC02 Surface CrackEC02 Off-center embedded crackEC03 Off-center embedded crack approaching a free
surfaceTC01 Off-center embedded through crackTC02 Surface through crack
Figure 4. Fracture Mechanics Crack Geometries andTransitions
Anomaly Distribution
The anomaly distribution is given in terms of anexceedance curve that describes the number (or probability) ofexpected anomalies as a function of the anomaly area for aspecified amount of material, for example, one million poundsor kilograms. An example of an anomaly exceedance curve isshown in Figure 5. The probability of having an anomaly is thetotal number of anomalies per the reference weight (e.g., onemillion pounds) times the weight of the disk or zone. Details onthe development of anomaly distributions by an industryworking group is described in [6]. A library of anomalydistributions is provided with DARWIN.
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Figure 5. Example of an Anomaly Distribution
From the exceedance curve, the cumulativedistribution function (CDF) of expected anomalies as a functionof anomaly area can be calculated as:
( ) ( )( ) 1
( ) ( )d d max
d min d max
N A N ACDF A
N A N A
−= −−
(3)
where A denotes anomaly area, ( )d maxN A is the number of
anomalies associated with the largest anomaly size defined in
the exceedance curve, and ( )d minN A is the number of
anomalies associated with the smallest anomaly size defined inthe exceedance curve.
Inspection Data
Nondestructive evaluation in DARWIN is simulatedusing probability of detection curves (PODs) and inspectionschedules. The POD gives the expected probability of detectinga flaw of a specified size. The inspection schedule determineswhen an inspection occurs, and when it does, what PODs are tobe applied. A library of industry-accepted POD curves isdistributed with DARWIN.
The time when an inspection occurs can be treated as arandom variable in DARWIN using either a normal randomvariable (user specifies mean and standard deviation) or a tableformat (user specifies cycles vs. probability of inspection).Figure 6 shows an inspection schedule consisting of twoinspections, both modeled by a normal distribution, and anassociated POD, see inset. This inspection schedule is assignedby the user to the appropriate zones. The user may define anumber of inspection schedules with associated PODs andapply them to the appropriate zones of the structure.
Figure 6. Example of Inspection Schedule
Material Properties
The material property options are shown in Table 5below. All input parameters are temperature (and if appropriateR ratio) dependent. Three options are available forinterpolating the material property data as a function oftemperature as shown in the last column of the table. A libraryof FCG properties is provided with DARWIN.
Table 5 Material Property Input (Air and Vacuum)
FCG Model R_Ratio Stress-StrainCyclic
• Paris• Bilinear Paris• Sigmoidal• Hyperbolic
Sine• Tabular
• None• Walker
Equation• Closure• Walker
Interpolation
• Ramberg-Osgood
• Tabular
Stress-StrainMonotonic
TemperatureInterpolation
• Ramberg-Osgood
• Tabular
• Nearest• Next Highest• Interpolation
Zone Definition
A major emphasis of the GUI is to assist the user in thedevelopment of risk zones. A zone is a grouping of materialsuch that all sub-regions in the zone have a generally uniformstress state, and the same fatigue crack growth properties,inspection schedules, probability of detection curves, andanomaly distribution. In other words, the risk computed for anysub-region of material of the zone will be the same; thus, thesubregions are grouped into a zone. The risk results obtainedby locating an anomaly in any particular portion of a zone are
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assumed representative of the zone in general; however, if thereis some variation within the zone, the most life limiting locationwithin the zone is typically chosen for the anomaly location.The results for a zone are obtained independently from anyother zone.
The definition of the zone requires visualization of thefinite element stress results. The FE stress results are input toDARWIN in the form of neutral files. A translator exists thatwill translate an ANSYS results file to the neutral file format.Figure 7 shows an example screen shot of a zone being defined.An axisymmetric finite element model (axis of rotation ishorizontal, parallel to bottom of the disk) is shown, color codedby hoop stress value, and zone 2 is being defined with zone 1previously defined. The white circle is the location of theanomaly. The darker shaded elements with white boundariesdefine the material comprising the zone. The rectangular plateis used to approximate the real disk geometry with the plategeometry being used for fracture mechanics calculations, seeinset. The arrow from one edge of the plate to the other edge ofthe plate is used to define the path along which stress gradientswill be extracted from the finite element model and used for thefracture mechanics calculations.
The GUI provides a number of tools for selectingelements to comprise the zone, locating the anomaly,defining/adjusting the plate, and for selecting material,probability of detection, anomaly distribution, and inspectionschedule properties of the zone.
Figure 7. Example of Zone Definition
Output Results
DARWIN computes the probability-of-fracture withand without inspection of the disk and the individual zones. Italso reports risk contribution factors which indicate the relativeimportance the individual zones to the probability-of-fracture ofthe disk.
Table 6 shows the available results which may beviewed graphically. The results are also available in a textoutput file. Figure 8 shows a screen image of probability-of-fracture vs. cycles for a disk with a plot of the risk contributionfactors shown as an inset.
Deterministic fracture mechanics results for each zonecan be displayed in terms of the maximum stress intensity factorvs. cycles and flaw area vs. cycles.
Stress processing results for each zone can bedisplayed in terms of the max-min load pairs of stress valuesalong the gradient of the plate. The length of the plate isnormalized from 0 – beginning of plate, to 1 – end of plate. Anexample is shown in Figure 9 for a particular zone. The stressprocessing results for any zone can be viewed.
Table 6. Available Postprocessing Graphical Displays
Risk Assessment• Overall disk risk assessment• Disk risk assessment per cycle• Zone risk assessments• Risk contribution factors
Fracture Mechanics (deterministic beginning with a(semi-) circular 10 mil flaw)
• Maximum stress intensity factor vs. cyclesfor each zone
• Crack area vs. cycles for each zoneStress Processing
• Max and min stress pairs for each load blockfor each zone
Risk Contribution Factors
Figure 8. Disk Assessment of Probability-of-Fractureand Risk Contribution Factors
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Figure 9. Stress Processing Results
CONCLUSIONSA probabilistically-based damage tolerance analysis
computer program for engine rotors, DARWIN, has beendeveloped under Federal Aviation Administration funding toaugment the traditional safe-life approach. This computerprogram quantifies the risk of rotor failure due to fatigue cracksinitiated at hard alpha anomalies in titanium.
Work is underway to extend DARWIN to deal withanomalies in cast/wrought and powder nickel disks, andmanufacturing and maintenance-induced surface anomalies inall disk materials.
ACKNOWLEDGEMENTSThis work was performed under project Turbine Rotor
Material Design, funded by the Federal Aviation Administrationunder grant no. 95-G-041. The authors would like toacknowledge the substantial contribution by members of theFAA: Joe Wilson, Bruce Fenton, Tim Mouzakis, and thecurrent and former program steering committee members:Honeywell: Mike Gorelik, Chet Date, Craig Balis, Joe Adams,Rolls-Royce Allison: Doug Hermann, Chuck Teague, NickProvenzano, General Electric Aircraft Engines: GaryMihlbachler, Nikki Howard, Jon Tschopp, Pratt & Whitney:Darryl Lehmann, Gary Peters.
REFERENCES1) “Turbine Rotor Material Design Final Report,” Southwest
Research Institute, Honeywell, General Electric, Pratt &Whitney, Rolls-Royce Allison, Scientific FormingTechnologies, Federal Aviation Administration Grant 95-G-041, August 1999.
2) Leverant, G.R., et al., “A Probabilistic Approach toAircraft Turbine Material Design,” Paper 97-GT-22,ASME International Gas Turbine & Aeroengine Congress,1997.
3) Wu, Y.-T., "An Efficient Method for Reliability Analysis ofStructures Subjected to In-Service Inspection," 13thAnnual ASCE Engineering Mechanics DivisionConference, Baltimore, MD, June 13-16, 1999.
4) McClung, R.C., et al., “Development of a ProbabilisticDesign System for Gas Turbine Rotor Integrity,” FATIGUE’99: The Seventh International Fatigue Conference,Beijing, China, June 8-12, 1999.
5) DARWIN User’s Manual, Southwest Research Institute,Version 3.2, July 1999.
6) “The Development of Anomaly Distributions for AircraftEngine Titanium Disk Alloys,” 38th
AIAA/ASME/ASCE/AHS/ASC SDM Conference, 1997,pp. 2543-2553.