Development of Risk Contours for Assessment of Aircraft Engine Components Michael P. Enright *, Wuwei Liang, Jonathan Moody Southwest Research Institute, San Antonio, TX, 78238, USA Simeon Fitch Mustard Seed Software, Charlottesville, VA 2290 The probability of fracture of an aircraft engine component is highly dependent on the geometry and stress values at the location of anomalies that may occur during the manufacturing process. The efficiency and accuracy of the method used to construct the spatial conditional failure oint probability density function (JPDF associated with fracture is dependent on the number of limit state evaluations at each location and on the number of locations used to describe it. Previous studies have focused on modeling this JPDF and associated risk contours as a multivariate histogram associated with regions within the domain that are referred to as zones. To ensure conservative estimates, the uniform probability density for each zone is set to the maximum possible value within the associated region. The primary disadvantage to this approach is that it requires human udgment for definition of zones and for identification of the maximum conditional failure probability within each zone. In this paper, an approximate approach is presented for estimating risk contours that is based on a series of crack growth life contour values associated with a range of initial anomaly sizes that are placed at the nodes of a finite element model. At each node, the PDF of initial anomaly size is applied to the deterministic life values associated with initial anomaly sizes to obtain the conditional crack growth life PDF. The conditional risk at each node is obtained by computing the area under the crack growth life PDF associated with the design life of the component. The methodology was illustrated for an aircraft gas turbine engine compressor disk, where the results were in close agreement with established reference solutions when fracture risk is based only on the variability in the initial anomaly size. The method provides a relatively fast approximate estimate of component risk for conceptual design that can also be used as a framework for traditional probabilistic computational methods. Nomenclature F(t = cumulative probability of failure over time t N = total life p i = probability of fracture at a specified location i p F = unconditional probability of fracture of component ( P d = probability of occurrence of anomalies of specified size d PFd ( = ( i conditional probability of fracture given the presence of an anomaly of a specified size d PFd = conditional probability of fracture at location i given the presence of anomalies of a specified size d P(t = cumulative probability of survival over time t * Principal Engineer, Reliability and Materials Integrity, 6220 Culebra Road, Member AIAA. Research Engineer, Reliability and Materials Integrity, 6220 Culebra Road, Member AIAA. Research Engineer, Reliability and Materials Integrity, 6220 Culebra Road, Member AIAA. Principal, 634 Brandywine Drive.
V i = volume associated with material at location i λ = average number of anomalies per unit volume at location i i I. Introduction robabilistic damage tolerance (PDT is becoming an industry practice for risk assessment of rotating P components in aircraft gas turbine engines. This approach addresses inherent and induced anomalies associated with the manufacturing process that can lead to catastrophic failures such as the incidents in Sioux City, Iowa and Pensacola, Florida 2. PDT is summarized in several recent FAA Advisory Circulars 3-5 that provide guidance on the use of this methodology for certification of aircraft engine components. The probability of fracture of an engine component is highly dependent on the geometry and stress values at the location of the anomaly. Inherent anomalies can be located anywhere in a component, so there often is a need to quantify risk as a function of spatial position. A oint probability density function (JPDF can be used to describe the conditional probability of failure versus spatial position which can be shown graphically in a risk contour plot. The efficiency and accuracy of the method used to construct the JPDF is dependent on the number of limit state evaluations at each spatial location and on the number of spatial locations used to describe it. Component failure probability is obtained by integrating the spatial conditional failure JPDF over the spatial domain, where computational efficiency and accuracy is dependent on the underlying conditional failure JPDF at the selected integration points. Previous studies 6-8 have focused on modeling the spatial conditional failure JPDF as a multivariate histogram associated with regions within the domain that are referred to as zones. The uniform probability density for each zone is set to the maximum possible value within the associated region to ensure a conservative estimate. The advantage to this approach is that a relatively coarse mesh can be used to estimate the conditional failure JPDF, and the mesh can be refined as necessary to satisfy computational accuracy requirements. The primary disadvantage to this approach is that it requires definition of the maximum conditional failure probability within each zone. In this paper, an approximate approach is presented for estimating the PDT risk contours. It is based on a series of crack growth life contour values associated with a range of initial anomaly sizes that are obtained at the nodes of a finite element model. At each node, the PDF of initial anomaly size is applied to the deterministic life values associated with initial anomaly sizes to obtain the conditional crack growth life PDF. The conditional risk at each node is obtained by computing the area under the crack growth life PDF associated with the design life of the component. Anomaly occurrence rates are assigned to nodes based on the volumes of the surrounding elements. Unconditional risk contours can then be obtained from the product of the conditional risk and anomaly occurrence rate values. II. Probabilistic Damage Tolerance A. Probabilistic fracture mechanics model Gas turbine engine materials may contain anomalies that can lead to fracture if undetected during routine manufacturing and shop visit inspections. The probability of failure of a component with multiple number and types of anomalies can be predicted using established system reliability methods 9-2 provided that the failure probabilities associated with individual anomaly types are known 3. The fracture event is dependent on ( the presence of an anomaly, and (2 the formation and growth of a crack that exceeds the fracture toughness of the material before the design life has been reached. The occurrence probability of an anomaly P(d can be measured by counting the number of anomalies of various sizes on the surface or within the volume of a component. P(d is typically modeled as a Poisson point process 4,5 : ( λivi P( d = exp( λivi (! The likelihood of fracture failure can be estimated using fatigue crack growth data. Since this value is dependent on the presence of an anomaly of a specified initial size, it is commonly expressed as a conditional probability P( F d, the probability of fracture given that an anomaly is present. P( F d is dependent on a number of random variables related to the applied stress values and the fatigue nucleation and growth processes 6. The occurrence of an anomaly in a component is a relatively rare event for some materials, such as hard alpha in titanium. For these materials, the probability of more than one anomaly is assumed to be negligible 6. The 2
probability of fracture at a specified location p i is therefore based on the occurrence of a single rare anomaly P F d : P( d and the probability of fracture at location i given that a single anomaly is present ( i i ( i ( p = P F d P d (2 Components made from some rotor-grade alloy materials may contain hundreds or even thousands of anomalies. For these multiple anomaly materials, the conditional probability P ( Fd is dependent on the number of anomalies present, and is expressed as P ( Fd i. Since cracks can form at the location of any anomaly, the failure of the component can be considered as a weakest-link (series system consisting of the fracture failure associated with each discrete number of anomalies. For conservative risk estimates, P ( F i d is set to the maximum probability of fracture associated with any one of the anomalies present. The resulting system model is expressed as 7 : ( i = P( Fi d P F d (3 The anomaly occurrence probability is also influenced by the number of anomalies in a component. This relationship is reflected in the Poisson model of Eqn. (. The dependence of P ( F i d and P( d on the number of anomalies is shown conceptually in Fig., where the members represent the discrete number of anomalies that could be present in a given component which are events that have a mutually exclusive relationship (i.e., it is not possible for a selected component to have only 2 anomalies and only 3 anomalies simultaneously. p i is equal to the sum of the failure probabilities associated with each discrete number of anomalies 0. To obtain p i, Eqns. ( and (3 are substituted into Eqn. (2 for each member and summed over the total number of members: { ( } ( λiv i exp( λ n pi = P Fi d ivi (4 =! Eqn. (4 requires specification of the number of anomalies n. It can be shown 7 that Eqn. (4 reduces to: ( pi = exp λivi P Fi d (5 The probability of fracture is also dependent on the location of an anomaly within a component. The component is often discretized into a number of subregions called zones, and the probability of fracture is assessed for anomalies located within each zone 7,6. Since component failure occurs when there is a failure within any zone, the component is modeled as a series system of zones with a probability of fracture that can be expressed as: m p = PF [ F F] = p ( (6 F 2 m i i= If multiple anomalies are present in each zone, p i can be estimated for each zone using Eqn. (5. Substitution of Eqn. (5 into Eqn. (6 yields the following expression for multiple anomalies at multiple locations 7 : m p = exp λ V P F d F i i i i= ( (7 Multiple anomaly types may also be present in one or more regions of a component. The multiple anomaly types can be modeled as additional members of the series system of zones, in which components with multiple anomaly types and locations can be modeled using several nested series systems to represent the relationship among the various failure events. 3
System P ( F i d P ( d PDF ( i P F d = PDF ( P d = = = 2 PDF ( i 2 P F d = d PDF P ( d = 2 PDF ( i n P F d = 2 PDF P ( d = n d = n Figure. For materials with multiple anomalies, both the probability of fracture given an anomaly P ( Fd i and the anomaly occurrence probability Pd ( are dependent on the number of anomalies present 3. B. Zone-based risk assessment Random inherent material anomalies can potentially be located anywhere within a component. To account for the location uncertainty, a zone-based risk integration approach is often used 6-8,6. For this approach, the component is subdivided into a manageable number of zones of approximately equal risk (Fig. 2. The risk is computed in each zone, taking into account the zone anomaly occurrence probability (i.e., the probability that an anomaly is present in a zone. The total risk for the component is based on the risks in all of the individual zones. To ensure conservative estimates, the uniform probability density for each zone is set to the maximum possible value within the zone. The advantage to this approach is that a relatively coarse mesh can be used to estimate the conditional failure JPDF, and the mesh can be refined as necessary to satisfy computational accuracy requirements. The primary disadvantage to this approach is that it requires human udgment for definition of zones and for identification of the maximum conditional failure probability within each zone. In addition, the mesh refinement process may require computation of conditional failures at more locations than would be needed to accurately describe the spatial conditional failure JPDF mentioned previously. Further details on the zone-based approach are provided in Refs. 6-8,6. III. Life Contour-Based Risk Assessment Stress contour plots are sometimes used to estimate the minimum life locations of a component under the assumption that high stress values typically result in short lives. However, fatigue crack growth lives are affected not only by stress but also by temperature, local geometry, and other factors. Fatigue life is also dependent on the influences of multiple stress values within the load spectrum, so it may be difficult to predict the combination of stress values that will produce the minimum overall life at a given location. Life contours can capture all of these effects in a single figure that provides direct visualization of the life limiting location. A new life contour plot capability was recently added to the DARWIN probabilistic fracture mechanics software 8 that allows the analyst to create and view life contours superimposed directly on the finite element model. In DARWIN, life contours are created using crack growth life values obtained at all of the finite element nodes associated with a specified initial anomaly size (Fig. 3. Life contours can also be used to generate risk contour plots, as discussed later in this section. n d 4
Figure 2. For the zone-based risk assessment approach, a component is subdivided into a number of zones of approximately equal risk to account for the uncertainty in the location of an anomaly. a N Figure 3. Life contours are created using crack growth life values that are obtained at all of the nodes associated with a finite element model. A. Automatic generation of fracture mechanics models An algorithm was recently implemented in DARWIN that automatically determines (without user input the orientation, size, and stress input for a fracture model that will produce accurate life results, given only the 2D model (or slice and the initial crack location. The new automatic fracture model generation algorithm emulates the udgment of an experienced user by orienting and sizing a rectangular plate fracture model to reflect the actual component boundaries in the vicinity of a surface, corner, or embedded crack. Special algorithms accommodate curved boundaries and non-normal corners. Plate models for embedded cracks near external boundaries are oriented to accommodate automatic transition to surface cracks. Embedded plate models are otherwise oriented to capture the most significant univariant stress gradient near the crack. The final plate model is not always fully contained within the component boundaries, since this may be excessively conservative, but plate width and thickness are always sized to preserve appropriate ligaments along the primary axes of the crack, and to prevent the crack itself from going outside the actual component boundaries. The algorithm estimates the critical crack size (based on stresses from user-specified time steps as an aid to making some sizing decisions, but requires no fatigue crack growth calculations, and so a large number of fracture models can be constructed in very little computational time. The automatic fracture model algorithm is illustrated in Fig. 4. In this figure, the boundaries of the fracture model plates generated by the algorithm are shown for an assortment of initial crack locations in an axisymmetric 5
(a (b Figure 4. Illustration of the DARWIN automatic fracture model algorithm. (a 2D axisymmetric finite element geometry, and (b automatically generated fracture models at various locations. component with arbitrary cross-sectional shape and a complex distribution of hoop stresses. The finite element geometry is shown in Fig. 4a, and the resulting fracture models are shown in Fig. 4b. The circles designate the estimated critical crack sizes near the crack center at each location (the crack origin is at the center of the circle. Further details regarding this algorithm are provided in Ref. 9. B. Generation of approximate risk contours The life contours capability in DARWIN performs a crack growth life computation for one or more userspecified anomaly sizes at each node in a finite element model. In theory, the probability of fracture could also be computed at each node, but use of Monte Carlo simulation would require tens or even hundreds of thousands of life computations at each of the finite element nodes (on the order 0 4-0 5 or more locations in a typical finite element model. Roughly 0 8 to 0 0 fracture life computations would therefore be required to compute the fracture risk for a typical finite element model. Even if the fracture life computation time could be reduced to 0.0 seconds, it would still require up to 3 years to obtain a risk result for a typical finite element model. Furthermore, since risk values change rapidly near the surface of a component, the discretization of the finite element mesh may not be fine enough to accurately model the risk values in this region, which may require even more computations. On the other hand, if an approximate model of risk could be obtained, it could be used to guide the selection of locations for full Monte Carlo simulation of fracture risk, which would substantially reduce the computation time. An alternative approach for estimating risk contours was developed that is based on life contours. As shown in Fig. 5, the anomaly distribution is discretized into a number of initial anomaly areas, and life contours are obtained for each of the anomaly area values. This transforms the PDF of initial anomaly area into a PDF of crack growth life values at each of the finite element nodes. The conditional risk at each node is obtained by computing the area under the crack growth life PDF associated with the design life of the component. Note that probabilistic fracture mechanics computations typically involve a number of random variables, and this approach only considers the initial anomaly area random variable. However, this approach appears to provide a reasonable approximate estimate of the conditional probability of fracture when the initial anomaly area is the dominant random variable, which is often the case. Further investigation is underway to explore the limitations of this approach. IV. Application to Risk Prediction of Gas Turbine Engine Disks The probabilistic methodology for estimation of risk contours was illustrated for a titanium aircraft gas turbine engine compressor disk. An anomaly exceedance curve for hard alpha anomalies 3 was used to assess the risk of fracture. The CDF complement (-CDF plot associated with this curve is shown in Fig. 6. Internal stresses and temperatures were based on finite element analysis results associated with four critical load steps in the flight cycle. 6
P f a P(N a c P(a Target N Figure 5. The approach for predicting approximate risk contour values was based on transforming the PDF of anomaly area into a PDF of crack growth life values and computing the area of the failure region. N 0 0 0 0 - Probability 0-2 0-3 0-4 0-5 0-6 0 0 0 0 2 0 3 0 4 0 5 Anomaly Area, mils 2 Figure 6. CDF complement plot for titanium hard alpha anomalies 7
A plot of the stress values associated with the peak operating condition is shown in Fig. 7a. Life contours were obtained at each of the initial area values associated with the anomaly exceedance curve. Life contours are shown in Figs. 7b,c, and d for several representative anomaly areas. Risk contours were computed using the approximate risk contour approach described in this section, shown in Fig. 8. The regions of maximum risk (indicated in red in Fig. 8 occurred at the outer edges of the lower left side of the component, which is consistent with results obtained using the zone-based approach. A comparison of the risk results obtained using the approximate risk contour approach with DARWIN zonebased reference solutions is shown in Fig. 9 for several representative zones. When fracture risk is based only on the variability in the initial anomaly size, the approximate risk contour results appeared to be in close agreement with the zone-based reference solutions. However, when additional random variables were considered such as stress scatter or life scatter, the approximate algorithm estimates were somewhat different from the reference solutions, and the extent of this difference was zone-dependent. Note that since stress and life scatter are not required for certification assessment of hard alpha anomalies in aircraft engine components 3, the fracture risk for this case is based primarily on the variability in the initial anomaly size. (a (b (c (d Figure 7. Stress and life contours for gas turbine engine impeller example. (a Stress contour based on finite element stress values at a single time point in the mission cycle; Life contours for selected initial anomaly areas: (b 5.04x0-4 in 2, (c 8.34x0-3 in 2, and (d 4.5x0-2 in 2. 8
Figure 8. Conditional risk contours for gas turbine engine impeller example. Conditional P f 0.20 0.5 0.0 0.05 Corner Zone (DARWIN Surface Zone (DARWIN Embedded Zone (DARWIN Corner Node (Life Contour Approach Surface Node (Life Contour Approach Embedded Node (Life Contour Approach (a - no scatter (b - 0% stress scatter (c - 30% life scatter (d - 0% stress & 30% life scatter (d (c (b (a (d (b (c (a 0.00 (abcd 0 5000 0000 5000 20000 Flights Figure 9. A comparison of the risk results obtained using the approximate risk contour approach with DARWIN zone-based reference solutions for several representative zones. 9
V. Conclusions An approximate approach was presented for estimating the risk contours associated with probabilistic damage tolerance assessment. It is based on transformation of the initial anomaly size PDF into a crack growth life PDF at each node within a finite element model. Conditional risk values were computed at each node for a specified design life, and risk contours were obtained from the product of the conditional risk and anomaly occurrence rate values at each node. The methodology was illustrated for an aircraft gas turbine engine compressor disk, where the results were in close agreement with established reference solutions when fracture risk is based only on the variability in the initial anomaly size. The method provides a relatively fast approximate estimate of component risk for conceptual design that can also be used as a framework for traditional probabilistic computational methods. Acknowledgments The support of the FAA through Grant 05-G-005 is gratefully acknowledged. Joseph Wilson of the FAA Technical Center and Tim Mouzakis of the FAA Engine and Propeller Directorate are thanked for their oversight and encouragement. The members of the proect steering committee, currently comprising Bob Maffeo (GE Aviation, Alonso Peralta (Honeywell, Johnny Adamson (Pratt & Whitney, and Jon Dubke (Rolls-Royce Corporation, have made many valuable contributions, along with their engine company colleagues. References National Transportation Safety Board. 990, "Aircraft Accident Report - United Airlines Flight 232 McDonnell Douglas DC-0-0 Sioux Gateway Airport, Sioux City, Iowa, July 9, 989," National Transportation Safety Board, NTSB/AAR-90/06, Washington, DC. 2 National Transportation Safety Board, 996, "Aircraft Accident Report Uncontained Engine Failure, Delta Air Lines Flight 288, McDonnell Douglas MD-88, N927DA, National Transportation Safety Board, NTSB/AAR-98/0, Washington, DC. 3 Federal Aviation Administration, 200, Advisory Circular - Damage Tolerance for High Energy Turbine Engine Rotors, U.S. Department of Transportation, AC 33.4-, Washington, DC. 4 Federal Aviation Administration, 2009, Advisory Circular - Guidance Material for Aircraft Engine Life-Limited Parts Requirements, U.S. Department of Transportation, AC 33.70-, Washington, DC. 5 Federal Aviation Administration, 2009, Advisory Circular - Damage Tolerance of Hole Features In High-Energy Turbine Engine Rotors, U.S. Department of Transportation, AC 33.70-2, Washington, DC. 6 Leverant, G. R., Millwater, H. R., McClung, R. C., and Enright, M. P., "A New Tool For Design and Certification Of Aircraft Rotors," Journal of Engineering for Gas Turbine and Power, Vol. 26, No., 2004, pp. 55-59. 7 McClung, R.C., Enright, M.P., Millwater, H.R., Leverant, G.R., and Hudak, S.J., 2004, A Software Framework for Probabilistic Fatigue Life Assessment, Journal of ASTM International, Vol., No. 8, Paper ID JAI563. Also published in Probabilistic Aspects of Life Prediction, ASTM STP 450, 2004, pp. 99-25. 8 McClung, R.C., Enright, M.P., Lee, Y-D., Huyse, L., and Fitch, S.H.K (200. Efficient Fracture Design for Complex Turbine Engine Components, Journal of Engineering for Gas Turbines and Power, ASME (accepted for publication. 9 Thoft-Christensen, P. and Murotsu, Y., 986, Application of Structural Systems Reliability Theory, Springer Verlag, New York. 0 Hoyland, A., and Rausand, M., 994, System Reliability Theory Models and Statistical Methods, John Wiley, New York, pp. 355-44. Enright, M.P., and Frangopol, D.M., 998, Failure Time Prediction of Deteriorating Fail-Safe Structures, Journal of Structural Engineering, 24 (2, pp. 448-457. 2 Melchers, R., 999, Structural Reliability Analysis and Prediction, Second Edition, Wiley, New York. 3 Enright, M.P., and McClung, R.C., 200, A Probabilistic Framework for Gas Turbine Engine Materials With Multiple Types of Anomalies, paper GT200-2368, Proceedings of the 55th ASME International Gas Turbine & Aeroengine Technical Congress, ASME, Glasgow, Scotland, June 4-8, 200. 4 Roth, P.G., 998, Probabilistic Rotor Design System Final Report, AFRL-PR-WP-TR-999-222, Air Force Research Laboratory, Cincinnati, OH. 5 Haldar, A. and Mahadevan, S., 2000, Probability, Reliability, and Statistical Methods in Engineering Design, John Wiley, New York. 6 Wu, Y.T., Enright, M.P., and Millwater, H.R., 2002, Probabilistic Methods for Design Assessment of Reliability with Inspection, AIAA Journal, Vol. 40, No. 5, pp. 937-946. 7 Enright, M.P., and Huyse, L., Methodology for Probabilistic Life Prediction of Multiple Anomaly Materials, AIAA Journal, 44 (4, 2006, pp. 787-793. 8 Southwest Research Institute, 200, DARWIN User s Guide. Southwest Research Institute, San Antonio, TX. 9 McClung, R.C., Lee, Y-D., Liang, W., Enright, M.P., and Fitch, S., 200, Automated Fatigue Crack Growth Analysis of Components, Proceedings, 0 th International Fatigue Congress, Prague, Czech Republic, June 6-, 200. 0