Preliminary Validation of Deterministic and Probabilistic Risk Assessment of Fatigue Failures Using Experimental Results

Transcription

1 9 th European Workshop on Structural Health Monitoring July -13, 2018, Manchester, United Kingdom Preliminary Validation of Deterministic and Probabilistic Risk Assessment of Fatigue Failures Using Experimental Results Ribelito F. Torregosa and Weiping Hu Aerospace Division, Defence Science and Technology Group, 506 Lorimer St., Fishermans Bend, Victoria, 3207, Australia Abstract The complexity in highly technological systems such as aircraft and the economic pressure on owning and operating such systems demand a more accurate assessment of their safety and reliability. For aircraft structures the safety is reflected in the predicted fatigue life and inspection intervals. In engineering practice, fatigue life may be calculated using either a deterministic or a probabilistic method, with the former being the dominant method historically used. Increasingly, probabilistic risk assessment methods are being used to evaluate aircraft safety and the availability of military assets. To provide confidence in the increasing use of the probabilistic approach to structural integrity assessment, this paper presents a comparison of numerical and experimental data to validate the accuracy of probabilistic assessment in comparison to the widely used deterministic method. The validation used the fatigue lives of aluminium test specimens from two tests, namely the experiment done at DST for variable amplitude loading and the data from the Virkler experiment for constant amplitude load. The initial inspection times specified in MIL-STD1530D for both the deterministic and the probabilistic approach were used to predict the time (i.e., number of cycles) of inspection times. As specified in the standard, the probabilistic approach requires the initial inspection to be conducted when the probability of failure exceeds -7 but is less than -5, whereas the deterministic approach requires the initial inspection to be conducted halfway through the predicted fatigue life. The predicted initial inspection times from each method were compared to the number of load cycles at which the first failure occurred in the experiment. Comparisons showed that the deterministic approach was over-conservative and predicted the initial inspection time much earlier than observed in the experiment. The probabilistic approach gave predictions which were both safe and closer to the experimental values. 1. Introduction Safety inspection against fatigue failure is an integral component in the structural integrity management of aircraft fleets. In spite of the advancement in materials and mechanical sciences, the accurate prediction of fatigue life of a given aircraft structural component under service load is still a significant challenge. This is because fatigue is inherently a stochastic process. Previous research showed that fatigue crack growth rates vary even for the same material and loading conditions. Virkler et. al., [1] conducted tests on 68 specimens of Aluminium 2024-T3 subjected to constant

2 amplitude loading and Torregosa and Hu [2] conducted tests on 85 specimens of Aluminium 7075-T7351 subjected to a variable amplitude spectrum loading. In both tests, the results showed a high degree of variability in crack growth rates. In common practice, fatigue life is calculated using either a deterministic method or a probabilistic method, with the former being the dominant one due to its simplicity. In the deterministic approach, all parameters are treated as constant. The uncertainties are accounted for through the application of a factor of safety. In the probabilistic or riskbased approach, the key inputs are treated as random variables, with appropriate probability distributions. In the context of fatigue, the safety inspection times can be determined using either approach. In a deterministic damage tolerance analysis following MIL-STD-1530D [3], a fatigue crack is assumed to exist at a critical structural location and is allowed to grow but the location must be inspected halfway through the predicted failure time in order to have another chance of detecting it before it leads to failure. Essentially, this gives a factor of safety of 2.0. The probabilistic structural integrity assessment of military aircraft also follows the MIL-STD-1530D [3] in determining the safety inspection time which is set at the probability of failure (PoF) between -7 and -5. Although both the deterministic and the probabilistic methods have been used in the field of structural integrity management of aircraft for many years, research comparing the predictions of the two methods can hardly be found. In recent years due to the increasing number of ageing aircraft and with the demand for extending the design lives of aircraft for economic reasons, the application of the probabilistic approach in fatigue life prediction has gained more popularity. In the future, it is envisaged that the probabilistic approach will also be applied to other aircraft operated by the Royal Australian Air Force. Thus the prediction using this approach needs to be verified in comparison with the deterministic approach. 2. Variability of Fatigue Crack Growth For many years, the variability of fatigue crack propagation has been the focus of much research. One of the earliest and probably the most widely cited document is the investigation conducted by Virkler et al. [1], in which 68 aluminium 2024-T3 specimens (see Figure 1) were subjected to constant amplitude loading of kn with R ratio of The test specimen had a width of 152 mm and thickness of 2.54 mm. The test specimens were pre-cracked to a crack size 9.0 mm. Torregosa and Hu [2] investigated the variability of fatigue crack growth of 7075-T7351 aluminium alloys by testing 85 middle tension specimens, shown in Figure 2, under a variable amplitude load spectrum. A fatigue starter was introduced through a centre notch, as shown in Figure 2. The load spectrum consisted of load turning points which were referred to as a load block. The maximum tensile load was 60 kn. Each test specimen was subjected to the same load spectrum applied repeatedly until the specimen failed by fracture. Crack sizes were measured at every 2000 load turning points (i.e., 00 cycles) using the direct current potential drop (DCPD) method [4]. Both studies showed that even under well controlled conditions, cracks grow stochastically, as illustrated in Figure 3 and Figure 4, respectively. In the DST experiment, the scale of the horizontal axis is given in load blocks. Figure 5 shows the typical crack sizes at the start of DST testing and the crack size shortly before fracture. It should be noted that at cycle zero, the crack sizes already exhibited variability because the measurement started after pre-cracking under constant amplitude load equal to 70% of the peak spectrum load value. 2

3 Figure 1 Virkler test specimen geometry Figure 2 DST test specimen geometry 50 Crack size, a (mm) Cycles Figure 3 Crack growth curves for aluminium 2024-T3 mid-tension specimens under constant amplitude loading from Virkler experiment [1] 3

4 Crack size, a (mm) Load blocks Figure 4 Crack growth curves for aluminium 7075-T7351 mid-tension specimens under variable amplitude load from DST experiment [2] (a) (b) Figure 5 PoF Test specimen showing crack size, a) at the start of the loading, b) before fracture 3. Deterministic and Probabilistic Approaches to Aircraft Fatigue Safety Inspection Following MIL-STD1530D For military aircraft, safety inspection follows MIL-STD1530 [3]. The standard specifies two methods in determining the inspection times. These are the deterministic and the probabilistic methods. The deterministic method mitigates the risk of a crack reaching its critical size unnoticed by requiring an inspection be conducted no later than halfway to the time when the crack is projected to fail. The idea is to have another chance of detecting the crack during the next inspection. This is illustrated in Figure 6. A master crack growth curve obtained from a damage tolerance analysis (DTA) is used to project the fatigue life of a crack from its initial size. In the probabilistic method, the inspection times are determined based on the acceptable risk of failure. For military aircraft as specified in MIL-STD1530D, the acceptable probability of (catastrophic) failure (PoF) during the next single flight is PoF= -7 [3]. When the PoF exceeds -7 4

5 but is less than -5, a safety inspection is required and beyond -5, the risk is considered unacceptable. This is schematically illustrated in Figure 7. Thus, in the probabilistic method, a risk curve is constructed and the flight hours corresponding to -7 and -5 are projected to determine the inspection times. Figure 6 Fatigue safety inspection time using the deterministic approach Figure 7 Fatigue safety inspection using the probabilistic approach [3] Given the way in which the two methods determine the safety inspection times, it is not difficult to compare which one is more accurate and conservative if a common data set is used. In this paper, the experimental results shown in Figure 3 and Figure 4 are used to validate the predictions from the two methods. 3.1 In-service versus experimental failure data Service load fatigue failure data of airframes and components are not easily available since airframes and their components are designed not to fail. Even with retired aircraft, failed components are hard to find. However, other data such as service load spectra and material properties are available. When these data are used in laboratory experiments, the resulting failure data can be a very good replacement for actual inflight data. For this reason, the comparison of failure prediction in this paper is based on experimental results obtained under laboratory conditions. In this investigation, each experimental specimen failure is treated as a failed airframe component which should not be allowed to happen in actual service. 3.2 Predicting and validating the initial inspection time using test data In actual fleet safety management, it is common to conduct a full scale fatigue test and use the test result as representative data for each fleet member. However, this exercise is very expensive and time consuming. Thus other sources of information such as teardown inspections, experiments and computational analyses are used to manage the safety of an aircraft. The accuracy of safety predictions, though, is very hard to validate since aircraft data of failed components are rare. In this paper, an attempt is made to verify the accuracy of both the deterministic and the probabilistic methods. The idea is to use the test results of specimens subjected to representative service loads experienced by actual airframe components. Hence, each specimen is thought to represent an aircraft 5

6 component, and the pool of all specimens essentially represents an aircraft fleet. From this assumption, the accuracy of predictions can be verified from the experimental outcome. Thus if the pool of test data represents the fleet, then the safe inspection time is the number of load cycles before the first failure among all specimens tested. For a prediction to be considered conservative, the predicted inspection time must be before the first failure of any specimen. When comparing two methods, the one that predicts a life that is closer but shorter than the time the first failure occurs is considered more accurate. In this paper, the predictions of the initial inspection time using the deterministic and the probabilistic methods are made according to following procedure: Deterministic procedure 1) Develop the initial crack size (ICS) distribution by randomly selecting five specimens. The crack size at 00 (treated as time zero) load cycles is treated as the initial crack size; 2) Randomly select a test specimen and get its crack growth curve obtained from the test; 3) Take the mean of the ICS in step 1 as the initial crack size, and use the crack growth curve from step 2) to project the mean to failure, to obtain the fatigue life t cr ; and 4) The safety inspection time is obtained by dividing t cr by Probabilistic procedure 1) Develop the initial crack size (ICS) distribution as in the deterministic procedure; 2) Randomly select a test specimen and get its crack growth curve obtained from the test. Use this as the master crack growth curve; 3) Develop the peak stress distribution based on the peak stresses of the load spectrum, and model the result using the Gumbel distribution; 4) Calculate the residual strength curve and the geometry correction factors based on the material property and geometric configuration of the test specimen; 5) Calculate the probability of failure, PoF [5, 6], using first a probabilistic fracture toughness and then a deterministic fracture toughness, and the following equations: (1) (2) In the above equations, f(a,t) is the crack size distribution at time t, where t is the time (in load cycles or load blocks) and is the peak stress distribution. The terms and represent the peak 6

7 stress exceedance probability. This is the probability that the applied stress exceeds the residual strengths and respectively for a particular crack size a and fracture toughness,. 6) On the PoF curve, project the load cycles corresponding to PoF = -7 and PoF = -5. The projected load cycles are the range in which the initial inspection time has to be conducted. The ICS distribution was represented by a bounded probability distribution model (i.e., Beta distribution) as proposed by Torregosa and Hu [7]. The minimum bound of the ICS is set at 0.0 mm and the upper bound is the half-width of the test specimen, for the obvious reason. The bounded distribution is considered a more realistic model for crack size distribution than the normal or the lognormal distribution. The probability density function of the Beta distribution with lower bound a and upper bound b is given by the equation [8]: (3) (4) The parameters of the distribution are q and r whereas by the equation; is the Beta function given (5) The mean and variance of the Beta distribution are (6) (7) 3.3 Comparison of the accuracy of the initial inspection time predictions based on the DST experiment In the DST experiment, the test specimens were subjected to a variable amplitude load spectrum. Since the peak load is 60 kn, the peak stress exceedance probability is P=1.0 for stresses less than 60 kn and P=0.0 for stresses greater than 60 kn. The initial crack 7

8 size (ICS) distribution is derived from the crack sizes at 00 cycles using five randomly selected specimens following the Beta distribution with lower bound of 0.0 mm and upper bound based on the half-width of the test specimen (see Figure 2). The parameters q and r are determined using equations (6) and (7) and the calculated mean and variance of ICS for the set of 5 specimens. Due to the use of crack sizes at load cycle 00, the time zero in the analysis refers to load cycle 00. Residual strengths are calculated assuming a fracture toughness K C =32 MPa m. According to MIL-HDBK-5J [9] the standard deviation of fracture toughness for 7075-T7351 aluminium alloy ranges from 1 to 8 MPa m. Since all the specimens were manufactured from the same aluminium plate, the minimum standard deviation of 1.0 MPa m was assumed in the analysis. A randomly selected crack growth curve from one of the test specimens was used as the master crack growth curve for both the deterministic and the probabilistic approaches. Using these input parameters, the fatigue lives of the test specimens were calculated using both the deterministic and probabilistic methods. Since most of the input parameters were randomly picked from the test results, multiple analyses were conducted (i.e., 5 trials) in order to get a range of predictions and to see the effect of random selection. For the probabilistic approach, separate trials were conducted for fixed K C using the mean value and for variable K C assuming a standard deviation of 1.0 MPa m. The resulting PoF curves are shown in Figure 8 and Figure 9 respectively. In each figure, the PoF curves of the probabilistic method are plotted in blue colour, and correspond to the right vertical axis. The experimental crack growth curves in black colour are plotted against the left vertical axis. 20 Constant Kc=32-3 a (mm) 15 5 Minimum life of all data PoF Block Figure 8 PoF curves using deterministic Kc plotted against actual crack growth curves to failure of test specimens Figure 8 shows the PoF curves assuming fixed value of K C of the five trials using randomly selected data from the experimental result. Figure 9 shows the PoF curves considering the variability of K C using randomly selected data from the experimental result. In the two figures, a horizontal dashed line corresponding to PoF= -7 is drawn intersecting the PoF curves. The intersection of each PoF curve and the line - 8

9 corresponding to PoF= -7, indicates the time (i.e., load blocks) for the initial inspection. A horizontal line corresponding to PoF= -5 is also shown in the figure to indicate the predicted time where the risk is no longer acceptable. It can be observed that using a fixed value of fracture toughness predicted a conservative initial inspection in 3 out of 4 trials. Using probabilistic fracture toughness (i.e., with standard deviation) led to predictions which were all conservative and safe. The times corresponding to PoF= -7 and PoF= -5 for each PoF curve are given in Table 1. In the same table the deterministic initial inspection time predictions (i.e., half-life) for the five trials are also given. 20 Variable Kc, mean =32 Stdev=1.0-3 a (mm) 15 5 Minimum life of all data PoF Block Figure 9 PoF curves using probabilistic Kc plotted against actual crack growth curves to failure of test specimens Table 1 Comparison of deterministic and probabilistic initial inspection time predictions, in load blocks, using DST Group test data Minimum specimen fatigue life (Load blocks) 12.1 Trial - Predicted inspection time Probabilistic (Fixed K C ) (Load block range) P= -7, -5 Deterministic (Load blocks) Probabilistic (Variable K C ) (Load block range) P= -7, , , , , , , , , , , Comparison of the accuracy of initial inspection time predictions between deterministic and probabilistic approaches as validated by Virkler data The Virkler [1] data which were obtained by testing aluminium 2024-T3 specimens using a constant amplitude load spectrum were used to compare the predictions for the initial inspection time using the deterministic and the probabilistic methods. Again the initial crack size distribution was derived using crack size data at load cycle 00 for 5 randomly chosen specimens. It should be noted that at load cycle 0, all the crack size data were identical at 9 mm thus the need to derive the initial crack size distribution at 9

10 higher load cycles. In this comparison, a master crack growth curve based on the geometry and loads used in the test was generated using CGAP [] and shown in Figure. This crack growth curve was used in both deterministic and probabilistic methods. Since the master crack growth curve was obtained using CGAP, the need for multiple trials is no longer necessary because for all test specimen with the same geometry and material properties, the resulting crack growth curve from CGAP will not vary. For this reason, the effect of varying the standard deviations of K C was investigated instead. In the probabilistic analysis using variable fracture toughness, the standard deviations of K C corresponding to 0.5, 0.8, 1.0 and 1.5 were used to predict the initial inspection time. The results of the comparison are shown in Figure 11. In the figure the four blue curves represent the risks PoF corresponding to the four different K C standard deviations. In the figure, the PoF curves in blue from left to right correspond to the assumed standard deviations of 1.5, 1.0, 0.8 and 0.5 respectively. The black curves are the crack growth curves to failure of the test specimens with vertical axis scale (i.e., crack length) shown to the left of the graph. The red vertical line indicates the time (i.e., in load cycles) when the first failure occurs among the specimen tested. Two horizontal lines were plotted to get the time where the PoF of -7 and -5 intersect the PoF curves. The exact times (in load cycles) the two horizontal lines intersect with the PoF curves are given in Table 2. It was noted that using a probabilistic fracture toughness with standard deviation of 0.8 or higher resulted in a safe prediction of the initial inspection time. In contrast, the deterministic approach gave an over-conservative prediction crack size (mm) Cycles Figure Master crack growth curve for the Virkler data test specimen Crack size (mm) Cycles Figure 11 PoF curves of the 4 trials using varying Kc standard deviations PoF Table 2 Comparison of deterministic and probabilistic prediction using Virkler data Deterministic Probabilistic Probabilistic inspection Fixed K C Mean K C = 25 MPa m (Cycles) Minimum specimen fatigue life (Cycles) Load cycles assumed corresponding to P= -7, -5 std. dev , Load cycles corresponding to P= -7, , , , ,

11 4. Discussion of results 4.1 Validation of Predictions verified using test specimen under variable peak stress spectra Based on the results of DST experiment shown in Table 1, the deterministic prediction of the initial inspection time as halfway through the predicted fatigue life was conservative since the first failure among test specimen occurred at 12.4 load blocks which is higher than the deterministic prediction of 7.3 ~ 7.8 load blocks. The deterministic prediction can be said to be overly conservative. Furthermore, the concept of conducting inspection halfway through the fatigue life in order to have a chance of another inspection before fatigue failure occurs as suggested by the deterministic approach seems validated by the experimental results because all the deterministic predictions of inspection times occurred before failure (i.e., 14.6 ~ 15.6 load blocks). As shown in Table 1, the probabilistic approach when considering the variability of the fracture toughness did a very good prediction on the initial inspection, predicting as early as 9.7 load blocks and as late as.4 load blocks which are not only safe but also closer to the experimental minimum fatigue life of 12.1 load blocks. This means that the probabilistic approach was conservative and more accurate than the deterministic approach. 4.2 Validation of Predictions Verified using Virkler data The second verification using the Virkler data given in Table 2 showed similar trend as in Table 1. The deterministic prediction of the initial inspection (i.e., half-life) was overconservative at cycles in comparison to the experimental result which was cycles. In practice, conducting an inspection too early may lead to cracks not being found due to their small size. On the other hand, the probabilistic approach considering the variability of fracture toughness gave a very good prediction which was closer to the experimental value. However, it is observed that the probabilistic approach using a deterministic (fixed value) fracture toughness slightly over-predicted the first inspection and gave a value of cycles. What can be observed in the table is that the highest assumed standard deviation at 1.5 MPa m gave the most conservative (i.e., safe) prediction among the probabilistic predictions. Furthermore the assumed Kc standard deviation from 0.8 to 1.5 MPa m resulted in safe and more accurate predictions in comparison to the deterministic predictions. Similar to the previous comparison using variable amplitude, the deterministic approach gave a prediction which was safe but over-conservative since the predicted inspection time of cycles was too early. The experimental fatigue life showed that inspection can be conducted as late as cycles (i.e., the minimum fatigue life of all the specimens). 5. Conclusions Based on the results of this investigation, conclusions are as follows: 1.) Following the guidance of MIL-STD1530D, both the deterministic and probabilistic approach in predicting the initial inspection time gave conservative predictions as validated by test specimens, but the probabilistic approach predicts a life closer to the experimental results; 11

12 2.) The initial inspection time from the deterministic approach is overly conservative; 3.) The application of both the deterministic and probabilistic approach in predicting the safe fatigue life and the initial inspection time may provide increased confidence and conservatism for the safe management of fleet. In the future, it is envisaged that the use of both deterministic and probabilistic methods in structural integrity management has many advantages over the use of only one method and will enable stakeholders to balance the requirements for safety and economy. Acknowledgements The authors would like to thank Dr Colin Pickthall and Dr Manfred Heller for reviewing this paper and for their valuable comments. References [1] Virkler, D.A., B.M. Hillberry, and P.K. Goel, The Statistical Nature of Fatigue Crack Propagation. 1978, Purdue University: West Lafayette, Indiana, USA. p [2] Torregosa, R. and W. Hu, Crack Growth Variability and Its Effect to Risk Analysis of Fracture Prediction, in International Committee on Aeronautical Fatigue. 2015: Helsinki, Finland. [3] Defence, U.S.D.o., MIL-STD-1530D Standard Practice - Aircraft Structural Integrity Program (ASIP). 2016: USA. [4] Černý, I., The use of DCPD method for measurement of growth of cracks in large components at normal and elevated temperatures. Engineering Fracture Mechanics, (4): p [5] Hovey, P.W., Berens, A.P., and Loomis, J.S., Update of the PROF Computer Program for Aging Aircraft Risk Analysis. 1998, Air Materiel Command, Air Force Research Laboratory, Wright-Patterson AFB, OH [6] Liao, M., Bombardier, Y., and Renaud, G., Quantitative Risk Assessment for the CC-130 Centre Wing Structure (Phase II). 2009, National Research Council Canada. [7] Torregosa, R. and W. Hu, Effect of models and derivation methods for initial flaw size distribution on probability of failure of airframes, in APCFS/SIF. 2014: Sydney, Australia. [8] Ang, A.H.-S. and W.H. Tang, Probability Concepts in Engineering Planning and Design. Vol. Vol. I. 1975, New York: John Wiley and Sons. [9] Defense, U.D.o., Metallic Materials and Elements for Aerospace Vevhicles Strcutures (MIL-HDBK-5J). 2003, US Department of Defense: USA. p [] Hu, W. and C. Wallbrink, CGAP Capabilities and Application in Aircraft Structural Lifing, in 15th Australian International Aerospace Congress (AIAC15). 2013: Melbourne. 12