Fatigue Reliability Analysis of Dynamic Components with Variable Loadings without Monte Carlo Simulation 1

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1 Fatigue Reliability Analysis of Dynamic Components with Variable Loadings without Monte Carlo Simulation 1 Carlton L. Smith., Chief Engineer, Structures and Materials Division, US Army, AMRDEC Redstone Arsenal, AL 35898, carlton.l.smith@us.army.mil Jung-Hua Chang, Ph.D., Lead Aerospace Engineer, Structures and Materials Division, US Army, AMRDEC Redstone Arsenal, AL 35898, junghua.chang@us.army.mil Martin H. Rogers, Ph.D., Branch Chief, Structures and Materials Division, US Army, AMRDEC Redstone Arsenal, AL 35898, martin.rogers@us.army.mil Keywords: Safe Life, Fatigue Reliability, Fatigue Spectrum, Monte Carlo Simulation, Load Variability ABSTRACT US Army helicopters have retirement times assigned for fatigue sensitive components. These retirement times are based on a deterministic approach. They are derived using the Prime Contractor s safe life methodology which typically includes fatigue testing of production components to determine a working S/N curve, measurements of top of scatter loads from flight load surveys, and construction of a composite usage spectrum from pilot surveys. The philosophy of the safe life approach is to maintain a level of reliability within the design life of a component. However, the level of reliability for each substantiated life has not been defined due to limitations of analytical tool availability and understanding of random variables for each parameter for its reliability evaluation. Monte Carlo Simulation is the most common approach for fatigue reliability analysis. Thompson and Adams [1] have used the simulation approach to demonstrate a six-nines reliability on several UH-60 dynamic components. In contrast to the simulation, this paper introduces an analytical methodology to evaluate the fatigue reliability of a spectrum with variable loading. The analytical approach provides fast execution to obtain a solution in seconds in comparison to long hours of the solution time from a Monte Carlo simulation run when a high degree of the reliability is required. Two assumptions were made in developing this approach. First, a Constant Damage Rate Tracking is assumed for each individual load step, and second, the fatigue reliability is based on a weighted average of the damage rates of the spectrum load steps. The analytical approach was validated by comparison to Monte Carlo simulations. This paper presents an accurate and efficient way to evaluate fatigue reliability analysis using the Constant Damage Rate Tracking approach. The approach allows a quick determination of a component s reliability at its retirement time when the fatigue strength, severity of load, load variability, and the usage are characterized. This paper presents the reliability solution of a two step spectrum problem and the AHS Round Robin problem [2] for verification. 1 Presented at the American Helicopter Society 63 rd Annual Forum, Virginia Beach, Virginia, May 1-3, Copyright 2006 by the American Helicopter Society International, Inc. All Rights reserved. 1

2 1.0 INTRODUCTION Rotorcraft fatigue design is traditionally based on the safe life methodology to reduce the chance of failure to an insignificant level within the design life of a component. The safe life methodology includes fatigue testing of production components to determine a working S/N curve, measurements of top of scatter loads from flight load surveys, and construction of a composite usage spectrum from pilot surveys. The safe life analysis uses Miner s Rule of linear cumulative damage. The individual damages are summed over the entire flight load spectrum to establish component fatigue life. The safe life approach is, in general, considered conservative. The design fatigue strength is based on a one in one-thousand probability of failure, along with the top of scatter loads and composite worst case usages. Component Reliability using the safe life methodology design is un-quantified due to two factors. First, there is no easy tool available for its reliability analysis; second, there is not enough effort to quantify severity and variability in loads and usages. In the late 1980s and early 1990s, the U.S. Army attempted to implement six-nines reliability for flight critical components. The Army, in conjunction with the American Helicopter Society (AHS) Subcommittee for Fatigue and Damage Tolerance, undertook a Round Robin Problem study to explore computational methods necessary to determine fatigue life as a function of the reliability [3]. The conclusion of that study indicated that more work is needed before reliability based fatigue design becomes standard industry practice. Monte Carlo Simulation is the most common way to manage the probabilistic and statistical complexities associated with load and usage variability for the reliability evaluation. Thompson and Adams [1] used the simulation approach on several UH-60 components and demonstrated that six-nines reliability is already in place using standard Sikorsky fatigue substantiation methodology. Monte Carlo simulation sometimes requires two or three days of computer run time to get a solution for the fatigue reliability in the six-nines range. There are other analytical approaches in evaluating fatigue reliability. Zion evaluated the implicit level of reliability and associated the confidence level of a mean- 3xSigma/TOS (Top of Scatter) safe life method [4]. Kececioglu and Zhang [5] developed formulas for transforming the operating cycles at the sequential stress levels into an equivalent number of operating cycles at the fixed stress level for reliability calculation. Kececioglu, Chester & Gardner [6] used the conditional probability approach to predict the reliability at the sequential stress level. Park and Tang [7] developed an algorithm, which combined the accumulated damage analysis with FORM (first-order reliability method), to evaluate fatigue reliability. It appears that none of the approaches had explicit expressions that can be easily automated for computation purpose. This paper provides an analytical approach for fatigue reliability analysis of dynamic components. The approach requires no Monte Carlo simulation. It allows the evaluation of fatigue reliability on a given spectrum with or without load variability. The analysis assumes that the fatigue reliability for each load step (of a multiple load step spectrum) will track at the same damage rate. The combined reliability from multiple load steps is calculated using the weighted reliability on its damage rate. The approach can take into account the load 2

3 variability by further distributing each load step into multiple load steps with prescribed statistical distribution to characterize the severity and variability of the input load step. Several examples are provided in this paper to validate the proposed fatigue reliability approach. Reliability solutions to the Round Robin Problem [2] with a normal distribution in strength and a Weibull distribution in loads are presented in Section 3.0. A two load step example, which has a normal distribution in both strength and loads, is also presented. The reliability results were compared to the Monte Carlo Simulation and show the analytical approach is slightly conservative for the Round Robin Problem. The analytical solutions agree very well with the simulation results when both strength and loads variations are in normal distributions. 2.0 FATIGUE RELIABILITY METHODOLOGY Two key steps in the fatigue reliability analysis are to evaluate the reliability for each load step of a given multi-load steps spectrum and to determine the appropriate way to combine reliability from each load step. At the beginning of this development, two assumptions were made. The first assumption involved constant damage rate tracking for each stress level. The second assumption was made, such that, overall fatigue reliability should be calculated using the weighted average of individual reliability on its damage rate. The following sections illustrate the proposed analytical approach. 2.1 ANALYTICAL FATIGUE RELIABILITY Figure 1 illustrates the methodology to calculate the reliability if the number of cycles that is changed while the stress or applied load is kept constant. A strength curve, with known fatigue reliability, is referenced for the load step σ i and n i. The damage rate for the i- step is n i /N i and has reliability established by the reference strength curve. (Note that if a mean-minus-three-sigma curve is chosen as the reference curve, the established point at (σ i, n i ) should have a reliability of ) The fatigue reliability for the reduced cycles n j at σ i in the figure can be established according to the following relationship, (a constant damage rate) n i /N i = n j /N j = Constant (Eq. 1) Once the N j is determined, the reliability at (σ i,n j ) can be determined by passing the S/N curve shape through this data point for endurance E j. The reliability for E j can be determined from the mean endurance E m and the standard deviation of the strength. E j 1 E x = COV m (Eq.2) The x is the random variable having the standard normal distribution which has a mean of zero and a standard deviation of 1. The reliability at (σ i,n j ) can be determined as follows: R ( x) = j 1 x e 2π 1 t 2 2 Microsoft Excel can be used to evaluate R j (x) as follows: dt R j (x) = NORMDIST(x, 0, 1, true) (Eq. 3) If the stress or applied load is changed for a given cycle, the change in stress or load will 3

4 reflect in the endurance as shown in Figure 2. The endurance change at σ j can be referred to a known reliability curve at σ i with the reference endurance E r such that σ j /E j = σ i /E r (Eq. 4) If both stress and cycles are changed, the change in endurance reliability is shown in the equation below. E j is the change in endurance if the stress is changed. Once E j is determined, the reliability can be determined using the cumulative normal distribution. E j (combined) = E j Er E r Cycles E E j r loads (Eq. 5) The reliability for E j can then be determined using the relationship established in Eq. 2 and Eq COMBINED RELIABILITY The proposed analytical approach for a multiple load step spectrum requires combining the reliability calculated from each load step. In the development, the reliability is combined using a weighted average of the damage cumulative rates for each load step. The equation to calculate the reliability for a multiple load steps spectrum is as follows: ni Ri N i R(t) = ni N i (Eq.6) in Equation 6 are validated through example problems presented in this paper. 2.3 FATIGUE RELIABILITY WITH LOAD VARIABILITY The methodology presented in Section 2.1 can be modified to include the load severity and variability if the statistical load distribution for each load step is characterized. Normal and Weibull distributions are the most common statistical models used to account for the load variability. Flight load conditions in the normal distribution can be represented by 2 1 z 2 p( Z) = e (Eq. 7) 2π Where z represents the standardized variable for the loads and p(z) is the probability function to represent the cycles. Flight conditions in the Weibull distribution are characterized by β t η β 1 βt f ( t) = e (Eq. 8) β η Where β is the Weibull slope, η is the characteristic load, t represents the load distribution, and f(t) is the probability function for the cycle distribution. Any load step in a given spectrum that is characterized by a probability function distribution for its severity and variation can be further characterized in a discrete form according to its statistical model. It can then be processed like a sub-step spectrum for easy computational implementation. Where R(t) is component reliability for the fatigue life at time t. The assumptions made 4

5 Constant Stress Reliability n i /N i = Constant n j /n i = N j /N i Note: n j /n i E j /E r σ i E m, Mean Curve E r, Reference Curve E j, Reliability Curve n j n i N j N i Figure 1: Reliability for Changing the Cycles Load Reduction Reliability σ j /σ i = E j / E r σ i σ j E m, Mean E r, Reference Curve E j, iability For ( σ i,n i ) point n i Figure 2: Reliability for Changing the Stress or Applied Load 5

6 3.0 VERIFICATION WITH EXAMPLE PROBLEMS Several example problems are provided in this section to validate the proposed methodology presented in Section 2.0. Fatigue reliabilities are calculated and compared to the Monte Carlo simulation results. It can be seen from this validation that the proposed tool is an accurate and efficient way to conduct a fatigue reliability analysis. 3.1 ANALYSIS WITH NO LOAD VARIABILITY The first problem considers a two load step spectrum with normal distribution in strength and no statistical variation in loads. The constant coefficient of variation (COV) is used for the entire S/N curve. The S/N curve for the fatigue strength is as follows: S E = N Where, N is the number of cycles in millions. The factor is a normalized factor for the endurance at 2,000,000 cycles. The fatigue mean strength (E) is 5,090 pounds, the standard deviation is at 545 pounds (or COV at 10.71%). The two-load step spectrum shown below represents a 1000-hr spectrum. Load Cycles Applied (lbs) 1 100,000 5, ,000 7,000 The fatigue life using the mean-3σ strength curve is hours. Without the load variability, the reliability is at the mean-3σ fatigue life. Figure 3 shows the mean S/N curve and the applied spectrum. Table 1 provides the analytical fatigue reliability results when the cycles and/or loads are changed. The analysis is based on selecting the mean-3σ strength as the reference curve. The results are compared to the Monte Carlo simulations. The solutions show strong agreement with each other. A sensitivity study was conducted to evaluate selection of the reference curve. As shown in Table 2, using the mean curve as the reference curve provides one-nine in error when compared to the simulation results. Using the Mean-3 x sigma strength curve as the reference curve provides very close results as the simulations predicted. 3.2 ANALYSIS WITH LOAD VARIABILITY INCLUDED The reliability analysis results are provided for the example problem presented in Section 3.1, with the load severity and variability included. As stated in section 2.3, the load severity variability for each load step was further modeled as a normal distribution. The analysis is conducted assuming the load spectrum represents the 50th through 90th percentile loads while the COV in loads is varied from 10%, 20% or 30%. Figure 4 shows how load severity and variability impact component fatigue reliability. When the spectrum represents the 50th percentile load for fatigue life substantiation, the fatigue reliability is between two-nines to three-nines. If the load spectrum represented the 95th percentile, the reliability increases by at least two-nines for the load COV at 20% or higher. The higher COV displays, the lower 6

7 reliability at the 50th percentile load. The higher COV increases the reliability when the percentile load is at 80% or higher. The results above provided the validation of the proposed methodology. The analytical results show strong agreement with the Monte Carlo simulation runs. Therefore, we believe the proposed analysis method using the constant damage rate tracking and the weighted average of damage rates, provides an innovative tool for fatigue reliability analysis Mean Failuire Load Spectrum Point Strength Cycles to Faiulure Figure 3: Mean S/N Curve and the Applied Spectrum Table 1: Reliability for Changes of Cycles and/or Loads (with no Load Variability). Cycle Case Load Reliability for Life Analytical Monte Carlo Spectru Description s Level Equivalent to Reliability Simulation m 1B B B B B B No Reduction % Load 75% Cycles 90% Load 100% Cycles 90% Load 75% Cycles 110% Load 100% Cycles 110% Load 75% Cycles

8 Table 2: Fatigue Reliability Sensitivity to Reference S/N Curves Case A Case B Cycles Loads Cycles Loads Reference Strength Reliability Reference Strength Reliability Simulation Simulation Fatigue Reliability Sensitivity Analysis Reliability is Calculated at Hrs Retirement Time % COV % COV 30% COV Failure Rate Simulation, 10%COV Simulation, 20% COV Simulation, 30% COV Load Step Cycles Loads 1 100, , % 60% 70% 80% 90% 100% Loads (Percentile) Figure 4: Fatigue Reliability Sensitivity Study on Loads and COVs of Loads 3.3 AHS ROUND ROBIN PROBLEM The AHS Round Robin problem [2] was solicited to evaluate usage monitor based 8

9 component reliability. A normal distribution is defined for the fatigue strength with the mean of 1000 psi and a standard deviation of 100 psi. S E 0.8 = N Where stress at endurance E is defined at 10 8 cycles, N is the number of cycles in millions. Six (6) flight conditions are defined. The flight loads are characterized by a Weibull distribution with a Weibull slope of 4. The 95% peak load values are defined at 2300, 1750, 1300, 900, 600, and 500 psi. Table 3 shows the cycle counting distribution. Table 3: Cycle Counting for the Round Robin Problem Cycles Distribution Load Distribution (Fraction of Total (Fraction of Peak Cycle) Load) The load frequency is 5 cycles/second. The load variability was implemented as a multiple load-step spectrum for computation to characterize the Weibull probability of occurrence and the load levels. Two spectra shown in Table 4 were analyzed for their fatigue reliability. Table 4: Severe and Moderate Usage Spectra for the Round Robin Problem Flight Condition Severe Spectrum Moderate Spectrum Total Table 5 summarizes the reliability versus the Monte Carlo simulation results. An Excel Macro was used to characterize the statistical load distribution. The constant stress tracking was automated using an Excel Macro. The run time was less than 1 minute for each analytical run. The average run-time on Monte Carlo simulation was 60- hours (for 10,000,000 samples). The fatigue reliability was calculated at its mean-3σ strength life. The solutions agree well with the Monte Carlo simulation. The analytical reliabilities show slightly conservative, in comparison to the simulation results. Table 5: Fatigue Reliabilities for the Round Robin Problem Case Fatigue Life Analytical Simulation Severe Spectrum,TOS* Severe Spectrum Moderate Spectrum *Top of scatter load used. 4.0 CONCLUSION This paper presented an analytical approach based on the constant damage rate tracking to calculate the fatigue reliability. The approach requires much shorter computation time to calculate fatigue reliability than using Monte Carlo simulation. This paper provided validation 9

10 of the proposed fatigue reliability approach through several example problems. The approach presented in this paper provides an innovative and effective tool for fatigue reliability analysis. 5.0 RECOMMENDATION This paper presented sensitivity analysis of reliability impacts associated with load severity, variation and usage. Most flight load surveys were conducted at one to two repetitions per maneuver. Flight load severity and variability have not been well defined from previous flight load surveys. In future load surveys, it should be planned such that the load variability can be characterized. Also, fatigue reliability is affected by the material S/N strength, cycle count methodology, maneuver binning, usage spectrum. All these parameters should be evaluated to accurately determine the component reliability. ACKNOWLEDGMENTS Thd authors would like to thank Dr. Robert Vaughan for his editorial comment, Mr. Kim Ong of ASRI Inc., for his help in providing the guides for Visual Basic programming, and creating the Macros for automating the process on Microsoft Excel. REFERENCES [1] Audbur E. Thompson and David O. Adams, A computation Method for the Determination of Structural Reliability of Helicopter Components, Presented at the AHS Annual Forum, May [2] AHS Round Robin Problem, Reliability of Usage Monitor Based Component Retirement, Phase 1, Hypothetical Data, Presented to Fatigue and Damage Tolerance Subcommittee, 62 nd AHS Forum, Phoenix, AZ, May 10, [3] Richard A. Everett, Jr. etc., Probabilistic Fatigue Methodology for Six Nines Reliability, Presented at the 47 th AHS Annual Forum, May 1991, Phoenix, AZ. [4] H. Lewis Zion, Safe Life Reliability: Evaluation of New Statistical Methods, Presented at the 47 th AHS Annual Forum, May 1991, Phoenix, AZ. [5] Dimitri B. Kececioglu and Jiliang Zhang,. Random Cumulative Damage Rules and Fatigue Reliability, AIAA , 1997 [6] Dimitri B. Kececioglu, Louie B. Chester, and Everett O. Gardner, Sequential Cumulative Fatigue Reliability, Proceeding 1974 Reliability and Maintenance Symposium, Los Angeles, CA, pp ,

11 [7] Young Ho Park and Jun Tang, An Efficient Methodology for Fatigue Reliability Analysis for Mechanical Components, Journal of Pressure Vessel Technology, Vol. 128, August,