Achieving Six-Nine’s Reliability Using an Advanced Fatigue Reliability Assessment Model

Dr. J. Zhao D.O. Adams

jzhao@sikorsky.com doadams@sikorsky.com Senior Staff Engineer, Structural Methods Technical Fellow, Test Engineering

Sikorsky Aircraft Corporation

Stratford CT

ABSTRACT The idea that the reliability of helicopter safe-life components could be statistically quantified, as opposed to estimating that the chance of failure was “extremely remote”, has been around for more than 20 years. Originally the work was done via tedious Monte-Carlo simulations that used a lot of (primitive) computer time. This was not practical for routine fatigue reliability evaluations. But now a suite of essentially analytical solutions - the Fatigue Reliability Assessment Model or FRAM - has been developed at Sikorsky. The FRAM provides numerically efficient and accurate solutions to easily determine fatigue reliability for comparison to the “six-nines” standard (i.e., less than one component in one million will fail within its established safe life). While this evaluation can only consider those statistical elements of fatigue that are known in advance, it can still serve as a standard for the component’s basic fatigue substantiation. This is the first step in the establishment of a new methodology that will provide reliability-based component retirement times. It also provides a means to evaluate the relative contributions and sensitivity of the strength, loads, and usage elements of fatigue reliability, so that emphasis can be placed on protecting the component from critical deviations in the original statistical characteristics assumed. Sensitivity to the occurrence rate of one specific flight condition would be an example. This paper reviews the advancement of statistical modeling and analysis in achieving six-nine’s fatigue reliability design for dynamic components. It also presents fundamentals of advanced probabilistic methodology and its application potential to establish more rational fatigue reliability designs. An evaluation of this capability is presented here, based on the fatigue substantiation of a main rotor shaft with measured flight loads and a simplified usage spectrum. BACKGROUND Helicopter fatigue substantiation was recognized to be a probabilistic discipline in the 1950’s when a few fatigue failures in service could not be explained otherwise – i.e., that “within tolerance” extremes of strength, loads, and/or usage elements that were not seen in testing could still occur and occasionally combine in a large service population to produce fatigue failures. So each element was described statistically with large arbitrary margins – strength by a “three-sigma” working curve, flight loads by the “max measured’ values, and usage by a “Composite Worst Case” assumption. However the absolute Reliability – one minus the Probability of Failure – could not be worked out with the statistical methods available at that time. So the term “extremely remote” was used to describe a degree of belief in an intuitively probabilistic sense for a component whose conventional retirement time was determined by

assuming that each element was simultaneously and continuously at its extreme statistical value. The statistical efforts to quantify component reliability began in the 1980’s, notably with the establishment of the 6-9’s requirement for the RAH-66 Comanche helicopter in 1988 by Bob Arden and documented by Immen and Arden in Reference 1. This standard has become the benchmark for all helicopter fatigue reliability studies since then. Thompson and Adams (1990), Reference 2, reported on a rudimentary Monte Carlo simulation that verified that conventional Sikorsky fatigue methodology generally produced the required six nines reliability for “conforming” substantiations. This study also produced the idea that the six nines are made up of approximately three from strength, two from loads, and one from usage. The statistical work that was done after that was largely crude Monte Carlo simulations. One of the major challenges associated with the basic Monte Carlo method is the use of extremely large sample sizes to estimate small probabilities of failure. This

Presented at the American Helicopter Society 66th Annual Forum, Phoenix Arizona, May 11-13, 2010. Copyright © 2010 by the American Helicopter Society International, Inc. All rights reserved.

limitation prevents practical application of Monte Carlo simulation. Attempts at a closed-form solution were generally unsuccessful due to the complexity of the problem – hundreds of flight regimes each with a different load variability characteristic and different rates of occurrence. But in 2009 the authors of this paper proposed an analytical solution, Reference 4, using recent advancements in probabilistic methodologies and structural reliability theory with more efficient numerical algorithms. This tool is called the Fatigue Reliability Assessment Model, or FRAM. It was used to verify the choice of a Usage Monitor Reliability Factor which maintained the baseline reliability of a component whose exact “counts” of damaging usage events was known. A great many other papers and articles have been written over the years on this topic and all have contributed to the progress made to where we are now. INTRODUCTION To continue the journey described above, the authors’ intent is to put the FRAM idea into more routine use and to try it out on some basic helicopter fatigue problems. The FRAM allows an easy and direct calculation of component structural reliability for comparison to the six nines standard established by the U.S. Army and since adopted by many others. The six nines criterion is that less than 1 part in 1 million will fail within its established safe-life retirement time. First it must be remembered the six nines criterion can only be a calculation goal or estimate that cannot be proven because the statistical distribution shapes at the very low probabilities involved cannot be physically verified. Secondly it is understood that this criterion applies only to “conforming” substantiations, that is, when the original statistical assumptions on the distributions of strength, loads, and usage are not violated. These deviations – very low strength parts, very high flight loads, and/or extreme severe usage – are not known in advance, and resist the assignment of statistical characterization in any event. It is also understood that it is precisely these deviations from our basic assumptions are what causes fatigue failures in service. As a result, the six-nines standard has its best use as a goal and reference value which assures that a robust minimum safety margin is provided by a

reliability-based component retirement time. FRAM offers a suite of advanced probabilistic methodologies for more accurate and efficient prediction of fatigue reliability. The inverse reliability analysis capability offers a vital path to determine component retirement times for a specified level of reliability requirement. Probabilistic analysis can also uniquely evaluate the relative contributions of the strength, loads, and usage elements of a fatigue substantiation, and to precisely identify critical low margin/high sensitivity flight conditions. So the connection of the six nines reliability standard to improvements in fatigue reliability is that it provides a basis for mitigating actions to build on. For example, Flaw Tolerant Safe Life methodology includes the strength-reducing effects of the inevitable flaws that can occur in manufacturing and in service by achieving the 6-9’s goal with flaws imposed on the fatigue substantiation components. In the same vein, improvements such as manufacturing process control, better materials, better inspections, improved maintenance practices, and usage monitoring can all provide additional margin so that 6-9’s can actually be achieved in service. A BRIEF REVIEW OF ADVANCED PROBABILISTIC METHODOLOGIES Probabilistic methodologies have been applied in various engineering fields and industries for uncertainty quantification and associated risk management. In recent decades, increasing demands for better understanding of the effects of variabilities and scatter in design, manufacturing, operation, and management triggers significant advancements of more robust and efficient probabilistic methodologies and further applications. In general, a reliability problem is defined by a so-called scalar performance function (also referred as limit state function) ( )Xg in an n-dimensional space

where ( ) 0<Xg denotes the failure domain F. As shown in Figure 1, the vector of random variables X is characterized by a joint probabilistic density function ( )xXf . The associated probability of failure can be estimated by

( )( )

( ) ( ) xxxxx XX X dfIdfpnR FgF ∫∫ == Eq. 1

Here ( )xFI is an indicator function of the failure domain F.

Figure 1 Illustration of Limit State and Joint Probability Density Function

Among the procedures developed for structural reliability assessment and failure probability prediction, a prominent position is held by simulation methods. The Monte Carlo simulation technique, as the basis of all simulation-based techniques, is the most widely applied numerical tool in probabilistic analysis. The associated estimator of probability of failure can be expressed as

( )( ) ( ) ( )xxx XfIN

p iN

i

iFF ~ i.i.d.,1ˆ

1∑=

=

Eq. 2

The convergent rate of the Monte Carlo estimator, in terms of mean square, is appropriately measured by the coefficient of variation of the estimated probability of failure, by

[ ] ( )F

F

F

FMC Np

pppVar

CoV −==

1ˆ Eq. 3

It is noted that the above equations are independent of dimensionality of the random vector X . The key benefit for Monte Carlo simulation is its easiness to understand and implement. It can provide an accurate prediction if the sample size for simulation is sufficiently large. The major disadvantage associated with Monte Carlo simulation is its inefficiency in estimating a small probability of failure due to the large number (roughly proportional to Fp10 ) of samples needed to achieve an acceptable level of accuracy (COV of 30% or lower). For probabilistic application to a rotorcraft structural problem, a target reliability of 6-9’s is generally required. This means that the probability of failure is less than or equal to 10-6 within the lifetime of the component. Therefore, 10 million samples are usually required to provide an acceptable estimator of failure probability.

In addition to the Monte Carlo simulation techniques, several emerging methodologies for reliability analysis have been developed in the last three decades. The fast reliability integration methodologies, including (1) first order reliability methods (FORM), (2) second order reliability methods (SORM), and (3) other hybrid methods (such as importance sampling), have been developed as effective alternates for fast probability assessments without compromising the accuracy of the results. The first order reliability method uses the closest point (also called most probable point) on the limit state surface ( ) 0=Xg to origin in the standard normal space as a measure of the reliability. The shortest distance, denoted as *x=β , specifies the

underlying reliability estimate. The probability of failure can be estimated by

( ) ( )*1

x−Φ=−Φ= βFp Eq. 4

Where ( )Φ is the cumulative distribution function in standard normal space (U-Space). For the case of a linear limit state function with Gaussian variables, FORM leads to an exact solution. If the limit state function becomes nonlinear, or non-normal random variables are encountered, the accuracy and efficiency of FORM needs to be further improved. The Second order reliability method (SORM) was developed to address the aforementioned concern. It approximates the limit state function by an incomplete second order polynomial with the assumption of maintaining the rotation symmetry requirement. In its simplest form, SORM can be expressed as:

( ) ( ) ∑=

ικ+−=≈n

iixxgg

2

21 2

1βxx Eq. 5

It has been proven that SORM asymptotically provides sufficiently accuracy for large β values. This implies SORM is a good candidate for a predicted small failure probability (~10-6), as generally required in the rotorcraft industry. The SORM-based failure probability estimate can be made by incorporating the β value obtained through FORM

with corrections considering the main curvature ικ obtained at the most probable point (MPP) through the following equation:

( ) ( )

( ) ( )( )∏

∏

=

−

ι

=

−ι

⎟⎟⎠

⎞⎜⎜⎝

⎛κ

−Φ−−Φ≈

κ−−Φ≈

n

i

n

iFp

2

21

2

21

1

12

ββφβ

ββ

Eq. 6

Where ( )φ denotes the standard normal density function. Various attempts to further improve the accuracy and efficiency of FORM and SORM can be found in literature (Reference 6, 7, and 10). The Importance Sampling technique is another attractive alternative. It has been regarded as one of the most prevalent approaches in the context of simulation-based methods for probabilistic analysis. Instead of drawing random samples arbitrarily as the way implemented in a Monte Carlo simulation, the Importance Sampling approach focuses on the region that contributes the most of failure points along the limit state surface. The important region can be identified by either MPP obtained through FORM/SORM solution or a prior estimate from pre-sampling. Recall the generic expression of probability of failure as defined by Eq. 1. The expression can be further rewritten as

( ) ( ) ( )( ) ( ) ( )

( )( ) ( )⎥

⎦

⎤⎢⎣

⎡=

== ∫∫

xxx

xxxxxxxx

XX

XXX

X

fhIE

dhfhIdfIp

F

RF

R FF nn

Eq.7

Where ( )xXh is the probability density function representing the identified importance region and [ ]E denotes the expected value.

In a similar fashion to the basic Monte Carlo simulation, the failure probability can be estimated as follows,

( )( )( )( )

( )( ) ( ) ( )xxxxx

XXX

hfhI

Np ii

N

ii

iF

F ~ i.i.d.,1ˆ1∑=

=

Eq. 8

Accordingly, the variance of Fp̂ is given by,

[ ] ( )( ) ( )

( ) ( )( ) ( ) ⎟⎟

⎠

⎞⎜⎜⎝

⎛−=

⎥⎦

⎤⎢⎣

⎡=

∫ d Fpdh

hfI

N

fhIVar

NpVar

F

FF

R XX

X

XX

xxxxx

xxx

22

21

1ˆ

Eq.9

In general, the efficiency of the Importance Sampling technique improves significantly with a large reduction of the variance of estimator, once the appropriate Importance Sampling density function is identified. The sensitivity of the reliability or failure probability with respect to the distribution parameters is useful in structural design, manufacturing and operations because it offers insights on the impact of each individual parameter on the underlying reliability of the structural component. Particularly, it establishes the sensitivity of probability of failure with respect to the mean and standard deviation of a random variable The sensitivity information can be used efficiently to identify the key attributors of system variability. Following the notion of Eq. 7, the sensitivity of failure probability with respect to a distribution parameter can be defined by,

( ) ( )

( )( ) ( ) ( )⎥

⎦

⎤⎢⎣

⎡=

∂∂

=∂∂

∫

xxxx

xθxx

θ

θXX

X

κfhIE

dfIp

F

R FF

n

Eq.10

Where θ is the vector of distribution parameters and ( )xθκ is the kernel function defined generally as (Reference 9),

( ) ( )( )

( )[ ]θxθx

xx

X

X

Xθ

∂∂

=

∂∂

=

f

ffln

1κ Eq. 11

If all the random variables considered are statistically independent, the evaluation of the kernel function with respect to a specific distribution parameter iθ for an input variable ix , can be further simplified as

( ) ( )( )

( )[ ]i

iX

iii

θxfθf

fx

i

∂

∂=

∂∂

=

ln

1 xx

X

Xθκ

Eq. 12

Where ( )ixfiX

is the marginal density function of

random variable iX . In the case that some of the random variables are dependent, a correlation matrix R is used to quantify the dependence. Following the concept of NATAF transformation, the joint probability density function ( )xXf can be expressed as

( ) ( )( )

( )

( ) ( )⎭⎬⎫

⎩⎨⎧ −−

⎭⎬⎫

⎩⎨⎧=

=

∏

∏

∏

=

=

=

zzAzzA

RZx 0X

TTn

iix

n

ii

n

iix

n

xf

z

xff

i

i

21exp

,

1

1

1

ϕϕ

Eq. 13

Where ( )izϕ is the standard normal density function

of the standard normal variable iz in the dependent

standard normal space, ( )0,Rznϕ is the n-dimensional normal density function of zero mean, unit standard deviation, and correlation matrix 0R ,

and A is the inverse matrix of 0R . The matrix 0R

can be obtained from matrix R , via NATAF model. Using Eq. 29, the partial derivative of marginal density function of variable ix with respect to distribution parameter iθ can be expressed as (Reference 8):

( ) ( )( )

( ) ( )

( ) ( )( )

⎥⎥⎦

⎤

⎢⎢⎣

⎡

∂∂

−∂∂

+∂∂

=

⎥⎦

⎤⎢⎣

⎡∂∂

−∂∂

+∂∂

=∂∂

Azzzzx

Azzzzxx

X

XXX

i

T

i

T

i

ix

ix

i

T

i

T

i

ix

ixi

i

xfxf

f

fxf

xffxf

i

i

i

i

θθθ

θθθθ

1

Eq. 14 Therefore, the kernel function can be further simplified as,

( ) ( )( )

( )( )

Azzzz

xx

X

X

i

T

i

T

i

ix

ix

iii

xfxf

θf

fx

i

iθθθ

κθ

∂∂

−∂∂

+∂

∂=

∂∂

=

1

1

Eq.15

Therefore, the sensitivity index of the probability of failure with respect to distribution parameter iθ of

variable ix can be evaluated using Eqs. 10, 13, and 15. Clearly, the algorithm highlighted is very efficient since it doesn’t require additional samples for sensitivity analysis. The evaluation of Eq. 13 and 15 is performed only at a failure domain where indicator function ( )xFI is a non-zero value. Once the failure sample in original physical space (x-Space) is obtained, the corresponding value in reduced standard normal space (z-Space) can be obtained through NATAF transformation. A similar algorithm of sensitivity analysis for crude Monte Carlo Simulations has been developed and implemented by UNIPASS Version 5.95 (Reference 8). APLICATIONS OF ADVANCED FATIGUE RELIABILITY ASSESSMENT METHODOLOGY The advanced Probabilistic Methodology, as outlined by Eqs. 1-13 has been used at Sikorsky for structural reliability assessments and quantified risk assessments for FT/DT applications. To address the impact of various sources of uncertainty on design and operation of fatigue-sensitive and safety-critical dynamic components, an engineering process, as depicted in Figure 2, has been developed to identify and quantify sources of uncertainty and understand their impact on component useful life and maintenance decisions. The figure shows the process of FT/DT design and management and highlights the key aspects of fatigue lifing process. This process map also provides a comprehensive understanding of the effects of process inputs and associated failure modes on the process outputs and propagated variability. It helps to identify the critical parameters and key drivers impacting fatigue design and life management.

Figure 2 Process Map of Uncertainty Identification for

FT/DT Reliability Design and Management As discussed earlier, the safe-life approach has been widely applied in the industry for fatigue design and substantiation. For the case under constant amplitude load, the safe-life approach can be expressed via a standard S-N relationship, such as:

γ

βNS

S

E

+= 1 Eq. 16

Where S is applied vibratory stress, N is fatigue life (in million cycles), β and γ are shape parameters,

and ES is the endurance limit representing the threshold below which no fatigue initiation would be expected. Fatigue strength parameters, β ,γ , and

ES , are obtained through statistical analysis of fatigue test data. As discussed earlier, there are three sources of uncertainty that contribute to the scatter of fatigue life, namely variation of usage, scatter associated with applied load, and inherent randomness of the fatigue endurance limit. The statistical models describing variability for usage, load, and fatigue strength can be established through analysis of field usage, flight loads testing, and laboratory fatigue testing. The details to establish statistical models were discussed in literature elsewhere (Zhao, 2008). A fatigue damage accumulation model, presented in The Reference 4 AHS paper (Adams and Zhao, 2009), has been used in this study. The advanced probabilistic methodologies have been applied to the safe-life approach as the means to estimate the underlying structural reliability. This paper focuses on the application of advanced structural reliability methodology to investigate the effect of usage

variability and fatigue strength reduction on the underlying fatigue reliability. FATIGUE RELIABILTY ASSESSMENT FOR A MAIN ROTOR SHAFT The original UH-60A main rotor shaft used as a reliability test case in the Reference 1 paper is used again here to evaluate the capabilities of the Fatigue Reliability Assessment Model. It is a good candidate for this since the complete flight loads data base statistical analysis was available that had determined Weibull parameters for the load distribution for each flight regime. The usage spectrum had 180 flight regimes. This component was replaced with an improved version because it had only a 1000-hour retirement time, based on the conventional safe-life calculation conducted in 1978. In the Reference 1 work, this retirement time calculation was confirmed to provide a baseline 6-9’s reliability based on a Monte Carlo simulation. The Reference 4 paper also obtained this result using the FRAM model on the same data set. However, a simplified version of this component substantiation was used to conduct the more extensive reliability studies reported in this paper. The usage spectrum was simplified to 31 line items and some modifications of loads were done to increase the calculated retirement time. As reported in Reference 1, the strength element of the fatigue evaluation is treated as a Normal distribution with the mean strength as determined by laboratory fatigue testing of this component. A standard deviation of 10% of mean value is also used, a value which has been confirmed in many substantiations of steel components. The flight loads characterization is a 2-parameter Weibull analysis of the results from 3 to 10 runs of each regime in an instrumented flight test. The scatter in flight loads is therefore statistically represented, rather than using just the “max measured” value employed in conventional calculations. The variation in usage for each flight regime is represented by a Weibull distribution with a beta value (slope) of 2.0 and a 90th percentile intercept equal to the Composite Worst Case Spectrum value. The value of 2 is an approximation taken from early usage surveys, and was also seen in usage monitor data for two Ground-Air-Ground (GAG) flight regimes in Reference 4. The three statistical treatments chosen for this study could potentially be improved with additional data. Log-normal or 3-parameter Weibull distributions could potentially be better fits. The usage assumption

especially will be considerably improved when large quantities of usage monitor data become available. In this example, there are 63 random variables, including thirty-one variables for the usage percentage of defined flight regimes, thirty-one variables associated with the peak loads, and the last random variable is to model variation of fatigue strength. The statistical characteristics for the random input variables for usage percentage, normalized peak load for each regime and normalized fatigue strength parameters are summarized in Table 1. In addition, 90th percentiles of usage rate and 99th percentile of peak load at each of the thirty-one flight regimes are provided. The percentage damage fractions for each of the 31 maneuvers based on a conventional damage calculation are presented in the table. In this example, seven maneuvers, including 45 degree AOB Turns, Run-on Landing, Dive, Hover Reversals, 60 Degree AOB Turns, Fwd Flt Reversals, and Severe Pullout, are identified as the damaging ones in the conventional calculation. Although Moderate Pullout is practically not a damaging maneuver based on this traditional approach, it is worthy of special attention since its 99th percentile peak load is in the vicinity of the fatigue limit defined at the mean minus three-sigma level.

Table 1. Statistical Models of Usage and Load for Main Rotor Shaft Example

A conventional safe-life damage calculation was conducted. This consists of the determination of “allowable cycles” on a “three-sigma” S-N working curve for each max load level and then determining the damage incurred by each one as a fraction of the applied cycles in the usage spectrum divided by the allowable cycles for a given number of flight hours.

The conventional retirement time is simply the number of flight hours divided by the amount of damage incurred in that number of flight hours (Miner’s Rule). In this example, the designed fatigue life of main rotor shaft is 4000 hours.

RELIABILITY ASSESSMENT AND PROBABILISTIC SENSITIVITY ANALYSIS FOR THE BASELINE CASE

To establish a reference for further sensitivity studies and comparisons, a baseline structural reliability assessment for the main rotor shaft is performed using the Fatigue Reliability Assessment Model described above. The FRAM provides 4 related methods to estimate the reliability. A First Order Reliability Model (FORM) resulted in a Probability of Failure at 4000 hours of 1.265×10-6. Various Second Order Reliability Model (SORM) algorithms produced results from 9.981×10-7 to 1.082×10-6 at 4000 hours. The Importance Sampling (IS) method with 200,000 samples predicts a failure probability of 1.246×10-6. A Monte Carlo Simulation with 15 million samples yields an estimated probability of failure of 1.04×10-6 with a CoV of 5%. Of these methods, the SORM at 1.08×10-6 is generally preferred, and verifies that the reliability of this component at 4000 hours achieves the 6-9’s goal. The newly-developed probabilistic sensitivity analysis capability for the Importance Sampling method provides a good means to evaluate the effect of each random variable and associated statistical parameters on the underlying fatigue reliability. The results of the probabilistic sensitivities using this method are shown in Figure 3. The vertical scale in this figure is the change in Probability of Failure in response to an incremental change in the mean value of each random variable, numbered 1 to 63. The changes in the mean values of each individual random variable are normalized by dividing by the associated standard deviations. The normalization eliminates the undesirable effect of the unit associated with each random variable in sensitivity analysis. It is observed that the mean value of fatigue strength (random variable #63) has the highest sensitivity to the estimated probability of failure, followed by the 450 AOB Turns Usage (#6) and Loads (#37). In this case, the 450 AOB Turns is the most important of the 31 flight regimes. This easily agrees with the deterministic lifing calculation in Table 1, where the 450 AOB Turns provide the highest damage contribution of the flight regimes.

Figure 3 Sensitivity Index of Probability of Failure With Respect to Mean Value

Figure 4 shows the effect on reliability due to changes of scatter, or the degree of variability, for each of the 63 random variables. The vertical scale in this case is the change in Probability of Failure for an incremental change in the standard deviation of each random variable, normalized by dividing through by the associated standard deviation for each of the corresponding random variables.

Figure 4. Sensitivity Index of Probability of Failure With Respect to Standard Deviation

The Figure 4 picture is very different. Now the 450 AOB Turn load (#37) has the highest contribution, even higher than the strength value and much higher than its usage contribution. This indicates that controlling the load scatter in this flight regime is more important than controlling average value of peak load or its occurrence rate. Note in Table 1 that 450 AOB turns have the lowest slope, or widest range, of the damaging maneuvers (except for the 600 turns which have the same slope with a higher load but a much lower usage rate). So when we consider the full statistical range of where the #37 load could go (FRAM calculation), not just the maximum that was measured in flight test (conventional calculation), it has a significant effect. This is illustrated in Figure 5 below. Changing the scatter on these maneuvers is essentially changing the slope of the Weibull line. So an improvement in reliability could be obtained by “straightening up” the lines, reducing the range of

loads that could occur. The Dive condition in the figure above is an example of relatively low scatter.

Figure 5 Weibull Illustration of Contributing Flight Regimes

A reduction in scatter could be obtained in aircraft that have Fly-By-Wire control systems by control law “shaping” to produce load alleviation, or by tactile cueing to limit extreme conditions. This capability has been successfully demonstrated in Flight Test, as described in Reference 5. The intent of that effort was to reduce the vibratory load values and hence provide longer retirement times. But the additional result of reducing scatter also improves the reliability of that retirement time. It should also be noted in Figures 3 and 4 that the highest in each category do not show the greatest sensitivity in affecting the reliability. The highest load is Severe Pullout (#32), but its usage rate is so low that any changes in it are a second order effect. The highest usage rate among the damaging conditions is Dive (#7), but while its load is just below the 450 AOB load, it is close enough to the endurance limit that changes in its usage rate do not have a significant effect. Efforts to improve the reliability on this component can concentrate the effort elsewhere. EFFECT OF USAGE RATE VARIABILITY ON UNDERLYING FATIGUE RELIABILITY Once the six-nine’s reliability-based CRT has been established and probabilistic sensitivity analysis has identified the most important random variables and associated distribution parameters, a further parametric study has been performed to investigate the effect of usage, load and fatigue strength on the underlying fatigue reliability.

First, we will investigate the effect of usage rate on the reliability of fatigue assessment. The effect of 450 AOB Turns, identified earlier as the most sensitive flight regime, is considered. As shown in Table 1, 1.33% time is the Composite Worst Case usage rate for this maneuver, which is taken to be a 90th percentile on a Weibull line with a slope of 2. Assuming the mean value of usage rate of 450 AOB Turns varies from 0.45% to 1.35% and the standard deviation is unchanged (0.25%). The associated 90th percentiles are 0.83% and 2.33%. FRAM analyses were conducted for usage rates with 90th percentile values ranging from .83% to 2.33% time, while the loads and strength distributions unchanged. The probability of failure at various component retirement times ranging from 1,750 hrs to 7,500 hrs were calculated at each of eight usage rates considered. These results are shown in Figure 6. In the plot, the functional relationship between calculated probability of failure and component retirement time at each of eight mean usage rates were depicted. For the consistency to the conventional six-nine’s reliability notion, the curves were categorized based on 90th percentile value of the maneuver, which is equivalent to the CWC usage rate. As expected, an increase of component retirement time results in an increase of probability of failure.

Figure 6 Effect of 450 AOB Turn Usage Rate On Reliability and Retirement Time

The figure shows the baseline 6-9’s retirement time of 4000 hours at the 450 AOB turn Composite Worst Case usage rate of 1.33% time. For any change is these values, the figure can be sliced – to show the relationship of retirement time to usage rate at a fixed 6-9’s reliability, to show the relationship of reliability to usage rate at a fixed 4000 hour retirement time, or to show the relationship of reliability to retirement time at a fixed usage rate of 1.33% time.

It can also be observed that a change of mean usage rate for the maneuver significantly impacts the probability of fatigue failure. With the increase of the usage rate of the maneuver investigated, the probability of failure increases rapidly. For example, if the equivalent CWC usage rate for the 450 AOB turns in service changed from 1.33% time to 2.33% time, due to a change in the fleet’s basic mission, say, the reliability would drop to approximately 5-9’s if the 4000 hour retirement time was retained. If it was desired instead to retain the 6-9’s reliability at the 2.33% usage rate, the retirement time would have to be reduced to 2300 hours. Another observation is that accepting a 5-9’s reliability would allow a 7000 hour retirement time at the reference CWC usage rate of 1.33%, and going to 7-9’s yields a 2500 hour retirement time. This verifies another rule of thumb that has been around for many years – “A factor of 2 in life is worth a factor of 10 in reliability”. Now it appears that this idea is a rough approximation, but we have seen it in several of these reliability evaluations, so it is still a useful rough guidance. The results can also be represented in a 2-dimensional contour plot, as shown in Figure 7. In this case constant reliability is represented by color bands with the variables of Component Retirement Time and 450 AOB turn usage percent time on the plot axes. To better represent the small probability of failure, the logarithm of PoF is used in the color bar. The color map ranges from -4.5 to -9, representing the range of PoF from 3.16×10-5 to 10-9, respectively. The 6-9’s reliability equivalent PoF curve is at the boundary between the yellow and orange bands.

Figure 7 Contour Plot of Log(PoF) as Function of

CRT and 450 AOB Turn Usage Rate Finally a 3-dimensional contour plot is shown in Figure 8 below which provides more insight into the

Standard Usage (90th percentile = 1.33%)

Baseline CRT 6-9’s Reliability Standard Usage

CRT = 4000 hrs

fatigue reliability problem. Again the Component Retirement Time, the Reliability (in terms of probability of failure), and the Usage Rate of the 450 AOB turns are plotted together. In the figure, the third dimension represents the logarithm of the estimated failure probability, while same color map is used for Log(PoF) as of the 2-D contour plot. The color coding allows visualization of the lines of constant reliability, where again the 6-9’s reliability is at the border between the yellow and orange bands. Now the shapes of the “slices” can be seen. For example it can be seen how rapid the improvement in reliability is with respect to a reduction in the rate of the occurrence of 450 AOB turns. The loss in reliability with an increase in the usage rate is less dramatic. It can also be seen that if a deliberate choice was made to temporarily operate at the 5-9’s level in order to obtain higher retirement times, the benefit of reducing the rate of steep turns is even greater, maybe providing an opportunity to recover some reliability in that scenario.

Figure 8 3D Plot of Probability of Failure as Function

of CRT and 450 AOB Turn Usage Rate Another exercise is to evaluate the effects of maneuvers whose load is high but the contribution to fatigue damage is very little because of very low rate of occurrence. One typical example of this for the Main Rotor Shaft is the Moderate Pullout. As summarized in Table 1, the usage rate of Moderate Pullouts is 0.28%, but its 99th percentile load is slightly above the working endurance limit and the damage is insignificant in the conventional calculation. Traditionally, the Moderate Pullout in this example will not be regarded as a damaging maneuver due to its little contribution to the fatigue damage. However, higher usage rates cause higher damage, and in the FRAM analysis, statistically high

loads above the 99th percentile are considered, and these can combine to lead to a damage contribution if the rate of occurrence is high enough. This effect is shown in Figure 9, where the reliability reduction is not significant until around 2% time. The rapid increase in damage after that eventually overpowers the 450 AOB turns as the most damaging maneuver. This is an important aspect of usage monitoring in that the rates of occurrence this type of high-load-low-occurrence maneuver must be tracked. This will protect the user who for whatever reason has a high occurrence rate of this flight regime.

Figure 9 Effect of Change of Usage Rate of Moderate Pullout on Probability of Failure

This is another significant finding which suggests the importance of monitoring all the critical and potentially critical maneuvers. EFFECT OF FATIGUE STRENGTH VARIABILITY ON UNDERLYING FATIGUE RELIABILITY Another interesting application of the advanced fatigue reliability methodology is to study the effect of fatigue strength on the underlying fatigue reliability. In any helicopter fatigue substantiation, the most important single element of any helicopter fatigue substantiation is the component fatigue strength. All of the flight loads together have the same effect, because they are on the same axis of the S-N curve, but since each regime load is generally independent of the others, the individual effect is less. Our second case will focus on quantification of fatigue reliability due to the change of fatigue strength. In a real world application, fatigue strength of an individual of fleet of dynamic components could be changed due to various reasons. One of the typical causes is the presence of flaw. It is well known that flaws could reduce fatigue strength significantly and the amount

of strength reduction depends on type of flaw, location of flaw, size of flaw, and type of material of the component. In general, the fatigue strength reduction increases with the increase of flaw size. Typically, small flaw sizes, such as 5 mils of gouge, will cause a marginal reduction of fatigue strength, while a large flaw may cause a significant amount of fatigue strength reduction. In this case study, the strength variation is described by a normal distribution, and for the UH-60A Main Rotor Shaft a Coefficient of Variation (CoV) of 10% is used. The conventional calculation assumes a “3-sigma” strength component, so uses a 70% working curve. In FRAM analysis, the entire distribution of fatigue strength is used in sampling and simulation. The most probable (most important) region for fatigue strength is in the vicinity of the working curve. Fluctuation around the position of working curve is expected. To determine the effect of strength variation on fatigue reliability, a strength reduction factor ranging from 0.7 to 1.0 is applied to adjust the mean value of the fatigue strength, while the CoV of the fatigue strength remains unchanged. The reduction factor of 1.0 represents an anticipated initial condition while a fatigue strength reduction factor of 0.70 on the mean is regarded as a significant strength reduction. FRAM analyses were conducted for mean strength adjustment factors of 100% down to 70%. In these analyses, Usage and Loads distributions were unchanged. These results are shown in Figure 10. The power of the strength variation is illustrated by noting that a 30% reduction in mean strength results in almost a 3-9’s reduction in reliability for the same 4000-hour retirement time. Or to retain 6-9’s reliability, the retirement time needs to be lowered to 1200 hours.

Figure 10 Plot of CRT vs. PoF as a Function of Fatigue Strength Reduction Factor

This result applies to a fleet of components operating at 70% of the original mean strength. Reducing the fleet retirement time to 1200 hours could be an appropriate action if a discrepancy or flaw is found which could potentially affect a large number of fielded components. While this approach would be extremely disruptive, it could be an attractive option compared to grounding the entire fleet. The results can also be represented in a 2-dimensional contour plot, as shown in Figure 11. In this case constant reliability is represented by color bands with the variables of Retirement Time and Fatigue Strength on the plot axes. 6-9’s reliability contour is at the boundary between the yellow and blue-green bands.

Baseline CRT6-9’s ReliabilityStandard Usage

Figure 11 Contour Plot of PoF as a Function of CRT

and Fatigue Strength Reduction Factor

The 3-dimensional contour plot for this case is also shown in Figure 12. The introduction of the third dimension visualizes additional change of PoF (in terms of slope and curvature) as the function of strength reduction and component retirement time.

Baseline CRT6-9’s ReliabilityStandard Usage

Figure 12 3D Plot of Probability of Failure as Function of CRT and Fatigue Strength Reduction Factor

The 3-D plot depicts a smooth increase of failure probability with increase of strength reduction for the entire range of CRT investigated. Again the 6-9’s reliability is at the border between the yellow and blue-green bands. In addition, the probabilistic results can also be represented in the form of a constant component retirement time curve. This is the special case for the 3-D plot at a series of fixed component retirement times. Shown in Figure 13, the probabilities of failure as a function of fatigue strength reduction at each of the selected component retirement times are presented. Each of the curves represents a constant retirement time in which the probability of failure can be determined as a function of strength reduction. In this case, a linear functional relationship between probability of failure and strength reduction factor can be observed, if the PoF is plotted in logarithmic scale.

1.0E-10

1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

1.0E-04

1.0E-03

1.0E-02

0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1

PoF

Fatigue Limit Reduction Factor

200030004000500060007000

Changed PoFReduced FatigueStrength

Constant CRT

Baseline CRT6-9’s ReliabilityStandard UsageNo Str. Reduction

CRT = 4000 hrs

Figure 13 PoF as a Function of Fatigue Strength Reduction Factor and CRT

The plot also provides a great insight into the effect of strength reduction and the associated scatter on underlying fatigue reliability. As depicted in Figure 13, there are three types of applications based on the plot. The horizontal arrow highlights potential application of determining appropriate component retirement time with known fatigue strength reductions. For example, at a target risk level of 10-6, the component retirement time can be easily determined for a specific strength reduction level. This application is ideal for component retirement time adjustments with known strength reductions to maintain a desired reliability level. The vertical arrows illustrate a potential application to perform a trade-off study between CRT and maximum allowable risk with a known strength reduction. Finally, the

arrow along the constant CRT curve shows the opportunity to quickly determine the probability of failure caused with a known strength reduction if the same component retirement time is maintained. CONCLUSIONS 1. A new Fatigue Reliability Assessment Model, or

FRAM, has been developed at Sikorsky that provides numerically efficient and accurate solutions to helicopter component safe-life fatigue reliability evaluations.

2. An evaluation of the method was conducted

using the fatigue substantiation of a helicopter main rotor shaft.

3. The FRAM is able to quantify the impact on

reliability due to variations in strength, loads, and usage.

4. An advanced probabilistic sensitivity analysis

capability provides additional means to identify the critical random inputs impacting a reliability-based component retirement time.

5. Components in service can be better managed

when the reliability-driving factors are known.

6. The methodology facilitates application potentials for risk-based fatigue design and it also opens the door to establish reliability-based dynamic component retirement times.

REFERENCES 1. Meeting Paper – Immen, F.H. and Arden, R.W., “U.S. Army Requirements for Fatigue Integrity”, National Technical Specialists’ Meeting on Advanced Rotorcraft Structures, Williamsburg, Virginia, October, 1988. 2. Meeting Paper - Thompson, A.E., and Adams, D.O., “A Computational Method for the Determination of Structural Reliability of Helicopter Dynamic Components,” 46th Annual Forum of the American Helicopter Society, Washington, D.C., May 1990. 3. Report - Zhao, J., Structures Technologies for CBM (2007-C-10-01.1-P3), Final Report Submitted to CRI for CRI/NRTC CBMT Program, July 2008. 4. Meeting Paper – Adams, D.O, and Zhao, J., “Searching for the Usage Monitor Reliability Factor Using an Advanced Fatigue Reliability Assessment Model,” 65th Annual Forum of the American Helicopter Society, Grapevine Texas, May 2009. 5. Meeting Paper – Wulff, O., Sahasrabudhe, V., Beale, R., Lamb, J., Rigsby, J., Fletcher, J. and Braddom, S., “Flight Test of Load Alleviating Control and Tactile Cueing System”, 66th Annual Forum of the American Helicopter Society, Phoenix AZ, May 2010. 6. Book – Ditlevsen, O. and Madsen H. O., Structural Reliability Methods, John Wiley & Sons, New York, 1996. 7. Book - Zhao, J. and Haldar, A. Reliability-based Structural Fatigue Damage Evaluation and Maintenance Using Non-destructive Inspections. Haldar A, Guran A, Ayyub BM. Uncertainty Modeling in Finite Element, Fatigue and Stability of Systems. World Scientific, New York, 1997. p. 159-214. 8. Lin, H-Z., PredictionProbe, private communication, 2010. 9. H.R. Millwater, “Universal Properties of Kernel Functions for Probabilistic Sensitivity Analysis,” Probabilistic Engineering Mechanics Vol. 24 (2009) p. 89–99. 10. Book - Der Kiureghian, A., “First- and Second-Order Reliability Methods,” Chapter 14 in Engineering Design Reliability Handbook, E. Nikolaidis, D. M. Ghiocel and S. Singhal, Eds., CRC Press, 2005.

AKNOWLEDGEMENTS The authors wish to recognize the significant contributions made to this work by Dr. Mike Urban.