Updating the Master S-N Curve to Account for Run-Out Data: Application to Piping Vibrations

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Procedia Structural Integrity 19 (2019) 711–718

Fatigue Design 2019 Fatigue Design 2019

Updating the Master S-N Curve to Account for Run-Out Data: Application to Piping Vibrations Updating the Master S-N Curve to Account for Run-Out Data: Application to Piping Vibrations Edrissa Gassamaa, Michael F. P. Bifanoa, Anthony J. Fellera, Daniel W. Springa * a Engineering Group, Inc., 20600 Chagrin a Heights, 44122, USA E G | The Equity Blvd., SuiteJ.1200, Shaker Edrissa Gassama , Michael F. P. Bifanoa, Anthony Feller , Daniel W. Springa * a

a

Abstract

2

E2G | The Equity Engineering Group, Inc., 20600 Chagrin Blvd., Suite 1200, Shaker Heights, 44122, USA

Abstract In the very high-cycle fatigue regime there is a well-known lack of data for the assessment of welded joints. This lack

of data has led to Codes and Standards with a wide range of recommendations. Most of these recommendations are In the very regime there is a well-known of data the assessment joints. This justified byhigh-cycle experiencefatigue and lack a well-documented technicallack basis. The for difficulty in testingofatwelded very high cycles is lack that of led to Codes andlong Standards with a wide of recommendations. Most these thedata testshas currently take too to be practical andrange are often suspended between 10 6ofand 107recommendations cycles. However,are in justified by experience and lack well-documented technical basis. difficulty in testing at veryofhigh cycles is that piping vibration applications theasystem may undergo more than 10 8The cycles per year. Since much the experimental 7 cycles. in the currently take too long to be practicalorand are oftendata, suspended between 10 6 andto10 datatests in this very high-cycle regime is run-out “censored” we explore approaches extract theHowever, information 8 cycles per year. Since much of the experimental piping vibration applications the system may undergo more than 10 contained in such data and infer the behaviour of the Master S-N curve in the very high-cycle regime. This paper data in this very high-cycle regime run-out data or “censored” wefor explore approaches to extract the information presents an approach to account for is censored to provide data, a basis extending the Master S-N curve to the very contained in such data and infer the behaviour of the Master S-N curve in the very high-cycle regime. This paper high-cycle fatigue regime for welded joints. presents an approach to account for censored data to provide a basis for extending the Master S-N curve to the very high-cycle fatigue regime for welded joints. © 2019 The Authors. Published by Elsevier B.V. © 2019 The Authors. Published by Elsevier B.V. Peer-reviewunder under responsibility of Fatigue the Fatigue Design 2019 Organizers. Peer-review responsibility of the Design 2019 Organizers. © 2019 The Authors. Published by Elsevier B.V. Peer-review underLife; responsibility of the Fatigue 2019 Organizers. Keywords: Fatigue Master S-N Curve; CensoredDesign Data; Very High Cycle Fatigue, Vibration Keywords: Fatigue Life; Master S-N Curve; Censored Data; Very High Cycle Fatigue, Vibration

1. Introduction 1. Introduction The Master S-N curve has long been used as the basis for determining the resistance of welded components to fatigue damage, and still serves as the state-of-the-art. For example, in the 2016 release of API 579-1/ASME FFS-1 The579), Master S-Npart curve has to long been damage used as was the basis for determining theMaster resistance welded to (API a new related fatigue introduced wherein the S-N of curve was components selected as the fatigue damage, and still serves as the state-of-the-art. For example, in the 2016 release of API 579-1/ASME FFS-1 (API 579), a new part related to fatigue damage was introduced wherein the Master S-N curve was selected as the * Corresponding author. Tel.: +1-216-658-4752. E-mail address: [email protected] * Corresponding author. Tel.: +1-216-658-4752. E-mail address: [email protected] 2452-3216 © 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Fatigue Design 2019 Organizers. 2452-3216 © 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Fatigue Design 2019 Organizers.

2452-3216 © 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Fatigue Design 2019 Organizers. 10.1016/j.prostr.2019.12.077

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recommended approach for determining the life of welded joints (API 579-1/ASME FFS-1 (2016)). In an upcoming release of API 579, a new part will be introduced that is specific to the assessment of piping vibrations (Breaux et al. (2016), Bifano et al. (2018)). The fatigue criteria in the 2016 release of API 579 ensures adequate protection against high cycle fatigue in the design stage, however, when applied to in-service equipment (Fitness-For-Service) the welded fatigue methodology can be overly conservative for vibrating equipment. For vibration applications, welded components have commonly been observed to successfully operate in service for several years at high frequencies (e.g., frequencies greater than 1 Hz), surviving for more than 109 cycles without failure. When assessing these components using the Master S-N curve, API-579 currently recommends extending of the curve into the high cycle regime, predicting lives much less than that observed in practice. The set of data used to fit the Master S-N curve primarily consists of cases where failure was observed in less than 107 cycles. Few data points in the set have lives beyond 107 cycles, and no data points have lives beyond 108 cycles. As a result, the utilization of the Master S-N curve in the very high cycle fatigue (VHCF) regime can lead to overlyconservative risk evaluations and costly mitigation measures. . Indeed, with the lack of available data in the VCHF regime, the true behavior is an unknown. As a result of the uncertainty related to fatigue life predictions in this regime, existing Codes and Standards differ in their recommendations. A very common approach, taken by many, is to prescribe a bilinear curve, accounting for longer lives in the VHCF regime by reducing the slope on the curve (PD 5500 (2006), BS7608 (2014), DNV-RP-C203 (2011), ASME B31.3 (2018), Nureau Veritas NT 3199 (2013), Hobbacher (2016)). While not always the case, often the bilinear curve is reserved for situations of variable amplitude loading, while endurance limits are implemented in cases of constant amplitude loading (BS7608 (2014), DNV-RPC203 (2011)). While adopting a bilinear curve to capture longer lives is a popular approach, there remains a debate as to what the second slope should be, and where the intersection of the two slopes should lie. In reality, it is not expected that there would be an abrupt kink in the fatigue life curve. Moreover, the transition from one slope to another introduces difficulties when trying to apply these bilinear curves in probabilistic approaches. The Master S-N curve, using equivalent structural stress as an index, has made tremendous strides to condense a wide range of fatigue tests to a narrow range, however, the variation in fatigue life at any level of equivalent structural stress can still span an order of magnitude. As a result of this inherent uncertainty, fatigue life predictions are often performed using probabilistic approaches (e.g. Monte Carlo). When performing a probabilistic analysis, any kink in a bilinear curve introduces a discontinuity in the standard deviation at this point. This discontinuity introduces either subjectivity or errors into the analysis, thus promoting the need for a smooth, continuous fit to the data. This paper explores one method of doing just that, in a statistically consistent manner, by incorporating the information stored within run-out test data. When performing an experimental fatigue test, it is common practice to stop the test after a fixed number of cycles. This is done for several reasons, but often it is for the practical purpose of reducing the time and costs associated with performing the test. The data points from tests that are stopped (run-outs) are often either ignored when fitting fatigue data or included as though they were points of failure. When either of these approaches is taken, the practitioner has, often unknowingly, introduced their own bias into the results and both approaches result in conservative estimates of fatigue life in the VHCF regime. Incorporating run-out or “censored” data (i.e., data in which the test was stopped so the true life is unknown) into a statistical analysis is not a novel approach (see, for example, Schmee and Hahn (1979), Chatterjee and McLeish (1981)). One of the most common approaches for performing a statistical fit of data containing censored lives is to use the maximum likelihood estimate approach. However, for developing standardized procedures that can be performed by all levels of practitioners, a simplified approach will be discussed herein that yields significantly the same results as the maximum likelihood estimate approach (Schmee and Hahn (1979)). In this paper, we explore one way in which run-out data points may be incorporated into a linear least-squares regression procedure, lessening the bias introduced by ignoring them or treating them as actual failures. Section 2, outlines the data fitting procedure, and discuss how the run-out data is included in the fit. In Section 3, the data collection procedure for refitting the Master S-N curve is described. Section 4 demonstrates how censored data can be included to develop a modified Master S-N curve and discusses the influence of the new fit on the predicted lives of components in the VHCF regime. Finally, Section 5 provides concluding remarks.



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2. Data Fitting In this section, we discuss the iterative procedure used to perform linear regression with censored data points. We first give a brief overview of least-squares regression and discuss the modifications needed to account for censored data points. Finally, we walk through the step-by-step procedure for implementation. 2.1. Least-squares regression Linear least-squares regression is often used to generate estimates and other statistics when fitting a set of data. For example, the standard linear model to predict a set of values, y , can generally be written as: $ y = XT

(1)





i 1 2 m where, $ y is the array of predicted values for a set of input values, X = x ( ) , x ( ) ,K , x ( ) , x ( ) ¡ n are arrays n containing the input values (or functions of the inputs),  ¡ is an array containing the parameters of the model, m is the number of observations being fit and n is the number of parameters in the model. For example, for a twoparameter model, the array of predicted values is evaluated as:

 1 x11 +  2 x12   x + x  $ y =  1 21 2 22    M   1 xm1 +  2 xm2 

(2)

The objective for least-squares regression is to find the values of  that yield the best fit to the observed data. Explicitly, least-squares regression aims to find the parameters of the model that minimize the L2 norm of the difference between the observed and predicted values:







f ( = y = min y − X T ) min y − $ 

2



2



(3)

Now, consider a set of data containing both uncensored and censored values. We can separate the input data as: y =  y1 , y2  and X = X1 , X2 

(4)

where subscript 1 corresponds to uncensored observations and subscript 2 corresponds to censored observations. The least-squares regression including censored data points can then be written as:



f ( ) = min y1 − X1T + E ( y2 ) − X2T 

2

2



(5)

where E ( y2 ) corresponds to the expected values of the censored data points. From an intuitive standpoint, if the tests that resulted in the censored data were allowed to run to failure, the true observed lives would be greater than the censored lives. Thus, we can write the expected value of the actual lives for the censored tests given that the lives are longer than the censored values as: E ( y y > y2= ) y2 + g ( )

(6)

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for some function g ( ) . Fundamentally, the idea behind fitting models that contain censored observations is to replace the censored points with their expected values. There have been many different forms of g ( ) discussed in the literature, however, one of the more prominent forms, and that selected in this study, follows from the use of the lower truncated normal expectation model (Schmee and Hahn (1979), Chatterjee and McLeish (1981)). In the lower truncated normal expectation model:

g (=  ) XT2  n −1 − y2 +  n −1

 (z)

1−  ( z)

(7)

where the superscript n − 1 indicates parameters from the previous iteration (see Section 2.2) and:

z=

y2 − X 2T

(8)



The probability density function (pdf) and cumulative distribution function (cdf) for the standard normal distribution are as follows:

=  (z)

1 exp ( − X22 2 ) 2 y2

(z) =   ( z ) dz

(9)

(10)

−

and  2 is the variance:

y) ( y − $ =

2

2

m−n

(11)

2.2. Solution procedure Hahn and Schmee proposed an iterative method to estimate the parameters of a linear model with censored data (Schmee and Hahn (1979)). In their method, the iterative solution procedure is described as follows: • STEP 1: Generate a least-squares fit of the data, including the run-out data, by treating the run-out data as if they failed at their censored times. • STEP 2: Substitute the estimated parameters and variance into Equation (6) to get an estimated failure time for the censored data. • STEP 3: Use the estimated lives from STEP 2, in place of the run-out points, and obtain a new least-squares fit of the data using Equation (5). • STEP 4: Repeat STEP 2 and STEP 3 until convergence of the fit is achieved. Hahn and Schmee recommend that convergence be based on the slope and intercept estimates agreeing to 3 decimal places on two consecutive iterations. 2.3. Master S-N fatigue curve The form of the Master S-N curve presented in the 2016 edition of API 579 is as follows:



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S = 1  N − h

715 5

(12)

where S is the alternating stress range and N is the number of cycles. In this study, to account for nonlinearity, an additional parameter is added to the model, yielding the following form: S = 1  N − h +  2

(13)

To solve the above nonlinear equation using linear least-squares regression, a two-step approach was taken, wherein the value of h was first determined by taking the logarithm of Equation (12) and performing least-squares regression on the dataset absent any run-out points. The resulting solution for h was then substituted into Equation (13) and the dataset containing run-out points was fit using the procedure outline in Section 2.2. 3. Data Collection The dataset used to fit the Master S-N curve in API 579 was extracted from reliable sources, with comprehensively documented experimental procedures (some of which include: Andrews (1996), Baptista et al. (2008), Bell and Vosikovsky (1993), Bell et al. (1989), Booth and Wylde (1990), Buitrago et al. (2003), ECSC (1975), Gioielli and Zettlemoyer (2007), Gioielli and Zettlemoyer (2008), Hechmet and Kuhn (1998), Hong (2010), Huo et al. (2005), Kang et al. (2011), Kang and Kim (2003), Kang et al. (2001), Kassner et al. (2010), Kihl and Sarkani (1997), Kihl and Sarkani (1998), Kim and Kang (2008), Kim and Yamanda (2005), Kim et al. (2006), Kim and Lotsberg (2005), Kirkhope et al. (1999), Kitsunai et al. (1998), Koenigs et al. (2003), Lindqvist (2002), Lotsberg (2009), Martinez and Blom (1997), Maddox (1982), Markl and George (1960), Mashiri et al. (2002), Mashiri et al. (2004), Mori et al. (2012), Mori et al. (2015), Ohta and Kudo (1980a), Ohta and Kudo (1980b), Okawa et al. (2013), Pook (1982), Rörup and Petershagen (2000), Sørenon et al. (2006), Spadea and Frank (2002), Tai and Miki (2014), Togasaki et al. (2010), Ting et al. (2009), Xiao and Yamada (2004), Xiao and Yamada (2005), Yagi and Tomita (1991), Yagi et al. (1991), Yagi et al. (1993), Yee et al. (1990)). However, run-out information was not originally published in the Master S-N Curve background documentation. In this study, we reviewed the same sources, both to verify the accuracy of the database and to document the run-out points that were left out of the original documentation. Some sources were not able to be retrieved as they were proprietary, internal reports which were not publicly accessible at the time of this publication. In total, approximately 1291 failure data points and 64 run-out data points were used in this analysis. 4. Results and Discussion The procedure outlined in Section 2.2 is applied to the collected fatigue data with the form of the model expressed in Equation (13). The converged mean values for the parameters in the model are as follows: = S 20475.8  N

−1

3.079355

+ 15.48078

(14)

The results of the fitting procedure are illustrated in Figure 1. The censored lives for the run-out points are shown in orange, and the estimated lives shown in green. From the inclusion of censored points alone, there is some indication that the lives at lower stresses deviate from the assumption of linearity, this becomes amplified when the lives at these stresses are estimated using the iterative solution procedure in 2.2. Interestingly, the nonlinear fit demonstrates good agreement with the bilinear curve with a slope of m = 5 after 107 cycles, a common approximation in various codes and standards. The lives predicted from the three forms of the curve (single slope, bilinear slope, and those from Equation (14)) are summarized for a select number of cases in Table 1.

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Table 1. Predicted lives for various levels of equivalent structural stress commonly found in VHCF applications  MPa Equivalent Structural Stress  2−mss   mm 2 mss

   

Single Slope

Bilinear Slope

Nonlinear Fit

( m = 3)

(m =5)

Equation (14)

114.45

1.00E7

1.00E7

1.35E7

100

1.50E7

1.96E7

2.20E7

75

3.55E7

8.28E7

6.47E7

50

1.20E8

6.28E8

3.46E8

25

9.59E8

2.0E10

1.83E10

Fig. 1. Predicted curve using the model in Equation (13) when including run-outs in the Master S-N dataset.

5. Concluding Remarks In this study, much of the dataset used to fit the Master S-N curve was investigated and many of the original sources were reviewed. The points within the dataset were verified and the run-out data points from the same sources were added (but flagged as being run-outs). An iterative least-squares approach was presented that statistically accounts for the effects of censored data. The data (with run-outs included) was then fit, and the resulting curve was compared to the single-slope Master S-N curve and a frequently recommended bilinear curve, often used for evaluating variable amplitude loading (BS7608 (2014), DNV-RP-C203 (2011)), wherein the slope is set to a value of m = 5 beyond 107 cycles. The nonlinear fit shows good agreement with the bilinear curve, however, the continuous nature of the nonlinear fit eliminates the ambiguities present in the bilinear curve (e.g. the slope is not prescribed and there is no need for an arbitrary intersection point to be selected). From an implementation perspective, the continuous curve is preferred over the bilinear curve as it avoids any discontinuities in the standard deviation when applying probabilistic approaches. The present work technically supports use of the bilinear curve and directionally improves the popular Master SN method to closer align with vibration fatigue legacy methods. Many industry recognized piping vibration assessment techniques either implement endurance limits or use piping vibration screening criteria that inherently incorporate assumptions of infinite life (Wachel (1990), Energy Institute (2008)). Further experimental data at stresses below 100 MPa ( ( 2 − mss ) 2mss ) along with the methods presented herein to incorporate runout data will provide further needed improvements to the Master S-N method.



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