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Assessment of industrial components in high temperature plant using the “ALIAS-HIDA” – A case study
Engineering Failure Analysis 13 (2006) 767–779 www.elsevier.com/locate/engfailanal
Assessment of industrial components in high temperature plant using the ‘‘ALIAS-HIDA’’ – A case study H. Deschanels a,*, C. Escaravage a, J.M. Thiry a, N. Le Mat Hamata b, D. Colantoni c a
AREVA/Framatome-ANP, 10 Rue J. Re´camier, 69456 Lyon, Cedex 06, France b ETD Ltd., 2 Warwick Gardens, Ashtead, Surrey KT21 2HR, UK c MPA, Pfaffenwaldring 32, D-70569 Stuttgart, Germany Received 31 January 2005; accepted 20 February 2005 Available online 17 May 2005
Abstract Prediction of the residual life of high temperature components is becoming a major issue. Key decisions, such as whether to repair or replace cracked components, have to be based on precise information. The cooperative program, HIDA Applicability, is aimed at improving these issues. This paper deals with the numerical application of probabilistic assessment to crack growth analysis. Based on the present studies, it has been shown that it is both possible and practical to use stratified samples to obtain increased accuracy or to reduce computation time. Stratified samples are particularly suitable when the desired probability of failure is very low. The ALIAS-HIDA software has been designed to calculate the probability of failure of cracked components under high temperature conditions. The Monte Carlo simulation technique was implemented successfully and the first tests seem to show it to be reliable. The ability of the software to deal with such calculations, which require a large number of random variables and simulations, was illustrated by a Creep Crack Growth calculation example. The HIDA test programme has provided numerous results, which have been supplemented by data from published and unpublished sources, and gathered in the HIDA database. Some information from the HIDA Database has been re-organized and analyzed in order to conduct probabilistic analysis. This task is still in progress, however, some results are available, and trends and technical issues are described for a 316 austenitic stainless steel. 2005 Elsevier Ltd. All rights reserved. Keywords: Probabilistic analysis; High temperature; Fracture mechanics; Creep Crack growth; Structural integrity
*
Corresponding author. Tel.: +33 4 72 74 72 18; fax: +33 4 72 74 73 25. E-mail address: [email protected] (H. Deschanels).
1350-6307/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfailanal.2005.02.017
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1. Introduction Assessing the residual life of high temperature components is of major importance. Key decisions such as the repair or replacement of cracked or damaged components have to be based on accurate information. The cooperative program, HIDA Applicability [1], is aimed at improving these technical issues. The paper deals with the numerical applications of probabilistic assessment to crack growth analysis.
2. Methods for probabilistic evaluation of failure 2.1. Presentation An iterative Monte Carlo simulation procedure is recommended for probabilistic fracture mechanics [2]. This is due to the rapid development of computational facilities and the simplicity of Monte Carlo simulation. In order to reduce the large number of simulations required by this technique, the use of the stratified sampling is also recommended. Paper [3] deals with the methodology and approaches used for the project that will be implemented and validated in the ALIAS-HIDA software. The fracture mechanics methods were specified in reference [4] and are mainly based on procedures from references [5,6]. 2.2. Monte Carlo simulation technique Within a given distribution function, a variable is selected at random. Usually, we randomly select a number in the range from 0 to 1, and the random variable is calculated as shown in the Fig. 1. We randomly select all of the variables of the probability space. An iterative Monte Carlo procedure has been implemented in the code ALIAS-HIDA [7], and have specified a maximum simulation number N = 104. This method is practical and suitable for moderate and low values of desired probability of failure, but it is not the best approach for very low probabilities, as the specified maximum number of simulations would not be sufficient. 2.3. Monte Carlo simulation using stratified samples When many different physical variables are considered to be random and since the simulation number remains limited, the Monte Carlo procedure becomes less accurate. Special sampling techniques have been
Fig. 1. Drawing lots a random variable with respect to the distribution function.
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developed to reduce statistical error or the time to carry out the simulations (see [8–11]). However, stratified sampling might lead to less accuracy on mean value estimation. One of these methods is illustrated by means of a calculational example (Section 5). The simulations are applied throughout the probability space but a larger number of simulations are concentrated on extreme values. This approach leads to an improved treatment of the tail of the distribution. In fact, when using the basic Monte Carlo method, most of the simulation is made with variable values close to their mean, and a very large sample is necessary in order to contain extreme values. Reducing the number of simulations can be achieved by using samples containing more values at low probability. The principle of the use of a stratified sample is to transform the uniform [0, 1] distributed random numbers by means of a continuous function, which promotes extreme values. So the new distribution could, for instance, have most of its values between 0 and 0.1 and between 0.9 and 1.
3. Random variables The most influential parameters are selected from the Sensitivity Analysis that was carried out in order to select the random variables to be considered [11]. These parameters can be divided into three kinds of data: Defects: Flaw sizing capabilities of NDE techniques are discussed in [12]. This type of data is linked to size distribution, existence frequency, and location. The probability of existence of embedded cracks is not well known as very little information is available on the distribution of the distance between the crack and the nearest surface [2]. Thus, only surface cracks are analysed in numerical calculations (c.f. Sections 5 and 6). Service conditions: These are temperature and load. Components fitted with an on-line monitoring system would be suitable for probabilistic applications, by measuring the variation in the strain level or the temperature variation. However, the stress data used in flaw assessment studies are generally obtained from design stress reports and are conservative. Material properties: This data will be provided from the HIDA database, after re-organisation and adaptation for probabilistic analysis. An example of the statistical treatment of data is shown for a 316 stainless steel in Section 4.2.
4. Statistical treatment of the data 4.1. Presentation Seven types of distribution function are available in the ALIAS-HIDA software (normal, lognormal, exponential, Weibull 2 parameters, Weibull 3 parameters, triangular and uniform) [7]. The normal and log-normal distributions are the most commonly used distributions for assessing Probability of Failure (Pof) [2]. One example of the statistical treatment of material data is described by Section 4.2. 4.2. Example for a 316 stainless steel 4.2.1. Presentation The creep crack growth (CCG) characteristics gathered from the HIDA database [13] are examined.
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4.2.2. (da/dt C*) correlation The creep crack growth law is formulated as: da da / ¼ DðC Þ or log10 ¼ /log10 ðC Þ þ log10 ðDÞ. dt dt
ð1Þ
It should be noted that C* is not a pure loading parameter. The C* parameter depends on the loading and the creep strain response of the material. The probabilistic aspect of the C* evaluation will be treated in Section 4.2.3. As the influence of temperature is taken into account through the creep strain response of the material, the correlation (da/dt C*) can be considered as independent of temperature. With these restrictions on the parameter C*, the probabilistic treatment of creep crack growth is restricted to the analysis of the linear regression of log10 (da/dt) against log10 (C*), which produces an average line with best fit values for / and D. To represent the scatter in the results, two curves are characterised which illustrate the confidence level accounting for the number of data. According to past experience (da/dt C*) data are not very different in weld metal from those in parent material. 4.2.3. Probabilistic aspect of C* evaluation As C* is a mixed parameter, its evaluation always requires some form of creep strain law. Creep strain is a relatively complex function of stress, time and temperature. In the past, investigation of this law was limited to one heat and one product. When a large amount of data concerning different products is considered, the practice in the past was to determine the best-fit values of the creep strain law parameters. There is no past experience of any statistical analysis of the creep strain results from a large database. However, the scatter band of creep strain data from product to product and batch to batch is known to be important and to have a considerable influence the C* parameter. 4.2.4. Data treatment The only tests taken into account were those that showed a domain where da/dt increases as C* increases. The number of tests considered at different temperatures, from 510 to 800 C, is indicated in Table 1. The 74 tests in parent material cover the temperature range 550–800 C. There are 10 tests in Heat Affected Zone material limited to 510, 525C and 560 C and 2 tests in weld metal at 600 C. The parent metal data, gathered in the HIDA database are shown in Fig. 2. For the probabilistic use of data, only that part of the table, which indicates increasing values of da/dt as C* increases for each individual test has been taken into account. Considering base material data, the results obtained at the different temperatures do not produce a common regression line. Table 2 indicates the different values found for / and D at the different temperatures. At 700, 750 and 800 C the values of / fitted to the data at each temperature are higher (0.8 < / < 1.04) than at lower temperature (0.68 < / < 0.76). There is no clear trend in the variation of / with temperature in the low temperature domain (525–650 C). Thus, it was not possible to define a relationship for the variation of / against temperature. However, for moderate temperatures (550–650 C), values are around 0.7 and for high temperature (700–800 C) the values are between 0.8 and 1. Table 1 Number of creep crack growth tests in HIDA database at the different temperatures Temperature (C)
Parent metal HAZ Weld metal
510
525
550
560
585
600
625
650
700
750
800
– 2 –
– 4 –
32 – –
– 4 –
2 – –
14 – 2
4 – –
12 – –
3 – –
3 – –
4 – –
da/dt (mm/h)
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CCG data for base material 316 - 550˚C 316 -650˚C 316 - 600˚C 316 - 700˚C 316 - 750˚C 316 -800˚C
1 0.1
0.01 0.001 2
Log 10 [C * (J/m .h)]
0.0001 0
1
2
3
4
5
6
Fig. 2. Data gathered in the Hida database. Table 2 CCG law – parent metal – da/dt = D(C*)/ T (C)
550
600
625
650
700
750
800
5
9.1 0.74
16.9 0.7
3.95 0.752
8.03 0.684
0.359 0.960
0.923 1.039
0.978 0.806
10 /
D
Finally, we propose a linear regression of da/dt versus C*, including all the data on base material from 550 to 800 C. The fitted values are: da ¼ 1.19 104 ðC Þ0.687 ; dt
ð2Þ
¼ 345 J=m2 h; where C* is in J/m2 h1 and da/dt in mm/h. This curve contains the mean point (C 3 da/dt = 6.58 · 10 mm/h). The distribution parameters are in log-normal space, the residual variance of the data around this line is equal to 0.103. This means that around the mean point, the creep crack growth rate da/dt has a logarithmic statistical distribution with an average value equal to log10 (6.58 · 103) = 2.182 and a standard deviation equal 0.321. The 90% confidence limits are shown in Fig. 2 (corresponding to 5% of expected data lower than the lower curve and 5% of expected data higher than the higher curve). For C* values far from the average point, the logarithmic distributions of da/dt is wider, as shown in Fig. 3, particularly for da/dt corresponding to low C* values (610 J/m2 h). This is a normal situation in the proper statistical treatment of data by linear regression to establish the confidence limits of a variable (log10 (da/dt) in this case), for different values of the one parameter (C*). As a result, the creep crack growth rate in the weld (Fig. 4), or in the HAZ, is of the same order of magnitude as in the base metal. 4.2.5. Requirement for statistical analysis It is statistically meaningless to process all the gathered data points when there is a large difference in the number of points between the different tests. Usually a statistical analysis starts with a database obtained by random sampling from the complete population to be studied. To date, no database of creep crack growth rates at random temperature exists. Nevertheless, some sources of error in the statistical analysis can be avoided: the number of tests should be similar at the different temperatures for the region to be covered, a similar number of data points, more or less equally spaced, should be retained from each test,
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da/dt (mm/h)
1000 90% confidence limit for CCGrate da/dt
100 10 1
316 -550˚C
316 - 650˚C
316 -600˚C
316 -700˚C
316 - 750˚C
316 -800˚C
Average curve
-5% confidence limit
+5% confidence limit
0.1 0.01 0.001 0.0001 0.00001
2
Log 10 [C* (J/m .h)]
0.000001 0
1
2
3
4
5
6
Fig. 3. 90% confidence limit for CCG rate da/dt.
da/dt (mm/h)
10 1 0.1
316 - 550˚C
316 - 650˚C 316 - 600˚C
Weld -600˚C
0.01 0.001 Log 10[C* (J/m2.h)]
0.0001 0
1
2
3
4
5
6
Fig. 4. Comparison weld metal/Parent metal.
alternatively a weighting should be attributed to the data of the different tests in order to compensate for the difference in the number of recorded data points from one test to another. A difficult problem is the definition of a rule for selecting, for each test, the first data point to be taken into account. In many of the tests, there is a trend of whereby low creep crack growth rates are obtained just after initiation of the crack growth. In many cases, this leads to an overestimation of the slope /. 4.3. Discussion of statistical analysis The data set required for rigorous probabilistic analysis of the lifetime of cracked components at high temperature is not available at the present. The absence of relevant data is directly linked to the fact that, when data are gathered, they usually focus on the pessimistic bound (deterministic approach), for example the material properties [14]. The methodology to be used for gathering relevant data, the reorganisation of existing databases and the development of a statistical treatment represent one of the major tasks within the HIDA Applicability project [1].
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5. FCG simplified numerical calculation 5.1. Presentation Reference [16] deals with the assessment of crack growth in pipes under fatigue conditions using the ALIAS-HIDA software. In the framework of the development of probabilistic tools, the aim of these studies is to highlight the main characteristics, strengths and weaknesses of two approaches based on Monte Carlo simulation techniques (see Section 2). From the initial size, crack growth is calculated up to 0.7 times the thickness of the component. The number of cycles necessary to reach this depth is then considered as a random variable. The goal of the study is to predict the distribution of this variable and finally to compare the results obtained by the two approaches (see Section 5.4). Calculations using the stratified samples are described in detail (see Section 5.3). 5.2. Main features and data The chosen example involves a circumferential semi-elliptical crack located in a straight pipe made of 316 L (N) austenitic stainless steel, with a wall thickness of 7 mm. Only the depth is taken into account, with the initial crack depth taken as 1.75 mm and the crack length considered to be constant (15 mm). The applied cyclic load results in through wall thickness membrane stresses of rmax = 77 MPa and rmin = 36 MPa. Growth is assumed to follow a ‘‘Paris law’’, which is defined as: da a ¼ C ðDK Þ ; dN
ð3Þ
where C = 108 and a = 3.3 (K is given in MPa m0.5, a in mm and da/dN in mm/cycles). DKeff is used instead of DK in order to take the R ratio effect (R = rmin/rmax) into account. The correction factor q, defined by DKeff = qDK, is given in the A16 procedure [6]. The stress intensity factor formula is given in reference [15]. Only two random variables, growth rate and load were chosen. C is taken as a random variable with a log-normal distribution function with a mean value of the logarithm of 18.42 and a standard deviation of the logarithm of 0.54. For the load, a normal distribution function is defined (see Table 3). 5.3. Monte Carlo simulation using stratified sample The transformation function bk used is defined on 2k + 1 points distributed between 0 and 1 as: 0, 10k+1, . . . 0.5, . . . 1–10k+1, 1. For example, the values of the function bk is given for k = 2 and k = 4 (see Table 4). One can see that the middle point 0.5 is kept unchanged. After transforming the uniform [0, 1] distribution random numbers through the sampling function bk, most of the random numbers are concentrated at the extreme values of the interval [0, 1]. Generally speaking, to each of the initial numbers, a, we associate a pair of numbers (a, da), where da is a weighting defined by the derivative of function b at point a.
Table 3 Load data – statistical parameters of normal distribution function Service conditions load
Mean value
Standard deviation
Maximum rmax (MPa) Minimum rmin (MPa)
77 36
10.3 4.8
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Table 4 Values of the bk function x
b(x)
k=4 0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1
0 0.001 0.01 0.1 0.5 0.9 0.99 0.999 1
k=2 0 0.25 0.5 0.75 1
0 0.1 0.5 0.9 1
For the case study, only two random samples are defined a and A, and transformed by the same function b, giving two sets of pairs (ba, da) and (bA, dA), where ba are the values of b(a) and da are the weighting. For the coefficient of the ParisÕs law we consider the distribution of Log C to be Gaussian. In fact we use the random variable kC which is defined by: C ¼ k C C nom .
ð4Þ
For the load, the multiplying coefficient kr is considered to be a random variable, having a Gaussian distribution: r ¼ k r rnom .
ð5Þ
Using the repartition functions of Kr and KC, the samples ka, kA, are defined. A parametric study was conducted using 100 values of each variable. Fig. 5 shows the number of cycles for which 70% of the thickness was reached with less than ‘‘x times N’’ simulations, (in other words x can be considered as the probability of failure (Pof) for N cycles). The stratified samples with k = 4 gave accurate results with a precision of 107, while the stratified sample with k = 2 is limited to an accuracy higher than 104. When stratified samples are used, the higher the value of k, the smaller the weighting used in the simulation. For example when k equals 4 the weighting is 102, and when three variables are used the simulation weighting decreases to 106. Handling stratified samples leads to very small numbers and requires special care to avoid numerical discrepancies. 5.4. Discussion of FCG numerical calculation Using 500 randomly selected values of each variable we obtain, through the non stratified sampling calculation, a mean value of 59.5 · 103 cycles and a standard deviation of 32 · 103 cycles. Fig. 6 shows the comparison of the results for stratified samples and non stratified samples for the maximum number of simulations specified in the ALIAS-HIDA software (104). As shown in the Fig., the stratified method is not necessary in our case for a Pof of 103, but for a Pof higher than 104 the stratified method gives considerably more accurate results. The present studies show that it is possible to use stratified samples to obtain greater accuracy or to reduce computation time. Stratified samples are required when the desired accuracy is lower than the inverse of the number of simulations.
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Ncycles 30000 stratified k=8 stratified k=7 stratified k=6 stratified k=5 stratified k=4 stratified k=3 stratified k=2 non stratified k=1
25000 20000 15000 10000 5000
PoF x 1.E-07
1.E-06
1.E-05
1.E-04
1.E-03
0 1.E-02
Fig. 5. Effect of stratification sampling – N = 100 · 100.
N cycles 25000 stratified (k=2) 20000
stratified (k=4) non stratified
15000 10000 5000 PoF x 1.E-06
1.E-05
1.E-04
1.E-03
0 1.E-02
Fig. 6. Comparison of non stratified and stratified samples calculations – N = 100 · 100.
6. CCG numerical calculation 6.1. Presentation At the time of writing this paper, further facilities such as stratified sample technique are being implemented and verified. Therefore, this section is a case study of probabilistic analysis using the ALIAS-HIDA software [7]. The component was a straight pipe made of 316 L(N) austenitic stainless steel which was submitted to mechanical loading at 550 C. The surface crack considered was located in the inner wall of the straight pipe. The probability space is described by 8 random variables, deterministic values being used for other parameters.
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6.2. Main features and data Constant values were taken for some variables, such as: Geometrical characteristics: internal radius 250 mm and the crack length 40 mm. Service conditions: temperature is 550 C. Material property: The Young modulus, E = 149,000 MPa. For the fracture toughness, the value considered in the case study is very low, KIC = 30 MPa m0.5. The data considered as random variables are summed-up in Tables 5 and 6. Although 316 L austenitic stainless steel exhibits a large primary creep stage, only secondary creep was taken into account for this case study of C* evaluation. Generally speaking, only extreme values are used in deterministic (pessimistic) analysis. For material characteristics, extreme and mean values are available, as given in [17]. Thus, assuming that extreme values correspond to the probabilistic distribution function at three standard deviations, we obtained an approximate function: geometrical: the normal distribution function is used for crack depth and thickness, service conditions: Only mechanical load is considered with temperature variations being investigated later. The variation in mechanical properties was taken into account through relevant links with the material Database [13]. material: an approximate CCG distribution function was characterized from data given by reference [6]. The variation in the creep strain rate law (see Section 4.2.3), and in creep rupture data is difficult to characterize. Thus, for this case study, we considered only the secondary creep stage and handled the variation through a multiplication coefficient. For mechanical properties such as the yield stress and the ultimate stress, minimum and mean values were provided by [17].
6.3. Calculation using the ALIAS-HIDA software The Monte Carlo based simulation technique was used to evaluate the probability of failure. The calculations take into account the failure mechanisms. The failure is considered when either: Table 5 Material data – statistical parameters of log-normal distribution function Material characteristics coefficient
Mean value of logarithm
Logarithmic standard deviation
CCG D (MPa m h1; mm/h) Creep strain rate As (MPa; h1) Creep damage B (MPa; h)
3.08 30.31 57.11
0.77 0.23 0.47
Table 6 Data – statistical parameters of normal distribution function Other characteristics
Mean value
Standard deviation
Ultimate stress ru (MPa) Yield stress ry (MPa) Bending moment M (kN m) Depth a (mm) Thickness t (mm)
415.89 136.96 280 2 10
15.42 12.78 9.33 0.17 0.33
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the crack size reaches 80% the wall thickness, or a fast fracture criterion is exceeded, or a time-dependent failure criterion is exceeded (creep rupture).
Crack depth [mm]
The results obtained with the base model, i.e., using mean values, are shown in Figs. 7–9. It can be seen that the creep damage accumulation in the ligament is moderate and fast fracture occurs (due to low fracture toughness), when the crack depth is about 4 mm (Da 2 mm). After propagation the crack size represents 40% of the wall thickness. The main result is the probability of failure (Pof). Fig. 10 shows the evolution of the Pof depending on the time for 3000 simulations. As a result, the remaining life is 1420 h for a Pof of 0.33%, with an accuracy of about 30% in the estimate of the Pof. For this case study (numerical example), it turns out that the fast fracture criterion is more relevant than the creep rupture and maximum crack size criteria (Figs. 7–9). Thus, the results describe a limited crack growth domain (20% of the wall thickness).
3.8 3.6 3.4 3.2 3 2.8 2.6 2.4 2.2
Time [h]
2
0
10000
20000
30000
40000
50000
Creep damage accumulation [-]
Fig. 7. Crack size evolution – base model (mean values).
0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02
Time [h]
0 0
10000
20000
30000
40000
Fig. 8. Ligament creep damage – base model (mean values).
50000
778
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Failure assessment diagram
1.2 1
Path -Depth Path -Surface Initial flaw - Depth
Kr [-]
0.8 0.6
Initial flaw - Surface Final flaw- Depth
0.4
Final flaw- Surface Assessment curve
0.2 0 0
0.5
1
Sr [-]
1.5
2
2.5
Fig. 9. Fast fracture – base model (mean values).
60 50 40
Probability [%]
70
30 20 10 0 100
Time [h] 1000
10000
100000
Fig. 10. Pof depending on the duration.
6.4. Discussion of CCG numerical calculation The calculation of the probability of failure for a cracked component at high temperature has been performed successfully using the ALIAS-HIDA software. Further studies could be carried out using the ALIAS-HIDA software under creep-fatigue conditions, taking the primary creep stage into account and using the data from the analysis and the statistical treatments, which are yet to be conducted (see Sections 3 and 4).
7. Conclusions Numerical calculations of crack growth have been conducted using the probabilistic Monte Carlo method. Based on the present studies, it has been shown that it is possible and practical to use stratified samples in
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order to obtain increased accuracy or to reduce computation time. Stratified samples are particularly suitable when the desired probability of failure is very low. The ALIAS-HIDA software has been designed to calculate the probability of failure of cracked components under high temperature. The Monte Carlo simulation technique was successfully implemented and the first steps of the verification seem to show it to be reliable. This was demonstrated by a calculational example which illustrates the ability of the software to deal with these calculations, which require a large number of random variables and simulations. In order to be able to conduct probabilistic analysis on cracked component subject to high temperatures, some of the data in the existing HIDA database has been re-organized and analyzed. This task is in progress but some results are available and trends and technical issues are outlined for a 316 austenitic stainless steel.
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Assessment of industrial components in high temperature plant using the ‘‘ALIAS-HIDA’’ – A case study H. Deschanels a,*, C. Escaravage a, J.M. Thiry a, N. Le Mat Hamata b, D. Colantoni c a
AREVA/Framatome-ANP, 10 Rue J. Re´camier, 69456 Lyon, Cedex 06, France b ETD Ltd., 2 Warwick Gardens, Ashtead, Surrey KT21 2HR, UK c MPA, Pfaffenwaldring 32, D-70569 Stuttgart, Germany Received 31 January 2005; accepted 20 February 2005 Available online 17 May 2005
Abstract Prediction of the residual life of high temperature components is becoming a major issue. Key decisions, such as whether to repair or replace cracked components, have to be based on precise information. The cooperative program, HIDA Applicability, is aimed at improving these issues. This paper deals with the numerical application of probabilistic assessment to crack growth analysis. Based on the present studies, it has been shown that it is both possible and practical to use stratified samples to obtain increased accuracy or to reduce computation time. Stratified samples are particularly suitable when the desired probability of failure is very low. The ALIAS-HIDA software has been designed to calculate the probability of failure of cracked components under high temperature conditions. The Monte Carlo simulation technique was implemented successfully and the first tests seem to show it to be reliable. The ability of the software to deal with such calculations, which require a large number of random variables and simulations, was illustrated by a Creep Crack Growth calculation example. The HIDA test programme has provided numerous results, which have been supplemented by data from published and unpublished sources, and gathered in the HIDA database. Some information from the HIDA Database has been re-organized and analyzed in order to conduct probabilistic analysis. This task is still in progress, however, some results are available, and trends and technical issues are described for a 316 austenitic stainless steel. 2005 Elsevier Ltd. All rights reserved. Keywords: Probabilistic analysis; High temperature; Fracture mechanics; Creep Crack growth; Structural integrity
*
Corresponding author. Tel.: +33 4 72 74 72 18; fax: +33 4 72 74 73 25. E-mail address: [email protected] (H. Deschanels).
1350-6307/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfailanal.2005.02.017
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1. Introduction Assessing the residual life of high temperature components is of major importance. Key decisions such as the repair or replacement of cracked or damaged components have to be based on accurate information. The cooperative program, HIDA Applicability [1], is aimed at improving these technical issues. The paper deals with the numerical applications of probabilistic assessment to crack growth analysis.
2. Methods for probabilistic evaluation of failure 2.1. Presentation An iterative Monte Carlo simulation procedure is recommended for probabilistic fracture mechanics [2]. This is due to the rapid development of computational facilities and the simplicity of Monte Carlo simulation. In order to reduce the large number of simulations required by this technique, the use of the stratified sampling is also recommended. Paper [3] deals with the methodology and approaches used for the project that will be implemented and validated in the ALIAS-HIDA software. The fracture mechanics methods were specified in reference [4] and are mainly based on procedures from references [5,6]. 2.2. Monte Carlo simulation technique Within a given distribution function, a variable is selected at random. Usually, we randomly select a number in the range from 0 to 1, and the random variable is calculated as shown in the Fig. 1. We randomly select all of the variables of the probability space. An iterative Monte Carlo procedure has been implemented in the code ALIAS-HIDA [7], and have specified a maximum simulation number N = 104. This method is practical and suitable for moderate and low values of desired probability of failure, but it is not the best approach for very low probabilities, as the specified maximum number of simulations would not be sufficient. 2.3. Monte Carlo simulation using stratified samples When many different physical variables are considered to be random and since the simulation number remains limited, the Monte Carlo procedure becomes less accurate. Special sampling techniques have been
Fig. 1. Drawing lots a random variable with respect to the distribution function.
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developed to reduce statistical error or the time to carry out the simulations (see [8–11]). However, stratified sampling might lead to less accuracy on mean value estimation. One of these methods is illustrated by means of a calculational example (Section 5). The simulations are applied throughout the probability space but a larger number of simulations are concentrated on extreme values. This approach leads to an improved treatment of the tail of the distribution. In fact, when using the basic Monte Carlo method, most of the simulation is made with variable values close to their mean, and a very large sample is necessary in order to contain extreme values. Reducing the number of simulations can be achieved by using samples containing more values at low probability. The principle of the use of a stratified sample is to transform the uniform [0, 1] distributed random numbers by means of a continuous function, which promotes extreme values. So the new distribution could, for instance, have most of its values between 0 and 0.1 and between 0.9 and 1.
3. Random variables The most influential parameters are selected from the Sensitivity Analysis that was carried out in order to select the random variables to be considered [11]. These parameters can be divided into three kinds of data: Defects: Flaw sizing capabilities of NDE techniques are discussed in [12]. This type of data is linked to size distribution, existence frequency, and location. The probability of existence of embedded cracks is not well known as very little information is available on the distribution of the distance between the crack and the nearest surface [2]. Thus, only surface cracks are analysed in numerical calculations (c.f. Sections 5 and 6). Service conditions: These are temperature and load. Components fitted with an on-line monitoring system would be suitable for probabilistic applications, by measuring the variation in the strain level or the temperature variation. However, the stress data used in flaw assessment studies are generally obtained from design stress reports and are conservative. Material properties: This data will be provided from the HIDA database, after re-organisation and adaptation for probabilistic analysis. An example of the statistical treatment of data is shown for a 316 stainless steel in Section 4.2.
4. Statistical treatment of the data 4.1. Presentation Seven types of distribution function are available in the ALIAS-HIDA software (normal, lognormal, exponential, Weibull 2 parameters, Weibull 3 parameters, triangular and uniform) [7]. The normal and log-normal distributions are the most commonly used distributions for assessing Probability of Failure (Pof) [2]. One example of the statistical treatment of material data is described by Section 4.2. 4.2. Example for a 316 stainless steel 4.2.1. Presentation The creep crack growth (CCG) characteristics gathered from the HIDA database [13] are examined.
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4.2.2. (da/dt C*) correlation The creep crack growth law is formulated as: da da / ¼ DðC Þ or log10 ¼ /log10 ðC Þ þ log10 ðDÞ. dt dt
ð1Þ
It should be noted that C* is not a pure loading parameter. The C* parameter depends on the loading and the creep strain response of the material. The probabilistic aspect of the C* evaluation will be treated in Section 4.2.3. As the influence of temperature is taken into account through the creep strain response of the material, the correlation (da/dt C*) can be considered as independent of temperature. With these restrictions on the parameter C*, the probabilistic treatment of creep crack growth is restricted to the analysis of the linear regression of log10 (da/dt) against log10 (C*), which produces an average line with best fit values for / and D. To represent the scatter in the results, two curves are characterised which illustrate the confidence level accounting for the number of data. According to past experience (da/dt C*) data are not very different in weld metal from those in parent material. 4.2.3. Probabilistic aspect of C* evaluation As C* is a mixed parameter, its evaluation always requires some form of creep strain law. Creep strain is a relatively complex function of stress, time and temperature. In the past, investigation of this law was limited to one heat and one product. When a large amount of data concerning different products is considered, the practice in the past was to determine the best-fit values of the creep strain law parameters. There is no past experience of any statistical analysis of the creep strain results from a large database. However, the scatter band of creep strain data from product to product and batch to batch is known to be important and to have a considerable influence the C* parameter. 4.2.4. Data treatment The only tests taken into account were those that showed a domain where da/dt increases as C* increases. The number of tests considered at different temperatures, from 510 to 800 C, is indicated in Table 1. The 74 tests in parent material cover the temperature range 550–800 C. There are 10 tests in Heat Affected Zone material limited to 510, 525C and 560 C and 2 tests in weld metal at 600 C. The parent metal data, gathered in the HIDA database are shown in Fig. 2. For the probabilistic use of data, only that part of the table, which indicates increasing values of da/dt as C* increases for each individual test has been taken into account. Considering base material data, the results obtained at the different temperatures do not produce a common regression line. Table 2 indicates the different values found for / and D at the different temperatures. At 700, 750 and 800 C the values of / fitted to the data at each temperature are higher (0.8 < / < 1.04) than at lower temperature (0.68 < / < 0.76). There is no clear trend in the variation of / with temperature in the low temperature domain (525–650 C). Thus, it was not possible to define a relationship for the variation of / against temperature. However, for moderate temperatures (550–650 C), values are around 0.7 and for high temperature (700–800 C) the values are between 0.8 and 1. Table 1 Number of creep crack growth tests in HIDA database at the different temperatures Temperature (C)
Parent metal HAZ Weld metal
510
525
550
560
585
600
625
650
700
750
800
– 2 –
– 4 –
32 – –
– 4 –
2 – –
14 – 2
4 – –
12 – –
3 – –
3 – –
4 – –
da/dt (mm/h)
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CCG data for base material 316 - 550˚C 316 -650˚C 316 - 600˚C 316 - 700˚C 316 - 750˚C 316 -800˚C
1 0.1
0.01 0.001 2
Log 10 [C * (J/m .h)]
0.0001 0
1
2
3
4
5
6
Fig. 2. Data gathered in the Hida database. Table 2 CCG law – parent metal – da/dt = D(C*)/ T (C)
550
600
625
650
700
750
800
5
9.1 0.74
16.9 0.7
3.95 0.752
8.03 0.684
0.359 0.960
0.923 1.039
0.978 0.806
10 /
D
Finally, we propose a linear regression of da/dt versus C*, including all the data on base material from 550 to 800 C. The fitted values are: da ¼ 1.19 104 ðC Þ0.687 ; dt
ð2Þ
¼ 345 J=m2 h; where C* is in J/m2 h1 and da/dt in mm/h. This curve contains the mean point (C 3 da/dt = 6.58 · 10 mm/h). The distribution parameters are in log-normal space, the residual variance of the data around this line is equal to 0.103. This means that around the mean point, the creep crack growth rate da/dt has a logarithmic statistical distribution with an average value equal to log10 (6.58 · 103) = 2.182 and a standard deviation equal 0.321. The 90% confidence limits are shown in Fig. 2 (corresponding to 5% of expected data lower than the lower curve and 5% of expected data higher than the higher curve). For C* values far from the average point, the logarithmic distributions of da/dt is wider, as shown in Fig. 3, particularly for da/dt corresponding to low C* values (610 J/m2 h). This is a normal situation in the proper statistical treatment of data by linear regression to establish the confidence limits of a variable (log10 (da/dt) in this case), for different values of the one parameter (C*). As a result, the creep crack growth rate in the weld (Fig. 4), or in the HAZ, is of the same order of magnitude as in the base metal. 4.2.5. Requirement for statistical analysis It is statistically meaningless to process all the gathered data points when there is a large difference in the number of points between the different tests. Usually a statistical analysis starts with a database obtained by random sampling from the complete population to be studied. To date, no database of creep crack growth rates at random temperature exists. Nevertheless, some sources of error in the statistical analysis can be avoided: the number of tests should be similar at the different temperatures for the region to be covered, a similar number of data points, more or less equally spaced, should be retained from each test,
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da/dt (mm/h)
1000 90% confidence limit for CCGrate da/dt
100 10 1
316 -550˚C
316 - 650˚C
316 -600˚C
316 -700˚C
316 - 750˚C
316 -800˚C
Average curve
-5% confidence limit
+5% confidence limit
0.1 0.01 0.001 0.0001 0.00001
2
Log 10 [C* (J/m .h)]
0.000001 0
1
2
3
4
5
6
Fig. 3. 90% confidence limit for CCG rate da/dt.
da/dt (mm/h)
10 1 0.1
316 - 550˚C
316 - 650˚C 316 - 600˚C
Weld -600˚C
0.01 0.001 Log 10[C* (J/m2.h)]
0.0001 0
1
2
3
4
5
6
Fig. 4. Comparison weld metal/Parent metal.
alternatively a weighting should be attributed to the data of the different tests in order to compensate for the difference in the number of recorded data points from one test to another. A difficult problem is the definition of a rule for selecting, for each test, the first data point to be taken into account. In many of the tests, there is a trend of whereby low creep crack growth rates are obtained just after initiation of the crack growth. In many cases, this leads to an overestimation of the slope /. 4.3. Discussion of statistical analysis The data set required for rigorous probabilistic analysis of the lifetime of cracked components at high temperature is not available at the present. The absence of relevant data is directly linked to the fact that, when data are gathered, they usually focus on the pessimistic bound (deterministic approach), for example the material properties [14]. The methodology to be used for gathering relevant data, the reorganisation of existing databases and the development of a statistical treatment represent one of the major tasks within the HIDA Applicability project [1].
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5. FCG simplified numerical calculation 5.1. Presentation Reference [16] deals with the assessment of crack growth in pipes under fatigue conditions using the ALIAS-HIDA software. In the framework of the development of probabilistic tools, the aim of these studies is to highlight the main characteristics, strengths and weaknesses of two approaches based on Monte Carlo simulation techniques (see Section 2). From the initial size, crack growth is calculated up to 0.7 times the thickness of the component. The number of cycles necessary to reach this depth is then considered as a random variable. The goal of the study is to predict the distribution of this variable and finally to compare the results obtained by the two approaches (see Section 5.4). Calculations using the stratified samples are described in detail (see Section 5.3). 5.2. Main features and data The chosen example involves a circumferential semi-elliptical crack located in a straight pipe made of 316 L (N) austenitic stainless steel, with a wall thickness of 7 mm. Only the depth is taken into account, with the initial crack depth taken as 1.75 mm and the crack length considered to be constant (15 mm). The applied cyclic load results in through wall thickness membrane stresses of rmax = 77 MPa and rmin = 36 MPa. Growth is assumed to follow a ‘‘Paris law’’, which is defined as: da a ¼ C ðDK Þ ; dN
ð3Þ
where C = 108 and a = 3.3 (K is given in MPa m0.5, a in mm and da/dN in mm/cycles). DKeff is used instead of DK in order to take the R ratio effect (R = rmin/rmax) into account. The correction factor q, defined by DKeff = qDK, is given in the A16 procedure [6]. The stress intensity factor formula is given in reference [15]. Only two random variables, growth rate and load were chosen. C is taken as a random variable with a log-normal distribution function with a mean value of the logarithm of 18.42 and a standard deviation of the logarithm of 0.54. For the load, a normal distribution function is defined (see Table 3). 5.3. Monte Carlo simulation using stratified sample The transformation function bk used is defined on 2k + 1 points distributed between 0 and 1 as: 0, 10k+1, . . . 0.5, . . . 1–10k+1, 1. For example, the values of the function bk is given for k = 2 and k = 4 (see Table 4). One can see that the middle point 0.5 is kept unchanged. After transforming the uniform [0, 1] distribution random numbers through the sampling function bk, most of the random numbers are concentrated at the extreme values of the interval [0, 1]. Generally speaking, to each of the initial numbers, a, we associate a pair of numbers (a, da), where da is a weighting defined by the derivative of function b at point a.
Table 3 Load data – statistical parameters of normal distribution function Service conditions load
Mean value
Standard deviation
Maximum rmax (MPa) Minimum rmin (MPa)
77 36
10.3 4.8
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Table 4 Values of the bk function x
b(x)
k=4 0 0.125 0.25 0.375 0.5 0.625 0.75 0.875 1
0 0.001 0.01 0.1 0.5 0.9 0.99 0.999 1
k=2 0 0.25 0.5 0.75 1
0 0.1 0.5 0.9 1
For the case study, only two random samples are defined a and A, and transformed by the same function b, giving two sets of pairs (ba, da) and (bA, dA), where ba are the values of b(a) and da are the weighting. For the coefficient of the ParisÕs law we consider the distribution of Log C to be Gaussian. In fact we use the random variable kC which is defined by: C ¼ k C C nom .
ð4Þ
For the load, the multiplying coefficient kr is considered to be a random variable, having a Gaussian distribution: r ¼ k r rnom .
ð5Þ
Using the repartition functions of Kr and KC, the samples ka, kA, are defined. A parametric study was conducted using 100 values of each variable. Fig. 5 shows the number of cycles for which 70% of the thickness was reached with less than ‘‘x times N’’ simulations, (in other words x can be considered as the probability of failure (Pof) for N cycles). The stratified samples with k = 4 gave accurate results with a precision of 107, while the stratified sample with k = 2 is limited to an accuracy higher than 104. When stratified samples are used, the higher the value of k, the smaller the weighting used in the simulation. For example when k equals 4 the weighting is 102, and when three variables are used the simulation weighting decreases to 106. Handling stratified samples leads to very small numbers and requires special care to avoid numerical discrepancies. 5.4. Discussion of FCG numerical calculation Using 500 randomly selected values of each variable we obtain, through the non stratified sampling calculation, a mean value of 59.5 · 103 cycles and a standard deviation of 32 · 103 cycles. Fig. 6 shows the comparison of the results for stratified samples and non stratified samples for the maximum number of simulations specified in the ALIAS-HIDA software (104). As shown in the Fig., the stratified method is not necessary in our case for a Pof of 103, but for a Pof higher than 104 the stratified method gives considerably more accurate results. The present studies show that it is possible to use stratified samples to obtain greater accuracy or to reduce computation time. Stratified samples are required when the desired accuracy is lower than the inverse of the number of simulations.
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Ncycles 30000 stratified k=8 stratified k=7 stratified k=6 stratified k=5 stratified k=4 stratified k=3 stratified k=2 non stratified k=1
25000 20000 15000 10000 5000
PoF x 1.E-07
1.E-06
1.E-05
1.E-04
1.E-03
0 1.E-02
Fig. 5. Effect of stratification sampling – N = 100 · 100.
N cycles 25000 stratified (k=2) 20000
stratified (k=4) non stratified
15000 10000 5000 PoF x 1.E-06
1.E-05
1.E-04
1.E-03
0 1.E-02
Fig. 6. Comparison of non stratified and stratified samples calculations – N = 100 · 100.
6. CCG numerical calculation 6.1. Presentation At the time of writing this paper, further facilities such as stratified sample technique are being implemented and verified. Therefore, this section is a case study of probabilistic analysis using the ALIAS-HIDA software [7]. The component was a straight pipe made of 316 L(N) austenitic stainless steel which was submitted to mechanical loading at 550 C. The surface crack considered was located in the inner wall of the straight pipe. The probability space is described by 8 random variables, deterministic values being used for other parameters.
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6.2. Main features and data Constant values were taken for some variables, such as: Geometrical characteristics: internal radius 250 mm and the crack length 40 mm. Service conditions: temperature is 550 C. Material property: The Young modulus, E = 149,000 MPa. For the fracture toughness, the value considered in the case study is very low, KIC = 30 MPa m0.5. The data considered as random variables are summed-up in Tables 5 and 6. Although 316 L austenitic stainless steel exhibits a large primary creep stage, only secondary creep was taken into account for this case study of C* evaluation. Generally speaking, only extreme values are used in deterministic (pessimistic) analysis. For material characteristics, extreme and mean values are available, as given in [17]. Thus, assuming that extreme values correspond to the probabilistic distribution function at three standard deviations, we obtained an approximate function: geometrical: the normal distribution function is used for crack depth and thickness, service conditions: Only mechanical load is considered with temperature variations being investigated later. The variation in mechanical properties was taken into account through relevant links with the material Database [13]. material: an approximate CCG distribution function was characterized from data given by reference [6]. The variation in the creep strain rate law (see Section 4.2.3), and in creep rupture data is difficult to characterize. Thus, for this case study, we considered only the secondary creep stage and handled the variation through a multiplication coefficient. For mechanical properties such as the yield stress and the ultimate stress, minimum and mean values were provided by [17].
6.3. Calculation using the ALIAS-HIDA software The Monte Carlo based simulation technique was used to evaluate the probability of failure. The calculations take into account the failure mechanisms. The failure is considered when either: Table 5 Material data – statistical parameters of log-normal distribution function Material characteristics coefficient
Mean value of logarithm
Logarithmic standard deviation
CCG D (MPa m h1; mm/h) Creep strain rate As (MPa; h1) Creep damage B (MPa; h)
3.08 30.31 57.11
0.77 0.23 0.47
Table 6 Data – statistical parameters of normal distribution function Other characteristics
Mean value
Standard deviation
Ultimate stress ru (MPa) Yield stress ry (MPa) Bending moment M (kN m) Depth a (mm) Thickness t (mm)
415.89 136.96 280 2 10
15.42 12.78 9.33 0.17 0.33
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the crack size reaches 80% the wall thickness, or a fast fracture criterion is exceeded, or a time-dependent failure criterion is exceeded (creep rupture).
Crack depth [mm]
The results obtained with the base model, i.e., using mean values, are shown in Figs. 7–9. It can be seen that the creep damage accumulation in the ligament is moderate and fast fracture occurs (due to low fracture toughness), when the crack depth is about 4 mm (Da 2 mm). After propagation the crack size represents 40% of the wall thickness. The main result is the probability of failure (Pof). Fig. 10 shows the evolution of the Pof depending on the time for 3000 simulations. As a result, the remaining life is 1420 h for a Pof of 0.33%, with an accuracy of about 30% in the estimate of the Pof. For this case study (numerical example), it turns out that the fast fracture criterion is more relevant than the creep rupture and maximum crack size criteria (Figs. 7–9). Thus, the results describe a limited crack growth domain (20% of the wall thickness).
3.8 3.6 3.4 3.2 3 2.8 2.6 2.4 2.2
Time [h]
2
0
10000
20000
30000
40000
50000
Creep damage accumulation [-]
Fig. 7. Crack size evolution – base model (mean values).
0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02
Time [h]
0 0
10000
20000
30000
40000
Fig. 8. Ligament creep damage – base model (mean values).
50000
778
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Failure assessment diagram
1.2 1
Path -Depth Path -Surface Initial flaw - Depth
Kr [-]
0.8 0.6
Initial flaw - Surface Final flaw- Depth
0.4
Final flaw- Surface Assessment curve
0.2 0 0
0.5
1
Sr [-]
1.5
2
2.5
Fig. 9. Fast fracture – base model (mean values).
60 50 40
Probability [%]
70
30 20 10 0 100
Time [h] 1000
10000
100000
Fig. 10. Pof depending on the duration.
6.4. Discussion of CCG numerical calculation The calculation of the probability of failure for a cracked component at high temperature has been performed successfully using the ALIAS-HIDA software. Further studies could be carried out using the ALIAS-HIDA software under creep-fatigue conditions, taking the primary creep stage into account and using the data from the analysis and the statistical treatments, which are yet to be conducted (see Sections 3 and 4).
7. Conclusions Numerical calculations of crack growth have been conducted using the probabilistic Monte Carlo method. Based on the present studies, it has been shown that it is possible and practical to use stratified samples in
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order to obtain increased accuracy or to reduce computation time. Stratified samples are particularly suitable when the desired probability of failure is very low. The ALIAS-HIDA software has been designed to calculate the probability of failure of cracked components under high temperature. The Monte Carlo simulation technique was successfully implemented and the first steps of the verification seem to show it to be reliable. This was demonstrated by a calculational example which illustrates the ability of the software to deal with these calculations, which require a large number of random variables and simulations. In order to be able to conduct probabilistic analysis on cracked component subject to high temperatures, some of the data in the existing HIDA database has been re-organized and analyzed. This task is in progress but some results are available and trends and technical issues are outlined for a 316 austenitic stainless steel.
References [1] Shibli IA, Le Mat Hamata N. Development of the HIDA procedure. ETD, Hida conferences: 21–22 September 2004, Cambridge, UK. [2] Probabilistic Approaches Applied to High Temperature Defect Assessment – Literature Review & Guidelines HIDA Applicability Project (Growth Project G1RD-CT-2002-00730) HA/WP2/1 Draft Version: 14 Aug-03 ETD report 1020-ec-21. [3] Le Mat Hamata N, ETD, UK, Balos D, MPA, Stuttgart, Germany, Deschanels H, Thiry Framatome JM, France. Probabilistic approach to defect assessment used in the HIDA KBS. Hida conferences: 21–22 September 2004, Cambridge, UK. [4] Nikbin K. A unified approach to a European high temperature defect assessment methodology and knowledge base system. In: 2nd HIDA conference 4–6 October 2000, Stuttgart, Session 5.1 paper S5.1, Imperial College, UK. [5] British Standards Institution. Guide on methods for assessing the acceptability of flaws in Fusion welded structures. BS 7910; 1999. [6] RCC-MR, Design and construction rules for mechanical component of FBR nuclear Island, Tome I – Volume Z, Appendix A16. Guide for leak before break and defect assessment, Edition 2002; AFCEN. [7] Colantoni D, Balos D, MPA, Stuttgart, Germany, Le Mat Hamata N, ETD, UK, Deschanels H, France. HIDA Knowledge Based System (KBS) for probabilistic & sensitive analysis of creep and fatigue crack growth in high temperature component. Hida conferences: 21–22 September 2004, Cambridge, UK. [8] Ibrahim Y. Observations on applications of importance sampling in structural reliability analysis. Struct Saf 1991;9:269–81. [9] Bucher C. Adaptative sampling, an iterative fast Monte Carlo procedure. Struct Saf 1988;5:119–26. [10] Melchers RE. Importance sampling in structural systems. Struct Saf 1989;6:3–10. [11] Joensson M, Siempelkamp, Germany, Le Mat Hamata N, ETD, UK, Concari S, CESI, Italy, Alburquerque JM, ISQ, Portugal, Deschanels H, Framatome, France. Sensitivity analysis of creep & fatigue crack growth in P22, P91, 1CrMo & 316SS steels using the HIDA KBS. Hida conferences: 21–22 September 2004, Cambridge, UK. [12] Boynard A, Vrohvac M, Albuquerque JM, Cruz AC, ISQ, Portugal. Comparison of flaw sizing capabilities of NDE techniques. Hida conferences: 21–22 September 2004, Cambridge, UK. [13] Concari S, ENEL, Italy, Fairman A, Shibli A, ERA Technology, UK, Escaravage C, Framatome, France. HIDA Databank. In: Int HIDA conference 15–17 April 1998, Saclay, France, Paper S8-53. [14] Holdsworth SR, ALSTOM Power, UK. The scope of material property data being generated in EC funded projects. In: 2nd HIDA conference 4–6 October 2000, Stuttgart, Paper S7-1. [15] Newman JC, Raju IS. An empirical stress intensity factor equation for surface crack. Eng Fract Mech 1981;15(1, 2):185–92. [16] Colantoni D, MPA, Stuttgart, Germany, Cerny I, Le Mat Hamata N, ETD, UK. Crack growth assessment of pipes under fatigue conditions – a comparison with knowledge-based system analysis. Hida conferences: 21–22 September 2004, Cambridge, UK. [17] RCC-MR. Design and construction rules for mechanical component of FBR nuclear Island, Tome I – Volume Z, Appendix A3 Edition 2002; AFCEN.