International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Fatigue reliability of casted wind turbine components due to defects Rafsanjani, Hesam Mirzaei; Sørensen, John Dalsgaard Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1 Fatigue Reliability of Casted Wind Turbine Components Due to Defects Hesam Mirzaei Rafsanjani PhD Fellow, Dept. of Civil Engineering, Aalborg University, Aalborg, Denmark John Dalsgaard Sørensen Professor, Dept. of Civil Engineering, Aalborg University, Aalborg, Denmark ABSTRACT: Manufacturing of casted components often leads to some (small) defects that are distributed in the volume of the components. These defects are different according to their size, type, orientation and etc. In this paper a probabilistic model is proposed for modeling manufacturing defects and their influence on the fatigue strength of the components. The fatigue life is dependent on the number, type, location and size of the defects in the component and is therefore quite uncertain and needs to be described by stochastic models. In this paper, the Poisson distribution for modeling of defects of component are considered and the surface and sub-surface defects categorized. Furthermore, a model to estimate the probability of failure by fatigue due to the defects is proposed. This model is used to estimate the failure location of component and it is compared to models of defect distributions and locations. Further, an upper bound of reliability is estimated using a modified Miner rule approach for fatigue damage accumulation.  1. INTRODUCTION Wind energy is a rapid growing industry in the renewable energy sector with large potentials for contributing significantly to the future energy production. A main focus for wind turbine manufactures and operators is to increase the reliability of wind turbines and to decrease the cost of energy. Offshore wind turbines are exposed to wave excitations, highly dynamic wind loads and wakes from other wind turbines. Therefore, most components in a wind turbine experience highly dynamic and time-varying loads (Mirzaei & Sørensen, 2014). These components may fail due to wear or fatigue and this can lead to unplanned shut down repairs that may be very costly. Design of mechanical components in the wind turbine drivetrain by deterministic methods using safety factors are generally not able to account rationally for the many uncertainties. Therefore, alternatively reliability assessments may be performed using probabilistic methods where stochastic modeling of failures and uncertainties is performed. Modeling of the fatigue failure of casted wind turbine components can be used for predicting the expected time-to-failure which is an important indicator to be used in planning of operation and maintenance. In order to estimate the probability of failure of casted components careful modeling of the aleatory (physical) and epistemic (model, statistical and measurement) uncertainties has to be performed (Sheng & Veers, 2011). Current fatigue designs are typically based on the safe life design approach (Shirani & Härkegård, 2010). In the safe life design, fatigue testing is carried out on baseline material to produce SN curves. However, the manufacturing process may affect SN-curves and the statistical uncertainties and different manufacturing processes may lead to the need to use different statistical models for the material strengths. The fatigue life of cast iron is often controlled by the growth of cracks initiated from defects such as shrinkage cavities and gas pores 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2 (Shirani & Härkegård, 2012). To predict the fatigue life, defects are considered as pre-existent cracks and fatigue life and fatigue limit are controlled by the crack propagation law and by the threshold of the stress intensity range, respectively. Early-life failures are often the result of poor manufacturing and inadequate design. A substantial proportion of early-life failure is also due to the presence of defects in the material. An important factor affecting the strength of components is the presence of defects due to processing, manufacturing or mechanical damage occurring during service (Todinov, 2006). Further, the fatigue life is highly dependent on the number, type, location, size of defects and the applied stress on the component during its design working life. Defects can thus be categorized in different groups. This paper focuses on statistical models of defect distribution in wind turbine components and development of probabilistic models of fatigue strength. To model the defect distribution, the location of defects (interior or surface) defects has to be modelled and the number of defects in each volume must also be modeled. It is noted that size, type, number, orientation and stress surface of defects affect the probability of failure. First, the distribution of defects in the specimen volume has to be modeled incl. clustering of defects in the interior and the surface of the specimens. The fatigue strength is estimated based on the defect distribution. Finally, the stochastic models for the defects and for the fatigue life given defects are used to estimate the probability of failure of components. 2. DEFECT DISTRIBUTION MODEL Manufacturing of components often leads to some (small) defects that are distributed in the volume of the components. These defects are different according their size, type, orientation and etc. and influence the load-bearing capacity of the components (Toft et al., 2011). Hence, an important factor affecting the strength of components is the presence of defects from processing and/or manufacturing. An important problem related to materials containing defects is to model the uncertainty by a defect density function for the component volume. In some cases, clustering of defects is important and strongly influencing the probability of failure. Clustering of two or more defects within a small volume often decrease dangerously the load-bearing capacity and increases the stress concentration. Hence, the number of defects in each volume and the size / dimension of the defects should be modeled according to their size, type, orientation and etc. In order to determine the probability of fracture (failure), all initiating defects are divided into categories depending on their type (Todinov, 2000). In the categories, each defect, according to its size, shape, and orientation etc., is characterized by a specific level of local maximum tensile stress, at which fatigue is triggered. Each type of defect i is characterized by a cumulative distribution function for the number of fatigue load cycles to failure, FN,I (n, R, σ), giving the probability that fatigue failure does not occur at a local maximum tensile stress  equal to σ and a given R-ratio, that is the ratio of the minimum stress experienced during a cycle to the maximum stress experienced during a cycle. Hence, suppose that in a specimen with volume VT, M types of defects exist (Todinov, 2000). It is important to emphasize that the nucleation of fatigue cracks in the groups of defects are assumed as statistically independent events. In other words, a fatigue crack in a particular group is not affected by fatigue crack in other groups. This assumption is related to the condition of nucleation on a particular defect depending only on the local maximum tensile stress and on the strength and orientation of the defect. According to this, the defects can be categorized in M groups of defects such that the size, type and orientation etc. of defects in each group are very close to each other. It is noted that each type of defects is characterized by a 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3 cumulative distribution function FN,I (n, R,σ) for the number of load cycles to failure, see below. Furthermore, it is assume that the number of defects of group j can be modeled by a multi-dimensional Poisson process. Thus, if D is any region in the multi-dimensional space for which the area or volume of the region and N(D) is the number of defects in D, then   P N D( ) = k( ) = λ D( )k e−λ Dk!  (1) is the probability that the number of defects in D is k. Equivalently, the density function of the number of defects in group j in volume V of component is (Ravi Chandran and Jha, 2005):  P k( ) = λ jV( )k e−λ jVk!  (2) where λj is the average number of defects of group j (there is M different groups of defects). The probability of occurrence of the number of defects in different parts of the specimen can now be assessed using the Poisson model. Eq. (1) and (2) are the general equations for modeling the defect distribution of the j-th group of defects. In the next step, the whole component is modeled. Suppose, that in a specimen with volume VT, a smaller volume dVi is stressed to a tensile stress σi,j which is assumed to be uniform inside the volume dVi, result in  VT = dVii=1NV∑  (3) where NV is the number of volumes dVi in volume VT. Fatigue cracks are assumed to initiate from surface cracks / defects or sub-surface cracks / defects. Volume dVi in Eq. (3) is rewritten as (Ravi Chandan and Jha, 2005):  dVi = Ai + dAi( )dl  (4) where, Ai is interior area of each smaller volume and dAi is the area of the surface rim of a certain width wrapping around Ai such that the total cross-sectional area of the sample is Ai+dAi (Figure 1). It is noted that the above equation can model other specimen’s volumes too.  Figure 1: The model segment of Specimen Volume.  The probabilities of defect occurrence in different parts of the specimen can now be assessed using the Poisson distribution. Whether a specimen fail by internal or surface crack initiation is determined depending on whether there is a cluster of defect at that location (Ravi Chandran and Jha, 2005). In this context, three cases are considered: • If there is a finite probability that there are cluster defects in the interior and no cluster defects in the surface rim volume, implying the specimens to fail only by internal initiation. • If there is a finite probability that there are cluster defects in the surface rim region and no cluster defect in the interior, implying the specimens to fail only by surface-initiation. • If cluster defects exist both in the interior as well as in the surface rim volumes, then, the specimen is likely to fail by surface-initiation only, because it is known that fatigue crack initiation at surface typically is accelerated by the air environment.  The probability of occurrence of interior defects, Pint,j , is the probability that one or more defects from group j in Ai with no such defects present in dAi. This can be written on the basis of Eq. (2) and (4) as  Pint,i , j k( ) = λ j Aidl( )( )k e−λ j Aidl( )k! e−λ j dAidl( )  (5) 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4 Similarly, the probability of occurrence of surface defects, Psurf,j , is equal to the probability that at least one or more defect from group j will occur in dAi regardless of a defect from group j being absent or present in the interior area Ai. In other words, the occurrence of a defect in the interior does not matter as long as one surface defect is present, since it will preferentially initiate a critical fatigue crack which is more critical than a defect in Ai due to environmental effects. The probability of surface-crack initiation is then given by the probability of presence of one or more defects in dAi  Psurf ,i , j k( ) = λ j dAidl( )( )k e−λ j dAidl( )k!  (6) In following, the probability of fatigue failure of component due to existence of defects will be modeled. 3. PROBABILTY OF FATIGUE FAILURE In the previous section, the defect location in the volume VT is modeled by a Poisson distribution. The volume VT is modeled as a “series system” of volumes dVi since failure in any of the volumes dVi is assumed to result in ‘collapse’. The probability of failure of each volume dVi, is denoted Pf,i . The probability Pf,i is dependent on • Probability of existence of defect in volume dVi. • Probability of fatigue failure if a defect exists in volume dVi.  As mentioned above, the defects are categorized in M groups. For each one of these groups, the probability of failure is to be evaluated. Hence, the probability of failure of each volume dVi  is written Pf ,i = P defect of j-th group exist in dVi( )!"j=1M∑×P fatigue failure defect from j-th type/size exist( )%&  (7) Eq. (7) is applied for both interior and surface defects and in each volume dVi.  As described above it is assumed that the defects follow a homogeneous Poisson process in the volume dVi. The distribution function FN,j (n,R,σi,j) is assumed to follow a Weibull distribution (Fjeldstad, Wormsen and Härkegård, 2008) FN , j n,R,σ i , j( ) =1− exp − nN j (R,σ i , j )"#$$ %&''bn , j()** +,--  (8) where Nj(R,σi,j) and bn,j are the shape and scale parameters of Weibull distribution and they are subject to statistical uncertainties if modelled on basis of test data and should be evaluated by statistical analysis methods such as Maximum Likelihood Method or Bootstrapping using available data sets. n is the actual number of cycles in the lifetime [0, TL] with stress range σ, where TL is life time of component. Hence n can be written as:  n =ν *TL  (9) where ν is the number of load cycles with the stress range σi,j per year. Eq. (8) should be determined for all M groups of defects. Let FN,j(n,R,σi,j) denote the conditional individual probability that the fatigue life characterizing a single defect from j-th group on the component’s volume will be smaller than n cycles, given that the defect resides in the component volume. The probability of failure in volume dVi can next be determined by subtracting from unity the probability of the complementary event that none of the defects fatigue lives will be smaller than n cycles (Todinov, 2006). This probability, 0 ),,( jirp  of the compound event: exactly r defects from group j-th exist in the volume dVi of the component and their fatigue lives will not smaller than n can be modeled by: p(r ,i , j )0 = P r  defects in dVi( )×P none fatigue failure less than n cycle r  defects( )  (10) This probability correspond to the probability of two events: 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5 • Exactly r defects from group j-th exist in volume dVi • None of fatigue lives associated with the r defects will be smaller than n cycles.  The probability 0, jip  that “component’s fatigue life will be greater than n cycle” is determined on basis of a union of disjoint probabilities 0 ),,( jirp . Consequently, the probability of the event 0, jip  is a sum of the probabilities defined by:  pi , j0 = p(r ,i , j )0r∑  (11) Eq. (11) can be generalized modeling interior and surface defects. If m defects are interior defects in volume dVi , then r-m defects will be surface defects. Hence, Eq. (11) can be written as (as mentioned before): pi , j0 (n,R,σ i , j ) = Pint,i , j (k)× 1− FN , j (n,R,σ i , j )#$ %&kk=0m∑ m = rPsurf ,i , j (l)× 1− FN , j (n,R,σ i , j )#$ %&ll=0r−m∑ r > m()**+**    (12) Then, the probability of failure due to defect from group j for the sub-components with volume dVi becomes  Pf ,i , j =1− pi , j0  (13) Eq. (13) can be generalized for multiple groups. Thus, the probability that component’s fatigue life will be greater than n cycle according to Eq. (13) is  pi0 (n,R,σ i , j ) = pi , j0 (n,R,σ i , j )j=1M∏  (14) Hence, the probability of failure of fatigue for volume dVi can be written as Pf ,i n,R,σ i , j( ) =1− pi0 n,R,σ i , j( ) =1− pi , j0 n,R,σ i , j( )j=1M∏  (15) The volume VT is assumed to be modeled as a “series system” of independent volumes dVi. Hence, the probability of failure of volume VT, can be written as Pf n,R,σ i , j( ) =1− 1− Pf ,i n,R,σ i , j( )( )i=0NV∏  (16) This failure probability is a function of: • The stress level σi,j • Time t because n=ν*t  Pf n,R,σ i , j( )  is the probability of failure in the life time [0,t] when n=ν*t. Eq. (16) is the probability of failure for volume VT. In these equations, the size, type, orientation, stress of defects are considered but when there are clusters of defects in components, the distance between each defects play very important role on fatigue life and should be introduced in the models, e.g. following the models in (Toft et al., 2011). 4. RESULTS In this section, the probabilistic model described in previous sections is illustrated using the data and defects observed in (Shirani 2012). The data considered are from specimen numbers 45, 49, 50 and 51. In Figure 2 is shown the geometry of the specimens and the coordinate system used.  Figure 2: The fatigue test Specimen.  Table 1 shows for each specimen, the location of the observed fracture surfaces with respect to the z-axis; for more details see (Shirani 2012). Moreover, dl in Eq. (4) is chosen to 2 mm, and the probability of failure is estimated by Eq. (15) with n chosen to 2,000,000 cycles for all specimens. Further, Nj(R,σi,j) and bn,j in Eq. (8) are estimated for each group of defects such that the Nj(R,σi,j) values are taken as the predicted values by (Shirani 2012), see the following tables. bn,j is taken as 1.5.  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6 Table 1: Location of fracture surface on z-axis for test specimens (Shirani 2012). Specimen Number Fracture surface observed (Shirani 2012) [mm] 45 98 49 138 50 53 51 96  Specimen number 45: The observed defects and the probability of failure estimated by Eq. (15) are shown in Figure 3. It is noted, that the probability of failure in Figure 3, and in the following figures, is the probability of failure for each sub-volume dVi estimated by Eq. (15) with dl = 2 mm, i.e. it is thus not the probability of failure of whole specimen.    Figure 3: Observed defects and probability of failure in dVi along the length of specimen 45  According to (Shirani 2012) the two largest defects (based on defects volumes) for specimen number 45 are located at z = 98.87 mm and 97.55 mm (see Table 2). The defect volumes are 26.44 mm3 and 24.72 mm3, respectively. According to Figure 3, the sub-volume at 97 - 99 mm has the highest probability of failure (upper bound estimated by Eq. (15)). Further, the smaller defect number 8 is also located in this sub-volume and contributes to the probability of failure. This sub-volume with the highest estimated probability of failure, and thereby the most critical sub-volume corresponds to the observed fracture surface, see Table 1.  Table 2: Specimen number 45 (Shirani, 2012). Num. Volume (mm3) X (mm) Y (mm) Z (mm) Nj(R,σi,j)  Predicted life (cycles) 1 26.44 7.37 6.11 98.87 690,134 2 24.72 4.29 3.03 97.55 774,977 3 5.39 3.74 3.14 101.5 1,946,904 4 4.65 8.03 2.59 95.13 1,963,542 5 7.48 5.22 1.27 94.36 2,046,626 6 3.92 3.96 1.93 99.42 2,774,349 7 1.93 4.73 2.70 69.04 3,758,844 8 2.21 5.06 1.27 97.55 4,065,269 9 2.28 5.33 2.48 96.67 4,501,964 10 1.24 4.4 0.39 96.78 4,898,411  Specimen number 49: The observed defects and the probability of failure (estimated by Eq. (15)) are shown in Figure 4. The largest probability of failure is in the sub-volume dVi between 139 and 141 mm (Table 3). According to Table 3, defect number 3, 4 and 10 are in the mentioned sub-volume. The observed fracture surface (at 138 mm) is very close to the sub-volume with the highest estimated probability of failure.  Table 3: Specimen number 49 (Shirani, 2012). Num. Volume (mm3) X (mm) Y (mm) Z  (mm) Nj(R,σi,j)  Predicted life (cycles) 1 4.2 -1.87 8.67 92.25 2,811,521 2 2.52 3.41 -4.31 72.55 3,886,311 3 2.41 -3.41 2.29 140.68 4,429,748 4 2.39 -8.92 2.07 139.25 5,088,111 5 2.02 6.82 3.61 102.93 4,106,853 6 2.23 9.35 0.31 102.05 4,409,628 7 1.43 0.33 -4.75 75.19 5,328,563 8 1.16 7.7 0.09 102.49 6,903,317 9 1.13 8.58 4.27 101.28 7,124165 10 1.04 -5.17 0.31 139.8 5,296,975  Specimen number 50: The two sub-volumes with the largest probabilities of failure are at the 53-55 mm and 57-59 mm sub-volumes, see Figure 5. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  7  Figure 4: Observed defects and probability of failure of dVi along the length of specimen 49.  According to Table 4, the defects number 3, 6, 7, 9 and 10 are inside the 53-55 mm sub-volume, i.e. 5 out of the 10 largest defects are located very close to where failure is observed (at 53 mm). Hence, failure is likely due to the cluster of defects that are very close to each other. It is noted that the largest defect (based on volume of defects) is located at 55.15 mm. Because this defect is located in the interior of the specimen, the probability of failure of this defect is lower than cluster of defects between surfaces 53-55 mm. These results are very similar to results in (Shirani 2012).  Table 4: Specimen number 50 (Shirani, 2012) Num. Volume (mm3) X (mm) Y (mm) Z  (mm) Nj(R,σi,j)  Predicted life (cycles) 1 12.82 -1.24 -7.32 55.15 1,174,747 2 7.89 -0.39 -1.98 57.82 1,734,182 3 6.37 0.68 -2.84 54.72 1,843,818 4 4.35 -3.7 -6.68 57.07 2,169,724 5 3.54 4.52 -7.75 52.91 2,355,450 6 3.41 -0.39 -5.61 54.08 2,529,115 7 1.47 4.42 -4.33 53.44 4,166,833 8 0.74 -2.2 -6.04 58.57 6,381,366 9 0.97 2.82 -3.9 53.76 6,588,717 10 0.96 6.98 -5.83 53.76 6,964,649  Specimen number 51: According to Table 5, the two largest defects are located at 97.85 mm and 98.29 mm on z-axis. Further, the defects number 4, 8, 9 and 10 are also located between or near the sub-volume 97-99 mm.   Figure 5: Observed defects and probability of failure of dVi along the length of specimen 50.  Table 5: Specimen number 51 (Shirani, 2012) Num. Volume (mm3) X (mm) Y (mm) Z  (mm) Nj(R,σi,j)  Predicted life (cycles) 1 1.86 7.7 -4.06 97.85 3,901,121 2 1.53 6.49 -4.73 98.29 4,312,266 3 1.99 -2.75 -8.51 102.04 4,761,026 4 2.1 -1.09 -9.17 98.29 5,617,233 5 0.74 -5.4 -6.75 122.86 6,307,730 6 0.69 4.3 -8.06 67 7,585,928 7 0.54 -5.98 0.52 123.74 8,320,367 8 0.53 7.05 -5.83 97.3 8,789,104 9 0.47 0.87 -4.31 98.18 8,789,104 10 0.4 0.76 -3.65 98.84 9,019,978   Figure 6: Observed defects and probability of failure of dVi along the length of specimen 51. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  8 The largest estimated probability of failure is in the sub-volume 97-99 mm, corresponding to the observed location of the fracture (at 96 mm). 5. CONCLUSION In this paper is presented generic models for estimation of the probability of fatigue failure due to manufacturing defects in casted components, e.g. for wind turbine components. The defects are categorized in different groups according to their size, type, orientation and etc. For each group, a Poisson distribution is used to model the distribution of defects. Further, the defects are divided in two sub-groups of interior and surface defects.  Next, the component volume is divided in smaller volume to obtain approximately homogeneous stresses in small sub-volumes. In each sub-volume sections, the probability of failure is modeled. The probability of failure is a function of i) existence of defect(s) in sub-volumes, ii) the conditional probability of failure due to existence of defect(s) in sub-volumes. In this model, the interior and surface defects are separated from each other. The model accounts for cluster of defects. For each sub-volume, the probability of failure is modeled. As the component volume is assumed to be considered as a series system of sub-volumes, the probability of failure of the whole components can be estimated according to a series systems probability of failure models.  Finally, the model is illustrated by application to test results. The results indicate that the probabilistic model is able to estimate a location of highest probability of failure quite close to the actual location of failure of the tests.  6. ACKNOWLEDGMENT The work is supported by the Strategic Research Center “REWIND – Knowledge based engineering for improved reliability of critical wind turbine components”, Danish Research Council for Strategic Research, grant no. 10-093966. 7. REFERENCES Fjeldstad, A., Wormsen, A., and Härkegård, G. (2010). “Simulation of fatigue crack growth in components with random defects” Engineering Fracture Mechanics, 75, 1184-1203. Mirzaei Rafsanjani, H., and Sørensen, J. D. (2014). “Stochastic models of defects in wind turbine drivetrain components?” In: Papadrakakis M, Stefanou G, editors. Multiscale Modeling and Uncertainty and Uncertainty Quantification of Materials and Structures. Switzerland: Springer; 287-298. Ravi Chandran, K. S., and Jha, S. K. (2005). ”Duality of the S-N fatigue curve caused by competing failure modes in a titanium alloy and the role of Poisson defect statistics” Acta Materialia, 53, 1867-1881. Sheng, S., and Veers, P. (2011, May). “Wind turbine drivetrain condition monitoring – An overview” Paper presented at Mechanical Failure Prevention Group: Applied Systems Health Management Conference, Virginia Beach, Virginia. Shirani, M., and Härkegård, G. (2010). “Fatigue life distribution and size effect in ductile cast iron for wind turbine components” Engineering Failure Analysis, 18, 12-24. Shirani, M., and Härkegård, G. (2012). “Damage tolerant design of cast components based on defects detected by 3D X-ray computed tomography” International Journal of Fatigue, 41, 188-198. Todinov, M. T. (2000). “Probability of fracture initiated by defects” Materials Science and Engineering, A276, 39-47. Todinov, M. T. (2006). “Equations and a fast algorithm for determining the probability of failure initiated by flaws” International Journal of Solids and Structures, 43, 5182-5195. Toft, H. S., Branner, K., Berring, P. and Sørensen, J. D. (2011). ”Defect distribution and reliability assessment of wind turbine blades” Engineering Structures, 33, 171-180. 

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