An approach to wind-induced fatigue analysis of wind turbine tubular towers

Journal of Constructional Steel Research 166 (2020) 105917

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Journal of Constructional Steel Research

An approach to wind-induced fatigue analysis of wind turbine tubular towers Tao Huo a,b,c, Lewei Tong a,b,⁎ a b c

State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China College of Civil Engineering, Tongji University, Shanghai 200092, China East China Architectural Design & Research Institute Co., Ltd, Shanghai 200011, China

a r t i c l e

i n f o

Article history: Received 23 October 2019 Received in revised form 20 December 2019 Accepted 21 December 2019 Available online xxxx Keywords: Wind turbine tubular tower structure Wind speed and direction joint distribution function Wind-induced fatigue Low stress range fatigue damage treatment Time domain and time-frequency domain fatigue analysis method

a b s t r a c t This study discusses the wind-induced fatigue analysis of typical pitch-controlled 1.25MW wind turbine structures, with the reasonable consideration of the influence of low stress range fatigue damage, wind direction and the rotating effect of the blades. An integrated finite element model consisting of blades, nacelle, tower and foundation is established in ANSYS software. The wind speed and wind direction joint distribution function at the construction site of the wind turbine structures based on the observation datum of the meteorological station in China is put forward. The wind-induced response time history analyses of the tubular tower structures corresponding to different wind speed and wind direction cases are conducted. Finally, the wind-induced fatigue analysis theory of the tubular tower structures based on the time domain and time-frequency domain method are systematically established. The results indicate that the rotating effect of the blades should be considered in the fatigue life analysis of the tubular towers. Neglecting the influence of the wind direction and the low stress range on the fatigue damage of the tubular towers could lead to the decreased fatigue life which is conservative but not economic. Compared with the time domain method, the fatigue life of the tubular tower based on timefrequency domain method decreases, but the calculation is simpler. Adopting the equivalent stress range method considering rain flow correction to conduct the fatigue analysis of the tubular tower is recommended in this study. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction The wind turbine tubular tower structures are not only subjected to the periodic excitation generated by rotating blades, but also the action of the complex alternating wind load，the case which could produce continuously large-scale vibration and fatigue issues. The fatigue cracks are easy to appear and expand, especially in the areas where the stress concentration phenomenon occurs and welding locations where the fatigue defects are easily produced. When the cracks are out of control, the fatigue failure and collapse of the wind turbine structures could happen, which can decrease the application reliability of wind turbine structures, i.e. the wind-induced fatigue issue. The fatigue failure, maintenance and collapse of the tubular tower structures could not only bring about significant economic losses, but also cause undesirable social impacts. Therefore, the assessment of the fatigue strength or fatigue life of tubular tower structures is an important issue in the structural design of wind turbine tubular towers [1]. In fact, the fatigue behavior assessment of the tubular towers generally involves two major aspects: fatigue load effect such as stress range, ⁎ Corresponding author at: State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China. E-mail address: [email protected] (L. Tong).

https://doi.org/10.1016/j.jcsr.2019.105917 0143-974X/© 2020 Elsevier Ltd. All rights reserved.

and structural fatigue strength. The fatigue strength calculation methods of the tubular towers are carefully described in the relevant guidelines such as GL guideline [2] in Germany and DS472 guideline [3] in Denmark. The main difficulty is the structural effect under the action of the wind load and other dynamic loads. Due to the randomness and complexity of the load exerted on the tubular tower, and additional influence of the rotating blades, the structural load effect for the fatigue assessment of the tubular tower is more complicated. At present, there is no reliable method to accurately calculate the wind-induced response of the tubular towers. Another issue that needs to receive attention is that most of the existing fatigue assessments of the wind turbine structures are concentrated in the foundation and the fatigue problems of the tubular tower are seldom involved. Moreover, although the finite element method, which has high simulation accuracy, can obtain the accurate response and stress concentration at the detail location, the rotating effect of the blades cannot be considered in the finite element software. Therefore, how to consider the rotating effect of the blades in the finite element software becomes the key to guarantee the simulation accuracy. As the rotating sampled spectrum considering the rotating effect of the blades was proposed [4], Veers [5] firstly adopted the rotating spectrum model to simulate the wind field around the blades, which transform the kinematics problem of the rotating blades into the statics

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T. Huo, L. Tong / Journal of Constructional Steel Research 166 (2020) 105917

problem. He [6] proposed a rotational Fourier spectrum model based on the physical mechanism. In the literature of [7], the rotational Fourier spectrum model considering the influence of phase angle in the cross power spectrum were put forward by the authors in this study. Based on rotational spectrum model, the finite element method has been widely used in the integrated modeling, dynamic response analysis and fatigue life assessment of wind turbine structures. Some researchers have conducted a series of the fatigue life of the tubular tower based on the finite element method. Labassas etal. [8] carried out the dynamic response analysis of a 1 MW wind turbine structure, and adopted the nominal stress method to estimate the fatigue life of the tower bottom, welds and pre-tightened bolts based on the stress cycle number statistical results, which obtained by the rain flow counting method, corresponding to the wind speed interval [6.3, 18.2] m/s. Harte et al. [9] put forward a dynamic equation to consider the nonlinear relationship between the loading sequence of the stress cycle and fatigue strength and derived the generalized expression of fatigue cumulative damage. Based on the proposed fatigue analysis method, the fatigue life calculation of a 3 MW wind turbine tubular tower in Germany was conducted. Paredes et al. [10] established a tension-pressure damage model, in which the bearing capacity of the structure is still maintained under the compressive stress cycle after fatigue damage of the tubular tower appears under the tension stress cycle, and set up the horizontal connection joints at the junction of the tower segments. Adopting the parallel mixed material theory, the fatigue damage distribution of the reinforced concrete tubular tower subjected to the periodic cyclic loading. Do et al. [11] proposed a fatigue damage model including the crack expansion and used in the fatigue life analysis of a 5 MW wind turbine tubular tower structure. The results show that compared with the conventional fatigue life analysis method according to the S\\N curve, this method, which has longer fatigue life, is safer and more reliable. Shan [12] obtained the stress response power spectrum through the stress time history measured at the top of the tubular tower, and calculated the fatigue life of the tubular tower based on the Dirlik method. Zwick et al. [13] developed a statistical regression model to reduce the workload of the fatigue life assessment based on the time domain fatigue analysis method and used the model in the actual wind turbine structure. It was concluded that the fatigue design load case was reduced from 21 to 3 successfully, and the total fatigue damage difference was approximately 6%, indicating the high accuracy and small amount of calculation. Wieghaus et al. [14] proposed a fourstep probabilistic framework to predict the wind-induced fatigue life of lightweight wind-excited steel structures, and compared the fatigue life prediction results with compiled inspection records for a large traffic signal structure population in Wyoming, USA. The results indicated that the efficacy of the proposed probabilistic approach was verified. In light of the above, Most of the existing fatigue life investigations of the tubular towers focuses on the time domain method currently. Moreover, the fatigue life calculation mostly uses the single slope S\\N curve including the authoritative wind turbine design software such as GH Bladed [15] and FAST software. That is to say, the fatigue damage caused by low stress range cycle was not taken into account reasonably. Furthermore, most of the existing wind turbine professional design software, codes, and finite element analysis did not consider the rotating effect of blades and the impact of the wind direction on fatigue damage. Therefore, it is necessary to take an in-depth look into the wind-induced fatigue analysis of the tubular towers considering the influence of the wind direction, low stress range fatigue damage and the rotating effect of the blades reasonably. In this study, the study commences with the introduction of the considered wind turbine structures and the corresponding integrated finite element model. The joint distribution function of wind speed and wind direction at the site of the wind turbine structures based on the observation datum of the Datong and Huade meteorological station in China. In addition, the wind-induced response time domain analyses of the tubular tower structures corresponding to different wind speed and wind

direction cases were conducted. Finally, the wind-induced fatigue analysis of the tubular tower structures adopting time domain and time-frequency domain method with the reasonable consideration of the influence of the rotating effect of the blades, wind direction and low stress range fatigue damage were carried out. 2. Integrated finite element model of considered wind turbine structure This study considers a typical pitch-controlled 1.25 MW wind turbine structure in Zhangbei country of China, based on which an integrated finite element model was built. In addition, the three blades were simplified into cantilevers with a rectangular cross section. The length, width and depth of each blade are 54.38 m, 2.86 m and 0.064 m, respectively. The mass of the rotor (including the blades and hub) is 45,000 kg. The nacelle and its internal components were treated as an integrated part in the model. The length, width and height of the nacelle are 13.6 m, 4.7 m and 4.7 m, respectively. The mass of the whole nacelle is 85,000 kg. The main body of the tower is comprised of four segments with varying cross section properties, and the height of each segment is 8 m, 18 m, 24 m and 30 m (from bottom to top), making a total height of 80 m. The corresponding tube thicknesses for the three segments are 52, 42, 30, and 18 mm, respectively. The diameter of the tower increases linearly from 4.2 m at the bottom to 2.58 m at the top. A 10 m × 10 m × 1.8 m reinforced concrete raft foundation is located at the bottom of the tower, and the yaw angle of the tower is 0°. In the literature of [16], the modeling strategy was illustrated by the authors in this study. Herein, the detailed modeling approach is omitted and only the integrated finite element model diagram is given, as shown in Fig. 1. 3. The joint distribution function of wind speed and wind direction at the site of wind turbine structures Most of the existing studies only consider the influence of the wind speed on the fatigue damage and neglect the influence of the wind direction on the fatigue damage, the case which could lead to inaccurate and conservative fatigue life of wind turbine tubular tower. Therefore, it is suggested to construct the joint distribution function of wind speed and wind direction at the construction site of the wind turbine structures to consider the combined effect of the wind speed and wind direction on the fatigue life of wind turbine tubular towers. At present, the commonly used method to determine the joint distribution function of wind speed and wind direction at the site of the wind turbine structures is based on the long-term observation datum of the meteorological station near the construction location of the wind turbine structures. 3.1. Probability statistics of extreme wind speed corresponding to different wind speed and wind direction This study considers a existing 1.25 MW wind turbine structure located at Zhangbei country, Hebei Province, China. On the basis of being unable to acquire the observation datum from the meteorological station in Zhangbei region, the observation data (including wind speed and corresponding wind direction) of Datong meteorological station in Shanxi Province of China (From January 1955 to April 2017) and Huade meteorological station in Neimenggu Province of China (From December 1952 to April 2017), which can be found in Meteorological Science Data Sharing Service Platform of China, were selected as the source of wind speed samples. It is worth noting that the selection of meteorological station near the construction location of the wind turbine structures not only considers the influence of distance, but also considers the altitude, terrain characteristics and predominant wind direction. This study adopts the method of extracting the extreme wind speed at interval of 8 days to compose the extreme wind speed sample. The wind rose

T. Huo, L. Tong / Journal of Constructional Steel Research 166 (2020) 105917

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Fig. 2. Wind rose diagram of Datong meteorological station. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

of the extreme wind speed, the commonly used joint distribution function of wind speed and wind direction can be obtained. For Extreme-I distribution, the joint distribution function of wind speed and wind direction is given in Eq. (2):

 x−bðθÞ P ðx; θÞ ¼ f ðθÞ exp − exp − aðθÞ

ð2Þ

where a is the scale parameter; b is the position parameter. For Extreme-II distribution, the detailed formula of the joint distribution function of wind speed and wind direction is given in Eq. (3): "
−γðθÞ # x−bðθÞ P ðx; θÞ ¼ f ðθÞ exp − aðθÞ

Fig. 1. Integrated finite element model.

diagrams of Datong and Huade meteorological station are given in Fig. 2 through Fig. 3. It can be seen from Figs. 2 and 3 that the probabilistic distribution trends of the two meteorological stations corresponding different wind direction intervals are generally similar. Therefore, the hypothesis that the wind directions of meteorological station and the construction site of wind turbine structures are consistent is reasonable. With the consideration of the similar altitude and terrain characteristics, the observation data of the Huade meteorological station is finally adopted to calculate the joint distribution function of wind speed and wind direction at the construction site of the wind turbine structures.

ð3Þ

where γ is the shape parameter.

3.2. Parameter estimation of the joint distribution probability model of wind speed and wind direction at the site of the meteorological station Ge et al. [17] proposed that the joint distribution function of wind speed and wind direction P(x, θ) can be determined through two independent functions, namely, wind direction probability function f(θ) and the extreme wind speed distribution function in each wind direction F(x). The detailed formula can be expressed in Eq. (1): P ðx; θÞ ¼ f ðθÞF ðxÞ

ð1Þ

In general, F(x) can be classified into three types: Extreme-I (Gumbel) distribution, Extreme-II (Frechet) distribution and ExtremeIII (Weibull) distribution. Based on the probability distribution model

Fig. 3. Wind rose diagram of Huade meteorological station. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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T. Huo, L. Tong / Journal of Constructional Steel Research 166 (2020) 105917

For Extreme-III distribution, the detailed expression of the joint distribution function of wind speed and wind direction is defined by Eq. (4): (

"
γðθÞ #) x−bðθÞ P ðx; θÞ ¼ f ðθÞ 1− exp − aðθÞ

ð4Þ

In this study, the parameter estimation of the joint probability distribution function of wind speed and wind direction is based on two following assumptions: 1) the extreme wind speed samples corresponding to different wind directions follow the same probability distribution type which can be obtained by fitting the extreme wind speed samples in all wind directions; 2) the parameter estimation of probability distribution type of extreme wind speed corresponding to different wind directions is independent, and the relevant parameter can be estimated by the extreme wind speed sub-samples in each wind direction. Based on the extreme wind speed samples of the two meteorological stations and the first assumption, different parameter estimation methods such as least square method, moment method, probability weight method and maximum likelihood method were used and different goodness of fit tests indexes were adopted to assess the fitness of the different methods. Finally, the extreme wind speed samples of Huade meteorological station are all closer to the Extreme-I distribution. According to the second assumption, the parameter estimation results corresponding to different wind directions can be obtained, as shown in Table 1. 3.3. Curve fitting of wind direction probability function and included parameter of extreme wind speed distribution It can be seen from Table 1 that both the wind direction probability function and parameters included in the extreme wind speed probability distribution function change with the wind direction angle. Coles et al. [18] proposed adopting the harmonic function to reflect the distribution law of the wind direction probability function and parameters included in the extreme wind speed probability distribution function on the circumference. As for Extreme-I distribution, the corresponding curve fitting expressions are given in Eq. (5): f ðθÞ ¼ Lθ þ

nθ X

aðθÞ ¼ L þ

na X i¼1

bðθÞ ¼ Lb þ

nb X

f ðθÞ ¼ 0:061 þ 0:096 cosðθ þ 1:355Þ−0:063 cosð2θ−0:925Þ −0:045 cosð3θ þ 0:036Þ−0:034 cosð4θ þ 1:339Þ þ 0:031 cosð5θ−0:7Þ aðθÞ ¼ 2:709 þ 0:999 cosðθ−0:075Þ þ 0:477 cosð2θ−0:004Þ þ 0:483 cosð3θ−33:062Þ−0:346 cosð4θÞ þ 0:096 cosð5θ−0:541Þ bðθÞ ¼ 8:973 þ 1:880 cosðθ þ 1:248Þ−0:919 cosð2θ−39:246Þ þ 0:577 cosð3θ þ 13:371Þ þ 0:309 cosð4θ−0:465Þ þ 0:262 cosð5θ−1:110Þ

ð6Þ

3.4. The joint probability distribution function of wind speed and wind direction at the site of wind turbines In fact, it can be considered that the gradient wind speeds of the meteorological station and construction location of wind turbine structures are equal because of the closer distance. Therefore, the relation between the extreme wind speed variable of the meteorological station U M at any height and that of wind turbine structures U W at any height meets Eq. (7): UM

  M θ cos iθ−Nθ

UW

i¼1 a

where L, M and N represent the undetermined parameters in the harmonic function；The corresponding superscript θ, a and b are used to distinguish the wind direction probability function, the shape parameter and the position parameter respectively; nθ, na and nb are the order of the wind direction probability function, the shape parameter and the position parameter respectively. In this study, when the order takes 5, the determination coefficient R2 fully meet the accuracy requirements. Based on the discrete values shown in Table 1, the curve fitting expressions of wind direction probability function and included parameters in the extreme wind speed distribution function for Huade meteorological station are given in Eq. (6):

  Ma cos iθ−Na

ð5Þ

  M b cos iθ−Nb

i¼1

Table 1 The parameter estimation results corresponding to different wind directions for Huade meteorological station. Wind direction

f(θ)

b

a

N NNE NE ENE E ESE SE SSE S SSW SW WSW W WNW NW NNW

0.0078 0.00586 0.00098 0.00341 0.00244 0.00195 0.00487 0.02244 0.0268 0.04729 0.05802 0.07117 0.1214 0.37006 0.1887 0.06681

10.1301 8.4380 6.8680 6.3274 8.6158 7.6961 8.2440 7.8023 8.0176 8.8670 9.2360 9.8797 10.4773 11.2014 11.4333 11.0828

3.9799 4.3768 3.9126 2.8156 1.3606 1.9715 2.4957 2.9101 1.6261 1.7671 1.9660 2.0337 2.2572 2.9481 3.4432 3.3027


¼

hM HM

α M


HW hW

α W

ð7Þ

where hM represents the meteorological observation height of the meteorological station, the value of which is set as 10 m; hW represents the reference height (the height of hub) of wind turbine structures, and its value is takes as 82.35 m in this study; HM and HW are the gradient wind height at the meteorological station and construction site of the wind turbine structures respectively; αM and αW are the surface roughness index of the meteorological station and construction site of the wind turbine structures respectively. As mentioned above, wind direction of the meteorological station and construction location of wind turbine structures can be assumed to be same. Based on the Eq. (7), the joint distribution function of wind speed and wind direction at the site of the wind turbine structures is given in Eq. (8):

α M
 α HM hW W U M Nx; θ hM HW
αM

 hM HW αW x; θ ¼ P UM N HM hW 

α M
  hM HW αW x ¼ f ðθw Þ 1− F HM hW

  P U W Nx; θ ¼ P

ð8Þ

Therefore, for Extreme-I distribution, the joint distribution function of wind speed and wind direction at the site of the wind turbine

T. Huo, L. Tong / Journal of Constructional Steel Research 166 (2020) 105917

structures can be illustrated in Eq. (9). 8 2 0
αM
139  > > hM HW αW > > > x−bðθÞC7> = < 6 B   HM hW 6 B C7 P U W Nx; θ ¼ f ðθÞ 1− exp6− expB− C 7  > > 4 @ A 5 a ðθw Þ > > > > ; : 


 ξx−bðθÞ ¼ f ðθÞ 1− exp − exp − aðθÞ

ð9Þ


α
 hM M H W α W . where ξ ¼ HM hW Substituting Eq. (6) into Eq. (9), the detailed formula of the joint distribution function of wind speed and wind direction at the site of the wind turbine structures can be obtained, based on the observation datum of Huade meteorological station. For the considered wind turbine structures in this study, the mean wind speed at the height of hub U h is adopted to calculate the joint distribution function of wind speed and wind direction. The mean wind speed interval ½ðU h Þ j ; ðU h Þ jþ1  under each given wind direction interval [(θw)i, (θw)i+1] is defined as the [i, j]th case. Moreover, the wind speed increment ΔU h is 2 m/s and the wind direction angle increment (Δθw)i is 22.5∘. According to the observation datum of Huade meteorological station, the occurrence probabilities of the extreme wind speed samples in different cases at the construction site of the wind turbine structures are shown in Fig. 4. 4. Wind-induced response analysis of wind turbine tubular towers At present, the time domain analysis method is the only recognized approach to carry out the wind-induced response calculation of wind turbine structures. In this study, due to the introduction of the improved rotational Fourier spectrum model (considering the influence of phase angle in the cross power spectrum) [7], the rotating effect of the blades is considered. Therefore, the traditional finite element method can be adopted to carry out the wind-induced response calculation of the wind turbine structures. According to the requirement of IEC61400–1 standard [19] and DS472 standard [3], the design cases including the normal operation, start up, normal shut down and parked (standing still or idling) state

Fig. 4. The three-dimensional columnar distribution diagram of the cumulative damage for each wind speed and wind direction case.

5

should be considered to carry out the wind-induced response analysis. However, Thomsen [20] presented that the fatigue damage mainly comes from the normal operation state, which accounts for 99.5% of the total fatigue damage. Therefore, the wind-induced response analyses corresponding to different wind speed and direction under the normal operation state (3m=sbU h b25m=s) are conducted in this study. 4.1. Wind-induced response dynamic model corresponding to different wind speeds and wind directions In the literature of [21] written by the authors in this study, the aerodynamic loads exerted on the rotating blades and the along-wind, cross-wind aerodynamic load exerted on the tubular towers were calculated using the Blade Element Momentum (BEM) theory and bluff body aerodynamics theory, respectively. The wind direction angle is defined in Fig. 5. The wind direction rotates along the anti-clockwise direction. The wind direction angle increment and analysis case definition of wind-induced response calculation are the same with the case of joint distribution function of wind speed and wind direction. When the wind turbine structure is in normal operation, the mean wind speed at the height of the hub U h must satisfy the inequality 3m=sbU h b25m= s. Because the tubular tower structure is nearly symmetrical about X and Y axis, and the blades are always facing the wind (without considering the influence of the wind direction on the blade), only the windinduced responses of wind turbine structure in the wind direction interval [0°, 90°] are calculated. 4.2. The influence of the rotating effect of the blades on the wind-induced response of the tubular tower Based on the field test datum, it can be found that the rotating effect of the blades could bring about the fundamental change to the tubular wind field around the blades [4–7]. Therefore, the rotating effect of the blades inevitably has a significant influence on the wind-induced response and fatigue life of tubular tower structures. 4.2.1. Height-wise stress response and stress responses corresponding to different wind speeds When the wind speed is 16 m/s and wind direction angle is 45°, the comparisons of the height-wise stress responses mean value and root mean square (RMS) values with and without considering the rotating effect of the blades are shown in Fig. 6. In addition, the influences of the rotating effect of the blades on the stress responses caused by different wind speeds at the bottom of the tubular tower are provided in Fig. 7. It is worth noting that the normal stress along the direction of the tower height σz is adopted as the basis for stress response comparison. It can be seen from the Fig. 6 that no matter whether the rotating effect of the blades is considered or not, with increase in tower height, the stress responses all exhibit a decreasing trend, although a certain level of fluctuations are observed. These fluctuating points corresponding to the stepped tube thickness variation as well as locations where the bending stiffness changes evidently. It can be observed in Fig. 7 that for all the considered case, with increase in wind speed, stress responses of the tubular towers exhibit an increasing trend. It is worth noting that due to the commercial secret, the wind turbine manufacturer cannot provide the blade airfoil parameter datum. Therefore, the influence of the blade pitch angle variation on the wind-induced response and fatigue life is not considered when the wind speed is greater than the rated wind speed and less than the cut-out wind speed. Furthermore, compared with the case without considering the rotating effect of the blades, the stress response statistic values induced by considering the rotating effect of the blades increase. In addition, compared with the mean value of the stress response, the influence of the

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T. Huo, L. Tong / Journal of Constructional Steel Research 166 (2020) 105917

Fig. 5. The schematic diagram of wind direction angle.

Tower Height /m

4.2.2. Stress response at the fatigue checking location In order to further investigate the influence of the rotating effect of the blades on the stress response of the tubular tower, the stress response of the fatigue checking location (‘A' point in Fig. 8) in the connection area of the tubular tower and the doorframe is selected. The corresponding results are provided in Fig. 9. Comparision results (see Figs. 7 and 9) indicate that compared with the other locations of the tubular tower, the influcence of the rotating effect of the blades on the stress response statistic values of the fatigue checking location is more significant, especially for RMS value of the stress response. For the welded steel structure studied in this study, compared with the mean value, the RMS value of the stress (an important parameter in frequency domain analysis of wind-induced fatigue life) can induce greater impact on the wind-induced fatigue life of the tubular tower. Another issue which needs to be addressed is that Powell and Connell [22] adopted the PNL model to consider the influence of

80 70 60 50 40 30 20 10 0

With considering the rotating effect of the blades Without considering the rotating effect of the blades

0

5 10 15 20 25 Mean value of the stress response /MPa

rotating blades on the wind field around the blades, and conducted the fatigue life comparison with and without considering the rotating effect of the blades. The results indicated that the fatigue life induced by neglecting the rotating effect of the blades can be 10 times of the corresponding values caused by considering the rotating effect of the blades. Therefore, in this study, it is proposed that the rotating effect of the blades must be considered when conducting the fatigue life prediction of the tubular tower. Otherwise, the fatigue life predication could lead to the unsafe results. It is worth noting that the accuracy of the proposed wind-induced response calculation approach has been verified by the field test in Reference [21] by the authors in this study. Moreover, the validated windinduced response analysis method was used to conduct the fatigue life prediction of wind turbine tubular towers in this study. 5. Wind-induced fatigue analysis of the tubular tower Under the combined excitation of the wind load and the rotating blades, the tubular towers could produce large amplitude vibration.

Tower height /m

rotating effect of the blades on the RMS value of the stress response is more significant.

80 70 60 50 40 30 20 10 0

With considering the rotating effect of the blades Without considering the rotating effect of the blades

0

1 2 3 4 5 6 RMS value of the stress response /MPa

Fig. 6. The influence of the rotating effect of the blades on the height-wise stress response.

30 24

RMS value of stress responseMPa

Mean value of the stress responseMPa

T. Huo, L. Tong / Journal of Constructional Steel Research 166 (2020) 105917

With considering the rotating effect of the blades Without considering the rotating effect of the blades

18 12 6 0

5

10 15 20 Wind speed/m/s (a) Mean value of the stress response

25

6

7

With considering the

5 rotating effect of the blades Without considering the

4 rotating effect of the blades 3 2 1 0

5

10 15 20 Wind speed/m/s (b) RMS value of the stress response

25

Fig. 7. The influence of the rotating effect of the blades on the stress response induced by different wind speeds.

After a large number of the stress cycles, the wind turbine structure could lead to high-cycle fatigue failure, which would greatly reduce the service life of the tubular tower and increase the maintenance cost. Therefore, it is necessary to systematically study the windinduced fatigue issue of wind turbine tubular tower structures. Combined with the joint distribution function of wind speed and wind direction at the construction site of the wind turbine structures obtained in Chapter 3 and wind-induced response calculation results corresponding to different wind speed and wind direction cases in Chapter 4, the time domain and time-frequency domain fatigue analysis theory of wind turbine tubular tower structures are systematical established considering the influence of wind direction, the rotating blades and low stress range fatigue damage based on the fatigue cumulative damage theory and random fatigue theory. 5.1. S\\N curve Studies have shown that beside the higher stress range, the stress range below the constant fatigue limit can also affect fatigue life.

Especially after the fatigue cracks are generated, the stress range below the fatigue limit could accelerate the expansion of the fatigue cracks. In view of this, it is essential to deal with the S\\N curve of the high life area where the structure is subjected to the low stress range levels. At present, the steel structure fatigue design codes in the world generally adopts the different treatment approaches. It is worth noting that the code for design of steel structures in China [23] uses the single-slope S\\N curve. For the connection of the wind turbine tubular tower structures, GL guideline [2] recommends to use a variable slope S\\N curve which consists of two oblique straight lines, as shown in Fig. 10. In this study, two connection regions of the tubular tower structures, namely, the connection region between the tubular tower and doorframe and the connection region between the tower bottom and the foundation, were selected to carry out the wind-induced fatigue life analysis of the tubular towers. This is because that the two areas mentioned above are inclined to produce the fatigue failure. After the detailed analysis and comparison, the A point is selected as the fatigue checking location for tubular towers in the connection area

Fig. 8. The diagram of the fatigue checking location A.

RMS value of the stress responseMPa

T. Huo, L. Tong / Journal of Constructional Steel Research 166 (2020) 105917

Mean value of the stress responseMPa

8

150 120 With considering the rotating effect of the blades Without considering the rotating effect of the blades

90 60 30 0

5

10 15 20 Wind speed/m/s

25

˄a˅Mean value of the stress response

24 18

With considering the rotating effect of the blades Without considering the rotating effect of the blades

12 6 0 5

10 15 20 Wind speed/m/s

25

˄b˅RMS value of the stress response

Fig. 9. The influence of the rotating effect of the blades on the stress response of the fatigue checking location.

Fig. 10. The S\ \N curve of the wind turbine tubular tower structures.

between the tubular tower and doorframe. The detailed constructional details are provided in Fig. 11(a). Since the doorframe is mainly used to prevent the local buckling near the door, the doorframe and weld joint are basically in an unstressed state and only play a reinforcing role. According to the GL guideline [2], this constructional detail is consistent with the Detail category 256. The corresponding constructional details of the fatigue checking location in GL guideline is shown in Fig. 11(b). In addition, the fatigue strength corresponding to the stress cycle number 2 × 106 is 90 MPa. As for the connection area between the tower bottom and the foundation, the B point was chosen as the fatigue checking location. The

Fig. 12. The detailed position of the fatigue checking location B.

detailed position of the fatigue checking location B is given in Fig. 12. The tower section at the bottom of the tubular tower and the reinforced concrete foundation are usually connected by the embedded ring. Taking the thick-shaped forging flange as an example, the constructional detail around the fatigue checking location B is shown in Fig. 13. It can be seen from the Fig. 13 that both the connection region between the tower section at the bottom of the tubular tower and the upper flange plate and the connection region between the lower flange plate and the bottom location of the embedded ring are connected by the butt welds. The existing failure cases have confirmed that the vicinity of the heat affected areas of the two butt welds are the weakest

Fig. 11. The constructional detail of the fatigue checking location A.

T. Huo, L. Tong / Journal of Constructional Steel Research 166 (2020) 105917

9

where Pij is the occurrence probability of each wind speed and wind direction case; fΔσ m is the occurrence frequency corresponding to the m-th stress range Δσm under the [i, j]th case; β and C are the constant of the low life area related to the properties of the material; p is the total number of wind direction angle, the value of which is 12 in this study; q is the total number of the wind speeds which participate in the calculation of wind-induced response and its value is taken as 11 in this study; fL ij is the average number of stress ranges per unit time and the detailed expression is given in Eq. (11): f Lij ¼

Fig. 13. The constructional detail around the fatigue checking location B.

locations of the tower bottom. The corresponding constructional detail in GL guideline [2], which is in accordance with Detail category 232, is shown in Fig. 14. Moreover, the fatigue strength corresponding to the stress cycle number 2 × 106 is 71 MPa.

5.2.1. The random fatigue analysis theory based on the time domain method After a series of derivation, the fatigue cumulative damage D of the structure in an arbitrary time length T is given in Eq. (10):

D¼T

p X q P f Λ X k X ij Lij ij i¼1 j¼1

C

f Δσ m ðΔσ m Þβ

m¼1

Fig. 14. The constructional detail of the butt weld area in GL guideline.

ð10Þ

ð11Þ

where NL ij is the total cycle number of the stress range obtained by the rain flow method during the time length Lij under the [i, j]th case. Λij is the damage correction coefficient considering the low stress range fatigue damage, i.e. the ratio of the cumulative damage calculated by the variable slope S\\N curve consists of two oblique straight lines over the cumulative damage obtained by the he single-slope S\\N curve. Generally, the damage correction coefficient is less than 1, indicating that the cumulative damage would decrease when the variable slope S\\N curve consists of two oblique straight lines is used. The detailed expression is provided in Eq. (12). ðΔσ l Þβ−β

5.2. Time domain fatigue analysis method of the tubular tower structures As mentioned above, the time domain analysis method is adopted to conduct the wind-induced response analysis. This method, which can be directly understand the dynamic behavior and fatigue information, is verified to have high accuracy and suitable to deal with the fatigue issue of the non-Gaussian stress process and the nonlinear structure. Based on the time domain response results, the wind-induced fatigue life analysis method for wind turbine tubular towers in this study can be divided into two types: time domain method and time-frequency method. It is worth noting that the time-frequency fatigue analysis method consists of the time domain wind-induced response analysis and frequency domain random fatigue analysis. For time domain fatigue analysis method, based on the stress response time history of the fatigue checking location, the rain flow method is adopted to obtain the frequency distribution histogram of the stress range. Furthermore, the fatigue life of the tubular towers are finally obtained according to the time domain fatigue analysis theory described in Section 5.2.1.

NLij Lij

0

X

0

Δσ m ≤ Δσ l

Λij ¼

f Δσ m ðΔσ m Þβ þ

Pk m¼1

X Δσ m NΔσ l

f Δσ m ðΔσ m Þβ

f Δσ m ðΔσ m Þβ

ð12Þ

where Δσl is the constant fatigue limit; C′ are the constant of the high life area related to the properties of the material; In this study, β is equal to 3; the relation between β′ and β meets the equation β′ = β + 2 = 5. In the fatigue analysis, the stress parameter Ω considering the fatigue damage correction of the low stress range can be defined in Eq. (13): p X q X

Ω¼

i¼1 j¼1

P ij f Lij Λij

k X

f Δσ m ðΔσ m Þβ

ð13Þ

m¼1

Therefore, Eq. (10) can be described in Eq. (14): D¼

TΩ C

ð14Þ

Moreover, the time length corresponding to fatigue failure is described with the variable Tf. Therefore, when the fatigue failure of the structure occurs, the corresponding fatigue life Tf of the structure can be obtained by Eq. (15). Tf ¼

C Ω

ð15Þ

5.2.2. Time domain fatigue analysis results of the tubular towers The empirical spectral moment correction method, which commonly used in the initial fatigue life estimation and described in Section 5.3.7, is adopted to compare the fatigue life at the fatigue checking location A and fatigue checking location B. The comparison results show that the fatigue checking location A is more unfavorable. It is worth noting that the fatigue checking position A is located at the maximum probability interval of the wind direction (as indicated in Chapter 3). Moreover, the normal stress σz along the tower height direction is used as nominal stress for the wind-induced fatigue analysis of the tubular towers. When the wind speed is 12 m/s and wind direction angle is 45°, the stress response results of the fatigue checking location A are given in Fig. 15. Based on the rain flow method, the stress range and

10

T. Huo, L. Tong / Journal of Constructional Steel Research 166 (2020) 105917

Stress response(Mpa)

136 128 120 112 104 96 0

100

200 300 Times

400

Fig. 15. The stress response of the fatigue checking location A.

Stress amplitude cycle number

Fig. 18. The fatigue cumlative damage distribution for each wind speed and wind direction case.

60 50 40 30 20 10 0

5

10 15 20 25 Stress amplitude MPa

30

Fig. 16. The stress range and corresponding cycle number.

5.2.3. The influence of wind direction, mean stress and low stress range on the fatigue life of the tubular towers As mentioned above, the influence of the wind direction is considered by introducing the joint distribution function of wind speed and wind direction. However, most of the wind-induced fatigue analysis did not consider the influence of wind direction, which leads to the fact that the accuracy of this practice needs to be further discussed. For the considered wind turbine structure in this study, the windinduced fatigue life Tf without considering the influence of wind direction is equal to 6.6 year. Obviously, the fatigue life without considering the influence of the wind direction does not meet relevant requirement that the minimum service life of the tubular tower is more than 20 years. In addition, the calculation results are relatively conservative, which finally affects the economy of the structural design of the wind turbine structures.

Fatigue cumulative damage

Fatigue cumulative damage

corresponding cycle number are illustrated in Fig. 16. Another issue that needs to be carefully addressed that as the research object of the fatigue analysis in this study is welded steel structure, the mean stress response has little influence on the fatigue life [23]. Therefore, this study did not perform the Goodman correction on the rain flow counting results. Based on the time domain fatigue analysis theory described in Section 5.2.1 and corresponding program developed by MATLAB software, the wind-induced fatigue life Tf of the wind turbine tubular towers corresponding to time domain method is equal to 34.73 year. In addition, the distribution of fatigue cumulative damage along the wind direction and wind speed are provided in Fig. 17.

It can be seen from Fig. 17 (a) that wind direction has significant influence on the wind-induced fatigue cumulative damage of the structure, and the greater fatigue cumulative damage occurred at the larger occurrence probability of wind direction. Moreover, as indicated in Fig. 17 (b), when the wind speed is greater than 16 m/s and less than 24 m/s, the corresponding fatigue cumulative damage is greater. In order to more fully describe the fatigue cumulative damage distribution of the tubular tower under different wind speed and wind direction cases, the three-dimensional columnar distribution diagram of the cumulative damage for each wind speed and wind direction case during the fatigue life Tf are shown in Fig. 18.

2.0E-7 1.5E-7 1.0E-7 5.0E-8 0.0 0

50 100 150 200 250 300 350 Wind direction angle /(degree)

(a) Fatigue damage distribution along the wind direction

1.2E-7 8.0E-8 4.0E-8 0.0 4

8

12 16 20 24 Wind speed /(m/s) (b) Fatigue damage distribution along the wind direction

Fig. 17. The fatigue cumulative damage distribution along wind direction and wind speed.

T. Huo, L. Tong / Journal of Constructional Steel Research 166 (2020) 105917

Another issue that needs to receive attention is the influence of the low stress range on the fatigue life. In this study, based on the GL guideline, variable slope S\\N curve consists of two oblique straight lines are used to reflect the impact of the low stress range fatigue damage. However, the code for design of steel structures in China adopts the single slope S\\N curve. In view of this, the damage correction coefficient Λij is introduced to correct the fatigue life results obtained by the single slope S\\N curve. In this study, in order to explicit the influence of the low stress range fatigue damage, the wind-induced fatigue life Tf of the tubular tower using the single slope S\\N curve is equal to 17.86 year. The comparison of Eqs.(16) and (18) shows that the windinduced fatigue life without considering the influence of the low stress range fatigue damage is 0.5 times of the corresponding values considering the influence of the low stress range fatigue damage. Therefore, if the single slope S\\N curve is used to calculate the fatigue life of the tubular tower in the actual design, the structural design cost would be greatly increased although the safety control of the tubular tower is stricter. In order to accurately reflect the influence of the mean stress on the fatigue life of wind turbine tubular towers, Goodman correction [24] is introduced to correct the rain flow counting results. For an asymmetric stress cycle, the equivalent stress range Δσe corresponding to zero mean stress can be described in Eq. (16).

11

5.3.1. The random fatigue analysis theory based on the time-frequency domain method It is generally considered that the stress range distribution of the tubular tower can be expressed by continuous probability density function. The fatigue cumulative damage of the structure at arbitrary time length T can be described by Eq. (17):   T XX P f Λ E Δσ β ij C i¼1 j¼1 ij Lij ij p

D¼

q

ð17Þ

where Λij is the damage correction coefficient, and the detailed expression is given in Eq. (18). Λij ¼

ðΔσ l Þβ−β

0

R Δσ l 0

0

ðΔσ Þβ f Δσ ij ðΔσ ÞdΔσ þ

R þ∞ Δσ l

ðΔσ Þβ f Δσ ij ðΔσ ÞdΔσ

EðΔσ β Þij ð18Þ

where fΔσ ij(Δσ) is the probability density function of the stress range under the [i, j]th case. In addition, stress parameter Ω considering the low stress range fatigue damage can be defined by Eq. (19): Ω¼

p X q X

  P ij f Lij Λij E Δσ β ij

ð19Þ

i¼1 j¼1

,
 σm Δσ e ¼ Δσ 1 1− σb

ð16Þ

where Δσ1 represents the stress range of asymmetric stress cycle; σm is the mean stress; σb is the tensile strength of material. After considering the influence of the mean stress, the wind-induced fatigue life of the wind turbine tubular tower is equal to 33.83 year. It can be seen that the influence of mean stress on the fatigue life for the welded steel structures is small. That is to say, neglecting the influence of mean stress on the fatigue life of welded steel structures is acceptable.

5.3. Time-frequency domain fatigue analysis method of the tubular tower structures The practice that stress cycles leading to the fatigue damage are identified by using the rain flow counting method needs a large amount of calculation, therefore it is not convenient for practical application. In this study, based on the wind-induced stress response time history of the tubular tower, the stress response power spectrum corresponding to the different wind speed and wind direction cases can be obtained by spectrum analysis. Furthermore, according to the obtained power spectral density function, the probability density function of the stress range can be gained through the crossing analysis. Finally, the fatigue life of the tubular tower can be calculated based on the frequency domain fatigue analysis theory. This practice is called the time-frequency domain analysis method. The time-frequency domain method usually has a relatively small amount of calculation and can obtain relatively accurate results. The time-frequency fatigue analysis method can be divided into the following types: the ideal narrow band method, the equivalent narrow band method, the equivalent stress range method, Dirlik method and so on. In this study, the fatigue life of the tubular tower is calculated based on these methods and compares with the fatigue life results based on the time domain fatigue analysis method. Finally, the reasonable method is recommended to guide the fatigue design of the wind turbine structure in the practical application.

Substituting Eq. (19) into Eq. (15), the fatigue life of the structure can be obtained. It can be seen from Eq. (19) that the key of the structural fatigue life calculation is how to obtain the stress range probability density function fΔσ ij(Δσ) and the average number of the stress range per unit time fL ij. (1) Stationary Gauss narrow band process (ideal narrow band method) When the stress process is the stationary Gauss narrow band process, the probability density function of the stress range can be described by Eq. (20). Δσ ðΔσ Þ2 f Δσ ðΔσ Þ ¼ exp − 8σ 2s 4σ 2s

! 0 ≤ Δσ b þ ∞

ð20Þ

where σs is the root mean square of the stress response. According to the cross analysis theory, the average frequency of the stress range fL ij is equal to the crossing zero rate ν0 or peak value rate νp of the stress process. The detailed expression of ν0 and νp are defined by Eq. (21): sffiffiffiffiffiffi λ2 ν0 ¼ λ0

sffiffiffiffiffiffiffiffiffiffiffiffi λ4 νp ¼ λ2

ð21Þ

where λ0 ¼ σ 2s ; λ2 ¼ σ 2_ ; λ4 ¼ σ €2 . s s Substituting Eq. (20) into Eq. (18), the damage correction coefficient Λij of the stationary Gauss narrow band process can be described by Eq. (22).
0 
 β β β−β0 γ þ 1; uij þ 1; uij Γ  2 2
 þ
 Λij ¼ uij 2 β β þ1 þ1 Γ Γ 2 2 

where uij ¼

ð22Þ



 ðΔσ l Þ2  β0 β þ 1; uij are the kinds of the þ 1; uij andΓ ;γ 2 2 2 8σ sij

incomplete Gamma functions. (2) Stationary Gauss broad band process

12

T. Huo, L. Tong / Journal of Constructional Steel Research 166 (2020) 105917

When the stress process is the stationary Gauss wide band process, there are two kinds of the definition of the stress cycle, namely, Δσ = 2σ p and Δσ = 2|σ p|. σp is the stress peak value. When the stress range Δσ and stress peak value σp meet the equation Δσ = 2|σp|, the total stress ranges including the negative stress ranges are considered. Obviously, the fatigue life calculation results are more accurate. Therefore, this definition of the stress cycle (Δσ = 2|σ p|) is used to obtain the probability density function of the stress range. The corresponding probability density function of the stress range is provided in Eq. (23). 2

ð24Þ

The damage correction coefficient Λij of the stationary Gauss wide band process can be expressed in Eq. (25). 2 pffiffiffiβ0 3
0      pffiffiffi β0
β0 2 2 β þ1 7 β 0 þ2 β0  0 ; uij εij σ sij þ 2 2σ sij γ þ 1; uij ηij 2Φij −1 5 4 pffiffiffi γ 2 2 π

β−β 0 6

Λij ¼

 11=β  pffiffiffiβ
β p q þ 1 ∑i¼1 ∑ j¼1 P ij f Lij Λij σ βsij ηij Φij C 2 2 Γ 2 C þ p q A ∑i¼1 ∑ j¼1 P ij f Lij

! ðΔσ Þ2 0 ≤Δσb þ ∞ 8σ 2s

λ2 η ¼ pffiffiffiffiffiffiffiffiffiffiffi λ0 λ4

EðΔσ β Þij  pffiffiffiβ
     pffiffiffi β
β 2 2 βþ1 þ pffiffiffi Γ σ βsij þ 2 2σ sij Γ ; uij εβþ2 þ 1; u0ij ηij 2Φij −1 ij 2 2 π þ EðΔσ β Þij

ð25Þ ðΔσ l Þ2 0 ðΔσ l Þ2 ; u ¼ . ij 2 8σ 2sij 8σ 2sij εij

The average frequency of the stress range fL ij is equal to the peak value rate νp of the stress process. 5.3.2. The equivalent stress range method Similar with the constant amplitude fatigue, the random fatigue issue still recommend to use a unified formula to calculate the fatigue life, namely, the equivalent stress range. After a series of derivation, the definition of the equivalent stress range can be described by Eq. (26).

Δσ e ¼

¼


1=β Ω ¼ fL

p

q

p

q

∑i¼1 ∑ j¼1 P ij Ωij

!1=β

∑i¼1 ∑ j¼1 P ij f Lij

  !1=β p q ∑i¼1 ∑ j¼1 P ij f Lij Λij E Δσ β ij p

q

∑i¼1 ∑ j¼1 P ij f Lij

∑i¼1 ∑ j¼1 P ij f Lij Δσ βeij p

¼

q

p

!1=β

q

∑i¼1 ∑ j¼1 P ij f Lij ð26Þ

ð28Þ

In fact, although the second definition of the stress cycle, namely, Δσ = 2|σp|, has produced relatively accurate results, its probability distribution of the stress range is quite different from the corresponding results based on the rain flow counting method, which leads to the great difference between the calculated fatigue life and the fatigue life obtained by the time domain fatigue analysis theory. Therefore, the relevant rain flow correction is needed and the detailed correction expression is given in Eq. (29). DW ij ¼ ξij DRij

ð29Þ

where DW ij is the fatigue cumulative damage considering the rain flow correction; DR ij is the fatigue cumulative damage of the broad band process when the stress range follows Rice distribution (as detailed in Eq. (23)); ξij is the rain flow correction coefficient. After considering the rain flow correction, the stress parameter Ω can be described by Eq. (30). Ω¼

p X q X i¼1 j¼1

where uij ¼

ð27Þ

For the stationary Gauss broad band process, the detailed expression is given in Eq. (28):

ð23Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffi where ε is the bandwidth parameter, ε ¼ 1−η2 ;η is the irregular coefficient, its detailed expression is defined by Eq. (24).

ðΔσ l Þ

 0
11=β β p q β pffiffiffiBΓ 2 þ 1 ∑i¼1 ∑ j¼1 P ij f Lij Λij σ sij C C Δσ e ¼ 2 2B p q @ A ∑i¼1 ∑ j¼1 P ij f Lij

0 pffiffiffiβ
 p q 2 2 β þ 1 XX B pffiffiffi P ij f Lij Λij εijβþ2 σ βsij Γ B B 2 π 2 i¼1 j¼1 B Δσ e ¼ B Xp Xq B P f B i¼1 j¼1 ij Lij @

!

ε ðΔσ Þ f Δσ ðΔσ Þ ¼ pffiffiffiffiffiffi exp − 2 2 8σ s ε 2πσ s # pffiffiffiffiffiffiffiffiffiffiffiffiffi " pffiffiffiffiffiffiffiffiffiffiffiffiffi! ðΔσ Þ 1−ε2 ðΔσ Þ 1−ε2 −1 2Φ þ 2σ s ε 4σ 2s exp −

For the stationary Gauss narrow band process, the detailed expression is given in Eq. (27):

  P ij Λij ξij f Lij E Δσ β R

ij

ð30Þ

Karadeniz [25] put forward to the empirical expression of the rain flow correction coefficient ξij, which is provided in Eq. (31).  β ξij ¼ 1 þ εij g 0 þ ε2ij g 1 þ ε3ij g 2

ð31Þ

where g0, g1and g2 are all the function of β. The relevant calculation expressions are shown in Table 2. 5.3.3. The equivalent narrow band method Most of the stress processes in the actual engineering are broad band process. However, the broad band Gauss processes still have the disadvantages of relatively complex calculation, unclear relationships between the stress peak value and stress range, and so on. On the contrary, when the stress process is the narrow band process, the calculation is more convenient and the applicability in the actual engineering is stronger. Therefore, the equivalent narrow band method is proposed to calculate the fatigue cumulative damage of the broad band stress process, i.e. the fatigue cumulative damage of the narrow band process multiplied by an correction coefficient. In order to guarantee the accuracy of the theoretical model, the theoretical calculation results can be properly corrected by a large number of the results based on the rain flow counting method, namely,

T. Huo, L. Tong / Journal of Constructional Steel Research 166 (2020) 105917 Table 2 The detailed calculation expression of the rain flow correction parameter. pffiffiffiffi β þ c2 β

β ≤ 6.0

g0 = c0 + c1β + c2β3

g1 = c0 + c1β + c2β3

g 2 ¼ c0 þ c1

c0 c1 c2

6.9147 × 10−2 −4.4101 × 10−2 1.3972 × 10−4

−2.8847 × 10−1 9.1178 × 10−2 −4.3161 × 10−4

−1.4279 1.0658 −0.2236

β N 6.0

g0 = c0 + c1β + c2β3

g1 = c0 + c1β + c2β3

g 2 ¼ c0 þ c1

c0 c1 c2

−2.79 × 10−2 −2.2944 × 10−2 4.9719 × 10−6

−3.0064 × 10−2 3.3988 × 10−2 −2.9019 × 10−5

−1.7032 × 10−1 5.8372 × 10−3 −5.1602 × 10−4

D0W ij ¼ μ ij DNij

ð32Þ

where DW ′ ij is the fatigue cumulative damage of the broad band process considering the rain flow correction; DN ij is the fatigue cumulative damage of the narrow band process which has the same stress root mean square value with the broad band Gaussian process; μij is the rain flow correction coefficient. After considering the rain flow correction, the stress parameter Ω can be described by Eq. (33).

Ω¼

  pffiffiffi β
β þ1 P ij Λij μ ij f Lij 2 2σ sij Γ 2 j¼1

p X q X i¼1

ð33Þ

where Γ(x) is the Gamma function. Wirsching and Light [26] proposed the empirical formula of the rain flow correction coefficient μij and the detailed form is given in Eq. (34).

   pffiffiffi β0
β 0 0 0 0 0 ðΔσ l Þβ−β D1 ð2Q Þβ γ ðβ 0 þ 1; uij Þσ βsij þ 2 2σ sij γ  þ 1; u0ij D2 jR1 jβ 2 Λij ¼ EðΔσ β Þij   pffiffiffi β0
β 0   0 þðΔσ l Þβ−β 2 2σ sij γ þ 1; u″ij D3 þ D1 ð2Q Þβ Γ β þ 1; uij σ βsij 2 þ EðΔσ β Þij


     pffiffiffi β
β pffiffiffi β β 2 2σ sij Γ þ 1; u0ij D2 jR1 jβ þ 2 2σ sij Γ þ 1; u″ij D3 2 2 þ EðΔσ β Þij

ð37Þ pffiffiffi β where EðΔσ β Þij ¼ σ βsij ½D1 Q β Γð1 þ βÞ þ ð 2Þ Γð1 þ βÞðD2 jR1 jβ þ D3 Þ， uij ¼

ð34Þ

where a = 0.926 − 0.033β; b = 1.587β − 2.323. 5.3.4. Dirlik fatigue damage formula Dirlik [27] proposed that the probability density function of the stress range is composed of two Rayleigh distributions and an exponential distribution. The specific expression is shown in Eq. (35). ! " !#
 1 D1 Z D2 Z Z2 Z2 þ 2 exp − 2 þ D3 Z exp − exp − 2σ s Q Q 2 R1 2R1

5.3.5. Zhao-Baker fatigue damage formula As for the stationary Gauss stress process, Zhao and Baker [28] proposed that the probability density function of the stress range can be composed of the Rayleigh distribution and the Weibull distribution. The detailed expression is provided in Eq. (38).

f Δσ ðZ Þ ¼ wa1 b1 Z

1:25ðη−D3 −D2 R1 Þ η−xm −D21 ；R1 ¼ . D1 1−η−D1 þ D21 The stress parameter Ω considering the low stress range fatigue damage is described by Eq. (36).  pffiffiffiβ
 β

 β  P ij f Lij Λij 2σ sij D1 Q β Γð1 þ β Þ þ 2 Γ 1þ D2 jR1 jβ þ D3 2 j¼1

p X q X i¼1

  Z2 exp −a1 Z b1 þ ð1−w1 ÞZ exp − 2

! ð38Þ

Δσ ; w1 is the weight coefficient, 0 ≤ w1 ≤ 1; a3 and b3 is 2σ s the Weibull coefficient. 1−η rffiffiffi
 2 1 −1=b3 a 1− Γ 1þ π b3 3 8 < 1:1 ηb0:9 a3 ¼ 8−7η b3 ¼ : 1:1 þ 9ðη−0:9Þ ηN0:9

ð36Þ For the Dirlik method, the average frequency of the stress range fL ij is equal to the peak value rate νp of the stress process.

ð39Þ

The stress parameter Ω with the consideration of the low stress range fatigue damage is given in Eq. (40). Ω¼

p X q X

P ij f Lij Λij σ βsij

i¼1 j¼1

2

β b3

−

6 6w1 a 3 4

sffiffiffiffiffiffi λ2 2ðxm −η2 Þ 1−η−D1 þ D21 ；D2 ¼ ； ；D1 ¼ 2 λ4 1−R1 1þη

D3 = 1 − D1 − D2；Q ¼

Ω¼

b1 −1

where Z ¼

ð35Þ Δσ λ1 ; xm ¼ where Z ¼ 2σ s λ0

Δσ l ðΔσ l Þ2 ðΔσ l Þ2 ，u0ij ¼ ， u″ij ¼ . 2 σ sij Q 2σ 2sij 2σ 2sij R1

w1 ¼

 b μ ij ¼ a þ ð1−aÞ 1−εij

f Δσ ðΔσ Þ ¼

In addition, the damage correction coefficient Λij of the stationary Gauss broad band process is given in Eq. (37).

pffiffiffiffi β þ c2 β

the rain flow correction. The rain flow correction relationship under the [i, j]th case is given in Eq. (32):

13

3

   p ffiffiffi β β β 7 7 2β Γ 1 þ þ ð1−w1 Þ 2 2 Γ 1 þ b3 2 5

ð40Þ

The damage correction coefficient Λij of the stationary Gauss broad band process is shown in Eq. (41). 2 ðΔσ l Þ

β−β0 6 6

β0 b3

−

4w1 a3

β0

2 γ

3    pffiffiffi β0
β 0 7 β0 β0 0 þ 1; uij σ sij þ ð1−w1 Þ 2 2σ sij γ þ 1; uij 7 5 b3 2




Λij ¼

EðΔσ β Þij β
  −  pffiffiffi β
β β b þ 1; u0ij þw1 a3 3 2β Γ þ 1; uij σ βsij þ ð1−w1 Þ 2 2σ sij Γ b3 2 þ EðΔσ β Þij

ð41Þ

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T. Huo, L. Tong / Journal of Constructional Steel Research 166 (2020) 105917

Table 3 The stress spectrum parameter of the tubular tower for different wind speeds when the wind direction is 45°. Wind speed /(m/s)

Bandwidth coefficient ε

Wind speed /(m/s)

Bandwidth coefficient ε

4 6 8 10 12 14

0.7506 0.8020 0.8468 0.9252 0.8677 0.8505

16 18 20 22 24

0.8078 0.8070 0.8267 0.8248 0.8398

For the Zhao-Baker fatigue calculation method, the average frequency of the stress range fL ij is equal to the peak value rate νp of the stress process. 5.3.6. Tovo-Benasciutti fatigue damage formula Benasciutti and Tovo [29,30] proposed that the fatigue cumulative damage can be considered as the linear combination of DN ij and DRM ij. The detailed expression under the [i, j]th case is given in Eq. (42): ð42Þ

where DN ij is the fatigue cumulative damage of the narrow band process; DRM ij is the fatigue cumulative damage based on the Range-Mean counting method, and the calculation formula is given in Eq. (43) [31].   pffiffiffi β
β þ 1 ¼ DNij ηβ DRMij ¼ P ij Λij f Lij 2 2σ sij η Γ 2

ð43Þ

The weight coefficient ℓ has two types of definitions. The first definition form is described by Eq. (44).

ℓ¼

 ðζ −ηÞ 1:112ð1 þ ζη−ðζ þ ηÞÞe2:11η þ ðζ −ηÞ ðη−1Þ2

ℓ¼

η20:75 −η2 1−η2

ð45Þ

λ0:75 where η0:75 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi. λ0 λ1:5




 β pffiffiffi β
− β β , whereEðΔσ β Þij ¼ σ βsij w1 a3 b3 2β Γ 1 þ þ ð1−w1 Þð2 2Þ Γ 1 þ b3 2
b3 2 Δσ l ðΔσ l Þ uij ¼ a3 ，u0ij ¼ . 2σ sij 8σ 2sij

Dij ¼ ℓDNij þ ð1−ℓÞDRMij

the weight coefficient expression was proposed, as shown in Eq. (45).

ð44Þ

λ1 where ζ ¼ pffiffiffiffiffiffiffiffiffiffiffi. λ0 λ2 Benasciutti and Tovo [32] considered that the weight coefficient ℓ given in Eq. (44) is not accurate enough. Therefore, another form of

5.3.7. Empirical spectrum moment correction method Tovo and Benasciutti [29] proposed to adopt the narrow band method to approximately estimate the fatigue cumulative damage of the broad band stress process. It is worth noting that a correction coefficient χij should be used to correct the fatigue cumulative damage results. The detailed expression is given in Eq. (46). D″W ij ¼ χ ij DNij

ð46Þ

where D″W ijis the fatigue cumulative damage of the broad band stress process; χij is the empirical spectrum moment correction coefficient, χij = η2ij0.75. 5.3.8. Time-frequency domain fatigue analysis results of the tubular towers The fatigue checking location A is still chosen to perform the timefrequency domain fatigue analysis of the tubular tower, which is the same with the case of the time domain fatigue analysis. Moreover, the bandwidth coefficient is used to judge whether the stress process of the tubular tower is narrow band process or broad band process. When the wind direction angle is 45°, the bandwidth coefficients of the stress process for different wind speeds are shown in Table 3. In practical application, it is generally considered that when the bandwidth coefficient ε is less than 0.4, the stress process can be regarded as the narrow band process. It is clearly seen in Table 3 that for all the considered cases, the bandwidth coefficients ε are greater than 0.4, that is, the stress processes at the fatigue checking location A are the broad band process. Therefore, the probability density function of the stress range follows Rice distribution, as detailed in Eq. (23). Based on the program developed by the MATLAB software, the timefrequency domain fatigue analysis method including the ideal narrow band method, the equivalent stress range method, the equivalent stress range method considering rain flow correction, the equivalent narrow band method, Dirlik method, Zhao-Baker method and the empirical spectrum moment method are adopted to calculate the fatigue life of the tubular tower. Furthermore, the fatigue life of the tubular tower obtained by the time domain analysis method and time-frequency domain analysis method are summarized, as shown in Table 4. It is worth noting that Tovo-Benasciutti fatigue damage formula −1(TB-1 formula for short)stands for the one using the definition expression of the weight coefficient ℓ detailed in Eq. (44). In addition, Tovo-Benasciutti fatigue

Table 4 The fatigue life calculation results collection of the tubular tower. Calculated methods

Time domain analysis method Time-frequency domain analysis methods

Ideal narrow band method Equivalent stress range method Equivalent stress range method considering rain flow correction Equivalent narrow band method Dirlik fatigue damage formula Zhao-Baker fatigue damage formula Tovo-Benasciutti fatigue damage formula −1 Tovo-Benasciutti fatigue damage formula −2 Empirical spectrum moment correction method

The calculated fatigue life /year

Time-frequency domain method / Time domain method

34.73

1

11.16 19.01 33.36

0.32 0.55 0.96

13.46 27.78 14.05 24.59 31.08 25.39

0.39 0.80 0.40 0.71 0.89 0.73

T. Huo, L. Tong / Journal of Constructional Steel Research 166 (2020) 105917

15

Fig. 19. The fatigue cumulative damage distribution comparisons along wind direction.

damage formula −2 (TB-2 formula for short) stands for the one adopting another expression of the weight coefficient given in Eq. (45). In addition, Figs. 19 and 20 show the comparisons of the fatigue cumulative damage distribution of the tubular tower along the wind direction and wind speed between the time-frequency domain analysis method and the time domain analysis method. It can be seen from Table 4, Figs. 19 and 20 that the fatigue life calculation results obtained by the wind-induced fatigue timefrequency domain analysis theory are smaller than the corresponding values induced by the time domain fatigue analysis theory. Compared with the fatigue life obtained by the time domain fatigue analysis method, the equivalent stress range method considering rain flow correction is the most accurate, followed by the TB-2 formula and Dirlik fatigue damage formula, and the ideal narrow band method is the least accurate. It is worth mentioning that the fatigue life obtained by the ideal narrow band method, equivalent narrow band method, Zhao-Baker formula and the equivalent stress range method cannot meet the relevant requirement that the minimum service life of the tubular tower is greater than 20 years. In particular, the fatigue life calculation results based on the ideal narrow band method and equivalent narrow band method, which are considered as the lower limit value of the fatigue life for the broad band stress process, are too conservative and the economy of the tubular tower structural design would be finally affected. In summary, in the actual engineering design, when it is required to obtain the fatigue life of the tubular tower with the high accuracy, the wind-induced fatigue time domain analysis method is recommended. The time domain analysis method can take into account the safety and economy requirements at the same time. However, the amount of the calculation is extremely large. Furthermore, when the fatigue life of the tubular tower is required to obtain conveniently and quickly, and the fatigue life prediction results

must satisfy high accuracy and safety, this study recommends to use the time-frequency domain fatigue calculation method, especially the equivalent stress range method considering the rain flow correction, TB-2 formula and Dirlik formula. If the fatigue life of the tubular tower is roughly estimated before design, the empirical spectrum moment correction method is recommended in this study. Although this method lacks the theoretical basis, the calculation is more convenient and the accuracy is higher. 6. Summary and conclusions In this study, an integrated finite element model consisting of rotor, nacelle, tower and foundation was established. Based on the observation datum of the meteorological station, the joint distribution function of wind speed and wind direction at the construction site of wind turbine structures was obtained. Furthermore, the wind-induced response analyses corresponding to the different wind speed and wind direction cases were conducted. Moreover, the influence of the rotating effect of the blades on the wind-induced responses of the tubular tower was investigated in detail. Finally, the time domain and time-frequency domain fatigue analysis theory of wind turbine tubular tower structures was systematical established considering the influence of wind direction, rotating blades and low stress range fatigue damage reasonably. The main conclusions are noted as follows: 1) The stress response statistic values induced by considering the rotating effect of the blades are greater than the corresponding values caused by the case without considering the rotating effect of the blades. Moreover, compared with the mean value of the stress response, the influence of the rotating effect of the blades on the RMS value of the stress response is more significant, especially for the fatigue checking location. Therefore, the rotating effect of the

Fig. 20. The fatigue cumulative damage distribution comparisons along wind speed.

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T. Huo, L. Tong / Journal of Constructional Steel Research 166 (2020) 105917

blades is proposed to consider when conducting the fatigue life analysis of the tubular tower. Otherwise, the fatigue life predication could lead to the unsafe results. 2) Wind direction has significant influence on the wind-induced fatigue cumulative damage of wind turbine structures. Therefore, it is essential to consider the impact of the wind direction on the fatigue life of the tubular tower structures; otherwise the smaller fatigue life result would be obtained. Moreover, neglecting the influence of the low stress range on the fatigue damage could lead to conservative but not economic result. 3) The fatigue life of the tubular tower based on time-frequency domain analysis method is smaller than the corresponding values based on the time domain fatigue analysis method. However, the calculation is simpler. Adopting the equivalent stress range method modified by the rain flow coefficient, TB-2 formula and Dirlik formula to carry out the fatigue analysis of tubular towers are recommended in this study.

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