Wind-Induced Response Analysis and Fatigue Reliability Study of a Steel–Concrete Composite Wind Turbine Tower

Abstract

Taking an actual 3MW steel–concrete composite wind turbine tower as an example, a finite element model of the tower structure was established, and static bearing capacity and dynamic time history response analyses were performed to identify the locations where the structure is prone to failure. On this basis, the fatigue lives of the turbine tower at the most unfavorable locations were predicted using linear cumulative damage theory, and the fatigue reliability at the corresponding locations of the structure was calculated using the kriging–subset simulation method. The most dangerous locations of the tower that are most prone to failure are as follows: the bottom of the leeward side of the upper steel tube, the flange of the steel tube, the bolt-hole imprinting surface of the flange, the leeward side of the transition tube, and the top of the leeward side of the concrete tube. The failure risk of the flange and bolt-hole imprinting surface of the upper steel tube is relatively high, followed by that of the transition tube. This indicates that special attention should be given to the design and daily maintenance of this part. The fatigue resistance of the tower can be enhanced by improving the strength of the flange plate or increasing the number of bolts and strengthening the transition tube.

1. Introduction

With the development of the social economy, the demand for energy from humans is also increasing day by day. However, traditional fossil fuels are facing gradual depletion, and their development and utilization are always accompanied by environmental pollution. Therefore, promoting the development and utilization of alternative and clean energy is inevitable. Wind energy, as an important alternative energy source, has attracted widespread attention due to its wide distribution, pollution-free nature, and renewable advantages. At present, the human utilization of wind energy mainly relies on wind turbines for power generation. As an important supporting structure for wind turbines, the main function of the tower is to fix the wind turbine at a working height to obtain sufficient wind energy for power generation. At the same time, it needs to withstand its own weight, wind loads under different working conditions, and additional loads generated by the rotation of the blades. Therefore, the tower must have sufficient strength, stiffness, stability and service life.

At present, the most commonly used forms of wind turbine towers include conical steel tubes, lattice steel towers, and concrete tubes. A conical steel tube tower has the advantages of a simple structure, a definite load-transferring mechanism, low design and installation costs, etc., and is the most commonly used structure for wind power towers below 80 m in height. However, when the tower height exceeds 100 m, the steel consumption and transportation and installation costs of conical towers also increase rapidly, making it difficult to adapt to the needs of taller towers. Compared with a conical tube tower, a lattice tower saves more steel and occupies less land area when its stiffness is similar. However, when the tower is higher, there are many bolt connections in the lattice tower, resulting in high installation costs and difficult maintenance in the later stage. The concrete tube tower mainly adopts cast-in-place reinforced concrete with a circular cross-section. Compared with steel tube towers, concrete tube towers have a greater strength and stiffness, but the construction period is longer, which cannot meet the development needs of rapid installation. In addition, due to fluctuations in the wind load direction and the tensile stress generated inside the tower, the concrete is prone to cracking, accelerating the corrosion of steel bars, and affecting the service life of the tower. Therefore, at present, pure concrete tube towers are less commonly used in newly built wind turbine towers.

Since the 1980s, with the rapid development of wind power technology worldwide, the installed capacity of mainstream wind power has increased from 200 kW to more than 3 MW. To install wind turbines with longer blades and larger diameters and to obtain more wind energy in a more sustainable and stable manner, it is necessary to construct wind turbine towers with greater heights. Among them, steel–concrete composite wind turbine towers are a new type of tower that have developed rapidly in recent years. This kind of tower is composed of an upper steel tube and a lower prestressed concrete tube, which can combine the advantages of a pure steel tube tower and a pure concrete tube tower. It has the advantages of a high stiffness, a strong stability, and low transportation and installation costs. Therefore, it is particularly suitable for popularization and use in the development of wind energy resources in areas with low and ultralow wind speeds. Researchers have also conducted relevant studies on steel–concrete composite wind turbine towers and their supporting structures. Through a scale model test, Zhu et al. [1] revealed the development law of the dynamic characteristics of steel–concrete composite wind turbine towers during service. Lu et al. [2] studied the relationship between the strain rate and dynamic strength of concrete, proposed a nonlinear uniaxial strength criterion for concrete, and verified its accuracy through experiments. Wang [3] studied the key factors affecting the mechanical performance of hollow steel–concrete tube towers and compared them with traditional steel tube towers. Chen et al. [4] proposed a hybrid tubular tower structure that outperforms traditional tower structures. Alves et al. [5] used computer-aided engineering (CAE) tools to optimize the design of a 100 m high prestressed concrete wind turbine tower system. Pan et al. [6] analyzed the dynamic characteristics of a wind turbine tower under wind loads and random blade loads. Robert et al. [7] evaluated the effect of along-wind aerodynamic damping values obtained using the Stepley and Holmes methods on the lifespan of wind turbines. Chen et al. [8] used FAST v7 software to compare and analyze the dynamic response of a pure steel tower and a steel–concrete composite tower under different working conditions. Huo et al. [9] proposed a calculation method for the wind-induced response of wind turbine structures under different wind speeds and wind directions, and the accuracy of this method was verified. Ren et al. [10] proposed a new type of steel–concrete composite adapter (SCCA) to improve the ultimate capacity of SCCAs. The above research mainly focused on the optimization of steel–concrete composite wind turbine towers, the study of the structure of the conversion section, and wind load and wind response analyses of the tower. Considering that wind power structures are inevitably affected by changing environmental wind loads during operation and that the rotation of wind turbine blades can also cause changes in structural stress, resulting in fatigue damage to the structure, fatigue is an important problem that many structures face. In practical engineering, tower collapse events caused by fatigue failure of components are inevitable, and the fatigue resistance determines the service life of a wind turbine tower to a large extent. Therefore, researchers have also conducted relevant studies on this issue. For example, Chen et al. [11] studied the propagation behavior of fatigue cracks with different initial shapes of welded steel bridgeheads considering welding participation stress and emphasized its importance in fatigue cracking. Luo et al. [12] investigated crack interactions in welded joints between ribs and decks in steel bridges by analyzing the effects of key parameters on fatigue crack growth. Li et al. [13] analyzed the wind-induced fatigue life of a wind turbine tower, taking into account the influence of the joint probability distribution of the wind speed and direction. Aitor et al. [14] proposed a method for predicting the service life of offshore wind turbines. A study by Stavridou et al. [15] indicated that the wall thickness of the tube was the decisive factor in the fatigue life of the structure. Alonso et al. [16] analyzed the effect of preloading forces on the fatigue damage of tower flange connection bolts. Fu et al. [17] analyzed the fatigue resistance performance of a wind turbine tower under random wind loads. However, the above studies focused mainly on factors such as the ambient wind load, tower material and wall thickness, and there is relatively insufficient research on the fatigue performance of steel–concrete composite wind turbine towers with complex connections. In fact, a steel–concrete composite wind turbine tower is prone to fatigue damage under long-term cyclic stress due to its many connecting parts (e.g., the upper steel tube is connected by inner flange bolts, and the lower concrete tube is connected by high-strength prestressed steel strands). On the other hand, the upper steel tube and the lower concrete tube of the composite tower are connected by a transition tube with an abruptly changing cross-section, which also causes stress concentration, thereby influencing the fatigue resistance of the transition tube. In severe cases, this may even lead to tower collapse accidents. In addition, considering the obvious randomness in the structural parameters and external loads of steel–concrete composite wind turbine towers, these factors have a great impact on the fatigue performance of tower structures. Therefore, reliability analysis based on probability is also an important part of structural analysis. Cui et al. [18] proposed an analytical framework for evaluating the occurrence probability of vortex-induced vibrations (VIVs). Combined with environmental wind information, a random bridge deck excitation amplitude and a nonlinear aeroelastic VIV model, the occurrence probability of VIVs on the bridge deck was evaluated. He et al. [19] used mechanical learning to establish an RF regression model through a large amount of data, which can predictably design and control the preparation of CD-based corrosion inhibitors. Dong et al. [20] predicted the fatigue reliability of welded multiplanar pipe joints of a fixed jacket offshore wind turbine support structure with a water depth of 70 m by considering the thickness thinning (loss) effect, corrosion and other influencing factors. However, research on the fatigue performance of tower structures from the perspective of reliability is still limited.

Based on the above analysis, an actual 3MW steel–concrete composite wind turbine tower was taken as an example to establish a full-scale finite element model of the tower, and a static bearing capacity analysis was performed to determine the locations where the structure is prone to failure. Then, the fluid–structure interaction method is adopted to analyze the wind-induced vibration response of the structure. The fatigue lives of the steel–concrete composite wind turbine tower at the most unfavorable positions were predicted using cumulative damage theory and fatigue strength curves. On this basis, the kriging-subset simulation method [21] was used to calculate the fatigue reliability of the structure. From a probability perspective, the fatigue resistance of the structure is evaluated, and design suggestions are recommended to provide a reference for the reasonable design and daily maintenance of similar structures.

2. Finite Element Modeling and Modal Analysis of the Steel–Concrete Composite Wind Turbine Tower

Figure 1 shows a prestressed steel–concrete composite wind turbine tower [22] with a height of 120 m. The main design parameters of the tower are shown in

Table 1. Main design parameters of the steel–concrete composite turbine tower.

Upper Steel Tube		Transition Tube		Lower Concrete Tube
Material	Steel (grade Q345)	Material	steel (Grade Q345)	Material	Concrete (grade C60)
Length	70 m	Length	15 m	Length	35 m
Diameter of the tube top	3.2 m	Diameter of the tube	4.3 m	Diameter of the tube top	4.5 m
Diameter of the tube bottom	4.3 m			Diameter of the tube bottom	7.0 m
Wall thickness	64 mm	Wall thickness	64 mm	Wall thickness at the top of the tube	400 mm
				Wall thickness at the bottom of the tube	300 mm
Type of element	Solid	Type of element	Solid, beam	Type of element	Solid, linear
Damping ratio	3.4%	Damping ratio	3.4%	Damping ratio	3.4%
Mass	270 t	Mass	30 t	Mass	402 t
Other	--	High-strength bolts	M36 (grade 10.9)	Steel strands	Φ15.2 (grade 1860)

. The upper part is a 70 m high variable cross-section steel tube with the typical WindPACT-3MW wind turbine, as referenced (Rinker et al.) [23], and the design parameters are shown in

Table 2. Detailed design parameters of the WindPACT-3MW wind turbine.

Type	Items	Value
Operating parameters	Rated speed	14.469 r/min
	Rated wind speed	12.9 m/s
	Cut-in wind speed	3 m/s
	Cut-out wind speed	25 m/s
Blades	Number	3
	Material	Fiberglass and wood
	Mass	13.2238 t
	Radius	49. 5 m
Hub and nacelle	Mass of hub	61.67 t
	Mass of nacelle	132.598 t
	Coordinates of the center of mass of nacelle relative to the center of the tower top	(−0.226 m, 0 m, 1861 m)
	Coordinates of the center of mass of hub relative to the center of the tower top	(4.650 m, 0 m, 2.270 m)

. The lower part of the tower is a 35 m high concrete tube, and 31 prestressed anchor cables are set along the circumference. Each anchor cable bundle is composed of thirteen low-relaxation prestressed steel strands with a diameter of 15.2 mm and a tensile strength of 1860 MPa. The middle of the tower is a 15 m high transition tube made of steel. The transition tube and the upper tube are connected by a flange with 88 M36 (grade 10.9) high-strength bolts.

ANSYS 2022 R1 software was used to establish a finite element model of the composite tower based on the above parameters. To improve the computational efficiency in the subsequent analysis, the following simplifications were made when establishing the finite element model. The wind turbine rotor, nacelle and blades at the top of the tower were simplified as lumped masses, which were coupled with the upper edge of the steel tube. The interaction between the soil and structure (SSI effect) was not considered, and the tower base was fixed. The Solid186 element was used to simulate the steel tube, transition tube, and concrete tube, with a mesh size of 100 mm, and the flange, with a mesh size of 20 mm. The Link8 element was used to simulate the steel strands, and the Beam188 element was used to simulate the high-strength bolts. The friction coefficient of the imprinting surface between the bolts and the flange was set to 0.2, and a bolt preload of 339.5 kN was added to the Beam188 element during analysis. The prestress of the steel strands embedded in the concrete tube was simulated using the cooling method by setting two predefined different temperatures. Due to the difference in the linear expansion coefficients between concrete and steel, the steel strands shrink and generate tensile stress. The final finite element model for the composite tower is shown in Figure 2. The number of meshed elements was 914,571, and the number of nodes was 433,840.

The natural frequencies and vibration modes of the first three modes of the structure were calculated using the Block Lanczos method, and the results are shown in Figure 3. As demonstrated in Figure 3, the first three vibration modes of the tower present typical bending shapes, and the corresponding natural frequencies are 0.272 Hz, 1.535 Hz and 3.919 Hz, respectively.

Table 3. A comparison of the simulated natural frequencies and the results given in the literature (Shi et al.) [22].

Mode Order	Simulated Frequency in the Present Study (Hz) f0	Reference Frequency Results (Shi et al.) (Hz) f	Relative Deviation (%) |(f0 − f)/f0|
1	0.272	0.258	5.1
2	1.535	1.584	3.2
3	3.919	3.823	2.4

presents a comparison of the results for the simulated natural frequencies in this study and the results in the literature (Shi et al.) [22]. As shown in Table 3, the finite element simulation results in this study are highly in agreement with the values in ref. [22], and the maximum deviation is 5.1%, which indicates that the modeling method and the established finite element model in this study are reliable and can be used for subsequent analysis.

In practical engineering, to avoid resonance between the wind turbine tower and the blades during rotation, it is necessary to keep the fundamental natural frequency of the tower far from the excitation frequencies caused by the rotor speed (1 P) or n P (where n is the number of blades) harmonics from the rotor [24]. Given that the rated speed of the WindPACT-3MW rotor is 14.469 rpm, the 1 P and 3 P frequencies can be calculated to be 0.241 Hz and 0.723 Hz, respectively, which are both far from the fundamental frequency of the tower, indicating that there will be no resonance when blades are running at the rated speed.

3. Wind Loads and Static Bearing Capacity of the Steel–Concrete Composite Wind Turbine Tower

3.1. Wind Loads on the Tower

The wind loads acting on the combined steel–concrete wind turbine tower mainly include the aerodynamic loads of the blades and the natural wind loads.

3.1.1. Aerodynamic Loads of the Blades

The aerodynamic loads of the blades refer to the aerodynamic force and aerodynamic torque generated by the wind turbine and blades. According to blade element momentum (BEM) theory, the axial force and torque acting on the wind turbine blade element are:

$$d{F}_{\mathrm{N}}=4\mathsf{\pi }\rho v^{2}a\left(1-a\right)rdr$$

(1)

$$dM=4\pi \rho \Omega v{a}^{\prime }\left(1-a\right)r^{3}dr$$

(2)

where ρ is the air density; a and a′ represent the axial and tangential velocity induction factors, respectively; Ω is the rotation angular velocity of the blade; and r represents the distance between the microelement segment and the blade root.

Based on the aforementioned design parameters of the 3MW-WindPCAT turbine, the time histories of the axial wind load and aerodynamic torque acting on the blades under a wind speed of 12.9 m/s at the hub height can be calculated according to the above theory. The results are shown in Figure 4.

3.1.2. Natural Wind Loads

The natural wind loads on the steel–concrete composite tower include the along-wind load and the cross-wind load. According to the “Load Code for the Design of Building Structures” (GB50009-2012) [25], the along-wind load perpendicular to the tower can be calculated by Equation (3):

$${\omega }_{\mathrm{k}}={\beta }_{\mathrm{z}}{\mu }_{\mathrm{s}}{\mu }_{\mathrm{z}}{\omega }_0$$

(3)

where ω_k is the characteristic value of the wind load (kN/m²); μ_s is the shape factor of the wind load; μ_z is the exposure factor for the wind pressure; β_z is the dynamic effect factor of the wind at a height of z; and ω₀ is the reference wind pressure (kN/m²). In this study, ω₀ = 0.35 kN/m².

For the simplicity of calculations, the tower tube can be divided into 12 sections vertically, and the corresponding wind load acting points are 5 m, 15 m, 25 m, 35 m, 45 m, 55 m, 65 m, 75 m, 85 m, 95 m, 105 m, and 115 m in height, as shown in Figure 5. Assuming that the wind speed in each section is uniform and that the annual average wind speed of the wind field where the structure is located is 8 m/s, the along-wind load F_k at the height of each representative point can be calculated and is shown in

Table 4. Along-wind loads at the height of each representative point of the steel–concrete composite wind turbine tower (kN).

Z (m)	5	15	25	35	45	55	65	75	85	95	105	115
Fk (kN)	0.197	0.237	0.315	0.435	0.571	0.730	0.921	1.165	1.551	1.805	2.157	2.493

.

A steel–concrete combined wind turbine tower is a typical high-rise structure with a circular cross-section. When the vortex shedding frequency of the incoming wind is close to the natural vibration frequency of the structure, vortex-induced resonance may occur in the structure, causing severe structural vibration. The vortex shedding frequency f_v can be calculated by Equation (4):

$$f_{\mathrm{v}}=\frac{S_{\mathrm{t}}v}{D}$$

(4)

where S_t is the Storoha number, which is determined by the Reynolds number and the cross-sectional shape of the structure, and 0.2 is generally used for a circular cross-section structure. v is the wind speed and D is the diameter of the cross-section.

When f_v approaches the jth-order natural frequency f_j of the structure, jth-order vortex-induced resonance may occur, and the wind speed at this time is called the critical wind speed v_cr,j. Within a certain wind speed range after the occurrence of vortex-induced resonance, the vortex shedding frequency is controlled by the natural frequency of the structure due to the feedback of structure vibrations on vortex shedding. Therefore, the vortex shedding frequency does not increase with increasing wind speeds, resulting in the phenomenon of “lock-in”. According to the “Load Code for the Design of Building Structures” (GB50009-2012) [25], the wind speed range corresponding to the “lock-in” phenomenon is v_cr,j/1.2~1.3v_cr,j. Thus, the starting height H₁ and ending height H₂ of the resonant region can be calculated using the following equations:

$$H_1=H\times {\left(\frac{v_{\mathrm{cr},\mathrm{j}}}{1.2v_{\mathrm{H}}}\right)}^{\frac{1}{a}}$$

(5)

$$H_2=H\times {\left(\frac{1.3v_{\mathrm{cr},\mathrm{j}}}{v_{\mathrm{H}}}\right)}^{\frac{1}{a}}$$

(6)

where v_cr is the critical wind speed, H is the height of the tower, v_H is the wind speed at the top of the tower, and α is a proportionality constant.

Based on the above analysis, the critical wind speeds, “lock-in” wind speed range, and height range of the resonant region can be obtained, as shown in

Table 5. Results for the critical wind speeds, “lock-in” wind speed range, and height range of the resonant region.

Natural Frequency (Hz)		Critical Wind Speed (m/s)	“Lock-In” Wind Speed Range (m/s)	Height Range of the Resonance Region (m)
First order	0.272	4.48	3.77~5.82	-
Second order	1.535	24	20~32.2	27.1~120
Third order	3.919	62.4	52~81.1	112.6~120

.

When the “lock-in” phenomenon occurs, the vortex shedding frequency is controlled by the natural vibration frequency of the structure. To simplify the calculation, the wind speed v (z) that varies in the “lock-in” wind speed range can be replaced by the critical wind speed v_cr,j. According to the Luhmann sinusoidal force model [26], the equivalent wind load caused by vortex-induced resonance on the structure can be calculated using the following formula:

$$p_{\mathrm{L},\mathrm{j}}\left(z,T\right)=\frac{1}{2}\rho v_{\mathrm{cr},\mathrm{j}}^{2}D{\mu }_{\mathrm{L}}\sin {\omega }_{\mathrm{j}}T$$

(7)

where ρ is the air density, D is the diameter of the cross-section, μ_L is the lift coefficient, ω_j is the vortex shedding circular frequency when jth-order vortex-induced resonance occurs, and T is the natural vibration period of the structure.

Given that the essence of vortex-induced resonance is the acceleration excitation effect of a structure, the dynamic effect of vortex-induced resonance can be simulated by applying the equivalent acceleration in the “lock-in” region [27]. The equivalent acceleration a_L,j(z,T) can be calculated using Equation (8), and the application of equivalent acceleration on the structure is shown in Figure 2f.

$$a_{\mathrm{L},\mathrm{j}}\left(z,T\right)=\frac{{\displaystyle {\int }_{H_2}^{H_1}{p}_{\mathrm{L},\mathrm{j}}\left(z,T\right)dz}}{{\displaystyle {\int }_{H_2}^{H_1}{m}_{0}dz}}=\frac{p_{\mathrm{L},\mathrm{j}}\left(z,T\right)}{m_0}=\frac{\rho v_{\mathrm{cr},\mathrm{j}}^{2}D{\mu }_{\mathrm{L}}\sin {\omega }_{\mathrm{j}}T}{2m_0}$$

(8)

where m₀ is the mass of the tube per unit height.

3.2. Static Bearing Capacity

Based on the analysis and results in the previous section, the loads and boundary constraints can be applied to the finite element model, and a static bearing analysis is performed. Figure 6 shows a diagram of the constraints and loads on the finite element model. Figure 7 presents the stress and deformation results of the structure.

As shown in Figure 7, the maximum stress occurred at the flange between the upper steel tube and the transition tube, indicating that the flange is a weak part of the tower structure and is most likely to fail under alternating external loads. Figure 7 also shows that the parts with high stress are as follows: the bottom of the leeward side of the upper steel tube, the flange of the upper steel tube, the bolt-hole imprinting surface of the flange, the leeward side of the transition tube, and the top of the leeward side of the concrete tube. The locations of the maximum stress in the composite structure under other wind load velocities are basically the same and are not specified here. In other words, these parts are the most dangerous locations of the tower structure and should receive special attention in design and daily maintenance.

4. Dynamic Response Analysis of the Steel–Concrete Composite Wind Turbine Tower Based on the Fluid–Structure Interaction Method

In the previous analysis, several critical locations of the steel–concrete composite wind turbine tower structure that were most prone to failure were identified. To obtain the stress time history response at these hazardous locations and provide a basis for subsequent fatigue analysis, a dynamic response analysis of the structure is needed. Considering the complexity of the tower structure in this study and the sensitivity of the structure to wind loads, it is better to conduct a dynamic analysis using the fluid–structure interaction method [28,29,30] to obtain results that are closer to engineering practice.

4.1. Fluid–Structure Interaction Model

Based on the coupled computational fluid dynamics and computational structural dynamics (CFD/CSD) method, ANSYS FLUENT and Mechanical APDL were chosen for computation. The process of the fluid–structure interaction analysis is shown in Figure 8.

To make the simulated turbulence characteristics closer to the actual situation, a hybrid grid is used for the flow field meshes. For the flow field near the structure, unstructured dynamic grids are adopted, and the grids near the structural boundary are densified. For the flow field far away from the structure, structured grids are adopted. To ensure information transfer between dense grids and sparse grids, the nodes at the interfaces of different grids are merged. The detailed meshing of the flow field is shown in Figure 9.

During analysis, the inlet boundary was set as the velocity inlet boundary, the outlet boundary was set as the pressure outlet boundary, and the upper and lower surfaces of the flow field were set as the nonslip walls. The side surfaces are set with symmetrical boundary conditions. The final established CFD analysis model is shown in Figure 10. The SIMPLE method was chosen to solve the pressure–velocity coupling problem. The least squares method was adopted for space discretization. The calculation time of the flow field was set to 60 s, and the time step size was set to 0.02 s.

In the CSD analysis, the base of the steel–concrete composite wind turbine tower was fully constrained; gravity, a bolt pretightening force, an aerodynamic force and aerodynamic torque were applied to the structure. The CSD model is shown in Figure 11. The calculation time is set as 60 s, with a time step size of 0.02 s, which is consistent with that of the CFD analysis model.

To ensure that the fluid–structure interaction simulation results in this study are insensitive to changes in grid density, grid independence verification is performed below. The Richardson extrapolation method [31] was chosen to estimate the error caused by grid changes. Three different mesh sizes were selected for the fluid–structure interaction calculations. The proportional coefficient Δ, as the convergence indicator of the mesh, can be calculated using Equation (9):

$$\Delta =\frac{S_3-S_2}{S_2-S_1}$$

(9)

where S_i represents the calculation results of the i-th group of meshes.

By adjusting the mesh size of the refinement zone and boundary layer, the overall mesh density of the flow field is obtained. In this section, the along-wind displacement at the top of the tower under the condition of a rated wind speed of 12.9 m/s was calculated to estimate the error due to grid variation. Three sets of flow field meshes of different sizes were selected: coarse, medium and fine grids. The comparison data are presented in

Table 6. Comparison of the along-wind displacement at the top of the steel–concrete composite tower.

Item	Boundary Layer Mesh Size (mm)	Mesh Size in Refinement Zone (mm)	Theoretical Value (mm) (a)	Calculated Value by Fluid–Structure Interaction Method (mm) (b)	Relative Error (%) |(a) − (b)|/(a)
Coarse grid	50	100	1148	1011.45	13.5
Medium grid	20	60		1087.11	5.6
Fine grid	10	40		1113.48	3.1

below.

Table 6 shows that with a decrease in the mesh size of the grids, the relative errors of the values calculated by the fluid–structure interaction method and the theoretical results do not exhibit large changes, and the errors are within the acceptable range under the conditions of fewer grids and more grids. Considering both the accuracy and computational cost, the medium grid is selected for subsequent calculations.

4.2. Wind-Induced Time History Response of the Steel–Concrete Composite Wind Turbine Tower

Based on the previous analysis, the displacement at the tower top and the stress at the most unfavorable locations of the structure are investigated. The following working conditions are considered: the rated wind speed condition (the wind speed is 12.9 m/s), the cut-out wind speed condition (the wind speed is 25 m/s), and the survival wind speed condition (the wind speed is 52.5 m/s).

4.2.1. Time History of the Displacement at the Top of the Tower

Figure 12 shows the displacement–time history curves at the top of the tower under the above three working conditions.

As shown in Figure 12, under the rated wind speed, the along-wind displacement of the tower was relatively large, while the cross-wind displacement was relatively small, approximately 1/15 of the corresponding along-wind displacement. This indicates that the aerodynamic thrust generated by the blades is the main factor causing the wind-induced vibration of the tower. Under the conditions of the cutting out wind speed and survival wind speed, the cross-wind displacement was relatively larger than the along-wind displacement, which was mainly due to the vortex-induced vibration of the tower.

A Fourier transform is performed on the abovementioned time history–displacement results to obtain the corresponding frequency-domain response results, as shown in Figure 13. It can be seen that under the cut-out wind speed and survival wind speed conditions, the higher-order vibration modes contribute more to the displacement response of the structure. Under these two conditions, the structure mainly undergoes cross-wind vibration, indicating that the structural damage caused by cross-wind vibration needs to be considered.

4.2.2. Time History of Stress at the Most Unfavorable Locations of the Structure

The stress–time history responses at the most unfavorable locations of the structure are shown in Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18.

Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18 show that the stress–time history responses of the most unfavorable locations of the structure under the rated wind speed are significantly greater than the responses under the other two wind speed conditions. A comparison of the stress levels of different parts of the structure shows that the stresses in the transition tube, the flange plates, and the imprinting surface of the bolt holes are significantly greater than those in the upper steel tube and the lower concrete tube.

5. Fatigue Life Analysis and Fatigue Reliability Assessment of the Steel–Concrete Composite Wind Turbine Tower

5.1. Fatigue Life Prediction of the Tower

Based on the stress–time history results mentioned above, the fatigue life of the structure was calculated using Palmgren–Miner theory [32]. The detailed analysis procedure is shown in Figure 19.

Taking the upper steel tube as an example, the rainflow counting method [33] was used for statistical analysis of the maximum stress at the most unfavorable location of the steel tube. In this way, the original stress–time history samples are converted into multiple groups of fatigue loads, as shown in Figure 20, and each group includes the mean stress, stress amplitude and corresponding number of cycles.

According to the “Standard for design of steel structures” (GB50017-2017) [34], the S_rhs-N_i curve (fatigue strength curve) for the steel tube was selected, as shown in Figure 21. The curve equations can be expressed as follows:

$$S_{\mathrm{rhs}}={\left(\frac{1.46\times {10}^{12}}{N_{\mathrm{i}}}\right)}^{\frac{1}{3}},\hspace{1em}\hspace{1em}\mathrm{if} N_{\mathrm{i}} \le 5\times {10}^6$$

(10)

$$S_{\mathrm{rhs}}={\left[66\times \frac{1.46\times {10}^{12}}{N_{\mathrm{i}}}\right]}^{\frac{1}{5}},\hspace{1em}\hspace{1em}\mathrm{if} 5\times {10}^6< N_{\mathrm{i}} \le 1\times {10}^8$$

(11)

$$S_{\mathrm{rhs}}=36,\hspace{1em}\hspace{1em}\mathrm{if} N_{\mathrm{i}}>1\times {10}^8$$

(12)

Because the fatigue strength curve is usually obtained through experiments without considering the mean stress, the Goodman correction method [35] is used to recalculate the effective stress range considering the influence of the mean stress. Based on the above stress statistical results and the S_rhs-N_i curve, the annual fatigue damage coefficient of the steel tube under different stress amplitudes considering the correction of the average stress can be obtained, and the results are shown in

Table 7. Annual fatigue damage coefficient corresponding to each stress range for the upper steel tube.

Stress Amplitude (MPa)	Mean Stress (MPa)	Corrected Stress Amplitude (MPa)	Actual Annual Number of Stress Cycles	Theoretical Annual Number of Stress Cycles	Annual Fatigue Damage Coefficient
2.32	7.22	2.29	32	-	-
14.94	14.03	14.53	26	-	-
24.01	25.90	22.83	19	-	-
35.41	34.39	33.13	17	-	-
45.52	45.07	41.76	18	3.50 × 1010	8.35 × 10−7
55.19	54.97	49.72	15	2.94 × 1010	1.19 × 10−6
64.64	62.60	57.45	28	2.54 × 1010	7.39 × 10−7
73.75	72.29	64.43	17	2.27 × 1010	1.36 × 10−6
84.25	0	84.25	11	1.26 × 108	3.79 × 10−4
94.08	0	94.08	4	8.12 × 107	1.62 × 10−3
104.38	0	104.38	7	5.36 × 107	1.40 × 10−3
114.21	0	114.21	6	3.74 × 107	2.34 × 10−3
123.86	0	123.86	3	2.70 × 107	6.48 × 10−3
Annual fatigue cumulative damage coefficient: 0.01223

.

As shown in Table 7, the annual cumulative fatigue damage factor of the upper steel tube under different stress amplitudes is 0.01223, and the fatigue life is calculated as 1/0.01223 = 81.76 (years), which is much longer than the required design service life (20 years) as required by the specifications.

Similarly, the fatigue lives of the flange of the upper steel tube, the imprinting surface of the bolt holes in the flange of the upper steel tube, the transition tube, and the lower concrete tube can also be obtained, and the results are shown in

Table 8. Fatigue life for the most unfavorable locations of the steel–concrete composite wind turbine tower.

Part of the Tower	Annual Cumulative Fatigue Damage Factor	Fatigue Life (Year)
upper steel tube	0.01223	81.76
transition tube	0.03141	31.84
lower concrete tube	0.00419	238.66
flange of the upper steel tube	0.04462	22.41
imprinting surface of the bolt holes in the flange of the upper steel tube	0.04855	20.60

.

Table 8 shows that most of the fatigue lives of the most unfavorable part of the tower are far greater than 20 years, which meets the design service life requirements. However, the fatigue lives of the flange and the imprinting surface of the bolt holes are only 22.41 years and 20.6 years, respectively, which are very close to the design service life. Therefore, in practical engineering, the stiffness and strength of flanges should be appropriately increased to enhance their fatigue resistance.

5.2. Fatigue Reliability Assessment of the Tower

As presented in the previous study, the fatigue life of the steel–concrete composite wind turbine tower was analyzed based on the deterministic stress–time history results of the key parts of the tower. Given the complex composition of such structures and the strong randomness of concrete materials, there may be some uncertainties that influence the fatigue life prediction results of actual tower structures. Therefore, in this section, considering the influence of random factors, a fatigue damage analysis is performed from the perspective of reliability.

The function for structural fatigue damage estimation is defined as

Z = D₁ − D

(13)

where D₁ is the fatigue damage of the structure within the design service life and D is the critical fatigue damage.

Commonly used methods for solving these functions include the Monte Carlo method, the subset simulation method [36], and the kriging–subset simulation method. The kriging–subset simulation method integrates the kriging model into the subset simulation method, which can replace the actual implicit function of the structure and greatly reduce the demand for multiple finite element calculations of structural response, thereby significantly improving the efficiency of reliability solving. The kriging surrogate model consists of two parts, a stochastic time history model and a regression model, and its general expression is as follows:

$$\widehat{y}\left(x\right)=f^{\mathrm{T}}\left(x,\beta \right)+z\left(x\right)$$

(14)

where $f^{\mathrm{T}}\left(x\right)=\left[f_1\left(x\right),f_2\left(x\right),\cdots ,f_{\mathrm{n}}\left(x\right)\right]$ is the n-term polynomial of variable x; β is the regression coefficient vector; and z (x) is a random time history function with a mean value of 0, whose covariance is expressed as

$$\mathrm{cov}\left[z\left(x_{\mathrm{i}}\right),z\left(x_{\mathrm{j}}\right)\right]={\sigma }^2\left[R\left(\theta ,x_{\mathrm{i}},x_{\mathrm{j}}\right)\right],\left(i,j=1,2,\cdots ,n\right)$$

(15)

where R(θ, x_i, x_j) is the spatial correlation function for sample points x_i and x_j and can be expressed as

$$R\left(\theta ,x_{\mathrm{i}},x_{\mathrm{j}}\right)={\displaystyle \prod _{\mathrm{k}=1}^{n_{\mathrm{dv}}}{R}_{\mathrm{k}}\left({\theta }_{\mathrm{k}},d_{\mathrm{k}}\right)},dk=\left|x_{\mathrm{i}}^{\mathrm{k}}-x_{\mathrm{j}}^{\mathrm{k}}\right|$$

(16)

The regression coefficient β and estimated variance of the random function z (x) are expressed as

$$\widehat{\beta }={\left(I{R}^{-1}I\right)}^{-1}{I}^{\mathrm{T}}{R}^{-1}Y$$

(17)

$${\widehat{\sigma }}^2=\frac{1}{n}{\left(Y-\widehat{\beta }I\right)}^{\mathrm{T}}{R}^{-1}\left(Y-\widehat{\beta }I\right)I$$

(18)

where I is an n × 1 vector whose element is 1.

The parameter θ in the correlation function can be obtained via maximum likelihood estimation and can be expressed as

$$L\left(\theta \right)={\left.-\left\{n\ln {\widehat{\sigma }}^2+\ln \left[\det \left(R\right)\right]\right\}\right|}_{\max }$$

(19)

Then, the following prediction equation can be obtained:

$${\mu }_{\widehat{\mathrm{y}}}\left(x_0\right)=\widehat{\beta }+r_0{R}^{-1}\left(Y-\widehat{\beta }I\right)$$

(20)

The calculation process of the kriging–subset simulation method is as follows:

(1)

The Latin hypercube sampling method is used to obtain n sample points, which are regarded as the initial points, and the response of the structure is obtained via the finite element method.

(2)

A kriging model is constructed based on the initial points, and a kriging surrogate model is obtained using the DACE toolbox in MATLAB.

(3)

N random samples are generated via the Monte Carlo method, and the response values of the samples are predicted via the kriging surrogate model to obtain the conditional probabilities P₀ and NP₀.

(4)

The samples obtained in step 3 are processed by the MCMC method, and the corresponding failure probability under given conditions can be obtained.

Taking the flange of the upper steel tube as an example, 100 sets of random time-history cyclic stress data can be obtained using the above method, as shown in Figure 22. Figure 23 lists the annual cumulative fatigue damage factors for the corresponding 100 groups of flanges. It is easy to see that the results show large dispersion, and among them, five factors exceed 0.05, which is the annual fatigue damage coefficient corresponding to the 20-year design service life. This indicates that it is not sufficient to predict the fatigue life of a tower through a single deterministic analysis, and a comprehensive evaluation from a probabilistic perspective is needed.

Using the above 100 sample examples, the kriging surrogate model was used to perform interpolation prediction, and 10,000 stress values of the flange were predicted. The probability distribution of the annual cumulative fatigue damage factor is shown in Figure 24. The distribution exhibits the characteristics of an approximately log-normal distribution. The failure probability of the structure is calculated by the subset simulation method, and the result is 10.8%. According to the first exceedance criterion, the fatigue reliability of the structure is 89.2%.

Similarly, using the same method, the fatigue reliability of the upper steel tube, the transition tube, the lower concrete tube, and the imprinting surface of the bolt holes in the flange of the upper steel tube can also be calculated, and the results are shown in

Table 9. Fatigue reliability analysis results for the tower.

Part of the Tower	Mean Value for the Annual Accumulative Fatigue Damage Coefficient of 100 Groups Results	Mean Fatigue Life (Years)	Failure Probability	Fatigue Reliability
upper steel tube	0.013257	75.43	3.26%	96.74%
transition tube	0.020197	49.51	5.35%	94.65%
lower concrete tube	0.008436	118.54	1.71%	98.29%
flange of the upper steel tube	0.026731	37.41	8.8%	91.2%
imprinting surface of the bolt holes in the flange of the upper steel tube	0.028754	34.78	9.2%	90.8%

.

As shown in Table 9, the mean fatigue life of each of the most unfavorable locations of the structure is greater than 20 years, which meets the design service life requirements. However, from the perspective of probability, the failure risk of the flange of the upper steel tube and the bolt-hole imprinting surface of the flange is relatively high, followed by that of the transition tube. This indicates that special attention should be given to these parts in design and daily maintenance. In practice, the fatigue resistance of a tower can be enhanced by improving the strength of the flange plate or increasing the number of bolts and strengthening the transition tube. A comparison of the transition tube, the upper steel tube and the lower concrete tube shows that the fatigue reliability of the transition tube is slightly lower than that of the adjacent steel tube and concrete tube. Therefore, special attention should also be given to this area. In contrast, the fatigue reliabilities of the steel tube and the concrete tube both exceed 95%, indicating that these two parts of the structure have a sufficient fatigue resistance under normal operation conditions. However, in practical engineering, it is still necessary to pay attention to its integrity to ensure that the processing and construction are flawless and to avoid eccentricity and uneven wall defects, thereby reducing the possibility of structural fatigue failure.

A comparison of Table 8 and Table 9 shows that the overall variation characteristics of the deterministic analysis results and the mean results of the fatigue life of each of the most unfavorable locations of the tower are similar, but there are still significant differences in the specific values of the fatigue life at the corresponding positions. This once again indicates that for structures such as concrete composite wind turbine towers, which are affected by many random factors during operation, it is necessary to estimate their fatigue resistance performance from the perspective of reliability.

6. Conclusions

In this study, numerical simulations and a theoretical analysis are performed to analyze the wind-induced response and fatigue reliability of a steel–concrete composite wind turbine tower. The main conclusions are as follows:

(1)

The parts of the steel–concrete composite wind turbine towers that are most likely to fail are as follows: the bottom of the leeward side of the upper steel tube, the flange of the steel tube, the bolt-hole imprinting surface of the flange, the leeward side of the transition tube, and the top of the leeward side of the concrete tube.

(2)

Under the rated wind speed, the along-wind displacement of the tower was relatively large, while the cross-wind displacement was relatively small, approximately 1/15 of the corresponding along-wind displacement. This indicates that the aerodynamic thrust generated by the blades is the main factor causing the wind-induced vibration of the tower. Under the conditions of cutting out wind speed and survival wind speed, the cross-wind displacement was relatively larger than the along-wind displacement, which was mainly due to the vortex-induced vibration of the tower.

(3)

For the tower analyzed in this study, the mean fatigue lives of the most unfavorable locations are all greater than 20 years, which meets the design service life requirements. However, from the perspective of probability, the failure risk of the flange of the upper steel tube and the bolt-hole imprinting surface of the flange is relatively high, followed by that of the transition tube. This indicates that special attention should be given to these parts in design and daily maintenance. In practice, the fatigue resistance of a tower can be enhanced by improving the strength of the flange plate, increasing the number of bolts, or strengthening the transition tube. A comparison of the transition tube, the upper steel tube and the lower concrete tube shows that the fatigue reliability of the transition tube is slightly lower than that of the adjacent steel tube and concrete tube. Therefore, special attention should also be given to this area. In contrast, the fatigue reliabilities of the steel tube and the concrete tube both exceed 95%, indicating that these two parts of the structure have sufficient fatigue resistance under normal operation conditions.

(4)

When designing a steel–concrete composite wind turbine tower in practical engineering, the following suggestions can be given: The flange plate should be subjected to a detailed fatigue analysis to ensure its safety. The material strength and quality of the transition tube should be appropriately improved. Attention should be given to the processing and construction of steel tubes and concrete tubes to reduce the risk of fatigue failure.

However, this topic still requires more in-depth research from the following aspects: a nonlinear analysis considering the nonlinearity of concrete and a CFD analysis taking into account the wind generator blades. These may be helpful for revealing the mechanical properties and loads of combined steel–concrete wind turbine towers.