**3. Finite element analysis**

## **3.1. Modeling**

The nonlinear finite element (FE) analysis code COM3D developed by Maekawa et al. [14, 15] are used in this study. The decrease of stiffness and the accumulation of plasticity of concrete subjected cyclic load are carefully formulated for concrete in the code. In particular, employing the logarithmic integral scheme that enabled to calculate fatigue damage of concrete is one of the advantages of this code. The properties for steel are expressed with bi-linear form.

An overview of the FE model is shown in **Figure 5**. The model was modified from the model in the previous study [13] through further material investigation and verifications. Mechanical properties of these constituent materials are summarized in **Table 1**.

In order to simplify structural model, the shape of nacelle and blades was not directly modeled. Alternatively, dead weight of them was applied to certain elements located at the top of with each material density. All the members except anchor bolts and the intermediate restraining reinforcements of the pedestal were modeled by solid element. Exceptions were expressed by line element; in particular, the torque on an anchor bolt was replaced by initial strain of the lines. The boundary condition between steel and concrete was modeled by joint element based on the Mohr–Coulomb theory with 0.6 as friction coefficient. Vertical displacement was restricted at the nodes of the footing bottom surface; however, confinement of surrounding soil was not considered on the side of the footing as same in literature [16].

**Figure 5.** Overview of FE model (based on [10]).

**Figure 4.** Fourier spectrum of acceleration of tower and strain on anchor bolt (based on [13]).

**Figure 3.** Acceleration response of tower and trajectory of its displacement (based on [13]).

The acceleration response of the tower in the time domain and trajectory of the tower displacement, which are derived through double integration of acceleration in the time domain are shown in **Figure 3**. To remove noise, a digital band pass filter with pass band between about 0.1 and 30 Hz was designed. The maximum displacement was about 0.5 cm at the top of the tower in the EW direction. Elliptical trajectories with different main axis were observed for each height in different scales when the wind turbine was operating. In particular, the trajectories of the top and middle of the tower were almost similar. This means that the pre-

The strain of nut for anchor bolt clearly depended on the acceleration variations, even though the value of the response was less than 1 μ. This can be explained that the location of strain gauges attached to anchor bolts was not consistent with wind direction which measured max

When taking a long-term measurement, time varying character of the wind can be captured in a spectrum. The Fourier spectrum exhibited the waveform shown in **Figure 4**. The natural

dominant vibration mode was the primary mode.

106 Stability Control and Reliable Performance of Wind Turbines

wind speed.

**2.3. Action to the foundation transmitted from the tower**


**Table 1.** Mechanical properties of each material in FE analysis.

#### **3.2. Verification of the model**

The model was verified by the comparison of the analytical result and data obtained during free vibration test. The natural frequency of tower and damping factor calculated by FE analysis were 1.84 Hz and 0.30%. Those obtained from the field test were 1.78 Hz and 0.27% [10].

In addition, the agreement of analytical results and data obtained in field measurement were examined. First, the displacement at the top of tower which was converted using double integration of acceleration in the time domain as mentioned in 2.1.2, was inputted to the model. Then, acceleration of tower, strain of tower body, and strain of anchor bolts calculated by FE analysis were compared with data obtained in the field for six different cases including data during typhoon and earthquakes. Samples of examinations in terms of maximum and minimum values are shown in **Figure 6**. The analysis results tended to show spikes due to the difficulty of convergence of calculation. However, acceptable agreement was seen in all six cases.

#### **3.3. Prediction of failure mode**

In order to determine the failure mode of this structure, monotonic horizontal displacement was applied to a node at the top of tower. The bending moment at the bottom of tower versus rotation angle derived from Eq. (1) [10] is shown in **Figure 7**.

$$\phi = \arctan\left(\frac{\delta\_{\rm at} - \delta\_{\rm w}}{B}\right) \tag{1}$$

where ϕ is rotation angle, δzt is vertical displacement of anchor plate in tension side, δzc is vertical displacement of anchor plate in compression side, and B is a diameter of anchor ring. The moment increased rapidly at the beginning of loading, and it gradually became mild. Then, the moment remained static after when 50% of anchor bolts reached their yielding strength. At the time, the horizontal displacement at the top of tower was 50 cm where the moment and rotation angle reached 7874 kNm and 0.0033 rad. **Figure 7** also shows the normalized stiffness by using the right axis. The stiffness mentioned here means the value that was obtained when moment was divided by the rotation angle. The normalized stiffness means the stiffness normalized by the initial stiffness. According to **Figure 7**, the normalized

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What did cause the significant drop to normalized stiffness? It was the cracking of concrete at the tip of anchor ring inside pedestal (see **Figure 8**). Consequently, the crack developed horizontally and came to the center of the pedestal in the end of analysis (see **Figure 9**). Regarding this FE analysis, the failure mode was a coupling mode of yielding 50% of anchor bolts and

development of horizontal crack to more than half of the pedestal's width.

stiffness significantly dropped in the first 0.0001 rad.

**Figure 8.** Principal strain distribution at the drop of normalized stiffness.

**Figure 7.** Bending moment at the bottom of tower and rotation angle.

**Figure 6.** Comparison of FE model and observed data during power generating (left: acceleration at the top of tower; middle: strain of tower body; right: strain of anchor bolt) (based on [10]).

**Figure 7.** Bending moment at the bottom of tower and rotation angle.

where ϕ is rotation angle, δzt is vertical displacement of anchor plate in tension side, δzc is vertical displacement of anchor plate in compression side, and B is a diameter of anchor ring.

The moment increased rapidly at the beginning of loading, and it gradually became mild. Then, the moment remained static after when 50% of anchor bolts reached their yielding strength. At the time, the horizontal displacement at the top of tower was 50 cm where the moment and rotation angle reached 7874 kNm and 0.0033 rad. **Figure 7** also shows the normalized stiffness by using the right axis. The stiffness mentioned here means the value that was obtained when moment was divided by the rotation angle. The normalized stiffness means the stiffness normalized by the initial stiffness. According to **Figure 7**, the normalized stiffness significantly dropped in the first 0.0001 rad.

What did cause the significant drop to normalized stiffness? It was the cracking of concrete at the tip of anchor ring inside pedestal (see **Figure 8**). Consequently, the crack developed horizontally and came to the center of the pedestal in the end of analysis (see **Figure 9**). Regarding this FE analysis, the failure mode was a coupling mode of yielding 50% of anchor bolts and development of horizontal crack to more than half of the pedestal's width.

**Figure 8.** Principal strain distribution at the drop of normalized stiffness.

**Figure 6.** Comparison of FE model and observed data during power generating (left: acceleration at the top of tower;

The model was verified by the comparison of the analytical result and data obtained during free vibration test. The natural frequency of tower and damping factor calculated by FE analysis were 1.84 Hz and 0.30%. Those obtained from the field test were 1.78 Hz and 0.27% [10].

**Compressive strength** 

Concrete 23.5 21.0 — 8.13 0.2 Anchor bolt 205 — 235 400 0.3 Other steels 205 — 325 490 0.3 Reinforcing bar 205 — 345 517.5 0.3

**Yield strength (MPa)**

**Tensile strength** 

**Poisson ratio**

**(MPa)**

**(MPa)**

In addition, the agreement of analytical results and data obtained in field measurement were examined. First, the displacement at the top of tower which was converted using double integration of acceleration in the time domain as mentioned in 2.1.2, was inputted to the model. Then, acceleration of tower, strain of tower body, and strain of anchor bolts calculated by FE analysis were compared with data obtained in the field for six different cases including data during typhoon and earthquakes. Samples of examinations in terms of maximum and minimum values are shown in **Figure 6**. The analysis results tended to show spikes due to the difficulty of convergence of calculation. However, acceptable agreement was seen in all

In order to determine the failure mode of this structure, monotonic horizontal displacement was applied to a node at the top of tower. The bending moment at the bottom of tower versus

*<sup>δ</sup>zt* <sup>−</sup> *<sup>δ</sup>* \_\_\_\_\_*zc*

*<sup>B</sup>* ) (1)

middle: strain of tower body; right: strain of anchor bolt) (based on [10]).

rotation angle derived from Eq. (1) [10] is shown in **Figure 7**.

*φ* = *arctan*(

**3.2. Verification of the model**

**Young's modulus** 

108 Stability Control and Reliable Performance of Wind Turbines

**Table 1.** Mechanical properties of each material in FE analysis.

**(GPa)**

**3.3. Prediction of failure mode**

six cases.

**Figure 9.** Principal strain distribution of the cross section of joint parts (left: at 1195 kNm of moment; right: at 7874 kNm of moment) (based on [10]).

The ultimate bending moment calculated by the FE analysis was 7874 kNm, it is much higher than the design moment in the case of earthquake in this area; 2430 kNm. This means that the targeted wind turbine has sufficient allowance of safety. However, the cracking moment 1195 kNm is close to the design moment owing to the wind in this area; 855 kNm. Therefore, evaluation of fatigue resistance of concrete foundation is needed.

#### **3.4. Identification of index**

Since the cracking inside pedestal was observed as the cause of the decrease of stiffness, evaluation of fatigue resistance of this structure focused on this event. The specific index, space averaged second invariant strain [17] was employed to determine the possibility of cracking. This index is independent of direction of stress or strain that is unsettled at each moment under the vibration of tower, but is a scalar obtained by Eqs. (2) and (3) [17].

There are no dependence on dimension or column: and  $\omega$  increases or common number under the vibration of tower, but is a scalar obtained by Eqs. (2) and (3) [17].

$$\sqrt{\frac{\gamma}{2}} = \sqrt[2]{\frac{2}{3}\left\{{\left(\frac{\varepsilon\_x - \varepsilon\_y}{2}\right)}^2 + {\left(\frac{\varepsilon\_y - \varepsilon\_z}{2}\right)}^2 + {\left(\frac{\varepsilon\_z - \varepsilon\_z}{2}\right)}^2\right\} + \left(\frac{\gamma\_y}{2}\right)^2 + \left(\frac{\gamma\_x}{2}\right)^2 + \left(\frac{\gamma\_x}{2}\right)^2 + \left(\frac{\gamma\_x}{2}\right)^2} \tag{2}$$

$$\overline{\sqrt{l\_2}} = \frac{\int\_{\mathbb{V}} \sqrt{l\_2} \cdot w(\mathbf{x})dV}{\int\_{\mathbb{V}} w(\mathbf{x})dV} \text{ } w(\mathbf{x}) \text{ } = \begin{cases} 1 - \mathbf{x}/L & \mathbf{x} \le L \\\ 0 & \mathbf{x} > L \end{cases} \tag{3}$$

**3.5. Evaluation of fatigue resistance of concrete foundation**

**Figure 10.** Horizontal displacement of tower versus the space-averaged second invariant of strain.

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design moment considers safety factor as 3.0.

The fatigue resistance of concrete foundation was examined by FE analysis. Since the horizontal reaction force at the top of tower when the moment was 1195 kNm was about 84kN in FE analysis, input horizontal load at the top of tower was offered as sine wave with different amplitudes. In addition, the number of cycles when the space-averaged second invariant of strain reach threshold value 0.00032 were calculated for each cases. The relationship between the normalized amplitude by 84kN and the number of cycle at threshold value was shown in **Figure 11**. It should be noted that the design moment owing to the wind in this is 855 kNm that is 71% of the calculated moment 1195 kNm. Thus, according to **Figure 11**, the cracking inside pedestal concrete possibly starts after 763,888 swings of tower due to strong wind. However, the

**Figure 11.** Normalized amplitude of horizontal load—cycles at cracking inside concrete based on FE analysis.

where, √ \_\_ *J* 2 ′ is the second invariant of strain, ¯ √ \_\_ *J* 2 ′ is the space-averaged second invariant of strain, ε and γ are normal and shear strains respectively, w(x) is a weighting function, x is a distance from the tip of anchor plate in tension side (mm), L is a radius of average volume (mm). For this analysis, L was determined as 200 mm based on [18].

The horizontal displacement of tower versus the space-averaged second invariant of strain is shown in **Figure 10**. The space-averaged second invariant of strain at the occurrence of the horizontal crack determined by **Figure 9** was identified as 0.000032. Thus, threshold value for first limit state of this structure was defined as 0.000032 by the space-averaged second invariant of strain.

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**Figure 10.** Horizontal displacement of tower versus the space-averaged second invariant of strain.

The ultimate bending moment calculated by the FE analysis was 7874 kNm, it is much higher than the design moment in the case of earthquake in this area; 2430 kNm. This means that the targeted wind turbine has sufficient allowance of safety. However, the cracking moment 1195 kNm is close to the design moment owing to the wind in this area; 855 kNm. Therefore,

**Figure 9.** Principal strain distribution of the cross section of joint parts (left: at 1195 kNm of moment; right: at 7874 kNm

Since the cracking inside pedestal was observed as the cause of the decrease of stiffness, evaluation of fatigue resistance of this structure focused on this event. The specific index, space averaged second invariant strain [17] was employed to determine the possibility of cracking. This index is independent of direction of stress or strain that is unsettled at each moment

\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \_\_<sup>2</sup>

1 − *x*/*L x* ≤ *L*

<sup>0</sup> *<sup>x</sup>* <sup>&</sup>gt; *<sup>L</sup>* (3)

is the space-averaged second invariant of strain,

(2)

evaluation of fatigue resistance of concrete foundation is needed.

under the vibration of tower, but is a scalar obtained by Eqs. (2) and (3) [17].

<sup>∫</sup>*<sup>V</sup> <sup>w</sup>*(*x*)*dV <sup>w</sup>*(*x*) <sup>=</sup> {

√ \_\_ *J* 2 ′

ε and γ are normal and shear strains respectively, w(x) is a weighting function, x is a distance from the tip of anchor plate in tension side (mm), L is a radius of average volume (mm). For

The horizontal displacement of tower versus the space-averaged second invariant of strain is shown in **Figure 10**. The space-averaged second invariant of strain at the occurrence of the horizontal crack determined by **Figure 9** was identified as 0.000032. Thus, threshold value for first limit state of this structure was defined as 0.000032 by the space-averaged second invariant of

\_\_ *J* 2 ′ <sup>∙</sup> *<sup>w</sup>*(*x*)*dV* \_\_\_\_\_\_\_\_\_\_

**3.4. Identification of index**

of moment) (based on [10]).

110 Stability Control and Reliable Performance of Wind Turbines

\_\_ *J* 2 ′ <sup>=</sup> <sup>√</sup>

¯

<sup>3</sup>{(

√ \_\_ *J* 2 ′ <sup>=</sup> <sup>∫</sup>*<sup>V</sup>* <sup>√</sup>

is the second invariant of strain, ¯

this analysis, L was determined as 200 mm based on [18].

*<sup>ε</sup><sup>x</sup>* <sup>−</sup> *<sup>ε</sup>* \_\_\_\_*<sup>y</sup>* <sup>2</sup> ) 2 + ( *<sup>ε</sup><sup>y</sup>* <sup>−</sup> *<sup>ε</sup>* \_\_\_\_*<sup>z</sup>* <sup>2</sup> ) 2 + ( *<sup>ε</sup><sup>z</sup>* <sup>−</sup> *<sup>ε</sup>* \_\_\_\_*<sup>x</sup>* <sup>2</sup> ) 2 } + ( *γxy* \_\_\_ <sup>2</sup> ) 2 + ( *γyz* \_\_\_ <sup>2</sup> ) 2 + ( *γ*\_\_\_*zx* <sup>2</sup> ) 2

√

\_\_ *J* 2 ′

where, √

strain.

**Figure 11.** Normalized amplitude of horizontal load—cycles at cracking inside concrete based on FE analysis.

#### **3.5. Evaluation of fatigue resistance of concrete foundation**

The fatigue resistance of concrete foundation was examined by FE analysis. Since the horizontal reaction force at the top of tower when the moment was 1195 kNm was about 84kN in FE analysis, input horizontal load at the top of tower was offered as sine wave with different amplitudes. In addition, the number of cycles when the space-averaged second invariant of strain reach threshold value 0.00032 were calculated for each cases. The relationship between the normalized amplitude by 84kN and the number of cycle at threshold value was shown in **Figure 11**.

It should be noted that the design moment owing to the wind in this is 855 kNm that is 71% of the calculated moment 1195 kNm. Thus, according to **Figure 11**, the cracking inside pedestal concrete possibly starts after 763,888 swings of tower due to strong wind. However, the design moment considers safety factor as 3.0.
