# Fatigue Reliability and Calibration of Fatigue Design Factors for Offshore Wind Turbines

^{*}

## Abstract

**:**

## 1. Introduction

^{−5}. For unmanned structures a maximum annual probability of failure is typically 10

^{−4}, see DNV-RP-C203 [1]. For offshore wind turbines a reliability level corresponding to unmanned structures or even lower can be relevant. This implies possibilities for cost reductions. Further, if inspections are performed even lower material costs can be obtained, but they have to be balanced with the costs to inspections. Such a reliability- and risk-based approach has been used for offshore oil & gas steel structures, see, e.g., Faber et al. [2] and Moan [3]. In this paper the reliability-based approach is used for support structures for offshore wind turbines, see also Sørensen [4].

## 2. Reliability Modelling of Fatigue Failure Using the SN-Approach

K_{1}, m_{1} | Material parameters for N ≤ N_{C} |

K_{2}, m_{2} | Material parameters for N > N_{C} |

Δs | Stress range |

N | Number of cycles to failure |

T | Material thickness |

T_{ref} | Reference thickness |

α | Scale exponent |

_{σ}groups/intervals such that the number of stress ranges in group i is n

_{i}per year. (ΔQ

_{i}, n

_{i}) is obtained by Rainflow counting and can e.g. be represented by “Markov matrices”. The code-based design equation using the Miner’s rule is written:

K_{i}^{C} | Characteristic value of K_{i} (logK_{i}^{C} equal to the mean of logK_{i} minus two standard deviations of logK_{i}) |

${s}_{i}=\frac{\mathrm{\Delta}{Q}_{i}}{z}{\left(\frac{T}{{T}_{ref}}\right)}^{\alpha}$ | Stress range in group i |

ΔQ_{i} | Action effect (proportional to stress range s_{i} in group i) |

z | Design parameter, e.g., a cross-sectional parameter |

T_{F} | Fatigue life |

_{L}is the design life, typically 25 years for offshore wind turbines. Note that for a linear SN-curve with slope m the FDF value is connected to partial safety factors for fatigue load, γ

_{f}and fatigue strength, γ

_{m}by $FDF={\left({\gamma}_{f}{\gamma}_{m}\right)}^{m}$.

Δ | Model uncertainty related to Miner’s rule for linear damage accumulation. Δ is assumed Log-Normal distributed with mean value = 1 and coefficient of variation COV_{Δ} |

${s}_{i}={X}_{W}{X}_{SCF}\frac{\mathrm{\Delta}{Q}_{i}}{z}{\left(\frac{T}{{T}_{ref}}\right)}^{\alpha}$ | Stress range for group i |

X_{W} | Stochastic variable modelling the uncertainty related to determination of loads. X_{W} is assumed Log-Normal distributed with mean value = 1 and coefficient of variation = COV_{W} |

X_{SCF} | Stochastic variable modelling the uncertainty related to determination of stresses given fatigue loads. X_{SCF} is assumed Log-Normal distributed with mean value = 1 and coefficient of variation = COV_{SCF} |

K_{i} | LogK_{i} is modeled by a Normal distributed stochastic variable according to a specific SN-curve |

t | Time (0 ≤ t ≤ T_{L}) |

_{F}(t) is defined by:

^{−3}and 10

^{−4}.

- One for a fatigue critical detail in air,
- One for a fatigue critical detail in marine conditions with cathodic protection, and
- One for a fatigue detail subject to free corrosion

_{SCF}and X

_{W}should be associated with specific recommendations for how detailed the estimation of stress concentration factors and wind/wave loads should be made.

_{Δ}and the standard deviation of logK

_{1}and logK

_{2}follows the recommendations in DnV-C203 [1]. It is noted that the uncertainties related to Δ and logK

_{i}should be modeled carefully. The uncertainty related to Δ (variable amplitude loading and linear damage accumulation by Miner’s rule) can be significant.

Variable | Distribution | Expected value | Standard deviation | Characteristic value | Comment |
---|---|---|---|---|---|

Δ | LN | 1 | 0.1/0.2/0.3 | 1 | |

X_{SCF} | LN | 1 | 0.05/0.10/0.15 /0.20 | 1 | |

X_{W} | LN | 1 | 0.10/0.20/0.30 | 1 | |

m_{1} | D | 3 | |||

logK_{1} | N | 12.564 | 0.10/0.15/0.2 | 12.164 | In air |

logK_{1} | N | 12.164 | 0.10/0.15/0.2 | 11.764 | With cathodic protection |

logK_{1} | N | 12.087 | 0.10/0.15/0.2 | 11.687 | Free corrosion |

m_{2} | D | 5 | |||

logK_{2} | N | 16.106 | 0.15/0.2/0.25 | 15.606 | In air |

logK_{2} | N | 16.106 | 0.15/0.2/0.25 | 15.606 | With cathodic protection |

logK_{2} | - | Free corrosion |

_{1}and logK

_{2}are assumed fully correlated.

## 3. Acceptable Reliability Level for Fatigue Failure

^{−5}(reliability index equal to 4.3) and 10

^{−4}(reliability index equal to 3.7) are generally accepted, see, e.g., DnV-C203 [1]. No explicit reference exists for the minimum reliability index required for the partial safety factors in IEC 61400-3:2009 [11] for offshore wind turbines. However, the implicit reliability level in the standards used for design of offshore wind turbines by DNV [7] and GL [8] can alternatively be estimated using the stochastic model presented in Section 2 together with the partial safety factors recommended in these standards. The First Order Reliability Method (FORM) was used and verified with Monte Carlo Simulations (MCS) in order to calculate the reliability indices obtained using the stochastic model proposed in Section 2.

Failure critical detail | Inspections | ISO 19902 | GL/DNV |
---|---|---|---|

Yes | No | 10 | 2.0 (3.0) |

Yes | Yes | 5 | 1.5 (2.0) |

No | No | 5 | 1.5 (2.0) |

No | Yes | 2 | 1.0 (1.0) |

^{7}. A linearized approximation of the stress distribution, which is characterized by the straight line, is also considered in order to illustrate the importance of the choice of stress range distribution, see Section 5.1.

- Mechanical load re-distribution may imply larger fatigue loads on other critical details.
- The directional distribution of wind speeds should be taken into account when assessing the fatigue load for the individual fatigue critical details. Using an omnidirectional distribution of wind speeds could be too conservative.
- Probabilistic parallel system effect may be important since failure of one detail does not necessarily imply total failure/collapse.

**Figure 1.**Representative model and linearized model for number of stress ranges as function of stress ranges (normalized).

**Table 3.**Reliability indices for different FDF values. xx/yy indicates reliability indices corresponding to cumulative and annual probability of failure, respectively.

FDF | In air | With cathodic protection | Free corrosion |
---|---|---|---|

1.0 | 1.3/2.4 | 1.2/2.4 | 1.3/2.3 |

2.0 | 2.0/2.8 | 1.9/2.8 | 2.3/3.0 |

3.0 | 2.5/3.1 | 2.4/3.1 | 2.9/3.4 |

_{0}of component 1 or of another component 2 with small or large correlation with component 1.

**Figure 2.**Illustration of updating of the reliability of a critical detail/component by inspection of the same component and by inspection of another component in the same wind turbine or in another wind turbine in a wind farm.

## 4. Reliability Assessment Taking into Account Inspections

#### 4.1. Reliability-Based Inspection Planning

_{F,MAX}is the maximum acceptable annual probability of failure. A similar requirement can be formulated based on the cumulative probability of failure:

**Figure 4.**Illustration of inspection plan where inspections are performed when the annual probability of failure exceeds the maximum acceptable annual probability of failure.

- A decision rule to be used in connection with inspections. The decision rule should specify the action to be taken of a crack is detected, this could be do nothing, grinding, repair by welding, replacement, etc.
- Costs of inspections
- Costs of repairs
- Costs of failure
- Discounting

_{I}and no cracks are detected then the probability of failure can be updated. In order to model no-detection event a limit state equation modelling the crack growth and a model for the reliability of the inspection method, e.g., by a Probability-Of-Detection (POD)-curve is needed, see below.

#### 4.2. Fatigue Assessment by Fracture Mechanics Approach

_{P}.

_{0}is the initial crack depth. The stress intensity range is obtained from

_{e}is an equivalent stress range determined from:

_{e}, i = 1, n are the stress ranges.

**Table 4.**Uncertainty modelling used in the fracture mechanical reliability analysis. D: Deterministic, N: Normal, LN: LogNormal.

Variable | Dist. | Expected value | Standard deviation |
---|---|---|---|

a_{0} | LN | 0.2 mm | 0.132 mm |

lnC | N | μ_{lnC} (reliability based fit to SN approach) | 0.77 |

m | D | m-value (reliability based fit to SN approach) | |

X_{SCF} | LN | 1 | See Table 1 |

X_{W} | LN | 1 | See Table 1 |

n | D | Total number of stress ranges per year | |

a_{C} | D | T (thickness) | |

Y | LN | 1 | 0.1 |

_{L}.

_{C}is the critical crack depth, typically the thickness T.

_{lnCC}and m can be fitted such that difference between the probability distribution functions for the fatigue live determined using the SN-approach and the fracture mechanical approach is minimized.

#### 4.3. POD Curves

_{0}and b are POD parameters. In DnV [21] the POD parameters in Table 5 are indicated for MPI and Eddy Current inspection techniques.

Inspection method | x_{0} | b |
---|---|---|

MPI underwater | 2.950 mm | 0.905 |

MPI above water, Ground test surface | 4.030 mm | 1.297 |

MPI above water, Not ground test surface | 8.325 mm | 0.785 |

Eddy current | 12.28 mm | 1.790 |

_{I}and no cracks are detected then the probability of failure can be updated by:

_{d}is smallest detectable crack length. c

_{d}is modeled by a stochastic variable with distribution function equal to the POD-curve:

## 5. Results

_{L}= 25 years and base values of coefficients of variation in Table 1. The reliability indices are shown for the three representative SN-curves mentioned in Section 2. It is seen that the bilinear SN-curves for “With cathodic protection” and “In air” results in almost the same reliability levels whereas the linear SN-curve for “Free corrosion” results in larger reliability indices, but at the same time also larger values of the design parameter z are obtained.

**Figure 5.**Annual reliability index as function of FDF for T

_{L}= 25 years and base values of coefficients of variation in Table 1.

**Figure 6.**Cumulative reliability index as function of FDF for life T

_{L}= 25 years and base values of coefficients of variation in Table 1.

#### 5.1. Required FDF Values with No Inspections

β | “With cathodic protection” | “In air” | “Free corrosion” |
---|---|---|---|

2.5 | 3.4 | 3.1 | 2.3 |

3.1 | 6.1 | 5.5 | 3.4 |

Δβ | “With cathodic protection” | “In air” | “Free corrosion” |
---|---|---|---|

3.1 | 3.0 | 2.9 | 2.3 |

3.7 | 6.5 | 6.1 | 3.9 |

_{SCF}, X

_{W}and Δ. It is seen as expected that relative changes of the COVs for X

_{SCF}and X

_{W}are more important than changes of COV for Δ.

**Table 8.**Required FDF values for SN-curve “With cathodic protection”. Sensitivity wrt COV [X

_{SCF}] and minimum cumulative reliability level.

β\COV[X_{SCF}] | 0.05 | 0.10 | 0.15 | 0.20 |
---|---|---|---|---|

2.5 | 2.7 | 3.4 | 4.8 | 7.3 |

3.1 | 4.5 | 6.1 | 9.3 | >10 |

**Table 9.**Required FDF values for SN-curve “With cathodic protection”. Sensitivity wrt COV[X

_{W}] and minimum cumulative reliability level.

β\COV[X_{W}] | 0.10 | 0.15 | 0.20 |
---|---|---|---|

2.5 | 3.4 | 4.8 | 7.3 |

3.1 | 6.1 | 9.3 | >10 |

**Table 10.**Required FDF values for SN-curve “With cathodic protection”. Sensitivity wrt COV[Δ] and minimum cumulative reliability level.

β\COV[Δ] | 0.10 | 0.20 | 0.30 |
---|---|---|---|

2.5 | 3.0 | 3.2 | 3.4 |

3.1 | 5.2 | 5.5 | 6.1 |

**Table 11.**Required FDF values for SN-curve “With cathodic protection”. Sensitivity wrt COV[X

_{SCF}] and minimum annual reliability level.

Δβ\COV[X_{SCF}] | 0.05 | 0.10 | 0.15 | 0.20 |
---|---|---|---|---|

3.1 | 2.5 | 3.0 | 4.0 | 5.3 |

3.7 | 4.9 | 6.5 | 9.7 | >10 |

**Table 12.**Required FDF values for SN-curve “With cathodic protection”. Sensitivity wrt COV[X

_{W}] and minimum annual reliability level.

Δβ\COV[X_{W}] | 0.10 | 0.15 | 0.20 |
---|---|---|---|

3.1 | 3.0 | 4.0 | 5.3 |

3.7 | 6.5 | 9.7 | >10 |

**Table 13.**Required FDF values for SN-curve “With cathodic protection”. Sensitivity wrt COV[X

_{Δ}] and minimum annual reliability level.

Δβ\COV[X_{Δ}] | 0.1 | 0.2 | 0.3 |
---|---|---|---|

3.1 | 2.7 | 2.8 | 3.0 |

3.7 | 5.6 | 5.9 | 6.5 |

_{1}] and COV[logK

_{2}]. It is seen as expected that relative changes of the COVs for logK

_{1}and logK

_{2}are less important than X

_{SCF}and X

_{W}but more important than changes of COV for Δ.

**Table 14.**Required FDF values for SN-curve “With cathodic protection”. Sensitivity wrt COV[logK

_{1}]/COV[logK

_{2}] and minimum cumulative reliability level.

β\COV[logK_{1}]/COV[logK_{2}] | 0.10/0.15 | 0.15/0.20 | 0.20/0.25 |
---|---|---|---|

2.5 | 4.1 | 3.7 | 3.4 |

3.1 | 6.7 | 6.3 | 6.1 |

**Table 15.**Required FDF values for SN-curve “With cathodic protection”. Sensitivity wrt COV[logK

_{1}]/COV[logK

_{2}] and minimum annual reliability level.

β\COV[logK_{1}]/COV[logK_{2}] | 0.10/0.15 | 0.15/0.20 | 0.20/0.25 |
---|---|---|---|

2.5 | 3.8 | 3.4 | 3.0 |

3.1 | 7.3 | 6.8 | 6.5 |

**Table 16.**Required FDF values using a linearized model for stress ranges for minimum cumulative reliability level. Base case values in ( ).

β | “With cathodic protection” | “In air” | “Free corrosion” |
---|---|---|---|

2.5 | 3.3 (3.4) | 2.9 (3.1) | 2.3 (2.3) |

3.1 | 5.9 (6.1) | 5.1 (5.5) | 3.4 (3.4) |

**Table 17.**Required FDF values using a linearized model for stress ranges for minimum annual reliability level. Base case values in ( ).

Δβ | “With cathodic protection” | “In air” | “Free corrosion” |
---|---|---|---|

3.1 | 3.0 (3.0) | 2.7 (2.9) | 2.3 (2.3) |

3.7 | 6.4 (6.5) | 5.6 (6.1) | 3.9 (3.9) |

#### 5.2. Required FDF Values with Inspections

- Inspection with Eddy Current, see POD-curve in Table 5
- Visual inspection with an exponential POD-curve with expected value of the smallest detectable cracks lengths equal to 50 mm.

**Table 18.**Required FDF values for minimum cumulative reliability level. SN-curve: “With cathodic protection”. Close visual inspection.

β/number of inspections | 0 | 1 | 2 | 4 | 10 |
---|---|---|---|---|---|

2.5 | 3.4 | 2.7 | 2.3 | 1.3 | 1 |

3.1 | 6.1 | 5.0 | 4.1 | 2.8 | 1 |

**Table 19.**Required FDF values for minimum cumulative reliability level. SN-curve: “With cathodic protection”. Inspections with the Eddy Current technique.

β/number of inspections | 0 | 1 | 2 | 4 | 10 |
---|---|---|---|---|---|

2.5 | 3.4 | 3.0 | 2.7 | 2.3 | 1 |

3.1 | 6.1 | 5.3 | 5.0 | 3.6 | 1.3 |

## 6. Conclusions

## Acknowledgments

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**MDPI and ACS Style**

Márquez-Domínguez, S.; Sørensen, J.D.
Fatigue Reliability and Calibration of Fatigue Design Factors for Offshore Wind Turbines. *Energies* **2012**, *5*, 1816-1834.
https://doi.org/10.3390/en5061816

**AMA Style**

Márquez-Domínguez S, Sørensen JD.
Fatigue Reliability and Calibration of Fatigue Design Factors for Offshore Wind Turbines. *Energies*. 2012; 5(6):1816-1834.
https://doi.org/10.3390/en5061816

**Chicago/Turabian Style**

Márquez-Domínguez, Sergio, and John D. Sørensen.
2012. "Fatigue Reliability and Calibration of Fatigue Design Factors for Offshore Wind Turbines" *Energies* 5, no. 6: 1816-1834.
https://doi.org/10.3390/en5061816