Calibration of Design Fatigue Factors for Offshore Wind Turbine Support Structure Based on Fatigue Test Database

The concept of Design Fatigue Factors (DFFs) was introduced for providing desired level of safety in structural fatigue design, often associated with damage calculated from S-N curves. Calculation of fatigue damage from S-N curves can be affected by multiple factors, e.g. types of weld class, corrosion condition, loading conditions, stress concentration on different geometries etc. Each of them can be subject to different level of uncertainties. This study intends to recalibrate the DFFs from a detailed reliability analysis by investigating the probabilistic models derived from the database of S-N curves that has been most frequently used in offshore wind industry. The results of such study indicate that the DFFs can be reduced substantially for the corrosive environmental fatigue models from current standards to the same level of target reliability.


Introduction
Structural fatigue in the offshore wind turbine foundation is a highly uncertain and complex phenomenon. To provide conservatism, the design S-N curve is usually taken as the values associated with mean-minus-2-standard-deviaiton S-N curve, representing a 2.3% probability of exceedance [1]. Additional considerations for a safe design include the following aspects:


Uncertainties of random loading and Miner's Rule fatigue damage calculation  Inspection plan and the reliability of the associate inspection method  Failure mechanism with or without corrosion protection  Failure consequences The DFF (Design Fatigue Factor) is introduced to be applied onto the fatigue damage calculated using the S-N curve based approach in standards such as DNVGL-RP-C203 [2], in order to provide a desired level of target reliability. Some standards use material factors to apply on stresses as an alternative to DFF, but in overall these factors enable the alignment of final design probability of failure or structural reliability to the target reliability.
Instead of calculating the failure probability of each individual design case separately, the use of DFF simplifies this process by a general level of calibration to ensure the design can achieve the target reliability. However, the generalized approach often results in over conservatism in individual cases. Since the values of DFF can constrain any fatigue related design and the foundation cost is a major component in an offshore wind farm development, it is of direct interest for an optimal design to use DFF values not to be overly conservative.
By performing a more detailed calibration of DFF through reliability analysis methods, e.g. FORM, SORM, or Monte Carlo simulation [3], based on probabilistic distribution models from test database for the frequently utilised S-N curve categories in offshore wind industry, less conservative DFFs can be developed which enables offshore wind farm cost reduction with the same levels of target safety.

Failure Function in Reliability Analysis
In this study, the First Order Reliability Method (FORM) has been selected to perform the reliability analysis. The detailed description of the method can be found from classic textbook such as Thoft-Christensen and Baker. [3]. For a FORM analysis, the failure function ( ) must be established in order to provide the failure curve/surface.
The failure function g(Z) for a limit state can be presented as the difference of Load (L) and Resistance (R) modelled as stochastic processes, where the variables are defined as probabilistic variables: And the probability of failure Pf is obtained by: The reliability index β is a measure of the distance from the origin to the failure surface and corresponds to the probability of failure when the limit state function is normally distributed: The S-N curve defines the fatigue life from the allowable number of cycles (N) for a given stress range (S) by: Under different weld categories, the allowable number of cycles N is governed by the Nintercept, i.e. log K and the S to the power of -m, i.e. the negative slope of S-N curve. The fatigue limit state function of a bilinear S-N curve, with the DFF implemented to increase the target design damage Δ, can be written as: where ni is the number of cycles for event group i, t is time; K1 and -m1 are the N-intercept and the negative slope for the low-cycle part of the S-N curve, K2 and -m2 are the N-intercept and the negative slope for the high-cycle part of the S-N curve, and ΔSc is the stress range at the intercept of two-slope S-N curve. As most of the fatigue S-N data are with less than 10 6 to 10 7 cycles, the limit state function can be further simplified following a single-slope S-N curve, and the uncertainties can be modelled as probabilistic models: where Δ is the uncertainty induced by Miner's rule damage calculation, XK is the uncertainty of S-N data representing the variation of the N-intercept in the S-N curve, i.e. log K, K1 c is the characteristic value ofthe S-N curve, from which the fatigue damage (=1) is calculated XS is the uncertainty of stress calculation and loading condition.
The effect of DFF to the reliability index  can be demonstrated in the normalised failure space as in Figure 1. Increasing DFF shifts the failure surface away from the origin, thus increasing the reliability index β.

Uncertainty Models
With the definition of fatigue limit state function, the uncertainties that may be encountered in the design process can be modelled as stochastic variables i,e. Δ, XK, and XS modelled as probabilistic distributions. A literature review of the uncertainty models that have been used throughout various studies is shown in subsections below. Moreover, in order to capture and quantify the uncertainty of the S-N data, a statistical analysis is performed to derive the probabilistic models from real data.

Uncertainty of Miner's Rule, Δ
There are a number of factors contributing to the variation of Miner's Rule such as: Environment (free corrosion, cathodic protected, or in-air)  Load (random, constant, high/low frequencies)  Interaction of the above DNVGL-RP-C203 [2] recommended the uncertainty of Miner's rule to be modelled as a log normal distribution with mean=1 and CoV=0.3. This is the most general assumption adopted in various studies ( [4], [5], [6], [7], [8], [9]).
Other assumptions based on mean=1 but with different CoVs can be found as well. Such as CoV=0.1 is assumed in [Error! Reference source not found.], and CoV=0.2 is assumed in [10].
Zhao et al. [11] found that the CoV varies from case to case, and recommends a log normal distribution with mean = 1 and CoV = 0.45 based on averaging results from a number of tests/models, as shown in Figure 2.

Uncertainty of S-N Curve, XK
In order to construct a sound probabilistic model to accurately represent the uncertainties brought by different design conditions, an overview of available fatigue S-N experiment data is conducted in this study, and data analysis is performed to interpret the data to an reliable probabilistic model, i.e. XK.
Data analysis is a very powerful tool to evaluate the fitness of current standards against the actual fatigue behaviour. As a result, a sound database with clearly defined attributes, sizable sample quantities and diverse variety of characteristics, is essential.
The database used in this study has taken input from a variety of research programme worldwide over the past half of a century in offshore oil & gas and renewable industry. Most of the fatigue experiments are conducted in the 80s and the 90s, where many tests have been produced in large scale programmes. There are relatively few tests conducted from the 90s to now, but nevertheless included in the database.
The entries in the database have been evaluated to ensure its data integrity with the following characteristics:  Quality of each individual test is satisfactory, i.e. certified weld procedure and weld quality  All entries in the database are based on clearly recorded test parameters, such as: a. Definition of loading, e.g. R-ratio, CA/VA loading, frequency b. Material c. Type/geometry and thickness of specimen d. Condition of cathodic protection e. Condition of corrosive environment f. Fabrication details, e.g. weld profile, treatment etc. g. Temperature  Definition of failure criteria and the associated number of cycles  Type of stress range, i.e. hot spot stress or nominal stress An overview list of the S-N test data sources is listed below in Table 1.  Table 2 presents the uncertainty models derived from the S-N database for different corrosion exposure category and weld class using a linear regression model. The linear regression method of the derivation can be referred to HSE report OTH 92 390 [1]. Note that these fatigue test data has been processed with thickness correction factors proposed in DNVGL-RP-C203 [2], as in practical design exercises using size and thickness correction.

Uncertainty of Stress and Loading, XS
The uncertainty in stress and loading can be decomposed into stress part and loading part, where the stress part can be represented by the uncertainty of Stress Concentration Factor, XSCF, and the loading part can be represented by the uncertainty of wind and wave, Xwind and Xwave.

Wind Load Uncertainty, Xwind
Sørensen [37] has suggested a three-band categorisation for modelling the uncertainty of wind load depending on the accuracy and the assessment method selected for the wind measurement/analysis. The standard deviations are assumed in associate with log-normal distribution of mean = 1 for different scenarios illustrated in Table 3.

Wave Load Uncertainty, Xwave
Ambuhl et al. [4] suggested to distinguish the uncertainty of wave loading by the level of assessment and the quantity of data available. A five-band rule of thumb for modelling wave load uncertainty assuming log-normal distributions with mean = 1 is suggested. The uncertainty bands and associated conditions are shown in Table 4.

Stress Concentration Uncertainty, XSCF
In DNVGL-RP-C210 [38], a guideline has been provided for the uncertainty modelling of stress concentration in a probabilistic model. With log-normal distributions of mean = 1, five bands of standard deviation suggested based on the SCF calculation method are shown in Table 5.

Combined Uncertainty of Environmental Loading
To simplify the modelling of wind and wave load uncertainty, XW is introduced as the combination of Xwind and Xwave, i.e.

Reliability Analysis and DFF Calibration
With the fatigue limit state function and the probabilistic models defined, the DFFs that enable designer to reach the dedicated target reliability can be calibrated using reliability tools. FORM is a reliable numerical iteration method that solves multi-dimensional reliability problems in a very efficient way.

DNVGL Uncertainty Model
In order to recalibrate the DFFs for the DNVGL S-N curves, it is essential to understand the DNVGL assumption of uncertainty models. According to Section 9 in DNVGL-RP-C203 [2], and Eq. 10.4 in DNVGL-RP-C210 [38], assumptions of uncertainty model are given in Table 6 and Table 7.  Table 8 that the probabilistic distribution of log K, i.e. XK are based on a one-off assumption, where the design curve is always assumed to be 2×0.2 below the mean value for each of the exposure category. The DNVGL XK distributions are compared to the revised distributions in Table 2, which the results are shown in Figure 3 to Figure 5. It has been found that in all exposure categories available in the fatigue database, the mean strengths of the revised uncertainty models are all higher than the DNVGL models, especially for corrosive categories in F curve.

Benchmark DNVGL Target Reliability
By introducing the fatigue failure function (Eq. 6) with the DNVGL uncertainty models (Table  7), a benchmark target reliability can be calculated for the DFFs specified in DNVGL-ST-0126 [36], Table 4-18, i.e. DFF=3, 2, or 1. The reliability level of DFF=3 is found to be very close to the JCSS Model Code recommended value for moderate consequence with large cost of safety measures, which is also suggested for application of OWT substructure by Sørensen [37].

DFF Calibration of DNVGL S-N Curve with Revised Uncertainty Model
The DNVGL uncertainty model of XK is replaced by the revised uncertainty model specified in Table 2 for the DFF recalibration exercise in this study.
The DFF calibration study is based on the following assumptions, and executed with a FORM analysis: The benefit of revised uncertainty model XK using the fatigue test database can be quantified by the change of required DFF to reach the same target reliability level.
According to Figure 6 to Figure 10, same level of DNVGL benchmark target reliability can be achieved using a similar or lower DFF value.
It can be seen that all three environmental conditions (In-air, CP, FC) are showing lower DFF for the same benchmarked reliability level for F curve. This is mainly because of the higher mean strength derived from data. The In-air D curve requires slightly lower DFF comparing to DNVGL specification, and the In-air T curve gives approximately the same DFF values comparing to DNVGL specification.
Therefore, the recommended practice given by DNVGL, of which the minimum DFFs are either 3, 2, or 1 depending on the exposure category and accessibility for inspection and repair, is revisited in Table 8.

Discussion and Conclusions
DFFs have been used in fatigue design of offshore wind turbine foundation to ensure a desirable level of safety without elaborate probabilistic analysis. However, it has been found that the DFFs in current standards are overly conservative especially for special cases such as fatigue with or without corrosion protection in sea water.
For the purpose of reducing the conservatism, reliability analysis and subsequent recalibration of DFFs have been conducted using probabilistic models based on an extensive fatigue test database. This recalibration of DFFs has shown that the revised DFFs can be substantially lower than those used in current design standards, especially in corrosive environment.
It is therefore proposed to revise the DFFs for D, F, and T S-N weld/joint class, especially for corrosion conditions with or without cathodic protection to reduce over conservatism in the fatigue evaluation while still providing the same target reliability of OWT substructures as in the standards.
The DFFs could be further improved by 1) further study, to refine the fatigue strength model to reduce model uncertainty in the fatigue analysis. 2) to expand the test database to consider other weld/joint categories that are not investigated in this study due to lack of data, with additional test programs.