The design of foundations for offshore wind turbines (OWTs) requires the assessment of long-term performance of the soil–structure interaction (SSI), which is subjected to many cyclic loadings. In terms of serviceability limit state (SLS), it has to be ensured that the load on the foundation does not exceed the operational tolerance prescribed by the wind turbine manufacturer throughout its lifetime. This work aims at developing a probabilistic approach along with a reliability framework with emphasis on verifying the SLS criterion in terms of maximum allowable rotation during an extreme cyclic loading event. This reliability framework allows the quantification of uncertainties in soil properties and the constitutive soil model for cyclic loadings and extreme environmental conditions and verifies that the foundation design meets a specific target reliability level. A 3D finite-element (FE) model is used to predict the long-term response of the SSI, accounting for the accumulation of permanent cyclic strain experienced by the soil. The proposed framework was employed for the design of a large-diameter monopile supporting a 10 MW offshore wind turbine.

Offshore wind turbines are slender and flexible structures which have to withstand diverse sources of irregular cyclic loads (e.g. winds, waves and typhoons). The foundation, which has the function of transferring the external loads to the soil, must resist this repeated structural movement by minimizing the deformations.

The geotechnical design of the foundation for an offshore wind turbine (OWT) has to follow two main design steps named static load design (or pre-design) and cyclic load design. A design step is mainly governed by limit states: i.e. the ultimate limit state (ULS), the serviceability limit state (SLS) and the fatigue limit state (FLS). The design of an offshore structure mostly starts with the static load design step in which a loop between the geotechnical and structural engineers is required to converge to a set of optimal design dimensions (pile diameter, pile length and can thickness). This phase is governed by the ULS in which it must be ensured that the soil's bearing capacity withstands the lateral loading of the pile within the allowable deformations (i.e. pile deflection and pile rotation at the mud-line).

Subsequently, the pre-design is checked for the cyclic load. The verification of the pre-design for the cyclic load design step regards three limit states: ULS, SLS and FLS. The cyclic stresses transferred to the soil can reduce the lateral resistance by means of liquefaction (ULS); can change the soil stiffness which can cause resonance problems (FLS); and can progressively accumulate deformation into the soil, leading to an inclination of the structure (SLS). If one of these limit states is not fulfilled, cyclic loads are driving the design and the foundation dimensions should be updated.

Performing the checks for the cyclic load design step is very challenging due to the following: (i) a high number of cycles is usually involved; (ii) soil subjected to cyclic stresses may develop non-linearity of the soil response, pore water pressure, changing in stiffness, and damping and accumulation of soil deformation (Pisanò, 2019); (iii) the load characteristics such as frequency, amplitude and orientation are continually varying during the lifetime; (iv) characteristic of the soil such as type of material, porosity and drainage condition can lead to different soil responses; (v) the relevant codes (BSH, 2015; DNV-GL, 2017) do not recommend specific cyclic load methods for predicting the cyclic load behaviour of structures, which leads to the development of various empirical formulations (Cuéllar et al., 2012; Hettler, 1981; LeBlanc et al., 2009) or numerically based models (Zorzi et al., 2018; Niemunis et al., 2005; Jostad et al., 2014; Achmus et al., 2007). Despite the different techniques used in these models, they all predict the soil behaviour “explicitly”, based on the number of cycles instead of a time domain analysis (Wichtmann, 2016). Time domain analysis for a large number of cycles is not convenient due to the accumulation of numerical errors (Niemunis et al., 2005).

In common practice due to the non-trivial task faced by the engineers, simplifications and hence introduction of uncertainties and model errors are often seen. The application of probabilistically based methods for designing offshore foundations is not a new topic (Velarde et al., 2019, 2020; Carswell et al., 2014), and it is mainly related to the static design stage. Very limited research has been developed regarding the probabilistic design related to the cyclic load design stage.

This current work focuses on the cyclic loading design stage and the
verification of the serviceability limit state. During the design phase, the
wind turbine manufacturers provide a tilting restriction for operational
reasons. The recommended practice DNV-GL-RP-C212 (DNV-GL, 2017) provides the
order of magnitude for the maximum allowed tilting of 0.25

Fluid levels and cooling fluid movement can vary.

Some reasons are due to aesthetics.

An advanced numerical method called the soil cluster degradation (SCD) method was developed (Zorzi et al., 2018). This method explicitly predicts the cyclic response of the soil–structure interaction (SSI) in terms of the foundation rotation. The main objective of this study is to use the SCD method within a probabilistic approach. The probabilistic approach along with the reliability framework was used to quantify the main uncertainties (aleatoric and epistemic), explore which uncertainty the response is most sensitive to and de-sign the long-term behaviour of the foundation for a specific target reliability level. In this paper, first the developed reliability-based design (RBD) framework is outlined in detail. Then, an application of the proposed RBD framework is presented for a large-diameter monopile supporting a 10 MW offshore wind turbine.

The rotation experienced by the foundation structure subjected to cyclic
loading is considered partially irreversible (irreversible serviceability
limit states) because the soil develops an accumulation of irreversible
deformation due to the cyclic loading action. For this reason, it is noted
that the accidental and environmental load cases for the SLS design are the
extreme loads that give the highest rotation. As for a deterministic
analysis, the first step in the reliability-based analysis is to define the
structural failure condition(s). The term failure signifies the infringement
of the serviceability limit state criterion, which is here set to a tilting
of more than 0.25

The design has to be evaluated in terms of the probability of failure. The
probability of failure is defined as the probability of the calculated value
of rotation

IEC 61400-1 (IEC, 2009) sets as a requirement with regard to the safety of
wind turbine structures an annual probability of failure equal to

IEC61400-1 does not specify the target reliability levels for the SLS
condition. Therefore, it can be argued that the target for SLS in this paper
should be in the range of

The calculation of the model response

The loading input for the model must be a design storm event simplified in a series of regular parcels. This loading assumption is also recommended by DNV-GL-RP-C212 (DNV-GL, 2017) and the BSH standard (BSH, 2015). The method is implemented in the commercial code PLAXIS 3D (PLAXIS, 2017).

Three stochastic input variables
(

The design of experiment (DoE) procedure is used to explore the most
significant combinations of the input variables

Methodology of reliability analysis.

Figure 1 summarizes the methodology for the reliability analysis design for lateral cyclic loading. The framework starts with the uncertainty quantification from the available data (CPT, cyclic laboratory tests of the soil, and metocean and aero-hydro-servo-elastic model) and the derivation of the stochastic input variables (soil stiffness, cyclic contour diagram and storm event). The chosen stochastic variables are the inputs of the SCD model. Based on the stochastic input variables, a response surface is then trained to yield the same output (in terms of structural tilting) of the 3D FE simulations. The response surface is then used to calculate the probability of failure passing through the formulation of the limit state equation and the MC simulation. If the calculated probability of failure does not meet the target probability, then the foundation geometry has to be changed, and the methodology is repeated to check whether the new design is safe.

In this section, firstly, the monopile pre-design (static load design step) is carried out in which the subsoil conditions of the case study and the ULS design of the monopile geometry supporting a 10 MW wind turbine are explained. The pre-design of the monopile is developed using the hardening soil model in finite-element model to predict the static response of the monopile.

Then the reliability framework for the cyclic load design shown in Fig. 1 is applied to the monopile to check if the pre-design satisfies the SLS criteria. The following subsections discuss the derivation of input uncertainties for the SCD method, derivation of the response surface and probability of failure, and reliability index calculation.

For the present case study, a tip resistance from the cone penetration test (CPT) and the boring profile are used to determine the geotechnical properties and soil stratigraphy at the site, where the monopile is assumingly installed. A CPT is basically a steel cone which is pushed into the ground and the tip resistance is recorded. Based on the recorded tip resistance, soil stratigraphy and soil properties can be empirically derived.

The CPT, shown in Fig. 2, features an increase in the tip resistance with
increasing depth, which is typical for sand. In combination with the
borehole profile, the tip resistance from the CPT suggests that the soil can
be divided into two different layers. At approximately

CPT profile.

To accurately predict the soil–structure interaction and incorporate the rigid behaviour of the large-diameter monopile, the ULS geotechnical verification of the preliminary design of the monopile is carried out, using the finite-element method in PLAXIS 3D.

The monopile is modelled in PLAXIS as a hollow steel cylinder using plate elements. For the steel, a linear elastic material is assumed with a Young's modulus of 200 GPa and a Poisson coefficient of 0.3. The interface elements are used to account for the reduced shear strength at the pile's surface.

The soil model used is the hardening soil model with small-strain stiffness (HSsmall) (PLAXIS, 2017). The hardening soil model with small strain stiffness can predict the non-linear stress–strain behaviour of the soil. It considers a stress and strain stiffness dependency, can predict the higher stiffness of the soil at small strain which is relevant for cyclic loading condition, and distinguishes between loading and unloading stiffness.

On the other hand, the Mohr–Coulomb model approximates the complex non-linear behaviour of the soil by a linear-elastic perfectly plastic constitutive law.

The soil model parameters for the two layers are derived from the tip resistance (Fig. 2) and listed in Table 1. The relative density (which is related to the soil porosity) of the two layers is calculated using the formula from Baldi et al. (1986) with the over-consolidated parameters (typical for offshore conditions), leading to a mean value of 70 % and 90 % for the first and second layers, respectively.

Soil model parameters.

The monopile design requires a loop between the structural and geotechnical
engineers to update the soil stiffness and loads at the mud-line level. A
fully coupled aero-hydro-servo-elastic model using HAWC2 (Larsen and Hansen,
2015) is developed to perform the time-domain wind turbine load simulations
(Velarde et al., 2020). The soil–structure interaction model is based on
the Winkler-type approach, which features a series of uncoupled non-linear
soil springs (so called

The final pile design consists of an outer pile diameter at the mud-line level of 8 m a pile thickness of 0.11 m, and a pile embedment length of 29 m. The natural frequency of the monopile is 0.20 Hz and is designed to be within the soft-stiff region. Fatigue analysis of the designed monopile is also carried out (Velarde et al., 2020). Figure 3a shows the horizontal displacement contour plot at 3.5 MN horizontal force, while Fig. 3b shows the horizontal load-rotation curve at the mud-line.

The application of the SCD model requires three inputs – soil stiffness (for the Mohr–Coulomb soil model), cyclic contour diagrams and a design storm event. The laboratory testing and field measurements are used to estimate the inputs for the model. In this estimation process, different sources of uncertainty of unknown magnitude are introduced (Wu et al., 1989). These parameters then have to be modelled as stochastic variables with a certain statistical distribution.

The uncertainties of the soil stiffness used in the SCD model are analysed.
The soil model employed in the SCD method is the Mohr–Coulomb model, with a
stress-dependent stiffness (i.e. the stiffness increases with depth). For
cyclic loading problems, the unloading–reloading Young's modulus

The design tip resistance is established by means of the best-fit line in
the data. A linear model is fitted to the data for each layer (Fig. 4,
green line). The maximum likelihood estimation (MLE) is used for estimating
the parameters of the linear model along with the fitting error (assumed to
be normally distributed and un-biased). From the MLE method, the standard
deviations and correlations of the estimated parameters (Sørensen, 2011)
are obtained. The linear model is expressed by means of Eq. (5) as below:

Average tip resistance.

Stochastic input variable for tip resistance.

The residuals are then plotted to check the assumption of the normality of the model error. For the first layer (Fig. 5a), the distribution of the residual is slightly skewed to the right. This means that the trend line under-represents the tip resistance due to the presence of high peaks at the boundary layer. For the second layer (Fig. 5b), a normal distribution about the zero mean is visible, implying that a better fit is achieved.

Histogram of residual for layer 1

An empirical linear relationship is used to calculate the drained constraint
modulus in unloading–reloading

To understand the uncertainty in the stiffness modulus,

Thus, the calculation of the drained constraint modulus in unloading–reloading, covering all possible uncertainties, is summarized as
follows:

Figure 6 shows the variability of the soil modulus

The drained constraint modulus in unloading–reloading

The soil stiffness depends on the depth. In the Mohr–Coulomb model, a
linear increase in the stiffness with depth is accounted for using the
following formula:

for the first layer at

for the second layer at

The aim of the contour diagrams is to provide a 3D variation in the
accumulated permanent strain in the average stress ratio (ASR), which is the
ratio of the average shear stress to the initial vertical pressure or
confining pressure; the cyclic stress ratio (CSR), which is the ratio of the
cyclic shear stress to the initial vertical pressure or confining pressure;
and the number of cycles (

For this work, a series of undrained single-stage two-way cyclic simple shear tests were performed at the Soil Mechanics Laboratories of the Technical University of Berlin. The tests were carried out on reconstituted soil samples. The samples were prepared by means of the air pluviation method. The initial vertical pressure was 200 kPa and no pre-shearing was considered.

The cyclic behaviour of the upper layer of sand was evaluated with samples prepared at a relative density of 70 %. For the lower layer sand, a 90 % relative density was used. Two-way cyclic loading tests were carried out, testing different combinations of ASR and CSR. All the tests were stopped at 1000 cycles or at the start of the cyclic mobility phase. For the results on the cyclic behaviour of various tests and relative densities, refer to Zorzi et al. (2019b).

All the data extracted from the laboratory tests were assembled in a 3D matrix (ASR, CSR,

Owing to this variability of the test, a mathematical formulation was fitted
to the raw interpolation. For this reason, different two-dimensional slices
(CSR vs.

Slice of the cyclic contour diagram.

The power-law function can be written in the form of Eq. (1).

Based on the results of the fitting procedure, a standard deviation of the
fitting error of 0.008 is chosen for the two diagrams for the two soils. The
parameters

It has to be noted that the fitting error, to some extent, reflects the uncertainties of repeatability of the tests. Moreover, the relative density of the soil samples is based on the empirical relation applied to the tip resistance (Sect. 3.1). To account for the uncertainty in the relative density, different sets of contour diagrams should have been derived from several tests performed with soil samples at different relative densities.

The contour diagrams for two different ASR slices are presented in Figs. 8 and 9 for the upper and lower layers, respectively.

Cyclic contour diagram for the first layer.

Cyclic contour diagram for the second layer.

The load input parameter for the SCD model is characterized by a regular loading package with a mean and cyclic amplitude load and an equivalent number of cycles (hereafter called “load inputs” for simplicity). In common practice, the structural engineer provides the irregular history at mud-line level by means of the aero-hydro-servo-elastic model. Therefore, a procedure is needed to transform the irregular design storm event to one single regular loading parcel. The environmental load used for the cyclic loading design relies on the chosen return period for the load. The statistical distribution of the environmental loads is then based on different return periods.

The design storm event is here defined as a 6 h duration of the extreme load (also called the peak of the storm) (DNV-GL, 2017). The underlying assumption in considering only the peak is that most of the deformations, which the soil experiences, happen at the peak of the storm. The considered design load case is DLC 6.1 (IEC, 2009; BSH, 2015), i.e. when the wind turbine is parked and yaw is out of the wind. The ULS loads are considered for the cyclic load design.

To derive the irregular load history at the mud-line level, the fully coupled
aero-hydro-servo-elastic model is developed in the wind turbine simulation
tool, HAWC2. Based on 5-year in situ metocean data from the North Sea, the
environmental contours for different return periods are derived as shown in
Fig. 10 (Velarde et al., 2019). The marginal extreme wind distribution is
derived using the peak-over-threshold method for wind speed above 25 m s

Environmental contour plot for extreme sea states (Velarde et al., 2019).

The five design sea states for maximum wind speed are summarized in Table 3. To account for short-term variability in the responses, 16 independent realizations are considered for each design sea state.

Design sea state for maximum wind speed.

Time-domain simulations provide an irregular force history of 10 min at the mud-line. To transform the 10 min irregular loading to a 6 h storm, each 10 min interval is repeated 36 times.

The irregular load histories have to be simplified to one equivalent regular package with a specific mean and cyclic load amplitude and an equivalent number of cycles that lead to the same damage accumulation (accumulation of soil deformation) as that of the irregular load series.

The following procedure is used (Andersen, 2015).

The rain flow counting method is utilized to break down the irregular history into a set of regular packages with different combinations of mean force

All the bins are ordered with increasing maximum force

3D contour diagrams in conjunction with the strain accumulation method are then used to calculate the accumulation of deformation. After scaling the loads to shear stresses, the result of this procedure gives the equivalent number of cycles for the highest maximum force

Rain flow matrix for 100-year-return-period wind speed.

To obtain a statistical distribution, the mean force, cyclic amplitude force and equivalent number of cycles are plotted versus the probability of non-exceedance for each return period.

The black points in the three following figures are, respectively, the mean load, cyclic amplitude and number of cycles of the regular packages obtained from the previous procedure and plotted vs. the probability of not exceedance for each return period. Assuming that for each return period the black points have a normal distribution, the 0.50 fractile (red circles) and the 0.95 fractile (blue circles) are obtained.

The statistical distributions for the loads are derived by fitting a Gumbel
distribution to the 0.95 fractile values (NORSOK, 2007). The MLM is employed
to fit the cumulative Gumbel distribution to the extreme response (blue
circles). The cumulative density function distribution is defined as

Gumbel parameters of the distribution for the load inputs.

Looking at the distribution of Fig. 12a–c, larger 0.50 fractiles (red circles) are present when increasing the return period. This is more pronounced for the mean force and is expected because the higher the return period, the higher the mean pressure on the wind turbine tower.

Distribution of the load inputs.

The scatter for each return period is more significant when the return period increases. This can have different reasons; for example, the “rare” storms with a lower probability of occurrence could cause more non-linearity problems, varying the wave and wind speeds in the aero-hydro-servo-elastic model. It could also depend on the model uncertainty in the time domain simulations.

The correlation coefficients

This type of error is difficult to estimate because it requires the
validation of the numerical error against different model tests. In the case
of the SCD model, this error arises due to the simplification of the
model for a much more complex behaviour of the soil–structure interaction
under cyclic loading. The model error

The stochastic variables are summarized in Table 5. For simplicity, a full correlation between the soil stiffness of the two layers and the loads is assumed.

Summary of the stochastic variables.

Once the stochastic variables are defined, the 3D FE model has to be substituted by a response surface.

The DoE is used to obtain the training point from the FE simulation. As most
of the variables are correlated, three stochastic variables are considered:
(i) the stiffness of the upper soil layer

Based on visual inspection of the output from the 3D FE model, a
second-order polynomial function is fitted to the sample data. The linear
regression method is used to estimate regression coefficients of the
polynomial function. The following function is the outcome of the linear
regression analysis:

Response surfaces.

Figure 13a shows the function at the CCD

The limit state function is written as

With the analysed monopile design, the annual probability of failure is

The sensitivity analysis of the stochastic input variables on the
reliability index is conducted by varying the coefficient of variation one
at a time for each input (0.5 and 2 coefficient of variation, CoV). The inclination of dashed lines in Fig. 14 marks the sensitivity of the stochastic variable. Mean force

Sensitivity plot.

During the lifetime of wind turbines, storms, typhoons or seismic action
are likely to cause permanent deformation of the structure owing to the
accumulation of plastic strain in the soil surrounding the foundation. The
serviceability limit state criteria require that the long-term structural
tilting does not exceed the operational tolerance prescribed by the wind
turbine manufacturer (usually less than 1

A discussion has to be started in the offshore community regarding the very
strict tilting requirement (i.e. 0.25

In this paper, a simplified model to calculate the permanent rotation (the SCD method) is implemented. It is noted that other models of varying complexity can also be used in the proposed probabilistic framework. If new inputs are introduced, the respective uncertainties should be considered in the reliability calculation and the function for the response surface should be adjusted accordingly.

The codes can be made available by contacting the corresponding author.

GZ, AM, JV and JDS designed the proposed methodology. GZ prepared the manuscript with the contributions from all co-authors.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Wind Energy Science Conference 2019”. It is a result of the Wind Energy Science Conference 2019, Cork, Ireland, 17–20 June 2019.

This research is part of the Innovation and Networking for Fatigue and Reliability Analysis of Structures – Training for Assessment of Risk (INFRASTAR) project. This project has received funding from the European Union's Horizon 2020 Research and Innovation programme under the Marie Skłodowska-Curie grant agreement no. 676139. The laboratory tests are provided by Chair of Soil Mechanics and Geotechnical Engineering of the Technical University of Berlin. The authors are grateful for the kind permission to use those test results.

This research has been supported by the Innovation and Networking for Fatigue and Reliability Analysis of Structures – Training for Assessment of Risk (INFRASTAR) (grant no. 676139).

This paper was edited by Athanasios Kolios and reviewed by Federico Pisano and one anonymous referee.