## Abstract

The loads induced as a result of seismic activities may jeopardize the serviceability of offshore wind turbines or may even lead the structure to reach the ultimate strength. The maximum load-carrying capacity of a support structure can be estimated by performing a structural assessment which accounts for the nonlinear effects arising from the material and geometry. The present work aims to analyze the fragility of a 5 MW monopile offshore wind turbine structure subjected to seismic activities accounting for soil interactions and time-variant structural degradation. The offshore wind turbine structure is subjected to different ground motions with different intensity. The nonlinear full transient dynamic structural analysis is carried out based on the finite element method, and the nonlinear monopile structural response during the different seismic activities is discussed. Finally, the fragility curves associated with the serviceability limit state design and the ultimate strength limit state are developed.

## Introduction

The majority of the fixed offshore wind turbines, currently operating in Europe, have been installed in the regions where no major earthquakes are expected. Therefore, the structural response and the safety of the offshore wind turbine (OWT) under seismic activities have not been a subject of many studies. It can be stated that performing the structural design based on a rare earthquake can lead to structural overdesign since such major earthquakes occur quite rarely. However, the earthquakes do occur, and the decision-makers need appropriate tools to know how to operate when such hazardous events occur as the serviceability can be at risk under a seismic event. Furthermore, there are several regions in the USA, Japan, and Southeast Asia that are quite resourceful in terms of wind energy and these regions are subjected to a significant seismic hazard.

The fragility analysis has been commonly used to evaluate the structural safety of the structures that are subjected to seismic loads. The fragility analysis varies due to the use of different numerical modelling techniques. The detailed numerical models are found to be computationally expensive and unpractical from the probabilistic analysis point of view.

The present study acknowledges the gap stated above and aims to perform a comprehensive assessment of a fixed offshore wind turbine subjected to seismic activities. It is targeted to have a numerical model, which evaluates the nonlinear structural response accurately for ageing OWTs.

The maximum load-carrying capacity of a support structure is estimated by performing a nonlinear finite element assessment. Moreover, the developed numerical model needs to account for the soil-structure interactions (SSI) and the time-variant structural degradation. In the present study, the fragility curves are derived based on the nonlinear dynamic finite element (FE) analysis. A sophisticated FE model is developed, taking into account non-uniform corrosion degradation, imperfections and SSI. The FE model of the monopile OWT structure is subjected to many different ground motions with different intensity.

Asareh et al. (2016) performed the fragility analysis of a 5-MW NREL wind turbine considering aero-elastic and seismic interaction using FEM. The study emphasized the importance of developing the fragility curves based on the nonlinear dynamic FE analysis. It is stated that the interaction between the wind and earthquake, and its impact on wind turbines are not yet well understood. Most of the research in this field considers multi-body dynamic linear models with limited degrees of freedom without considering the earthquake excitation.

Shinozuka et al. (2000) examined the fragility curves developed through two different analytical approaches. The first one uses the time-history analysis, and the latter one uses the capacity spectrum method, which is one of the simplified nonlinear static procedures. The results indicated that the agreement is reasonably good for the minor damage state, however not as good for the state of significant damage where nonlinear effects play a significant role, which emphasizes the importance of the nonlinear time-domain analysis.

Alam et al. (2017) attempted to tackle the issues arisen from the expensive computational effort given for the numerical simulation in developing. The possible uncertainty reduction of the median seismic fragility curve is investigated using the Bayesian Inference and Markov Chain Monte Carlo simulation.

Rota et al. (2010) adopted a methodology which was based on the Monte Carlo simulations. The Monte Carlo simulations generated input variables from the probability density functions of mechanical parameters. In particular, nonlinear pushover (static) analyses were used to define the probability distributions of each limit state while nonlinear dynamic analyses enabled to determine the probability density function of the displacement demand corresponding to different levels of the ground motion. However, the study still lacks the complexity of the full nonlinear dynamic analysis, by doing so; it is possible to carry out a Monte Carlo simulation without having to face the computational effort dilemma.

Martins et al. (2018) studied the variability of the collapse probability and the influence of this variability on the development of risk-targeted hazard maps. Each structure was modeled as a 3D FE model and tested against a set of ground motion records using a nonlinear dynamic analysis. Several fragility functions were derived for yielding and collapse states and combined with the seismic hazard. Moreover, Ioannou et al. (2015) assessed the impact of the ground-motion variability and uncertainty on empirical fragility curves as they observed considerable differences in the developed fragility curves for similar buildings.

Recently, Rossetto and Ioannou (2018) presented an overview of some significant observations made by the authors while working on earthquake and tsunami empirical fragility and vulnerability functions over the last ten years. Common biases in the post-event damage and loss datasets were presented, the consequences of ignoring the biases were discussed, and possible ways of dealing with them were suggested. The impact of statistical model fitting assumptions was also described and illustrated with examples drawn from the empirical earthquake and tsunami fragility and vulnerability studies. Throughout, areas for further research were highlighted, and it was observed that some sources of uncertainty remain mostly unexplored.

Despite the fragility analysis has been commonly adopted in the performance-based assessment of buildings under seismic loads, it is still possible to encounter the use the similar analysis for other structure types.

Golafshani et al. (2011) assessed the performance of a jacket offshore platform under extreme waves. Golafshani et al. (2011) based the assessment on a framework named probabilistic incremental wave analysis, which has considerable similarity with the incremental dynamic analysis and the applied framework considers the uncertainties associated with the sea state parameter, structural response, and collapse capacity.

Zeinoddini et al. (2018) also assessed the structural integrity of offshore platforms against extreme irregular waves where the stochastic nature and the uncertainties in the ocean waves were taken into account. Further, Ajamy et al. (2018) developed seismic fragility curves of an existing jacket type offshore platform subjected to possible ground motions in the Persian Gulf. The effects of both aleatory and epistemic uncertainty were considered using the incremental dynamic analysis method, and the seismic fragility curves related to an extensive damage state and a collapse state were defined in terms of cumulative distribution functions.

As far as the OWTs supported by monopile structures are concerned, Hallowell et al. (2018) analyzed the risk of failure of OWTs to hurricane-induced wind and waves in a study where a well-established framework in a performance-based earthquake engineering was adopted.

Kim et al. (2014) investigated seismic fragility of offshore wind turbine given emphasis on the application of the ground motion. To this end, an interface between ground soils and piles were modeled via nonlinear spring elements on whose boundaries the ground motions were applied as in the present study. Two methods were considered for applying ground motion. For the first one, different time histories from free field analysis were applied to each boundary in the first loading plan, and for the latter one, the same ground motion was applied to all boundaries. In conclusion, the paper suggested the application of ground motions layer by layer in the seismic design of OWT.

Mo et al. (2017) reported a study where an FE model of a 5 MW monopile OWT was generated accounting for the soil-pile interactions. The study aimed to develop fragility curves for different operational conditions, and it was concluded that wind turbines during the operational condition related to the rated wind speed had a higher level of probability of exceeding the damage limit states than those in other operational conditions including the idling condition.

De Risi et al. (2018) focused on the structural performance assessment of a typical offshore wind turbine subjected to strong ground motions. The ground motions varied, considering different earthquake type. It was pointed out that monopile OWTs are particularly vulnerable to extreme crustal and interface earthquakes, and the vulnerability increases when the structure is supported by soft soils. Moreover, refined structural modelling is generally necessary to avoid overestimation of the seismic capacity of OWTs. Further, Zheng et al. (2015) showed that the coupling of earthquake and wave loads might intensify a total structural response. However, the wind-induced loads were not mentioned in the analysis in the experimental study.

## Methodology

### Fragility analysis

The fragility analysis is carried out to assess the vulnerability of a structure for a given hazard. The outcome of the fragility is to be combined with the hazard identification and the loss estimation so that the risk can be calculated. Therefore, it can be stated that the development of the fragility curves is an essential part of the probabilistic risk assessment, code development and the expected loss estimation, which are part of the risk-based decision-making processes.

The purpose of the present study is to develop fragility curves associated with the identified hazard in terms of predefined limit states; therefore, much effort is given to the numerical modelling, advance structural assessment and development of the fragility curve. Several parameters are affecting the fragility of a structure such as the type of hazard, strength of hazard, structural type, load types acting on the structure, construction material, soil-structure interactions, etc. It is also intended here to address these parameters while creating numerical modelling and performing the simulations.

The seismic activity as a result of an earthquake is considered as the hazard type, and different intensity levels are considered for the structural assessments. There can be found various examples of an intensity measure for the severity of the hazard. The intensity measure could be given in terms of the magnitude of the earthquake, ground motion measures such as acceleration, velocity and displacement, or some other intensity measure that can be defined through regression analysis taking into account different characteristics of an earthquake. Moreover, the intensity measure can differ depending on the limit state as well as the loss estimation associated with the limit state.

These investigations go beyond the scope of the present study, where the main aim is to obtain the fragility curves based on the advance structural assessment. For this reason, the present study considers a simplification on the definition of the hazard intensity measure and accounts for the ground motion records related to the moderate and high-intensity earthquakes in the ultimate strength limit state. This assumption is based on an earthquake with a 475-year return period for which there is a 10% probability of exceedance in 50 years (British Standard 2005; Solomos et al. 2008). Here the intensity measurement is described in terms of a peak ground displacement, and for each intensity different ground motions are selected from the PEER database (Pacific Earthquake Engineering Center, 2018@@). The ground motions with diverse characteristics such as the magnitude, *R*_{rup}, Arial intensity, etc. are chosen.

The probability of failure, or fragility, can be estimated by comparing the demand to the capacity. One common approach is executing separate analyses for both demand and capacity, and then comparing the results. By doing so, it is aimed to avoid tedious computational effort. A widely used example of this approach can be the incremental dynamic analysis, where the ground motion is scaled until the structure experiences a deformation exceeding the ultimate limit. The ultimate strength of the structure is estimated by performing a pushover analysis, which reveals the maximum displacement that the structure can withstand. Regardless of the dynamic analysis being linear or nonlinear, there is a non-negligible advantage in terms of computational time; however, this approach overlooks so many aspects regarding the nonlinear structural response, the multi-axial loading, and local failure phenomena and so on and the results can be misleading.

The present study intends to follow what has been a common practice for the fragility analysis; however, it also aims to make amendments to address the limitations of the common practice by introducing a more advanced analysis. For the probabilistic aspect of the fragility analysis, apart from the ground motions with different dynamic characteristics, a probabilistic wind-induced loading and non-uniform corrosion depth are incorporated into the analysis. The methodology introduced to perform the fragility analysis is shown in Fig. 1.

The present study aims to address the overlooked areas developing an advanced numerical model based on FEM. The developed FE model is capable of incorporating inelastic behavior of the material in a large displacement analysis where imperfections, non-uniform corrosion and soil-structure interactions are present as well.

Furthermore, the dynamic characteristic of the ground motion and the wind-induced loading are critical because the amplitude of the structural vibration can be amplified when the excitation frequency gets close to the natural structural frequency. The plausible resonance may induce unexpected local deformation. Thus, for more realistic results, it is necessary to perform the nonlinear dynamic structural analysis and ultimate strength assessment in one single numerical simulation. The details of the FE model and the nonlinear full-dynamic analysis are provided in “Finite element analysis” section.

### Support structure and wind turbine data

A monopile offshore wind turbine consists of a wind turbine (blades, hub, rotor, and nacelle), a tower (T), and a transition piece (TP), and a foundation (F). The monopile foundation is connected to the transition piece, which is bolted to the tower structure holding the wind turbine. Figure 2 shows a typical monopile offshore wind turbine.

The foundation transfers the vertical and horizontal loads acting on the OWT to the subsoil and constrains the excessive motion of the OWT using the soil pressure acting on the foundation. Furthermore, the present study adopts the monopile OWT optimized by Yeter et al. (2017), whose dimensions are presented in Table 1. The specification of the wind turbine is given in Table 2.

The load assumption is based on the studies in which the maximum load was estimated for the parked wind turbine condition during a storm. Thus, the wind-induced load is considered to be a random Gaussian process. Consequently, the wind-induced load is described by a Normal distribution with a mean value of 1E+5 N and a standard deviation of 0.2E+5 N.

### Site data (soil profile)

In the present study, an FE model is developed to carry out the time-variant nonlinear analysis accounting for the soil-structure interactions (SSI). The Winkler spring model is employed, which uses the subgrade reaction forces associated with the soil to define the stiffness of the implemented spring based on the *p-y* method. The subgrade reaction represents the overall soil characteristics such as the soil density, soil porosity, shear modulus, and Young’s modulus of the soil.

The nodes are connected to the horizontal Winkler spring elements, forming a circular layer. The stiffness coefficients of the springs are calculated for each layer depending on the pile penetration depth, the soil profile and the unit length of the pile. The unit length of the pile depends on the pile length. The number of layers with a series of the Winkler springs is assumed 25. Figure 3 shows the model of the soil-structure interactions on which the boundary conditions are based in the nonlinear FE analysis.

The soil stiffness increases as moving towards more profound in the soil. The present study employs an approach regarding the change in the soil profile with the depth since the soil of a possible offshore wind farm site does not necessarily need to be linear it can be of a parabolic shape. In this study, the standard logistic sigmoid function is employed to address this issue. Figure 4 illustrates the soil characteristics varying through the depth.

The function is described as a bounded differentiable real function that is defined for all real input values and has a positive derivative at each point (Han and Moraga 1995). The sigmoid function is expressed as:

where *x* is the distance from the seabed surface. *g*(*x*) is the function of the variation of the soil characteristics along with the depth, and it varies from 0 to 1. *W* is the shape factor and *B* is the scale factor. The shape factor *W* is associated with the steepness of the curve denoting how the soil stiffness changes with the distance from the seabed surface. The scale factor *B* is associated with the soil stiffness at the seabed surface.

This means that to achieve solid support, the pile has to be driven until it reaches a soil layer of very dense sand or rock. The shape factor *W* is designed in such a way that the coefficient of subgrade can be estimated around 8 MN/ m^{3}, which can be considered as impenetrable soil. The monopile foundation must be driven into the subsoil as much as it is needed to guaranty the support to the offshore wind turbine as required by the design criteria.

If the behavior of each spring is assumed linear, and the soil reaction corresponding to the lateral displacement of the pile can be formulated as (Reese 1983):

where *p* is the soil resistance regarding force per unit length, *y* is the lateral pile deflection, and *E*_{py} is termed as a reaction modulus and represents the slope of the *p – y* curve. As far as a linear soil profile is concerned, the reaction modulus *E*_{py} may be expressed as:

whereas for the nonlinear soil profiles defined based on the sigmoid function can be expressed as (Yeter et al. 2019):

where *z*, *W* and *B* are the distance below the soil surface, the shape factor of the sigmoid function and the scale factor of the sigmoid function, respectively. *K*_{hs} is the coefficient of subgrade reaction.

### Seismic data

The elastic strain energy is accumulated and suddenly releases a rupture as the groundmasses move together. As the distorted plates crack for the energy equilibrium, an earthquake ground motion is created. When the earthquake occurs due to the sudden fault slip, seismic waves travel from the focal point to the observed site. The slip of the plates can take place horizontally, vertically or both. As a result, two types of seismic waves are generated, which are known as body and surface waves.

Body waves (primary waves) are longitudinal (primer) and transversal (secondary) waves. The primary waves travel very fast; however, they have little damage potential, whereas the secondary waves lead to horizontal and vertical motions, and both can cause substantial damage. At the earth, body waves manifest as surface waves such as the Love waves and Rayleigh waves. While the body waves equally represented in the seabed at all depths, the surface waves most likely occur in shallow earthquakes. The ground motions are the combination of the mentioned waves occurred; as a result, the energy release, and they can be measured regarding the displacement, velocity and acceleration (Elnashai and Di Sarno 2008) as can be seen in Table 3.

## Finite element analysis

### Finite element model

The element type SHELL181 is used to model the tubular monopile structure. The element has four nodes with six degrees of freedom at each node: translations in the X, Y, and Z-axes, and rotations about the X, Y, and Z-axes. The element type includes the stress stiffness terms by default and supports the nonlinear material models, which makes the element type well-suited for nonlinear, large rotation, and large strain nonlinear applications. Furthermore, the material model used in the nonlinear FE analysis is a bilinear elastic-perfectly plastic stress-strain relationship where the yield stress is of 355 MPa.

The monopile offshore wind turbine is analyzed based on the FEM employing the commercial software ANSYS (2009). The FE model of the monopile OWT involves the tower, transition piece and pile. A very thick plate that is as heavy as the blades, hub and nacelle are modeled at the top of the tower structure to account for the overall weight of the wind turbine.

A structure is subjected to buckling as a result of the imbalanced loading or displacement. Introducing an imperfection to the geometry of a structure is a common practice, and it results in out of balance load or displacement so that more realistic results can be achieved. Therefore, the present study introduces the geometrical imperfection to create a more accurate FE model by modifying the vertical and horizontal position of the nodes, as can be seen in Fig. 5.

The imperfections are applied based on a function generating superimposed periodic buckling wave shapes throughout the structure. The modified nodal location is defined based on a function superimposing sine and cosine functions onto a 2-D surface. The number of half wave around the circumference of the monopile and the amplitude of the half wave are defined under the consideration of the plate thickness and the length of the support structure.

### Load and boundary conditions

As far as the load application is concerned, the present study a master node is created at the center of the tower top, and then the master node is coupled with the nodes located around the circumference. By doing so, it is aimed to apply the wind-induced loadings at tower top as a concentrated load. The nodes located at the circumferences have the same translation and rotation at all dimensions. By using the given modeling technique, possible stress concentrations and local failures are avoided. Figure 6 shows the load and boundary conditions applied in the nonlinear FE analysis.

The nodes are connected to the horizontal Winkler spring elements, forming a circular layer. The stiffness coefficients of the springs are calculated for each layer depending on the pile penetration depth, the soil profile and the unit length of the pile. The unit length of the pile is also a mesh density depends on the pile length. There are 25 layers created to apply the Winkler springs, as can be seen in Fig. 7.

The one side of the spring elements are connected to the pile structure, and the other side is where the seismic loads are applied as displacements. Also, the nodes which are subjected to the seismic load are constrained in terms of rotations in all dimensions.

The intensity of the seismic loads varies depending on the characteristics of the soil characteristic through which the seismic waves travel. Thus, this is modeled in the present study by scaling the displacements given in the ground motion record according to the different depth levels. The scaling is done by the soil profile defined by the sigmoid function, which is also used to define the stiffness of the Winkler springs (see Fig. 4).

The liquefaction phenomenon involving soil deformation caused by monotonic or transient disturbance of saturated cohesionless soils is not taken into account in the soil-structure interactions. The reason behind this assumption is to isolate as much as possible the joint effect of wind and seismic loads on the dynamic structural response rather than the numerical difficulties.

### Nonlinear transient dynamic analysis

It is common to carry out quasi-static analysis when the frequency of the excitation forces is very low comparing to the natural frequency. As well, when the frequency of the excitation force is very high, then the initial effect cannot be significant as the structure, or mass, cannot respond to the force applied. However, when the ratio of the external load frequency to the natural frequency approximates around 1, and then the significance of performing a transient analysis increases significantly because it is essential to account for the inertial and damping effect. Moreover, the excitation loads that have a vague dynamic behavior such as earthquakes are needed in the analyzed, especially when it is applied simultaneously with an environmental load such as wind-induced loading.

In the present study, the nonlinear time domain analysis is carried out by using one of the methods offered by ANSYS (2009) called the full transient method. The full transient method does not involve matrix reduction or mass matrix approximation, to calculate the transient response; it uses the full system matrices and allows the nonlinearities associated with both material and geometry. However, due to all displacements and stresses are calculated, the full transient analysis can be considered computationally expensive. Some parameters can be adjusted to ease the cumbersome of the computation without having to sacrifice from the accuracy of the analysis´ results such as finite element size and the time step size. In this regard, appropriate finite element size and time step size are found to be 0.167 m (*L*_{pile}/150) and 0.2, respectively, after performing a sensitivity analysis. As for the load step, the ramp-type is chosen as it is more convenient for the transient processes, and the damping parameters are defined such way that the structural damping 4%.

The OWT support structures are not only subjected to tensile, but also compressive stresses, which may result in local or global instability of the structure, leading the structure to reach the ultimate strength. In the scope of the present study, the ultimate strength of the monopile OWT structure is numerically assessed by performing a nonlinear FE analysis. The large deformation option is activated to solve the geometric and material nonlinearities. In the nonlinear structural response, it is possible to observe the phases that the structure goes through under progressive load such as proportional limit, buckling, ultimate strength and post-collapse.

The damage mechanism such as corrosion degradation may affect the ultimate strength of the structure considerably; therefore, it is necessary to perform ultimate strength assessment for ageing structures (Garbatov et al. 2007; Saad-Eldeen et al. 2011; Silva et al. 2013). Three fundamental approaches are typically applied for corrosion deterioration modelling. The conventional approach is considered that corrosion grows linearly, which may lead to a considerable overestimation of the corrosion deterioration or underestimation of the corrosion effects in early life.

Garbatov and Guedes Soares (2008) developed the corrosion model that is based on a non-linear time-dependent function of general corrosion wastage. The model is based on the solution of a differential equation for the corrosion wastage, which leads to the mean value of the corrosion depth as:

The parameters of the corrosion depth as a function of time are determined under the assumption that it is approximated by an exponential function. The long-term corrosion wastage for monopile structure near the splash zone as *d*_{∞} = 0.004 m. The time without corrosion *τ*_{c} is assumed to be 5 years, and the transition period *τ*_{l} is assumed to be 12 years.

### Seismic structural analysis

The wind-induced load as concentrated load and the seismic load as displacement are applied to the monopile offshore wind turbine simultaneously. Although the ground motions are applied in three dimensions, the wind-induced loads are applied in the dimension of the most dominant ground motion. The nonlinear structural assessment is performed in the time-domain for which the applied loads are coupled. In the following, the results of the nonlinear structural analysis are presented and discussed.

Figure 8 shows the deformation shape obtained for the load case E4 (see Table 3) with “peak ground displacement” PGD = 0.125 m (upper figure) and PGD = 0.175 m (lower figure). The structure that is subjected to the load case E4 with PGD = 0.125 m does not fail; however, the maximum displacement is estimated to be 1.06 m. The structure that is subjected to the E4 with PGD = 0.175 m does collapse; however, the maximum displacement is calculated to be 0.508 m. In both load cases, the structure that experiences local failure situated in the mid-tower section. Further, the deformation shape occurs such a way that resembles the mode shape of the second natural frequency of a monopile structure. This indicates that the excitation load triggers a resonance, which results in a magnification of the structural displacement. The high level of rotation occurs at the mid-tower section, and structure fails under compressive stress as a result of buckling.

The results presented in Fig. 8 point out that the maximum displacement cannot be the only indicator for the definition of the failure. The failure most likely to occurs due to a local stress concentration, which causes local plastic deformation and buckling, especially, for the slightly thinner structural components. After the local buckling occurs, if the load continues to be high, eventually, the structure loses its load-carrying capacity.

Other forms of structural failure are observed in the analyses performed for different load cases. Figure 9, up shows the displacement shape at the failure for the load case E5 with the maximum displacement of 1.11 m and Fig. 9, down shows the displacement shape for the load case E6 with the maximum displacement of 1.34 m. Although the peak ground displacement (PGD) for both earthquakes is the same as 0.175 m, the structure subjected to E5 fails, and the structure subjected to E6 does not. Moreover, the magnitude of the earthquake E5 is lower than the one of the E6, which rules out any conclusion that can be driven from the magnitude of the earthquake. The effect of the earthquake varies with the distance to the rupture, soil characteristics, and the dynamic characteristics of the ground motion.

Failure may occur due to the stress concentration emerging at the bottom of the pile structure where the maximum bending moment is expected. Also, the stress concentration is observed in both tensile and compressive side, which might indicate the failure is rather yielding than buckling as usually observed for the thick-walled structures. Also, it can be noted that torsion and shear are other factors that are contributing to the maximum stress at the bottom of the pile, which happens due to the three-dimensional load application of the earthquake.

The structural responses shown in Fig. 9 have similarities in terms of the displacement shape. However, only the load case (E5) fails. This is merely due to the random wind-induce load applied at the tower top coupled with the ground motions. The randomness in the load is the reason where in some cases the local failure resulting in the global failure, whereas in other cases the failure remains in the local domain.

Figure 10 shows the displacement shapes associated with the load cases E5 and E9 with PGD = 0.175 m. The applied wind-induced load on the top is far less than the maximum load carrying capacity. However, the displacement in the direction of the wind-induced load is amplified once the wind-induced loads are coupled with the earthquake ground motions. The displacement shapes given in Fig. 10 are quite similar to the first mode shape of vibration of a monopile. In the load case E9, the monopile fails from the monopile at the seabed level due to buckling.

Lastly, Fig. 11 shows other possible displacement shapes that might emerge at the collapsed state. Both displacement shapes are taken from the analysis where the structure is subjected to E10. Figure 11 shows several highly stressed regions, where the structure might first experience local failure due to buckling, and the global failure follows. These regions are the bottom of the pile, monopile section at the seabed and the mid-section of the tower structure.

It is worth mentioning that the non-uniform corrosion degradation introduced to the monopile does not cause any direct effect on the structural failure such as a local failure. Nevertheless, corrosion degradation does affect the overall stiffness of the monopile, which is important to be taken into account for ageing structures. Further, the conclusion given for the corrosion degradation concerning its effect on the structural failure is also a valid argument for the imperfection applied on the monopile.

## Fragility assessment

The fragility curve is a mathematical expression which denotes the relationship between the frequency of the structural failure and the seismic load that the structure is exposed for a given limit state. The seismic load can be peak ground acceleration, velocity or displacement (Kennedy et al. 1980). Zentner et al. (2017) recently reviewed the existing approaches regarding the fragility analysis and their applications, where four methods, in particular, are discussed. The first one is the safety factor method, in which the fragility curve is estimated based on safety margins with respect on an existing deterministic design; the second one is the numerical simulation method, in which the parameters of the fragility curve are obtained. The third one is the regression analysis or maximum likelihood estimation from a set of nonlinear time history analysis at different seismic levels, and the last method is the incremental dynamic analysis method where a set of accelerograms is scaled until failure or capacity spectrum method.

In the present study, the empirical data obtained through the nonlinear full-transient analysis are later on fitted to a mathematical expression through regression analysis. The relationship between the peak ground displacement and the corresponding probability of failure are defined based on the Weibull cumulative distribution function as follows:

where E[P_{f}(*x*)] denotes the expected probability of failure for the earthquake motion with a peak ground displacement *x*. *k* and *λ* are the shape and scale of the fragility curve, respectively.

The empirical data are obtained through numerical simulations for serviceability limit state design and the ultimate strength limit state, as presented in Table 4.

The serviceability limit state design is defined as the failure states during the normal operation, where the structure is subjected to deterioration of the common functionality, including corrosion degradation and deformations that are not acceptable, and this state does not lead to collapse. The established practice defines the serviceability limit state criterion with a major concern about the efficiency and cost performance without leading to major repair and downtime.

The ultimate limit state design reflects the collapse of the structure as a result of the reduction of the stiffness and strength, which leads to repair of the entire structure.

The fragility curves are fitted to the empirical data based on the least square method. Table 5 represents the statistical descriptors of the developed fragility curves. Figures 12 and 13 represent the fragility curves associated with the serviceability limit state design and the ultimate limit state.

For the peak ground displacement higher than 0.2 m, the probability of the structure exceeding the defined limit states is quite the same, whereas for the peak ground displacement of 0.06 m and 0.18 m the fragility curve differs substantially. These results imply that the structure has not collapsed after an earthquake, it is quite probable that the structure experienced a plastic deformation. In the case of plastic deformation occurs as a result of cyclic load, a defect may grow up into a crack as a result of the low-cycle fatigue, which emphasizes a regular inspection. Also, the fragility curves are presented in Figs. 12 and 13 with the upper and lower bounds of a 95% confidence interval, which are later on used to estimate the reliability index as a function of the peak ground displacement.

The results presented in Figs. 12 and 13 show how reliable is the monopile OWT at the end of its service life when it is subjected to an earthquake with a specific peak ground displacement. The results can be extended to a recommendation for design or inspection purposes. For this reason, Figs. 14 and 15 show the Beta reliability indices for the serviceability limit state design and the ultimate strength limit state.

The reliability indices are compared to the target reliability assumed here as *β* = 3.71 respecting a normal safety class where the structures with reserve capacity may suffer ductile failures causing fatalities, pollution and significant economic consequences.

From the ultimate strength point of view, the peak ground displacement should not exceed 0.024 m based on the considered target reliability index. Also, the upper and lower bounds of the peak ground displacement that might affect the serviceability are estimated 0.011 m and 0.25 m. The lower and upper bound associated with the ultimate strength limit state is wider compared to the one associated with the serviceability limit state design, which can be explained by the different uncertainties involved in the fragility analysis.

It is also important to point out that fragility analysis is to be merged with the hazard analysis and the loss analysis to serve as a complete risk-based decision-making tool.

## Conclusion

In the present work, a fragility analysis was performed for an ageing monopile OWT structure subjected to seismic activity. The nonlinear full-transient structural analysis was performed based on the nonlinear FE analysis to evaluate structural capacity explicitly. In the FE model, the ground motions during the earthquake and wind-induced loadings were coupled, and the soil-structure interactions were accounted for. The time-variant corrosion degradation and imperfection were also included in the FE model of the ageing monopile structure. A number of different ground motions with different dynamic characteristics were chosen, and the analytical fragility curves for the serviceability limit state design and the ultimate strength limit state were derived.

The nonlinear time-domain analyses indicated that the structural failure might occur in different forms depending upon the dynamic characteristics of the applied ground motion. The displacement shapes at failure showed similarities to the mode shapes of the first and second natural mode frequencies. Moreover, it was demonstrated that the displacement estimated at the collapsed might be far less than what the pushover analysis indicates, which raises questions on the accuracy of the seismic structural assessments performed employing quasi-static (pushover) analysis. Also, there are found to be several critical regions, at which the local stress concentration occurred, leading to local buckling and then the global failure.

Finally, the data generated by the nonlinear full-transient analysis were fitted to fragility curves. Subsequently, the fragility curves were translated to the reliability indices. The peak ground displacements that should not be exceeded were identified based on the predefined target reliability to provide guidance offshore wind turbine operators about the seismic structural safety.

Although the developed curves can provide a risk-based decision-making tool for the OWT structures subjected to earthquake hazard, the number of the full transient analysis must be augmented to have more confidence in the assessment.

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## Acknowledgements

This work was performed within the Strategic Research Plan of the Centre for Marine Technology and Ocean Engineering (CENTEC), which is financed by Portuguese Foundation for Science and Technology (Fundação para a Ciência e Tecnologia - FCT) under contract UID/Multi/00134/2013-LISBOA-01-0145-FEDER-007629.

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Yeter, B., Tekgoz, M., Garbatov, Y. *et al.* Fragility analysis of an ageing monopile offshore wind turbine subjected to simultaneous wind and seismic load.
*Saf. Extreme Environ.* (2020). https://doi.org/10.1007/s42797-020-00015-9

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### Keywords

- Fragility curve: Nonlinear dynamic analysis: FEM: Offshore wind turbine: Seismic load: Soil-pile interactions