Long-term behaviour of offshore wind turbine monopiles in sand

A_n

loop area of the nth cycle

A₁

loop area of the first cycle

C

cyclic tests

C_u

uniformity coefficient

D

pile external diameter

D_r

relative density

d

pile internal diameter

d₁₀, d₅₀, d₆₀

particles size diameter

E_pI_p

pile bending stiffness

$Es$

soil modulus at level of pile tip

e_max

maximum void ratio

e_min

minimum void ratio

e_x

load eccentricity

f_f

cyclic frequency force

f_n

first natural frequency in the nth cycle

f_str

fixed-base natural frequency

f₀

first natural frequency at the initial condition (after installation)

f_0,1g

first natural frequency at the initial condition at 1g

f_0,Ng

first natural frequency at the initial condition at Ng

G_s

specific gravity of soil particles

G₀

small-strain shear modulus

g

gravitational acceleration

H

lateral load

H_max

maximum cyclic lateral load

H_min

minimum cyclic lateral load

H_r

lateral reference force at 0·1D displacement

K_n

secant stiffness in the nth cycle

K₀

initial secant stiffness

K₁

secant stiffness in the first cycle

k_r

pile relative stiffness

L

pile embedment length

L_L

pile free length

l_b

distance between the lower part of the pile and the bottom of the cylindrical box

M

monotonic tests

m₁

lumped mass on the top of the tower

m₂

monopile and tower mass

N

scale factor of centrifuge modelling

n

number of cycles

S

distance between the pile and the cylindrical box wall

t

pile wall thickness

V

gravity force acting on the wind turbine

Y

lateral pile displacement measurement at ground level

Y_H

lateral pile displacement measurement at load level

Y_He and Y_e

recoverable component of the lateral pile displacement

Y_Hp and Y_p

irrecoverable ratcheting component of the lateral pile displacement

Y_n

lateral pile displacement at ground level in the nth cycle

Y₀

initial lateral pile displacement at ground level

Y₁

lateral pile displacement at ground level in the first cycle

$α¯$

fitting function/coefficient of normalised secant stiffness

β

soil–structure interaction parameter

γ

unit weight

γ_max

maximum unit weight

γ_min

minimum unit weight

δY_n

rate of increase of displacement

ζ_b

parameter characterising cyclic load magnitude

ζ_c

parameter characterising cyclic load amplitude

ν

Poisson’s ratio

$σ′v$

soil vertical effective stress

ϕ

angle of friction

The use of wind energy has increased significantly in recent years. In this context, offshore wind farms are gaining prominence due to their advantages over onshore wind farms, including access to more consistent wind speeds and fewer restrictions on site location, with recent growth particularly notable in Europe, Asia and North America (REN21, 2023).

Offshore wind turbines (OWTs) are slender structures subjected to cyclic lateral environmental loads (wind, waves and currents) and cyclic operational loads (commonly referred to as 1P and 3P), which vary in magnitude and frequency over their 20–25 year service life (approximately 10⁷ cycles). These loads may induce permanent displacements and changes in soil–foundation stiffness, which can significantly affect the dynamic response and long-term performance of the structure.

From a geotechnical perspective, the selection of foundation type for OWTs depends primarily on soil conditions, turbine dimensions, environmental loads and water depth. Among available solutions, monopile foundations have become the most widespread worldwide and are expected to maintain their predominance due to their proven technical and economic efficiency (NREL, 2022). These consist of large-diameter tubular steel piles, currently ranging from 3 to 10 m in diameter, typically used in waters up to 35 m deep, with an embedded length-to-diameter (L/D) ratio between 3 and 6. Despite their conceptual simplicity, accurately modelling the soil–structure interaction (SSI) behaviour of monopile-supported turbines remains a major challenge (LeBlanc et al., 2010), particularly under long-term cyclic loading.

Two main types of lateral SSI mechanisms are generally recognised: cyclic interaction and dynamic (or vibratory) interaction (Bhattacharya et al., 2021). Understanding these mechanisms is essential for predicting the long-term behaviour of foundations and ensuring structural reliability. Cyclic loading can cause progressive tilting of the foundation and modification of soil stiffness, which in turn alters the system’s first natural frequency (Cuéllar et al., 2012). This issue is particularly relevant for soft–stiff designs, where such changes may induce resonance and potential failure under serviceability limit state conditions.

Experimental and physical modelling studies indicate that the long-term cyclic lateral response of piles in sand is governed by coupled near-field grain-scale mechanisms. Conventional model tests by Cuéllar (2011) and Cuéllar et al. (2012) showed that cyclic loading around piles induces two dominant deformation phases: an initial stage dominated by grain rearrangement and densification, followed by a quasi-static ratcheting process organised in convective cells bounded by a direct-shear surface separating mobilised and static soil zones. At the pile–soil interface, cyclic reversals promote small gap openings that allow downward grain migration during unloading and lateral forcing during reloading, leading to incremental permanent deformation. At the macro scale, this behaviour manifests itself as simultaneous accumulation of permanent displacement, progressive increase in secant stiffness and reduction of hysteresis loop area, as reported in cyclic pile tests (Abadie et al., 2019). These coupled mechanisms therefore provide a physical framework to interpret the long-term evolution of stiffness, damping and natural frequency under cyclic loading.

Ensuring that the first natural frequency (f₀) remains outside the excitation load frequency range is a key requirement in OWT design. The DNV (2021) guidelines recommend that f₀ should be at least 5% away from both the 1P (rotor effects) and 3P (blade-passing effects) operational frequencies, a constraint that becomes increasingly narrow as turbine capacity increases (Arany et al., 2016). Consequently, accurately estimating the natural frequency and its long-term evolution under operational loads is a critical aspect of design. Because f₀ depends not only on the structural properties but also on the stiffness of the foundation and surrounding soil, it is essential to investigate how long-term cyclic loading influences the overall system response.

The geotechnical uncertainties associated with the design of monopile foundations, particularly in sandy soils, have motivated several research programmes, primarily involving conventional (1g) and centrifuge (Ng) physical modelling tests. A brief review of the literature on lateral displacement behaviour has been presented by Frick & Achmus (2022) and Lemos (2024). These studies have typically either included only a limited number of cycles in centrifuge tests (e.g. Klinkvort, 2012; Truong et al., 2019; Richards et al., 2021; Li et al., 2022; Almeida et al., 2024) or adopted 1g conditions without adequately reproducing the soil stresses (e.g. LeBlanc et al., 2010; Cuéllar, 2011, Cuéllar et al., 2012; Nikitas et al., 2016; Abadie et al., 2019; Richards et al., 2020; Xu et al., 2020; Liang et al., 2021; Abdullahi et al., 2022; Wang et al., 2022).

Moreover, most researchers have investigated only the soil–foundation interaction without accounting for the influence of the upper part of the structure, which is particularly significant for the dynamic response of OWTs. Futai et al. (2018, 2021) are the only researchers known to have examined the entire wind turbine in a series of centrifuge tests in sand; however, the effects of cyclic loading on the natural frequency were not investigated.

Other contributions, such as the pile soil analysis (PISA) project, have introduced new design methodologies for monopiles through large-scale field testing campaigns, focusing primarily on the monotonic behaviour of monopiles (Byrne, 2020), with limited cyclic loading tests (Byrne et al., 2020), which involved fewer tests and a smaller number of cycles.

To address the difficulty of predicting OWT behaviour, researchers have relied on field monitoring data. However, as Abdullahi et al. (2022) note, challenges such as limited accessibility and repeatability restrict this approach. As an alternative, scaled model tests have been adopted. Richards et al. (2021) emphasised the importance of reproducing full-scale stress conditions, which are often neglected in 1g physical modelling of cyclic effects on f₀ (e.g. Nikitas et al., 2016; Xu et al., 2020; Liang et al., 2021; Abdullahi et al., 2022).

In this paper, a series of centrifuge tests is presented (Lemos, 2024), designed to investigate variations in horizontal displacement, secant stiffness, hysteresis and natural frequency of a monopile foundation in sand under long-term cyclic loading, a study that has not previously been carried out for up to 10⁵ cycles under comparable prototype stress–strain conditions. Unlike most studies, which focus solely on the foundation and disregard the upper structure, this research adopts an integrated modelling approach that represents the OWT as a complete system under macrogravity conditions. This methodology enables the stress–strain field to be simulated realistically and allows the natural frequency evolution to be evaluated accurately under cyclic loading.

To this end, a novel experimental set-up was developed to measure the natural frequency at both the beginning and end of each cyclic test, thereby enabling analysis of its evolution – an aspect not previously addressed in physical modelling studies. Furthermore, a new approach is proposed for estimating the natural frequency of OWTs, based on a series of free-vibration tests with increasing confining stress around the monopile, offering a new insight into the influence of soil confinement on the system’s dynamic response.

The granular soil used in the centrifuge tests was a dry, uniform quartz sand, which has been previously employed in many studies (e.g. Almeida et al., 2024). The primary reason for using dry sand was to ensure a drained behaviour during the lateral loading tests (Li et al., 2010; Klinkvort et al., 2013), which is essential for correct interpretation of the results. However, this choice implies a higher effective stress field than that in a saturated prototype, a factor that must be accounted for when considering this condition.

Table 1 summarises the main parameters of this soil. The effects of grain size were minimised by using D/d₅₀ = 107·5, which is greater than 88, the minimum value proposed by Klinkvort et al. (2013).

The sand layers were prepared using an automatic sand-pouring device capable of maintaining a constant sand flow rate as well as a constant fall height, to ensure a uniform relative density D_r = 80% (ASTM 4254; ASTM, 2016) in the present research, corresponding to a unit weight γ = 15·85 kN/m³. Cone penetration tests confirmed a uniform relative density of 80 ± 1% throughout the samples.

The small-strain shear modulus (G₀) of the soil is a key parameter in SSI analyses. It is well established that G₀ varies with effective stress, increasing with depth due to the corresponding rise in confining pressure. The G₀ profile was evaluated on sand using air hammer tests (Ghosh & Madabhushi, 2002) performed in the COPPE/UFRJ geotechnical centrifuge at varying acceleration levels, under the same relative density as the monopile tests (D_r = 80%). Fig. 1 presents a comparison between the air hammer results and additional laboratory data obtained from resonant column and bender element tests (Almeida et al., 2024). These experimental results are compared with predictions from three widely used empirical models: Hardin & Drnevich (1972), Oztoprak & Bolton (2013) and Seed & Idriss (1970). The predictions were computed directly from the equations proposed in the respective reference papers. Among them, the model by Seed & Idriss (1970) shows the closest agreement with the air hammer data across the tested stress levels.

The OWT centrifuge model adopted in the present study was based on the typical dimensions of a 3·5 MW OWT, as seen in Fig. 2(a) (Byrne & Houlsby, 2015). However, an intermediate virtual model, scaled 1:2 from the 3·5 MW prototype (Fig. 2(b)) was necessary because the full-scale dimensions of the wind turbine were too large to be modelled in the COPPE/UFRJ mini-beam centrifuge. This approach is not unprecedented, as other authors have adopted similar strategies in the literature (Tobita & Iai, 2016; Seong et al., 2017; Futai et al., 2018, 2021; Freitas et al., 2024).

The 1:2 virtual model was modelled on a 1:100 scale using an aluminium tube to represent the foundation and the tower (m₂ = 50·5 g), with a mass concentrated at the top (m₁ = 181·8 g) to represent the turbine elements: generator, blades, nacelle and rotor (Fig. 2(c)). The total mass (m₁ + m₂ = 232·3 g) was approximately 1/N³ of the assumed 1:2 virtual model mass (300 g = 300 t × 1/100³) and the corresponding 3·5 MW prototype wind turbine mass (V = 6 MN).

Regarding the main OWT parameters presented in Table 2, the ratio of embedment length to monopile diameter (L/D) was kept constant, as shown in Fig. 2(b). The model bending stiffness E_pI_p was assumed to be 1/N⁴ of the 1:2 virtual model value (Madabhushi, 2014). The model geometry was adjusted accordingly to compensate for the aluminium’s lower stiffness and preserve flexural stiffness similarity between the 1:2 virtual model and the centrifuge model. The pile relative stiffness values (k_r) were calculated using equation (1), as proposed by Poulos (1982). The E_s was obtained using equation (2), assuming Poisson’s ratio ν = 0·3 (drained conditions). The G₀ profile was obtained using Seed & Idriss’s (1970) empirical approach, consistent with the air hammer tests (Fig. 1), yielding G₀ = 127 MPa at 10 cm depth in the model scale at 100g, G₀ = 127 MPa at 10 m depth in the 1:2 virtual model scale, and G₀ = 180 MPa at 20 m depth in the prototype scale for the 3·5 MW wind turbine.

The eccentricity between the load application point (top of the monopile) and the displacement measuring point (12 mm above ground level) does not strictly represent the external work. However, these locations reflect a condition that closely replicates the combined loading mechanisms acting on these structures, capturing the dynamic behaviour of OWT monopiles realistically. In addition, the measuring point just above the soil is of high interest, since the soil–monopile interaction is strongly associated with these displacements.

Although the pile relative stiffness of the centrifuge model and the 1:2 virtual model were around 10 times higher than that of the real 3·5 MW wind turbine, both were within the intermediate range of behaviour between flexible and rigid, along with several other wind turbines (Abadie et al., 2019).

The wind turbine centrifuge model was statically jacked at 1g with a constant rate of 0·1 mm/s until an embedment of 100 mm was reached. In-flight installation was not considered due to the complexities of developing a suitable device for this purpose in a mini-beam centrifuge.

It is acknowledged that in-flight (Ng) installed jacked piles might exhibit greater initial stiffness than those jacked at 1g. This is mainly due to the higher lateral stresses induced during installation under augmented acceleration (Dyson & Randolph, 2001; Hajialilue-Bonab et al., 2007; Klinkvort et al., 2013). El Haffar (2018) further showed that the reduction in lateral displacement due to in-flight installation is more pronounced in the initial phase but tends to diminish after several hundred cycles, with soil stiffness converging to values similar to those of 1g installed piles after 1000 cycles. Fan et al. (2021) also found that in loose sand, in-flight installed jacked piles have approximately a 10% increase in secant stiffness and load capacity compared to 1g installed ones.

Regarding the influence of the installation process on the surrounding soil, hollow piles with most of their buried inner part filled with soil after jacking show less soil disturbance compared to closed-end or plugged piles, where more soil is displaced during installation (Kirkwood, 2015). Measurements taken directly after static jacking under 1g conditions indicate that approximately 94% of the buried pile length was filled with sand, demonstrating reduced interference from the installation method.

The tests were carried out in the COPPE/UFRJ mini-beam geotechnical centrifuge (Almeida et al., 2014), which has a diameter of 1·6 m and a rectangular swing basket with internal dimensions of 0·10 m × 0·30 m × 0·18 m with a load capacity of 9 g-tonne. Adjustments were required to accommodate the installation of the test set-up, including an oscilloscope and test automation and control components.

Regarding the centrifuge test box (Fig. 3), the problem of a laterally loaded pile is not axisymmetric and must be analysed using a three-dimensional (3D) approach. However, to consider all aspects involved in this type of simulation, a cylindrical box made of high-strength aluminium with an internal diameter of 220 mm and a height of 145 mm was adopted and installed inside the rectangular box (Fig. 3(a)). This arrangement minimises the differences in the ratio between the pile diameter (D) and the distance between the pile and the box wall (S). Boundary effects were minimised with S/D = 5·2, which is above the minimum value of 4·75 evaluated by Kirkwood (2015), and I_b/D = 2·3, which in turn is greater than the minimum value of 1·5 used by Bayton et al. (2018), where I_b is the distance between the lower part of the monopile and the bottom of the centrifuge box.

Two independent actuation systems were mounted on the centrifuge box: one for the cyclic tests to evaluate the displacement and secant stiffness evolution over cycles, and the other for the free vibration tests to assess the system’s natural frequency, as shown in Fig. 3. The free vibration tests were conducted at the beginning and end of the cyclic tests, as well as during the N-varying tests, which are discussed later in this paper.

Free vibration tests were used to evaluate the first natural frequency of the wind turbine model. This involved generating an impact on the model structure and identifying its first natural frequency. He & Zhu (2019) and Xu et al. (2020) compared the impact-induced free vibration test with other model excitation methods (shaker or ambient vibration), obtaining similar laboratory performance and confirming the reliability of this type of test.

For the cyclic lateral load (H ) on the pile, an actuator consisting of a high-frequency, low-friction pneumatic cylinder controlled by solenoid valves (Fig. 3(b)) – positioned next to the cylinder inlets – was used to maximise the loading frequency. For the free vibration tests, an impact actuator consisting of a mini-pneumatic cylinder was employed.

Both the cyclic and free vibration tests were performed in flight, with the model disconnected from the cyclic actuator during the vibration tests. This procedure was implemented using an electromagnet device positioned at the top of the monopile, which was engaged to connect the model to the cyclic actuator and disengaged to perform the free vibration tests (Fig. 3(b)).

The test actuation sequence was automatically driven by a microcontroller, and a power circuit activated the valves. Custom software was developed to operate the microcontroller, allowing activation of the components (cyclic actuator, impact actuator and electromagnet) as well as the force-limiting parameters: maximum (H_max) and minimum (H_min) values, frequency force (f_f) and number of cycles (n).

The instrumentation consisted of a load cell (Omega Engineering LCM202), two laser displacement sensors (Baumer OADM 12U6460/S35A and OADM 12U6430/S35A) and two microelectromechanical systems (MEMS) accelerometers (Analog Devices ADXL1001). Custom software was developed for force and displacement data acquisition.

The two MEMS-type accelerometers were diametrically attached to the top of the monopile model (Fig. 3(b)). The data were acquired using a digital oscilloscope (Pico Technology 4000 A Series PicoScope 4824 A) with a sampling rate of 80 million samples per second, which provided time and frequency domain views through dedicated software. The data were processed using the fast Fourier transform technique with both PicoScope and OriginPro software.

Validation of the free vibration tests, conducted to assess the natural frequency of the wind turbine model, was performed in the centrifuge box in two stages: the first stage at 1g (outside the centrifuge) and the second stage in flight at Ng. Two monopile models, ME-1 (Fig. 4(a)) and ME-2 (Fig. 4(b)), both attached to a rigid metal base, were used, with different masses placed on top of the piles. The impact was applied from a height of 85 mm relative to the base of the models. Readings from the two MEMS accelerometers, with a typical duration of less than 1 s, were acquired using the PicoScope oscilloscope, and the data were processed in the frequency domain, allowing easier identification of the natural frequency.

The fixed-base natural frequency (f_str) of a wind turbine can be assessed by assuming a simplified model, in which the structure is treated as a shaft attached to a fixed, rigid base, with a mass positioned at its top. In this model, the shaft represents the tower, and the mass represents the turbine elements. Using this simplification, the fixed-base natural frequency can be calculated using the formulation by Van der Tempel & Molenaar (2002), presented in equation (3), where E_pI_p and m₂ are the bending stiffness and mass of the shaft (tower free length mass), m₁ is the top mass and L_L is the free length (Fig. 4).

Since the problem involves a mass on top of a cantilever, the experimental fixed-base natural frequency (f_str) can be compared with the analytical formulation for the structural frequency proposed by Van der Tempel & Molenaar (2002), where E_pI_p = 242·12 Nm², m₁ = 54·93 g (ME-1) or 181·83 g (ME-2), m₂ = 28·60 g and L_L = 122·22 mm (ME-1) or 147·40 mm (ME-2).

Table 3 presents the results of the tests conducted at 1g and 100g, where the former shows a difference of less than 1% between the experimental and calculated values, while the latter exhibits a variation of less than 2% from the theoretical values.

Cyclic lateral loading can be characterised in terms of two normalised parameters: magnitude, ζ_b, and amplitude, ζ_c, proposed by LeBlanc et al. (2010). H_max and H_min are the maximum and minimum applied lateral loads, while H_r corresponds to the reference lateral load required to reach a ground-level displacement of 0·1D, as proposed by Byrne et al. (2015) and adopted by several authors, including Abadie et al. (2019), Richards et al. (2020, 2021) and Wang et al. (2022). The ζ_b parameter varies between 0 and 1, while ζ_c varies between −1 and 1, where ζ_c = 0 for one-way loading, ζ_c = −1 for symmetrical two-way loading and ζ_c = 1 for a static or monotonic test.

Figure 5 presents a typical schematic illustration of the response of a monopile subjected to a cyclic lateral load with constant amplitude (ζ_c), depicting the progressive increase in lateral displacement (Y_n) with the number of cycles (n).

The secant stiffness at each cycle (K_n) can be expressed as the difference between the maximum (H_max) and the minimum (H_min) lateral forces, divided by the difference between their corresponding maximum (Y_n) and minimum displacements at ground level.

Hysteresis behaviour can also be quantified through the loop area (A_n) at each cycle, as shown in Fig. 5. This loop area is directly related to the energy dissipated during cyclic loading and provides relevant information for fatigue assessment.

Table 4 shows the experimental programme for monotonic (M) and cyclic (C) tests, with ζ_c = 0 (one-way loading) for all tests. K₀ and K₁ denote the secant stiffnesses at cycles 0 and 1 (Fig. 5), Y₀ and Y₁ represent the soil-level monopile displacements at cycles 0 and 1 (Fig. 5), and f_n/f₀ is the ratio of the model’s natural frequency at cycle n (final condition) to its initial value (before cycle 0). The cyclic tests were conducted at 0·1 Hz (prototype scale) under force-controlled conditions.

The monotonic load tests, M1 and M2, were conducted at 100g under displacement-controlled conditions, in which a fixed horizontal displacement of 30 mm was applied at a rate of 0·32 mm/s at the top of the model. The test results are presented in Fig. 6, alongside the normalised frameworks proposed by Abadie et al. (2019) and Wang et al. (2022), which cover a wide range of pile geometries, load eccentricities, sand densities and testing methodologies (1g, centrifuge and field). The results show good agreement with the formulation proposed by Wang et al. (2022) up to 0·1D, while exhibiting a slightly softer response than that of Abadie et al. (2019) over the same displacement range. This consistency with established references supports the adoption of a reference lateral load, H_r = 930 kN, defined as the load required to produce a lateral displacement of 0·1D at the soil surface, and used for normalisation purposes in this study.

The cyclic tests C1 to C26 were performed at 0·1 Hz (prototype scale) under force-controlled conditions, where the model was loaded at a predetermined maximum constant force (H_max). The lateral displacement at the load level (Y_H) and at a point located 12 mm above the sand surface (Y ) was monitored throughout the tests. These two displacement signals, illustrated in Fig. 7 for test C26, allow the total displacement to be divided into two components: a recoverable component (⁠$YHe$ and Y_e), which stabilises after approximately 1000 cycles, and an irrecoverable ratcheting component (⁠$YHp$ and Y_p), which accumulates permanently as the number of cycles increases. This separation is particularly important when assessing long-term effects under non-zero mean cyclic loads. As highlighted by Abadie et al. (2019), permanent strain accumulation (ratcheting) is difficult to capture with conventional constitutive models, while stabilisation after a certain number of cycles – through accommodation or adaptation – can be modelled using combined isotropic and kinematic hardening.

The variation of the resulting maximum lateral displacement at the soil level for cycle n (Y_n), associated with the maximum lateral load H_max, can be quantified more directly by the rate of increase of displacement (δY_n = Y_n – Y_n−1). According to Abadie et al. (2019), the evaluation of this parameter helps to quantify whether the accumulated deformations are still occurring after several cycles.

Figure 8 presents the evolution of the normalised rate of increase of displacement (δY_n/D) over cycles for tests C8 (100 cycles), C11 (1000 cycles), C17 (10 000 cycles) and C24 (100 000 cycles) with D_r ≈ 80%, ζ_b ≈ 0·87 and ζ_c ≈ 0. The results indicate a high rate of increase during the early cycles, notably within the first 1000 cycles, followed by a marked reduction in δY_n/D after 10 000 cycles, which approaches but does not reach zero in 100 000 cycles. The behaviour is consistent with remarks by Abadie (2015) in 1g tests that included the results from Cuéllar (2011), based on over 10⁶ cycles at 1g.

A power function (equation (6)) was obtained by fitting the normalised rate of increase (δY_n/D) from all tests (C1 to C26) in the present study. The results for tests C8, C11, C17 and C24, shown in Fig. 8, illustrate the agreement with this trend.

This trend suggests that, while some ratcheting persists, it slows significantly after the initial phase. For design purposes, this observation is relevant given that a real wind turbine may undergo approximately 10⁷ cycles over its service life – 100 times the 100 000 cycles of this study. Thus, understanding early-cycle behaviour is critical for predicting long-term foundation performance under cyclic loading.

Figure 9 shows the experimental data from tests C5 and C24 for ζ_b = 0·5 and 0·9, respectively, alongside the equations proposed by Truong et al. (2019), Richards et al. (2021), Wang et al. (2022) and Li et al. (2022). For values of n ≤ 1000, Li et al. (2022) showed good agreement for ζ_b = 0·5, while the other equations tended to overestimate the normalised displacement. This discrepancy may be due to differences in the pile relative stiffness (k_r) assumed by each author, as analysed by Wang et al. (2022). For values of n > 1000, all the proposed equations strongly overestimate normalised displacements, mainly due to the previously mentioned reduction in the rate of increase of Y_n/Y₁ at approximately n = 1000 cycles.

Figure 10(a) shows the relationship between normalised initial displacement Y₀/D and ζ_b, based on tests C1 to C26, together with the normalised polynomial fit given by equation (7). Y₁ can be calculated directly using equation (8), the linear best fit of the Y₀/D and Y₁/D data through the origin (Fig. 10(b)).

Figure 11 illustrates a typical monopile lateral load–displacement signal for test C26, with cycles 1, 10, 10², 10³, 10⁴ and 10⁵ highlighted. The reduction in the area enclosed between the first and subsequent cycles indicates decreased energy dissipation due to soil grain rearrangement, leading to an increasingly elastic response, as observed by Kirkwood (2015). Abadie et al. (2019) reported that this dissipation, represented by the hysteresis loop area and directly proportional to the damping ratio, decreases with the number of cycles following an exponential decay. They also noted that the magnitude of the applied load has little influence on the evolution of the hysteresis and damping ratio.

Figure 12 shows the evolution of the normalised hysteresis loop area (A_n/A₁) as a function of cycles for tests C14 and C20. The fitted relationship from equation (9), based on all tests (C1 to C26) in the present study, slightly underestimates the normalised hysteresis for n < 100. The data suggest a rapid decrease during the first cycles, followed by stabilisation, indicating reduced plastic rearrangements and a progressive shift toward elastic behaviour. This trend, similar to the displacement rates in Fig. 8, highlights that the most significant changes occur in the early stages of cyclic loading, with the long-term response being predominantly elastic.

The secant stiffness at the nth load–displacement cycle, K_n, has previously been described using logarithmic functions, as shown by LeBlanc et al. (2010) and Abadie et al. (2019), for 1g tests up to 10⁵ cycles, and by Klinkvort & Hededal (2013) and Kirkwood (2015) for centrifuge tests up to 10⁴ cycles. By contrast, Li et al. (2022) proposed a power function based on tests of up to 150 cycles to describe the evolution of the normalised secant stiffness (K_n/K₁), where K₁ is the secant stiffness at the first cycle. This power function was also preferred by Richards et al. (2021) for consistency of interpretation with ratcheting and applied in the present study. Given the similarities (installation method, L/D = 5·2; e_x/D = 8; D_r = 80%, ζ_c = 0) between the tests conducted by Li et al. (2022) and those of the present study, the former were plotted alongside the experimental data to extend the ζ_b results range, leading to the following equations.

The results from Figs 8 and 12 and equation (10) show that the decrease in normalised displacement rate (δY_n/D), the reduction in hysteresis loop area (A_n/A₁) and the increase in secant stiffness (K_n/K₁) are interrelated responses to cyclic loading. Lower δY_n/D is associated with higher K_n/K₁ and smaller A_n/A₁, indicating that reduction in ratcheting, increase in stiffness and damping decrease are coupled phenomena. Therefore, numerical models should incorporate these linked mechanisms to reproduce accurately the observed cyclic behaviour in monopile foundations. This coupled evolution is consistent with the densification–ratcheting mechanisms described in the literature, in which early grain rearrangement and subsequent quasi-static ratcheting control both deformation accumulation and stiffness mobilisation under cyclic loading.

Figures 13(a) and 13(b) show the predictions of equation (10) for the normalised secant stiffness evolution over the number of cycles for tests C4 and C22 with ζ_b = 0·5 and ζ_b = 0·9, respectively. The equations proposed by Klinkvort & Hededal (2013), validated by Kirkwood (2015), and by Li et al. (2022) are also plotted for comparison. Both equations are in good agreement for ζ_b = 0·5. By contrast, for ζ_b = 0·9, the Li et al. (2022) equation tends to overestimate the secant stiffness while the Klinkvort & Hededal (2013) equation tends to underestimate it, although the latter shows some agreement for n < 300.

Regarding the evaluation of the secant stiffness at cycles 0 (K₀) and 1 (K₁), Fig. 14(a) shows the relationship between K₁ and ζ_b data through which equation (11) was obtained. The equation proposed by Li et al. (2022) to evaluate K₀ based on ζ_b tends to underestimate the secant stiffness values as seen in Fig. 14(a). In addition, K₁ can be evaluated by equation (12), which is the best fit to the K₀ and K₁ data that passes through the origin and is shown in Fig. 14(b).

This section analyses centrifuge test data on the long-term behaviour of the natural frequency under cyclic loading conditions. In the experimental programme presented in Table 4, free vibration tests were performed to determine the natural frequency of the model in its initial condition (f₀) and in its final condition (f_n) after the application of cyclic loading.

The results shown in Fig. 15(a) indicate that the normalised natural frequency (f_n/f₀), defined as the ratio of the final natural frequency (f_n) to the initial natural frequency (f₀), varies with the number of cycles (n > 100) according to a power function. Equation (13) applies for ζ_b = 0·9, and equation (14) applies for ζ_b = 0·5.

As shown in Fig. 15(a), the normalised natural frequency decreases slightly – by approximately 1·0% – during the first 1000 cycles for both load magnitudes. In contrast, Yu et al. (2015), Guo et al. (2015) and Liang et al. (2021) observed an initial increase in the f_n/f₀ ratio, followed by a long-term decrease that levelled off at f₀. For the 1g models investigated by these and other authors, natural frequency variations ranged from 5 to 40%. By comparison, the results presented here indicate a variation of only about 3% after 100 000 cycles, highlighting the influence of stress level on long-term dynamic behaviour.

According to Cuéllar et al. (2011, 2012), who conducted conventional (1g) physical modelling tests, the sand surrounding the pile exhibits two distinct phases of deformation during lateral loading cycles: an initial phase, characterised by densification, and a subsequent phase characterised by convection, which is a constant-volume cyclic movement of the particles. During the initial phase, pronounced subsidence is observed, attributed to the displacement and rearrangement of grains. This behaviour can account for the small decrease of around 1% in the natural frequency observed during the first 1000 cycles, as grain displacement affects pile confinement, thereby reducing the embedment (L). The larger lateral displacements observed in this early stage support this hypothesis. In the second phase, a progressive recovery may be linked to the convection of sand grains.

Since the natural frequency of a wind turbine is directly related to the relative stiffness of the pile, which in turn depends on the secant stiffness, a relationship between the ratio f_n/f₀ and the ratio K_n/K₀ is proposed in equation (15), based on a linear best fit applied to the experimental data shown in Fig. 15(b). Although some scatter is present in the experimental data, a clear trend of linear increase in the ratio f_n/f₀ with respect to the ratio K_n/K₀ is evident. However, it is important to note that equation (15) is valid only for the studied interval 2 ≤ K_n/K₀ ≤ 10.

As previously discussed, the evolution of secant stiffness is intrinsically linked to changes in hysteresis loop area (or damping, as reported by Abadie et al. (2019)), implying that variations in normalised natural frequency (f_n/f₀) reflect not only the increase in stiffness but also the reduction in damping associated with cyclic soil behaviour. Therefore, constitutive models for cyclic soil behaviour should consider the coupled evolution of stiffness and damping, enabling the changes in natural frequency of monopile foundations to be estimated reliably.

The installation process, which was jacked at 1g in the present case, may disturb the sand surrounding the pile and alter its density, as any installation method would. However, although this variation might affect the initial density, it does not significantly change the evolution of the natural frequency described by equation (15). Nevertheless, it is important to note that these results are valid only for jacked piles with similar installation conditions.

Figure 15(b) shows that, in the present study, a 300% increase in normalised secant stiffness (K_n/K₀ from 2 to 8) results in an approximately 2% increase in normalised natural frequency (f_n/f₀ from 1·00 to 1·02). Arany et al. (2017), using an analytical model to predict the natural frequency of an OWT with a monopile foundation (Arany et al., 2015, 2016), concluded that a 10% change in soil stiffness (coefficient of subgrade reaction) results in an approximately 0·5% change in the natural frequency of such a turbine. A similar analysis was carried out in the present study using the approach of Darvishi Alamouti et al. (2017), resulting in a 10% variation in soil stiffness and a consequent change of around 2% in the natural frequency of the prototype wind turbine. Discrepancies between results obtained in conventional (1g) and centrifuge (Ng) physical modelling tests are expected, since differences between the two types of tests directly affect the soil–foundation relative stiffness, which in turn affects its natural frequency. For this reason, as reported by Richards et al. (2021), centrifuge tests tend to be more reliable in quantifying the foundation behaviour because they more accurately simulate the real stress–strain field.

Using the power functions given by equations (13) and (14), the estimated variation in natural frequency over approximately 10⁷ cycles would be around 3–5%. This result falls within a range that is typically considered acceptable for accounting for installation tolerances and minor operational uncertainties, although it should not be interpreted as direct compliance with any prescribed safety margin. In comparison, Darvishi Alamouti et al. (2020) observed that a 200% increase in secant stiffness, obtained by extrapolating a 130 cycle experimental centrifuge study using the Abaqus Lanczos method, increased the natural frequency by approximately 31%, far exceeding the recommended safety margin.

This section presents the results of N-increasing free vibration tests conducted in the centrifuge and examines the effects of foundation and soil effective stress conditions on the natural frequency of the wind turbine model.

The sand-embedded monopile models QZ-1 and QZ-2, shown in Figs 4(c) and 4(d), together with models ME-1 and ME-2 (attached to a rigid metal base), were subjected to free vibration tests at various centrifuge accelerations: N (g) = 1, 20, 40, 60, 80 and 100. Models ME-1 and QZ-1 share the same geometry (D, L_L), tower L_L mass (m₂) and top masses (m₁), as do models ME-2 and QZ-2. The masses of the ME-1 = QZ-1 and ME-2 = QZ-2 models are presented in Fig. 4, and the main dimensions of the QZ-1 and QZ-2 models are presented in Table 5.

For each model and acceleration, 3–10 natural frequency measurements were performed. The experimental data are summarised as mean values in Table 6.

Figure 16 shows the effect of soil stress level for different top masses, where f_0,Ng is the initial natural frequency at Ng, and f_0,1g is the initial natural frequency at 1g. The ratio f_0,Ng/f_0,1g is greater than 1 in all tests and increases with both m₁ and N. The results indicate that increasing the centrifuge acceleration level (N) leads to an increase in the natural frequency of the system due to higher soil effective stress and confinement around the pile, with this effect being more pronounced for the model with the larger top mass (m₁ = 181·83 g). The stronger dependence on N for the heavier model suggests that soil–structure interaction becomes increasingly significant as turbine mass rises, which is likely to be due to higher inertial forces and enhanced soil confinement effects.

Figure 17 presents the ratio of the soil–structure initial natural frequency (f₀) of the sand layer models QZ-1 and QZ-2 to the fixed-base natural frequency (f_str), that is the β soil–structure interaction parameter (equation (16)), across different centrifuge accelerations (N), which correspond to varying soil effective stress levels. As expected, the β parameter is less than 1 for all tests and varies as a function of N, which directly influences the soil surrounding the pile.

As shown in Fig. 17, for the QZ-1 model, the β parameter ranged from 0·40 (n = 1) to 0·59 (n = 100), while for the QZ-2 model, with a heavier mass on top of the pile, the ratio exhibited a broader range, varying from 0·35 (n = 1) to 0·70 (n = 100). Although the structural stiffness of the pile is independent of the stress level in the soil, the soil–foundation stiffness increases with greater confinement, thereby raising the natural frequency of the soil–structure. The data indicate that the very low confining stresses in the 1g model tests are unrealistic, as confirmed by the centrifuge tests. While the structural stiffness of the model pile remains constant across stress levels, the foundation stiffness increases substantially under higher soil confinement. The same behaviour was observed in wind turbine models subjected to centrifuge tests conducted by Futai et al. (2018, 2021) on loose (D_r = 35%) and dense (D_r = 70%) sand models, with L/D ratios of 5·26, 6·58 and 7·89.

Figure 18 shows the variation of the β soil–structure interaction parameter, modified by the ratio D/L_L, where most data converge to a single fitted curve (equation (17)). The soil effective stress (σ’_v) was assessed at one-third (1/3) of the embedded pile length, where the maximum soil pressure occurs according to the λ-based pivot point defined by Darvishi Alamouti et al. (2017). The apparent independence of the modified β values from the mass on top of the pile may be attributed to the D/L_L ratio. While the mass clearly influences both the fixed-base natural frequency (f_str), as shown in equation (3), and the soil–structure initial natural frequency (f₀), the D/L_L ratio appears to offset this effect in relation to the β parameter. Data from Futai et al. (2018, 2021) were also plotted in Fig. 18, showing good agreement with the present results and helping to reconcile differences between the models.

Equation (18), which combines equation (17) with equation (16), expresses the initial natural frequency (f₀) as a function of the soil vertical effective stress (σ’_v), the fixed-base natural frequency (f_str), the pile diameter (D) and the free pile length (L_L).

By combining equation (18) with equation (3) from Van der Tempel & Molenaar (2002), the initial natural frequency of the soil–structure can be estimated based solely on the structural mass, the geometric characteristics of the pile and the soil vertical stress.

A series of 26 centrifuge tests was conducted to simulate the cyclic lateral loading of a wind turbine supported by a monopile foundation in sand. The tests focused on the long-term evolution of horizontal displacements, secant stiffness and natural frequency.

With respect to horizontal displacements, the results showed a high rate of increase during the first 1000 cycles, which accounted for most of the total displacements, followed by a substantial decrease in this rate after 10 000 cycles (n). Existing proposals in the literature for estimating the horizontal displacements over time do not adequately predict the displacement evolution for n > 1000, as they fail to account for the marked reduction in the rate of displacement increase after the initial 1000 cycles.

Regarding the variation of secant stiffness with the number of cycles, the proposal of Li et al. (2022) exhibited a better performance for lower values of the cyclic load magnitude ζ_b compared to other methods in the literature. However, for higher values of ζ_b under long-term conditions, a new equation was proposed to describe this parameter adequately. In addition, an equation is provided to calculate the initial secant stiffness based on the magnitude of the cyclic load. The hysteresis analysis revealed a rapid reduction in loop area during the early cycles, followed by stabilisation, indicating reduced damping and energy dissipation as the response became progressively more elastic. This behaviour is intrinsically linked to the evolution of secant stiffness and, therefore, to the dynamic stiffness of the monopile–soil system.

Regarding the variation of natural frequency, the results showed a smaller final increase (3%) with the number of cycles than those found in conventional (1g) physical modelling tests available in the literature, including an initial decrease associated with the first 1000 cycles, followed by an increase for a greater number of cycles. This difference may be attributed to the more realistic stress–strain conditions in centrifuge testing (Ng), which provide a more accurate simulation of soil stiffness behaviour and densification under cyclic loading, thereby reducing the variation in natural frequency compared to 1g tests. Overall, the long-term behaviour observed in the centrifuge tests can be interpreted within a unified physical framework, in which early-cycle grain rearrangement and densification is followed by a quasi-static ratcheting mechanism in the near-field region around the monopile. This process provides a physical explanation for the coupled evolution of permanent displacement, stiffness increase, damping reduction and the resulting changes in natural frequency.

The N-increasing free vibration tests conducted in the centrifuge show that the natural frequency of the OWT increases with the soil’s effective stress around the pile, leading to an equation to estimate the natural frequency based solely on known design parameters and the soil vertical stress.

Finally, it should be noted that all the tests presented here were carried out on a specific type of monopile, using dry sand with D_r = 80%, under one-way loading conditions, and employing the 1g jacking installation method. Therefore, the results are only valid for these conditions. Further studies are required to extrapolate the findings to other cases.

The authors would like to express their gratitude to the staff of LM²C (COPPE/UFRJ) for their technical support, and to the Brazilian funding agencies CNPq (proc. no. 409695/2018-1 and no. 303427/2022-1), FAPERJ (proc. nr. E-26/200.886/2021), MCT/INCT-REAGEO and CAPES-COFECUB for their financial support.