1 Introduction

Offshore wind turbines (OWTs) have developed rapidly as an alternative to fossil fuel as an energy source. Monopiles are both cost-effective and easy to implement; thus, they are the most common foundations for offshore wind turbines. Monopiles are large-diameter piles with an embedment length-to-diameter ratio (L/D) of less than 10 [1]. OWTs tolerate different load combinations, including both horizontal and vertical loads. Horizontal loads resulting from wave action and wind are the primary cause of concern when designing OWT foundations.

The conventional p–y method used by API [2] and DNV [3] is based on the results of experiments conducted with slender piles having a diameter of less than 1 m and with a small load eccentricity [4, 5]. This means that the reliability of this method for the design of large-diameter piles is in question [6,7,8]. Furthermore, these guidelines have been developed for the offshore oil and gas sector, from which the loading characteristics of OWTs differ.

An OWT is subjected to millions of cycles during its lifespan. This cyclic loading could cause accumulated rotation of the structure. Current guidelines use a simple approach for considering the effect of cyclic loading. API [2] considers a different calibration factor for lateral resistance in the case of cyclic loading. This factor is independent of the number of load cycles, load magnitude and symmetry. An accurate prediction of accumulated rotation and changes in stiffness caused by cyclic loading is vital when designing monopile foundations. For this reason, many researchers have begun to study the cyclic behaviour of piles, particularly monopiles [9,10,11,12,13,14,15,16].

Little and Briaud [17] introduced a power function based on pressuremeter tests as a degradation factor for soil resistance. This function considers the effect of the number of cycles. Other studies have proposed degradation functions which considered parameters such as the soil density, installation method and cyclic loading ratio [18, 19]. These studies have been done on long flexible piles, and the number of load cycles was limited. Their results suggested that one-way loading caused greater displacement than two-way loading; however, recent studies have reported that asymmetric two-way loading generated more displacement than one-way loading [10, 11, 20, 21].

LeBlanc et al. [20] studied the effect of long-term cyclic loading on stiff piles in dry, loose sand using small-scale laboratory modelling. Their results indicated that the power function was the best fit for the accumulated rotation of piles under cyclic lateral loading. They introduced \({\upzeta }_{b}\) and \({\upzeta }_{c}\) to describe the loading characteristics as follows:

$${\upzeta }_{\mathrm{b}}=\frac{{M}_{\mathrm{max}}}{{M}_{\mathrm{r}}}=\frac{{F}_{\mathrm{max}}}{{F}_{\mathrm{r}}},$$
(1)
$${\upzeta }_{\mathrm{c}}=\frac{{M}_{\mathrm{min}}}{{M}_{\mathrm{max}}}=\frac{{F}_{\mathrm{min}}}{{F}_{\mathrm{max}}},$$
(2)

where \({M}_{\mathrm{min}}\) and \({M}_{\mathrm{max}}\) are the minimum and maximum moments at the mudline, respectively, and \({M}_{\mathrm{r}}\) is the moment at failure or at a specific rotation at the mudline. They proposed the following equation for accumulated rotation:

$$\frac{\Delta \theta (N)}{{\theta }_{\mathrm{s}}}={T}_{\mathrm{b}}\left({\zeta }_{\mathrm{b}} .{R}_{\mathrm{d}}\right){T}_{\mathrm{c}}\left({\zeta }_{\mathrm{c}}\right).{N}^{\alpha },$$
(3)

where N is the number of cycles, Rd is the relative density of the soil, \({\theta }_{\mathrm{s}}\) is the rotation under static loading, \({\Delta \theta }_{N}\) is the increase in rotation after N cycles, α is an accumulation parameter and Tb and Tc are dimensionless functions that depend on the loading characteristics and relative density.

Their results showed that the largest number of rotations occurred at \({\upzeta }_{c}\) =  − 0.6, which indicates asymmetric two-way loading. Klinkvort and Hededal [22] used a similar model for predicting the accumulated rotations. They investigated the behaviour of monopiles in saturated and dry sand using centrifuge tests. In contrast to the previous research, their results showed that one-way loading was the most damaging load condition. Furthermore, they noted that the foundation stiffness increased with an increase in N, as has been documented by Refs. [20, 23]. This contradicts the findings of Achmus et al. [13], API [2] and DNV [3]. The primary cause for this contradiction could be that each researcher used a different definition of foundation stiffness [24, 25].

Albiker et al. [26] carried out a series of 1 g model tests on rigid and flexible piles. Their results indicated that, in the case of rigid piles, asymmetric two-way loading caused the maximum displacement accumulation, and the loading function (Tc) was not affected by system parameters such as the pile stiffness and relative density of the sand. In the case of flexible piles, one-way loading was the most damaging load condition.

Nicolai et al. [14] investigated the effect of cyclic loading on the post-cyclic capacity of monopiles in dense silica sand. In both the centrifuge and 1 g tests, the post-cyclic capacity increased with the magnitude of the cyclic load and the number of loading cycles. This increase was more significant when the loading symmetry approached balanced two-way loading. On the other hand, centrifuge tests conducted by Truong et al. [27] indicated that the post-cyclic capacity was equal to or less than the static capacity.

The results of various studies show inconsistency related to issues such as soil-pile stiffness, displacement prediction functions, the effect of load symmetry on displacement and the post-cyclic capacity. All studies discussed above were carried out on siliceous sand, although about 40% of the ocean floor is covered by calcareous sediment [28]. These biogenic materials, formed mainly from shell fragments and coral reef detritus, are commonly found at low latitudes in tropical areas such as the Persian Gulf.

Because of its high particle angularity, high void ratio and particle crushability, the engineering properties of calcareous sand differ from those of siliceous sand [29]. The pile capacity in calcareous sediment has been a focus of study in geotechnical engineering. Most studies have concentrated on the axial capacity of the piles, but a few researchers have addressed the problem of monotonic lateral loading of piles in calcareous sand [30, 31]. These studies suggest that the existing method underestimates pile resistance in the case of large displacement in calcareous sand; however, no studies were found that investigated the cyclic lateral behaviour of monopiles in this type of sand.

The present study explored the cyclic behaviour of monopiles in calcareous sand by means of centrifuge modelling. The effect of cyclic loading and its characteristics on the accumulated deformation and soil-pile stiffness have been compared with the findings of previous works. The effect of cyclic loading on the ultimate resistance of the monopiles also has been investigated, and a model is proposed to predict the accumulated displacement of the pile head based on the method introduced by LeBlanc et al. [20].

2 Experimental Procedure

All testing were carried out using the geotechnical centrifuge provided by the University of Tehran. The rotation radius and maximum acceleration of the beam centrifuge were 3 m and 130 g, respectively. More details about this apparatus can be found in Moradi et al. [32]. All centrifuge tests were conducted at a gravity level of 40 g, which translates to a scaling factor of 40. A servo-motor was used to generate the cyclic loads, which were applied at 5.75D (D = diameter of the monopile) above the mudline. The rotary motion was converted to linear motion using a ball-screw mechanism. To allow free rotation at the loading point, a hinge connection between the monopile and transition piece was designed that is similar to the one employed by Choo and Kim [33]. A sketch and a photo of the setup are shown in Figs. 1 and 2, respectively.

Fig. 1
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Sketch of the test setup

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Fig. 2
figure 2

Centrifuge model setup

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The model monopile had a stainless-steel tubular profile that was 50.8 mm in diameter and 1 mm in thickness. The pile diameter (D), after coating, was 52 mm (208 cm at prototype scale). It had an embedment length of 5D and was instrumented with ten pairs of strain gauges (TML-FLA-5-11; Tokyo Sokki Kenkyujo). Each pair of gauges was arranged in a half-bridge Wheatstone configuration to measure the bending moment. The first pair was installed at 10 mm above the pile tip and the centre-to-centre spacings were 30 mm.

Poulos and Hull [34] introduced pile-soil relative stiffness (\({E}_{\mathrm{P}}{I}_{\mathrm{P}}/{E}_{\mathrm{s}}{L}^{4}\)) to consider the effect of pile rigidity on the pile response where Es and Ep are the elastic modulus of the soil and pile, respectively, Ip is the moment of inertia of the pile and L is the embedment length. By considering Es = 50 MPa, based on estimates of Abadie [35], the ratio for the monopile in the present study would be approximately 0.04, which is similar to some monopiles used in industry [35].

The lateral displacement of the pile shaft was measured by three linearly variable differential transformers (LVDTs) at levels of 16 cm, 30 cm and 38 cm above the mudline (model scale). Two of them were positioned above the loading point to allow calculation of the rotation of the monopile head. After obtaining the deflection and rotation of one point as well as the bending moment diagram, displacement along the monopile length could be calculated using classic beam theory, as shown in Eq. (4). In this research, classic beam theory was only used to calculate the pile deflection and rotation at the mudline. It should be noted that the bending moment diagram along the free length of the monopile was linear:

$${E}_{\mathrm{P}}{I}_{\mathrm{P}}\frac{{\mathrm{d}}_{y}^{2}}{{\mathrm{d}}_{x}^{2}}=M\left(x\right).$$
(4)

All samples were prepared in a strongbox with a width of 60 cm, a length of 70 cm and a height of 55 cm. The box length was greater than 10D, which was sufficient to allow the boundary effect to be disregarded, as demonstrated in previous studies [36,37,38]. Calcareous sand from Hormuz Island in the northern Persian Gulf was used for the experiments. All tests were carried out in dense sand because monopiles are usually placed under similar conditions [1]. Several studies have examined the behaviour of Hormuz sand, primarily using element test results [39,40,41,42,43]. The sand properties are presented in Table 1, and more detail can be found in Rasouli et al. [44].

Table 1 Engineering properties of Hormuz calcareous sand
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Compressibility and particle shape are the primary sources of the behaviour of uncemented calcareous sand. Nauroy et al. [45] defined the tangent compressibility index (Cpi) as the slope of the e-log(p) curve at the pressure pi. Le Tirant and Nauroy [46] suggested a reduction coefficient for Pu (ultimate lateral capacity of piles) as calculated using the API method, which is equal to one at Cpi < 0.02. Figure 3 shows the isotropic compression test results on Hormuz calcareous sand performed with a triaxial apparatus. The Cpi of dense Hormuz calcareous sand was less than 0.02 for the stress range of this study. This means that it is logical to expect the overall cyclic performance of the monopile in such sand to be similar to that of a monopile in siliceous sand.

Fig. 3
figure 3

Isotropic compression test results for Hormuz calcareous sand

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The ratio of pile diameter to median grain size was approximately 160, which is large enough to ignore the effect of grain size [47, 48]. The sand was pluviated into the container from a constant drop height. Each layer thickness was approximately 30 mm and, after every stage, the surface was levelled. The relative density was calculated by measuring the sand weight and the dimensions of the soil container.

The pile was driven into the sand at 1 g using a rubber hammer to reach the target embedment of 5D. Although installation at 1 g acceleration level (instead of 40 g) could significantly change the pile lateral resistance [49], its effect on the comparison between cyclic and monotonic loading, the focus of the present study, was neglected [23]. During installations, conical depressions formed around the pile. A similar phenomenon has been reported by Richards et al. [15].

2.1 Testing Program

The details of the tests are presented in Table 2. All tests were conducted at an acceleration level of 40 g and lateral loads were exerted at a height of 5.75D above the mudline. To determine the ultimate lateral capacity of the system and confirm the repeatability of the experiments, two monotonic tests were performed. These two tests were conducted displacement-controlled with a rate of 0.1 mm/s to ensure sufficient data collection. The sand was dry; therefore, there was no concern about the drainage situation. In monotonic tests, the purpose was to obtain the load corresponding to a displacement or rotation value; therefore, it was easier to measure the pile resistance to an implied displacement as is done in displacement-controlled tests.

Table 2 Test program
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The cyclic testing program was designed to investigate the effect of the cyclic load magnitude and symmetry on the accumulated rotation or displacement. The cyclic loading parameters were selected to cover a range of cyclic loading conditions. However, because of discrepancies in the literature regarding the effect of asymmetric two-way loading, more focus was placed on asymmetric two-way loading. In the cyclic tests, the purpose was to simulate wind and wave loading and measure the accumulated displacement and rotation. This made it more convenient to use load-controlled tests.

All the cyclic tests were designed to execute 600 cycles, but test C3 stopped after 145 cycles for unknown reasons. The cyclic loading was conducted at a frequency of 0.4 Hz, which is higher than the peak frequency of offshore waves. This frequency was chosen to apply a greater number of cycles within a certain period of time. Rate dependency is low in dry sand; therefore, it was unlikely that this change of frequency significantly affected the results [16, 23]. Test C6 was conducted to investigate the effect of loading sequence on the accumulated deformation of the monopile; however, only the first part of this test has been considered in the article.

3 Results

3.1 Monotonic Tests

The results of two monotonic tests are presented in Fig. 4. The horizontal load was normalized according to the framework suggested by LeBlanc et al. [20]. The two curves nearly overlap, which indicates the repeatability of the test procedure. Different criteria were used to determine the lateral capacity of the pile. Refs. [20, 22] defined failure at a pile rotation of 4°, while Truong et al. [27] used a ground rotation of 0.5° as the reference lateral load criterion. In the present study, a lateral load corresponding to 1° of rotation at the soil surface was chosen as the reference load. This criterion, as shown in Fig. 4, produced a normalized reference force of 0.625.

Fig. 4
figure 4

Monotonic tests results showing repeatability of experiments

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3.2 Cyclic Tests

Cyclic loading was applied in a load-controlled mode. In the first three tests (C1, C2, C3), the maximum magnitude of loading was changed, but the load reversal was kept constant. Three more tests were allocated to investigation of the effect of loading symmetry and loading reversal. The results of six cyclic tests based on rotation at the soil surface and displacement at the loading point are presented in Figs. 5 and 6, respectively. The calculated rotation at the mudline experienced more noise than did displacement at the loading point; therefore, the latter was used in the analysis.

Fig. 5
figure 5

Pile lateral load-rotation response: a \({\upzeta }_{\mathrm{c}}\) ≈ 0; b \({\upzeta }_{\mathrm{b}}\) ≈ 0.45

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Fig. 6
figure 6

Pile lateral load–displacement response: a \({\upzeta }_{\mathrm{c}}\) ≈ 0; b \({\upzeta }_{\mathrm{b}}\) ≈ 0.45

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As stated, parameters \({\upzeta }_{\mathrm{b}}\) and \({\upzeta }_{\mathrm{c}}\), were used to describe the cyclic loading characteristics. Parameter \({\upzeta }_{\mathrm{b}}\) is the ratio of the maximum load to Fr and denotes the magnitude of the maximum load, while \({\upzeta }_{\mathrm{c}}\) is the symmetry or reversal of cyclic loading and varies between 1 and − 1. For example, \({\upzeta }_{\mathrm{c}}\) = − 1 indicates balanced two-way loading and \({\upzeta }_{\mathrm{c}}\) = 0 indicates one-way loading.

Figure 7 shows the results of monopile displacement and rotation under cyclic loading. It can be seen that the change in rotation and displacement resemble each other. In all tests except for C4, in which balanced two-way loading was applied, the accumulated displacement increased as the number of cycles increased. In test C4, the accumulated displacement remained nearly constant. This indicates that the direction of the first quarter of balanced two-way loading did not determine the direction of the net accumulated displacement. Similar results for balanced two-way loading have been reported by Klinkvort and Hededal [22], but these contradict the results of Refs. [23, 24].

Fig. 7
figure 7

a Maximum accumulated displacement at loading point; b maximum accumulated rotations at soil surface

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The slope of accumulated displacement increased with an increase in \({\upzeta }_{\mathrm{b}}\), but the effect of \({\upzeta }_{\mathrm{c}}\) was more complex. The maximum slope occurred at \({\upzeta }_{c}\) = − 0.36, while the slope of accumulated displacement was almost identical at \({\upzeta }_{\mathrm{c}}\) values of − 0.53 and 0. Although a logarithmic function provided a better fit to the results, especially for initial cycles, the power function proposed by LeBlanc et al. [20] was used in the present study because it ensures higher accuracy at higher cycles.

Figure 8 shows the values of Δy/y1, where y1 is the maximum displacement during the first cycle for all cyclic tests except C5. The normalized accumulated displacement values have been fitted with the power function presented in Eq. (5). A value of α = 0.25 was determined by averaging the fitted values of all tests, although these values were nearly identical. This value is less than the value of 0.31 reported by LeBlanc et al. [20] and greater than the values reported by Albiker et al. [26] (0.23) and Nicolai and Ibsen [10] (0.13). The fitted lines are shown in Fig. 8. The main purpose of the deformation function was to predict the accumulated deformation at higher cycles. Therefore, the first ten cycles were excluded from fitting analysis and the focus was on higher cycles. As was expected, the accumulated displacement caused by cyclic loading (\(\Delta y\)) increased with an increase in the maximum magnitude:

Fig. 8
figure 8

Normalized pile displacements at loading point. Lines obtained using Eq. (5)

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$$\frac{{\Delta y}_{N}}{{y}_{1}}=B.{N}^{\alpha = 0.25},$$
(5)
$$B={T}_{\mathrm{b}}\left({\zeta }_{\mathrm{b}}.{R}_{\mathrm{d}}\right){T}_{\mathrm{c}}\left({\zeta }_{\mathrm{c}}\right).$$
(6)

In these equations, Tc takes into account the effect of loading symmetry or reversal. When one-way loading (\({\upzeta }_{\mathrm{c}}=0\)) was applied, the output of Tc was assumed to be equal to 1; therefore, by calculating the fitting curve coefficient (B) for tests in which one-way loading was applied (C1, C2, C3), the value of Tb could be calculated. Figure 9 shows the values of Tb based on \({\upzeta }_{\mathrm{b}}\), where \({\upzeta }_{\mathrm{c}}\) is approximately zero. The fitted line was forced to have a zero intercept because it is not possible to have accumulated displacement without a cyclic load. The results were also compared with those of other studies [10, 20]. It should be noted that LeBlanc et al. [20] defined pile lateral capacity at a pile rotation of 4° while Nicolai and Ibsen [10] used ultimate resistance. In the present study, the 1° criterion was used; therefore, the \({\upzeta }_{\mathrm{b}}\) values in Fig. 9 are not directly comparable. The relationship between Tb and \({\upzeta }_{\mathrm{b}}\) in the present research can be described as:

Fig. 9
figure 9

Tb function in terms of \({\upzeta }_{\mathrm{b}}\)

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$${T}_{\mathrm{b}}=0.212 {\zeta }_{\mathrm{b}}.$$
(7)

The effect of loading reversal on accumulated displacement was investigated by changing \({\upzeta }_{\mathrm{c}}\) while holding \({\upzeta }_{\mathrm{b}}\) constant. The value of Tc was calculated by dividing the fitting curve coefficient extracted from Fig. 8 [B in Eq. (6)] by the Tb value estimated from Eq. (7). The results are presented in Fig. 10. By definition, the value of Tc in the case of static loading (\({\upzeta }_{\mathrm{c}}=0\)) should be equal to zero. In test C4, when loading approached the balanced two-way condition, the accumulated displacement remained constant; therefore, the value of Tc for \({\upzeta }_{c}\)=  − 1 was considered to be zero. It can be seen that a third-order polynomial fits the data in Eq. (8):

Fig. 10
figure 10

Tc function in terms of \({\upzeta }_{\mathrm{c}}\)

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$${T}_{\mathrm{c}}=0.97{\zeta }_{\mathrm{c}}^{3}- 1.02{\zeta }_{\mathrm{c}}^{2}-0.97{\zeta }_{\mathrm{c}}+1.01.$$
(8)

The shape of the derived Tc function is similar to the one reported by Refs. [10, 20, 26]; however, the maximum value was much less than the one described by LeBlanc et al. [20]. The results show that partially unbalanced loading (\({\upzeta }_{\mathrm{c}}\) = − 0.35) was the most damaging condition. This is in line with the observations of Refs. [20, 26, 50], but contradict the results of Klinkvort and Hededal [22].

The foundation stiffness is an essential aspect of the design of OWT foundations. There are discrepancies in the definitions of secant stiffness in the literature. In the present study, secant stiffness has been defined as the ratio of the difference between the peak values of the lateral load in a cycle to the difference between the corresponding displacements (see Fig. 11). Other studies [16, 51] have used a similar definition.

Fig. 11
figure 11

Definition of cyclic parameters [27]

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In Fig. 12, the calculated secant stiffness (prototype scale) is presented against the number of cycles in the logarithmic scale. As can be seen, stiffness always increased with an increase in the number of cycles; however, the amount of increase was less than that reported by some researchers. For example, the maximum increase in stiffness was observed in test C2, but this was only about 15%, which is much less than the amount reported by Refs. [22, 27]. Part of this dissimilarity could be from the use of different definitions.

Fig. 12
figure 12

Calculated secant stiffness: a \({\upzeta }_{\mathrm{c}}\) ≈ 0; b \({\upzeta }_{\mathrm{b}}\) ≈ 0.45

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As shown in Fig. 12, the change in stiffness could be described by a logarithmic law. The slope of the line was governed by the magnitude of cyclic loading (\({\upzeta }_{\mathrm{b}}\)) and increased with an increase in \({\upzeta }_{\mathrm{b}}\). On the other hand, the load reversal condition only slightly affected the slope. These results show that the Y-interval (K1) was dependent on both loading parameters. K1 decreased with an increase in the cyclic loading magnitude as a result of the nonlinear behaviour of the soil-pile system, as shown in Fig. 12a. For \({\upzeta }_{\mathrm{c}}\), K1 was nearly constant when \({\upzeta }_{\mathrm{c}}\) varied from 0 to − 0.55 and then dropped rapidly at \({\upzeta }_{\mathrm{c}}\) = − 1, indicating balanced two-way loading. The change in stiffness can be expressed as:

$${K}_{N}= {K}_{1}+{A}_{k}\left({\zeta }_{\mathrm{b}}\right).\mathrm{ln}\left(N\right),$$
(9)
$${K}_{1}= {k}_{\mathrm{b}}\left({\upzeta }_{\mathrm{b}}\right).{k}_{\mathrm{c}}\left({\upzeta }_{\mathrm{c}}\right),$$
(10)

where K1 and KN are the stiffness in the first cycle and in cycle number N, respectively (Fig. 11), and Ak is a dimensionless coefficient. LeBlanc et al. [20] assumed Ak to be constant, but the results of the present study show that it is a function of \({\upzeta }_{\mathrm{b}}\).

Figure 5 shows that, in all tests except C3, monotonic loading was applied after cyclic loading. Figure 13 presents the post-cyclic response. Similar trends were observed for the rotation-based response. The post-cyclic response of the monopiles in all tests except for C2 and C6 was nearly the same as for the virgin monotonic response at high loading magnitudes. In tests C2 and C6, the post-cyclic capacity did not reach the static capacity. In these two tests, the monopile was subjected to higher loading magnitudes; therefore, cyclic loading caused more permanent displacement than in the other tests.

Fig. 13
figure 13

Post-cyclic response

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There could be a threshold in the permanent displacement or the cyclic load magnitude beyond which post-cycle resistance decreases. Truong et al. [27] observed a similar phenomenon. In their tests with a permanent rotation of greater than 0.5°, post-cyclic resistance was lower than for monotonic resistance. On the other hand, Nicolai et al. [14] stated that the monotonic resistance increased after cyclic loading. Although the permanent rotation of the monopile in tests C2 and C6 was greater than 0.5°, more data are needed to reach a firm conclusion on this. The decrease in resistance could be due to experimental error.

Figure 14 shows the bending moment diagrams measured by ten pairs of strain gauges along the pile length. The results show that the maximum bending moment increased slightly after 600 cycles; however, the depth of the maximum moment remained almost constant. It should be noted that part of this increase was due to small differences in the lateral load magnitude. In most tests, the maximum bending moment at the point of zero lateral loads increased significantly.

Fig. 14
figure 14

Bending moment profiles along monopile in cyclic tests

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Kirkwood and Haigh [12] used the term “locked-in moments” to describe non-zero bending moments in piles when no loading was applied. As shown in Fig. 14, the maximum locked-in moment occurred around the middle of the pile embedment length. The change in the locked-in moment magnitude with the number of cycles is shown in Fig. 15. The slope of increase in the locked-in moments appears to be dependent on \({\upzeta }_{\mathrm{c}}\). In tests C1 and C2, where \({\upzeta }_{\mathrm{c}}\) was approximately zero, the increases in the locked-in moments had similar slopes. In test C5, where balanced two-way loading was applied, the locked-in moments were almost constant. In the tests where \({\upzeta }_{\mathrm{c}}<0\), there were two moments in every cycle when the applied force was equal to zero. The locked-in moments for both of these moments are presented in Fig. 14. The results showed that, after a negative peak, the locked-in moment was negative and, after a positive peak, it was positive. However, in both cases, the magnitude of the locked-in moments was less than for the one-way loading test at equal values of \({\upzeta }_{\mathrm{b}}\).

Fig. 15
figure 15

Bending moment in monopile at 5.2 m in depth at zero lateral load

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The existence of a locked-in moment indicates a change of density and stress caused by cyclic loading. Kirkwood and Haigh [12] attributed the greater pile displacement in the case of unbalanced two-way loading to the fact that, in that case, the locked-in moment was smaller than when one-way loading was applied. However, in the current study, the lower residual moment in balanced two-way loading did not increase displacement.

4 Conclusions

A series of centrifuge model tests were performed to investigate the effects of the loading characteristics on monopiles in dense calcareous sand. Parameters \({\upzeta }_{\mathrm{b}}\) and \({\upzeta }_{\mathrm{c}}\) were used to describe the cyclic loading magnitude and reversal, respectively. The following conclusions have been drawn:

  • The accumulated displacement increased as a power function of the number of cycles. This function was strongly affected by the loading characteristics. Partially unbalanced loading was the most damaging loading condition. The model proposed by LeBlanc et al. [20] was adopted to predict the accumulated displacement.

  • For balanced two-way loading (symmetric loading), the accumulated displacement almost remained constant; however, the soil-pile stiffness increased.

  • The secant stiffness of the soil-pile system increased moderately as cyclic loading proceeded. A logarithmic function has been used to express the variation in stiffness. The slope of this function increased with an increase in the maximum load magnitude (\({\upzeta }_{\mathrm{b}}\)), although stiffness was lower for tests with higher \({\upzeta }_{\mathrm{b}}\) values. Nevertheless, \({\upzeta }_{\mathrm{c}}\) did not affect the slope of this function. Furthermore, in the test with balanced two-way loading (\({\upzeta }_{\mathrm{c}}\) = − 1), the secant stiffness was considerably lower than for the other tests.

  • After cyclic loading, monotonic loading was applied. In most tests, except for tests with a higher magnitude of cyclic loading, the post-cyclic capacity was almost equal to the static capacity. This indicates that there could have been a threshold for the maximum cyclic load magnitude or permanent displacement beyond which the post-cyclic resistance decreased.

  • The magnitude of locked-in moments in tests with one-way loading increased but remained relatively constant in the balanced two-way loading test. The maximum locked-in moments occurred at the middle of the pile in all tests.

The overall behaviour of the monopile in calcareous sand was similar to that previously reported for siliceous sand, and it appears that crushability did not affect the monopile behaviour, probably because of the low level of stress.