Effect of Loading Rate on Lateral Pile-Soil Interaction in Sand Considering Partially Drained Condition

Reliable assessment of the lateral pile-soil interaction is of pronounced importance for the design of mono-pile foundations of offshore wind turbines. As the offshore engineering moves to deeper waters, the diameter of mono-piles is getting larger, usually about 5 m and could be up to 8 m, which may lead to partially drained behaviors of sand in the vicinity of the pile and thus imply limitations of conventional design methods in which fully drained conditions were assumed. To shed light on this issue, a fully-coupled finite element model was established using an in-house developed finite element code DBLEAVES, incorporating a cyclic mobility constitutive model that is capable of describing the instantaneous contractive and dilative response of sands simultaneously. Triaxial and centrifuge model tests were conducted to calibrate the constitutive model and validate the pile-soil interaction model respectively. This is followed by a parametric study primarily focusing on the effects of loading rates. The initial stiffness of the p-y curve was found to increase with the loading rate whilst the bearing capacity showed the inverse, and the mechanism behind this phenomenon is examined in detail. Then an explicit model was developed to evaluate the development of excess pore pressure in the pile front upon lateral loading, and an upper boundary of normalized loading rate was identified to distinguish fully and partially drained conditions.


Introduction
Monopiles are one of the most popular foundation types in offshore developments owing to their well understood loading behavior and the rich experience in design (Chen, 2011;Negro et al., 2017;Zhang et al., 2019). As the offshore wind energy industry moves to deeper waters and the size/capacity of turbines get significantly larger, the monopiles that support the wind turbines have experienced considerable increase in size. The diameter (D) of these piles could be up to 8 m (Zhu et al., 2015), e.g., 7 m in London Array and 7.8 m in Vega Mate offshore wind farm (Negro et al., 2017). The comparatively long drainage path around the pile section poses new challenges to engineers in that these large diameter piles may exhibit partially drained responses in sand even under slow loadings. This differs from the behaviour of conventional small diameter piles that are typically assumed to behave under fully drained conditions. The pile-soil interaction is therefore strongly affected by the development of excess pore pressures. Moreover, these large diameter piles are always of significantly higher relative stiffness and thus can develop different soil failure mechanisms from those of smaller ones.
The monopile foundations for offshore wind turbines are typically subjected to large lateral and overturning loads caused by winds, waves, currents, etc., compared with the vertical dead load of the foundation and the upper struc-tures. Extensive attention has been paid to lateral pile-soil interaction, primarily by means of the p-y curve method (e.g. Matlock, 1970;Reese et al., 1974;Poulos, 1988Poulos, , 1989Liang et al., 2018), which is considered to be one of the most effective and practical approaches to account for the nonlinear pile-soil behavior in industry. However, the p-y curves used in most design codes were obtained from field tests using piles of diameter smaller than 1.5 m, rendering its utility to large diameter piles under question (Achmus et al., 2009;Choo et al., 2014;Byrne et al., 2015;Zhu et al., 2016).
The effect of excess pore pressure accumulation on pile-soil interaction has attracted enthusiastic research interest, by means of in situ trials (Weaver et al., 2005;Rollins et al., 2005) and model tests (Liyanapathirana and Poulos, 2005;Tang et al., 2010;Ashour andArdalan, 2011, 2012;Chang and Hutchinson, 2013;Maula et al., 2014;Su et al., 2016). In these studies, the soil was presumably under fully undrained condition and the partial drainage effect was not specifically examined. This issue was explored by Hansen (2012) in which sands of distinct permeability was used to reproduce both fully and partially drained conditions, and by Ashour et al. (2009) in which a simplified model using the strain wedge method was proposed to evaluate the pile-soil interaction in partially or completely liquefied sand. In this model the undrained stress-strain relationship of soil element has been implemented to enable the correlation between the hardening/softening of pile-soil interaction and the dilative/constructive behavior of sand. More detailed investigation into the partially drained behavior of sand using triaxial tests has also been pursued (e.g. Koga et al., 1989;Igarashi and Yamada, 1993;Yamamoto et al., 2009).
However, studies specifically on the effect of loading rates on pile-soil interactions considering partially drained conditions are still scanty. Among them, Yang (2019) conducted centrifuge tests of a large diameter mono-pile (D=5.9 m in prototype) with the bending moment along the pile and the excess pore pressure in the vicinity of pile being carefully studied. It was found that the initial stiffness of loaddisplacement curve increased with the loading rate whilst the bearing capacity decreased with it, owing to initial accumulation-dominant and the latter dissipation-dominant development of excess pore pressure; however assessment models on the pile-soil interaction were not developed. These expensive centrifuge tests are rather rare and numerical solutions considering the development of excess pore pressures have also been explored by fully coupled effective stress approaches (e.g. Zienkiewicz et al., 1990;Elgamal et al., 2002). However challenges exist when it comes to find proper constitutive models to describe the complex behavior of sands of different confining pressures and densities, particularly when a soil element experiences both dila-tion and contraction during the loading. This paper investigates the effect of loading rate on laterally loaded large diameter monopiles. An effective stressbased finite element model was established to examine the effect of excess pore pressure on lateral pile-soil interactions. A cyclic mobility soil model developed based on the classic modified Cam-clay (MCC) model was employed to describe the simultaneously contractive and dilative soil behavior under different stress levels . Two centrifuge tests with different loading rates were conducted to provide useful insights into this problems as well as to confirm the validity of the present numerical model. Following that an extensive parametric study is introduced, based on which an analytical model was developed to assess the effect of loading rates on the lateral pile-soil interactions for large diameter monopiles in sand.

Cyclic mobility soil model
The numerical model was established using an effective stress-based finite element program which incorporates the cyclic mobility soil model proposed by Zhang et al. (2007). Quadrilateral first-order element (C3D8P) was used to model the soil, in which the void is saturated with incompressible water. The cyclic mobility constitutive model used in this paper has proved its success in simulating the excess pore pressure (EPP) responses of soil upon liquefaction Xia et al., 2010). This soil model is originally proposed to capture the cyclic mobility phenomenon of medium dense sand and liquefaction of loose sand, but also works well in simulating the contractive and dilative behavior of sand under monotonic loading. In addition to the five routine parameters in the classic modified Cam-clay (MCC) model (i.e. M, e 0 , λ, κ and υ), six extra parameters were introduced, including defining the collapse rate of the structure, m R defining the losing rate of over-consolidation ratio (or the change in soil density) and b r determining the developing rate of stress-induced anisotropy, and three parameters, R 0 , and ζ 0 defining the initial state of them. In the parametric study one parameter varied with the remaining, as well as other parameters (i.e. D r ), being fixed despite that these parameters are intrinsically correlated. This setting is deemed reasonable here as, e.g., two different sands with the same D r could have different permeability, and vice versa. The advantage of the present model is that the soil under different stress or void conditions could be described in a unified way. This model has three yield surfaces, i.e., a subloading one, a normal one and a superloading one, which respectively represent the present state, the normally consolidated state and the constructive state. In this manner, the cyclic mobility of sands as reported by a number of studies (Castro and Poulos, 1977;Ishihara, 1985;Elgamal et al., 2002) can be captured in that the sub-loading surface can ro-tate according to the stress-induced anisotropy that is reflected in the over-consolidation ratio. The sand used was Fujian sand, whose properties are listed in Table 1. Drained triaxial tests at different relative densities (D r =40% and 60%) and confining pressures were conducted to validate as well as to calibrate the soil model (see Figs. 1a and 1b), where good agreement can be seen. The stress-strain response at D r =40% exhibits hardening at all confining pressures, whilst both hardening and softening behavior could be witnessed at D r =60%, illustrating strong dependency on confining pressures. The observations here demonstrate the capability of the soil model to capture the different loading characteristics of sands with different relative densities.

Lateral pile-soil interaction model
Only half of the soil domain and the pile was considered due to symmetry and the model was discretized by 4 289 three-dimensional eight-node elements. The dimensions of the soil block were determined as 50 m (length)×25 m (width)×65 m (depth) based on preliminary simulations to mitigate boundary effects. Both loose (D r =40%) and medium dense (D r =60%) sands were considered and more soil properties are given in Table 2. Zero pore pressure boundary conditions were applied to the soil surface. Fully bonded interface was assumed. Hansen (2012) reported that this treatment can reasonably describe the lateral pile-soil interaction behavior, whilst it should be borne in mind that the mechanical and hydraulic behavior near the pile-soil interface is rather complex as both normal and tangential flow may occur once the pile separates from the soil. Detailed inspection into this issue is beyond the scope of the present study, however the validation presented later demonstrates proper functioning of the present numerical model. The pile was modeled using solid purely elastic material with an elastic modulus E of 29.6 GPa (i.e. giving equivalent bending stiffness for a steel pipe pile with the thickness ratio of 0.018) and a Poisson's ratio of 0.3. The pile with three diameters being considered, namely 2.5 m, 5 m and 7.5 m, was assumed to be wished-in-place at an embedment depth (d) of 45 m. The lateral load was modeled as the concentrated force acting on the pile with a load eccentricity e (i.e. the loading height) of 10 m. Various loading rates were simulated to capture pile-soil interactions under differing drainage conditions. In tests LS-0, MS-0, MS-5 and MS-10 (see Table 3) no pore pressure response was allowed to develop, to serve as base cases for simulations with different loading rates where excess pore pressure developed. More details of the numerical model are provided in Tables 3 and 4.

Validation
Two sets of centrifuge tests on monopile, one (Case A) reported by Zhu et al. (2016) and one recently conducted (Case B), are used to explore the validity of the numerical model in mimicking both fully drained and partially drained pile-soil interactions. The two simulations were performed in the same manner, except for the difference in pile diameter D, embedment depth L, loading eccentricity e and loading rate v h . More information is given in Table 5 (in prototype scale). The soil used was Fujian standard sand,   Table 1. In order to satisfy the time-scaling of seepage process, silicon oil with viscosity n times that of water was used to saturate the soil. The model pile was driven into the sand at 1g, which may produce softer lateral response than that installed inflight (Dyson and Randolph, 2001;Taylor, 2014). The centrifuge was then operated at 100g for one hour before the loading tests to achieve full settlement of the surrounding soil, in order to somewhat mitigate this effect.

Case A
In the centrifuge test, pore pressure cells were installed around the pile but little development of EPP was monitored, thus the soil response was taken to be fully drained. Fig. 2a compares the experimental and numerical lateral load-displacement curves. Both exhibit hardening response and a very good match between them can be seen. This load-displacement response is further examined in Fig. 2b, in terms of distribution of pile deformation y along embedment depth at different levels of lateral load. The numerical model slightly overestimates the pile deflection, exhibiting softer pile-soil response, but generally good agreement can still be seen. For more detailed inspection, the p-y curves at different depths were compared in Fig. 2c. In the centrifuge test p was acquired by twice differentiating of bending moment along the pile length, but in the simulation p was acquired by differentiating the shear force, which is associated with higher precision. Overall agreement can be seen between the numerical and experimental results despite that the numerical data show somewhat softer soil reaction. The results calculated by API code are presented as well and show much stiffer performance than both.

Case B
To further explore the validity of the model to capture the EPP response, a centrifuge model test on the monotonic lateral loaded monopile was conducted. The layout of the   testing programme is illustrated in Fig. 3a. The lateral load was applied by means of a loading rig under displacementcontrol at a relative high rate of 20 mm/s, so that the effect of excess pore pressure is not negligible. The lateral load was monitored by an in-house designed load cell with the precision of 0.02 kN. The lateral displacement at the loading position was measured by a laser displacement transducer. Thirteen pairs of strain gauges (CF350-2FB(23)C24) were installed to measure the bending moments along the pile shaft, and the pore pressure at depths of 1D and 2D in front of the pile was acquired by two pore pressure transducers (PPT) made by MEASUREMENT in France. Fig. 3b compares the experimental and numerical lateral load-displacement curves. No yield point is observed for both, exhibiting hardening behavior (c.f. Fig. 2a). Moderate divergences can be seen between the experimental and numerical curves. It is worth mentioning that the parameters of soil properties used in the numerical model were calibrated against triaxial tests rather than the centrifuge model tests presented here. Clearly closer match between the numerical and centrifugal curves can be achieved if the latter was conducted but this was not further explored in the present study. The distributions of bending moments at different levels of lateral load are shown in Fig. 3c. Very good agreement between the numerical and experimental results can be seen. The maximum bending moment corresponding to each loading level takes place at a depth between 3D and 4D. EPP was witnessed in both centrifuge test and numerical simulation, represented by u. It can be seen in Fig. 3d that the excess pore pressure at z = 1D accumulates considerably faster than that at z = 2D during the initial loading stage, most probably owing to the greater pile deflection rate at shallower depth. However with further loading the excess pore pressure at 1D starts to drop rapidly and fall below that at 2D due to shorter drainage path and thus more dissipation, which is in agreement with the real situation and implies an upward seepage. Despite the moderate difference in the amplitude of u, the overall trend observed in the centrifuge test is satisfactorily captured by the numerical model, given the complexity of the response examined here. The numerical curve in Fig. 3b is moderately higher than the experimental one, probably owing to the facts that: (1) no separation was allowed at the rear of the pile, leading to stronger soil response, and (2) the pile in the centrifuge test was driven under 1g, as discussed above.
Above all, the numerical results show qualitative and quantitative agreement with the centrifuge test data in respect of various key loading responses, thus providing confidence to the use of the proposed numerical model to perform an extensive parametric study on the loading behavior of large diameter monopiles, as presented below.

Effect of EPP on load-displacement response
Figs. 4a and 4b illustrate the influence of loading rate on the load-displacement responses for loose sand (LS) and medium dense sand (MS) respectively. All curves exhibit a hardening response without showing any yielding even at pile deflection beyond 0.1D. The MS curve is always asso- ciated with a higher stiffness than LS. The initial stiffness in general increases with loading rate, whilst the reverse can be seen with relatively larger y h . It appears that there exists a threshold value of loading rate below which all loading curves tend to converge. The development of excess pore pressure seems to have more impact on loose sand, as evidenced by the notable difference between LS-0 and LS-1 curves in Fig. 4a which cannot be seen for their counterparts in Fig. 4b. Fig. 5 compares the lateral pile deflection along depths for the slowest and rapidest loading cases at two representative lateral load amplitudes of 3 MN and 10 MN. For both loose and medium dense sands considered here, less pile deflection can be seen in LS-4 and MS-4 than in LS-0 and MS-0 respectively at H =3 MN, and this trend is reversed at greater load of H=10 MN. This is confirmed in Fig. 6, where the soil in the rapidest loading case shows significantly stiffer response than the slowest one but then experiences severe degradation and falls below the latter with further displacement.
To explain the above observation, the development of the excess pore pressure u and the mean effective stress σ' m in front of the pile are shown in Fig. 7. It can be seen that for all depths examined here u accumulates rapidly upon lateral loading and remains almost constant thereafter, and as a result a drop in σ' m can be seen for both LS and MS. Differ-ent response between the two sands can be seen after y/D = 0.01, as σ' m in LS keeps decreasing, owing to the continuous contraction, and then stabilizes at about 50% of the initial value, whilst that in MS starts to gain a rebound and even greatly surpasses the initial value. This can be confirmed by the stress paths in p-q plane as shown in Fig. 8. In cases LS-0 and MS-0 where excess pore pressure is not considered, the soil elements at differing depths show similar behaviour, i.e., starting linearly from the k 0 consolidated line to the critical state line. On the contrast, when pore pressure is simulated, contraction can be seen for all depths throughout LS-4 while there is a transition point in MS-4 where the contractive response shifts to a dilative one for   ZHU Bin et al. China Ocean Eng., 2020, Vol. 34, No. 6, P. 772-783 777 soils at deeper depth. This transition point also happens to be the one distinguishing the accumulation-dominant and dissipation-dominant stages as shown in Fig. 7. By comparing Figs. 8a and 8c, it is found that the value of q on the critical state line in LS-4 is significantly less than that in LS-0, indicting soil softening due to excess pore pressure accumulation. Similar phenomenon can be seen in the simulations using medium dense sand, although less pronounced. Fig. 8 also demonstrates the capability of the soil constitutive model used, in that the same soil parameters were used throughout the depths and both contractive and dilative responses at shallow and deep depths, respectively, can be simultaneously captured, depending on the initial density and instantaneous stress condition of soil. Based on the results discussed so far, it is found that both the effective soil stress and the excess pore pressure affect the lateral pile-soil interaction greatly. Prior to any lateral loading, a pile section at an arbitrary depth is subject to static soil pressure σ' n0 and static pore pressure u 0 , both distributing evenly around its perimeter. When lateral load is applied, the pile section is subject to normal soil pressure σ' n , shear stress τ' n and pore pressure u+u 0 , where u 0 is the static pore pressure and u is the excess pore pressure. As u 0 distributes evenly around the pile, its influence on the overall pile-soil interaction is limited. The reaction force per length, denoted by p, on the pile can thus be obtained by θ in which denotes the angle between the loading direction and the normal direction of pile shaft. For cases where the

778
ZHU Bin et al. China Ocean Eng., 2020, Vol. 34, No. 6, P. 772-783 p s = σ ′ n cosθ + τ ′ n sinθ + ucosθ u y = ucosθ pile is loaded at extremely low rates and full drainage can be exploited, u is taken as zero (e.g. in MS-0). The distribution of p s ( ) and p w ( ) around the pile in the fully drained (MS-0) and the partially drained (MS-4) cases are shown in Fig. 9. It can be seen that p s in front of the pile contributes the most to p and the distribution of p s in front of the pile in cases MS-0 and MS-4 are rather similar at y/D = 0.03. However, p s at the back of pile in MS-4 is greater than that in MS-0 owing to the development of u, which may explain why p is greater in MS-4 at the early stage. It is of interest to note that the negative u at the pile rear has positive effect on p. When y/D = 0.2, p s in MS-4 is less than that in MS-0. The reason for this phenomenon lies in the fact that the accumulation of excess pore pressure reduces effective stress, causing softening of soil, as demonstrated in Figs. 8b and 8d.

Assessment model of EPP
As seen in Fig. 10, with the increase in loading rate, the accumulation rate of excess pore pressure increases as expected. The excess pore pressure responses in MS-3 and MS-4 are very similar, indicating that a nearly fully undrained condition has been achieved. Since the lateral displacement rate of pile is not constant along the depths, here the concept of loading rate v = dy/dt at a given depth is proposed. In accordance with the definition of p in p-y curves, a normalized term is introduced as: The calculated results are plotted in Fig. 11 against the normalized loading rate vD/c v . Linear relations can be seen between η and vD/c v in the semi-logarithmic coordinates, which can be expressed as: and K(D) = 20.4D/D 0 -17, in which D 0 is the reference diameter taken as 1 m. Since the model is developed based on simulations with pile diameters of 2.5 m, 5 m and 7.5 m, its utility to piles with diameter outside this range is under question (e.g. negative value of η can be obtained at very small values of D). When the loading rate is very slow (i.e. approaches zero), there is no accumulation of excess pore pressure (i.e. η=0). There is a clear upper boundary of vD/c v beyond which little increase in η can be achieved for all pile diameters considered. This finding is consistent with most previous studies that a normalized loading rate over 20 develops fully undrained soil behaviour (e.g. Randolph and Hope, 2004). For slower loadings, the accumulation rate of pore pressure can be determined using Eq. (4).

Assessment model of initial pile-soil stiffness
p-y curve method is one of the most effective methods ZHU Bin et al. China Ocean Eng., 2020, Vol. 34, No. 6, P. 772-783 779 to consider the nonlinear behavior of pile-soil interaction, and a hyperbolic shaped form originally proposed by Kim et al. (2004) is adopted here: in which k ini represents the initial stiffness of p-y curve and p u represents the ultimate soil-pile interaction. Since the rotation of large diameter piles in the field are restricted to very small values, and p u could not be reached, detailed inspection into this parameter was not achieved here. The cal-culated results for D = 2.5 m and 7.5 m are examined in Fig. 12. It can be seen that the model proposed by Zhu et al. (2016), given as k ini = n h z can reasonably describe the relation between k ini and z, where n h is the constant of horizontal subgrade reaction, though moderate divergence is witnessed between cases with different loading rates. For the analyses with D = 2.5 m, there is little difference between MS-1 and MS-2, implying little effect of excess pore pressure. However, for that with D = 7.5 m, remarkable difference exists between MS-11 and MS-12. Of interest to note is that though the loading rates of tests MS-1 and MS-11 are the same, the normalized loading rate is greater in MS-11, owing to the larger D. The observation in Fig. 13 thus implies that pile diameter, loading rate and soil depth could be the most important factors that affect k ini . A clear trend is found between k ini /σ' mo and vD/c v : where p D accounts for the effect of pile diameter and β the effect of vD/c v . In this study, p D = 0.1 D/D 0 + 0.75, α(D) = 0.2(10 -D/D 0 ) (Fig. 13). β can be expressed by: vD/c v > 1000 when or , β=0.16 and 0.25, respectively.

Discussion: cyclic loading
This section presents a preliminary study on the pile-soil interaction under cyclic loadings. The parameters ≤ considered are tabulated in Table 4. Here a degradation factor ξ N = (k ini ) N /k ini is proposed to characterize the degradation of secant stiffness with loading cycles, in which (k ini ) N is the initial stiffness of p-y curve in the N-th cycle. Fig. 14 shows the calculated p-y curves for different soil types and loading conditions. Generally p increases with z and loose sand experiences more degradation than medium coarse sand when the development of EEP was taken into account, as p approaches zero during the last few cycles in LC-4. Also, when comparing Figs. 14a and 14c, it is found that the stiffness of p-y curve experiences considerable reduction when EPP is simulated. Fig. 15 presents the relation between ξ N and N. Since the shape of p-y curves at shallow depth (z 1D) changes dramatically as the sand is nearly liquefied (Ashour and Ardalan, 2012), only the data at z=4D are presented. Severe degradation is witnessed to take place Fig. 11. Relationship between pore pressure accumulation rate and loading rate. in the first few cycles for all cases. The decreasing trend of curves tends to stabilize in LC-0 and MC-0 while develops continuously in LC-4 and MC-4, most probably owing to accumulation of excess pore pressure.

Conclusions
A pile-soil interaction model is established using the finite element program DBLEAVES coupled with a cyclic mobility soil model that can properly capture the behavior of sands at the presence of significant accumulation of excess pore pressure. The effect of loading rate on the lateral response of monopiles of different diameters in sand is examined, accounting for the partially drained conditions. Main findings are summarized below.
(1) The capability of the cyclic mobility model to capture the instantaneous contractive and dilative behavior of Fujian standard sand was successfully demonstrated, through comparison of numerical results with both triaxial compression tests and centrifuge model tests.
(2) With the increase in loading rate, excess pore pressure around the pile accumulates more rapidly, resulting in higher initial stiffness of p-y curves, while the lateral bearing capacity of the pile decreases due to more severe degradation of the surrounding soil.
(3) Throughout the lateral loading, an accumulationdominant stage and a dissipation-dominant one are identified. During the first stage the pile-soil interaction is governed by both effective soil stress and excess pore pressure, while at the second the load borne by the pore water greatly transfers to the soil.
(4) An explicit model was proposed to assess the accumulation of excess pore pressure in the passive area of laterally loaded piles and a boundary is identified, beyond which the pile-soil response can be taken as fully undrained. The relation between the loading rate and the initial stiffness of p-y curve was analyzed and an empirical model was proposed.
Despite the capability of the numerical model to capture the development of excess pore pressure around the piles undergoing lateral loading, more realistic behaviours such as the normal and tangential gap flow were not reasonably mimicked. Further studies regarding this issue (Cerfontaine et al., 2015(Cerfontaine et al., , 2016 would certainly extend the understanding of partially drained pile-soil interactions presented in this paper.  ZHU Bin et al. China Ocean Eng., 2020, Vol. 34, No. 6, P. 772-783