Analysis of Cyclic Response of Bucket Foundations Based on Simplified Kinematic Hardening Model

)e response of bucket foundations for offshore wind turbines subjected to cyclic loading in saturated clay is explored through three-dimensional finite element numerical analyses. In the analyses, nonlinear cyclic hysteretic behavior of clay under undrained condition is modeled through a simple kinematic hardening constitutive model embedded in ABAQUS.)e finite element model is validated against published in situ tests of bucket foundations under quasistatic cyclic loading in Bothkennar clay. )e computed results agreed generally with those from in situ tests. )e behavior of bucket foundations with different aspect ratios under displacement-controlled cyclic loading mode is investigated. )en, the evolution of foundation displacement with increasing number of cycles is studied subjected to wind and wave combined loading. )e results show that, for the cycles of lowamplitude rotation, dimensionless moment-rotation curve is approximately elastic; however, the curve engenders obvious hysteresis loop, whose shape is influenced by soil-sidewall interface condition, during high-amplitude cycles. Under thousands of loading cycles, for bucket foundations of low aspect ratio, the oscillatory displacement component is smaller; however, the residual component will accumulate gradually until the serviceability rotation is exceeded. For foundations of high aspect ratio, the oscillatory component is relatively larger, but the accumulation rate of residual displacement decreases gradually.


Introduction
With the gradual consumption of nonrenewable resources and the increasing power shortage, international efforts are being made to develop offshore wind power. As a novel type of marine foundation, bucket foundation has gradually been widely utilized in construction of offshore wind turbines [1]. e forces exerted on the bucket foundation for offshore wind turbines vary greatly from those of offshore platform. e vertical load from the upper tower and turbine blade is relatively small, while the horizontal load and moment component is larger. Moreover, the bucket foundation for offshore wind turbines is subjected to repeated loads for a large number of cycles during its service life [2]. e soil around the foundation may produce plastic strain accumulation and excessive displacement, thus deteriorating the normal operation of turbines. It is noted that long-term cyclic response is typically important design situation in the wind industry, whereas bearing capacity is dominating the design in the oil and gas industry. e stability of bucket foundation for offshore wind turbines under combined loading was studied by Fan et al. and Liu et al., and the characteristics of failure envelope of bucket foundations under different load components were analyzed [3,4]. However, these works are limited to static loading and do not take into account the mechanical behavior of the foundation under cyclic loading. Kourkoulis et al. explored the cyclic response of bucket foundation in clay, but more attention is paid to seismic response and the number of cycles is not more than 10 times [5]. e behavior of bucket foundation under cyclic moment loading and inclined loading was studied, respectively, through full scale in situ test and model test in the pressure chamber [6,7]. e relationship between stiffness and load level of the coupled system of bucket foundation and soil is obtained from these tests [6,7]. Zhu et al. carried out high-cycle loading of bucket foundation in sandy soil and obtained the relationship between horizontal displacement as well as rotation angle of foundation and the number of cycles [8].
e long-term dynamic loading tests of bucket foundation in sand were carried out by Bhattacharya et al. to analyze the dynamic characteristics of soil-foundation coupling system in frequency domain [9]. To study the mechanical response of bucket foundation for offshore wind turbines under cyclic loading in clay, in the paper, the kinematic hardening model is used to simulate the stress-strain relationship of saturated clay under undrained conditions. e displacement development law of bucket foundation for 3.5 MW offshore wind turbines under combined wind and wave loads is discussed preliminarily.

A Brief Introduction of Constitutive Models
e stress-strain relationship of saturated clay under cyclic loading is very complex. e advanced elastoplastic constitutive model based on effective stress is perfect in theoretic aspect, but it has many constitutive parameters to be measured. In the undrained analysis, Biot's consolidation equation or the constrained condition of volume change of soil body should be combined with application of such type of constitutive model. To simplify the analysis of interaction between soil and structure, saturated clay under undrained condition can be regarded as a single-phase medium, and the total stress constitutive model is adopted. Among them, Lemaitre and Chaboche proposed a simple kinematic hardening model, which has been widely used recently [10][11][12].
e linear elastic stress-strain relationship is adopted in the elastic part of the model, and only two parameters of elastic modulus E and Poisson's ratio v are required. Under the undrained condition, Poisson's ratio is selected as v � 0.49. e yield function of the model is where σ ij is stress tensor, α ij is back stress tensor, and it is also the kinematic hardening parameter of the model, which controls the movement of the yield surface in the stress space. H is the isotropic hardening parameter of the model, which denotes the size of the yield surface. Under undrained conditions, the yield of saturated clay is only related to deviatoric stress, but not to hydrostatic pressure. erefore f(σ ij − α ij ) may be expressed as follows: where s ij is tensor of deviatoric stress and α dev ij is the deviatoric component of back stress. e plastic strain increment is determined by the associated flow rule, as shown in where dε p ij is the increment of plastic strain tensor and dε p is the equivalent plastic strain increment and can be expressed as dε p � e yield stress in the model is controlled by the isotropic hardening law and the kinematic hardening law. e isotropic hardening component changes with the accumulation of equivalent plastic strain during loading, as shown in where σ 0 is size of the initial yield surface when the plastic strain just occurs and Q and b are model parameters. e possible maximum variation of yield surface size is defined by the parameter Q. e size of the yield surface can enlarge when Q is a positive value, whereas negative value can be used to indicate strain softening of clay [12]. e rate of change of yield surface size with equivalent plastic strain is controlled by the parameter b. e expression of the kinematic hardening component in the stress space is where C is initial kinematic hardening modulus which can be taken as elastic modulus of soil E and η controls the rate of back stress that recovers with the accumulation of equivalent plastic strain. e term of ηα ij _ ε p in equation (5) where σ y is maximum yield stress of clay and it can be determined as σ y � � 3 √ s u and s u is undrained shear strength of clay, since the Mises strength failure criterion is adopted in this model. e size of initial yield surface, σ 0 , can be selected as σ 0 � λσ y � λ � 3 √ s u following the recommendation of Anastasopoulos et al. [11]. e parameter λ can be determined by fitting the relation curve between dynamic shear modulus G and shear strain c, which is obtained through dynamic triaxial test or cyclic direct shear test. e range of value of the parameter λ is usually between 0.1∼0.3. erefore the following equation can be used to calculate the value of η:

Verification of In Situ Test
To verify the applicability of the kinematic hardening model used in this paper, the in situ cyclic loading test of bucket foundation carried out by Houlsby et al. [6] is numerically simulated. In this test, the bucket foundation diameter is D � 3 m and embedment depth is L � 1.5 m. e approximate mass of the bucket (including appurtenances) is 2000 kg, and the vertical load on the bucket throughout the test is augmented by a 2400 kg concrete block. e quasistatic cyclic horizontal loads are applied at the top of the A-frame by means of a hydraulic jack placed 4.23 m above the lid of the caisson. e effective unit weight of soil at Bothkennar is c ′ � 6.8 kN/m 3 . e best estimate of the undrained shear strength is s u � 11.4 + 1.9z, where s u is in kPa and z is the depth in meters below the base of the test site. Following the recommendation of Houlsby et al. [6], the elastic modulus of soil is E � 525s u . In the absence of other soil investigation results, during the numerical simulation, the parameter is selected as λ � 0.1, Q � 0, and b � 1.
en, according to equation (7), the kinematic parameter can be obtained as η � 336.8.
In the finite element model, the friction contact pair algorithm in ABAQUS package [13,14] is utilized to simulate contact response at the soil-bucket interface. When the normal tensile stress develops at the soil-bucket interface, separation between soil and pile occurs. When the normal stress is compressive, the Coulomb friction law is used to describe the tangential friction stress as follows: where τf is the limiting friction stress on the interface, σn is the normal contact pressure, and μ is the coefficient of friction. In the paper, a value of μ � 0.7 is selected. Considering the effect of construction disturbance, the soil strength at the interface of the inner and outer sides of bucket foundation is taken as αs u , and α is reduction coefficient. Using inverse analysis for the penetration resistance of the bucket foundation in the test site, Houlsby et al. suggested that the reduction factor is taken as α � 0.5 [6]. e comparison between the calculation results and the test results is shown in Figure 1. In this figure, M and θ are separately the bending moment and rotation angle at the center of the top of the foundation. rough comparison, it can be seen that the numerical results are generally in good agreement with that of the in situ loading test, although the kinematic hardening constitutive model used in this paper is relatively simple. e numerical results reasonably predict the hysteretic response and rotational stiffness reduction of the M-θ relationship curve. It is shown that the kinematic hardening model proposed by Lemaitre and Chaboche [10] is applicable in simulating the interaction between soil and foundation subjected to cyclic loading.

Finite Element Model
For the typical 3.5 MW offshore wind turbines, the mass of tower and wind turbine is individually m t � 200t and m nr � 220t, and the height of the rotor center from the caisson lid is h � 80 m. e soil is saturated hard clay, and its effective unit weight is c ′ � 8 kN/m 3 . For uniform soil, the undrained shear strength is s u � 60 kPa, whereas for heterogeneous soil, the undrained shear strength increases linearly with depth z, s u � s um + kz, where s um is the undrained strength at the seabed level, and k is linearly increasing gradient. In this paper, it is assumed that s um � 30 kPa; k � 2.5 kPa/m. In the kinematic model, the parameter inputted is listed as follows: and η � 384.9. It can be seen that the soil parameter η does not change with the depth below seabed level and is a constant for both soil profiles. e bucket foundation with diameter D � 20 m and wall thickness t � 0.025 m is considered.
e developed finite element model is displayed in Figure 2 taking advantage of problem geometry symmetry. In order to reduce the boundary effect of finite element model, the diameter and thickness of soil domain are taken as 6D and 3D. Soil is modeled with eight-noded hexahedral continuum elements, concentrated near the foundation.
To illustrate the influence of soil properties on the cyclic response of bucket foundation, the deformation of bucket and upper tower is not considered in the model. e bucket foundation is simulated by rigid body shell elements, and the tower is simulated by beam elements. e wind turbine is replaced by a lumped mass element at the top of the tower, as shown in Figure 2.
Considering the effect of construction disturbance and long-term cyclic loading, a layer of frictional thin element is introduced into the soil surrounding the bucket [15][16][17]. e undrained strength of this layer of elements is αs u , and reduction coefficient α is taken as 0.3. Frictional contact pairs are arranged between the bucket wall and the layer of thin element, which allows the interface to potentially break away and slip. e interface friction coefficient μ � 0.7 is assumed. For the sake of comparison, the case with full interface adhesion, i.e., α � 1, is also studied. In this case, the bucket wall and soil can not be permitted to separate and slip.

Displacement-Controlled
Monotonic Loading e ultimate capacity of bucket foundation under monotonic horizontal loading is analyzed. e finite element model is subjected to displacement-controlled pushover loading at the mass element; i.e., the height of application of the lateral displacement is h � 80 m. Two cases of L/D � 0.2 It is shown from Figure 3 that, considering the interface disturbance, the moment capacity is much lower than that when the interface is fully bonded. is shows that, in analyses of bucket foundation, the capacity obtained without considering the interface disturbance is too large, which leads to the unsafe foundation design. rough comparing Figure 3(a) with 3(b), it can be seen that the moment capacity increases with the increase of embedment. For example, when the aspect ratio increases from 0.2 to 0.6, the dimensionless moment capacity increases by about 80% for the fully bonded case. According to rough load estimates for an anticipated 3.5 MW wind turbine offshore the UK, the maximum overturning moment occurring at the seabed level is of the order of 120 MN [6]. It is noted that, for L/D � 0.2, the moment can be obtained as M � 0.78ADs u0 � 294 MN taking into account the imperfect interface. erefore, e foundation design of low aspect ratio such as L/D � 0.2 meets the requirement in terms of bearing capacity. By comparing Figure 3(b) with 3(c), the dimensionless capacity of foundation in heterogeneous soil is 13% and 26% lower than that in homogeneous soil for the case of full bonding and imperfect interface. e mechanism that the capacity is lower in heterogeneous soil relative to homogeneous soil is illustrated in Figure 4.
As shown in Figure 4, the instability mode of bucket foundation under lateral loads is generally in the form of rotation. In homogeneous soil, the arc-shaped slip surface is mainly located on the outside of the foundation and passes through the deep soil beneath the bucket. However, in the heterogeneous soil, the strength of shallow soil is relatively low; therefore the slip surface is limited to the shallow soil in the bucket, but not extended to the deep soil with higher strength.

Displacement-Controlled Cyclic Loading
To investigate the influence of interface conditions on the response of bucket foundation under cyclic loading, displacement-controlled cyclic loading is carried out for two aspect ratios of L/D � 0.2 and 0.6, respectively. Constantamplitude displacement cycles are applied at the nacelle level of the 3.5 MW turbine, i.e., the mass element shown in Figure 2. e displacement cycle varies according to sinusoidal law with period T � 10 s. Two sets of analyses consisted of imposing displacement amplitudes that produced rotations of θ � 0.01 rad and 0.05 rad, respectively. Figures 5 and 6 depict the cyclic moment-rotation graphs for both aspect ratios and interface conditions. As shown in Figures 5 and 6, under small cyclic loading, the hysteresis loop formed by the moment-rotation relation curve is close to the elastic response, whereas the area of the hysteresis loop increases obviously under large cyclic loading. It is shown that there is a significant plastic deformation in the soil under large cyclic loading, which results in a larger energy dissipation in the soil-bucket coupling system, but the shape of the hysteresis loop is related to the interface condition. is conclusion is consistent with that of Kourkoulis et al. [5].
For the full bonding interface condition, the loading and unloading paths generally obey the Masing rule and follow the skeleton curve, i.e., the monotonic loading curve. However, under the imperfect interface condition, the loading and unloading paths after the first cycle deviate from the skeleton curve and tend to be flattened. e area surrounded by the hysteresis loop is lower than that produced under full bonding condition. is is attributed to the fact that, during the first cycle, there forms an unrecoverable gap between the loading side of the bucket and the surrounding soil, and then, under the subsequent cycles, the existence of these cracks leads to the reduction of the overall resistance.

Response to Environmental Loading
e bucket foundation for offshore wind turbines is subjected to long-term cyclic loading from wind and wave during the whole service period, and the soil around the foundation may produce strain accumulation or stiffness weakening, which will lead to excessive displacement of the foundation and affect the normal operation of the turbines. To ensure the normal operation of the fixed-bottom offshore wind turbines, the cumulative rotation angle of the foundation should not exceed 0.2°, i.e., approximately 3.5 × 10 −3 rad [9] . So the main issue with bucket foundations under lateral cyclic loading is to predict how much rotation (or lateral displacement) will occur at given amplitude of cyclic load. e wave loads have a period of the order of 10 seconds and can be simulated in a slow-cyclic manner. e wind load is generally acknowledged to be of much larger period; therefore it can be simulated in a quasistatic behavior.
In this section, for a typical 3.5 MW offshore wind turbine, it is assumed that the bucket foundation is founded at 15 m water depth. According to the current codes of practice, a design wind load of F wind � 1 MN is assumed, acting on the center of the rotor-nacelle assembly, whose distance from mud-line is h � 80 m. e wind load is combined with a design wave load of F wave � 2 sin ωt(MN), acting on a height of h � 7.8 m from the mud-line. e circular frequency of wave load is ω � 2π/T, in which T is wave period.
In the finite element analysis, the following loading scenario is adopted: monotonic loading due to wind, followed by one thousand cycles of loading due to waves. According to available results [18][19][20], the one-way cyclic loading has been generally shown to cause larger accumulation of plastic deformation than two-way cyclic loading, so the loading scenario adopted in this paper is more conservative. us, the rotation accumulation of bucket foundation with the number of cycles is obtained.
It can be seen that, under thousands of loading cycles, the oscillatory component Δθ of the bucket foundation with low aspect ratio is smaller than the residual component θ, and the residual component θ gradually accumulates until it exceeds the allowable value. e amount of oscillatory component Δθ is only 10% of the residual component θ. However, for bucket foundations of high aspect ratio, the ratio of oscillatory component to residual component increases to 23%∼26%, but the accumulation rate of residual component decreases gradually before it tends to zero.  Advances in Civil Engineering From Figure 7(a), considering the case of imperfect interface, the accumulated rotation of bucket foundation with L/D � 0.2 subjected to one thousand cycles of lateral loading has exceeded the allowable rotation limit. In the case of full bonding, although the rotation under one thousand cycles of lateral loading does not reach the serviceability limit, the curve representing residual component shows a rising trend. Taking into account that an offshore wind turbine is expected to sustain approximately 10 8 cycles of loading over a period of 20 years of service [21][22][23], the rotation of bucket foundation with aspect ratio L/D � 0.2, in the case of full bonding, is still possible to exceed the allowable deformation limit. From Figure 8, it can be recognized that whether the interface is full bonding or not, the L/D � 0.6 footing maintains almost constant rotation despite the increasing number of cycles. e foundation with L/D � 0.2 and D � 20 m meets the requirement in terms of bearing capacity; however the accumulated rotation may exceed the allowable deformation limit during the service period; therefore it should be cautiously utilized in engineering practice.

Conclusion
e response of bucket foundations for offshore wind turbines subjected to cyclic loading in saturated clay is explored through three-dimensional finite element numerical analyses. e finite element model is validated against published in situ tests of bucket foundations under quasistatic cyclic loading in Bothkennar clay. e following conclusions can be drawn: (1) Under displacement-controlled cyclic loading, for the cycles of low-amplitude rotation, dimensionless moment-rotation curve is approximately linear elastic; however the curve engenders significant hysteresis loop, whose shape is influenced by soilsidewall interface condition, during high-amplitude cycles.
(2) Under thousands of loading cycles, for bucket foundations of low aspect ratio, the ratio of oscillatory displacement component to residual component is only 10%; however the residual component Advances in Civil Engineering will accumulate gradually until the serviceability rotation is exceeded. For foundations of high aspect ratio, the ratio of oscillatory component to residual component is relatively larger, but the accumulation rate of residual displacement decreases gradually. (3) e bucket foundation of aspect ratio L/D lower than 0.2 meets the design requirement in terms of bearing capacity; however, the accumulated rotation may exceed the allowable deformation limit during the service period.

Data Availability
All the data in this paper are obtained from tests in this study, and no other data in the literature are used to support this study.

Conflicts of Interest
e authors declare no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.