Horizontal Cyclic Bearing Characteristics of Bucket Foundation in Sand for Offshore Wind Turbines

Abstract

During the service period, the offshore wind turbine foundation mainly bears the wind load from the upper structure and the periodic loads such as wave load and sea currents from the lower structure. Long-term cyclic loads have an important impact on the cumulative deformation, foundation stiffness changes, and horizontal ultimate bearing capacity of the offshore wind turbine bucket foundation. This paper conducts cyclic loading tests on mono-bucket foundations under unidirectional and multidirectional cyclic loading conditions based on the multidirectional intelligent cyclic loading system of offshore wind turbine foundations and analyzes the cumulative effects of the loading direction, vertical load, and drainage status on the mono-bucket foundation during the cyclic loading process, effects of rotation angle, cyclic stiffness, and monotonic bearing capacity of foundation after cycles. The research results show that multidirectional cyclic loading significantly reduces the cyclic cumulative rotation angle of the mono-bucket foundation, and the maximum reduction rate can reach more than 50%. After unidirectional and multidirectional cyclic loading, the bearing capacity of the bucket foundation can be increased by up to 30% and 20%, respectively. At the same time, this paper proposes calculation formulas for vertical load, number of cyclic loading, and normalized cumulative rotation and establishes a calculation method for vertical load and bearing capacity of the foundation after cycles.

1. Introduction

Suction bucket foundation is also called suction foundation, bucket foundation, suction anchor, etc. It is a top-sealed, bottom-open structure similar to an inverted cylinder (Figure 1). The construction method of the suction bucket foundation is to first let the foundation sink to a certain depth on the seabed under its own weight, then apply suction to the bucket to create a pressure difference between the top and bottom of the foundation, and then use the pressure difference between the top and bottom of the bucket foundation to penetrate the foundation into the seabed surface. Compared with other offshore wind turbine foundation types, the suction bucket foundation has the advantages of simple construction, recyclability, low construction cost, and self-floating and towing and is suitable for soft foundations. In recent years, this foundation type has gradually been used in offshore wind farm construction.

The offshore wind turbine foundation is mainly composed of wind load and wave load transmitted from the top of the foundation (Figure 2). With the development of offshore wind turbines toward large-capacity wind turbines (10 MW and above) and from shallow water (within 30 m) to medium water depth (30–50 m), bucket foundations have gradually begun to be used. However, there are few studies on the bearing characteristics of bucket foundations, and the bearing mechanism is still unclear. It is necessary to conduct in-depth research on the bearing characteristics of multi-bucket foundations [1,2]. At present, the general design service life of wind turbines is about 25 years, while the ultimate bearing capacity design of wind turbine foundations is usually based on a load that occurs once in 50 years, and the impact of cyclic loads on the bearing capacity of the foundation is not considered. During the service life of an offshore wind turbine foundation, the number of cyclic loads it bears is as high as 107 times or more [3,4]. In addition, offshore wind turbines in different locations are subjected to different cyclic loading patterns. The effects of cyclic loading on the bearing characteristics of a mono-wind turbine and the development law of foundation deformation need to be further clarified.

Many scholars have conducted extensive research on the cyclic bearing capacity of bucket foundations. Jia N (2018) et al. [5] innovatively adopted a seven-cabin composite bucket-shaped foundation structure similar to a honeycomb structure. They conducted a field test with a water depth of 0.4 m in a silty natural pool and estimated the horizontal bearing capacity of the structure. The results showed that increasing load will cause the soil pressure in the passive zone outside the bucket skirt to increase significantly while the soil pressure on the corresponding active zone skirt wall remains basically unchanged. Tran X N (2017) et al. [6] conducted a series of three-dimensional finite element analyses using an elastic-plastic model to model sand, following the Mohr–Coulomb failure criterion, changing the spacing between each bucket, the embedding depth of the bucket, the bucket diameter, and the size of the vertical load to analyze the bearing capacity of the tripod bucket foundation in medium-dense sand. Finally, a bearing capacity formula that considers the effects of bucket spacing, embedding depth, foundation diameter, vertical load, and soil density was proposed. Chen X (2016) et al. [7] proposed an incremental elastic-plastic finite element method, encoding the total stress-based boundary surface model developed by the authors into ABAQUS to simulate the deformation process of suction caissons in soft clay under cyclic loading. The method was verified by scaled model tests and centrifuge tests to be effective in analyzing deformation and determining cyclic bearing capacity, which is more advantageous than traditional methods. Hung (2018) et al. [8] conducted an experimental study on the cyclic bearing capacity of bucket foundations in clay and found that the foundation unloading stiffness increases with the number of cycles. Jiang M (2024) [9] et al. studied the monotonic and cyclic performance of composite bucket foundation breakwaters in clay using a centrifuge model. The application of monotonic loads simulates extreme wave conditions, while cyclic loads correspond to long-term service conditions. The test results show that in the monotonic test, the center of rotation of the foundation continuously moves downward during loading, indicating that the deeper soil will be activated to resist the horizontal loads. Byrne and Houlsby [10] (2004) found that the loading history has a large impact on the cumulative deformation of the foundation. When the foundation experiences a storm load during its service life, the deformation it produces is the main part of the cumulative deformation of the foundation. The stiffness of the soil gradually decreases under the action of cyclic loads. Zhu (2013) et al. [11] conducted unidirectional and bidirectional horizontal cyclic bearing characteristics tests on the bucket foundation in loose sand. They studied the cumulative rotation angle, foundation settlement, and cyclic stiffness of the bucket foundation during loading. They found that the cumulative rotation angle of the foundation increased with the number of cycles, while the cyclic stiffness was independent of the number of cycles.

Lee [12] (2016) studied the cumulative rotation angle of bucket foundations under cyclic load at different burial depths in dry sand with two different densities and proposed a rotation angle deformation formula based on the experimental results. Wang (2018) et al. [13] conducted model tests and finite element analysis on the horizontal monotonic and cyclic bearing capacity of mono-bucket foundation and tripod-bucket foundation in medium-density sand. The study found that under the action of horizontal load, the triple-bucket foundation increased the stiffness of the foundation and reduced the angular deformation of the foundation. The rotational stiffness and dynamic stiffness of the triple-bucket foundation changed greatly under the initial cyclic load, while the changes were smaller in the later period. Wang X (2024) et al. [14] studied the offshore wind turbine suction bucket foundation. Through centrifuge tests and finite element analysis, they explored the tensile and compressive characteristics of the bucket foundation under different aspect ratios, analyzed the soil stress response, proposed and verified the bearing capacity calculation method, and provided a reference for engineering design. Kumar T (2023) et al. [15] conducted numerical studies to analyze the bearing characteristics of a mono-pile bucket foundation in a sandy foundation under lateral and vertical loads; explored the influence of various factors on its displacement, bearing capacity, etc.; and proposed relevant prediction expressions as a preliminary design guide. Zhu B (2014) et al. [16] conducted a series of model tests on silt and sandy seabeds for single-caisson foundations of offshore wind turbines; studied the effects of size, eccentricity, and other factors on their bearing capacity; proposed a deformation-based calculation method; and provided design diagrams to provide a basis for engineering design.

At present, the loading devices for applying cyclic loads to the foundation mainly include three types: loading rod cyclic loading, counterweight rotation cyclic loading, and servo motor-controlled cyclic loading. The pull rope loading device controlled by PAC can realize automatic application cycle and loading amplitude. However, the wind turbine foundation cyclic loading test has the characteristics of a long test period, so this paper develops a multidirectional horizontal cyclic loading system for offshore wind turbine foundations. The new loading system can load the test foundation model with multiple complex loading modes such as sinusoidal loading, square wave loading, and load time history curve. Combined with the characteristics of the wind rose diagram, the setting of the loading parameters and loading angle of the bucket foundation is realized through the visual control software. Mono-bucket unidirectional and multidirectional horizontal cyclic bearing capacity model tests are carried out in saturated sand to analyze the influence of multiple factors, such as the number of cyclic loading, vertical load, aspect ratio, cyclic amplitude ratio, and cyclic direction on the mono-bucket cumulative deformation, cyclic stiffness, and post-cyclic bearing capacity. A prediction model for the relationship between the cumulative rotation angle of a mono bucket and the number of cyclic loading, the cyclic load amplitude ratio, and the vertical load is established, and a calculation method for the vertical load and the bearing capacity of the foundation after cyclic loading is established.

2. Cyclic Loading Analysis

2.1. Definition of Cyclic Loading-Related Parameters

Leblanc [17,18] proposed the definition of cyclic load amplitude ratio ζ_b and cyclic load symmetry ratio ζ_c (see the Formula (1)). Most of the related cyclic bearing characteristics of bucket foundations and pile foundations refer to this definition method [11,19,20,21]

$${\mathsf{\zeta }}_{\mathsf{b}}={\displaystyle \frac{{\mathrm{M}}_{\max }}{{\mathrm{M}}_{\mathrm{ult}}}},{\mathsf{\zeta }}_{\mathrm{c}}={\displaystyle \frac{{\mathsf{{\rm M}}}_{\mathsf{\mu }\mathsf{\iota }\mathsf{\nu }}}{{\mathrm{{\rm M}}}_{\mathsf{\mu }\mathsf{\alpha }\mathsf{\xi }}}}$$

(1)

where M_max and M_min are the maximum and minimum values of the moment load in a mono cycle; Mult is the ultimate moment-bearing capacity of the pile foundation in the horizontal monotonic loading test; ζ_b is the ratio of the maximum value of the cyclic load to the horizontal ultimate-bearing capacity, ranging from 0 to 1; and ζ_c is the ratio of the minimum value of the cyclic load to the maximum value, ranging from −1 to 1. When ζ_c < 0, it is horizontal bidirectional cyclic loading, and when ζ_c ≥ 0, it is horizontal unidirectional cyclic loading.

The number of cyclic loads that an offshore wind turbine foundation is subjected to during its entire service life is as high as 107 times. Existing experimental studies have shown that the cumulative deformation of the bucket foundation develops rapidly in the early stage [3,4], and the increase in the number of test cyclic loading will also greatly increase the test time. Combined with existing experimental studies, this test sets the number of loading times to 20,000 times and the loading cycle to 2 s [11]. This study mainly focuses on the comparative analysis of the cyclic loading direction, the aspect ratio of the bucket foundation, and the cyclic bearing characteristics of mono and four buckets and studies the influence of different working conditions on the cumulative rotation angle of the foundation, cyclic stiffness, changes in the position of the foundation rotation point, and monotonic bearing capacity of the bucket foundation after cyclic loading.

For the cumulative rotation of a bucket foundation during loading, LeBlanc [18,22] proposed a dimensionless expression of the horizontal rotation of a mono-pile foundation and proposed a theoretical calculation method. Many scholars have used this method to process test results [19,20,21,23]. The expression is as follows:

$${\displaystyle \frac{{\mathsf{\theta }}_{\mathrm{N}}-{\mathsf{\theta }}_0}{{\mathsf{\theta }}_0}}={\mathrm{T}}_{\mathrm{b}}\times {\mathrm{T}}_{\mathrm{c}}\times {\mathrm{N}}^{\mathsf{\alpha }}$$

(2)

$${\displaystyle \frac{{\mathsf{\theta }}_{\mathrm{N}}-{\mathsf{\theta }}_0}{{\mathsf{\theta }}_0}}=\mathsf{\beta }\times {\mathrm{N}}^{\mathsf{\alpha }}$$

(3)

where θ₀ and θ_N are the maximum rotation angles corresponding to the initial loading and the Nth loading to the maximum load M_max, respectively. T_b and T_c are dimensionless coefficients, which are related to the cyclic load ratio ζ_b and the cyclic load symmetry ratio ζ_c, respectively. α is obtained by regression analysis of test data, and β is simplified from T_b and T_c.

The stiffness calculation method of the bucket foundation is the ratio of the difference between the maximum and minimum values of the horizontal load bending moment of the bucket foundation in a cycle to the difference in the rotation angle in this cycle. The stiffness calculation method of the bucket foundation at the Nth cycle is shown as follows:

$${\mathrm{k}}_{\mathrm{N}}=\left({\displaystyle \frac{{\mathrm{M}}_{\mathrm{maxN}}-{\mathrm{M}}_{\mathrm{minN}}}{{\mathsf{\theta }}_{\mathrm{maxN}}-{\mathsf{\theta }}_{\mathrm{minN}}}}\right)$$

(4)

where k_N is the stiffness of the bucket foundation at the Nth loading cycle; M_maxN and M_minN are the maximum and minimum values of the moment load on the bucket foundation at the Nth loading cycle, respectively; and θ_maxN and θ_minN are the foundation rotation angles corresponding to the maximum/minimum loads, respectively.

2.2. Intelligent Cycle Loading System

In order to ensure accurate loading during the test and to achieve precise control of the applied load, a set of intelligent cyclic loading systems was designed in combination with the load characteristics of the wind turbine foundation, namely the multidirectional intelligent cyclic loading system for offshore wind turbine foundations [24] (Figure 3).

This multidirectional cycle intelligent loading device is mainly composed of computer control system software, PAC control cabinet, and servo loading device. The PAC control system mainly includes motion controller, control card, communication card and motor driver, etc. Its main function is to receive computer setting parameters and convert the setting parameters into signals to achieve precise control of the horizontal displacement and angle of the horizontal servo loading motor and the angle servo motor. The loading system adopts the parameter interface precise closed-loop control, which can realize the precise control of the force loading and displacement loading of the test model. At the same time, the loading device has the function of adjusting the height of the loading point, which can realize the loading of various scale test models.

Table 1. Multidirectional intelligent cyclic loading system parameters.

Parameter Name	Parameter Value	Parameter Name	Parameter Value
Loading amplitude	0~1000 N	Displacement positioning accuracy	0.02 mm
Loading accuracy	0.05 N	Loading height range	700~1700 mm
Loading frequency range	0~10 Hz	Loading angle range	0~300°
Displacement loading interval	0~200 mm	Motor rotation self-locking force	318 N·m
Maximum loading speed	250 mm/s	Angular velocity loading range	0~60°/s

shows the main technical parameters of this test loading device.

In addition, the intelligent loading system has the functions of test data collection and data storage with multiple collection frequencies. Before the test starts, the initial data can be reset to zero, and the collection frequency of the test data can be set. The loading program can record the loading time, load amplitude, displacement amplitude, angle, and number of loading times during the loading process, and the data is saved in the file format of .xls file.

3. Test Equipment and Test Conditions

3.1. Test Soil Preparation and Parameters

The height of the test bucket foundation is 250 mm, and the diameter is 300 mm. The model scale is 1:50, the diameter of the prototype mono-bucket foundation is 15 m, and the height of the bucket skirt is 12.5 m. In actual projects, the wall thickness of the bucket foundation is about 15~35 mm, which is less than 1 mm after reduction, so the production of the test model cannot be realized. Referring to the existing literature research, the material and thickness of the bucket wall can be determined based on the method that the prototype foundation and the model foundation have the same equivalent elastic modulus Ee. After calculation, it can be concluded that when the thickness of the prototype foundation bucket wall is a 25 mm steel plate, the test model bucket wall is made of an aluminum plate with a thickness of 1.5 mm. The top cover of the mono-bucket foundation skirt is made of plexiglass, and four drainage holes are set on the top cover of the bucket foundation. The outlet holes of the pore water pressure gauge and the earth pressure gauge are also set. A 45# high-strength rigid hollow loading rod with a length of 1200 mm is installed on the upper part of the mono-bucket foundation (Figure 4). The diameter of the mono-bucket foundation B1–B3 is 300 mm, the foundation height is 60 mm, 150 mm and 250 mm respectively, the wall thickness is 1.5 mm, and the model weight is 59 N, 61.2 N, 78.4 N respectively.

The main sensors in this test include laser displacement meter, wire displacement meter, weighing sensor, pore water and soil pressure meter, inclinometer, and pressure sensor (Figure 5). In the test, the wire displacement meter is arranged at the same height as the loading point, the inclinometer is arranged at the top of the foundation loading rod, and the pore water pressure meter is located at the top and bottom of the bucket foundation and the four-bucket skirt connection structure. It is used to measure the upper and lower pressure difference of the bucket foundation and the skirt connection structure during foundation sinking and loading. According to the height of the fan head in the actual project and the research on the existing literature, the height of 3.5D is taken as the loading height of the horizontal load for the mono-bucket foundation, and D is the diameter of the bucket foundation. The particle grading curve of Fujian standard sand is used in the test (Figure 6). From the results of the screening test, it can be found that the sand is mainly composed of medium sand particles (particle size range 0.25~0.5 mm) and coarse sand particles (particle size range > 0.5 mm), which account for 37% and 27%, respectively, and the median particle size d₅₀ is 0.17 mm. From the particle grading test curve, the uneven coefficient of sand C_u = 1.81 < 5 and the curvature coefficient C_c = 1.02 > 1 are obtained. Therefore, the sand used in this test is homogeneous fine sand with poor particle grading.

3.2. Cyclic Loading Test Steps and Test Conditions

After completing the test installation of the equipment, the test steps are as follows:

(1)

After completing the sand preparation, complete the suction sinking installation of the bucket foundation B₃ and further install sensors, such as laser displacement meters, and set their parameters.

(2)

Preload the bucket foundation B₃ with a horizontal load of M_max = 180 N·m. After preloading, set the loading mode of the cyclic loading system to the sinusoidal loading mode; set the maximum cyclic load value to 180 N·m, the minimum value to 18 N·m, the test loading period to 2 s, the number of loading cycles to 20,000, and the test data acquisition frequency to 20 Hz during the entire loading process.

(3)

After the cyclic loading is completed, the foundation is unloaded, and the direction of the loading turntable is changed so the loading direction is consistent with the initial loading direction. The monotonic horizontal ultimate bearing capacity test of the bucket foundation is carried out. The test process is consistent with the horizontal monotonic loading test process of the bucket foundation in the previous chapter. The influence of cyclic loading on the monotonic bearing capacity of the foundation is analyzed based on the test results.

When performing multidirectional cyclic loading, first complete the preloading of the basic horizontal load, then enter the number of loading times and angle change in each loading direction, and set the minimum horizontal load of the loading rod during the rotation of the angular motor to ensure the pull rope remains in an extended state during the rotation. In this test, the rotation horizontal load value is set to 2 N. When the cyclic loading system completes the set number of loading times in one direction, it starts to unload to the rotating horizontal load, and then the loading turntable starts to rotate to the next loading direction. After the rotation is completed, the cyclic load loading in this direction is carried out. In this study, the multidirectional loading directions are divided into two-way loading and four-way loading. The relationship between the horizontal load direction and the number of loading times during the loading process is shown in Figure 7. In the multidirectional cyclic loading conditions, except for one-directional cyclic loading condition, the number of cyclic loading times in each direction of the remaining multidirectional loading conditions is 1000.

Table 2. Horizontal cyclic bearing capacity test conditions of mono-bucket suction foundation.

Test No.	Model No.	Cyclic Load Amplitude Ratio ζb	Loading Direction (°)	Drainage Conditions	Vertical Load (N)
CDT311	B3	0.8	0	drain	0
CDT312	B3	0.8	0	drain	40
CDT313	B3	0.8	0	drain	80
CDT314	B3	0.8	0	drain	120
CDT321	B3	0.8	0–180	drain	0
CDT322	B3	0.8	0–180	drain	40
CDT323	B3	0.8	0–180	drain	80
CDT324	B3	0.8	0–180	drain	120
CDT33	B3	0.8	0	No drainage	0
CDT34	B3	0.8	135+45–45–135	drain	0

shows the test conditions of the mono-bucket suction foundation in this study. A total of 10 test conditions were carried out, and the cyclic bearing characteristics of mono-bucket suction foundations of different sizes under different directions and vertical load conditions were studied. In the test, the cyclic load amplitude ratio ζ_b was taken as 0.8, and the cyclic load symmetry ratio ζ_c was taken as 0.1.

4. Unidirectional Horizontal Cyclic Load Characteristics of Mono-Bucket Foundation

4.1. Analysis of Cyclic Cumulative Rotation Angle Changes Under Unidirectional Cyclic Loading

Figure 8 is the relationship curve between the cumulative rotation angle of the mono-bucket foundation B3 under different vertical loads and the number of cyclic loadings. The test results show that the cumulative rotation angle of the bucket foundation under cyclic load increases with the increase of the number of cyclic loadings, and the greater the vertical load, the greater the cumulative rotation angle of the foundation under cyclic load.

According to the normalized cumulative rotation angle and loading number curves of bucket foundations under different working conditions, the relationship between the cumulative rotation angle and the number of cyclic loadings of the mono-bucket foundation under CDT311 working conditions is obtained by regression analysis method, as shown in Formula (5).

$${\displaystyle \frac{\Delta {\mathsf{\theta }}_{\mathrm{N}}}{{\mathsf{\theta }}_0}}=0.036{\mathrm{N}}^{0.36}$$

(5)

The same method is used to fit the normalized cumulative rotation angle and number of cyclic loadings of the mono-bucket foundation under different vertical loads, and the cumulative deformation calculation parameters of the mono-bucket foundation under different working conditions are obtained, as shown in

Table 3. Parameter values of cumulative deformation calculation formula under different working conditions.

Test Conditions	Β	α
CDT311	0.036	0.36
CDT312	0.115	0.25
CDT313	0.067	0.26
CDT314	0.166	0.24

. The α value obtained when the vertical load (the wind turbine head’s own weight) is applied is between 0.24 and 0.26, which is smaller than the value obtained when no vertical load is applied. It is close to the α = 0.28 obtained by Zhu [19], but smaller than the test results obtained by Leblanc [18] (α = 0.31), Cox [25] (α = 0.3), and Zhu [11] (α = 0.39) and larger than the test result parameter value obtained by Foglia [26] (α = 0.18).

Table 4. Loading times and total cumulative rotation angles under different working conditions.

Load Times	CDT311	CDT312	CDT313	CDT314
500	0.45	0.70	0.79	0.68
1000	0.50	0.72	0.82	1.08
5000	0.71	0.84	0.92	1.26
20,000	0.80	0.87	1.10	1.59

shows the cumulative rotation angles corresponding to different numbers of cycles of the bucket foundation. The test results show that when the mono-bucket foundation is loaded 5000 times, the cumulative rotation angle of the foundation has reached more than 80% of the total cumulative rotation angle. The final cumulative rotation angle of the mono-bucket foundation under cyclic loading (the rotation angle corresponding to 20,000 cyclic loading) and the vertical load are dimensionless. The cumulative rotation angle is θ_m/θ₀, and the vertical load of the mono-bucket foundation is m/M, where θ_m is the cyclic cumulative rotation angle of the mono-bucket foundation under different vertical loads, θ₀ is the cyclic cumulative rotation angle of the mono b foundation when the vertical load is 0, m is the vertical load of the mono-bucket foundation, and M is the deadweight of the mono bucket foundation. Based on the above data, the relationship curve between the final cumulative rotation angle of the mono-bucket foundation and the vertical load is shown in Figure 9.

By analyzing the relationship between the cumulative rotation angle and vertical load of the mono-bucket foundation, it is found that the cyclic cumulative rotation angle and the vertical load are approximately nonlinear. The calculation formula for the cumulative rotation angle and the vertical load of the mono-bucket foundation under the action of unidirectional horizontal cyclic load of the mono-bucket foundation is established through data fitting as follows.

$${\displaystyle \frac{{\mathsf{\theta }}_{\mathrm{m}}}{{\mathsf{\theta }}_0}}=0.082{\mathrm{e}}^{1.682({\displaystyle \frac{\mathrm{m}}{\mathrm{M}}})}+0.91$$

(6)

4.2. Analysis of the Change of Rotation Point Position and Cyclic Stiffness Under Unidirectional Cyclic Load

Figure 10 shows the relationship between the rotation point height of the mono-bucket foundation and the number of cycles. During the entire cyclic loading process of working condition CDT311, the rotation point height of the bucket foundation is located at a position 0.45 times the bucket skirt height from the top cover inside the bucket foundation, and the position of the foundation rotation point basically does not change during the cyclic loading process. When there is a vertical load on the upper part of the mono-bucket foundation, the rotation point height shows a linear increase trend within the initial cyclic loading of 1500 times, after which the rotation point height of the bucket foundation stabilizes within the range of 0.5 times the bucket skirt height.

Figure 11 shows the relationship between the horizontal position of the rotation point and the number of cyclic loadings under different vertical load conditions of the mono-bucket foundation. The test results show that after the cyclic load is applied, the horizontal position of the rotation point of the mono-bucket foundation moves backward to a certain extent and remains stable. When the vertical load of the foundation is less than 80 N, the horizontal backward distance of the rotation point is less than 0.25D. When the vertical load of the mono-bucket foundation is 120 N, the horizontal backward distance of the rotation point is about 0.5D. Therefore, when the vertical load of the mono-bucket foundation is less than the deadweight of the foundation, the horizontal position of the rotation point of the bucket foundation basically does not change; when the vertical load exceeds 0.5 times the deadweight of the foundation, the horizontal backward displacement of the rotation point of the bucket foundation increases.

Figure 11a shows the relationship between the foundation cyclic stiffness and the number of cyclic loadings under different working conditions of mono-bucket foundation cyclic load. The test results show that the cyclic stiffness of the mono-bucket foundation in sand increases first and then decreases, and the greater the vertical load of the foundation, the greater the peak stiffness value. When there is no upper vertical load on the foundation, the peak stiffness appears earlier than when there is a vertical load. In the CDT311 working condition, when the foundation cyclic loading number is about 500 times, the cyclic stiffness of the bucket foundation reaches the peak value and then decreases rapidly; when the foundation has a vertical load, the cyclic stiffness of the bucket foundation reaches the peak value when the cyclic loading reaches 2000 times, and then the foundation stiffness decreases. When the mono-bucket foundation is cyclically loaded 5000 times, the cyclic stiffness of the bucket foundation tends to be stable, and the cyclic stiffness of the mono-bucket foundation shows the rule that the cyclic stiffness decreases as the vertical load increases. In addition, in the stable stage of cyclic stiffness, except for working condition CDT314, the cyclic stiffness values of the bucket foundation in other working conditions are higher than the foundation stiffness in the initial loading stage.

Figure 11b shows the result of normalizing the cyclic stiffness of the mono-bucket foundation. When the bucket foundation has no upper load, the cyclic stiffness of the bucket foundation reaches 1.15 times the initial stiffness after 5000 cycles of load, and the cyclic stiffness of the bucket foundation no longer changes. When the mono-bucket foundation has a vertical load, the cyclic stiffness of the mono-bucket foundation tends to be stable after 5000 cycles of load. The cyclic stiffness of CDT312 and CDT313 foundations are 1.45 and 1.20 times the initial stiffness, respectively, while the cyclic stiffness of CDT314 is about 0.83 times the initial stiffness. The test results show that when the vertical load of the mono-bucket foundation is less than 1.5 times the deadweight of the foundation, the cyclic stiffness of the foundation is greater than the initial stiffness; when the vertical load exceeds 1.5 times the deadweight of the foundation, the cyclic stiffness of the mono-bucket foundation will decrease.

4.3. Analysis of Monotonic Horizontal Bearing Capacity Results After Unidirectional Cyclic Loading

Figure 12 shows the relationship between the horizontal monotonic bearing capacity and the rotation angle of the cylindrical foundation under different vertical loads when cyclic loads are applied and when cyclic loads are not applied. According to the test results, it was found that the horizontal bearing capacity of the mono-bucket foundation after cyclic loading was significantly higher than that without horizontal cyclic load. In addition, after cyclic loading of different vertical loads, the development pattern of the relationship between monotonic horizontal load and rotation angle of the mono-bucket foundation was found, which is basically the same.

Figure 12 shows the horizontal monotonic bearing capacity and rotation angle relationship of the bucket foundation under different vertical loads when cyclic loads are applied and when cyclic loads are not applied. According to the test results, the horizontal bearing capacity of the mono-bucket foundation after cyclic loads is significantly higher than that without horizontal cyclic loads. In addition, after cyclic loads with different vertical loads, it is found that the development law of the monotonic horizontal load and rotation angle relationship of the mono-bucket foundation is basically the same.

Table 5. Ultimate bearing capacity of mono-bucket foundation under different working conditions.

Test Conditions	DT30	CDT311	CDT312	CDT313	CDT314
Horizontal ultimate bearing capacity (N m)	39.4	53.5	55.2	54.3	53.5
Ultimate bearing capacity ratio	1	1.36	1.40	1.38	1.36

shows the horizontal ultimate bearing capacity and ultimate bearing capacity ratio of the bucket foundation under different working conditions. The bearing ratio of the monotonic horizontal ultimate bearing capacity of the working condition DT30 is defined as 1. The test results show that the horizontal ultimate bearing capacity of the mono-bucket foundation increases significantly after cyclic loading, and its value is 1.36 to 1.40 times the bearing capacity under horizontal monotonic conditions. This is mainly due to the cyclic loading of the sand around the mono-bucket foundation. It is caused by the significant increase in post-density, while the difference in the horizontal ultimate bearing capacity values after cyclic loading of the foundation under different vertical loads is small. The test results show that the vertical load has basically no effect on the horizontal bearing capacity of the bucket foundation after cyclic loading.

4.4. Unidirectional Cyclic Load-Bearing Characteristics of Mono-Bucket Foundation in Undrained State

Figure 13a shows the relationship between the foundation cumulative rotation angle and the number of foundation cyclic loadings when the mono-bucket foundation is unidirectionally cyclically loaded in the undrained and drained states. When the foundation is in an undrained state, cyclic cumulative pore pressure is generated inside the mono-bucket foundation, and the cyclic cumulative rotation angle of the mono-bucket foundation is significantly greater than the drainage condition. When the number of cyclic loadings is 5000, the cumulative angle of the undrained cycle condition CDT33 is approximately twice the cumulative angle of the drained cycle condition CDT311; however, during the entire loading process, the angle change of the working condition CDT33 is smaller than that of the working condition CDT311 during cyclic loading. The accumulated rotation angle changes during the cycle. Figure 13b shows the relationship between the normalized cumulative rotation angle and the number of cyclic loadings. According to the relationship between the two curves, it can be found that under undrained conditions, the cumulative rotation angle of the mono-bucket foundation in the initial loading stage tends to be stable after 5000 cyclic loadings; in the drainage condition, the cumulative rotation angle of the mono-bucket foundation still has an increasing trend after 5000 cyclic loadings. The test results show that at the end of the cyclic loading, the cyclic cumulative deformation in the undrained cyclic condition is 47.5% lower than the cumulative angle in the drainage condition.

According to the test results of the mono-bucket foundation undrained cycle, the relationship between the cumulative normalized cumulative rotation angle and the number of cycles of the mono-bucket foundation during the undrained cycle is as follows:

$${\displaystyle \frac{\Delta {\mathsf{\theta }}_{\mathrm{N}}}{{\mathsf{\theta }}_0}}=0.033{\mathrm{N}}^{0.31}$$

(7)

Figure 14 shows the relationship between the cyclic stiffness and normalized cyclic stiffness of the mono-bucket foundation and the number of cyclic loadings of the foundation in the CDT33 and CDT311 working conditions, respectively. The results in Figure 14a show that the cyclic stiffness of the mono-bucket foundation in working conditions CDT311 and CDT33 is basically the same in the initial cyclic loading stage. In the later loading stage, the cyclic stiffness of the foundation in CDT311 working condition is greater than that in CDT33 working condition. The results in Figure 14b show that during the entire loading process, the cyclic stiffness of the undrained cyclic working condition CDT33 remains basically stable; while in the cyclic loading of the drainage working condition CDT311, the cyclic stiffness of the foundation fluctuates with the number of loadings during the entire loading process, and the cyclic stiffness increases during the cyclic loading process.

Figure 15 shows the relationship between the rotation point height and the number of cyclic loadings for the two working conditions. The test results show that the rotation point of the mono-bucket foundation in both working conditions moves upward during the cyclic loading process. The rotation point of the undrained cyclic condition CDT33 is closer to the top cover of the foundation, and the change of the center position of its rotation point during the cyclic loading process is less than that of the drainage condition. The rotation point of the drainage condition rises from the end of the bucket skirt to 0.4 times the height of the bucket skirt within 2500 cyclic loadings after the cyclic loading occurs, and then the height of its rotation point changes in a basically consistent manner with the undrained condition. The test results show that in the undrained state, the stiffness of the upper soil of the foundation is increased due to the excess pore pressure generated by the soil in the bucket; in the undrained state, the stiffness of the upper soil increases with the increase in the number of cyclic loadings, and finally, the stiffness of the soil in the bucket of the two working conditions tends to be consistent.

Figure 16 and Figure 17, respectively, show the relationship between the pore water pressure value and the cyclic loading time under the cylindrical foundation top cover of the CDT33 mono-bucket foundation under the undrained working condition in the initial cyclic loading stage, as well as the maximum and minimum pore water under the foundation top cover during the entire cyclic loading process. Relationship between pressure and number of cycles: In the initial loading stage, the maximum and minimum pore water pressures at the lower part of the bucket foundation roof both increase as the loading time increases. The cyclic cumulative pore pressure normally occurs and continues to increase. The maximum pore water pressure value is approximately five times the minimum value. The generation of accumulated positive pore pressure causes a layer of water film to be produced in the soil below the sea area of the foundation roof. Therefore, the cyclic accumulation angle of the bucket foundation in the working condition CDT33 is significantly larger than that in the CDT11 working condition. According to the results shown in Figure 17, it is shown that after 5000 cyclic loadings of the mono-bucket foundation, the maximum/minimum cumulative pore pressure at the bottom of the top cover of the bucket foundation tends to be stable. The maximum pore water pressure value is 0.433 kPa, and the minimum pore water pressure value is −0.06 kPa.

Figure 18 shows the monotonic horizontal bearing capacity bending moment rotation curve of the bucket foundation under different working conditions. The test results show that the monotonic bearing capacity of the bucket foundation under cyclic loading in the undrained working condition is greater than that under the drainage condition. The horizontal ultimate bearing capacity values of the mono-bucket foundation under different working conditions are shown in

Table 6. Ultimate bearing capacity of mono-bucket foundation under different working conditions.

Test Conditions	DT30	CDT311	CDT33
Horizontal ultimate bearing capacity (N·m)	39.4	53.5	58
Ultimate bearing capacity ratio	1	1.36	1.47

. The test results show that the horizontal bearing capacity of the mono-bucket foundation after cyclic loading in the undrained working condition is 9.5% higher than that under the drainage condition and 47% higher than that under the monotonic horizontal loading condition when no cyclic loading is applied.

5. Multidirectional Horizontal Cyclic Load-Bearing Characteristics of Mono-Bucket Foundation

5.1. Analysis of Cyclic Cumulative Rotation Angle Changes Under Multidirectional Cyclic Loading

In view of the influence of cyclic loading direction on the cyclic bearing characteristics of mono-bucket foundation, the mono-bucket foundation was subjected to bidirectional symmetrical loading and four-directional symmetrical loading, respectively. The bidirectional loading of the mono-bucket foundation is symmetrical loading in the directions of 0 degrees and 180 degrees, and the number of loading times in each direction is 1000 times; the four-directional loading of the mono-bucket foundation is loading the foundation in directions with an interval of 90 degrees, and the number of loading times in each direction is 1000 times.

Figure 19 shows the relationship between the cumulative angle of the mono-bucket foundation and the number of loadings under two-way and four-way cyclic loads. The cumulative rotation angle in the figure is the cumulative rotation angle in the same direction as the initial direction. The test results show that under multidirectional cyclic load conditions, the cumulative rotation angle of the foundation is smaller than that under unidirectional cyclic load conditions. Under the same number of cyclic loadings, the cumulative rotation angle of the foundation under vertical loading conditions is smaller than that under four-way loading conditions. Taking the nineteen thousandth loading cycle as an example, the cumulative rotation angle under unidirectional loading conditions is 0.86 degrees, the cumulative rotation angle under bidirectional loading conditions is 0.27 degrees, and the cumulative rotation angle under four-way loading conditions is 0.39 degrees. The bidirectional loading and four-way loading conditions are 69% and 55% lower than those under unidirectional loading conditions, respectively. Therefore, the cumulative rotation angle of the bucket foundation under multidirectional cyclic loading is smaller than that under unidirectional cyclic loading conditions; in multidirectional loading conditions, the cyclic loading history has a greater impact on the change of the cumulative rotation angle, and the cumulative rotation angle generated by two-way symmetrical cyclic loading is smaller than that of the four-way cyclic loading condition under the same number of loadings.

Figure 19 shows the relationship between the normalized cumulative rotation angle and the number of loading times of the mono-bucket foundation at the end of each cycle under two-way and four-way cyclic loads (as shown in Figure 20). The normalized cumulative rotation angle and the number of cyclic loading times have the following relationship. Mono-bucket foundation circulation bidirectional circulation working conditions:

$${\displaystyle \frac{\Delta {\mathsf{\theta }}_{\mathrm{N}}}{{\mathsf{\theta }}_0}}=-1.48\ln (0.092\ln (\mathrm{N}))$$

(8)

Mono-bucket foundation circulation four-way circulation working conditions:

$${\displaystyle \frac{\Delta {\mathsf{\theta }}_{\mathrm{N}}}{{\mathsf{\theta }}_0}}=-2.44\ln (0.112\ln (\mathrm{N}))$$

(9)

5.2. Analysis of Cyclic Stiffness Changes Under Multidirectional Cyclic Loading

Figure 21 shows the relationship between the foundation stiffness and the number of loadings during the multidirectional cyclic loading of a mono-bucket foundation. Figure 21a shows that the intermediate cyclic stiffness of the bucket foundation during the bidirectional cyclic loading process fluctuates around the unidirectional cyclic stiffness value, but the cyclic stiffness is not less than the initial cyclic stiffness (as shown in Figure 21b); the cyclic stiffness during the four-directional cyclic loading process is less than the unidirectional cyclic stiffness, and the cyclic stiffness increases first and then decreases and tends to be stable. As shown in Figure 21b, the final cyclic stiffness value is close to the initial stiffness value.

The normalized stiffness of the mono-bucket foundation increased from 1.0 to 1.12 before 500 unidirectional cycles; after 500 cyclic loadings, the foundation cyclic stiffness remained basically unchanged. For multidirectional cyclic loading conditions, the cyclic stiffness of the foundation in the four-directional cyclic loading condition remained basically unchanged after 7000 cyclic loadings, and its normalized stiffness was about 1.08. In this test, the cyclic stiffness of the mono-bucket foundation under bidirectional cyclic loading showed some fluctuations, and the final normalized stiffness value was 1.10.

5.3. Analysis of Monotonic Horizontal Results of Foundation After Multidirectional Cyclic Loading

Figure 22 is the load rotation curve of the monotonic horizontal ultimate bearing capacity test after different cyclic loading tests of the mono-bucket foundation.

Table 7. Ultimate bearing capacity of mono-bucket foundation after horizontal cycles.

Test Conditions	DT30	CDT311	CDT321	CDT34
Horizontal ultimate bearing capacity (N·m)	39.4	53.5	50	47
Ultimate bearing capacity ratio	1	1.36	1.27	1.20

shows the corresponding horizontal ultimate bearing capacity and ultimate bearing capacity ratio under different working conditions, where the monotonic horizontal ultimate bearing capacity ratio of the mono-bucket foundation is defined as 1.

The horizontal ultimate bearing capacity of the foundation under the three cyclic loading conditions is 1.36 times, 1.27 times, and 1.19 times that of the mono-bucket foundation B3 monotonic horizontal loading condition, respectively. Under the same total number of cycles, the ability of mono-direction cyclic loading to improve bearing capacity is greater than that of multidirectional cyclic loading. The test results show that the horizontal ultimate bearing capacity of the mono-bucket foundation after two-direction cyclic loading (CDT321) is greater than that of four-direction cyclic loading (CDT34) and horizontal monotonic loading DT30 conditions, indicating that the horizontal ultimate bearing capacity of the mono-bucket foundation in a certain loading direction under multidirectional cyclic loading is related to the cumulative number of cycles of the mono-bucket foundation in that direction. The greater the number of cycles, the denser the sand, and the greater the horizontal bearing capacity of the bucket foundation after the cycle.

5.4. Analysis of Bidirectional Cyclic Bearing Characteristics of Mono-Bucket Foundation Under Different Vertical Loads

Figure 23 shows the relationship between the cyclic cumulative rotation angle of the mono-bucket foundation and the number of cyclic loadings under different vertical loads. In the first loading direction loading stage, due to the vertical load on the upper part of the foundation, the cumulative rotation angle of the mono-bucket foundation is significantly greater than that without vertical loads. Vertical load conditions (CDT321 and CDT311) and the greater the vertical load, the greater the initial cumulative rotation angle of the foundation. When a foundation undergoes a reverse loading cycle, its cumulative rotation angle decreases significantly. Except for the working condition CDT323, the cumulative rotation angles of the other test conditions are consistent with the cumulative rotation angles of the one-way cyclic loading condition. After the second loading direction alternation, the cumulative rotation angle of the two-way cyclic loading condition was significantly smaller than the cumulative rotation angle of the one-way cycle. After that, the cumulative rotation angle gradually decreased and became stable. The test results show that the cyclic cumulative deformation of the foundation increases with the action of the two-way cyclic loading. The vertical load increases and decreases.

Figure 24 is a curve showing the relationship between the cyclic stiffness of the bucket foundation under bidirectional cyclic load and the number of cyclic loadings. The cyclic stiffness curves of the foundation under different vertical load conditions show a trend of first increasing and then decreasing, and the cyclic stiffness values are all near the cyclic stiffness curve of the CDT311 condition, indicating that the vertical load has little effect on the foundation stiffness. Except for condition CDT322, the cyclic stiffness of the foundation during cyclic loading is greater than the initial stiffness.

Figure 25 shows the bending moment and angle relationship curve of the mono-bucket foundation under different working conditions. It is obvious that the monotonic bearing capacity of the mono-bucket foundation is significantly improved after cyclic loading. The bearing capacity limit values and ultimate bearing capacity ratios corresponding to different working conditions are as shown in

Table 8. Ultimate bearing capacity of mono-bucket foundation after horizontal cycles.

Test Conditions	DT30	CDT311	CDT321	CDT322	CDT323	CDT324
Horizontal ultimate bearing capacity (N·m)	39.4	53.5	50	55	60	62
Ultimate bearing capacity ratio	1	1.36	1.27	1.40	1.52	1.65

. As shown in the figure, the monotonic ultimate bearing capacity ratio of the foundation under the DT30 working condition of the mono-bucket foundation is defined as 1. Under the two-way cyclic working condition, the increase in vertical load increased the monotonic bearing capacity of the foundation after cyclic loading. The bearing capacity of working conditions CDT322, CDT323, and CDT324 increased by 10%, 20%, and 24%, respectively, compared with CDT321.

Figure 26 shows the relationship between the monotonic bearing capacity and vertical load of a mono-bucket foundation after bidirectional cyclic load under different vertical loads. Obviously, the vertical load has a linear relationship with the monotonic bearing capacity of the foundation after bidirectional cycle, and the relationship is as follows:

$${\displaystyle \frac{{\mathrm{M}}_{\mathrm{G}}}{{\mathrm{M}}_0}}=0.247{\displaystyle \frac{{\mathrm{m}}_{\mathrm{G}}}{\mathrm{m}}}+1.27$$

(10)

where M_G is the bearing capacity after bidirectional cycles under different vertical loads, M₀ is the monotonic bearing capacity of the mono bucket foundation, m_G is the vertical load, and m is the deadweight of the foundation.

The above-mentioned cyclic test research results of mono-bucket foundations show that the monotonic bearing capacity of mono-bucket foundations significantly increases after being subjected to cyclic loads in sand foundations. It increases by more than 30% after unidirectional cyclic loads and by more than 30% under multidirectional cyclic loads. Increased by more than 20%. When the bucket foundation is subjected to multidirectional cyclic load, the cumulative rotation angle of the foundation is reduced by more than 50% compared with the one-way cyclic condition. In the practical application of bucket foundation, the design can be carried out based on factors such as soil conditions and with reference to the research results of this paper.

6. Conclusions

This paper studies the cyclic loading characteristics of the mono-bucket foundation under unidirectional and multidirectional cyclic loading conditions and analyzes the influence of loading direction, vertical load and drainage state on the cumulative rotation angle, cyclic stiffness, rotation point position, and monotonic bearing capacity of the mono-bucket foundation during cyclic loading. The main research conclusions are as follows:

(1)

Under the action of unidirectional cyclic loading, the density of the sand around the foundation increases significantly, and the bearing capacity of the mono-bucket foundation can be increased by more than 30% after cyclic loading. The cyclic cumulative rotation angle of the mono-bucket foundation increases with the increase of vertical load. A calculation formula for the normalized cumulative rotation angle and the number of cyclic loadings of the mono-bucket foundation under different vertical loads was established. During the unidirectional cyclic loading process, the cyclic stiffness of the bucket foundation first increases and then decreases and becomes stable. In the stable stage of cyclic stiffness, the cyclic stiffness of the foundation decreases as the vertical load of the foundation increases. The height of the rotation point of the bucket foundation basically does not change during the cyclic loading process, and the height is within 0.5 times the height of the bucket skirt at the bottom of the foundation top cover; the horizontal position of the rotation point of the bucket foundation is basically stable during the cyclic loading process, and its position is located within the range of 0.25D behind the center of the bucket foundation; when the vertical load exceeds 1.5 times the self-weight of the foundation, the horizontal rotation point position is approximately 0.5D behind the center of the foundation.

(2)

Under the undrained cyclic loading condition of the mono-bucket foundation, an asymmetric cumulative pore pressure is generated under the foundation during the cyclic loading process, and the cumulative pore pressure is basically stable after 5000 cyclic loadings. Under this condition, the final cumulative rotation angle is 47.5% lower than that under the drainage condition, and the relationship between the normalized cumulative rotation angle of the mono-bucket foundation under the undrained condition and the number of cyclic loadings is established. The cyclic stiffness of the mono-bucket foundation under the undrained unidirectional cyclic condition is slightly lower than that under the drainage condition but remains stable during the cyclic loading process. The height of the rotation point of the mono-bucket foundation under the undrained condition develops rapidly from the bottom of the foundation to the top after the cyclic loading begins, until it stabilizes at 0.4 times the height of the bucket skirt. During the entire loading process, the height of its rotation point is lower than that under the drainage cyclic loading condition.

(3)

The results of the study on the bidirectional and four-way cyclic loading of the mono-bucket foundation show that the bidirectional and four-way symmetrical cyclic loading reduces the cumulative rotation of the bucket foundation during the cyclic loading process, and the reduction can reach more than 50%. The relationship between the cumulative rotation and the number of cyclic loading during bidirectional and four-way cyclic loading is established. The stiffness of bidirectional cyclic loading is greater than that of unidirectional cyclic loading, but the stiffness of four-way cyclic loading is slightly less than that of unidirectional cyclic loading. The four-way cyclic loading condition produces a stiffness degradation phenomenon. After multidirectional cycles, the horizontal bearing capacity of the foundation increases by more than 20%. The horizontal ultimate bearing capacity of the mono-bucket foundation in a certain loading direction under multidirectional cyclic loading is related to the cumulative number of cycles of the mono-bucket foundation in that direction. The greater the number of cycles, the greater the horizontal bearing capacity in that direction.

(4)

The results of two-way cyclic loading tests under different vertical loads show that before the first cyclic load turns, the vertical load increases the cyclic cumulative rotation angle of the foundation, but during the subsequent loading process, the cyclic cumulative rotation angle increases with the vertical load and decreases as the load increases. When the vertical load on the upper part of the foundation increases, the density of the soil around the bucket foundation increases significantly, and the bidirectional cyclic stiffness of the bucket increases with the increase of the vertical load. The test results of the horizontal monotonic bearing capacity after two-way cyclic loading show that the vertical load increases the monotonic bearing capacity of the mono-bucket foundation after cyclic loading by more than 10%. The horizontal bearing capacity after cyclic loading has a linear relationship with the vertical load of the foundation. It is established based on the test results. The relationship between the horizontal bearing capacity and the vertical load of the mono-bucket foundation after the bidirectional cyclic load is applied.