Numerical Study on the Responses of Suction Pile Foundations under Horizontal Cyclic Loading Considering the Soil Stiffness Degradation

Abstract

This paper analyzes the deformation responses of single-cylinder and four-cylinder suction pile foundations in soft clay under horizontal cyclic loading. Based on the existing stiffness degradation model of soft clay, a more suitable model for marine soft clay is proposed by carrying out dynamic triaxial experiments under different stress levels. The stiffness degradation and cumulative displacement of soil under cyclic loading are implemented in ABAQUS by compiling a USDFLD subroutine. The rationality is verified by comparison with the displacement response of the pile measured by the centrifugal model test and the scaled model test reported in the literature. With the aid of the current model, the stress state and deformation of suction pile foundations under horizontal cyclic load are further analyzed. The numerical results reveal the path of soil stiffness degradation and the deformation response of foundations during horizontal cycling. The cumulative displacement generated by the cyclic load is quantitatively displayed. It is proved that this model has the capability to reflect the cumulative displacement under cyclic loading and shows promise in analyzing the long-term deformation response of offshore foundations.

1. Introduction

With the proposal of the strategy to develop marine resources, the bearing capacity of single-cylinder foundations has been unable to meet the design requirements, which makes the multi-cylinder composite structure widely used. In the extreme marine environment, a foundation will not only be subjected to the working load transmitted by the structure but also to the low-frequency cyclic loading caused by the waves and wind. It is of great significance to evaluate the stability and deformation of the foundation under cyclic loading by using reasonable methods.

There are many researchers who have carried out a series of experimental studies on the response of a foundation under cyclic loading [1,2,3]. Peralta and Achmus [4] conducted horizontal cyclic load tests on rigid and flexible piles under 1 g conditions, respectively. They studied the effects of the cyclic load amplitude and frequency on the horizontal cumulative deformation of the foundation. Liao et al. [5] conducted a series of 1 g cyclic loading model tests to analyze the mechanical characteristics of the flexible pile foundation in over-consolidated soft clay under lateral cyclic loading. They measured the cyclic deflection change in the pile head and the degradation of system stiffness. Zhu et al. [6] carried out a high-gravity model test of large-diameter single pile under cyclic loading. They obtained the p-y curves of cyclic loads and introduced cyclic degradation factors to account for cyclic effects.

The classical constitutive models can only reflect the plastic strain generated by the first loading, such as the Mohr–Coulomb model, Drucker–Prager model, modified Cam–Clay model, and so on. These models cannot reflect the cumulative responses of soil under long-term multiple cyclic loading. To this end, researchers used advanced constitutive models to calculate the cyclic responses of soil [7,8]. Cheng and Wang [9] simulated the cyclic responses of the foundation by establishing an elastic–plastic boundary surface model and implementing it into the ABAQUS program. Pasten et al. [10] and Chong [11] used a hybrid model to track the accumulation of plastic deformation during cyclic loading by combining the modified Cam–Clay model and the semi-empirical cumulative function. They used ABAQUS to analyze the long-term response of single pile foundations and shallow foundations under cyclic loading. Hong et al. [12] used the advanced hypoplastic constitutive model in the numerical study to analyze the horizontal cyclic loading responses of semi-rigid piles in soft clay. Carstensen et al. [13] also used the hypoplastic model to simulate the degradation of soil stiffness and the cumulative displacement under cyclic loading, but only the case of lower number of cycles can be considered.

Advanced constitutive models can reflect the responses of soil under cyclic load to a certain extent, but the complexity and accuracy of model parameters will seriously affect the computational efficiency. Based on the existing stiffness degradation model [14], a more suitable model for marine soft clay is proposed by carrying out dynamic triaxial experiments under different stress levels. By compiling a USDFLD subroutine, it is implemented into the ABAQUS main program. The concise constitutive model based on the Tresca yield criterion is used so that the soil parameters can be determined by the basic soil mechanics test in the laboratory. This avoids the complex parameters used in the advanced constitutive model, which can greatly improve the calculation efficiency. Based on this, the above soil stiffness degradation model is innovatively applied to the cyclic loading deformation calculation of single-cylinder and four-cylinder suction pile foundations. The cumulative deformation of the soil is analyzed. The deformation characteristics of the suction pile foundations under horizontal cyclic loading are displayed. The main contribution of this paper is to provide a simple calculation method and reference for the cyclic bearing characteristics of offshore suction pile foundations.

2. Stiffness Degradation Model

Relevant research results show that the stiffness of soft clay will degrade under cyclic loading, and the degree of degradation is affected by the initial static stress and cyclic stress level of the soil element. Based on the results of strain-controlled cyclic triaxial tests, Idriss et al. [15] proposed an exponential expression of the soil softening index and the number of cyclic loadings, which quantitatively characterizes the attenuation rule of the soft clay modulus. However, the above studies do not reflect the influence of the initial deviatoric stress.

Based on the expression proposed by Idriss et al. [15], Cheng et al. [14] carried out a series of dynamic triaxial tests. They introduced the cyclic stress level parameter and fitted the relationship between the soil degradation coefficient $\xi$ and the number of stress cycles N. The expression is:

$$\xi =N^{-\lambda }$$

(1)

$$\lambda =0.3353S,$$

(2)

where $\lambda$ is the degradation parameter, which is related to the static and cyclic stress level of the soil; N is the number of stress cycles; S is the stress cycle parameter, which is expressed as the parameter of initial static stress and cyclic stress, as follows:

$$S=\frac{{\tau }_{8,cy}}{{\tau }_{8,f}}(1-0.5\frac{{\tau }_{8,a}}{{\tau }_{8,f}}),$$

(3)

where ${\tau }_{8,a}$ is the initial octahedral static shear stress of the soil; ${\tau }_{8,f}$ is the peak octahedral shear stress when the soil is destroyed; ${\tau }_{8,cy}$ is the octahedral cyclic shear stress of the soil. ${\tau }_{8,a}/{\tau }_{8,f}$ represents the octahedral static shear stress ratio; ${\tau }_{8,cy}/{\tau }_{8,f}$ represents the octahedral cyclic shear stress ratio. According to the weakening rule of the elastic modulus obtained above, the initial elastic modulus is set to be $E_0$. After N cycles, the elastic modulus after degradation is:

$$E_N=\xi E_0$$

(4)

$$\xi =N^{-0.3353\frac{{\tau }_{8,cy}}{{\tau }_{8,f}}\left(1-0.5\frac{{\tau }_{8,a}}{{\tau }_{8,f}}\right)}.$$

(5)

Idriss et al. [15] defined the cyclic degradation coefficient as:

$$\xi =\frac{{\epsilon }_{\max }^1-{\epsilon }_{\min }^1}{{\epsilon }_{\max }^N-{\epsilon }_{\min }^N},$$

(6)

where the superscript represents the number of stress cycles; ${\epsilon }_{\max }$ and ${\epsilon }_{\min }$ are the maximum and minimum axial strain of the N_th cyclic loading, respectively; ${\epsilon }_{\max }-{\epsilon }_{\min }$ represents the secant modulus of the N_th cyclic loading curve, as shown in Figure 1.

To further apply the exponential degradation expression to the practical offshore engineering, the degradation coefficient proposed by Idriss et al. [15] and Cheng et al. [14] should be modified based on the characteristics of marine soft clay. To this end, dynamic triaxial experiments under different stress levels based on stress control were carried out in our lab by using clay samples taken from the central zone in the South China Sea. The initial static deviatoric stress was applied to the sample first. After the deformation was relatively stable, a 0.1 Hz sinusoidal constant amplitude cyclic deviatoric stress was applied. Considering the influence of the initial deviatoric stress and referring to the definition of the degradation coefficient, the test scheme is formulated as shown in

Table 1. The scheme of cyclic triaxial experiments.

Test Number	Depth (m)	Unit Weight (kN/m3)	${\mathit{\tau }}_{8,\mathit{a}}$	${\mathit{\tau }}_{8,\mathit{c}\mathit{y}}$	${\mathit{\tau }}_{8,\mathit{a}}/{\mathit{\tau }}_{8,\mathit{f}}$	${\mathit{\tau }}_{8,\mathit{c}\mathit{y}}/{\mathit{\tau }}_{8,\mathit{f}}$	N
T09-1	15.6	18.62	0	37.85	0	0.53	450
T09-2	15.5	18.69	0	49.2	0	0.69	40
T11-1	15.38	18.83	0	58.4	0	0.85	9
T11-2	18.48	18.52	43.8	21.3	0.636	0.31	100
T11-3	18.7	18.29	43.8	47.45	0.636	0.69	40

.

Through the test results, the relationship between the degradation coefficient and the number of cycles under different stress levels was obtained. The relationship between the degradation coefficient and the number of cycles was fitted, as shown in Figure 2.

The evolution rule between the cyclic degradation parameter $\lambda$ and the cyclic stress level parameter S under different stress levels is summarized, and the coefficient recommended by Cheng et al. [14] is modified. As shown in Figure 3, $\lambda$ increases linearly with S under different octahedral static shear stress ratios, which can be fitted by the following formula:

$$\lambda =0.3058S$$

(7)

Through the results of the dynamic triaxial test, the degradation parameter $\lambda$ of clay samples from the central zone in the South China Sea was modified. It was found that the improved $\lambda$ value is smaller, which means that the stiffness degradation coefficient of marine clay decreases more slowly under the same cyclic stress level and number of cycles. It shows that its structure is relatively stronger. The current expression could reflect the stiffness degradation characteristics of marine clay in the central South China Sea under cyclic loading more specifically.

Therefore, Equation (5) can be expressed as:

$$\xi =N^{-0.3058\frac{{\tau }_{8,cy}}{{\tau }_{8,f}}\left(1-0.5\frac{{\tau }_{8,a}}{{\tau }_{8,f}}\right)}.$$

(8)

By compiling the USDFLD subroutine, the stiffness degradation model of marine soft clay was applied to ABAQUS. The function of the subroutine is to capture the stress state of the soil after each cycle and calculate the degradation coefficient according to the above stiffness degradation model. The field variable is used as the bridge between the degradation coefficient and the elastic modulus, so that the iterative calculation is implemented into the main program.

3. Validation of the Stiffness Degradation Model against the Horizontal Cyclic Loading Test of the Pile

In this paper, the rationality is verified by comparing with the displacement responses of the pile measured by the centrifugal model test and the scaled model test published in journals. Selected tests include the centrifuge test reported by Khemakhem [16] and the scaled model test reported by Liao et al. [5]. Firstly, the experimental scheme is briefly introduced, and then the cyclic displacement response simulation is described using the developed numerical simulation method. The main parameters of pile and soil used in the analysis are summarized in

Table 2. Pile and soil parameters.

	Parameters	Khemakhem’s Centrifuge Test	Liao’s Scaled Model Test
Pile	Diameter, D (m)	0.954	0.05
	Length, L (m)	18	0.7
	Wall thickness, T (mm)	40	5
	Elastic modulus (GPa)	74	3.8
Soil	Undrained shear strength su (kPa)	1.17 z + 1.27	0.034 z + 1.1
	Elastic Modulus (kPa)	1000 su	1000 su
	Effective unit weight ${\mathsf{\gamma }}^{\prime }$ (kN/m3)	7	7

.

3.1. Khemakhem’s Centrifuge Test

Khemakhem [16] carried out several centrifugal model experiments of piles under cyclic load in kaolin foundations. The unidirectional sinusoidal cyclic loading was used. The total number of cycles was 1000. As shown in Figure 4, the evolution of the pile top displacement calculated in this work was compared with the test results. It can be observed that compared with the test results, the pile top displacement calculated in this work is smaller, but the overall developmental trend is consistent. This is because the parameters of this model are based on the dynamic triaxial tests of the undisturbed South China Sea clay sample. Compared with the reconstituted kaolin, the undisturbed South China Sea marine clay has a stronger structure, which may be the reason why it is relatively insensitive to cyclic loading.

3.2. Liao’s Scaled Model Test

Liao et al. [5] conducted several horizontal cyclic loading model tests of a single pile in in situ marine soft clay. The one-way cyclic loading method was used. A total of 5000 cycles were loaded in the test. Considering the computational cost and efficiency, a total of 1000 cycles were calculated in this work. As shown in Figure 5, the change in the pile top displacement with the number of cycles calculated in this work was compared with the test results. At the initial stage of loading, the calculation results are slightly lower than the test results. In general, the calculated results are in good agreement with the experimental results, indicating that the model used in this work can reasonably reflect the response of marine soft clay under long-term cyclic loading.

4. Numerical Examples

4.1. Single-Cylinder Suction Pile Foundation

4.1.1. Numerical Model

The FE model of the single-cylinder suction pile foundation was established as shown in Figure 6. The diameter and the wall thickness of all models was set to 10 m and 25 mm respectively, which is the usual size of a steel cylinder foundation [17]. The whole system was modeled using a semi-structured method, and the symmetric boundary was applied on the symmetric plane. The vertical and horizontal displacements were constrained by the bottom boundary of the soil, while the horizontal displacement was constrained by the side boundary [18]. The three-dimensional hexahedron eight-node complete integral element was used, which is widely used in the relevant finite element numerical calculation [19,20,21]. Based on the mesh size sensitivity analysis, the minimum size of the element was determined to be 0.1 D. The mesh size at the edge of the model increased slightly to improve the computational efficiency. The total number of elements was about 80,000.

The foundation material was steel. The elastic modulus was E = 210 GPa, and the Poisson’s ratio was $\upsilon$ = 0.3. The constitutive model based on Tresca yield criterion was adopted for the soil constitutive model, which has been widely used for the finite element numerical study of soft clay [17]. The shear strength was set to 10 kPa, and the internal friction angle was 0 to improve the computational efficiency and avoid convergence difficulties. The elastic constants were E = 500 s_u and $\upsilon$ = 0.499.

The contact interface of the soil in the cylinder was set to be rough and bonded to reflect the negative suction [18,22,23]. The friction contact was adopted for other contact interfaces. The friction coefficient was set to $\mu$ = 0.55 [24]. It is assumed that the suction pile was embedded in the foundation in advance, and the in situ stress balance operation was carried out to eliminate the vertical displacement caused by self-weight. The horizontal cyclic loading was applied to the reference point (RP) of the top surface of the suction pile.

4.1.2. Load Application

Cyclic loading methods are usually one-way or two-way cyclic loading. For offshore foundations, horizontal cumulative displacement is a design control factor. Considering that one-way cyclic loading will produce greater displacement accumulation [25,26,27], this work adopted a one-way cyclic loading model (Figure 7) to simulate the most unfavorable conditions of displacement accumulation. The cyclic loading frequency was 0.1 Hz.

The elastic modulus was set as shown in

Table 3. Elastic modulus and field variable value.

Elastic Modulus (MPa)	Field Variable Value
5	1
0.5	0.1

. According to the field variables calculated by the degradation coefficient in the subroutine, the corresponding elastic modulus was obtained by interpolation and extrapolation. The initial elastic modulus was 5 MPa, and the corresponding field variable value and the degradation coefficient were 1. With the decrease in the degradation coefficient during the cyclic loading, the corresponding field variable value and the elastic modulus also decreased gradually.

4.1.3. Analysis of Soil Degradation Rules

Figure 8 shows the evolution curve between the horizontal displacement of the single-cylinder suction pile top and the number of cycles under the horizontal cyclic loading of 0.8 H_ult. The total number of cycles was 1000. As shown in the figure, the cumulative horizontal displacement of the pile top developed rapidly at the initial loading stage. After 100 cycles, the cumulative horizontal displacement of the pile top was about 0.18 m. With the increasing number of cycles, the growth rate of the cumulative displacement gradually slowed down. After 1000 cycles, the cumulative horizontal displacement of the pile top was about 0.27 m.

Figure 9 shows the degradation coefficient of soil under different cycles. The view plane is the pile diameter section plane along the direction of the horizontal cyclic loading. In the legend, SDV5 is the degradation coefficient of soil stiffness mentioned above. It can be seen from the figure that in the process of cyclic loading, different soil elements had different degrees of degradation due to the different stress levels of the soil near the pile body. With the increase in the number of cycles, the degradation coefficient decreased gradually, and the degradation degree and area increased gradually. Under the action of a horizontal cyclic load, the soil on the passive side of the pile body appeared to experience stiffness degradation first. After 1000 cycles, the minimum degradation coefficient was about 0.12. In addition, with the increase in the number of cycles, the degradation area of the pile end and the internal soil of the suction pile expanded gradually, which developed from the pile end to the inside of the pile.

As shown in Figure 10, the typical soil element around the suction pile is selected as the characteristic element. The evolution curves of the degradation coefficients of different soil elements around the foundation are shown in Figure 11. The soil element with a large degree of stiffness degradation around the pile was chosen as the characteristic element to extract the evolution curve of the degradation coefficient during the cycle. It can also be seen that the degradation degree of the passive side soil was more significant. The minimum degradation coefficient of the passive side soil was about 0.17, and the minimum degradation coefficient of the pile end soil was about 0.34.

Figure 12 shows the development of the horizontal displacement of the pile top under different amplitude horizontal cyclic loadings. The cumulative displacement in the initial stage of loading accounted for a large proportion, while the cumulative displacement in the later stage of loading tended to be gentle. The development of the cumulative horizontal displacement was greatly affected by the amplitude of the cyclic load. When the load amplitude exceeded a certain value, the cumulative displacement developed rapidly.

Figure 13 shows the distribution of the horizontal displacement along the depth of the suction pile. It can be found that with the development of a cyclic process, the cumulative displacement increased gradually. Taking the horizontal displacement of the pile top as an example, the horizontal displacement was 0.18 m at the 100th cycle. The horizontal displacement was 0.27 m after 1000 cycles, with an increase of about 50%. The inclination of the pile and the cumulative displacement along the pile also increased with the development of the cycle. Since the material of the suction pile was steel, its elastic modulus was much larger than that of the soil; thus, the displacement of the pile along the depth distribution curve was a straight line.

4.2. Four-Cylinder Suction Pile Foundation

4.2.1. Numerical Model

Furthermore, the numerical calculation model of the four-cylinder suction pile foundation was established. The plane layout of the four piles is shown in Figure 14a. It is assumed that the piles are rigidly connected by the top surface to constrain their relative displacement. The whole system was modeled by semi-structure, and symmetrical boundaries were applied on the symmetry plane [28]. The FE model and element division are shown in Figure 14b. The modeling process and method were similar to that of the single-cylinder foundation. The total number of elements was about 100,000.

4.2.2. Analysis of Soil Degradation Rules

Figure 15 shows the evolution curve between the horizontal displacement of the four-cylinder suction pile top and the number of cycles under the horizontal cyclic load of 0.8 H_ult. It can be seen that the foundation accumulated horizontal displacement, and it developed rapidly in the initial stage of loading. With the development of a cyclic process, the cumulative displacement growth rate slowed down gradually.

The degradation coefficient of soil under different cycles is shown in Figure 16. The view plane is the pile diameter section plane along the direction of the horizontal cyclic loading. Similar to the single-cylinder suction pile, the soil on the passive side also had a large degree of degradation area. In addition, the soil between the adjacent two piles also had a certain degree of degradation. It can also be seen that the soil at the bottom of the compressive pile was more affected by the cycle than that of the uplift pile.

As shown in Figure 17, the typical soil element around the suction pile is selected as the characteristic element. Figure 18 shows the evolution curve of the degradation coefficient of the four-cylinder suction pile. Figure 18a shows the degradation rule of the soil on the active side and the passive side of the suction pile. It can be observed that the degradation degree of the soil on the passive side was more significant. Figure 18b shows the degradation rule of soil elements at different depths along the suction pile. It can be determined that the degree of degradation decreased with increasing depth. Figure 18c shows the degradation rule of the soil element at the bottom of the compressive pile and the uplift pile, and the degradation degree of the soil at the bottom of the compressive pile was greater.

The distribution of the horizontal displacement of the suction pile along the depth under different cyclic times and load amplitudes was extracted, as shown in Figure 19. Taking the horizontal displacement of the pile top as an example, the horizontal displacement in the 100th cycle was 0.24 m. After 1000 cycles, the horizontal displacement was 0.32 m, and the growth rate was about 33%. That is to say, the cumulative displacement of the foundation after the first 100 cycles was 75% of that after 1000 cycles. Therefore, it is necessary to focus on the cumulative displacement of the foundation in a short time after the completion of construction, and then the long-term service safety performance of the foundation can be evaluated accordingly.

Figure 20 shows the evolution of the horizontal displacement of the four-cylinder suction pile top under different amplitude horizontal cyclic loadings. Similar to the single-barreled suction pile, the development of cumulative horizontal displacement was greatly affected by the amplitude of the cyclic load. When the load amplitude exceeded a certain value, the cumulative displacement developed rapidly.

5. Discussion

Based on the above research, the main findings are as follows:

(1)

The USDFLD subroutine based on the current stiffness degradation model can be implemented in ABAQUS. Compared with the centrifugal model test and scaled model test results, it can well reflect the stiffness degradation of marine soft clay and the cumulative deformation of soil around the foundation under cyclic loading.

(2)

Under the action of horizontal cyclic load, the soil on the passive side of the suction pile foundation appears to experience stiffness degradation first, and the degree of degradation increases gradually. The degradation area of the pile end and the internal soil of the suction pile are expanded gradually, developing from the pile end to the interior of the pile. The soil between two adjacent piles also has a certain degree of degradation.

(3)

In the early stage of cyclic loading, the horizontal displacement of the pile top develops relatively rapidly, while the growth rate of displacement slows down in the later stage. The evolution of cumulative horizontal displacement is greatly affected by the amplitude of cyclic loading. When the load amplitude exceeds a certain value, the cumulative displacement develops rapidly.

6. Conclusions

In this work, the cumulative displacement of single-cylinder and four-cylinder suction pile foundations in marine soft clay under horizontal cyclic load was numerically simulated. To further apply the stiffness degradation model to the practical offshore engineering, several dynamic triaxial tests were conducted using the clay samples taken from the central zone in the South China Sea. The model was implemented in the ABAQUS main program by compiling the USDFLD subroutine.

The numerical calculation method proposed in this paper can better simulate the cumulative displacement of marine soft soil under long-term horizontal cyclic loading. By simulating the cyclic response of single-cylinder and four-cylinder suction pile foundations, a broad understanding of the cyclic characteristics of this type of foundation can be provided, which helps to provide safer and more economical design guidelines for seabed suction pile foundations. It is hoped that the research in this study can provide some reference for the displacement monitoring a short time after completion, as well as the evaluation of the long-term service stability of the subsea foundation.

It should be noted that the above soil parameters used in the numerical simulation and the parameters in the stiffness degradation model were obtained from the laboratory test of marine clay samples. In the future, the model can be improved by more tests on soils with multiple soil properties.