Predicting the cyclic behaviour of suction anchors based on a stiffness degradation model for soft clays

Computers and Geotechnics 122 (2020) 103552

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Research Paper

Predicting the cyclic behaviour of suction anchors based on a stiffness degradation model for soft clays

T

Xinglei Chenga,b, Piguang Wangb, , Na Lia, Zhongxian Liua, Yadong Zhoua ⁎

a b

Key Laboratory of Soft Soil Engineering Character and Engineering Environment of Tianjin, Tianjin Chengjian University, Tianjin 300384, China Beijing University of Technology, Beijing 100124, China

ARTICLE INFO

ABSTRACT

Keywords: Stiffness degradation Soft clays Suction anchors Cyclic loads Numerical simulation

The behaviour of suction anchors subjected to combined average and cyclic loads in soft clays is an essential consideration in their design. A new numerical computation method was developed based on a stiffness degradation model for soft clay that was proposed by performing a series of undrained cyclic triaxial tests and embedded in the ABAQUS software package by encoding the USDFLD subroutine and defining relevant field variables. The numerical method predicts the cyclic behaviour of suction anchors by coupling the elastic-perfectly plastic model with the Mohr-Coulomb yield criterion and the proposed stiffness degradation model that can reflect the stiffness degradation and the accumulation of plastic deformation of soils around the anchor during cyclic loading. The numerical method was verified by a comparison with the model test results of suction anchors subjected to combined average and cyclic loads in soft clays. The proposed method can predict the cyclic deformation as well as the bearing capacity and capture the nonlinearity, hysteresis and cyclic accumulation characteristics of the load-displacement responses of suction anchors in soft clays.

1. Introduction Suction anchors have been widely used in taut mooring systems for floating facilities, such as floating production storage and offloading (FPSO) units. In many cases, the suction anchor needs to be installed in soft clay foundations, which are subjected to average loads caused by the floatage of the superstructure as well as cyclic loads induced from various environmental factors such as winds, waves and flows in the marine environment [1–4]. The strength and stiffness of the soil around the suction anchor will be degraded under long-term cyclic loading, which will induce the cumulative displacement of the anchor along the mooring direction. The failure of suction anchors occurs when the cumulative displacement reaches a certain failure standard. Therefore, proposing an appropriate method to analyse the cyclic deformation process and evaluate the cumulative displacement is of great significance for the design and safe service of suction anchor foundations in soft clays. The finite element method has been widely used for the study of suction anchor behaviours in soils. Some studies have also been reported on the finite element analysis of suction anchors under static or cyclic loads. Many attentions have been focused on the simulation of the behaviours of suction anchors under static loads. The selection of a soil constitutive model is very important for finite element analysis. ⁎

Sukumaran et al. [5] determined the capacity of a suction anchor subjected to lateral loads in soft clays under undrained conditions using a linear elastic model with the Von Mises strength criterion. Cao et al. [6] simulated the behaviour of suction anchors subjected to vertical loading in normally consolidated clays based on the modified Cam-Clay model of porous soil materials. A computational procedure was proposed by Maniar [7] to simulate the installation process and the loaddisplacement response of anchors subjected to axial and inclined loads, and a bounding surface model was used to describe the nonlinear behaviour of the clayey soil. Monajemi and Razak [8] investigated the failure mechanism and ultimate capacity of an anchor under combined V-H-M loading by conducting nonlinear analysis using a simplified elastic-perfectly plastic model for saturated clays. Ahn et al. [9] estimated the holding capacities of optimally loaded suction caisson anchors embedded in cohesive soils with a linear strength distribution by the total stress analysis based on the Von Mises model of soil. Kim et al. [10] investigated the performance of suction anchor groups subjected to horizontal pull-out loading by performing centrifuge model tests and numerical simulations, and the soil was modelled using the MohrCoulomb failure criterion with the non-associated flow rule. Wang and Guo et al. proposed a novel two-dimensional static model for a mooring cable during pretensioning and a three-dimensional quasi-static model for a cable in service [11]. In addition, Guo et al. performed model tests

Corresponding author. E-mail address: [email protected] (P. Wang).

https://doi.org/10.1016/j.compgeo.2020.103552 Received 3 November 2019; Received in revised form 14 March 2020; Accepted 15 March 2020 Available online 20 March 2020 0266-352X/ © 2020 Elsevier Ltd. All rights reserved.

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with different loading angles and cyclic parameters to investigate the failure mode and capacity of suction caissons under inclined short-term static and one-way cyclic loadings [12]. A series of model tests were performed by Chen and Randolph [1] and Zhu et al. [13] to investigate the responses of suction anchors subjected to cyclic loading. However, research on numerical simulations of the cyclic behaviours of suction anchors is limited. The main difficulty is the lack of constitutive models that can not only reasonably describe the nonlinear cyclic stress-strain responses of soft clays but also be easily implemented in 3D nonlinear finite element analysis codes. These models, such as the elastic-perfectly plastic model, modified Cam-Clay model, and Mohr-Coulomb model encoded into commercial finite element software, are incapable of calculating the cyclic responses of soil. Therefore, researchers have performed numerical simulations of the cyclic behaviours of suction anchors using advanced constitutive models. Gelagoti et al. [14] proposed a simplified equivalent linear iterative approach to account for the nonlinear lateral stiffness and bearing capacity of suction caissons by a simplified kinematic hardening model in the context of Von Mises associative plasticity. Kourkoulis et al. [15] simulated the behaviour of suction caissons subjected to wave and earthquake loading using a simple kinematic hardening model. Zhang et al. [16] predicted the cyclic behaviour of suction caissons using a finite element method based on a thermodynamics-based constitutive model. Cheng et al. [17,18] simulated the cyclic behaviours of suction anchors by developing an elastoplastic bounding surface model and encoding it into ABAQUS software. Although the advanced constitutive model with different kinematic hardening rules can describe the cyclic responses of soil to some extent, the complexity of the model leads to inefficient numerical calculation, especially for long-term cyclic loads. The cyclic loading can lead to the degradation of stiffness for soft clays, and the stress-strain curve shows significant nonlinear and hysteresis properties. In this paper, the stiffness degradation model for soft clays was proposed by performing a series of cyclic triaxial tests, and then the model was embedded in the ABAQUS software package by encoding the USDFLD subroutine and defining relevant field variables. The behaviours of suction anchors subjected to combined average and cyclic loads in soft clays were simulated by coupling the elastic-perfectly plastic model with the MohrCoulomb yield criterion and the proposed stiffness degradation model. This numerical method has higher computational efficiency than previous methods using advanced constitutive models.

Fig. 1. Definition of the secant shear modulus and cyclic degradation.

characteristic of clay is described by the decay of the secant shear modulus with the increase in the number of cycles. As shown in Fig. 1, the secant shear modulus of the unloading curve is defined as the slope of a straight line connecting the two ends of the unloading curve, the longitudinal coordinate is the axial deviatoric stress represented as R , and the transverse coordinate is the axial strain represented as y . The secant shear modulus is represented by Eq. (1).

GSN =

N R,max N y,max

N R,min N y,min

(1)

where N represents the number of stress cycles, is the secant shear modulus for the Nth cycle, RN,max and RN,min are the maximum and minimum deviatoric stresses for the Nth cycle, respectively, and yN,max and yN,min are the corresponding axial strain, as shown in Fig. 1. Referring to the definition of the degradation index by Idriss et al. [19] based on the results of cyclic triaxial tests under the strain-controlled mode, the expression for the cyclic degradation index under a constant amplitude stress-controlled mode is defined as follows:

GSN

GN = S1 = GS

N R,max N y,max 1 R,max 1 y,max

N R,min N y,min 1 R,min 1 y,min

=

1 y,max N y,max

1 y,min N y,min

(2)

where

N is the cyclic degradation index. RN,max R,min is equal to 1 1 for constant amplitude stress-controlled cyclic tests. R,max R,min The secant shear modulus for each cycle number can be determined by the tested stress-strain curve according to the definition of Eq. (1), and then the cyclic degradation index for each cycle number can be determined based on Eq. (2). The functional relationship between the cyclic degradation index and the cycle number N can be determined.

2. Stiffness degradation model of soft clays Many studies have revealed that the stiffness degrades and the strength decreases for soft clay under cyclic loading [19–22]. The degree of degradation is related to the number of cycles, the level of initial static stress and the level of cyclic stress, and it is affected by many factors, such as the overconsolidation ratio, principal stress direction, strain rate and vibration frequency [23–27]. The results of soil dynamic tests show that the cyclic degradation of soft clays means that the shear modulus of the stress-strain hysteresis curve decreases gradually; meanwhile, the cyclic deformation increases gradually as the number of cycles increases. The plastic strain of soil accumulates when there is an initial static deviate stress.

2.2. Cyclic degradation model Idriss et al. [19] defined the concept of the degradation index and proposed the exponential relationship of the degradation index versus the cycle number based on the results of cyclic triaxial tests the under strain-controlled mode but did not give the definition of the degradation index under the stress-controlled mode or consider the effect of the initial deviatoric stress on the exponential relationship. In this paper, the relationship between the degradation index and the number of cycles for soft clays subjected to the combination of initial static deviatoric stress and cyclic deviatoric stress is studied. The undrained cyclic triaxial tests on soft clay under the joint actions of different static and cyclic deviatoric stresses are performed under the stress-controlled mode. In these tests, an axial static deviatoric stress was first applied to the soil sample, and then a 0.1 Hz sinusoidal constant-amplitude cyclic deviatoric stress was exerted under undrained conditions. The stress level of the soil elements in the general stress states can be represented

2.1. Cyclic degradation index The test results for soft clay subjected to combined static and cyclic deviatoric stresses show that the soil stiffness degradation is not significant with no cyclic stress reversal; otherwise, significant stiffness degradation will occur. The greater the cyclic stress is relative to the static deviatoric stress, the more obvious the stress reversal is, and the more significant the stiffness degradation is [20,21]. In this paper, the secant shear modulus of the unloading curve is used to describe the stiffness of soft clay under cyclic loads, and the stiffness degradation 2

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ζ

X. Cheng, et al.

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

Fig. 2. Typical stress-strain curve of soil under the joint actions of static and cyclic deviatoric stress. Table 1 Different combinations of static and cyclic stress levels for the cyclic triaxial tests. 8, cy 8, f

0 0 0 0

0.781 0.667 0.483 0.322

8, a 8, f

0.3 0.3 0.3 0.3

8, cy 8, f

0.549 0.412 0.323 0.263

8, a 8, f

0.5 0.5 0.5 0.5

8, cy 8, f

0.494 0.423 0.358 0.310

ζ

8, a 8, f

=N

ζ

using the octahedral shear stress 8 . Here, the octahedral static shear stress 8, a is defined as the octahedral shear stress of the soil element after static loading. The octahedral cyclic shear stress 8, cy is defined as one half of the difference between the maximum and minimum octahedral shear stress of the soil element under cyclic loading, as shown in Fig. 2. In addition, the octahedral peak shear stress 8, f is defined as the octahedral shear stress when the soil sample reaches the deformation failure criteria of 10% axial strain. The normalized octahedral static and cyclic shear stresses were defined as 8, a 8, f and 8, cy 8, f , respectively, which represent the static deviatoric stress level and cyclic stress level that were applied to the sample during the experiment. For each 8, a 8, f , different 8, cy 8, f values were selected for the tests, as shown in Table 1. According to the definition of the degradation index in Section 2.1 above, the relationships between the degradation index and the number of cycles under different stress levels can be determined based on cyclic triaxial test results, as shown by the discrete points in Fig. 3. It is found that Eq. (3) proposed by Idriss et al. [19] can be used to express the relationship between the degradation index and the number of cycles. where t is the cyclic degradation parameter and reflects the rate of cyclic degradation as the cycle number increases. The value of t is closely related to the static and cyclic stress levels of soils. To reflect the change rule of the degradation parameter t with the stress level, the cyclic stress parameter S is defined as Eq. (4) or (5). 8,cy (2 8, f

2

8,f

100

150

N

τ8,cy / τ8, f =0.549 τ8,cy / τ8, f =0.412 τ8,cy / τ8, f =0.323 τ8,cy / τ8, f =0.263

τ8,a / τ8, f =0.3 0

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

50

100

N

150

200

10

20

30

N

40

50

8,f

1

0.5

70

0.20 0.15

(4)

τ8,a / τ8, f =0 τ8,a / τ8, f =0.3 τ8,a / τ8, f =0.5

0.05

8,a 8,f

60

0.25

0.10 8,cy

250

τ8,cy / τ8, f =0.494 τ8,cy / τ8, f =0.423 τ8,cy / τ8, f =0.358 τ8,cy / τ8, f =0.310

τ8,a / τ8, f =0.5 0

200

0.30

Or

S=

50

0.35

8,a ) 2

0

t

S=

τ8,a / τ8, f =0

Fig. 3. Cyclic degradation index ς versus cycle number N.

(3)

t

1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

τ8,cy / τ8, f =0.781 τ8,cy / τ8, f =0.667 τ8,cy / τ8, f =0.483 τ8,cy / τ8, f =0.322

(5)

0.00 0.0

t is associated with different combinations of static and cyclic deviatoric stresses and can be determined by fitting the test results in Fig. 3 using Eq. (3), and then t against different values of S can be plotted as the discrete points shown in Fig. 4. It can be seen that t increases monotonically with S under different 8, a 8, f , and the two basically follow a linear relationship, which can be expressed in Eq. (6) as follows.

0.1

0.2

0.3

0.4

0.5

S

0.6

0.7

0.8

0.9

1.0

Fig. 4. Cyclic degradation parameter t versus cyclic stress parameter S.

3

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(6)

t = 0.3353S

Su/kPa

1, and 1, In Eq. (4) or Eq. (5), 8, cy 8, f 8, a 8, f ( 8, a + 8, cy ) 8, f 1. Hence, it can be considered that 0 S 1. S = 0 when 8, cy 8, f = 0 , and S = 1 when 8, cy 8, f = 0 and 8, cy 8, f = 1. Eq. (4) or Eq. (5) also show that for a given 8, a , the larger 8, cy is, the larger S is, and then the larger t correspondingly is through Eq. (6), which indicates that when the static deviatoric stress is constant, the larger the cyclic deviatoric stress is, the more significant the cyclic degradation will be. In addition, the equations also show that for a given 8, cy , the smaller 8, a is, the larger S is, and then the larger t is, which means that when cyclic deviatoric stress is constant, the smaller the static deviatoric stress is, the more significant the cyclic degradation is. It can also be seen from the test results in Fig. 3 that the cyclic degradation is the most significant when 8, a 8, f is 0. The cyclic degradation gradually becomes insignificant with increasing 8, a 8, f . Based on the above Eqs. (3)–(6), the degradation index can be expressed as a function of the static and cyclic deviatoric stress levels and the number of cycles, as shown in Eq. (7).

=N

0.3353

(

8,cy 8,a 1 0.5 8,f 8,f

)

0

Depth/cm

20

EN = E1

(9)

6

9

12

15

30 40

S1 S2 S3 S4 S5 S6

50 60 70 80

Fig. 5. Shear strength of the soft clay along the depth.

Several LVDT displacement sensors were used to measure the displacement of the anchor along multiple directions, and the load sensor was used to measure the force along the loading direction.

The curves of the cyclic degradation index versus the number of cycles can be plotted based on Eq. (7) after determining the values of 8, a 8, f and 8, cy 8, f , as shown by the solid line in Fig. 3. It can be seen that the solid line generally agrees with the experimental data, which indicates that Eq. (7) can predict the change in the degradation index with the number of cycles. The secant shear modulus for the Nth cycle GSN can be determined by Eq. (8) which is deduced from Eq. (2) after obtaining the value of . In this research, to easily apply the degradation relationship to the numerical calculation, the elastic modulus of the soil is used to represent the stiffness of the soil, and the degradation relationship for the secant shear modulus is used to represent the degradation of the elastic modulus of the soil. Therefore, we can obtain a more applicable degradation relationship, as shown by Eq. (9). (8)

3

10

(7)

GSN = GS1

0

3.2. Model test steps A static tensional load (average load) with a magnitude of Fa was first applied to the anchor using the grading loading mode. When the displacement of the anchor was relatively stable under the static load, sinusoidal cyclic loads with a peak value of Fcy and a frequency of 0.1 Hz were applied to the anchor in the load control mode. The static and cyclic loads resulted in static and cyclic cumulative displacements of the anchor. It is considered that the failure of the anchor occurred when the sum of the static and cyclic cumulative displacements along the loading direction at the loading point reached the failure criterion of 0.6 times the anchor plate width (static loading and cyclic loading follow the same displacement failure criterion; therefore, the failure criterion is determined by first performing static monotonic tests, and the determining method can be seen in reference [18]). Four groups of model tests were carried out. The test programs are shown in Table 2. The number of load cycles corresponding to the failure of the anchors is defined as the number of load cycles to reach failure, denoted as Nf . The cyclic bearing capacity corresponding to Nf is determined based on Fa + Fcy . Ff is the ultimate bearing capacity of the anchor under monotonic static loading. Fa Ff is the normalised average load that specifies the magnitude of the average load. Fcy Ff is the normalised cyclic load that specifies the magnitude of the cyclic load. (Fa + Fcy ) Ff is the normalised cyclic bearing capacity that specifies the magnitude of the cyclic bearing capacity. The test results for the cyclic responses of the suction anchors will be compared with the numerical simulation results in the following sections.

where N represents the number of stress cycles, EN is the elastic modulus for the Nth cycle, and E1 is the elastic modulus for the first cycle or initial elastic modulus. 3. Model tests on suction anchors subjected to cyclic loads in soft clays 3.1. Model test apparatus In the following section, numerical simulations for the suction anchor model tests will be performed based on the above stiffness degradation model of soft clays. Here, the model tests of the suction anchor are first introduced. Model tests were conducted in a test tank with a length of 1.5 m, width of 1 m and height of 1.2 m. The soft clays for the model tests are prepared using the vacuum preloading method and clay slurry collected from Bohai Bay Beach of Tianjin, China. The unit weight, plastic limit, liquid limit, plastic index and sensitivity of the clay are 17.5 kN/m3, 27.01, 44.44, 17.43 and 4.0, respectively. The vane shear strength of the soft clay Su is basically uniform along the depth of the stratum with an average value of 6 kPa, as shown in Fig. 5. The model anchor was made of stainless steel with a diameter of 0.152 m, a height of 0.456 m and a wall thickness of 0.002 m. The loading system used in the model tests consists of a loading frame, an oriented plate with pulleys, and a multifunctional electric servo control loading device, as shown in Fig. 6. Cyclic loading tests on the suction anchor were carried out using the loading system in the load control mode. The upper pulley in the oriented plate was fixed, and the loading direction was changed by regulating the location of the pulley below.

4. Numerical simulation for cyclic degradation behaviours of suction anchors in soft clays 4.1. Geometries and meshes The 3D finite element model was established using the ABAQUS software package to simulate the above model tests on suction anchors. Considering the symmetries of the geometries and loading conditions during the tests, only half of the foundation was meshed to improve the calculation efficiency, as shown in Fig. 7. The finite element results dependent on the mesh density and the incremental step size. It is shown that more accurate calculation results can be obtained with higher mesh density and smaller incremental step size, while 4

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LVDT 1

Upper leading pulley

Loading actuator

Guide plate

Below leading pulley

Guide plate

Load sensor

Loading frame

Load sensor

LVDT 3

LVDT 2

Displacement sensor Loading actuator

Suction anchor

Displacement gauge Suction anchor

Below leading pulley

(a) Real loading system

(b) Schematic diagram of the loading system Fig. 6. Loading system of the model tests.

calculation efficiency is lower. After many trial calculations, we found that the displacement of anchors increases as the mesh density increases and the incremental step size decreases, and the calculation results will have an acceptable accuracy when the number of elements is more than 10,000 and the incremental step size is not greater than 0.01 (100 incremental steps per loading cycle). Hence, the anchors and the soils were all simulated using 14,320 8-node linear brick elements and the incremental step size is set as 0.01 in this research. Normal horizontal constraints were applied to the vertical boundaries, and fixed constraints were applied to the bottom boundary. The top boundary was fully free. The anchor was treated as a rigid body during the calculations. The model test results showed that the soil plug within the anchor remained in close contact with the inner wall of the anchor during loading due to a large negative pressure; therefore, the tie-contact condition was set between the soil plug and the inner wall. Besides, relative displacements and the separation could occur between the outer wall of the anchor and the stratum, so the contact conditions of the tangential slip and normal separation were set between the outer wall and the stratum. The tangential slip conditions were set according to the Coulomb friction model with a friction coefficient of 0.26.

degradation index of 1. As the degradation index decreases, the modulus degrades more significantly, the value of the field variables increases, and the elastic modulus decreases gradually. The degradation index is divided into several intervals. The increment of each interval is 0.1. The corresponding elastic modulus is the product of the initial elastic modulus and the average value of the field variable for each interval. The function of the field variable is to establish the corresponding relationship between the elastic modulus and degradation coefficient. In addition, the degradation index is closely related to the initial static stress level, the cyclic stress level and the cyclic number. The main function of the USDFLD subroutine is to determine the cyclic degradation index according to Eq. (7) based on the static and cyclic stress levels and the cyclic number, and then the corresponding degraded elastic modulus can be determined. The main steps of simulating the cyclic behaviours of suction anchors in soft clays based on the stiffness degradation model are as follows: (1) The first step is the geostatic analysis step. The purpose of this step is to impose self-weight loads on the soil so that the soil domain has an initial geostress field; meanwhile, the initial displacement field is virtually zero. (2) The second step is to apply a static tensile load (average load) to the suction anchor. At the end of this step, six stress components of any soil element in the whole soil domain are output to calculate the corresponding octahedral shear stress of the soil element, which is defined as the initial octahedral static shear stress 8, a , and then the normalized octahedral static shear stress 8, a 8, f can be obtained for any soil element in the whole soil domain at the end of static loading. (3) The third step is to apply a tension sinusoidal cyclic load to the suction anchor. The numerical calculation in this step is performed by calling the USDFLD subroutine. Six stress components of the soil element at the end of any time increment step in each cycle are output, and the corresponding octahedral stress is calculated by calling the subroutine. The peak value of the octahedral stress 8,max in each cycle is tracked, judged and then output. The octahedral cyclic shear stress 8, cy can be determined by the formula of 8,max 8,a , and then the normalized octahedral cyclic shear stress

4.2. Numerical implementation of the stiffness degradation model The elastic-perfectly plastic model with the Mohr-Coulomb yield criterion was used to analyse the deformation of the soil. The cohesion force was set as 6 kPa. The Poisson’s ratio was = 0.49 and the friction and dilation angles of = = 0 were set to simulate the undrained condition of soft clays (the Mohr-Coulomb yield criterion reduces to the Tresca yield criterion in the case of = 0 ). However, the model cannot describe the stiffness degradation and cumulative deformation characteristics of soils under cyclic loads. Therefore, the cyclic stress-strain responses of soil were described by combining the above stiffness degradation model with the ideal elastic-plastic model. The stiffness degradation model was implemented in ABAQUS by encoding the USDFLD subroutine. The elastic modulus and the corresponding field variables and cyclic degradation index are shown in Table 3. The elastic modulus varies with the field variables that are determined by the cyclic degradation index . The initial maximum elastic modulus E1 is 1.8 MPa, corresponding to a field variable of 0 and Table 2 Programs for the constant amplitude cyclic loading tests on suction anchors. Test number

Ff/kN

Fa/Ff

Fcy/Ff

(Fa + Fcy)/Ff

Nf (test)

Nf (simulation)

(Nf

T1 T2 T3 T4

2.10 2.00 1.95 2.07

0.5 0.5 0.5 0.5

0.45 0.38 0.33 0.24

0.95 0.88 0.83 0.74

52 195 268 No failure

47 218 286 No failure

9.6 11.8 6.7 \

5

Nf ) Nf \%

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0.152m 0.456m

Load

0.8m 0.456m

1m Fig. 7. 3D finite element model of the suction anchor in soft clays.

degradation responses of soils can be calculated by combining the ideal elastic-plastic soil model with the stiffness degradation model. The main calculation process is shown in Fig. 9.

Table 3 Elastic modulus and field variables. Elastic modulus (MPa)

Field variables

Degradation index

1.80 1.71 1.53 1.35 1.17 0.99 0.81 0.63 0.45 0.27 0.09

0 1 2 3 4 5 6 7 8 9 10

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

< < < < < < < < < <

8000

≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤

5. Comparisons of the predictions and model test results

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

5.1. Cyclic degradation behaviours Fig. 10 shows the contours of the cyclic degradation index as the cycle number increases (N = 5, 15, 30, 45). It can be seen that the range of degradation in the passive soil domain of the anchor wall and the soil domain of the anchor bottom expands and the degradation degree increases as the number of cycles increases. The degradation degree for the same cycle number is different in different soil domains around the anchor. The degradation indexes are different for each soil element in the whole soil domain around the anchor due to different values of 8, a 8, f and 8, cy 8, f for the same cycle number N; meanwhile, the degraded elastic modulus is also different. Therefore, the degraded elastic modulus for each soil element corresponding to different cycle numbers is determined according to the static and cyclic stress levels of each element. Degradation mainly occurs in the passive soil domain of the anchor wall and the soil domain at the anchor bottom. The degradation in the active soil domain of the anchor wall is relatively insignificant. The most significant degradation positions are the soil domains near the lower part of the anchor wall and near 1 times the anchor diameter from the anchor bottom. Fig. 11 shows the degradation index versus the cycle number N for several typical soil elements around the anchor. The selected several types of soil elements are shown in Fig. 11(a). Fig. 11(b) shows the degradation rules of Element A in the active soil domain and Element E in the passive soil domain. It can be seen that the two soil elements experience a certain stiffness degradation as the cycle number increases; however, the degradation of Element E is more significant than that of Element A. This is because the soil elements in the passive soil domain are constantly and forcefully squeezed by the anchor wall during cyclic loading, while the soil elements in the active soil domain only undergo slight lateral unloading. Fig. 11(c) shows the degradation rules of Elements B ~ H along the stratum depth in the passive soil domain. The stiffness degradation of soft clays becomes more obvious with increasing depth along the anchor wall. The elastic modulus near the anchor top degrades to approximately 90% of the initial maximum elastic modulus, while the elastic modulus near the anchor bottom

Element K Element J Element E

7000 6000

τ8

5000 4000 3000 2000 1000 0 0

1

2

3

4

5

Time/s

6

7

8

9

10

Fig. 8. Octahedral stress-time history curves for three typical soil elements (elements can be seen in Fig. 11(a)). 8, cy 8, f can be determined for any soil element in the whole soil domain during cyclic loading. The magnitude of the octahedral stress varies periodically with time, as shown in Fig. 8 (indicated by only 10 cycles), and the period of the stress cycle is virtually the same as the period of the sinusoidal cyclic loads, so the number of stress cycles N can be determined according to the number of sinusoidal loads applied to anchors. The modulus degradation index corresponding to N cycles can be determined after obtaining the values of 8, a 8, f , 8, cy 8, f and N, and then the corresponding degraded elastic modulus can be determined. Therefore, the cyclic

6

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Geostatic analysis

Define field variables

Static monotonic tension load analysis

Output τ 8,cy / τ8,f and N

Output the value of τ 8,a / τ8,f for each soil element in the whole soil domain at the end of static loading

Calculate ζ Calculate EN

Perform cyclic load analysis by calling the USDFLD subroutine based on the stiffness degradation model

Calculate soil deformation using EN based on the ideal elastic-plastic model

0.8m

0.8m

Fig. 9. Diagram of the calculation program.

1.0m

(a) N=5

(b) N=15

0.8m

0.8m

1.0m

1.0m

1.0m

(c) N=30

(d) N=45

0.00 0.10 0.21 0.31 0.41 0.52 0.62 0.73 0.83 0.93 1.00 Fig. 10. Contours of the cyclic degradation index with increasing cycle number.

7

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Fig. 11. Modulus degradation rules for different soil elements around the anchor as the number of cycles increases.

degrades to approximately 35% of the initial elastic modulus. Fig. 11(d) shows the degradation rules of Elements E ~ L along the horizontal direction in the passive soil domain. The soil element with the most significant degradation is not Element E nearest to the anchor wall but Element J. The degradation degree of Element L farthest from the anchor wall is also relatively small. Therefore, the degradation degree of the soil element along the horizontal direction is not inversely proportional to the distance from the anchor wall but is most significant at a certain distance from the anchor wall, and the distance should be related to the direction of the loading and movement of the anchor.

5.2. Cyclic deformation process The typical test result and prediction for the anchor at failure are shown in Fig. 12(a) and (b), respectively. The translational motion of the anchor occurs, and the vertical cumulative displacement is greater than the horizontal cumulative displacement in Fig. 12(b). The separation occurs between the outer wall of the anchor and the soils in the active regions; meanwhile, the soils are compressed and uplifted in the passive regions. These predictions are consistent with the test results.

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Fig. 12. Comparison between the numerical simulation and model test results at anchor failure.

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(7) is relatively higher than the true value of the T2 test, which indicates that the stiffness of the soil is overestimated as the number of cycles increases during cyclic loading. This is possibly because the clay in the T2 test is relatively softer than the clay in the other tests. In addition, there are some other reasons for the deviation. One reason is that the strengths of the soft clays along the depth in different locations of the model tank show certain differences; however, the strength of the soft clay used in determining the model parameters is constant. Another reason might be that cracking of the soils in passive regions cannot be captured using the FEM; however, it has an effect on the deformation of the caisson. Actually, due to the strong nonlinearity of the deformation of the soft clays and the soil-anchor contact under cyclic loading, it is still challenging to predict the cyclic deformation process of the anchor perfectly under three-dimensional stress conditions. Although accurately predicting the test results is still a challenge, the consistency of the experimental results in the overall trend proves the feasibility of the numerical method to a certain extent. Fig. 15 shows a comparison of the load-displacement curves of the anchor between the predictions and test results. It can be seen from the figure that the finite element method based on the stiffness degradation model can describe the nonlinear, hysteretic and displacement accumulation characteristics of the suction anchor in soft clays under cyclic loads, and the predictions are basically consistent with the test results. It should be pointed out that the ideal elastic-plastic constitutive model itself cannot reasonably describe the cyclic stress-strain response of soft clays; however, the introduction of the stiffness degradation model causes the elastic modulus of soils to decrease and the plastic deformation of soils occurs as the cycle number increases. In addition, ABAQUS automatically linearly interpolates the elastic modulus set in Table 3 during the calculation for each time-increment step in each cycle, so the elastic modulus used in each time-incremental step will be different. As a result, the nonlinear and hysteretic characteristics of soil during each loading and unloading cycle can be captured.

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Fig. 13 shows a comparison of the predicted and measured displacement-time curves along the mooring direction at the mooring point. The predicted results show that the displacements increase rapidly at the beginning of cyclic loading, and then the increasing rate gradually decreases as the number of load cycles increases. The displacement in a single cycle changes less compared to the cumulative displacement along the tension direction during cyclic loading. Excessive cumulative displacement is the main cause of anchor failure. The predictions are generally in agreement with test results. Fig. 14 shows a comparison of the cumulative displacement between the predictions and the model test results under different load levels. The predictions indicate that the higher the cyclic load level is, the faster the increase in the cyclic cumulative displacement, and the smaller the number of load cycles required to reach the failure displacement. The predictions are consistent with the overall trend of the test results, although there are some deviations in the magnitude. Especially for the T2 test, the numerical method underestimates the real displacement value of the anchor, so the predicted displacement is smaller than the experimental displacement, which results in predictions for the T2 and T3 tests crossing over each other. The underestimation of the displacement means that the cyclic degradation index determined by Eq.

5.3. Cyclic bearing capacity The cyclic bearing capacity can be predicted based on the failure 9

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Establishing the stiffness degradation model and its numerical implementation are crucial for the numerical method. The stiffness degradation model is proposed first referring to the definition of the degradation index by Idriss et al. [19]. by performing a series of undrained cyclic triaxial tests on soft clays under the joint actions of different static and cyclic deviatoric stresses. Subsequently, the proposed stiffness degradation model for soft clays is embedded in the ABAQUS software package by encoding the USDFLD subroutine and defining relevant field variables. The function of the stiffness degradation model is to establish the relationship between the elastic modulus of the soil elements and the initial static stress level, cyclic stress level and stress cycle number. When applied to the analysis of the suction anchor, it can reflect the degradation of the stiffness and the accumulation of the plastic deformation of soils around the anchor during cyclic loading. The new numerical computation method predicts the cyclic behaviour of suction anchors by coupling the elastic-perfectly plastic model with the Mohr-Coulomb yield criterion and the proposed stiffness degradation model for soft clays. The numerical method can simulate the cyclic deformation process of suction anchors in soft clays to some extent and capture the nonlinearity, hysteresis and cyclic accumulation characteristics of the load-displacement responses of anchors. The cyclic bearing capacity of anchors can also be predicted well using the method based on an appropriate displacement failure criterion. As a simplification, only the applicability of the new numerical method for harmonic loads was discussed in this paper. However, cyclic loads such as wave loads or earthquake loads are more irregular variable-amplitude cyclic loads than constant-amplitude harmonic loads in the marine environment. “The number of load cycles” was introduced into the stiffness degradation model for soft clays in the paper. However, the number of load cycles is only applicable to constantamplitude harmonic loads but not to irregular cyclic loads. Ordinarily, the laboratory tests are run with one constant cyclic shear stress amplitude throughout each test. Therefore, the concept of an “equivalent

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criterion of 0.6 times the anchor plate width after obtaining the displacement-time curves. The predicted and measured number of load cycles to failure is denoted as Nf and Nf , respectively. The specific values of both are shown in Table 2. The maximum relative error between the predictions and test results for the number of load cycles to failure is less than 15%, as shown in Table 2. The normalised cyclic bearing capacity versus the number of load cycles to failure is shown in Fig. 16. They can be fitted using the same straight line in semilogarithmic coordinates based on Origin 9.0 software, and the coefficient of determination R2 is 0.91, which indicates that the predictions generally agree with the test results, so the cyclic bearing capacity can also be predicted well using the finite element method. 6. Conclusions The main contribution of the paper is to present a new numerical computation method that can predict the cyclic behaviour of suction anchors based on a stiffness degradation model for soft clays. 10

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number of load cycles” [3,20] should be introduced into the research to consider the effects of the cyclic loading history for irregular variableamplitude cyclic loads. According to the concept, the equivalent number of load cycles is the number of cycles at the loading level of a constant-amplitude load packet that would produce the same permanent deformation (or permanent pore pressure) as that produced by irregular cyclic variable-amplitude cyclic loads. The applicability of irregular cyclic variable-amplitude loading will be further studied and discussed in future research. The cyclic behaviours of the reduced-scale suction anchor in soft clays are simulated well using the proposed numerical computation method in this study. However, due to some limitations of the reducedscale model tests, such as incorrect stress levels, the applicability of the numerical method to real offshore conditions should be further validated by simulating prototype tests or centrifuge model tests of suction anchors in future work.

[4] Ravichandran V, Maji VB, Gandhi SR. Field Testing of suction anchors for mooring applications. Indian Geotech J 2015;45(3):267–77. [5] Sukumaran B, McCarron WO, Jeanjean P, Abouseeda H. Effcient finite element techniques for limit analysis of suction anchors under lateral loads. Comput Geotech 1999;24:89–107. [6] Cao J, Phillips R, Popescu R. Numerical analysis of the behavior of suction anchors in clay. Int J Offshore Polar 2003;13(2):154–9. [7] Maniar DR. A computational procedure for simulation of suetion anchor behavior under axial and inclined loads. Austin: University of Texas at Austin; 2004. [8] Monajemi H, Razak HA. Finite element modeling of suction anchors under combined loading. Mar Struct 2009;22(4):660–9. [9] Ahn J, Lee H, Kim YT. Holding capacity of suction caisson anchors embedded in cohesive soils based on finite element analysis. Int J Numer Anal Met 2014;38(15):1541–55. [10] Kim S, Choo YW, Kim JH, Kim DS, Kwon O. Pullout resistance of group suction anchors in parallel array installed in silty sand subjected to horizontal loadingcentrifuge and numerical modeling. Ocean Eng 2015;107:85–96. [11] Wang L, Guo Z, Yuan F. Quasi-static three-dimensional analysis of suction anchor mooring system. Ocean Eng 2010;37(13):1127–38. [12] Guo Z, Jeng DS, Guo W, Wang LZ. Failure mode and capacity of suction caisson under inclined short-term static and one-way cyclic loadings. Mar Georesour Geotec 2018;36(1):52–63. [13] Zhu FY, O'Loughlin CD, Bienen B, Cassidy MJ, Morgan N. The response of suction caissons to long-term lateral cyclic loading in single-layer and layered seabeds. Géotechnique 2017;68(8):729–41. [14] Gelagoti F, Georgiou I, Kourkoulis R, Gazetas G. Nonlinear lateral stiffness and bearing capacity of suction caissons for offshore wind-turbines. Ocean Eng 2018;170:445–65. [15] Kourkoulis RS, Lekkakis PC, Gelagoti FM, Kaynia AM. Suction caisson foundations for offshore wind turbines subjected to wave and earthquake loading: effect of soilfoundation interface. Géotechnique 2014;64(3):171. [16] Zhang Z, Cheng X. Predicting the cyclic behaviour of suction caisson foundations using the finite element method. Ships Offshore Struct 2017;12(7):900–9. [17] Cheng X, Wang J, Wang Z. Incremental elastoplastic FEM for simulating the deformation process of suction caissons subjected to cyclic loads in soft clays. Appl Ocean Res 2016;59:274–85. [18] Cheng X, Yang A, Li G. Model tests and finite element analysis for the cyclic deformation process of suction anchors in soft clays. Ocean Eng 2018;151:329–41. [19] Idriss IM, Dobry R, Singh RD. Nonlinear behavior of soft clays during cyclic loading. J Geotech Geoenviron 1978;104(12):1427–47. [20] Anderson KH, Pool JH, Brown SF, Pool JH. Cyclic and static laboratory tests on Drammen clay. J Geotech Eng Div ASCE 1980;106(5):499–529. [21] Vucetic M, Dobry R. Degradation of marine clays under cyclic loading. J Geotech Eng 1988;114(2):133–49. [22] Huang MS, Shuai LI. Degradation of stiffness and strength of offshore saturated soft clay under long-term cyclic loading. Chinese J Geotech Eng 2010;32(10):1491–8. [23] Guo L, Wang J, Cai Y, Liu H, Gao Y, Sun H. Undrained deformation behavior of saturated soft clay under long-term cyclic loading. Soil Dyn Earthq Eng 2013;50:28–37. [24] Li LL, Dan HB, Wang LZ. Undrained behavior of natural marine clay under cyclic loading. Ocean Eng 2011;38(16):1792–805. [25] Hyodo M, Yamamoto Y, Sugiyama M. Undrained cyclic shear behaviour of normally consolidated clay subjected to initial static shear stress. Soils Found 1994;34(4):1–11. [26] Zhou J, Gong X. Strain degradation of saturated clay under cyclic loading. Can Geotech J 2001;38(1):208–12. [27] Yasuhara K, Hirao K, Hyde AF. Effects of cyclic loading on undrained strength and compressibility of clay. Soils Found 1992;32(1):100–16.

CRediT authorship contribution statement Xinglei Cheng: Conceptualization, Methodology, Software, Writing - original draft. Piguang Wang: Investigation, Validation, Software. Na Li: Data curation, Formal analysis. Zhongxian Liu: Supervision. Yadong Zhou: Writing - review & editing. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements Financial support from the National Natural Science Foundation of China (Grant No. 51878434, 51678014) and China Postdoctoral Science Foundation funded project (Grant No. 2019M650411) and Beijing Postdoctoral Research Foundation (Grant No. ZZ2019-101) is gratefully acknowledged. References [1] Chen W, Randolph MF. Uplift capacity of suction anchors under sustained and cyclic loading in soft clay. J Geotech Geoenviron 2007;133(11):1352–63. [2] Anderson KH. Bearing capacity under cyclic loading-offshore, along the coast, and on land. Can Geotech J 2009;46(5):513–35. [3] Andersen KH, Lauritzsen R. Bearing capacity for foundations with cyclic loads. J Geotech Geoenviron 1988;114(5):540–55.

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