Influence of the Spudcan Angle on the Ultimate Bearing Capacity of Jack-up Platform

Jack-up platforms of the Ocean engineering structures always withstand the vertical gravity loads which are applied to the seabed by spudcan, so it is important to determine the bearing capacity and the penetration depth of the spudcan for its geometry. In fact, it is up to the deformation law and the failure modes of soil surrounding the spudcan which can calculate the ultimate bearing capacity of the spudcan foundation on the soil seabed. By using the finite element analysis software Abaqus, the deformation law of soil around the spudcan is analyzed in detail, and the failure modes of soil surrounding the spudcan foundation are achieved. At the same time, based on the limit equilibrium theory, by use of static permissible slip-line field, the ultimate bearing capacity of the spudcan foundation is analyzed and the lower limit solution is derived theoretically, and the effect of the spudcan angle on the ultimate bearing capacity is investigated. The numerical results are compared with those obtained by the theoretical formulas deduced in this paper. On the basis of the lower limit solutions in this paper, the effect of the spudcan angle on the ultimate bearing capacity is revealed, and a practical bearing capacity formula is given to take the effect of the spudcan angle into consideration.


Introduction
In the process of offshore petroleum exploration and development, the jack-up platform is a commonly used mobile structure. The common features of such marine structures are their spudcan foundations, which are supported on the seabed. In order to accurately position the jack-up platform and its pile legs at the operating sea area and easily penetrate the surface hard soil layer, the bottom of the spudcan is usually not a plane, but having a certain angle generally called the spudcan angle in engineering. For the reason of the existence of the spudcan angle, the contact surface between the spudcan and seabed is no longer a traditional plane. At present, the theoretical hypothesis of the ultimate bearing capacity, which is suggested by Prandtl or Terzaghi, is based on the assumption that the contact surface between the rigid foundation and deformed seabed is plane. So the traditional formulas to calculate the bearing capacity of foundation have some limitations in calculating the bearing capacity of platform pile legs with the spudcan angle.
At present, in view of the research on the ultimate bearing capacity of spudcan on the seabed, most of the theoretic-al results mainly focus on the geometric simplification of the spudcan or the appropriate simplification of the failure mechanism of the seabed soil, and then a more practical, simplified calculation formula is proposed. Many scholars have analyzed the ultimate bearing capacity of the spudcan and studied the soil deformation law around the spudcan foundation. By adopting centrifuge model tests, Craign and Chua (1990) and Okamura et al. (1997) studied the effects of the soil parameters in detail such as the gravity, stiffness and soil thickness on the soil deformation surrounding the spudcan. De Santa Maria (1988), Martin (1994) and Tan (1990) also studied the effect of loading paths on the failure mode and the ultimate bearing capacity of spudcan using the centrifuge test. Using the centrifuge test and numerical analysis methods, Hossain et al. (2010) studied the influence of the pile leg length and preloading ratio on the bearing capacity in detail. The physical properties and mechanical parameters of the seabed soil are extremely complex, so the finite element numerical analysis method has been widely used in geotechnical engineering. Hu and Randolph (1998) used the large deformation algorithm of RITSS to analyze the interaction between the rigid spudcan and the elastiplastic deformed soil, and studied the flow deformation mechanism of soil surrounding the spudcan. Using the finite element method, Wang and Carter (2002) analyzed the gradual development of plastic soil around the spudcan, and studied the influence of the bulk density of soil on the ultimate bearing capacity. Houlsby and Martin (2003) analyzed the influence of the foundation buried depth and the change of soil shear strength on the ultimate bearing capacity, meanwhile modified the existing calculation formulas reasonably. Yi et al. (2012) studied the change law of excess pore pressure in soil around the spudcan using the Euler algorithm, and reached a conclusion that there was a strong intrinsic relationship between the bearing capacity and the soil void ratio. Zhang et al. (2011) studied the fluidsolid coupling mechanism between the pile legs and the undrained saturated soft clay in detail, and gave a formula to calculate the ultimate bearing capacity and the penetration depth of spudcan. Zhang et al. (2012) used the finite element method to study the deformation law of soil when the spudcan was penetrated into the seabed, and analyzed the forming process of the pile pit and the reflux law of soil around the pile leg. Zhang and Zhang (2015) adopted numerical analysis method to study the influence of the soil physical parameters on the deformation law of soil around the pile leg, and drew the conclusions that the friction angle had a significant effect on the bearing capacity of spudcan. Hossain and Randolph (2009) analyzed the failure mechanism of the spudcan foundation and its bearing capacity on single layer clay. Bienen et al. (2009) analyzed the ultimate pull-out loads of the spudcan foundation during its extraction. Zhang et al. (2014) studied the bearing capacity of spudcan in soft clay. Using experimental test, Vlahos et al. (2008) studied the bearing capacity of a three-legged jackup on clay. Hossain et al. (2011) studied the penetration mechanism of spudcan in multi-layered fine-grained soils. δ In this paper, the general finite element software Abaqus is used to study the deformation law of soil surrounding the spudcan, and the failure mechanism of soil is revealed in detail. Finally, in consideration of the published results, a static permissible stress field is constructed in this paper based on the plastic equilibrium principle, and the ultimate bearing capacity of spudcan is deduced theoretically by using the lower limit analysis method, meanwhile, the influence of the spudcan angle on the ultimate bearing capacity is analyzed in detail. The model of spudcan for the numerical calculation and theoretical analysis is shown in Fig. 1, where is the spudcan angle.

FEM model
In order to study the ultimate bearing capacity of spudcan of the jack-up platform, the general finite element soft-ware Abaqus is used to analyze the deformation law and failure mechanism of soil surrounding the spudcan foundation.

R δ
Considering that the spudcan model is axisymmetric in geometry, the finite element model is established as shown in Fig. 2. Assuming that the spudcan with the radius and angle is located on the undrained saturated seabed, and an ideal Mohr-Coulomb yielding criterion is adopted, the spudcan-soil interface is assumed to be completely smooth or completely rough which is simulated by using contact pairs and tie constraint in Abaqus respectively. Under smooth interface, the interaction is expressed using contact pairs with normal hard contact and tangential frictionless, and in the case of rough contact, the bottom surface of spudcan is tied with the top surface of seabed soil.

10R 20R
In order to reduce the boundary effect of the mesh model during the numerical simulation, the width and depth of seabed soil are and , respectively, the left and right displacement boundaries of this mesh are set to be fixed in horizontal and vertical directions, the bottom boundary of mesh is fixed in three degrees of freedom which are the horizontal, vertical and moment, and the top boundary of mesh is free. The finite element mesh model is shown in Fig. 2.
In view of the volume incompressible characteristics of undrained saturated soft clay, the second-order hybrid elements and reduction integral technique are adopted to elim-  inate the shear self-locking phenomenon during the numerical calculation process. In the actual operation process of the jack-up platform, the material strength of the spudcan is very large compared with the seabed soil, so the spudcan can be considered as a rigid body. During the process of the numerical calculation, the seabed soil is modeled as a uniform isotropic elastic-perfectly plastic continuum with the shear failure described by the Mohr-Coulomb yielding criterion, characterized by the limiting shear strength denoted as S u . And the soil elastic behavior is defined by a Poisson's ratio and a elastic modulus to shear strength of E/S u = 500. The physical and mechanical parameters of the saturated soft clay are listed in Table 1.

Numerical results
In order to study the soil deformation law and ultimate bearing capacity of the spudcan foundation when the rigid piles were penetrated into the seabed soil with different spudcan angles , some researches were carried out to study the bearing capacity of spudcan with the angles satisfying = 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 1.75, 2.00. The equivalent plastic shear strain of soil surrounding the rigid spudcan is shown in Fig. 3 when the contact surface between the spudcan and soil is smooth, and Fig. 4 shows the soil deformation law around the spudcan foundation under the condition that the contact surface is rough.

cot δ
The bearing capacity and soil deformation law are simulated by the finite element method, the soil deformation law around the spudcan is described and summarized in Fig. 3 and Fig. 4. The soil deformation surrounding the rigid spudcan is as follows. (1) With the increase of the cotangent of spudcan angle , the geometric contact interface between the spudcan and seabed soil enlarges gradually, the constraint soil deformation zone at the bottom of spudcan becomes smaller, and the passive failure zone increases gradually.
(2) In the case of the smooth interface, the tangential shear stress on the contact interface does not exist, so this interface can be considered as a principal stress plane, and the constraint soil deformation zone shrinks gradually until it disappears, and then the soil failure mode develops into active failure zone and passive failure zone. The failure mode of surrounding soil is similar to Prandtl mechanism.
(3) In the case of the rough interface, the constraint deformation zone enlarges gradually until it extends to the bottom of spudcan, and then the soil failure mode develops into the constraint deformation zone and passive failure zone. (4) Both in the smooth interface and rough interface, the passive general shear zone always appears the rigid translation which is not affected by the spudcan angle.
(5) Detailed comparion and analysis are made between the smooth and rough interface cases, and the contrast results indicate that the tangential friction stress on the spudcan surface obviously affects the soil deformation surrounding it. Under the tangential friction stress, the constraint soil together with the spudcan penetrated into the seabed is similar to the Terzaghi failure mode. So a conclusion can be drawn from the above numerical simulation that the failure modes of soil around the spudcan consist of the passive general shear zone, plastic shear deformation zone and constrained deformation zone. Then a reasonable static permissible stress field can be proposed to solve the slip-line equations, and a lower limit solution of the ultimate bearing capacity can be deduced theoretically.

Lower limit solution
On the basis of the equivalent plastic strain shown in Fig. 3 and Fig. 4, it can be found that the soil surrounding the spudcan is compressed and the plastic shear deforma- tion occurs. The shear failure mode is composed of three continuous zones which are the constrained deformation zone under the spudcan, the passive failure zone below the mudline and the plastic shear deformation zone in the middle. In view of the failure mode mentioned above, a static permissible stress field is established in this paper, which is based on the limit equilibrium theory and obeys the Mohr-Coulomb yielding criterion. Then a lower limit solution of the bearing capacity of the spudcan can be obtained theoretically.

Equations of the plastic limit equilibrium
In order to simplify the mathematical difficulty during the theoretical derivation, the bulk density of soil around the spudcan is not considered in this paper. So the stress equilibrium equations are as follows: , and are the soil stress state of a point in the seabed.

Plastic yielding criterion
In the process of the theoretical derivation, the Mohr-Coulomb yielding criterion of soil is adopted as follows: τ max S u where is equal to the undrained saturated shear strength .

Constitutive equations
In this paper, it is assumed that the Levy-Mises soil constitutive equation obeying the plastic increment theory is applied to simulate the soil deformation when the spudcan is penetrated into the seabed.
where , and are the soil strain rates of a point in the seabed respectively.

Condition of the incompressible volume
In the process of the spudcan penetration, the saturated soil satisfies the incompressible condition as follows: uv where and are the soil displacement rates of a point in the seabed, respectively.

Smooth contact interface
When the contact interface between the spudcan and soil is smooth, there is only the normal stress acting on the interface and no shear stress in the tangential direction, the contact surface is a principal stress plane shown in Fig. 5. CA stress boundary condition:

Ω
x θ x q f where represents the angle between the normal direction and axis; represents the angle between the maximum principal stress and axis; represents the maximum normal stress acting on the contact surface.

Rough contact interface
When the contact interface between the spudcan and soil is rough, there exist both the normal stress and shear stress in the tangential direction acting on the interface. The con- ZHANG Qi-yi, LIU Zhi-jie China Ocean Eng., 2018, Vol. 32, No. 4, P. 476-481 tact surface is no longer the principal stress plane, and the slip lines in the stress field are shown in Fig. 6. BG stress boundary condition: BO stress boundary condition: Ω = π/2, θ = π/4. (8)

Lower limit solution of the bearing capacity
x A simplified slip line equation can be obtained by establishing artificially the coordinate system in which the axis is along with the tangential direction of the contact surface, as shown in Fig. 5 and Fig. 6.

DAE
For the stress boundary conditions shown in Fig. 5, the zone of is the Riemann difference problem mathematically, and the difference equation can be described as: The solution of the normal stress of spudcan with the smooth surface is as follows: According to the diagram shown in Fig. 5b, the ultimate bearing capacity of the spudcan can be obtained.
where is the spudcan angle, is the undrained saturated shear strength of soil, and is the coefficient of the bearing capacity.

Rough surface β BOF
Based on the stress boundary condition shown in Fig. 6, the slip-line equation of Zone is established as: The solution of the spudcan surface pressure with the rough contact is calculated to be According to the stress diagram shown in Fig. 6b, the ultimate bearing capacity of spudcan can be written as:

Comparison and analysis
The lower limit solutions derived in this paper are compared with the finite element numerical results. The contrast analysis results are shown in Fig. 7 and Fig. 8.
By analyzing the ultimate bearing capacity curves shown in Fig. 7 and Fig. 8, it can be found that: (1) In both cases of the smooth interface and rough interface, the finite element numerical solution is higher than the lower limit solutions. In the course of the numerical calculation, the seabed which possesses infinite degree of freedom is simulated to be a numerical mesh artificially with finite degree of freedom, so the stiffness of numerical mesh is larger than the practical engineering, and then the numerical results of the ultimate bearing capacity are slightly larger than the theoretical solution.
(2) The numerical results of the ultimate bearing capacity of spudcan are consistent with the theoret-  (2 + π) ical solution, and the lower limit solutions given in this paper for the smooth and rough interface are practical calculation formulas.
(3) With the increase of the cotangent of the spudcan angle , the ultimate bearing capacity of spudcan shows a nonlinear decreasing trend, which indicates that the resistance of soil around the spudcan decreases gradually. (4) When the spudcan angle and the interface is smooth, the bottom surface of spudcan is plane, and the ultimate bearing capacity coefficient is equal to which is similar to the Prandtl solution shown in Fig. 7; when the interface is rough, the bearing capacity coefficient is equal to 5.71 shown in Fig. 8.

Conclusions
In this paper, the ultimate bearing capacity of spudcan and the soil deformation law are studied numerically by using the finite element analysis software Abaqus, and the failure mechanism of soil is analyzed from the microscopic point. At the same time, the ultimate bearing capacity of spudcan is derived by using the lower limit analysis method and solving the slip-line equations. Some conclusions are drawn as follows.
(1) In view of the smooth and rough interface, considering the change of the spudcan angle, the failure mode and deformation law of soil around spudcan are studied using the finite element method, which consists of the passive general shear zone, plastic shear deformation zone and constrained deformation zone.
(2) On basis of the equivalent plastic strain from the Abaqus numerical results, a static permissible stress field is given. By solving the slip-line equations, a lower limit theoretical solution is deduced in this paper as Eqs. (13) and (17), which can express the effect of the spudcan angle on the bearing capacity. (3) The ultimate bearing capacity coefficient proposed in this paper can indicate the change of spudcan angle , which can provide a certain reference value for further theoretical research on this problem.