*3.1. E*ff*ect of An Existing Footprint*

At first, the flat base footing is taken as an example to show how an existing footprint affects the resistance profile during installation and to show the soil flow mechanism. Zero depth is defined as the maximum cross-section area of footing touching seabed level. *V, H*, and *M* denote the vertical force, horizontal force, and bending moment acting on the reference point, respectively.

The *V, H*, and *M* profiles are plotted in Figure 7 and the soil flow mechanisms are shown in Figure 8. The positive *H* value means a horizontal force towards the footprint. The positive *M* value means an anti-clockwise bending moment acting on the reference point (RP).

In this case, the horizontal force comes from two parts: (1) When *z*/*D* < 0.15, with the penetration goes by, the soil under the footing is pushed into the footprint while the footing is constrained without horizontal movements. The relative motion provides a friction force towards the footprint (as the yellow arrow shown in Figure 8a. When *z*/*D* < 0.15, the maintained zero *H*/*AS*<sup>u</sup> value of case TB-2*D*-1.0*D* (smooth) as shown in Figure 7a confirms this conclusion. (2) As the footing penetrates deeper (*z*/*D* > 0.15), the soil on the footing's right side is compressed, which provides a leftward earth pressure (as the green arrows shown in Figure 8c. When the footing reaches the toe of the footprint, 0.33*D*, the soil on the footing's left side heaves up and provides a rightward earth pressure. After that, the total horizontal force reduces. When the penetration depth reaches ~0.8*D*, the soil on both the left and right sides show a symmetric fully flowing back mechanism, as shown in Figure 8d. The symmetric soil flow mechanism results in that the *H* and *M* values are much smaller.

*M* peaks as soon as the footing touches the seabed (around *z*/*D* = 0.02). This is because that at this depth the eccentric distance of the resultant vertical resistance force is very large, as shown in Figure 8a, although the vertical resistance is far from the peak value at this depth. As the penetration depth increases, the eccentric distance of vertical force reduces and as a result the corresponding bending moment obviously reduces.

Compared with the centrifuge test results from Kong [25], the horizontal force profile of Kong's lies between the smooth and rough cases of this study (Figure 7a) because the friction characteristic of the centrifuge test on the interface of aluminum footing and the soil is between rough and smooth. It can be seen that the numerical results of this study and the centrifuge test results from Kong [25] have very similar *H* and *M* profile trends.

**Figure 7.** Effect of existing footprint on the *V*, *H*, and *M* responses of the flat base footing.

#### *J. Mar. Sci. Eng. J. Mar. Sci. Eng.***2019 2019**, *7* , , 175 *7*, 175 7 of 19

(**d**) *z*/*D* = 0.8

**Figure 8.** Soil flow mechanism of cases FS (**left**) and TB-2*D*-1.0*D* (**right**) for flat base footing. **Figure 8.** Soil flow mechanism of cases FS (**left**) and TB-2*D*-1.0*D* (**right**) for flat base footing.

*3.2. Effect of An Existing Footprint* 

#### *3.2. E*ff*ect of An Existing Footprint J. Mar. Sci. Eng.* **2019**, *7*, 175 8 of 19

In Abaqus/CEL, the basic coulomb friction model defines the maximum allowable friction (shear) stress across an interface to the contact pressure stress, τmax, as a function of the contact pressure: In Abaqus/CEL, the basic coulomb friction model defines the maximum allowable friction (shear) stress across an interface to the contact pressure stress, *τ*max, as a function of the contact pressure:

max

$$
\pi\_{\text{max}} = \mu \,\, p \tag{1}
$$

in which *p* is the contact pressure and µ is a friction coefficient that can be any non-negative value. In some special cases, the contact pressure *p* might be so large that τmax = µ *p* exceeds the yield stress in the material beneath the contact surface, thus a shearing limit value, τlimit, is adopted to avoid this situation. Regardless of the magnitude of the contact pressure stress, sliding will occur if the magnitude of the equivalent shear stress reaches τlimit. When both τmax and τlimit exceed *s*<sup>u</sup> (yield stress), the maximum allowable friction (shear) stress equals *s*u. All in all, the µ value only affects the friction force before the contact pressure reaches *p*<sup>2</sup> = *s*u/µ. After that, the friction force would be equal to ~*s*<sup>u</sup> due to the yielding of clay. The relationship between equivalent shear stress and the contact pressure is plotted in Figure 9. in which *p* is the contact pressure and *μ* is a friction coefficient that can be any non-negative value. In some special cases, the contact pressure *p* might be so large that *τ*max *= μ p* exceeds the yield stress in the material beneath the contact surface, thus a shearing limit value, *τ*limit, is adopted to avoid this situation. Regardless of the magnitude of the contact pressure stress, sliding will occur if the magnitude of the equivalent shear stress reaches *τ*limit. When both *τ*max and*τ*limit exceed *s*u (yield stress), the maximum allowable friction (shear) stress equals *s*u. All in all, the *μ* value only affects the friction force before the contact pressure reaches *p*2 = *s*u/*μ*. After that, the friction force would be equal to ~*s*<sup>u</sup> due to the yielding of clay. The relationship between equivalent shear stress and the contact pressure is plotted in Figure 9.

**Figure 9.** Behavior of the contact element in Abaqus/CEL. **Figure 9.** Behavior of the contact element in Abaqus/CEL.

A simple test model, as shown in Figure 10, is created to verify the accuracy of calculating friction force by Abaqus. All the parameters are detailed in Figure 10. The three anchors are disconnected and go right at a speed of 0.1 m/s at the same time. Only the friction force of anchor 2 on the contact surface surf 2 are considered, anchor 1 and anchor 3 are created to eliminate the influence of backflow soil. An empty element layer of 5 m thick at the top of the soil is set to allow soil heaving. A simple test model, as shown in Figure 10, is created to verify the accuracy of calculating friction force by Abaqus. All the parameters are detailed in Figure 10. The three anchors are disconnected and go right at a speed of 0.1 m/s at the same time. Only the friction force of anchor 2 on the contact surface surf 2 are considered, anchor 1 and anchor 3 are created to eliminate the influence of back-flow soil. An empty element layer of 5 m thick at the top of the soil is set to allow soil heaving.

In the simple test model, the friction coefficient is set to *μ* = 10,000 and the shearing limit value is set to *τ*limit = 5.5 kPa (larger than *s*u = 5 kPa). According to Equation (1), *τ*max = *μp* = 10,000 × 70 = 700 MPa, which is far greater than *τ*limit. The cases with different mesh size and calculated results are listed in Table 2 and plotted in Figure 11. It can be seen that the calculated friction force is a little lower than the theoretical solution, which may be because of the fractional volume method in CEL. The numerical friction force is getting close to the theoretical solution as the mesh density increases. When the minimum element size is *b*min/*B* =1/30, the calculation error is 6%, which is selected in the following analyses. In the simple test model, the friction coefficient is set to µ = 10,000 and the shearing limit value is set to τlimit = 5.5 kPa (larger than *s*<sup>u</sup> = 5 kPa). According to Equation (1), τmax = µ*p* = 10,000 × 70 = 700 MPa, which is far greater than τlimit. The cases with different mesh size and calculated results are listed in Table 2 and plotted in Figure 11. It can be seen that the calculated friction force is a little lower than the theoretical solution, which may be because of the fractional volume method in CEL. The numerical friction force is getting close to the theoretical solution as the mesh density increases. When the minimum element size is *b*min/*B* = 1/30, the calculation error is 6%, which is selected in the following analyses.

contributes to *H*.

*J. Mar. Sci. Eng.* **2019**, *7*, 175 9 of 19

**Figure 10.** The test model. **Figure 10.** The test model. **Figure 10.** The test model.



 mesh=*B*/80 **Figure 11.** Numerical friction force in test model. **Figure 11.** Numerical friction force in test model.

0.0

**Figure 11.** Numerical friction force in test model. After investigating the behavior of the friction element in CEL, the effects of soil–structure friction on the *V H* and *M* of a spudcan penetrating near an existing footprint are carried out. The friction coefficient is set to *μ* = 0 and 10,000 to represent smooth and rough conditions respectively. After investigating the behavior of the friction element in CEL, the effects of soil–structure friction on the *V H* and *M* of a spudcan penetrating near an existing footprint are carried out. The friction coefficient is set to *μ* = 0 and 10,000 to represent smooth and rough conditions respectively. The shearing limit value is set as the undrained shear strength of the surrounding clay. After investigating the behavior of the friction element in CEL, the effects of soil–structure friction on the *V H* and *<sup>M</sup>* of a spudcan penetrating near an existing footprint are carried out. The frictioncoefficient is set to <sup>µ</sup> <sup>=</sup> 0 and 10,000 to represent smooth and rough conditions respectively. The shearing limit value is set as the undrained shear strength of the surrounding clay.

The shearing limit value is set as the undrained shear strength of the surrounding clay. Comparing the smooth and rough cases, it can be seen that some certain friction has a significant effect on *H* profile, but no obvious effects on *V* and *M*, as shown in Figure 12. The friction condition does not affect the location where *H*max and *M*max occur. The soil flow mechanism in Figure 13 explains how the friction affects *H* profile. For the smooth case, *H* is only from the lateral pushing force of the soil on the right side of the footing. While for the rough case, the friction on the footing bottom also Comparing the smooth and rough cases, it can be seen that some certain friction has a significant effect on *H* profile, but no obvious effects on *V* and *M*, as shown in Figure 12. The friction condition does not affect the location where *H*max and *M*max occur. The soil flow mechanism in Figure 13 explains how the friction affects *H* profile. For the smooth case, *H* is only from the lateral pushing force of the soil on the right side of the footing. While for the rough case, the friction on the footing bottom also contributes to *H*. Comparing the smooth and rough cases, it can be seen that some certain friction has a significant <sup>e</sup>ffect on *<sup>H</sup>* profile, but no obvious effects on *<sup>V</sup>* and *<sup>M</sup>*, as shown in Figure 12. The friction conditiondoes not affect the location where *<sup>H</sup>*max and *<sup>M</sup>*max occur. The soil flow mechanism in Figure <sup>13</sup> explains how the friction affects *<sup>H</sup>* profile. For the smooth case, *<sup>H</sup>* is only from the lateral pushing force of thesoil on the right side of the footing. While for the rough case, the friction on the footing bottom also contributes to *H*.

*z* /

*D*

**Figure 12.** The *V*, *H,* and *M* profile during smooth and rough conditions. (TB−2*D*−0.25*D*). **Figure 12.** The *V*, *H,* and *M* profile during smooth and rough conditions. (TB−2*D*−0.25*D*).

**Figure 12.** The *V*, *H,* and *M* profile during smooth and rough conditions. (TB−2*D*−0.25*D*).

(**a**) smooth (**a**) smooth

(**b**) rough (**b**) rough

**Figure 13.** The soil flow mechanism of (**a**) smooth and (**b**) rough conditions. (TB−2*D*−0.25*D*). **Figure 13.** The soil flow mechanism of (**a**) smooth and (**b**) rough conditions. (TB−2*D*−0.25*D*). **Figure 13.** The soil flow mechanism of (**a**) smooth and (**b**) rough conditions. (TB−2*D*−0.25*D*).

The maximum normalized values of *H* and *M* of flat base footing are summarized in Figure 14. It can be seen clearly that the maximum *H* value of rough cases is around three times of that of smooth case, while the friction condition has a much smaller effect on *M*max values (increasing 1.2 to 1.4 times). The maximum normalized values of *H* and *M* of flat base footing are summarized in Figure 14. It can be seen clearly that the maximum *H* value of rough cases is around three times of that of smooth case, while the friction condition has a much smaller effect on *M*max values (increasing 1.2 to 1.4 times). The maximum normalized values of *H* and *M* of flat base footing are summarized in Figure 14. It can be seen clearly that the maximum *H* value of rough cases is around three times of that of smooth case, while the friction condition has a much smaller effect on *M*max values (increasing 1.2 to 1.4 times).

**Figure 14.** Maximum normalized *H* and *M* values against eccentricity ratio (flat base footing). **Figure 14.** Maximum normalized *H* and *M* values against eccentricity ratio (flat base footing). **Figure 14.** Maximum normalized *H* and *M* values against eccentricity ratio (flat base footing).

### *3.3. Effect of the Location of the Reference Point (Working Leg Length) 3.3. E*ff*ect of the Location of the Reference Point (Working Leg Length)*

*3.3. Effect of the Location of the Reference Point (Working Leg Length)*  For the convenience of discussion, the *V*, *H,* and *M* discussed above are obtained using a reference point (see RP in Figure 4). If the reference point is located at the leg–hull connection section (see RP0 in Figure 15), an additional moment (*M*a) will be mobilized by *H* and its eccentricity (i.e., the working leg length), while the horizontal and vertical forces are not affected by the location of RP. In For the convenience of discussion, the *V*, *H,* and *M* discussed above are obtained using a reference point (see RP in Figure 4). If the reference point is located at the leg–hull connection section (see RP0 in Figure 15), an additional moment (*M*a) will be mobilized by *H* and its eccentricity (i.e., the working leg length), while the horizontal and vertical forces are not affected by the location of RP. In For the convenience of discussion, the *V*, *H,* and *M* discussed above are obtained using a reference point (see RP in Figure 4). If the reference point is located at the leg–hull connection section (see RP<sup>0</sup> in Figure 15), an additional moment (*M*a) will be mobilized by *H* and its eccentricity (i.e., the working leg length), while the horizontal and vertical forces are not affected by the location of RP. In practical engineering cases, the leg–hull connector section could be the most dangerous section.

practical engineering cases, the leg–hull connector section could be the most dangerous section. Assuming that the top head of the leg is fully fixed and the footing is considered as a rigid body, the additional bending moment (*M*a) at RP0 due to the horizontal force acting on the footing can be calculated as *M*a = *H* \* *L*w-leg. The bending moment at RP0 (*M*hull) varies with the working leg length. The total moment at RP0 (*M*hull) can, therefore, be calculated as *M*hull = *M* + *M*a. The maximum value of both horizontal force (*H*max) and bending moment (*M*max) are taken as the most unfavorable combination of loads to calculate the bending moment on the leg–hull connection at different working leg lengths. As an example, the profile of *M*hull of the flat base footing reinstalling near the TA footprint is plotted in Figure 16. It can be seen that *M*hull is within a positive value at a small leg length, which means an anticlockwise moment. With the increasing of the working leg length, *M*<sup>a</sup> practical engineering cases, the leg–hull connector section could be the most dangerous section. Assuming that the top head of the leg is fully fixed and the footing is considered as a rigid body, the additional bending moment (*M*a) at RP0 due to the horizontal force acting on the footing can be calculated as *M*a = *H* \* *L*w-leg. The bending moment at RP0 (*M*hull) varies with the working leg length. The total moment at RP0 (*M*hull) can, therefore, be calculated as *M*hull = *M* + *M*a. The maximum value of both horizontal force (*H*max) and bending moment (*M*max) are taken as the most unfavorable combination of loads to calculate the bending moment on the leg–hull connection at different working leg lengths. As an example, the profile of *M*hull of the flat base footing reinstalling near the TA footprint is plotted in Figure 16. It can be seen that *M*hull is within a positive value at a small leg length, which means an anticlockwise moment. With the increasing of the working leg length, *M*<sup>a</sup> increases linearly and, as a result, the total moment *M*hull decreases. When *L*w-leg is less than ~30 m, the Assuming that the top head of the leg is fully fixed and the footing is considered as a rigid body, the additional bending moment (*M*a) at RP<sup>0</sup> due to the horizontal force acting on the footing can be calculated as *M*<sup>a</sup> = *H* \* *L*w-leg. The bending moment at RP<sup>0</sup> (*M*hull) varies with the working leg length. The total moment at RP<sup>0</sup> (*M*hull) can, therefore, be calculated as *M*hull = *M* + *M*a. The maximum value of both horizontal force (*H*max) and bending moment (*M*max) are taken as the most unfavorable combination of loads to calculate the bending moment on the leg–hull connection at different working leg lengths. As an example, the profile of *M*hull of the flat base footing reinstalling near the TA footprint is plotted in Figure 16. It can be seen that *M*hull is within a positive value at a small leg length, which means an anticlockwise moment. With the increasing of the working leg length, *M*<sup>a</sup> increases linearly and, as a result, the total moment *M*hull decreases. When *L*w-leg is less than ~30 m, the total

value of the clockwise *M*hull would be larger than the anticlockwise *M*hull at *L*w-leg = 0.

increases linearly and, as a result, the total moment *M*hull decreases. When *L*w-leg is less than ~30 m, the total moment is within a negative range (clockwise). With further increasing of *L*w-leg, the absolute

total moment is within a negative range (clockwise). With further increasing of *L*w-leg, the absolute

moment is within a negative range (clockwise). With further increasing of *L*w-leg, the absolute value of the clockwise *M*hull would be larger than the anticlockwise *M*hull at *L*w-leg = 0. *J. Mar. Sci. Eng.* **2019**, *7*, 175 12 of 19 *J. Mar. Sci. Eng.* **2019**, *7*, 175 12 of 19

Considering working leg length, the bending moment, *M*hull, is a combination of the *H* and *M* at RP. To simplify the discussions, only the moments at the lower end of the leg (Section 1.1), M1-1, are presented. Considering working leg length, the bending moment, *M*hull, is a combination of the *H* and *M* at RP. To simplify the discussions, only the moments at the lower end of the leg (Section 1.1), M1-1, are presented. Considering working leg length, the bending moment, *M*hull, is a combination of the *H* and *M* at RP. To simplify the discussions, only the moments at the lower end of the leg (Section 1.1), M1-1, are presented.

**Figure 15.** Shift load reference point. **Figure 15.** Maximum normalized *H* and *M* values against eccentricity ratio (flat base footing). **Figure 15.** Shift load reference point.

**Figure 16.** The bending moment on the leg−hull connection section (flat base footing, TB). **Figure 16.** The bending moment on the leg−hull connection section (flat base footing, TB). **Figure 16.** Maximum normalized *H* and *M* values against eccentricity ratio (flat base footing).

#### *3.4. Effect of Footprint Geometry 3.4. Effect of Footprint Geometry 3.4. E*ff*ect of Footprint Geometry*

The resistance profiles of spudcan penetrating through the edge of footprints TA, TB, and TC are presented in Figure 17*,* in which the offset distance is 0.75*D*. The resistance profiles of spudcan penetrating through the edge of footprints TA, TB, and TC are presented in Figure 17*,* in which the offset distance is 0.75*D*. The resistance profiles of spudcan penetrating through the edge of footprints TA, TB, and TC are presented in Figure 17*,* in which the offset distance is 0.75*D*.

As expected, the deeper the footprint is, the more effect it has on the reinstallation resistance profiles. All three *H* Profiles have the same trend, but the case with steeper slope causes higher *H* values. The *H*max value for TC is about 3–5 times higher than that for TA. The deeper the footprint is, the longer it takes for *H* to reduce to zero. As expected, the deeper the footprint is, the more effect it has on the reinstallation resistance profiles. All three *H* Profiles have the same trend, but the case with steeper slope causes higher *H* values. The *H*max value for TC is about 3–5 times higher than that for TA. The deeper the footprint is, the longer it takes for *H* to reduce to zero. As expected, the deeper the footprint is, the more effect it has on the reinstallation resistance profiles. All three *H* Profiles have the same trend, but the case with steeper slope causes higher *H* values. The *H*max value for TC is about 3–5 times higher than that for TA. The deeper the footprint is, the longer it takes for *H* to reduce to zero.

The bending moment at Section 1.1 (*M*1-1) can be derived according to the *V, H,* and *M* values acting on the Reference Point. The vertical force acting on the RP has no contribution on the bending moment at the section, *M* has a positive contribution, and *H* times distance has a negative contribution. The maximum *M*1-1 values occur at a very shallow depth. With further penetration, the horizontal force becomes larger and plays a leading role in *M*1-1 value. This results in that the positive *M*1-1 reduces gradually to negative in Figure 17, with an increasing penetration depth. The bending moment at Section 1.1 (*M*1-1) can be derived according to the *V, H,* and *M* values acting on the Reference Point. The vertical force acting on the RP has no contribution on the bending moment at the section, *M* has a positive contribution, and *H* times distance has a negative contribution. The maximum *M*1-1 values occur at a very shallow depth. With further penetration, the horizontal force becomes larger and plays a leading role in *M*1-1 value. This results in that the positive *M*1-1 reduces gradually to negative in Figure 17, with an increasing penetration depth. The bending moment at Section 1.1 (*M*1-1) can be derived according to the *V, H,* and *M* values acting on the Reference Point. The vertical force acting on the RP has no contribution on the bending moment at the section, *M* has a positive contribution, and *H* times distance has a negative contribution. The maximum *M*1-1 values occur at a very shallow depth. With further penetration, the horizontal force becomes larger and plays a leading role in *M*1-1 value. This results in that the positive *M*1-1 reduces gradually to negative in Figure 17, with an increasing penetration depth.

(**a**) Effect of footprint geometry on flat base footing (*β =* 0.75*D*)

(**b**) Effect of footprint geometry on fusiform spudcan footing (*β* = 0.75*D*)

(**c**) Effect of footprint geometry on skirted footing (*β* = 0.75*D*)

**Figure 17.** Effect of footprint geometry on three types of footings. **Figure 17.** Effect of footprint geometry on three types of footings.

#### *3.5. Effect of Footings' Geometry Shape and Offset Distance 3.5. E*ff*ect of Footings' Geometry Shape and O*ff*set Distance*

The *H* and *M* profiles of the three footings reinstalling at selected typical offset distances are shown in Figure 18. The *H* can be separated into two parts. The first is the horizontal component of normal contact force between the footprint slope and the right side of footing, which is the primary cause of the first peak shown in Figure 19. The second is the lateral pushing force from the right-side The *H* and *M* profiles of the three footings reinstalling at selected typical offset distances are shown in Figure 18. The *H* can be separated into two parts. The first is the horizontal component of normal contact force between the footprint slope and the right side of footing, which is the primary cause of the first peak shown in Figure 19. The second is the lateral pushing force from the right-side soil caused by the asymmetry soil flowing, which is the primary cause of the second peak. For a flat base footing or a skirted footing, the horizontal component of normal contact force is relatively small, since the footing base is horizontal. However, for a fusiform spudcan footing, due to the inverted conical shape, the first part of *H* force plays a leading role in *H* profile. After deep penetration, the geometry shape has a minor effect on the resistance, since the soil flow mechanisms are both fully back flow left and right. base footing or a skirted footing, the horizontal component of normal contact force is relatively small, since the footing base is horizontal. However, for a fusiform spudcan footing, due to the inverted conical shape, the first part of *H* force plays a leading role in *H* profile. After deep penetration, the geometry shape has a minor effect on the resistance, since the soil flow mechanisms are both fully back flow left and right.

(**c**) Skirted footing

**Figure 18.** The soil flow mechanism for different footings (TB-2*D*-0.25*D*). **Figure 18.** The soil flow mechanism for different footings (TB-2*D*-0.25*D*).

The *H*max and *M*1-1max values of all the cases in this study are listed in Table A1 in Appendix A and plotted in Figure 20. For flat base footings and skirted footings, both *H*max and *M*1-1max are significant when *β*/*D* = 0.25 to 1.25. When *β*/*D* ≥ 1.5, the value of *M*1-1max reduces to zero, while *H*max still remains at significant values. For fusiform spudcan footings, both *H*max and *M*1-1max are significant when *β*/*D* = 0.25 to 0.5. From the perspective of the footing shape, the flat base footing gives the lowest *H*max but the largest *M*1-1max, and the performances of the fusiform spudcan footing and the skirted footing are similar. The *<sup>H</sup>*max and *<sup>M</sup>*1-1max values of all the cases in this study are listed in Table A1 in Appendix <sup>A</sup> andplotted in Figure 20. For flat base footings and skirted footings, both *<sup>H</sup>*max and *<sup>M</sup>*1-1max are significant when β/*D* = 0.25 to 1.25. When β/*D* ≥ 1.5, the value of *M*1-1max reduces to zero, while *H*max still remains at significant values. For fusiform spudcan footings, both *H*max and *M*1-1max are significant whenβ/*D* = 0.25 to 0.5. From the perspective of the footing shape, the flat base footing gives the lowest *H*max but the largest *<sup>M</sup>*1-1max, and the performances of the fusiform spudcan footing and the skirted footingare similar.

It is worthwhile to note that the thickness of the skirt for the skirted footing of the numerical

It is worthwhile to note that the thickness of the skirt for the skirted footing of the numerical model is higher than in situ skirted footing in order to mitigate numerical divergence. That might cause an overprediction on resistance loads. The effect of the skirt thickness can be ignored when the base level (with the maximum cross-section area) of the skirted footing fully touches the soil. *J. Mar. Sci. Eng.* **2019**, *7*, 175 15 of 19 cause an overprediction on resistance loads. The effect of the skirt thickness can be ignored when the base level (with the maximum cross-section area) of the skirted footing fully touches the soil.

(**b**) The *H* and *M*1-1 profile of fusiform spudcan footing at offsets 0.5*D*, 0.75*D,* and 1.0*D*

(**c**) The *H* and *M* profile of skirted footing at offsets 0.5*D*, 1.0*D,* and 1.5*D*

**Figure 19. Figure 19.** The The *H* and *H M*1-1 profile of three kind of footings at different offsets. and *M*1-1 profile of three kind of footings at different offsets.

(**c**) Skirted footing

**Figure 20.** Maximum values of *H* and *M* against eccentricity ratio. **Figure 20.** Maximum values of *H* and *M* against eccentricity ratio.

#### *3.6. Resultant Force of V H and M 3.6. Resultant Force of V H and M*

spudcan footings.

The above analyses are based on *V H* and *M* values at the reference point. To provide another view, the *V, H,* and *M* values of each case can be transformed into one resultant force acting on a point at the footing base level. The resultant force has an inclination of *α =* tan−1 (*H*/*V*) to the vertical line and an offset ratio of *e*/*D = M*/*VD* to the central line of the footing. From Figure 21, it can be seen that when *z* > 0m, the load inclination *α* and eccentricity *e*/*D* of skirted footing is smaller than that of fusiform spudcan footing. When *z* < 0 m, although both *α* and *e*/*D* of the skirted footing are larger, the vertical force is relatively small and the force acting on the footing may not be sufficient to cause structure failure. That is to say, the skirted footings may have a certain potential in resisting the damage during reinstallation near existing footprints, by comparing with commonly used fusiform The above analyses are based on *V H* and *M* values at the reference point. To provide another view, the *V, H*, and *M* values of each case can be transformed into one resultant force acting on a point at the footing base level. The resultant force has an inclination of α = tan−<sup>1</sup> (*H*/*V*) to the vertical line and an offset ratio of *e*/*D* = *M*/*VD* to the central line of the footing. From Figure 21, it can be seen that when *z* > 0 m, the load inclination α and eccentricity *e*/*D* of skirted footing is smaller than that of fusiform spudcan footing. When *z* < 0 m, although both α and *e*/*D* of the skirted footing are larger, the vertical force is relatively small and the force acting on the footing may not be sufficient to cause structure failure. That is to say, the skirted footings may have a certain potential in resisting the damage during reinstallation near existing footprints, by comparing with commonly used fusiform spudcan footings.

**Figure 21.** Variations in the (**a**) load inclination and (**b**) load eccentricity during the reinstallation process of the β = 0.5*D*, 1.0*D* cases. **Figure 21.** Variations in the (**a**) load inclination and (**b**) load eccentricity during the reinstallation process of the β = 0.5*D*, 1.0*D* cases.

#### **4. Conclusions 4. Conclusions**

This paper carried out large deformation finite element analyses to investigate the effect of an existing footprint on the stability of jack-ups' reinstallation. The following conclusions can be drawn according to the present numerical analyses: The friction condition of the soil–footing interface has a significant effect on *H* profile but much This paper carried out large deformation finite element analyses to investigate the effect of an existing footprint on the stability of jack-ups' reinstallation. The following conclusions can be drawn according to the present numerical analyses:

less effect on *M* profile. The deeper is the footprint, the more effect it has on both *H* and *M* profiles. The eccentricity ratio is a key factor to evaluate *H*max and *M*1-1max. For flat base footings and skirted The friction condition of the soil–footing interface has a significant effect on *H* profile but much less effect on *M* profile. The deeper is the footprint, the more effect it has on both *H* and *M* profiles.

footings, both *H*max and *M*1-1max are significant when *β*/*D* = 0.25 to 1.25. The value of *M*1-1max reduces to zero when *β*/*D* ≥ 1.5, while *H*max still remains at a significant value. For fusiform spudcan footings, both *H*max and *M*1-1max are significant when *β*/*D* = 0.25 to 0.5. The geometry shape of the footing also has a certain effect on the *V, H,* and *M* profiles. The flat The eccentricity ratio is a key factor to evaluate *H*max and *M*1-1max. For flat base footings and skirted footings, both *H*max and *M*1-1max are significant when β/*D* = 0.25 to 1.25. The value of *M*1-1max reduces to zero when β/*D* ≥ 1.5, while *H*max still remains at a significant value. For fusiform spudcan footings, both *H*max and *M*1-1max are significant when β/*D* = 0.25 to 0.5.

base footing gives the lowest *H*max but the largest *M*1-1max, and the performances of the fusiform spudcan footing and the skirted footing are similar. From the view of the resultant forces, both *α* and *e*/*D* of the skirted footing are only large before the base level (with the maximum cross-section area) fully touches the soil, which shows a certain potential in resisting the damage during reinstallation near existing footprints by comparing with commonly used fusiform spudcan footings. The bending moment on the leg–hull connection (*M*hull) at different working leg lengths (*L*w-leg) is discussed. When *L*w-leg is less than ~30 m, the total moment is within a negative range (clockwise). The geometry shape of the footing also has a certain effect on the *V, H,* and *M* profiles. The flat base footing gives the lowest *H*max but the largest *M*1-1max, and the performances of the fusiform spudcan footing and the skirted footing are similar. From the view of the resultant forces, both α and *e*/*D* of the skirted footing are only large before the base level (with the maximum cross-section area) fully touches the soil, which shows a certain potential in resisting the damage during reinstallation near existing footprints by comparing with commonly used fusiform spudcan footings.

With further increasing of *L*w-leg, the absolute value of the clockwise *M*hull would be larger than the anticlockwise *M*hull at *L*w-leg = 0. In this study, the artificial footprints were adopted to simplify the problem neglecting the disturbance of the soil during initial spudcan penetration. In the further study, the soil profiles, soil The bending moment on the leg–hull connection (*M*hull) at different working leg lengths (*L*w-leg) is discussed. When *L*w-leg is less than ~30 m, the total moment is within a negative range (clockwise). With further increasing of *L*w-leg, the absolute value of the clockwise *M*hull would be larger than the anticlockwise *M*hull at *L*w-leg = 0.

In this study, the artificial footprints were adopted to simplify the problem neglecting the disturbance of the soil during initial spudcan penetration. In the further study, the soil profiles, soil properties, geometry of footprints and spudcans, leg details, use of spigots (or not) etc. should be noted as a factor to consider in site-specific analyses.

**Author Contributions:** Data curation, X.W.; formal analysis, H.Z.; software, J.L.; supervision, L.Y.

**Funding:** This study was supported by the Chinese National Natural Science Foundation (51890915, 51639002, 51539008 and 51679038).

**Conflicts of Interest:** The authors declare no conflict of interest.
