#### Abstract

This paper proposed a method for analysis of a drilled pile under vertical load at the crest of rock slope. Based on wedge theory, a modified model of normal stiffness of socket wall affected by the slope is obtained. Analyze the shear behaviors of the pile-rock interface, an analytical solution of load transfer of pile at the crest of rock slope is obtained. To evaluate the accuracy of the new method, this method is compared with the results of finite difference analysis. Finally, the method is used to analyze the effect of slope, pile, and rock properties on the unit side resistance and axial force.

#### 1. Introduction

Piles, which are widely used in offshore drillings, bridges, and other structures, are often embedded in rocks. Foundations constructed in level ground near continental, nearshore slopes, or man-made slope are inevitable in engineering practice. Because of slope, increase in settlement is observed [1, 2].

To analyze load transfer of pile under vertical loads, a series of method have been proposed. Laboratory test [3–7] could evaluate the vertical behaviors of pile; however, there are differences between small scale model pile test and engineering practice. Numerical method [1, 2, 8, 9] could simulate various engineering conditions, but parameters of sophisticated soil constitutive relations are hard to obtain. Theoretical method [10–23] could quickly obtain the relationship between load and settlement under various working conditions.

Jiang et al. [1, 2] used theoretical method to investigate the behaviors of rigid piles in sloping ground and obtained the effect of slope on the settlement. Jesmani et al. [24] investigated the effect of different types of clayey slopes on vertical bearing capacity based on finite element. All these methods applied to the soil slope; however, there are differences between soil and rock. For short pile in soil, structural loads are carried by base resistance, but for long pile embedded in rock, the bearing capacity almost depends on side resistance.

Johnston and Lam [25] analyzed pile-rock surface and proposed the theory of dilation energy. Based on the theory of dilation energy, many scholars [26–29] have studied the shear behavior of pile-rock interface. Zhao MH et al. [30], Xing HF et al. [31], and Zhao H [32] investigated the principal mechanisms of shear transfer between piles and rock, and the failure surfaces were assumed to line and slip-line field, respectively. All these methods obtained the effect of the concrete-rock surface on the behaviors. But for the pile at the crest of rock slope, because of slope, the normal stiffness of socket wall will decrease. To accurately evaluate the behaviors of pile at the crest of rock slope, the decrease of normal stiffness of socket wall is the key to the principal mechanisms of shear transfer between piles and rock.

To obtain the effect of slope on normal stiffness of socket wall, the wedge theory is used to illustrate it. An analytical solution of load transfer of pile at the crest of rock slope is obtained by analyzing the shear behaviors of the pile-rock interface. To evaluate the accuracy of the new method which is analysis of a drilled pile under vertical load at the crest of rock slope, this method is compared with the results of finite element analysis. Finally, the method is used to analyze the effect of slope, pile, and rock properties on the unit side resistance and axial force.

#### 2. An Analytical Solution of Load Transfer of Pile at the Crest of Rock Slope considering the Shear Behaviors of the Pile-Rock Interface

##### 2.1. Initial Normal Stiffness Equation Based on Wedge Theory

The basic assumptions of this paper are as follows: For the pile at the crest of rock slope, because of slope, the normal stiffness of socket wall will decrease. The schematic of the slope-pile model analyzed is illustrated in Figure 1. is the Cartesian coordinate system, and the direction is perpendicular to the - plane. The stress of the rock around the pile under the self-weight is .

Based on the wedge stress theory, the three-direction stress relationship of rock under the self-weight at the crest of slope is:

where ,, and are the elastic stress solutions of the rock in the Cartesian coordinate system, is Poisson’s ratio of rock, and is the complementary angle of .

The initial tangent modulus equation of the pile at the crest of slope is:

where is the initial tangent modulus, is the compression modulus, and and are the parameters defined as:

The initial tangent modulus equation of the pile in level ground is:

The normal restraint stiffness of the pile in level ground can be expressed as:

where is the radius of pile.

For the pile at the crest of rock slope, the normal stiffness of pile can be obtained by Eqs. (2)–(6):

where is the ratio of the initial tangent modulus at the crest of slope to that in level ground; it can be expressed as:

is a function related to the Poisson’s ratio , slope angle , and embedded depth . In order to obtain a more concise expression for , calculate the value of in various situations.

Figure 2 shows the corresponding value of when Poisson’s ratio , 0.25, 0.3, 0.35, 0.4; slope angle , 2.5°, 5°, 7.5°,...,30°; and embedded depths m, 4 m, 6 m,...,20 m. The distribution of has a certain regularity, and it can obtained.

is negatively correlated with the slope angle and positively correlated with the Poisson’s ratio of the rock. The equation for was fitted by Matlab as:

For the pile at the crest of rock slope, the normal stiffness of pile can be obtained by Eqs. (7)–(9):

##### 2.2. Shear Behavior of the Pile-Rock Interface

###### 2.2.1. Assumption

Piles will be embedded in weathered rock of slopes, and its bearing capacity almost depends on side resistance. Relative slippage of concrete-rock interface occurs when the pile embedded in rocky slope is subjected to vertical load. Since the concrete-rock interface is rough, the slippage is often accompanied by radial expansion of the pile. The normal stress increases subsequently, and unit side resistance of pile is working. When pile embedded in slope is loaded, the sketch of relative shear motion of asperity is shown in Figure 3.

To formulate shear behaviors of pile-rock interface and obtain solution of load transfer of pile at the crest of rock slope, assumptions of this paper are made in advance. (1)The pile-rock interface with regular triangular asperity is consistent, and the inclination of asperity is (2)The cohesion of the pile-rock interface is small and cannot be accurately measured; the cohesion of the pile-rock interface is ignored. The initial normal stress could be ignored in the case of pile embedded in rocky slope because of a release of the initial geostatic stresses [33](3)The elastic modulus of the pile is greater than that of the rock; the surrounding rock is destroyed before the pile. The failure surface is curved, and the critical normal pressure of interface can be expressed as

where is the cohesion of rock, is the internal friction angle of rock, and is vertex angle of regular triangle.

###### 2.2.2. The Principal Mechanisms of Shear Transfer

The progress of shear behavior can be divided into dilation and residual periods. Dilation occurs when the shear displacement is small. With the development of the shear dilation, the pile-rock interface carries loads on the reach of critical stress, and the regular triangular asperity would be shorn off.

For the pile subjected to vertical load at the crest of rock slope, the slippage is often accompanied with radial expansion of the pile .

where is the dilation angle of rock.

And the normal stress will increase in the process of dilation; the incremental normal stress of pile-rock interface can be expressed as:

As mentioned in the previous assumption, the initial normal stress could be ignored. Therefore, stress acting normal to the direction along the pile can be expressed as:

During progress of dilation, with the development of shear displacement, the normal stress increases continuously. Shear dilation for pile-rock interface is illustrated in Figure 4.

A classic theory for shear behavior is proposed, which was proposed by Patton et al. [34]. Shear stress for dilation can be expressed as:

where is the shear stress for dilation period and is the base friction angle of rock.

For the pile subjected to vertical load at the crest of rock slope, the unit side resistance for dilation can be obtained by Eqs. (12)–(15):

where is shear resistance for dilation at the depth and is shear displacement at the depth .

With the development of the shear dilation, the pile-rock interface reaches the critical stress, and the regular triangular asperity would be shorn off. The condition of static equilibrium can be expressed as:

where is the triangular half-chord length, is the normal stress perpendicular to the pile-rock interface, and is the slippage.

When the normal stress perpendicular to the pile-rock interface reaches the critical normal pressure of interface , the condition of static equilibrium can be rewritten as:

where is the critical shear displacement.

As a result, the critical shear displacement can be obtained by Eqs. (12)–(15) and (18):

The interface is plastic when the relative shear displacement is more than the critical shear displacement . And the shear resistance for residual periods at the depth would depend on residual parameters; the formulation of side resistance can be expressed as:

where is the residual friction angle of rock.

As a result, the equation of shear behavior for the pile subjected to vertical load at the crest of rock slope is generated as:

##### 2.3. A Closed-Form Solution of Vertical Load Transfer of Pile at the Crest of Rock Slope

###### 2.3.1. The Analytical Method of the Load Transfer

Generally speaking, the applied vertical load is supported by base resistance and side resistance. A number of related tests were conducted by scholars, and the experimental results illustrate that the bearing capacity almost depends on side resistance for long pile embedded in rock. For simplicity, the base resistance could be neglected in the case of pile embedded in rocky slope.

In this investigation, shear behavior can be distinguished into two periods. As illustrated by Eq. (21), the pile-rock interface would be behave plastic when the relative shear displacement is more than the critical shear displacement . In contrast, the pile-rock interface would be in elastic.

To depict the distribution of side resistance exactly, the analytical method of the load transfer has been used. It can combine the shear behavior of pile-rock interface with side resistance of pile, and the static equilibrium of unit pile under the vertical load can be expressed as:

where is the shear displacement at the depth , is the perimeter of pile, is the elastic modulus of the pile, and is the sectional area of the pile.

As illustrated in Figure 5, is the plastic depth, and is the elastic depth; is settlement of pile top, and is vertical load of pile top.

###### 2.3.2. Solution of Plastic State

The pile-rock interface would be behave plastic when settlement of pile top is more than the critical shear displacement . The plastic solution for load transfer could be obtained by Eqs. (21) and (22):

so, a general solution is expressed as:

where and are constants.

The boundary conditions can be expressed as:

Substitution of Eq. (25) into Eq. (24) produces a solution:

The plastic solution of vertical load in the case of pile embedded in rocky slope could be shown as:

The pile-rock interface is in conformity with continuity in displacements at the plastic depth . It could be seen

The plastic depth can be obtained by Eqs. (19), (27), and (30):

Substituting Eq. (31) into Eq. (29) produces

###### 2.3.3. Solution of Elastic State

The pile-rock interface would be in elasticity below the plastic depth. As the previously analysis of shear behavior proposed, the solution of elastic state for load transfer could be obtained by Eqs. (21) and (22):

According to the statements introduced above, a general solution is expressed as:

where , , and are constant.

Substituting Eq. (34) into Eq. (33) produces

The pile-rock interface is in conformity with continuity in axial force. And it can be expressed as:

This leads to

The pile-rock interface is in conformity with continuity in displacements at the plastic depth . It could be seen

The elastic depth can be expressed as

The elastic solution of vertical load in the case of pile embedded in rocky slope could be shown as:

#### 3. Verification and Discussion

The mechanism of vertical load transfer is nonlinear and sophisticated for the pile at the crest of rock slope. It is difficult to study the progress of load transfer by conducting relative tests. To evaluate the accuracy of the new method, this method should be compared with the results of finite element analysis. The model of pile-slope is established with FLAC^{3D}. A pile embedded in the level ground for situ test was reported by Dong P et al. [35]. The basic parameter values of pile and slope for verification are showed in Table 1.

The reliability of the numerical simulation results should be demonstrated first. In case of slope angle , the comparison of the results of numerical analysis, field measured results, and predict solution is shown in Figures 6 and 7.

**(a)**

**(b)**

**(a)**

**(b)**

Figure 6(a) illustrates the finite element results and the measured results of axial force distributions in depth with 53 MN vertical load. It is obvious that the regular triangular asperity has been shorn off, and part of the pile-rock interface has been in plasticity with 53 MN vertical load. The field measured result of the plastic depth is 5 m. By resolving the above equation, the plastic depth is 5.6 m, which matches closely with the field measured results. The numerical results of axial force distributions match more closely with the field measured results in the upper part of the pile. Figure 6(b) shows that the numerical results and the measured results of axial force distributions in depth with 32 MN vertical load. The numerical results of axial force distributions match closely with the field measured results. Since the base resistance has been neglected in analytical solution for simplicity, the base axial force of numerical results is larger than analytical results. Numerical simulation is more suitable to the actual working conditions, and the base resistance exists.

For the pile subjected to vertical load at the crest of rock slope, there are few relevant tests. Thus, the numerical method would be used to verify the accuracy of the theory.

In case of slope angle , the comparison of the results of finite element analysis and predict solution is shown in Figure 7. As illustrated in Figure 7(a), the analytical results of axial force distributions match closely with the numerical results in depth with any load. It is similar to the case of that the base axial force of numerical results is larger than analytical results. As explained in the previous section, the base axial force is 0 in analytical results while the base axial force exists in the actual engineering. The pile-rock interface would be in elastic when the vertical load is minimal. It would behave plastic when the vertical load is large enough, and the plastic depth would be larger with the increase of vertical load.

The highest axial force occurs in the upper part of the pile, which means the highest load transfer in the upper part of the pile correlates with highest values of shear displacements. The reduction of axial force is related to the shear displacement on the pile-rock interface. Figure 7(b) shows that the comparison of the results of finite element analysis and predict solution with different diameter of pile in case of . It is obvious that the plastic depth has been larger with the reduction of diameter. And it seems that the higher load transfer in the upper part of the pile is associated with the smaller diameter. In other words, larger diameter piles have higher load carrying capacity.

In summary, the rationality of presented method has been explained by a series of numerical study for load transfer. And the analytical results of axial force distributions have a good match with the numerical results within the reasonable error range.

#### 4. Parametric Study

There are many related parameters that affect the distribution of side resistance and axial force, especially for the vertical loaded pile at the crest of rock slope. Many scholars have focused on the issue of load transfer for pile at the level ground, and some conclusions are drawn. The effect of slope on the distribution of side resistance is an emphasis in this study. Therefore, a series of parametric analysis were conducted to further descript the presented method. The parametric study is investigated with the different parameters reported by O’Neill, as shown in Table 2.

##### 4.1. Residual Period

The side resistance in plastic depth is constant, and it depends on the residual parameters and interface roughness. Figure 8 shows the effect of slope angle and roughness parameters on side resistance of residual periods in condition with 1296 kN vertical load. The values of half-chord length, dilation angle, and slope angle are varied from 8 mm to 32 mm, 5° to 40°, and 0° to 75°.

**(a)**

**(b)**

Figure 8(a) shows a tendency that the side resistance of residual periods decreases with the raised slope angle. At the same time, the higher slope angle, the greater reduction of the side resistance. This is because the larger slope angle, the smaller the normal constraint of pile. Figure 8(b) illustrates that the side resistance of residual periods increases and then decreases with the increase of dilation angle. It means that the simple increase of the dilation angle cannot enhance the side resistance; this may be caused by the quicker reduction of the contact area with the higher dilation angle. And it is obvious that the half-chord length is positively correlated with the side resistance.

##### 4.2. Dilation Period

The value of side resistance in elastic depth is mobile on the method proposed previously, and the elastic depth is not immobile under different working conditions. As to the vertically loaded pile in dilation period, it is more difficult to descript the distribution of side resistance.

Figure 9 shows the effect of dilation angle on side resistance of dilation periods in condition of 4000 kN vertical load and 0.8 m diameter of pile. With the increase of dilation angle, the side resistance of dilation period increases initially and then decreases. This is similar to the behavior of residual period. As indicated previously, the smaller the dilation angle, the more uniform the distribution of side resistance. The values of dilation angle are varied from 8° to 36°.

Figure 10 illustrates the effect of slope angle on side resistance of dilation periods The values of slope angle are varied from 0° to 75°. The smaller the slope angle, the more concentrated the distribution of pile lateral friction at the top of the pile, the greater the decreasing rate of the side resistance, and the deeper the plastic depth. This may be caused by the normal constraint of pile is small when the slope angle is large. As a result, the slope angle would give engineers a guidance to optimize designs.

To further study the load transfer, the term of “efficiency of load transfer ” is referred in this paper. It can be expressed as . According to the previous analysis of load transfer, is a comprehensive parameter of slope pile. And it is a macroscopic composite index reflecting the bearing performance of pile embedded in the crest of the slope. Figure 11 illustrates the effect of comprehensive parameters of slope-pile on the efficiency of load transfer , and the values of are varied from 0.1 to 0.6. The greater the value, the greater the value, and the upper pile bears more load. Therefore, the smaller the load ultimately transferred to the pile tip, the weaker the bearing capacity of the slope pile. As a result, the smaller is beneficial to enhance the bearing capacity of piles.

#### 5. Conclusion

In order to obtain the response of drilled pile under vertical load at the crest of rock slope. Efforts have been made in this paper to analysis the effect of slope on the normal stiffness of socket wall. The following conclusions can be drawn:
(a)A modified model of normal stiffness of socket wall affected by the slope is obtained; the effect of slope angle and Poisson’s ratio on the reduce factor of normal stiffness of pile was proposed(b)Analyze the shear behaviors of the pile-rock interface, an analytical solution of load transfer of pile at the crest of rock slope is obtained. The response of the drilled pile at the crest of rock slope was obtained from the closed-form solution(c)The results of the new method were compared with the results of Flac^{3D}, which shows remarkable agreement(d)The simple increase of the dilation angle cannot enhance the side resistance, and the half-chord length is positively correlated with the side resistance(e)The side resistance of residual periods decreases with the raised slope angle, and the slope is a disadvantage for engineering. And the smaller comprehensive parameter of slope pile is beneficial to enhance the bearing capacity of piles

#### Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant Nos. 51978665 and 51678570) and the Focus on research and development plan of Hunan province (No. 2015SK2053).