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.

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 [

To analyze load transfer of pile under vertical loads, a series of method have been proposed. Laboratory test [

Jiang et al. [

Johnston and Lam [

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.

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

Schematic view of the slope-pile model.

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

where

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

where

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

For the pile at the crest of rock slope, the normal stiffness of pile can be obtained by Eqs. (

where

Figure

Fitting curve of

For the pile at the crest of rock slope, the normal stiffness of pile can be obtained by Eqs. (

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

Shear motion of asperity.

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.

The pile-rock interface with regular triangular asperity is consistent, and the inclination of asperity is

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 [

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

where

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

where

And the normal stress will increase in the process of dilation; the incremental normal stress of pile-rock interface

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

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

Schematic view of shear dilation for pile-rock interface.

A classic theory for shear behavior is proposed, which was proposed by Patton et al. [

where

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

where

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

When the normal stress perpendicular to the pile-rock interface

where

As a result, the critical shear displacement

The interface is plastic when the relative shear displacement

where

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

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. (

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

As illustrated in Figure

The sketch of plastic depth and elastic depth.

The pile-rock interface would be behave plastic when settlement of pile top

so, a general solution is expressed as:

where

The boundary conditions can be expressed as:

Substitution of Eq. (

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

The plastic depth

Substituting Eq. (

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. (

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

where

Substituting Eq. (

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

The elastic depth

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

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. [

The basic parameter values for verification.

Shaft diameter ( | 2.5 m |

Shaft length ( | 32 m |

Elastic modulus of pile ( | 30 GPa |

Friction angle of interface | 35° |

Triangular half-chord length | 3 mm |

Cohesion of rock ( | 200 kPa |

Internal friction angle of rock ( | 25° |

Elastic modulus of rock ( | 2.0 GPa |

Poisson’s ratio | 0.25 |

Dilation angle of rock | 10° |

The reliability of the numerical simulation results should be demonstrated first. In case of slope angle

Verification in case of slope angle

Verification in case of slope angle

Figure

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 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

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.

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

The basic parameter values for parametric study.

Shaft diameter ( | 0.61 m |

Shaft length ( | 6.1 m |

Elastic modulus of pile ( | 27.6 GPa |

Friction angle of interface | 30° |

Triangular half-chord length | 8 mm |

Cohesion of rock ( | 1.2 MPa |

Internal friction angle of rock ( | 24.8° |

Elastic modulus of rock ( | 232 MPa |

Poisson’s ratio | 0.3 |

Dilation angle of rock | 10° |

The side resistance in plastic depth is constant, and it depends on the residual parameters and interface roughness. Figure

The influence of slope angle and interface roughness on residual unit side resistance: (a) the influence of slope angle; (b) the influence of the half-chord length.

Figure

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

The influence of dilation angle on dilation unit side resistance.

Figure

The influence of slope angle on dilation unit side resistance.

To further study the load transfer, the term of “efficiency of load transfer

The influence of comprehensive parameters of slope-pile

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 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

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

The results of the new method were compared with the results of Flac^{3D}, which shows remarkable agreement

The simple increase of the dilation angle cannot enhance the side resistance, and the half-chord length is positively correlated with the side resistance

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

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

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

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).