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# Three-Part Wedge Method for the Stability Calculation of Embankment Supported on Rigid Pile Foundation

## Abstract

Rigid piles have been widely used to reinforce soft subgrade. However, some studies do not consider the bending failure of piles and wedge sliding surfaces, which may lead to an overestimation of stability. In this study, a stability calculation method for a multi-section linear sliding surface is first derived and then simplified to three-part wedge method based on the numerical modeling of the failure evolution process. The real safety factor is replaced by the average value of the upper and lower bound solutions of the embankment stability safety factor. Through calculations, then the average value or the approximate solution of the real value, of both the upper and lower bound solutions can be obtained. The accuracy of the approximate solution can be verified through an analysis of the relative error of the average and true values. Finally, the proposed method was verified by a three-dimensional numerical simulation method and compared with the traditional limit equilibrium method and the equivalent shear strength parameter method. The results indicate that the support contribution of rigid piles to the embankment can be effectively reflected by considering wedge sliding surfaces, and a more reasonable stability safety factor can be obtained through the proposed method.

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## Abbreviations

B :

Width of the crest of the embankment (m)

c :

Cohesive force of the soil in the wedge sliding surface (kPa)

c′:

Cohesive force of the soil at the wedge sliding surface (kPa)

C h :

Cohesive force of the soil in the wedge element at the sliding surface (kN/m)

C v :

Cohesive force between the wedge element and adjacent wedges (kN/m)

e :

Absolute error of the approximate solution compared with the real stability safety factor of the embankment

e max :

Upper limit of the absolute error

Ei, Ei+1(i=1,2,3…) :

Normal forces exerted by adjacent wedges (kN/m)

E 0 :

Modulus of elasticity (MPa)

F i(i=1,2,3…) :

Tangential force of the bottom of the slip surface acting on the wedge element (kN/m)

H 1 :

Height of the Wedge 1 (m)

H 2 :

Height of the Wedge 3 (m)

J :

Multiple of the soil deformation modulus

K :

Overall stability and safety coefficient of the embankment

K ave :

Average value of Kmax and Kmin

K max :

Upper bound solution of the embankment stability safety factor

K min :

Lower bound solution of the embankment stability safety factor

K true :

Real safety factor

K v :

Safety coefficients of the interfaces between the wedges

L :

Pile length above the wedge sliding surface (m)

m :

Number of piles in the wedge element

M :

Bending strength of the pile (kN·m/m)

M 1 :

Anti-sliding moment contributed by the bending strength of the pile (kN·m/m)

M 2 :

Anti-sliding moment contributed by the equivalent resistance of the pile at the sliding surface (kN·m/m)

n :

Number of polyline segments of the sliding surface

N i(i=1,2,3…) :

Normal force of the bottom of the slip surface acting on the wedge element (kN/m)

P :

Anti-sliding concentrated force of the pile (kN/m)

Q i(i=1,2,3…) :

Equivalent resistance of the pile in the wedge element at the sliding surface (kN/m)

Q t :

Equivalent resistance of the t-th pile in the wedge element at the wedge sliding surface (kN/m)

R :

Radius of the inscribed circle of the wedge sliding surface (m)

T :

Ratio between the maximum shear force and the maximum bending moment of the pile in the whole model (m−1)

W i(i = 1,2,3…) :

Weight of the wedge element (kN/m)

Xi, Xi+1(i=1,2,3…):

Tangential forces exerted by adjacent wedges (kN/m)

α :

Angle between the sliding surface and horizontal direction of Wedge 1 in the three-part wedge model (°)

β :

Angle between the sliding surface and horizontal direction of Wedge 2 in the three-part wedge model (°)

γ :

Unit weight of the soil in the wedge sliding surface (kN/m3)

δ :

Internal friction angle of the soil at the wedge sliding surface (°)

ε :

Relative error of the approximate solution compared with the real stability safety factor of the embankment

εmax :

Upper limit of the relative error

θi(i=1,2,3…) :

Angle between the slip line and the horizontal direction (°)

μ :

Poisson’s ratio

φ :

Internal friction angle of the soil in the wedge sliding surface (°)

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## Acknowledgements

This study is financially supported by the National Natural Science Foundation of China (No. 41572253).

## Author information

Correspondence to Fu-quan Chen.

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Reprints and Permissions

Liu, P., Xiong, C. & Chen, F. Three-Part Wedge Method for the Stability Calculation of Embankment Supported on Rigid Pile Foundation. KSCE J Civ Eng 24, 794–806 (2020). https://doi.org/10.1007/s12205-020-1513-0

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### Keywords

• Embankment
• Stability calculation
• Bending failure
• Three-part wedge
• Rigid pile