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Three-dimensional method for slope stability with the curvilinear route of the main sliding

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Bulletin of Engineering Geology and the Environment Aims and scope Submit manuscript

Abstract

The route of the main sliding is considered to be a linear line in the existing three-dimensional limit equilibrium methods for slope stability analysis. However, there is a curvilinear route for the main sliding caused by the translational movement and the rotation of the sliding body in slope failures. In this study, a three-dimensional limit equilibrium method was proposed for slope stability with the curvilinear route of the main sliding. The basic physical quantities for slope stability analysis in the orthogonal curvilinear coordinate system were obtained by the mapping between the orthogonal curvilinear coordinate system and the Cartesian coordinate system. The four equilibrium equations (a force equilibrium equation and three moment equilibrium equations) were established by the equilibrium conditions of the entire sliding body. The three-dimensional factor of safety for the slope with the curvilinear route of the main sliding was deduced by an analytical or a numerical method. Some examples proved that the present method can calculate the factor of safety and evaluate the stability of the slope. The results of examples showed that the factor of safety of the curvilinear route was significantly smaller than that of the linear route. The three-dimensional limit equilibrium method for slope stability analysis was further developed in this study, which provided a theoretical basis for the scientific and reasonable evaluation of slope stability.

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Abbreviations

x, y, z :

Are the Cartesian coordinates

q 1, q 2, q 3 :

Are the curvilinear coordinates

i, j, k :

Are the unit coordinate vectors for the Cartesian coordinate system

e i :

Is the unit coordinate vector for the curvilinear coordinate system

g i :

Is the covariant basis vectors for the curvilinear coordinate system

δ ij :

Is the Kronecker delta

r :

Is the radius vector of the any point

D :

Is the transformation matrix for the coordinate systems

H i :

Is the Lame coefficient

ds :

Is the arc length element

φ :

Is any scalar

▽:

Is the gradient operator

β i :

Is the direction angle of shear stress in xy plane

τ 0 :

Is the component of shear stress in xy plane

σ :

Is the normal stress over the slip surface

τ :

Is the normal stress over the slip surface

α :

Is the direction angle of shear stress in curvilinear coordinate system

n σ :

Is the unit vector normal to slip surface

n τ :

Is the unit vector tangential to slip surface

n i :

Is the components of the unit vector normal to slip surface

m i :

Is the components of the unit vector tangential to slip surface

Δ:

Is a coefficient

dS :

S the infinitesimal area element on the slip surface

da 3 :

Is the components of dS on the xy plane

n γ :

Is the unit vector of soil weight

γ :

Is the unit soil weight

F e :

Is the active force vector acting on the sliding body

w :

Is the weight of the infinitesimal element

p(q 1, q 2):

Is the ground surface equation

h(q 1, q 2):

Is the slip surface equation

M e :

Is the active moment vector acting on the sliding body

r 1 :

Is the radius vector of the center point of the infinitesimal element

r 2 :

S the radius vector of any point on the slip surface

F s :

Is the factor of safety

u :

Is the pore pressure

q :

Is the known vector

λ :

Is the unknown parameter vector

σ 0 :

Is the initial normal stress over to the slip surface

c :

Is the cohesion

ϕ :

Is the internal friction angle

N σ, N τ, N γ :

Are the coefficient vectors

A, A', B, B':

Are the coefficient matrices

F N :

Is the matrix function

x :

Is the unknown parameter matrix

ε :

Is the iteration accuracy value

S:

Is the linear route of the main sliding

L:

Is the curvilinear route of the main sliding

L':

Is the mapped curvilinear route of the main sliding

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Funding

The research was supported by the Natural Science Foundation of China (Grant No.: 52079121).

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Authors and Affiliations

  1. School of Civil Engineering, Hefei University of Technology, Hefei, China

    Kunlin Lu, Linfei Wang, Yang Yang & Guigui Zhou

Authors
  1. Kunlin Lu
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  2. Linfei Wang
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  3. Yang Yang
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  4. Guigui Zhou
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Corresponding author

Correspondence to Kunlin Lu.

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Conflict of interest statement

The authors declare no conflict of interest.

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Lu, K., Wang, L., Yang, Y. et al. Three-dimensional method for slope stability with the curvilinear route of the main sliding. Bull Eng Geol Environ 81, 67 (2022). https://doi.org/10.1007/s10064-021-02551-5

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  • DOI: https://doi.org/10.1007/s10064-021-02551-5

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