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Strength reduction strategy for rock slope stability using the variation principle based on the Hoek–Brown failure criterion

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Abstract

Rotational failure may exhibit in homogenous rock slopes, and the critical sliding surface of a rock slope based on the rotational failure mechanism is not a single log-spiral under nonlinear failure criterion. This work proposed an advanced nonlinear analysis method for estimating the seismic stability of homogenous slope in rock masses which is governed by the generalized Hoek–Brown failure criterion. According to the virtual power principle, the critical slope height is obtained using particle swarm algorithm. Strength reduction technique is further introduced to explore the reduction strategy of the strength parameters of the Hoek–Brown failure criterion. The outcomes indicate that the factor of safety obtained by simultaneously reducing the unconfined compressive strength σci and material parameter mi is in good agreement with the results of other methods. In addition, two cases are re-analyzed to illustrate the applicability of the proposed method, and the maximum discrepancy with existing results is about 8%. The effects of strength parameters, slope angle, and seismic quasi-static coefficients on the slope stability factor and critical sliding surface are analyzed, which shows that the seismic load and the Hoek–Brown parameters have a significant effect on the slope stability factor. There is no need to assume the expression of the potential sliding surface, which can provide theoretical support and a useful reference for the nonlinear analysis of slope stability.

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Data Availability

All of the data, models, or code that support the findings of this study are available from the corresponding author upon reasonable request.

Abbreviations

σ 1 :

Major principal stress

σ 3 :

Minor principal stress

σ ci :

Unconfined compressive strength

m b, s, a :

Dimensionless parameters in Hoek–Brown failure criterion

D :

Disturbance factor

GSI :

Geological Strength Index

m i :

Intact rock type-dependent constant

σ :

Normal stress

τ :

Shear stress

A, B :

Dimensionless parameters in Hoek–Brown failure criterion

σ tm :

Unconfined tensile strength of rock mass

y(x):

Potential sliding surface function in the Cartesian coordinate system

r(θ):

Potential sliding surface function in the polar coordinate system

α, β :

Slope angle

H :

Vertical distance from entry point to slope toe

H 1 :

Slope height

L :

Distance from entry point to slope crest

L 1 :

Horizontal distance from entry point to slope crest

L 2 :

Horizontal distance from slope crest to slope toe

O :

Center of rotation

\(\dot{\Omega }\) :

Angular velocity

φ t :

Friction angle

\(\dot D\)  :

Internal energy dissipation power per unit area on the sliding surface

δw :

Linear velocity

\(\Sigma \dot{D}\) :

Total internal energy dissipation on the sliding surface

r 0, θ 0 :

Initial radius and initial angle

r n, θ n :

Terminate radius and terminate angle

k x, k y :

Horizontal and vertical seismic acceleration coefficients

x c, y c :

Coordinates of the center of rotation O

γ :

Unit weight of homogenous rock mass

\({\dot{W}}_{e}\) :

Total work rate of external forces

A 1, B 1, C 1 :

Coefficients of univariate quadratic equation

H f, V f :

Horizontal and vertical resultant forces

M f :

Moment of the rotational block to the center of rotation

I :

Total virtual work rate

δu, δv, δΩ:

Horizontal, vertical, and rotational virtual displacements

F :

Core of integral functional I

X, Y :

Cartesian coordinate system with rotation center as origin

y′:

First derivative of sliding surface with respect to x

σ′:

First derivative of normal stress with respect to x

τ ′ :

First derivative of shear stress with respect to x

λ :

Euler-Lagrange coefficient

G :

Function of strength envelope

f :

Hoek-Brown failure criterion with respect to normal stress

\({F}_{{y}^{^{\prime}}}\) :

First derivative of F with respect to y′

y 1, y 1′:

Slope surface function and its first derivative with respect to x

F s :

Safety factor of slope

Δθ :

Increment of θ

σ 0 :

Initial normal stress at entry point

β′:

Calculated slope angle

H 1′:

Calculated slope height

M :

The size of particle swarm group

P :

The number of variables

x ji, v ji :

The position and velocity of particles

p best, g best :

Individual's historical optimal value and the group's optimal value

c 1, c 2, ω :

Parameters of particle swarm algorithm

k :

The current iteration number

\({\sigma }_{ci}^{cr}\),\({m}_{i}^{cr}\) :

Reduced strength parameters

τ cr, F' s :

Reduced shear strength and its corresponding reduction coefficient

N :

Slope stability factor

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Acknowledgements

This study was financially supported by National Natural Science Foundation of China (Nos. 51878668, 51978666), Distinguished Young Scholar Foundation of Hunan Province (No. 2021JJ10063), Scientific and technological progress and innovation project of Department of Transportation of Hunan Provincial (No. 202115), and Postgraduate Innovative Project of Central South University (No. 150110068). The Natural Science Foundation of Hunan Province (2023JJ40078) and the Scientific Research Project of Hunan Provincial Education Department (No. 22C0573). Additionally, the fourth author, Shi Zuo, led both of these projects. All financial supports are greatly appreciated.

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Authors and Affiliations

  1. School of Civil Engineering, Central South University, Changsha, 410075, Hunan Province, China

    Shihong Hu, Liang Li, Lianheng Zhao, Shi Zuo & Dongliang Huang

  2. Key Laboratory of Heavy-Haul Railway Engineering Structure, Ministry of Education, Central South University, Changsha, 410075, Hunan Province, China

    Lianheng Zhao

  3. Hunan Provincial Key Laboratory for Disaster Prevention and Mitigation of Rail Transit Engineering Structure, Central South University, Changsha, 410075, Hunan Province, China

    Lianheng Zhao

Authors
  1. Shihong Hu
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  2. Liang Li
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  3. Lianheng Zhao
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  4. Shi Zuo
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  5. Dongliang Huang
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Contributions

All authors contributed to the study conception and design. Methodology, code, data calculation and analysis, and comparison were performed by Shihong Hu, Liang Li and Lianheng Zhao. Figures and Tables were performed by Shihong Hu, Shi Zuo and Dongliang Huang. The first draft of the manuscript was written by Shihong Hu and Lianheng Zhao, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Lianheng Zhao.

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The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

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Hu, S., Li, L., Zhao, L. et al. Strength reduction strategy for rock slope stability using the variation principle based on the Hoek–Brown failure criterion. Bull Eng Geol Environ 82, 297 (2023). https://doi.org/10.1007/s10064-023-03303-3

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