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Fast stability assessment of rock slopes subjected to flexural toppling failure using adaptive moment estimation (Adam) algorithm

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Abstract

Stability assessment of anti-dip bedding rock slopes (ABRSs) remains a troublesome issue especially when there are hundreds or thousands of rock layers. The adaptive moment estimation (Adam) algorithm proposed in recent years has been widely used in artificial intelligence research and is a powerful tool for solving the above-mentioned problem. In this work, the limit equilibrium method (LEM) is combined with the Adam algorithm (referred to as the LEM–Adam method) to determine the failure surface of an ABRS undergoing shearing–flexural toppling failure. Two tests reported in previous studies (a 1 g model and a centrifuge model) were analyzed to validate the LEM–Adam method. The critical failure surface and the safety factor agree well with the experimental observations, validating the LEM–Adam method in the stability analysis of ABRSs. Finally, the LEM–Adam method was compared with the LEM combined with a genetic algorithm (GA). The results show that the LEM–Adam method has higher computational efficiency and eliminates the randomness of the solution. The LEM–Adam method proposed in this work offers a practical and fast approach for the stability analysis and design of ABRSs.

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Abbreviations

H :

Height of the slope

β :

Angle of the plane normal to the joints

β s :

Angle of the slope

c :

Cohesion of the intact rock

φ :

Friction angle of the intact rock

φ d :

Friction angle of the joints

σ t :

Tensile strength of the intact rock

\({\sigma }_{i,\mathrm{max}}^{t}\) :

Maximum tensile stress on the base of the ith rock layer

F s :

Safety factor of the slope

h i :

Height of the ith rock layer above the failure surface

h si :

Total height of the ith rock layer above the reference plane in the slope (that passes through the toe of the slope and is normal to the joints)

\(\overline{h}_{i}\) :

Distance from the failure surface in the ith rock layer to the reference plane

i :

Number of rock layers, numbered from the toe to the crest of the slope

I i :

Inertia of the ith rock layer

l i :

Height of left-hand side of the ith rock layer

M i :

Moment acting on the base of the ith rock layer

Ni,s,Ni,t :

Normal forces acting on the base of the ith rock layer when it undergoes shearing failure and flexural toppling failure, respectively

\({P}_{i}^{r}\) :

Normal force acting on the right side of the ith rock layer

\({P}_{i}^{l}\) :

Normal force acting on the left side of the ith rock layer (given by \({P}_{i}^{l}=\mathrm{max}\left({P}_{i,s}^{l},{P}_{i,t}^{l},0\right)\))

\({P}_{i,s}^{l},\ {P}_{i,t}^{l}\) :

Normal forces acting on the left side of the ith rock layer when it undergoes shearing failure and flexural toppling failure, respectively

\({Q}_{i}^{r}\) :

Tangential force acting on the right side of the ith rock layer

\({Q}_{i,s}^{l},\ {Q}_{i,t}^{l}\) :

Tangential forces acting on the left side of the ith rock layer when it undergoes shearing failure and flexural toppling failure, respectively

\({r}_{i}\) :

Height of right-hand side of the ith rock layer

\({t}_{i}\) :

Thickness of the ith rock layer

S i,s,S i,t :

Tangential forces acting on the base of the ith rock layer when it undergoes shearing failure and flexural toppling failure, respectively

\({W}_{i}\) :

Weight of the ith rock layer

χ :

Dimensionless parameter related to the distribution of normal stresses

x i :

Dimensionless parameter used to determine the depth of the failure in the ith rock layer (given by \({x}_{i}=\overline{h}_{i}/{h}_{si}\))

x :

Vector used to specify the position of the failure surface (given by \(\mathbf{x}=\left[{x}_{1},{x}_{2},\cdots ,{x}_{n}\right]\))

γ :

Unit weight of the intact rock

α i :

Gradient of Fs with respect to xi

m i,t :

First moment of the ith rock layer in the tth iteration

\(\widehat{{m}_{i,t}}\) :

Corrected first moment

v i,t :

Second moment of the ith rock layer in the tth iteration

\(\widehat{{v}_{i,t}}\) :

Corrected second moment

η :

Coefficient related to the iteration number determining the step length of optimization

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Funding

The research was financially supported by the National Natural Science Foundation of China (Grant nos. 12072358 and 12102315).

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Authors and Affiliations

  1. State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan, 430071, Hubei, China

    Yun Zheng, Runqing Wang, Congxin Chen & Fei Meng

  2. University of Chinese Academy of Sciences, Beijing, 100049, China

    Yun Zheng, Runqing Wang & Congxin Chen

  3. Hubei Key Laboratory of Roadway Bridge and Structure Engineering, Wuhan University of Technology, Wuhan, 430070, Hubei, China

    Fei Meng

  4. College of Geosciences and Engineering, North China University of Water Resources and Electric Power, Zhengzhou, 450046, China

    Yun Zheng

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  1. Yun Zheng
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  2. Runqing Wang
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Correspondence to Fei Meng.

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Zheng, Y., Wang, R., Chen, C. et al. Fast stability assessment of rock slopes subjected to flexural toppling failure using adaptive moment estimation (Adam) algorithm. Landslides 19, 2149–2158 (2022). https://doi.org/10.1007/s10346-022-01902-x

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  • DOI: https://doi.org/10.1007/s10346-022-01902-x

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