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Stability Analysis of Anti-dip Rock Slopes with Flexural Toppling Failure Based on Deformation Compatibility

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Abstract

Flexural toppling often occurs in layered rock slopes, and rock columns behave like inclined superimposed cantilever beams that bend under their own weight. However, anti-dip rock slopes always have nonuniform joint spacing, and rock columns do not have the same ability of deformation due to different stiffness. To have the same deflection, based on the mechanics of materials, the moment and shear force at the failure surface should satisfy a certain proportional relationship. An analytical model based on deformation compatibility against the flexural toppling failure has been established. In the proposed model, the slope is divided into free deflection zones and compatible deflection zones by shear forces on the failure surface. The safety factor of each block can be calculated by the bending moment, and the least factor is used to represent the safety of the slope. In addition, for an equal thickness model, the average stiffness method is suggested to determine the calculated thickness for foliated rock slopes with nonuniform joint spacing. Finally, a case study is used for practical verification of the proposed method.

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Abbreviations

E i :

Elastic modulus of the beam or block i

I i :

Cross-sectional moment of inertia of the beam or block i

L i :

Length of the beam i

h i :

Height of the block i

b i :

Thickness of the beam i or block i

\(\bar{b}\) :

Average thickness of blocks

F i :

Shear force of the beam or block i under their own weight

N i :

Normal force of the beam or block i under their own weight

M i :

Bending moment of the beam or block i under their own weight

\(\gamma\) :

Unit weight of blocks

\(\omega (x)\) :

Vertical deflection of beams or blocks

F * i :

Shear force of the beam or block i under the interaction of blocks

\(F_{m - j}^{ * }\) :

Total shear force of blocks m to j

M * i :

Bending moment of the beam or block i under the interaction of blocks

\(\lambda (x)\) :

Vertical deflection difference of beams 1 and 2

\(P\left( x \right)\) :

Function of interbeam force

\(\varphi\) :

Frictional angle at the common boundary of blocks

\(\rho\) :

Curvature of beams

m :

Number of the highest block in the compatible zone

n :

Number of the lowest block in the compatible zone

\(\beta_{\text{c}}\) :

Angle of the cut slope with respect to the horizontal direction

\(\beta_{\text{g}}\) :

Angle of the natural ground with respect to the horizontal direction

H :

Slope height

\(\alpha\) :

Inclination of rock columns

\(\sigma_{\text{t}}\) :

Tensile strength of blocks

k :

Number of the rock column at the toe

P i :

Interbeam or interlayer resultant force acting on beam or block i

\(\theta\) :

Angle of the failure plane with respect to the horizontal

F si :

Safety factor of the block i

F s :

Safety factor of the slope

\(\psi\) :

Equivalent length

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Acknowledgements

This research was supported by the National Natural Science Foundation of China (51579032, U1765107). Their support is gratefully acknowledged.

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Authors and Affiliations

  1. School of Civil Engineering and State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology (DUT), No. 2 Linggong Road, High-tech Zone, Dalian, 116024, Liaoning, China

    Wei Zhao, Runqing Wang & Tingkai Nian

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  1. Wei Zhao
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  2. Runqing Wang
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  3. Tingkai Nian
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Correspondence to Runqing Wang.

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We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work. There is no professional or other personal interest of any nature or kind in any product, service or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled, “Stability Analysis of Anti-dip Rock Slopes with Flexural Toppling Failure Based on Deformation Compatibility”.

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Appendix

Appendix

In order to enable readers to use this method, authors provide main codes. Users should first prepare a text with writing the height h, width b, modulus E and tensile strength \(\sigma_{t}\) of each rock layer from top to bottom. The bending moment of each rock layer and the safety factor of slope can be obtained by the MATLAB program.

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Zhao, W., Wang, R. & Nian, T. Stability Analysis of Anti-dip Rock Slopes with Flexural Toppling Failure Based on Deformation Compatibility. Rock Mech Rock Eng 53, 3207–3221 (2020). https://doi.org/10.1007/s00603-020-02098-z

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