Rock Slope Stability Analysis Incorporating the Effects of Intermediate Principal Stress

Abstract

This paper proposes an analytical approach for assessing rock slope stability based on a three-dimensional (3D) Hoek–Brown (HB) criterion to consider the effects of intermediate principal stress. The 3D HB criterion, considering an associate flow rule, is utilized to describe the perfectly plastic behavior of rock mass under a plane strain condition. To reflect the change of friction angle on the failure surface, the potential failure surface (PFS) is divided into small segments with each segment being assigned a unique friction angle. The upper bound theorem of limit analysis is combined with the strength reduction method to determine the factor of safety (FOS) of a rock slope with a defined PFS. By optimizing the PFS, the minimum FOS and the critical failure surface (CFS) of the rock slope are obtained by the customized genetic algorithm. The proposed approach is validated by comparing it with an HB criterion-based solution and numerical simulations. Parametric studies are also performed to investigate the effects of rock mass properties, slope geometry, and loading conditions on the FOS and CFS. The results indicate that ignoring the 3D strength of rock leads to underestimation of FOS and it is important to consider the various factors when evaluating the stability of a rock slope. For the effortless application of the proposed approach, a Python-based graphical-user-interface application is developed as a stand-alone executable app and is successfully applied to analyze a rock slope.

Highlights

• Three dimensional (3D) Hoek-Brown criterion is applied to analyze rock slope stability

• Customized-genetic algorithm is utilized to accelerate finding most critical condition

• Ignoring 3D strength of rock mass leads to underestimation of the factor of safety for a rock slope

• Graphical-user-interface app was developed using Python for effortless application of the proposed approach

Appendices

Appendix

Appendix 1

Details of Eq. (10)

$$\frac{\partial {f}_{\mathrm{NMGZZ}}}{\partial {\sigma }_{i}^{*}}=\frac{\partial {f}_{\mathrm{NMGZZ}}}{\partial {I}_{1}^{*}}+\frac{\partial {f}_{\mathrm{NMGZZ}}}{\partial {q}_{\mathrm{d}}}\frac{3{\sigma }_{i}^{*}-{I}_{1}^{*}}{2{q}_{\mathrm{d}}}+\frac{\partial {f}_{\mathrm{NMGZZ}}}{\partial {I}_{3}^{*}}\frac{{I}_{3}^{*}}{{\sigma }_{i}^{*}};i=\mathrm{1,2},3$$

(A1a)

$$\frac{\partial {f}_{\mathrm{NMGZZ}}}{\partial {I}_{1}^{*}}=\left[{\left(\frac{{q}_{\mathrm{d}}}{{\sigma }_{\mathrm{c}}}\right)}^{2/a-2}+{m}_{\mathrm{b}}{\left(\frac{{q}_{\mathrm{d}}}{{\sigma }_{\mathrm{c}}}\right)}^{1/a-1}\right]\frac{3{I}_{1}^{*2}-{q}_{\mathrm{d}}^{2}}{3}$$

(A1b)

$$\frac{\partial {f}_{\mathrm{NMGZZ}}}{\partial {q}_{\mathrm{d}}}=-\left[{\left(\frac{{q}_{\mathrm{d}}}{{\sigma }_{\mathrm{c}}}\right)}^{2/a-2}+{m}_{\mathrm{b}}{\left(\frac{{q}_{\mathrm{d}}}{{\sigma }_{\mathrm{c}}}\right)}^{1/a-1}\right]\frac{2{q}_{\mathrm{d}}{I}_{1}^{*}}{3}+\left[\frac{2-2a}{a}{\left(\frac{{q}_{\mathrm{d}}}{{\sigma }_{\mathrm{c}}}\right)}^{2/a-2}+{m}_{\mathrm{b}}\frac{1-a}{a}{\left(\frac{{q}_{\mathrm{d}}}{{\sigma }_{\mathrm{c}}}\right)}^{1/a-1}\right]\left(\frac{{I}_{1}^{*3}-{I}_{1}^{*}{q}_{\mathrm{d}}^{2}}{3{q}_{\mathrm{d}}}-\frac{9{I}_{3}^{*}}{{q}_{\mathrm{d}}}\right)$$

(A1c)

$$\frac{\partial {f}_{\mathrm{NMGZZ}}}{\partial {I}_{3}^{*}}=-9\left[{\left(\frac{{q}_{\mathrm{d}}}{{\sigma }_{\mathrm{c}}}\right)}^{2/a-2}+{m}_{\mathrm{b}}{\left(\frac{{q}_{\mathrm{d}}}{{\sigma }_{\mathrm{c}}}\right)}^{1/a-1}\right]-2{m}_{\mathrm{b}}^{2}$$

(A1d)

Appendix 2

Details of Eq. (15)

Considering the failure surface in Fig. 2

$${\int }_{{\theta }_{0}}^{{\theta }_{\mathrm{b}}}{\int }_{{r}_{\mathrm{s}}}^{{r}_{\mathrm{fr}}}{r}^{2}\mathrm{cos}\theta \mathrm{d}r\mathrm{d}\theta ={\int }_{{\theta }_{0}}^{{\theta }_{\mathrm{b}}}\frac{{r}_{\mathrm{fr}}^{3}}{3}\mathrm{cos}\theta \mathrm{d}\theta -{\int }_{{\theta }_{\mathrm{a}}}^{{\theta }_{\mathrm{o}}}{\int }_{0}^{{r}_{\mathrm{s}}}{r}^{2}\mathrm{cos}\theta \mathrm{d}r\mathrm{d}\theta -{\int }_{{\theta }_{\mathrm{o}}}^{{\theta }_{\mathrm{b}}}{\int }_{0}^{{r}_{\mathrm{s}}}{r}^{2}\mathrm{cos}\theta \mathrm{d}r\mathrm{d}\theta$$

(A1)

$${\int }_{{\theta }_{\mathrm{a}}}^{{\theta }_{\mathrm{o}}}{\int }_{0}^{{r}_{\mathrm{s}}}{r}^{2}\mathrm{cos}\theta \mathrm{d}r\mathrm{d}\theta =\frac{{r}_{\mathrm{o}}^{3}}{6}\left(\mathrm{tan}{\theta }_{o}+\mathrm{tan}\beta \right){\mathrm{cos}}^{3}{\theta }_{\mathrm{c}}\left[{\left(\frac{\mathrm{tan}{\theta }_{\mathrm{o}}+\mathrm{tan}\beta }{\mathrm{tan}{\theta }_{\mathrm{a}}+\mathrm{tan}\beta }\right)}^{2}-1\right]$$

(A2)

$${\int }_{{\theta }_{\mathrm{o}}}^{{\theta }_{\mathrm{b}}}{\int }_{0}^{{r}_{\mathrm{s}}}{r}^{2}\mathrm{cos}\theta \mathrm{d}r\mathrm{d}\theta =\frac{{r}_{\mathrm{o}}^{3}}{3}{\mathrm{sin}}^{3}{\theta }_{\mathrm{o}}\left(\mathrm{cot}\alpha +\mathrm{cot}{\theta }_{\mathrm{o}}\right)\times \left\{\left(\frac{\mathrm{cot}\alpha +\mathrm{cot}{\theta }_{\mathrm{o}}}{\mathrm{cot}\alpha +\mathrm{cot}{\theta }_{\mathrm{b}}}-1\right)\left(\mathrm{cot}\alpha +\mathrm{cot}{\theta }_{\mathrm{o}}\right)-\frac{\mathrm{cot}\alpha }{2}\left[{\left(\frac{\mathrm{cot}\alpha +\mathrm{cot}{\theta }_{\mathrm{o}}}{\mathrm{cot}\alpha +\mathrm{cot}{\theta }_{\mathrm{b}}}\right)}^{2}-1\right]\right\}$$

(A3)

$${\int }_{{\theta }_{0}}^{{\theta }_{\mathrm{b}}}{\int }_{{r}_{\mathrm{s}}}^{{r}_{\mathrm{fr}}}{r}^{2}\mathrm{sin}\theta \mathrm{d}r\mathrm{d}\theta ={\int }_{{\theta }_{0}}^{{\theta }_{\mathrm{b}}}\frac{{r}_{\mathrm{fr}}^{3}}{3}\mathrm{sin}\theta \mathrm{d}\theta -{\int }_{{\theta }_{\mathrm{a}}}^{{\theta }_{\mathrm{o}}}{\int }_{0}^{{r}_{\mathrm{s}}}{r}^{2}\mathrm{sin}\theta \mathrm{d}r\mathrm{d}\theta -{\int }_{{\theta }_{\mathrm{o}}}^{{\theta }_{\mathrm{b}}}{\int }_{0}^{{r}_{\mathrm{s}}}{r}^{2}\mathrm{sin}\theta \mathrm{d}r\mathrm{d}\theta$$

(A4)

$${\int }_{{\theta }_{\mathrm{a}}}^{{\theta }_{\mathrm{o}}}{\int }_{0}^{{r}_{\mathrm{s}}}{r}^{2}\mathrm{sin}\theta \mathrm{d}r\mathrm{d}\theta =\frac{{r}_{\mathrm{o}}^{3}}{3}\left(\mathrm{tan}{\theta }_{\mathrm{o}}+\mathrm{tan}\beta \right){\mathrm{cos}}^{3}{\theta }_{\mathrm{o}}\times \left\{\frac{\mathrm{tan}\beta }{2}\left[1-{\left(\frac{\mathrm{tan}{\theta }_{\mathrm{o}}+\mathrm{tan}\beta }{\mathrm{tan}{\theta }_{\mathrm{a}}+\mathrm{tan}\beta }\right)}^{2}\right]-\left(\mathrm{tan}{\theta }_{\mathrm{o}}+\mathrm{tan}\beta \right)\left(1-\frac{\mathrm{tan}{\theta }_{\mathrm{o}}+\mathrm{tan}\beta }{\mathrm{tan}{\theta }_{\mathrm{a}}+\mathrm{tan}\beta }\right)\right\}$$

(A5)

$$\int\limits_{{\theta_{{\text{o}}} }}^{{\theta_{{\text{b}}} }} {\int\limits_{0}^{{r_{{\text{s}}} }} {r^{2} \sin \theta {\text{ d}}r{\text{ d}}\theta = \frac{{r_{{\text{o}}}^{3} }}{6}} } \sin^{3} \theta_{{\text{o}}} \left( {{\text{cot}}\theta_{{\text{o}}} + {\text{cot}}\alpha } \right)\left[ {\left( {\frac{{{\text{cot}}\theta_{{\text{o}}} + {\text{cot}}\alpha }}{{{\text{cot}}\theta_{{\text{b}}} + {\text{cot}}\alpha }}} \right)^{2} - 1} \right]$$

(A6)