Direct expressions for linearization of shear strength envelopes given by the Generalized Hoek–Brown criterion using genetic programming

doi:10.1016/j.compgeo.2012.04.008

Computers and Geotechnics

Volume 44, June 2012, Pages 139-146

https://doi.org/10.1016/j.compgeo.2012.04.008 Get rights and content

Abstract

The non-linear Generalized Hoek–Brown (GHB) criterion is one of the most broadly adopted failure criteria used to estimate the strength of a rock mass. However, when limit equilibrium and shear strength reduction methods are used to analyze rock slope stability, the strength of the rock mass is generally expressed by the linear Mohr–Coulomb (MC) criterion. If the GHB criterion is used in conjunction with existing methods for analyzing the rock slope, methods are required to determine the equivalent MC shear strength from the GHB criterion. Deriving precise analytical solutions for the equivalent MC shear strength from the GHB criterion has not proven to be straightforward due to the complexities associated with mathematical derivation. In this paper, an approximate analytical solution for estimating the rock mass shear strength from the GHB criterion is proposed. The proposed approach is based on a symbolic regression (SR) analysis performed by genetic programming (GP). The reliability of the proposed GP solution is tested against numerical solutions. The results show that shear stress estimated from the proposed solution exhibits only 0.97% average discrepancy from numerical solutions using 2451 random sets of data. The proposed solution offers great flexibility for the application of the GHB criterion with existing methods based on the MC criterion for rock slope stability analysis.

Introduction

The stability of rock slopes is significant for various rock engineering projects, such as open pit mining and dam construction. One of the most popular methods for analyzing slope stability is the limit equilibrium method (LEM) where rock mass strength is generally expressed in terms of the linear Mohr–Coulomb (MC) criterion.

The principles of LEM can be applied to determine the factor of safety (FOS) of a given slope by the method of slices as shown in Fig. 1a. The FOS can be defined as a function of resisting force f_R divided by driving force f_D. The forces of f_R and f_D can also be expressed in terms of shear stress τⁱ and normal stress $σ_{n}^{i}$ acting on the base of an arbitrary element i as shown in Fig. 1b [1].

Fig. 2 illustrates the MC failure envelope. The slope of the tangent to the MC envelope gives angle of friction $ϕ$ and the intercept with the shear stress axis gives cohesion c. The MC criterion is linear, therefore the values of shear strength parameters c and $ϕ$ are unchanged under various normal stress σ_n values. Traditional LEM only need unique values of c and $ϕ$ to calculate FOS of a given slope. That means, arbitrary slice (as shown in Fig. 1b) with various normal stress σ_n has the same c and $ϕ$ values.

The Hoek–Brown (HB) criterion was originally proposed by Hoek and Brown [2]. Over the past 30 years the HB criterion has been widely adopted in rock engineering to estimate the strength of rock masses. If the HB criterion is used with LEM for assessing rock slope stability, it becomes necessary to determine the equivalent MC shear strength for a failure surface under a specified normal stress σ_n in a rock mass governed by the HB criterion. That means, slices (as shown in Fig. 1b) with different values of normal stresses σ_n have various c and $ϕ$ values.

As Brown noted [3], deriving accurate analytical solutions for estimating the equivalent MC parameters at a given normal stress from the Generalized Hoek–Brown (GHB) criterion [4] has proven to be a challenging task.

In this research, an approximate analytical solution for estimating the rock mass shear strength from the GHB criterion [4] is proposed. The proposed approach is based on a symbolic regression (SR) analysis performed by genetic programming (GP).

Genetic programming [5] is a promising approach which attempts to find an explicit solution to explain the relations between the variables. GP is well suited to geotechnical problems and it is increasingly used by researchers in geotechnical engineering [6], [7], [8], [9]. However, there is no evidence in the literature that GP based approaches are used to estimate the rock mass shear strength from the HB criterion.

In this paper, review of existing methods for the determination MC shear strength from the HB criterion is described in Section 2. The GP approach is introduced in Section 3. The GP modeling for the GHB criterion is described in Section 4. Validation of the GP results is given in Section 5.

Section snippets

Introduction of the HB criterion

The non-linear Hoek–Brown (HB) criterion was initially proposed by Hoek and Brown [2] in 1980. The latest version is the Generalized Hoek–Brown (GHB) criterion presented by Hoek et al. [4] in 2002. The equations are expressed as follows: $σ_{1} = σ_{3} + σ_{ci} {(\frac{m_{b} σ_{3}}{σ_{ci}} + s)}^{a}$ m_b, s and a are the Hoek–Brown input parameters that depend on the degree of fracturing of the rock mass [1], [2], [3], [4] and can be estimated from the Geological Strength Index (GSI), given by : $m_{b} = m_{i} e^{(\frac{GSI - 100}{28 - 14 D})}$ $s = e^{(\frac{GSI - 100}{9 - 3 D})}$ $a = 0.5 + e^{(\frac{- GSI}{15})} -$

Overview of genetic programming

In this section, genetic programming (GP) will briefly be introduced; further information about GP can be found from Koza [5].

GP modelling for the GHB criterion

GP is composed of functions and terminals appropriate to the characteristics of the problem. If the functions and terminals selected are not appropriate for the problem, the desired solution cannot be achieved [9]. Therefore, in order to overcome the limitation of GP and achieve satisfactory results, it is crucial to have a deep understanding of the problem to choose the appropriate GP model.

In this research, there are two GP models available for finding a function for τ expressed by input

Validation of the GP results

200 alternative expressions were generated by GP. Given lower AAREP value and the simplicity of the function generated, Eq. (19) was selected as the winning function. $\frac{σ_{3}}{σ_{ci}} = \frac{a \frac{σ_{n}}{σ_{ci}}}{\sqrt{a (1 + \sqrt{m_{b}}) - \frac{σ_{n}}{σ_{ci}}}}$ Substituting Eq. (19) into Eq. (13), the angle of friction $ϕ$ can be calculated as follows: $P = 2 + {am}_{b} {(m_{b} \frac{σ_{3}}{σ_{ci}} + s)}^{a - 1}$ $ϕ = \arcsin (1 - \frac{2}{P})$ Finally, with the help of Eq. (11) the shear stress τ can be expressed as follows: $τ = σ_{ci} \frac{\sqrt{P - 1}}{P} \frac{{(m_{b} \frac{σ_{n}}{σ_{ci}} + s)}^{a}}{{(\frac{Pa + P - 2}{aP})}^{a}}$ where P is the intermediate parameter. The proposed Eq. (22) differs from the

Conclusions

Existing numerical methods in conjunction with symbolic regression (SR) analysis preformed by genetic programming (GP) have been used to derive analytical solutions for estimating the Mohr–Coulomb (MC) shear strength of rock masses from the non-linear Generalized Hoek–Brown (GHB) criterion.

The reported research used Eq. (14) to build a GP model as the basis for calculating the intermediate parameter σ₃/σ_ci expressed by input parameters m_b, s, a and σ_n/σ_ci. After obtaining analytical solution

Acknowledgements

PhD Scholarship provided by China Scholarship Council (CSC) is gratefully acknowledged. The Sections 1 and 2 of the manuscript were developed with Prof. Stephen Priest as co-author. Thus, many thanks go to Prof. Stephen Priest for his contribution. The authors also would like to express their gratitude to anonymous reviewers for their constructive comments on the paper.

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Technical CommunicationDirect expressions for linearization of shear strength envelopes given by the Generalized Hoek–Brown criterion using genetic programming

Abstract

Introduction

Section snippets

Introduction of the HB criterion

Overview of genetic programming

GP modelling for the GHB criterion

Validation of the GP results

Conclusions

Acknowledgements

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