Rock slope stability assessment based on the critical failure state curve for the Generalized Hoek ‒ Brown criterion

The strength reduction method (SRM) based on the Generalized Hoek ‒ Brown (GHB) criterion has become an important and popular means to analyse the stability of rock slopes. Various reduction strategies have been proposed and applied by civil engineering. This paper proposed a new SRM for rock slopes with the GHB criterion based on the critical failure state curve (CFSC). The existence of the CFSC has been proven by theoretical analysis, and the explicit expression of the CFSCs for different parameters m i and slope angles β , considering the in�uence of disturbance factor D, has been obtained by curve �tting based on a great deal of simulation data. The new SRM provides a graphic method to determine the parameters at the critical failure state from the initial state and proposes a de�nition of the factor of safety (FOS) based on the parameters of the two states. This method was applied to 8 slope examples to verify its validity and accuracy. The relative errors between the critical state parameters obtained from the graphic method and that from the simulation analysis are less than 10%, which proves the accuracy of the CFSCs. The FOSs obtained by the proposed de�nition are compared with that obtained by the Bishop simpli�ed method and the local linearization method (LLM), and the results are very close. The relative error is less than ± 5% compared with the LLM, and the stability state predicted is perfectly accurate. However, the calculation procedure is largely simpli�ed, and the calculation speed is largely improved.


Introduction
The stability assessment and analysis of rock slopes is a signi cant task for rock engineering projects, such as open pit mining.The strength reduction method (SRM) is widely accepted and used by researchers to analyse slope stability because of its advantage over the traditional limit equilibrium method (LEM) (Zhao et al. 2005, Krahn 2007, Liu et al. 2015).Presently, SRM research is mainly based on the conventional Mohr-Coulomb (MC) criterion, which is incapable of explaining the nonlinear deformation and failure characteristics of rock masses.The Hoek-Brown (HB) criterion, rst proposed in 1980 as an empirical nonlinear failure criterion, has developed into a rigorous and complete strength criterion and has been widely applied in rock mechanics and engineering (Hoek et al. 1980(Hoek et al. , 2002)).The Generalized Hoek-Brown (GHB) criterion, as the latest version of the HB criterion presented in 2018 (Hoek and Brown 2019), is capable of estimating the strength and deformation characteristics of homogeneous and isotropic rocks with few discontinuities and heavily jointed rock masses.Combining the advantage of SRM on slope stability analysis and the nonlinear GHB criterion to explore the new strength reduction strategy has become an important and popular research eld (Melkoumian et al. 2009, Shen et al. 2014, Zong 2008).Due to the complexity of the parameters in the GHB criterion, various reduction strategies have been proposed and applied by many scholars, but none of them has gained wide acceptance by the engineering eld.Presently, the representative reduction methods are classi ed into the following four types: 1) Some or all of the GHB parameters are directly reduced by the reduction factor is usually called the "direct reduction method", and the global safety factor is traditionally de ned by the reduction ratio of the original parameters to the reduced parameters at failure.For example, Wu SY (2006) and Han LQ (2016) suggested reducing the uniaxial compressive strength of the intact rock (σ ci ) and the material constant for the intact rock (mi).Song (2012) discussed seven cases of the direct reduction method and found that directly reducing σ ci and the geological strength index (GSI) could obtain a reasonable global safety factor.However, the direct reduction method could lead to distortion of the GHB failure envelope after the parameters are reduced.
2) The integral strength envelope of the GHB criterion is lowered by a reduction factor until the critical failure state, proposed by Hammah (2005).The determination of the equivalent GHB curve that best approximates the lowered envelope is obtained through minimization of the total squared error by the simplex method, of which the analysis procedure is complex.Thomas et al. (2008) adopted the spatial mobilized plane (SMP) concept to realize intrinsic material strength factorization.However, this method requires iterative computations to obtain the shear strength at every reduction step, which greatly lowers the calculation e ciency.
3) The nonlinear GHB criterion is transformed into the linear MC criterion, and the slope stability is analysed based on the MC criterion, called the equivalent linearization method (Priest 2005, Shen et al. 2012, Yang et al. 2010).This method was classi ed into two types: the global approach (Hoek 2002, Zong et al. 2008) and the local approach (Fu et al. 2010, Xu et al. 2018, Wei et al. 2021).The former is to obtain a global linear optimization of the shear strength curve of the GHB criterion; the latter is to obtain the local stress and strength values.Sukanya et al. (2012) believed that the local approach was physically more correct than the global approach.However, the local MC parameters need to be solved by the Newton iterative formula, and the calculation is heavy for a complex model.Recently, a new SRM strategy based on the GHB criterion (Yuan et al. 2020), whose core concept is to search an optimal reduction pathway for the GHB criterion parameters by establishing the critical failure state curve of different slopes.Inspired by its strategy, this paper establishes a more accurate and detailed critical failure state curve equation based on a large amount of numerical simulation data, and an improved parameter reduction scheme and a simpler safety factor de nition method are proposed.This method is applied to several classical examples to verify the validity and reliability of the parameter reduction scheme and the de nition of the safety factor.This strategy provides a fast and accurate method to obtain the safety factor of rock slopes based on the GHB criterion without relying on mechanical modelling analysis.
2. The General Hoek-Brown criterion and the slope critical failure state curve

The General Hoek-Brown Criterion
The GHB strength criterion is usually given in principal stress space with the following expression.
where σ 1 and σ 3 are the maximum and minimum principal stresses of the rock mass at failure (compressive stress is taken to be positive), and σ ci is the uncon ned compressive strength of the intact rock according to the 2018 edition (Hoek 2018).m b , s, and α are empirical parameters re ecting rock mass characteristics related to the fracturing degree of the rock and can be estimated by the functions of the geological strength index (GSI), the disturbance factor D and the material constant of intact rock mi.

2
where GSI is estimated by the structure (or blockiness) and the surface conditions of the jointed blocky rock masses, of which the maximum value is 100 (for intact rock).D is the disturbance factor subjected to blasting damage and stress relaxation of the rock mass, with a value range of 0.0 (undisturbed) ~ 1.0 (disturbed).m i is a material constant for the intact rock, which represents the rock type and hardness Balmer et al. (1952) proposed that the normal stress and shear stress could be expressed as functions of principal stresses as follows: The differential expression ∂σ 1 ∂σ 3 can be derived from Eq. (1) as follows: The expressions of normal stress and shear stress based on the GHB criterion can be obtained by substituting equations (4) into (3), given as follows: As noted in the GHB criterion, the uniaxial compressive strength of the rock mass (σ cmass ) is expressed by setting σ 3 = 0, and the tensile strength of the rock mass (σ tmass ) can be obtained by setting σ 1 = σ 3 = σ tmass , given below: The stability of the slope is determined by the unit weight of the rock mass (γ), the slope height (H), the slope angle (β) and all the parameters involved in the failure criterion of the rock mass.Based on the GHB criterion, a general functional relationship related to all the parameters to describe the critical failure state of any slope could be established.According to the classic de nition of the factor of safety (FOS), a slope at the critical failure state indicates an equilibrium between the total resistant shear force and the total sliding force on the potential sliding surface.Thus, this equilibrium state for any slope can be deduced as follows: where l denotes the potential sliding surface; in each zone, τ s is the resistant shear stress; and τ m is the driving shear stress, which greatly depends on the gravity of the overlying rock mass above the potential sliding surface (γH).Thus, Eq. ( 7) can be expressed as: From formula (5), the following equation can be derived.
Therefore, the minor principal stress σ 3 σ ci can be solved by an iteration of σ n σ ci with given m b , s and α. τ σ ci is the function of σ 3 σ ci , m b , s and α.Thus, τ σ ci can be expressed as follows: The value of σ n depends on the gravity of the overlying rock mass γH and the slope angle β only under the condition of gravity.Thus, Eq. ( 10) can be expressed as: where the parameters m b , s and α can be expressed by Eq. (2).Thus, Eq. ( 11) can be expressed as follows: According to the equilibrium equation Eq. ( 8), the expression below can be derived: According to Eq. ( 6), s α is the function of GSI and D. Thus, Eq. ( 13) is nally expressed as: In the literature of Yuan (2020), σ ci and s α are combined and replaced by σ cmass from Eq. ( 6).In addition, Eq. ( 14) is deduced below.
γH is a dimensionless parameter.The parameters λ, β and mi, which satisfy the equilibrium Eq. ( 15) at the slope critical failure state, could establish the slope critical failure state curve (CFSC).

Explicit expression of the CFSC by numerical simulation
It is a great challenge to establish the explicit expression of CFSC from the implicit function ( 15) by theoretical analysis.Thus, a numerical simulation method based on FLAC 3D software is used to achieve this goal.Referring to reference (Hammah 2005, Fu 2010), the geometry and grid layout of the slope model used in this numerical simulation is shown in Fig. 1.
To simplify the simulation procedure, the slope angle is assigned to 45°, 60° and 75°.The height of the slope is assigned to 10m, 25m, 50m, 100m and 200m (the heights are ne adjusted in accordance with the slope angles shown in Table 1).The horizontal displacements of the left and right boundaries are xed, while both horizontal and vertical displacements are xed along the bottom boundary.The unit weight γ is set to 25.0 kN/m 3 .The disturbance factor D is initially set to 0, and its in uence on the CFSC is discussed in section 2.4.Only the gravity of the rock mass is considered as an external load in the numerical simulation.The convergence criterion to judge whether the slope has reached a mechanical stable state is setting the maximum unbalance force ratio R to 10 − 5 .The slope model was considered to be homogeneous and isotropic.Considering the mesh sensitivity, the amount of grids remains the same for different slope models(approximately 4000 ~ 5000 grids).
The elasticity modulus (E) and Poisson's ratio (ν) of the rock mass remain constant when the GSI, D and σ ci vary in this study.Because this study pays more attention to the parameters causing plastic failure than the parameters involved in elastic deformation.The simulation results has proven that the changes in the elastic modulus and Poisson coe cient do not affect the parameters of the slope critical failure state.The procedure of the numerical simulation is as follows: 1. Slope models with different slope angles and heights are established, and the rock mass is assigned to be homogeneous and isotropic.2. For each slope model, the unit weight of the rock mass remains the same, and the disturbance factor D is initially set to 0. m i is xed as the value in Table 1.GSI is varied with the range of Table 1, and nally, σ ci is adjusted to lead the slope to the critical failure state.The material parameter values (m b , s and α) of the slope model are transformed by the Fish function in FLAC 3D with the values of GSI, m i and D, and σ ci is repeatedly reduced by an interval of 0.2 until the numerical calculation does not converge, which means that the slope has reached the critical failure state.3. The uniaxial compressive strength of the rock mass σ cmass and the dimensionless parameter λ are calculated according to the parameters at each critical failure state.The simulation results show that the dimensionless parameter λ at the critical failure state remains constant for different heights and unit weights, while it varies with GSI, m i and slope angle β. Figure 2 shows the critical failure state curve of the dimensionless parameter λ varying with GSI for different vales of m i and slope angle β.To better analyse the theoretical relation, the dimensionless parameter λ can be decomposed as the multiplication of two parameters, σ ci γHand s α , as shown in Eq. ( 16), where σ ci γH was termed the strength ratio (SR) of a rock slope and was strongly related to the safety factor, proposed by s α is taken as the x-coordinate, and σ ci γHis taken as the y-coordinate.The points paired by s α and σ ci γH at the critical failure state for different vales of m i and slope angles β are drawn in the s α ∼ σ ci γH coordinate system, which form the slope CFSC.
The Curve Fitting Tool (cftool) of MATLAB was used to t the CFSC.After comparing various tting methods, it was found that the tting type power 2 (a * x b + c) had the highest tting accuracy and the minimal tting error.The tting expressions of the CFSCs and tting accuracy are given in Table 2.The points paired by s α and σ ci γH at the critical failure state and the tting curve for different slope angles and values of m i are shown in Fig. 3.  Therefore, the CFSCs of D≠0 can be obtained by multiplying the CFSCs of D=0 by the raito R D .The estimation of the raito R D for different β and can be solved by simulation analysis.The procedure is as follows: for β=45°, mi is set to 5, 10, 15, 20 and 25.The parameters at the critical failure state for D=0, 0.5, 0.7, 0.9 and 1.0 are searched for the given , and the dimensionless parameter λ and the raito R D for 0.5, 0.7, 0.9 and 1.0 are calculated with Eq. ( 17).Fig. 5 shows that the dimensionless parameter λ varies with mi for different D at β=45°, 60° and 75°.Fig. 6 shows that the ratio R D for D=0.5, 0.7, 0.9, and 1.0 varies with at β=45°, 60° and 75°.With the ratio R D and the CFSCs for D=0 obtained from section 2.3, the CFSCs of D=0.5, 0.7, 0.9, and 1.0 for different mi and β can be obtained.
3. SRM based on the slope CFSC

Reduction strategy for the GHB criterion
With the theoretical existence of CFSC proved and the explicit expression of CFSC obtained, a new reduction strategy based on the compressive strength of rock mass for the GHB criterion was proposed, which can better re ect the physical meaning of strength reduction than the direct reduction of material parameters.In this strategy, the compressive strength of intact rock σ ci and the parameter combination s α are reduced by the same ratio, where s α is numerically equivalent to JP(Jointing Parameter) of the RMi system (Russo 2008), which represented the rock mass quality and structure.m i , as a parameter representing the degree of intact rock hardness, should not be involved in reduction.However, with the decrease in GSI, the parameter of rock mass m b will decrease correspondingly.According to the GHB criterion of the 2018 edition, the compressive to tensile ratio and the parameter m i satisfy the approximate relationship (18).Thus, the same reduction ratio of the compressive strength and the tensile strength of intact rock can be realized by keeping m i constant.
σ ci σ t = 0.81m i + 7 18 The disturbance factor D, which subjected to blast damage and stress relaxation, should not participate in the reduction.The reduction of s α is only determined by the reduced GSI.The reduction strategy can be expressed by the formula (19).
where K r is the reduction ratio and σ t ci , s t and α t are the parameters of the target slope.σ r ci , s r and α r are the parameters of the reduced slope.
According to the CFSC, the reduction strategy can be replaced by reducing the parameter σ ci γH and the parameter combination s α by the same ratio represented by formula (20), because γH is constant during the reduction procedure.4) were applied by this reduction strategy to predict the critical state parameters.The disturbance coe cient D was assigned to 0 to simplify the procedure.The parameters predicted by this strategy were compared with those obtained by simulation model analysis.The results are shown in Table 3.
Table 3 The comparison results of the predicted parameters and simulation parameters for the critical state From the results of Table 3, it can be seen that the relative errors between the parameters of the critical state predicted by the CFSC and the parameters obtained by simulation analysis are lower than 3%, which proves the validity and accuracy of this strategy.

of the in uence of reduciton
determine whether m i should be reduced, the 3 slope examples given in this paper are reduced and analysed by applying the strength reduction above.The parameters of the 3 slope examples are shown in Table 4.The parameters at the slope critical failure state are obtained by numerical simulation with the decrease in m i from the initial value.With the critical state parameters, the curves of shear stress to normal stress of the slope are drawn to analyse the in uence of m i decreasing on the strength envelope of the GHB criterion.The curves of shear stress to normal stress of the three slopes at the initial state and the critical failure state are shown in Fig. 8(a).The shear stress ratio of the initial state to the critical state varies with the normal stress is shown in Fig. 8(b).It can be seen that the strength envelopes decrease with decreasing mi.When the value of mi decreases by 1 ~ 1.5 (Eq.21), The shear stress ratio increases less than 0.12.According to the conservative principle, the strength envelope is the highest and the shear stress ratio is the smallest when mi is not reduced, corresponding to the lowest safety factor.Ratio mi = m i − initial m i − reduced (21) Figure 8 The curve of shear stress and the shear stress ratio of the three slopes

The de nition of factor of safety
The de nition of factor of safety (FOS) is the key of slope stability evaluation.In slope engineering, the strength reserve safety factor is often used to evaluate the slope stability, which is de ned by the ratio of anti-sliding force and sliding force on the sliding surface of the slope.This method relies heavily on the determination of the sliding surface, for example, the global safety factor de ened by Yuan (2020).Some scholars directly used the reduction ratio of the GHB criterion parameters as the safety factor to simplify the analysis procedure, which is obviously unreasonable.According to the strength reduction strategy in section 3.1, the strength parameters of the initial state and critical state have been obtained, based on which, a new factor of safety (FOS) de nition method independent of sliding surface determination was proposed.In this method, the ratio of the average shear strength between the target state and the critical state was taken as the FOS of the slope, where the average shear strength at the critical failure state represents the sliding force per unit area, and the average shear strength at the initial state represents the sliding resistance force per unit area, which satis es the meaning of strength reserve.The average shear strength with the range of the normal stress shows overall mechanical behaviour.
The speci c calculation process is as follows: 1) The range of minimum principal stress σ 3 at the critical state is determined by Eq. ( 22) based on the literature (Hoek 2002) where σ cm represents the global "rock mass strength" re ecting the overall behaviour of a rock mass.σ 3max is the upper limit of the minimum principal stress for slopes.σ ci , m b , s, and α are the parameters at the critical state.
2) The normal stress σ n and the shear stress τ at the initial state and the critical state, respectively, are calculated with the corresponding parameters by Eq. (5), and the upper limit of σ n at the initial state and the critical state (σ t nmax and σ r nmax ) were calculated with the upper limit σ 3max at the critical state.
3) The arithmetic mean value of shear stress τ within the range σ n obtained by step 2 is taken as the average shear strength, and the ratio of the average shear strength of the target state to the critical state is taken as the safety factor.The formula is as follows:

The analysis of slope examples
To verify the validity and accuracy of the method in slope stability analysis, several slope examples selected from the preexisting literature were analysed and discussed.The examples were classi ed into two groups for D = 0 and D ≠ 0. The slope parameters at the initial state are shown in Table 5 and Table 7. First, the parameters at the critical failure state obtained by the graphic method based on CFSCs were compared with those obtained by numerical simulation, of which the results are shown in Table 6 and Table 8.From the table, it can be seen that the relative error of dimensionless parameter λ is less than 8% for D = 0 and less than 10% for D ≠ 0. This proves that the CFSCs for D = 0 and mi ≠ 5,10,15,20,25 obtained by cubic spline interpolation have high accuracy, and the CFSCs for D ≠ 0 obtained by the D ratio are relatively accurate.According to the de nition of the factor of safety in section 3.3, the FOS of these slope examples can be calculated with the initial and critical parameters.The FOS obtained by graphic analysis and simulation analysis were compared with the FOS obtained by the local linearization method and the Bishop simpli ed (limit equilibrium method).The results are shown in Table 9.

× 100%
The FOS obtained by the proposed method (Graphic method) is very close to that of the simulation analysis with the same FOS de nition (relative error < ± 5%).Moreover, the error between the proposed method and the local linearization method is also very low (relative error < ± 5%).The results of the proposed method are usually smaller than that of the Bishop simpli ed method, with a relative error lower than 20%.

Discussion
To verify the validity and accuracy of the reduction strategy and the de nition of FOS, the local linearization method and the Bishop simpli ed method were used to analyse the stability of the example slopes and compare the results.The local linearization method (LLM), as a sophisticated SRM, has been applied in FLAC3D for solving the FOS of the HB criterion slope.This method is physically more correct than the global approach (Sukanya 2012).Among the limit equilibrium methods (LEM), the Bishop simpli ed method is the most widely used in slope stability analysis.However, limited by the concepts of circular sliding surfaces and slice, the LEM has lower accuracy than SRM.The FOSs obtained by the proposed method is very close the results of the LLM with the absolute relative error smaller than 5%, has an acceptable error limit compared with the Bishop simpli ed method.Moreover, the relative error decreases as the FOS approaches 1.Based on the slope stability state classi cation in Table 10, the proposed method has the same stability state as the other two methods.The calculation speed and e ciency of the proposed method have a large improvement compared with simulation analysis and local linearization method, where searching for an appropriate reduction ratio takes considerable calculation time.

Conclusions
This paper proposed a new SRM (graphic method) based on the CFSCs.First, the CFSCs are established based on a large amount of simulation data, the in uence of m i and D on the CFSCs are analysed, and the explicit expressions of the CFSCs are determined by curve tting.Then, the new strength reduction scheme with the same reduction ratio for σ ci and s α based on the CFSCs, and the new de nition method of FOS were proposed.This new method was applied to 8 slope examples to verify its validity and accuracy.The main conclusions are as follows: 1.The parameters σ ci γH , s α , mi, β satisfy a relationship at the critical failure state by theoretical analysis.For given m i and β, the relation between γH and s α can be represented by an exponential function.These curves are called CFSCs. 2. The variations of the CFSCs with m i and D are analysed.The CFSCs for any value of m i can be obtained by cubic spline interpolation.The CFSCs for D ≠ 0 can be obtained by multiplying the CFSCs of D = 0 by the ratio R .
3. A new reduction strategy based on CFSCs is proposed, where σ ci and s α are reduced by the same ratio, m i remains constant to guarantee the same reduction ratio of the compressive strength and the tensile strength of intact rock, and D subjected to blast damage and stress relaxation should not be reduced.The reduction in σ ci and s α , representing the strength of intact rock and joint parameters (JP) of the rock mass, respectively, can better re ect the strength reduction of the rock mass.The parameters at the critical state can be obtained by the graphic method based on the CFSCs.4. By comparing the critical state parameters obtained by the graphic method and that from the simulation analysis of slope examples, the CFSCs for different m i and D have a high accuracy.
5. A new de nition method of FOS, independent of sliding surface, based on the parameters of the initial and critical states, was proposed.By the analysis of 8 slope examples, the results showed that FOSs obtained by the graphic reduction method and the new de nition are close to the results of the local linearization method and the Bishop simpli ed method, with ± 5% relative error compared with the local linearization method.
The accuracy of FOS increases when FOS approaches 1, and the stability state predictions are perfectly accurate.The comparison results veri ed the validity and reliability of the SRM and the FOS de nition proposed.
. The calculation speed and e ciency of the graphic method have a large improvement compared with simulation analysis and local linearization method, where searching for an appropriate reduction ratio takes considerable calculation time.

Declarations Figures
Page 15/  The geometry and grid layout of the slope model   The CFSCs with different D and the ratio R D varies with for β=45° and m i =10 The dimensionless parameter λ varies with m i for different D at β=45°, 60° and 75°.
The graphic representation of the reduction strategy and the example for m i =5, β=60 °

4 )
Applying stability charts for rock mass slopes satisfying the Hoek-Brown criterion to obtain the safety factor of the slope (Li et al. 2008, Shen et al. 2013, Sun et al. 2016).For example, Shen (2013) and Sun (2016) proposed a slope stability chart for a speci ed slope angle β = 45° and disturbance factor D = 0 and then combined the weighting factors f D and f β to calculate the FOS of a slope assigned various slope angles under different blasting damage.The graph method relies heavily on the accuracy of chart data.

2
Theoretical relationships of slope parameters at the critical failure state Shen et al.(2013), and s α determined by GSI and D, is numerically equivalent to JP(Jointing Parameter) of the RMi system(Russo 2008), which represented the rock mass quality and structure.

Figure 3
Figure 3 The tting curve of CFSCs for different m i and β values When mi takes values other than 5, 10, 15, 20 and 25, the corresponding CFSCs can be obtained by interpolation.Since the curve of λ varying with mi does not satisfy the linear relation, the CFSCs of other mi values can be obtained by cubic spline interpolation.2.4The in uence of disturbance factor D on the CFSCsThe value of disturbance factor D has great in uence on slope stability(Li et al. 2011).In the latest version of the GHB criterion, the suggested values of D are given for different engineering slopes, that is, D = 0.5 for controlled presplit or smooth wall blasting, D = 0.7 for mechanical excavation effects of stress reduction damage, and D = 1.0 for production blasting.The in uence of D on the CFSCs is presented by setting D = 0.5, 0.7, 0.9 and 1.0.Fixing β and mi, search for the parameter combination (s α ∼ σ ci γH) of the critical failure state for D = 0.5, 0.7, 0.9 and 1.0.The in uence of D on the CFSCs for β = 45° and m i = 10 is shown in Fig.4, from which it can be found that the CFCSs rises with the increasing of D. The ratio of the CFSCs (D ≠ 0) to the CFSCs (D = 0) remains almost constant when s α varies, which can be represented by the equation below.

Figure 2 The
Figure 2

Figure 3 The
Figure 3

Table 1
Slope model and values set for the parameters

Table 2
The tting expressions and accuracy of the CFSCs for differentm i andβ Note: a RMSE: Root mean squared error, b R-square: Coe cient of determination, which re ects the tting accuracy.

Table 4
The geometry and material parameters of the 3 slope examples

Table 5
The slope geometry and rock mass parameters at the initial state for D = 0

Table 7
The slope geometry and rock mass parameters at the initial state for D ≠ 0

Table 9
The comparison of FOS of slope examples by different methods

Table 10
The relation between the FOS and stability state