A chart-based seismic stability analysis method for rock slopes using Hoek-Brown failure criterion

    A chart-based seismic stability analysis method for rock slopes using HoekBrown failure criterion Xing-yuan Jiang, Peng Cui, Chuan-zheng Liu PII: DOI: Reference:

S0013-7952(16)30161-2 doi: 10.1016/j.enggeo.2016.05.015 ENGEO 4304

To appear in:

Engineering Geology

Received date: Revised date: Accepted date:

29 September 2015 3 May 2016 28 May 2016

Please cite this article as: Jiang, Xing-yuan, Cui, Peng, Liu, Chuan-zheng, A chart-based seismic stability analysis method for rock slopes using Hoek-Brown failure criterion, Engineering Geology (2016), doi: 10.1016/j.enggeo.2016.05.015

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ACCEPTED MANUSCRIPT A chart-based seismic stability analysis method for rock slopes using

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Hoek-Brown failure criterion

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Xing-yuan Jiang a, b, Peng Cui a, *, Chuan-zheng Liu a, b

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a

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Hazards and Environment, Chinese Academy of Sciences, Chengdu, Sichuan 610044,

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China

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b

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* Corresponding author: [email protected] (Peng Cui)

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Key Laboratory of Mountain Hazards and Surface Process, Institute of Mountain

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University of Chinese Academy of Sciences, Beijing 100049, China

List of Symbols α Hoek-Brown input parameter c Cohesion D Disturbance factor of rock mass fβ Scaling factor of slope angle fKh Scaling factor of horizontal seismic acceleration coefficient fD Disturbance weighting factor FD Driving force FN Normal force FoS Factor of safety FoSLEM Factor of safety in limit equilibrium method FoSLAM Factor of safety in limit analysis method GSI Geological strength index H Slope height HB Hoek-Brown criterion JCond8 Joint Condition 9

Kh

LAM

Horizontal seismic acceleration coefficient Vertical seismic acceleration coefficient Limit analysis method

LEM

Limit equilibrium method

Kv

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1

mb

Hoek-Brown input parameter

mi MC

Hoek-Brown constant Mohr-Coulomb

N PS

Stability number Pseudo-static method

RQD S SCAM SR β

Rock Quality Designation Hoek-Brown input parameter Stability charts analysis method Non-dimensional strength ratio Slope angle

φ

Friction angle

γ σ1 σ3 σci σn

Unit weight of rock mass Maximum principal stress Minimum principal stress Uniaxial compressive strength of the Intact rock Normal stress

τ

Shear stress

λ

Number of discontinuities per meter

Abstract

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The parameters of rock mass structures, strength and seismic effect are critical

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factors for the seismic stability analysis of rock slopes. This paper demonstrates the

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ACCEPTED MANUSCRIPT use of a new form of a chart-based slope stability method that satisfies the

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Hoek-Brown (HB) criterion. The limit equivalent method is used to assess the

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stability of rock slopes subjected to seismic inertial force. First, stability charts for

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calculating the factors of safety (FoS) with a slope angle of β=30° in static and

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pseudo-static states were proposed by using Slide 6.0 software. Next, scaling factors

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of the horizontal seismic acceleration coefficient (fKh) and slope angle (fβ) were

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established to illustrate the influence of the horizontal seismic load and slope angle on

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the stability of rock slopes, respectively. Using regression analyses of fKh and fβ, a fast

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calculation model was proposed to solve the slope safety factors based on the stability

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charts. Finally, the stability charts analysis method (SCAM) was verified against the

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numerical solutions; the results showed that 70.63% of the data had discrepancies of

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less than ±10%, and the data with discrepancies greater than ±10% were associated

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with high values of geological strength index (GSI) and horizontal seismic

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acceleration coefficient (Kh). The proposed model calculating the FoS of rock slopes

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is simple and straightforward to use for seismic rock slope design and stability

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evaluation.

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Keywords: Rock slope; Hoek-Brown failure criterion; Factor of safety; Scaling factor;

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Stability chart analysis method

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1. Introduction

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In seismically active areas, earthquakes are a major trigger factor for the failure

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of natural and man-made slopes (Li et al., 2009). Hence, predicting the dynamic

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stability of rock slopes is a significant task for civil engineers with respect to dams,

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open pit excavations, roads and other engineering projects. Determining the factor of 2 / 36

ACCEPTED MANUSCRIPT safety (FoS) is the most common way to assess the stability of rock slopes

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(Michalowski, 2010). The limit equilibrium method (LEM) is the most extensive

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means for evaluating slope stability; however, the rock masses are inhomogeneous

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and characterized by several discontinuities including joints, fractures, bedding planes

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and faults. However, most commercial software and theoretical formulas based on the

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LEM require the conventional Mohr-Coulomb (MC) shear strength parameters

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cohesion c and friction angle φ to estimate the FoS of slopes, which completely ignore

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the non-linear nature of the rock mass strength; therefore, the linear MC criterion do

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not agree with the rock mass failure envelope (Sheorey et al., 1989; Jimenez et al.,

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2008; Zheng et al., 2009; Fu and Liao, 2010; Shen et al., 2012).

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The application of the Hoek-Brown (HB) criterion surmounts the shortcomings

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of the conventional MC criterion. Hoek and Brown (1997) and Hoek et al. (2002)

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proposed a method for converting the rock mass strength parameters into the

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equivalent MC parameters. However, Li et al. (2008, 2011) found that this conversion

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could produce inconsistent estimates; the difference between using equivalent

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parameters and the native yield criterion was found to be up to 64% for slope stability.

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This suggests that the best way to address the rock and rock mass problems is to use

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the HB failure criterion directly in the calculations. Over the past 30 years, the HB

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criterion has been applied successfully to a wide range of intact and fracture rock

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types. The latest version of the HB criterion proposed by Hoek et al. (2002) is written

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as follows:

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1   3   ci (mb 3 /  ci  S )

(1)

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where σ1 and σ3 are the maximum and minimum principal stresses, respectively; σci is

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the uniaxial compressive strength of the intact rock; and mb, S, and α are the HB

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parameters, which represent the fracturing degree of rock masses.

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mb  mi exp((GSI 100)/(2814 D))

(2)

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S  exp((GSI 100)/(93D))

(3)

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

1 1  (exp( GSI /15)  exp( 20/3) ) 2 6

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(4)

ACCEPTED MANUSCRIPT From Eqs. (2)–(4), we can see that parameters mb, S, and α all depend on the

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geological strength index (GSI), which ranges from 5 (for highly fractured and poor

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rock masses) to 100 (for intact rock masses); mi is the HB constant for intact rock, and

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its value (1.0–35.0) reflects the hardness of the rock mass. D is the disturbance factor;

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its value ranges from 0 (for undisturbed rock masses) to 1 (for disturbed rock masses).

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The values of GSI and mi can be estimated using the method introduced by Hoek and

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Bray (1981) ,Marinos and Hoek (2001) and Hoek et al.(2013).

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The SCAM is a technique for rapid or preliminary analysis of slope stability, and

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it has been broadly used to estimate the stability of slopes; examples include the

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works of Taylor (1937), Hoek and Bray (1981), Zanbak (1983), Gens et al. (1988),

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Michalowski (2002), Siad (2003), Loukidis et al. (2003), and Li et al. (2008, 2009).

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However, developing suitable stability charts to estimate the slope FoS directly from

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the HB criterion is challenging because at least six parameters (GSI, mi, σci, γ, H and β)

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must be considered for a dry slope with D=0 (Shen et al., 2013).

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Hoek and Bray (1981) and Zanbak (1983) proposed charting solutions for

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stability and toppling problems for rock slopes, respectively. However, these methods

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were based on statistical analyses, and none considered seismic effects. The

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pseudo-static (PS) method is a popular technique for evaluating seismic slope stability

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and has been used by many researchers, such as Newmark (1965), Ling et al. (1997),

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Hong et al. (2005) and Baker et al. (2006). In the PS method, the earthquake effects

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are simplified to horizontal and/or vertical seismic coefficients (Kh and Kv). Terzaghi

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first applied the PS method to assess seismic slope stability (Hong and Xu, 2005), and

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Newmark (1965) applied and extended the PS method to estimate the ground

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displacements caused by earthquakes. Subsequently, the PS method has been accepted

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and extensively used for the study of earthquake-induced landslides and rockslides

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(Huang et al., 2001; Sepúlveda et al., 2005). Although this method is generally

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considered to be conservative, due to the simplicity of the PS approach, it is still used

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in research.

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The slope stability charts proposed by Carranza-Torres (2004) and Li et al. (2008,

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2009, 2011) are among the few charts that can be used to estimate the FoS directly 4 / 36

ACCEPTED MANUSCRIPT 93

from the HB failure criterion. Li et al. (2009) put forward the seismic stability charts

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for rock slopes via the limit analysis method (LAM) for the first time; the stability

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number was defined as in Eq. (5):  ci  HFOS LAM

(5)

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where N is the stability number of the slope, γ is the unit weight of the rock mass and

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H is the parameter of the slope height. FoSLAM is the factor of safety obtained by using

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the limit analysis method.

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Fig. 1 shows seismic slope stability charts for a slope angle of β=30°. Because

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the upper and lower boundaries of the results bracket a narrow range of values of N,

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Li et al. (2009) adopted the average value solution to generate the charts for simplicity.

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The stability number N can be calculated using the parameters GSI and mi and by

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specifying D=0. By obtaining the value of N, the FoS can be calculated based on Eq.

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(5). The definition of FoSLAM for Eq. (5) is different from that of FoSLEM obtained

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from the limit equilibrium method (LEM). Values of FoSLAM are not equal to FoSLEM,

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which was illustrated by Li et al. (2012) and Shen et al. (2013).

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Carranza-Torres (2004) revealed that when the HB parameter α=0.5, the FoS of a

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given slope only depends on the three independent variables  H , S/mb2 and β. The

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limitation of the proposed stability chart is that it is based on α=0.5 with a slope angle

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of β=45°. Shen et al. (2013) first proposed a slope angle scaling factor fβ (Eq. (6)) to

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illustrate the influence of the slope angle on the slope stability and developed a model

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for calculating the FoS (see Eq. (7)) by proposing a chart for estimating the

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disturbance weighting factor fD based on the HB criterion.

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f   2.66exp0.022 

(0< FoS <4)

FoS  f 45  f   f D

(6) (7)

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However, the limitation of the HB criterion is that Hoek and Brown (1980) only

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provided a range of values (0–1.0) for the disturbance factor D, and it is difficult to

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determine the exact value under various slope conditions. In general, the rock

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ACCEPTED MANUSCRIPT 120

mechanics parameters that are generated by the HB strength reduction method are less

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reliable because of this uncertainty. In this research, we aim to evaluate the rock slope stability based on the LEM by

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using the pseudo-static method and propose a new chart-based technique for

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analyzing rock slope stability under earthquakes with D=0. The FoS of slopes is

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calculated directly based on the parameters of the HB criterion (GSI and mi), slope

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geometry (H and β), rock mass properties (σci and γ) and seismic effect (Kh).

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2. Determination of model parameters

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With the aforementioned motivation, we applied the simplified Bishop method in

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the Slide 6.0 software to calculate the FoS of rock slopes. The boundaries of the

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slipping surface generated using the simplified Bishop method should locate between

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the upper plane of crest and lower plane of toe in the ideal geometry model. Under

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this condition, the boundaries should be far from the slope to reduce the impact of

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boundary effects on the slope stability factors (Zheng and Zhao, 2004). The accuracy

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of the calculating result is reasonable if the distance from the toe of the slope to the

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left boundary is 1.5 times the slope height, the distance from the top of the hill to the

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right boundary is 2.5 times the slope height, and the upper and lower boundaries are

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more than 2 times the slope height. The model is shown in Fig. 2.

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Dai et al. (2011) used the statistical analysis method to investigate the slope

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geometry information of nearly 20,000 landslides triggered by the Wenchuan

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Earthquake; the result showed that slopes exceeding 35° are more susceptible to

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landsliding. Li et al. (2008, 2009) also used a lower limit slope angle of 30° for

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rockslide stability analysis. In our research, a slope angle varying between 30° and 75°

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is adopted. The values of the horizontal seismic acceleration coefficient ranging from

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0.0 to 0.375 proposed by Hynes-Griffith and Franklin (1984) are used. The impacts of

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rock slope elevation amplification and geomorphologic factors are not considered in

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this paper.

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3. Results and analysis 6 / 36

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3.1 Model development In this research, the HB criterion was put into the software to calculate the

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instantaneous shear stress τ under the normal stress σn on the slope failure surface

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based on Balmer's equation; the general forms are expressed as follows:

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  n 3 1  3 amb (mb ( 3 /  ci )  S ) a 1     mb  S  1  a 1   ci  ci 2   ci   2  amb (mb ( 3 /  ci )  S ) 

(8)

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  1  amb (mb ( 3 /  ci )  S ) a 1     mb 3  S  a 1  ci   ci  2  amb (mb ( 3 /  ci )  S )

(9)

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where the values of the input parameters mb, S, α, σci and σn are given. The shear stress

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τ and the minimum principal stress σ3 can be calculated using the equations above; the

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simplified forms are as follows:

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3   f1 ( n , mb , S , a)  ci  ci

(10)

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   f 2 ( n , mb , S , a)  ci  ci

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(11)

When horizontal seismic inertia force is considered, the FoS can be defined as a

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function of the sliding normal force FN and the driving force FD, which were divided

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by the weight of the slide γH and the seismic inertia force KhγH as shown in Eq. (12).

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 FD   H sin   K h H cos    FN   H cos   K h H sin 

(12)

The FoS can be calculated as follows: FoS 

 FN FD



  H (sin   K h cos  )

   ci  f3      ci  H (sin   K h cos  ) 

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   ci   f3  f 2 ( n , mb , S , a)   H (sin   K cos  )  h ci  

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The normal stress σn on the slope failure surface is determined by the parameters

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γH and Kh. Eq. (13) can be expressed as:

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(13)

ACCEPTED MANUSCRIPT   H F  f5  ci f 4 ( , mb , S , a, kh ,  )   ci  H 

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function can be defined as follows: F  f6 (

 ci , mb , S , a, kh ,  )  f 6 ( SR, GSI , mi , D, kh ,  ) H

(15)

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The parameters (mb, S, and α) are calculated by using Eqs. (2)–(4); the FoS

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(14)

Eq. (15) shows that when the values of D, GSI, mi and Kh are given for a slope,

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the FoS is uniquely related to the non-dimensional strength ratio parameter SR

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(SR=σci / (γH)) regardless of the magnitude of σci, γ and H. When D=0, the number of

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independent parameters for calculating the FoS is reduced to five (SR, GSI, mi, Kh and

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β) (see Eq. (16)).

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F  f6 (SR, GSI , mi , kh ,  )

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(16)

To investigate the effect of parameters in Eq. (16) on the FoS of slopes, two

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tables with different cases/groups were studied, and those cases are not the specific

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natural examples. Table 1 presents three different cases of σci, γ and H, associated

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with the same SR, GSI, mi, Kh and β values of slopes, to calculate the FoS using five

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limit equilibrium methods. The results show that the FoS values are all nearly the

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same. Once more, three additional group cases corresponding to the same values of Kh,

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β and SR in Table 1 and using the Bishop method over a range of GSI, and mi were

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used to calculate the FoS of slopes. The comparison of the FoS calculated in Table 2

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reveals that the FoS depends only on the magnitude of SR when the values of GSI, mi

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and β are the same. Eq. (16) is a theoretical expression that is difficult to use directly

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to calculate the FoS of rock slopes. In the next stage, based on the relationship

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between the FoS and the parameters SR, GSI, mi, Kh and β, charts for estimating the

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FoS with a slope angle β=30°, D=0 are proposed under static and pseudo-static

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conditions.

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3.2 Charts of rock slope stability under static and pseudo-static conditions

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As mentioned in section 2, a slope angle ranging from β=30° to β=75° was

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ACCEPTED MANUSCRIPT adopted in this paper. First, rock slope stability charts for a slope angle of β=30° under

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the static condition were constructed. Fig. 3 shows that the FoS clearly increases as

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GSI and SR increase. For example, in Fig. 3a, when GSI increases from 10 to 100 with

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SR=1, the values of FoS increase from 0.614 to 3.576. The parameter SR has an

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important effect on the FoS especially with high values of GSI, but at low values of

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GSI (GSI≤60), the FoS increases slightly as SR increases. For instance, when

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GSI=100, the FoS is 5.483 for SR=2 and increases to 18.481 for SR=10; when GSI=20,

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the FoS is 1.037 for SR=2 and increases to 1.781 for SR=10. The stability chart for

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β=30° (Fig. 3) forms the basis for the flowing part of this research. To reveal the

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influences of β and Kh on the FoS of rock slopes, it is important to choose a

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benchmark for comparison in the following research.

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Fig. 4 elucidates the relationship between SR, mi and FoS for a slope with Kh

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(0.1–0.3) and β=30°. The results show that at low values of GSI (GSI≤60), the FoS

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increases slightly as SR increases and that the minimum value of mi=5 and the

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maximum value of mi=35 generate a narrow range of FoS. In contrast, the FoS

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increases dramatically for GSI>60; the FoS for different values of mi meet at one

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point, which is shown in rectangles A–D in Fig. 4a. When the values of SR are smaller

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than the values of the intersection points, the FoS increases as mi increases. However,

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when the values of SR increase, the FoS decreases as mi increases. A comparison of

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the data in Fig. 3c and Fig. 4 shows that Kh plays an important role in the stability of

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rock slopes. For example, when mi=15, GSI=60, SR=10 (Fig. 3c), the values of the

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FoS change from 4.775 to 3.926, 3.303 and 2.913, with Kh ranging from 0.0 to 0.3; the

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decrease in magnitude is 0.849, 1.472 and 1.862, respectively.

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An alternative form of the stability charts is shown in Fig. 5. In general, the FoS

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increases as mi increases. However, for high values of GSI and SR, the FoS decreases

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as mi increases (Fig. 5d). This suggests that the stability of the slope is independent of

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the HB parameters. For further investigation, the stress conditions in the region of the

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ACCEPTED MANUSCRIPT slip surface were extracted and observed more closely. The obtained information

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associated with the Hoek-Brown yield envelope for the rock slope material is shown

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in Fig. 6. When GSI=100, mi=35 and Kh=0, most of the stress points along the failure

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surface are under compression. However, for GSI=100, mi=35, and Kh=0.3, most of

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the stress points fall in the region with small values of σ1/σci-σ3/σci and are in a tension

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state. This result is also true for GSI=100, mi=5, and Kh=0.3 (triangular symbols). This

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suggests that under seismic forces, the tensile strength of the material will control the

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overall stability. It is also found in Fig. 6 that the tensile strength is greater for mi=5

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than for mi=35 under the condition of GSI=100; when the values of GSI are low

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without seismic forces, the slip surface is mainly under compression. This means that

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the collapse of the seismic rock slope with high values of GSI is due to tensile failure.

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Rock masses with low values of mi have high tensile strength and are therefore more

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stable.

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3.3 Scaling factor analysis of β and Kh

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The slope angle and seismic inertia force are important factors with respect to the

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stability of seismic rock slopes. The principal aim of this section is to propose the

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scaling factors fKh and fβ for use in refining the influence of Kh and β in analyzing rock

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slope stability.

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3.3.1 The seismic scaling factor fKh

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The seismic scaling factor fKh, which is defined as the ratio of the FoS under

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seismic disturbance to that of the undisturbed slope, was used to evaluate the

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influence of Kh on the stability of rock slopes. The first step for obtaining the factor of

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fKh is to analyse the FoS under different seismic acceleration coefficients from 0 to 0.3

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with the same values of GSI, mi, SR and slope angle β=30°. The statistical data

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analysis result in Fig. 7 shows that when Kh=0.1, the maximum and minimum values

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of the scaling factors are 0.9302 and 0.7271, respectively, the average value is 0.8178,

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and 84% of the scaling factors are distributed in the region of the average values. It is

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clear that the scaling factor fKh decreases as Kh increases (0.1–0.3); the average scaling 10 / 36

ACCEPTED MANUSCRIPT factors are 0.8178, 0.6809 and 0.5759, respectively. By using the curve fitting method,

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an exponential function with a high fitting degree of over 0.9 is established. Then, the

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seismic scaling factors, with different slope angles (β=45°–75°), are acquired in the

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same way; the regression equations are shown in Table 3.

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Fig. 8 illustrates the relationship between fKh and Kh; the minimum and

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maximum slope angles (β=30°–75°) generate a narrow range of fKh values, which

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means the slope angle has inappreciable influence on fKh. To eliminate the influence of

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the slope angle on fKh, the average values of the scaling factors are adopted and the

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simplified fitting equation is developed, as shown in Eq. (17).

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3.3.2 The slope angle scaling factor fβ

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(17)

Actually, the slope angle influence effect on the seismic scaling factors in Eq. (17)

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f kh  0.353( Kh )2 - 1.390( Kh )  1.0

was eliminated, with errors ranging from -6.3 to 6.7%. After hundreds of software

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calculation with a wide range of rock mass properties and slope geometries, a chart

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(Fig. 9) that represents the relationship between the slope angle scaling factor fβ and

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the slope angle β was presented based on the data of 0≤FoS≤4. Using a curve fitting

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method, a simplified logarithmic function (Eq. (18)) was developed. The equation is

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applicable to slope angles from 30° to 75°.

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f   0.68 ln( )  3.323

(0≤FoS≤4)

(18)

3.4 Charts application and validation

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The proposed rock slope stability charts can be easily used to calculate the FoS

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of a given slope. First, for given values of GSI, SR and mi, the values of the FoS can

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be calculated by using the static stability chart (Fig. 3). Second, the seismic

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attenuation factor for any horizontal seismic acceleration coefficient Kh can be

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obtained from Fig. 8. Then, for a given slope angle β, the slope angle scaling factor fβ

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can be calculated from Fig. 9. Finally, the FoS of the rock slope can be calculated

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using Eq. (19).

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ACCEPTED MANUSCRIPT F  FoSS  f   f Kh

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(19)

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F  FoSS   0.68 ln(  )  3.323  0.353( Kh ) 2 - 1.39( Kh )  1.0

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The parameter GSI is the most important and the first point of entry into the

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proposed chart-based method. Hoek and Brown (1997) and Hoek et al. (2013)

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introduced a quantitative method to relate field observations to the rock mass quality

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(Fig. 10). Fig. 10 shows the chart in which the horizontal and vertical axes were

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defined by 1.5JCond89 and RQD/2 respectively. Horizontal axis represents

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discontinuity surface conditions while the vertical axis represents the blockiness of

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the rock mass. The value of GSI is given by the sum of these two scales in the

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following relationship:

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GSI  1.5JCond89  RQD / 2

(21)

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The Joint Condition (JCond89) rating defined by Bieniawski (1989) and the Rock

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Quality Designation (RQD) defined by Deere (1963) have been in use for many years

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and found it to be both simple and reliable to apply in the field. The rating parameters

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are given in Appendix 1. When no core is available, Priest and Hudson (1976) found

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that RQD could be obtained from the discontinuity spacing measurements of slope

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faces utilizing the negative exponential equation:

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RQD  100e0.1 (0.1  1.0)

(22)

where λ is the number of discontinuities per meter.

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In order to present the practical issue of the proposed seismic stability charts, the

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authors, though field rock slope investigations, illustrated the detailed process for GSI

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determination and rock slope FoS calculations. Two cases, Zipingpu (ZPP) reservoir

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rock slope and Huangnigang (HNG) landslide were chosen with different rock mass

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properties and slope geometries respectively.

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3.4.1 Zipingpu Reservoir rock slope

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Reservoir Sichuan, China. The lithology of the slope was mainly dolomite with slope

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height 100m and slope angle 50°. Unit weight of dolomite rock mass was 28kN/m3;

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HB parameter mi was 10 and the uniaxial compressive strength of intact rock mass

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was 100MPa based on the tables 2 and 3 in appendix 1. Three dominant joint sets (J1,

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J2 and J3) and one damage zone were found though detailed mapping work (Fig. 11).

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The discontinuities are slightly weathered planar surfaces, whose persistence ranging

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from 0.2 to 3.0m with apertures＜1.0mm and soft clay in it. The physical measuring

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tape was used and held in front of the rock slope face. The length of the intact rock

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segments greater than 10cm falling between natural fractures intersecting the tape are

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summed in a fashion similar to the core-based RQD. The measuring result showed

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that the discontinuities are closely spacing with λ ranges from 10 to 15.

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3.4.2 Huangnigang landslide

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The Huangnigang (HNG) landslide is a rock slope failure that occurred in

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Wenchuan earthquake in May 2008 in northwest Chengdu, China (Fig. 12). It

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involved the failure of 1.5M m3 of grey sandstone material with shallow surface loose

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deposits. Slope height is 140m in original slope angle 53°. Unit weight of rock mass

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was 24kN/m3, HB parameter mi was 18 and the uniaxial compressive strength of

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intact rock mass was 50MPa. Five dominant discontinuity sets were recognized

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during field investigation along the failure plane and the side-scarps (See Table 4).

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Joint set 1 and 2 are oblique intersected steeply dipping tension cracks forming the

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rear and basal-release surfaces respectively. The rock mass on the rear surface was

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poorly interlocked, heavily broken with mixtures of angular and rounded rock pieces.

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Joint set 3 is related to the stratification and dips into the slope at approximately 65°.

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Joint sets 4 provided the lateral sliding surface and is steeply dipping and

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perpendicular to J3 and J5. These three orthogonal discontinuities make the slope to

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be extremely blocky and interlocked rock masses. The discontinuity persistence

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ranges from 0.2 to 8.0m with apertures >1.0mm and soft clay filling. The liner density

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of the discontinuity λ ranges from 17 to 28. Based on the measuring result and rating parameters in appendix 1, the JCond89

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values for these two cases range from 10 to 24 and 5 to 13, the RQD ranges from 56

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to 72 and 20 to 50, respectively. The final GSI values range from 43 to 72 and 17 to

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45 based on the Eq. (21). The lower values of GSI were used and other parameters for

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calculating the FoS for these two slopes are listed in Table 5. The HNG landslide was

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re-analyzed with the developed model and the detail steps are as follows: to begin

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with, the FoS for mi=15, GSI=17, SR=15 and mi=20, GSI=17, SR=15 from static

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stability chart in Fig. 3 were used to estimate the range of values for mi=18, and the

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values for mi=15 and mi=20 are 2.35 and 2.45, respectively. Next, values of fKh from

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seismic scaling factor chart in Fig.8 or Eq. (17) and fβ from slope angle scaling factor

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chart in Fig. 9 or Eq. (18) were obtained with slope angle β=53° and Kh=0.3, the result

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was fβ=0.62 and fKh=0.61. Finally, the upper and lower values of FoS for case1 can be

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calculated in Eq. (19) and the results were 0.889 for mi=15 and 0.927 for mi=20, the

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result calculated in slide 6.0 was 0.908 with mi=18. The results showed that there is a

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close agreement between the proposed stability charts analysis method and software

346

result with discrepancies of 1.3% and -0.3%.

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Compared with the values of FoS in Slide 6.0, the values estimated from the

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charts in Fig. 3, Fig. 8 and Fig. 9 show some discrepancy. The results of analysis of

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2480 sets of data show that 70.63% of the data errors are lower than ±10%. A

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discrepancy greater than ±10% appears with slope angles greater than 60° and lower

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GSI values with high seismic acceleration coefficients(the values of FoS are lower

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than 1.0).

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4. Conclusions and discussion

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A novel charts-based analysis method for calculating the FoS of seismic rock

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slopes based on the HB failure criterion has been put forward using the limit

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equilibrium method. This method follows the general assumption of the earthquake

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effect which only takes the horizontal seismic coefficient Kh into consideration, and

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range values of Kh are adopted, which are consistent with most design codes. Based 14 / 36

ACCEPTED MANUSCRIPT 359

on this study, the main conclusions are as follows: (1) The theoretical relationship between FoS and SR was derived. The seismic

361

horizontal acceleration coefficient Kh and the disturbance factor D=0 were adopted to

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evaluate the stability of seismic rock slopes. The function has five parameters (GSI,

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SR, mi, β and Kh), and FoS is only related to SR with the same values of GSI, mi, β,

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and Kh and has no relationship with individual parameters σci, γ, and H.

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(2) Based on the relationship between FoS and SR, stability charts were proposed

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in Fig. 3 and Fig.4 to calculate the FoS of rock slopes with slope angle β=30° and Kh

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ranging from 0.1 to 0.3. It was found that the FoS increases obviously as GSI and SR

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increases for small values of mi. When GSI≤60, the increase in FoS is small with

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increasing SR; FoS increases significantly for GSI>60. The values of FoS decrease

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with increases in β and mi for high values of GSI (GSI>60) and Kh (Kh≥0.3). This is

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explained by the fact that the tensile strength of the material controls the overall

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stability. The collapse of the rock slope is attributed to tensile failure; rock masses

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with smaller values of mi have high strengths and are therefore more stable.

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(3) The scaling factors fKh and fβ were proposed to take stock of the influence of

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the horizontal seismic acceleration coefficient Kh and the slope angle β on the stability

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of rock slopes. Eqs. (17)–(18), which represent relationships between fKh and Kh, fβ

377

and β, respectively, were proposed for 30°≤β≤75° and 0.0≤Kh≤0.3. Combined with the

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stability charts for slope angle β=30° (Fig. 3), values of fKh and fβ can be used to

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determine the FoS of rock slopes with various slope angles and different values of Kh.

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The FoS can be calculated as F=FoSs×fβ×fKh.

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The predictive model (Eq. (20)) was tested using 2480 sets of data. The values of

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FoS had some discrepancies with those obtained using Slide 6.0. The results showed

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that 70.63% of the data had discrepancies of less than ±10%, and the data with higher

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discrepancies had slope angles greater than 60° and lower GSI values with high

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seismic acceleration coefficients. The proposed stability chart analysis method

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including the earthquake effects is simple and straightforward to use for estimating

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seismic rock slope stability in the initial design phase and slope stability evaluation.

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All of the data in the stability charts were based on calculations using the Slide 15 / 36

ACCEPTED MANUSCRIPT 6.0 software; the FoS are nearly the same using SCAM or any other software on the

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HB criterion. However, the software calculations require much time in the laboratory,

391

which generally cannot be used directly in fieldwork, and are unsuitable for the fast

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stability evaluation of a large amount of rock mass slopes after earthquakes. The

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limitation of this method is that the vertical seismic force, impacts of rock slope

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elevation amplification and geomorphologic factors, interactions between the attitudes

395

of rock mass discontinuities and the propagation directions of seismic waves are not

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considered in this study.

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The vertical acceleration which can largely influence the acceleration

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components normal and tangential to the sliding surface, is often overlooked in

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stability analysis of slopes stability during earthquakes. The values and direction for

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Kv, the combination relationships between Kv and Kh are key factors that affect the

401

FoS of slopes. Elnashi and Papazolou(1997), Coller and Elnashai(2012) studied the

402

relationship among Kv/Kh, earthquake magnitudes and epicentral distance (Fig.13),

403

and found that the ratio of Kv/Kh increases with the increasing of earthquake

404

magnitude, and decreases with the increasing epicentral distance. Guo (2003) also

405

found that the value of Kv was about 1/3~2/3 the value of Kh, and suggested the Kh

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control the stability of slopes under the far-filed earthquakes. For further study, three

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group ratios of Kv/Kh (Kh=0.3g, Kv=±0.45g,±0.3g,±0.1g, Minus value mean the

408

direction of Kv is downward ) were used to calculate the FoS of slopes under different

409

parameters of GSI, SR and mi with slope angle β=30° and β=75°，respectively. The

410

result comparing to Kh=0.3, Kv=0 were analyzed (Figs.14-15). In Fig.14, when slope

411

angle β=30°，the FoS of slopes under Kv=±0.45 show some differences and with the

412

decreasing of Kv/Kh,, the FoS become nearly equal. In Fig.15, when slope angle β=75°,

413

the FoS become bigger for Kv=-0.45g than that for Kv=0.45g with high values of GSI.

414

This is because the values and the direction of the vertical component affect the

415

driving and the resistance force on the sliding surface, which influence the FoS of

416

corresponding slopes. With the decreasing of Kv/Kh, these differences become smaller.

417

Thus, it is seen that, in the near-field earthquakes, where the ratio of the Kv/Kh was

418

bigger than 1.0, (Epicentral distance less than 30 km), both of the horizontal and

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ACCEPTED MANUSCRIPT vertical seismic accelerations control the stability of slopes. But for the far-field

420

earthquakes, the effect of vertical seismic acceleration can be neglected. From this

421

point of view, the stability charts analysis method that we developed in this research

422

is suitable for slopes that near the far-field earthquakes.

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The original Geological Strength Index (GSI) chart was developed on the

424

assumption that the observations of the rock mass would be made by qualified and

425

experienced geologist or engineering geologists. When such individuals are available,

426

the utilization of GSI charts based on the descriptive categories of rock mass structure

427

and discontinuity surface conditions have been found to work well. Although the

428

quantitative method for GSI had been introduced, there are many situations where

429

engineering staff rather than geological staff are assigned to collect data, which may

430

result in differences in the GSI values. The lower value of GSI was adopted in this

431

paper to calculate the FoS of rock mass slopes, and this is reasonable for slope

432

stability assessment in spite of being conservative.

433

Acknowledgements

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This research was financially supported by the Key Project of the Chinese

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Academy of Sciences (Grant No. KZZD-EW-05-01), External Cooperation Program

436

of BIC, Chinese Academy of Sciences (Grant No. 131551KYSB20150009) and the

437

Support Project of Science and Technology in Sichuan Provence (Grant No.

438

2014SZ0163). The authors are indebted to Dr. Amar Deep Regmi for his helpful

439

revision and discussion for regarding this manuscript.

440

Appendix 1

441

Table 1. Definition of JCond89 from Bieniawski (1989) Very rough Slightly Slightly surfaces rough rough Not surfaces surfaces Condition of continuous Separation Separation discontinuities No ＜1mm ＜1mm separation Slightly highly Uneathered weathered weathered wall rock walls walls Rating 30 25 20

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Guidelines for classification of discontinuity conditions Discontinuity length ＜1m 1–3m 3–10m

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Slickensided surfaces or Gouge＜5 mm thick or Separation 1-5mm Continuous

Soft gouge ＞5mm thick or Separation ＞5mm Continuous

10

0

10–20m

＞20m

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Separation (aperture) Rating

None 6

Roughness Rating

Very rough 6 None

Infilling (gouge) Rating

6 Unweath ered

Weathering Rating

＜0.1mm 0.1–1.0m 5 m 4 Rough Slightly 5 rough 3 Hard Hard infilling infilling ＜5mm ＞5mm 4 2 Slightly Moderat weathere e d weathere 5 d 3

1

0

1–5mm 1

＞5mm 0

Smooth 1

Slickensi des 0 Soft infilling ＞5mm 0 Decompo sed

Soft infilling ＜5mm 2 Highly weathered 1

0

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(persistence) Rating

Table 2. Values of constant mi for intact rock, by rock group (Hoek et al. 1992) Grain Sedimentary Metamorphic Igneous size Carbon Detrital Chemi Carbon Silicate Felsic Mafic Maf ate cal ate ic Dolomi Conglome Granit Gabbr Nori Coars Marble Gneiss te rate e o te e 9.3 29.2 10.1 20 32.7 25.8 21.7 Amphibo Doleri Medi Chalk Sandstone Chert lite te um 7.2 18.8 19.3 31.2 15.2 Limest Andes Bas Siltstone Gypsto Quartzite Rhyol Fine one ne ite ite alt 9.6 23.7 8.4 15.5 20 18.9 17 Anhydr Very Claystone Slate ite fine 3.4 11.4 13.2 Values shown were derives from statistical analysis of triaxial test data for each rock type. Values in parenthesis have been estimated.

444 445

Table 3. Estimates of uniaxial compressive strength σci for intact rock (Hoek et al. 1992) Uniaxial Point Comp. Load Field estimate of Term Examples* strength Index I strength /MPa /MPa Rock material only Basalt, chert, diabase, Extremely chipped under gneiss, granite, quartzite ＞250 ＞10 strong repeated hammer blows Requires many blows Amphibolite, andesite, of a geological basalt, dolomite, Very 100-250 4-10 hammer to break gabbro, gneiss, granite, Strong intact rock granodiorite, limestone, speciments marble, rhyolite, tuff Hand held speciments Limestone, marble, broken by single phyllite, sandstone, Strong 50-100 2-4 blow of geological schist, slate hammer Medium 25-50 1-2 Firm blow with Claystone, coal,

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ACCEPTED MANUSCRIPT geological pick concrete, schist, shale, indents rock to 5mm, siltstone Knife just scrapes surface Knife cuts material Chalk, Rock salt, Potash but too hard to shape Weak 5-25 ** into triaxial speciments Material crumbles Highly weathered or Very under firm blows of altered rock 1-5 ** Weak geological pick, can be shaped with knife Extremely Indented by Clay gouge 0.25-1 ** Weak thumbnail *All rock rypes exhibit a broad range of uniaxial compressive strengths which reflect heterogeneity in composition and anisotropy in structure. Strong rocks are characterized by well interlocked crystal fabric and few voids. **Rocks with a uniaxial compressive strength below 25MPa are likely to yield highly ambiguous results under point loading testing.

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ACCEPTED MANUSCRIPT List of figures

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Fig. 1. Slope stability charts based on limit analysis method (Based on Li et al. 2009)

565

Fig. 2. Rock slope geometry model in Slide 6.0

566

Fig. 3. Static stability charts for β=30°, Kh=0

567

Fig. 4. Pseudo-static slope stability charts for β=30°

568

Fig. 5. Comparison of the FoS for different values of SR, GSI, and Kh

569

Fig. 6. The stress state under different values of GSI and mi

570

Fig. 7. The seismic scaling factors with slope angle β=30°

571

Fig. 8. Seismic scaling factor chart

572

Fig. 9. Slope angle scaling factor chart

573

Fig. 10. Quantification of GSI by Joint Condition and RQD (Modified after Hoek et al.

574

2013)

575

Fig. 11. Topographical and joint distribution conditions of Zipingpu rock slope

576

Fig. 12. Topographical and joint distribution conditions of Huangnigang landslide

577

Fig. 13. Statistical relationship among Kv/Kh, earthquake magnitude and epicentral

578

distance

579

Fig. 14. Stability charts for Kh=0.3, β=30° (Kv/Kh =±1.5, ±1.0, ±0.3)

580

Fig. 15. Stability charts for Kh=0.3, β=70° (Kv/Kh =±1.5, ±1.0, ±0.3)

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100

100

Average Tangential Method SLIDE-HB Model

Average SLIDE HB Model

Average Tangential Method SLIDE HB MODEL

10

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GSI=50

GSI=50

0.1

0.1

0.001 5

10

15

20

25

30

10

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(b) β=30°, Kh=0.2

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(a) β=30°, Kh=0.1

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H

β

GSI=100

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1

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Stability number-N

GSI=10

SC R

Stability number-N

Stability number-N

10

5

10

15

20

25

30

35

mi

(c) β=30°, Kh=0.3

Fig. 1. Slope stability charts based on limit analysis method (based on Li et al.

585

2009)

D

584

Toe

AC

589

β

H

L 2.5H

Khg mg

CE P

TE

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S 1.5H

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Jointed Rock σci，GSI, mi，γ

Fig. 2. Rock slope geometry model in Slide 6.0

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3.5

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mi=10

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100

100

90

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mi=10

mi=5

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50 40 30

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100

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15

90

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80

10

70 60 50 40 30 20 10

1.5 1

5

0.5 0

1

1.5

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SR=(σci /γH)

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Fig. 3. Static stability charts for β=30°, Kh=0

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10

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20

25

SR=(σci /γH)

(g) 20

4

0

0

2

SR=(σci /γH)

SR=(σci /γH)

100

0.5

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(f)

FOS

CE P 593

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0

25

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mi=35

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60 50 40 30 20 10

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15

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FOS

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5

SR=(σci /γH)

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SR=(σci /γH)

(e)

1.5

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SR=(σci /γH)

(c) mi=15

0.0

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SR=(σci /γH)

SR=(σci /γH)

70

5

1

10

IP

0.5

10

1.5

1.037

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5.483

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FOS

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ACCEPTED MANUSCRIPT

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40 0 5

A

C

25

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3.926

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2.913

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SR: σc i / (γH) mi=5 mi=25

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Fator of safety

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0 0

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Fator of safety

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Fator of safety

45

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T

30 10 30

SC R

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20

0

0

10

35 20 30 SR: σc i / (γH) mi=5 mi=25

40

mi=15 mi=35

* In this figure, the biaxial coordinate was used to distinguish the trends of the curves with the parameters GSI, SR and mi in one chart; the squares represent the intersection of curves for different values of mi.

D

(b) Kh=0.2

TE

Fig. 4. Pseudo-static slope stability charts for β=30°

CE P

595

(a) Kh=0.1

AC

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(c) Kh=0.3

5.0

SR=1.0

SR=10

4.5

Kh=0.1 Kh=0.2 Kh=0.3

Kh=0.0

4.0 3.5

Kh=0.1

3.0

Kh=0.2

T

Kh=0.0

Kh=0.3

2.5

IP

2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

Factor of safety

2.0 1.5 1.0

5

10

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25

30

35

5

mi

5

Kh=0.2

4

Kh=0.3

3

TE

2 25

30

40

Kh=0.0

30

Kh=0.1 Kh=0.2

20

Kh=0.3

10 0 5

35

10

15

20

25

30

35

mi (d) β=30°, GSI=100

CE P

(c) β=30°, GSI=60

Fig. 5. Comparison of the FoS for different values of SR, GSI, and Kh

598 599 4.0 3.5

AC

597

35

50

mi

596

30

SR=40

MA

Kh=0.1

Factor of safety

6

20

25

60

D

Factor of safety

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15

20

NU

Kh=0.0

10

15

(b) β=30°, GSI=40

70

SR=20

5

10

mi

(a) β=30°, GSI=20 8

SC R

Factor of safety

ACCEPTED MANUSCRIPT

GSI=100 mi=5,Kh=0.0 GSI=100 mi=5,Kh=0.3 GSI=100 mi=35,Kh=0.3 GSI=100 mi=35,Kh=0.0

σ1/σci

3.0

GSI=100,mi=35

2.5 2.0

GSI=100,mi=5 Tensile

1.5 1.0

Compressive

0.5

GSI=50,mi=5

0.0 -0.25

600 601 602

-0.15

-0.05

0.05

σ3/σci

0.15

0.25

0.35

Fig. 6. The stress state under different values of GSI and mi (revised from Li et al. 2009)

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ACCEPTED MANUSCRIPT 1.0

Seismic attenuation factor fkh

F
0.9

The average values of attenuation factors-Fave 12.5%

F>Fave 3.5%

0.8

Fmin=0.7271 Fave=0.8178

Kh=0.2

Fmax=0.7647

Fmin=564785 Fave=0.6809

Kh=0.3

Fmax=0.6851

Fmin=0.4562 Fave=0.5759

T

Fmax=0.9302

0.4

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126 131 136 141 146 151 156 161 166 171 176 181 186 191 196

0.5

IP

0.6

SC R

603

Kh=0.1

0.7

604

NU

No. of calculated cases

Fig. 7. The seismic scaling factors with slope angle β=30°

MA

1.0

seismic effect factor for β=30° seismic effect factor for β=45° seismic effect factor for β=60° seismic effect factor for β=75° average values of seismic effect factors Fitting equation for seismic effect factors

D

0.9

-3.1~2.7% 0.7

TE

0.8

-4.9~3.3%

-6.3~6.7%

0.6 )2

fKh =0.353(Kh - 1.390(Kh) + 1.0

CE P

607

Seismic attenuation factors fkh

605 606

0.5 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Horizontal acceleration coefficient Kh

Fig. 8. Seismic scaling factor chart

610

Slope angle weighting factor fβ

609 1.2

AC

608

Statistical data Fitting curve

1.0 0.8 0.6 0.4

fβ = -0.68 ln(β) + 3.323 （0≤FoS≤4）

0.2 0.0 30

35

40

45

50

55

60

65

70

Slope angle/°

611

Fig. 9. Slope angle scaling factor chart

612

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75

VERY POOR

IP

T

Slickensided, highly weathered surfaces with soft clay coatings or fillings

Slickensided, highly weathered surfaced with compact coating or fillings of angular fragments

POOR

NU

SC R

FAIR Smooth, moderately weathered and altered surfaces

GOOD Rough, slightly weathered, iron stained surfaces

VERY GOOD Very rough, fresh, unweathered surfaces

GEOLOGICAL STRENGTH INDEX FOR JOINTED ROCKS From the lithology, structure and surface conditions of the discontinuities, estimate the average values of GSI. Do not try to be too precise. Quoting a range from 33 to 37 is more realistic than stating that GSI=35. Note that the table does not apply to structurally controlled failures. Where weak planar structural planes are present in an unfavourable orientation with respect to the excavation face, these will diminate the rock mass behaviour. The shear strength of surfaces in rocks that are prone to deterioration as a result of changes in moisture content will be reduced if water is present. When working with rocks in the fair to very poor categories, a shift to the right may be made for wet conditions. Water pressure is dealt with by effective stress analysis

SURFACE CONDITIONS

ACCEPTED MANUSCRIPT

DECREASING SURFACE QUALITY

50

N/A

N/A

45 40

TE

CE P

VE RY BLOCKY-Interlocked, partially disturbed mass with multi-faceted angular blocks formed by 4 or more joint sets

BLOCKY/DISTURBED/SEA MY-Folded with angular blocks formed by many intersecting discontinuity sets. Persistence of bedding planes or schistosity

AC

DISINTE RGRATE D- Poorly interlocked, heavily broken rock mass with mixtures of angular and rounded rock pieces.

L AMI NATE D/SHE AR ED Lack of blockiness due to close spacing of the weak schistosity or shear planes

30 25 20

15

Huangnigang landslide

10

5 0 45

40

35

N/A

30

25

20 15

10

5

0

N/A

1.5 JCond89

613 614 615 616 617 618 619 620 621

RQD/2

35

D

B L O C K Y - Well interlocked undisturbed rock mass consisting of cubical blocks formed by three intersecting discontinuity sets

DECREASING INTERLOCKING ROCK PIECES

INTACT OR MASSIVE-Intact ro ck speciments or massive insitu-roc k with fe w wide ly spaced discontinuities

MA

STRUCTURE

Fig. 10. Quantification of GSI by Joint Condition and RQD (Modified after Hoek et al. 2013)

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ACCEPTED MANUSCRIPT 622 623

(b )

J 2

Fault and damage zone

Failure block

Rockfall

CE P

TE

(a ) Fig. 11. Topographical and joint distribution conditions of Zipingpu rock slope: (a) Overview of Slope; (b) Damage zone in the middle of the slope; (c) Three domain joint sets distribute characteristic on the slope surface.

AC

624 625 626 627

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MA

NU

J 1

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J 1

J 3

J 2

(c )

ACCEPTED MANUSCRIPT N

(a)

Surface loose deposits

T

Sidescarp

IP

Headscarp

SC R

J3

D

MA

NU

J3

(b)

TE

CE P

J4

AC

Compressive and crushed zone

628 629 630 631 632 633

J3 J3

J4

J5

J1

J2 20cm

Potential sliding surface

J2

Fig. 12. Topographical and joint distribution conditions of Huangnigang landslide: (a) Overview of the landslide; (b) Joint sets distribution on the sidescarp; (c) Joint sets distribution on the headscarp.

1.2 1.1

Kv/Kh

1.0 0.9 Ms=8.0

0.8 0.7

Ms=7.5

0.6

Ms=6.5

0.5

Ms=5.5

0.4 0.3 0

634

(c)

Tension crack

J3

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Epicentral distance/km

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ACCEPTED MANUSCRIPT Fig. 13. Statistical relationship among Kv/Kh, earthquake magnitude and epicentral distance

11 10

8

9

7

8

GSI=100

GSI=80

4

7

6 5

GSI=80

4 GSI=50

3

2

GSI=50

2

GSI=10

1

GSI=10

1

0

0 5

15

25

35

6 5

GSI=80

4

NU

3

5

15

mi

25

GSI=50

3 2

GSI=10

1 0

35

5

15

mi

mi

639 640 641

(a) Kv/Kh=±1.5 Kv/Kh=±0.3

642

Fig. 14. Stability charts for Kh=0.3, β=30° (Kv/Kh=±1.5, ±1.0, ±0.3)

8

FoS

6

AC

FoS

10

GSI=100

7

5 4

Kv=0.0 Kv=0.3g Kv=-0.3g

9

9 GSI=100

8

8

7

7

6

6

5 4

1 GSI=10

GSI=10

GSI=10

0

0

0 5

GSI=50

GSI=50

1

1

GSI=80

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2 GSI=50

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3

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GSI=100

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GSI=80

GSI=80

3

Kv=0.0 Kv=0.1g Kv=-0.1g

10

FoS

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15

25

35

mi

(c)

D

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35

11

11

Kv=0.0 Kv=0.45g Kv=-0.45g

25

(b) Kv/Kh=±1.0

TE

11

Kv=0.0 Kv=0.1g Kv=-0.1g

GSI=100

8

FoS

5

FoS

FoS

9 GSI=100

7

6

10

SC R

9

11 Kv=0.0 Kv=0.3g Kv=-0.3g

T

Kv=0.0 Kv=0.45g Kv=-0.45g

IP

10

MA

635 636 637 638

5

15

25

35

5

mi

15

25

643 644 645

(a) Kv/Kh=±1.5 Kv/Kh=±0.3

646

Fig. 15. Stability charts for Kh=0.3, β=70° (Kv/Kh=±1.5, ±1.0, ±0.3)

(b) Kv/Kh=±1.0

647 648

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35

mi

(c)

ACCEPTED MANUSCRIPT List of tables

650

Table 1 Comparison of the FoS for a given slope with the same values of SR

651

Table 2 Comparison of the FoS for a given slope with different values of GSI and mi

652

Table 3 Regression equations for the seismic scaling factors

653

Table 4 Orientation and characteristics of the discontinuity sets observed at the ZPP

654

and HNG rock slopes

655

Table 5 Rock slope cases analyzed using the proposed charts

IP

SC R

NU

656

Table 1 Comparison of the FoS for a given slope with the same values of SR Case 1 Case2 Case3 Input 40 40 40 GSI parameters 10 10 10 mi 0.1 0.1 0.1 Kh (g) 45 45 45 β (°) 20 50 100 σci (MPa) 23 24.6 26.5 γ (kN/m3) 30 70 130 H (m) 28.985 28.985 28.985 SR (σci/γH) FoS 2.698 2.700 2.700 Fellenius 2.871 2.873 2.873 Bishop 2.627 2.629 2.628 Janbu simplified 2.882 2.884 2.883 Spencer simplified 2.864 2.866 2.866 Morgenstern Price Table 2 Comparison of the FoS for a given slope with different values of GSI and mi

658

AC

CE P

TE

D

MA

657

T

649

HB GSI mi parameters 10 5 10 15 10 25 10 35 50 5 50 15 50 25 50 35 100 5 100 15 100 25 100 35

Group 1 FoS 1.269 1.811 2.123 2.358 4.018 4.355 4.666 4.952 44.878 30.392 25.769 23.418

659 660 661

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Group FoS 2 1.269 1.811 2.123 2.359 4.022 4.358 4.669 4.955 44.95 1 30.44 0 25.80 6 23.45 0

Group FoS 3 1.269 1.811 2.123 2.358 4.021 4.257 4.668 4.955 44.939 30.462 25.800 23.444

ACCEPTED MANUSCRIPT Table 3 Regression equations for the seismic scaling factors slope regression fitting angle-β equations-fKh degree-R2 f Kh  0.99exp1.8 Kh 30° 0.9981 f Kh  1.02exp1.5 Kh 45° 0.9997 1.4 Kh f  1.0exp 60° 0.9994 Kh f Kh  1.03exp1.7 Kh 75° 0.9777

IP

T

662

Table 4 Orientation and characteristics of the discontinuity sets observed at the ZPP and HNG rock slopes Dip Dip Surface Surface Landslide Joint Set (°) direction(°) shape roughness J1 54 201 Planar Smooth ZPP J2 65 110 Planar Smooth Rock Slope J3 48 350 Undulating Rough J1 84 211 Stepped Rough J2 47 235 Undulating Rough HNG J3 85 21 Planar Smooth landslide J4 61 146 Planar Smooth J5 35 220 Planar Smooth

D

TE

Table 5 Rock slope cases analyzed using the proposed charts Input ZPP HNG landslide Parameters slope σci/(MPa) 100 50 GSI 40 17 mi 10 18 3 γ/(kN/m ) 28 24 H/(m) 100 140 β/(°) 50 53 Kh 0.2 0.3 SR 35 15 fKh 0.73 0.61 fβ 0.66 0.62 Slide 2.474 0.911 FoS 6.0 Eq. (20) 2.505 0.908 Discrepancy 1.3% -0.3%

AC

CE P

667 668 669

MA

NU

664 665 666

SC R

663

670 671

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ACCEPTED MANUSCRIPT Highlights

673



A chart-based model for calculating the FoS of rock mass slopes is proposed.

674



Theoretical relation between FoS and HB parameters is developed based on the

675

LEM.

676



IP

T

672

The dependability of the proposed model is test using 2480 sets of data.

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CE P

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D

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