Pseudo-static stability analysis of rock slopes reinforced by passive bolts using the generalized Hoek–Brown criterion

Accepted Manuscript Pseudo-static stability analysis of rock slopes reinforced by passive bolts using the generalized Hoek-Brown criterion Mounir Belghali, Zied Saada, Denis Garnier, Samir Maghous PII:

S1674-7755(16)30194-9

DOI:

10.1016/j.jrmge.2016.12.007

Reference:

JRMGE 347

To appear in:

Journal of Rock Mechanics and Geotechnical Engineering

Received Date: 24 October 2016 Revised Date:

21 December 2016

Accepted Date: 24 December 2016

Please cite this article as: Belghali M, Saada Z, Garnier D, Maghous S, Pseudo-static stability analysis of rock slopes reinforced by passive bolts using the generalized Hoek-Brown criterion, Journal of Rock Mechanics and Geotechnical Engineering (2017), doi: 10.1016/j.jrmge.2016.12.007. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Pseudo-static stability analysis of rock slopes reinforced by passive bolts using the generalized Hoek-Brown criterion Mounir Belghali1, Zied Saada1, Denis Garnier2, Samir Maghous3,* 1

Laboratoire de Génie Civil, ENIT, Université de Tunis El Manar, Tunis, Tunisia

2

Université Paris-Est, Laboratoire Navier (UMR8205), ENPC-IFSTTAR-CNRS, Marne-la-Vallée, France

3

Department of Civil Engineering, Federal University of Rio Grande do Sul, Av. Osvaldo Aranha 99, Porto Alegre-RS, 90350-190, Brazil

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Received 24 October 2016; received in revised form 21 December 2016; accepted 24 December 2016

Abstract: The stability analysis of passive bolt-reinforced rock slopes under seismic loads is investigated within the framework of the kinematic approach of limit analysis theory. A pseudo-static method is adopted to account for the inertial forces induced in the rock mass by seismic events. The strength properties of the rock material are described by a modified Hoek-Brown strength criterion, whereas the passive bolts are modeled as bar-like inclusions that exhibit only resistance to tensile-compressive forces. Taking advantage of the ability to compute closed-form expressions for the support functions associated with the modified Hoek-Brown strength criterion, a rotational failure mechanism is implemented to derive rigorous lower bound estimates for the amount of reinforcement strength to prevent slope failure. The approach is then applied to investigating the effects

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of relevant geometry, strength and loading parameters in light of a preliminary parametric study. The accuracy of the approach is assessed by comparison of the lower bound estimates with finite element limit analysis solutions, thus emphasizing the ability of the approach to properly predict the stability conditions and to capture the essential features of deformation localization pattern. Finally, the extension of the approach to account for slipping at the interface between reinforcements and surrounding rock mass is outlined.

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Keywords: reinforced rock slope; modified Hoek-Brown criterion; seismic loading; limit analysis; rotational failure mechanism

Lin et al., 2013; Wang et al., 2015), yield design or limit analysis (de Buhan,

1. Introduction

1986; Juran et al., 1990; de Buhan et al., 1989; Michalowski, 1997, 1998a, b; Ausilio et al., 2000; Porbaha et al., 2000; Siad, 2001; He et al., 2012; Clarke et

Rock slope stability analysis still remains an important problem in rock

al., 2013), and other theoretical/numerical methods (e.g. Kim et al., 1997;

engineering and many contributions have been made to address this issue (e.g.

Koyama et al., 2012; Seo et al., 2014) such as the discrete element method

Zhang and Chen, 1987; Drescher and Christopoulos, 1988; Michalowski, 1995;

(Kim et al., 1997).

Referring to the theoretical framework of limit analysis and related lower

techniques have been developed in the last decades with the purpose to

and upper bound theorems, the stability of reinforced slopes has been

improve stability of slopes. Among the most attractive reinforcement

investigated considering gravitational forces (Michalowski, 1997, 1998a;

techniques, one may quote the technique of soil nailing and rock bolting. In

Porbaha et al., 2000; Siad, 2001) or gravitational forces combined with

this respect, inserting passive fully grouted bolts leads to a net improvement of

seismic acceleration (Michalowski, 1998b; Ausilio et al., 2000; He et al.,

the overall strength of rock mass, which in turn results in an increase in slope

2012). In most of these works, the strength capacities of the constitutive

stability. Both experimental and theoretical analyses were performed in order

geomaterial were described by the Mohr-Coulomb failure criterion. However,

to assess the efficiency of such kind of slope reinforcement.

it was experimentally proved that the strength envelopes of rocks are nonlinear

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Yu et al., 1998). From an engineering viewpoint, several methods and

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On one hand, several works based on laboratory tests have been performed

(Hoek and Brown, 1980, 1997; Hoek, 1983; Agar et al., 1987; Goodman, 1989; Marinos and Hoek, 2001; Jiang et al., 2003). Among the nonlinear failure

Sengupta, 2009, 2010; Rawat et al., 2014). Shaking table tests were performed

criteria proposed in the literature, the Hoek-Brown failure criterion appears

to examine the dynamic behavior of nailed slopes (e.g. Hong et al., 2005; Giri

well suitable for describing the strength resistance of rocks. In this context,

and Sengupta, 2009, 2010). Centrifuge model tests of nailed soil slopes were

recent studies implemented the latter failure criterion to assess the stability of

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to study the stability of nailed soil slopes (Hong et al., 2005; Giri and

also conducted under several loading conditions (e.g. Tei et al., 1998; Zhang et

rock slopes (Collins et al., 1988; Sonmez et al., 1998; Yang et al., 2004a, b;

al., 2013, 2014). Parametric studies were performed in these works to optimize

Serrano et al., 2005; Yang and Zou, 2006; Yang, 2007; Li et al., 2008, 2009,

the efficiency of slope nailing, and field experiments were also conducted with

2012; Zheng et al., 2009; Saada et al., 2012), while equivalent Mohr-Coulomb

the aim of assessing the same problem (e.g. Zhou et al., 2009; Guo and

parameters were evaluated from the Hoek-Brown failure criterion to

Hamada, 2012; Blanco-Fernandez et al., 2013). The experimental results

investigate the stability of rock slope reinforced by bolts by means of the finite

clarified the effects of several parameters on the stability of nailed slopes.

difference code FLAC3D (Wang et al., 2015).

On the other hand, several theoretical and numerical methods were

Sonmez et al. (1998) developed a practical procedure for back

developed to study the stability of reinforced slopes by nails or bolts. They

determination of shear strength parameters mobilized in closely jointed rock

could be roughly categorized into five groups: limit equilibrium (Patra and

slopes that obeys the Hoek-Brown failure criterion. The same nonlinear failure

Basudhar, 2005; Wei and Cheng, 2010a; Maleki and Mahyar, 2012; Wei et al.,

criterion was used by Zheng et al. (2009) in order to search the critical slip

2012), elastoplastic finite element (FE) (Sharma and Pande, 1988; Zhu and

surface of a slope based on the strength reduction technique. Collins et al.

Zhang, 1998; Zhu et al., 2003; Maleki and Mahyar, 2012; Kim et al., 2013;

(1988) evaluated the stability of homogeneous rock slopes with the original

Kato et al., 2014; Sahoo et al., 2015) or finite difference approaches (Wei and

Hoek-Brown failure criterion. Serrano et al. (2005) conducted a theoretical

Cheng, 2010a, b; Halabian et al., 2012; Wei et al., 2012; Cheuk et al., 2013;

analysis of the stability of infinite rock slopes with different hypotheses of simplified seepage flow nets. They used the original and modified Hoek-

*Corresponding author. Fax: +55-51-33-08-39-99; E-mail address: [email protected]

Brown failure criteria. The important effect of groundwater flow in the slope

ACCEPTED MANUSCRIPT T

was underlined. Yang et al. (2004a, b) investigated the stability of rock slopes whose strength properties were described by a modified Hoek-Brown failure

H/n0

criterion. Yang et al. (2004b) considered a rock slope subjected only to its weight, and compared their results to previous results provided by Collins et

bound theorem of limit analysis was applied to the seismic and static stability

γ

‒

e

‒3

the safety factor of the slope. Besides, the seismic displacement of rock slopes

e

β

‒2

problems of homogeneous rock slope with the modified Hoek-Brown failure criterion. This approach enabled the authors to evaluate a least upper bound of

H

x2

khγe‒ 2

al. (1988) for the original Hoek-Brown failure criterion. In addition, Yang et al. (2004a) considered a rock slope subjected to seismic loads. The upper

x1

α

e ‒1 Rinforcement layers

Fig. 1. Geometry and loading mode of studied rock slope.

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with nonlinear Hoek-Brown failure criterion was studied by Yang (2007). More recently, Li et al. (2008, 2009) improved the results provided by Yang et

Regarding the loading mode, the material system is basically submitted to

al. (2004a, b). These authors derived FE upper and lower bound solutions in

gravitational forces as well as inertial forces developed in the rock mass by the

order to provide stability charts for rock slopes. Within the same context, Li et

passage of seismic waves. In the present simplified analysis, the acceleration

al. (2012) performed reliability assessments for rock slopes based on the

distribution within the rock mass is accounted for through the concept of

Hoek-Brown failure criterion. They used a new form of safety factor for rock

average seismic coefficient (Seed, 1979). In addition, only the horizontal

slope design and presented its use in probabilistic assessment.

component of earthquake acceleration will be considered in the analysis, and it is assumed that the horizontal distribution of acceleration is uniform within the

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This paper focuses on the face stability analysis of rock slopes reinforced by passive bolts, considering that the rock strength properties could be modeled

rock mass. Denoting by γ the rock unit weight, the material system is therefore

by means of a modified Hoek-Brown failure criterion. The analysis proposed

subjected to body forces induced by gravity γ = γ e 1

herein for the rock slope stability under the combined loading defined by

accelerations k h γ e 2 , where kh is an average seismic coefficient. It should be

gravitational and earthquake-induced forces is based on the implementation of

finally pointed out that the effect of pore water pressure is not considered.

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and seismic

the kinematic approach of limit analysis. The analysis relies upon the pseudo-

The reinforcement scheme of the structure consists of several layers of

static method, thus implicitly assuming that dynamic effects of earthquake

passive fully grouted rock bolts, uniformly distributed along the height of the slope with a constant spacing ∆H. It is assumed that all the layers are parallel

addition, only the horizontal pseudo-static force during an earthquake

and inclined at an angle with respect to the horizontal direction. Referring to

sequence is accounted for. Indeed, Chen and Liu (1990) reported that the

T-x1-x2 coordinate system, the intersection of the i-th reinforcement layer with

effects due to the vertical component of seismic forces are generally

the slope wall is defined by depth:

negligible. However, when the horizontal seismic acceleration becomes relatively large (compared to gravitational forces for instance), this effect

1 H  x 1 (i ) =  i −   2 n 0

should be included in design analysis. The effects of seepage forces are not

where n0 denotes the number of layers. The vertical spacing between the

considered in the present analysis. The principle of how these driving forces

layers is therefore defined by ∆H = H/n0. The orientation of any bolt layer is

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motions on the variations of rock strength properties are disregarded. In

could be accounted for in the stability analysis can be found in Saada et al. (2012).

It should be noted that within the assumption above-mentioned, the main

(i ∈ {1, 2,L , n 0 })

defined by the unit tangent vector e b to the neutral axis: e b = −e 1 sin α + e 2 cos α

(1)

(2)

For application purposes, the unit vector t = −e b defining the orientation of

objective of the paper is to assess lower bound estimates for the so-called

the reinforcements from the right side to the left side will also be used.

“required reinforcement strength” defining the reinforcement ratio that is

2.2. Strength properties One of the main characteristics of rock masses is the presence at different

subsequent analysis follows from the upper bound theorem based on the

scales of discontinuities of various sizes and orientations. At a macroscopic

kinematic approach of limit analysis.

scale, the rock material may be regarded as a homogeneous and anisotropic

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required for preventing failure. The methodology implemented in the

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2. Statement of the problem and method of analysis

medium, whose strength properties can be assessed experimentally or resorting to upscaling methods (de Buhan and Maghous, 1997; Maghous et al., 1998, 2008; de Buhan et al., 2002).

This section provides a description of the geometry of the problem together

In what follows, it is first assumed that the strength properties of the rock

with the loading mode and the strength properties of the reinforced rock

material are isotropic. At the scale of stability analysis (i.e. macroscopic

material. The general framework of kinematic approach of limit analysis is

scale), this assumption proves reasonable for intact or heavily jointed rocks.

also briefly presented.

Moreover, the strength capacities of the rock material are defined by a

2.1. Geometry and loading mode

modified Hoek-Brown yield condition (Hoek and Brown, 1997):

The plane strain stability analysis considered herein refers to a homogeneous and isotropic rock slope reinforced by a series of passive bolts. The slope is defined by the angle β (inclination with respect to the horizontal plane) and height H, as shown in Fig. 1. Referring to the coordinate system (x1, x2, x3) associated with the orthonormal frame (T , e 1 , e 2 , e 3 ), where the origin T is located on the top of the slope, the stability analysis shall be addressed within the formulation of a two-dimensional (2D) plane strain problem parallel to x1-x2 plane.

n

  σ F r (σ ) = σ 1 − σ 3 − σ c  −m 1 + s  ≤ 0 σc  

(3)

where σ1 and σ3 stand respectively for the major and minor effective principal stresses (stresses are counted positive in tensile), and σc ≥ 0 is the uniaxial compressive strength of rock. The magnitudes of material parameters m, s and n depend on the geological strength index (GSI) characterizing the quality of the rock mass (Hoek et al., 2002): m  GSI − 100  = exp   mi  28 − 14 D 

(4)

ACCEPTED MANUSCRIPT  GSI − 100  s = exp    9 − 3D 

(5)

1 1  GSI   20   n = + exp  −  − exp  − 3   2 6  15   

(6)

at any point x when crossing a possible discontinuity surface Σ following its normal ν (x ). The contribution related to the virtual motion of the reinforcement layers results from the axial strain d = (d U / d s )e b (extension

rate) and the velocity jump [U (Pk )] across point Pk following the bolt

where D is a disturbance coefficient which varies from 0 for undisturbed in

direction e b .

situ rock masses to 1 for very disturbed rock masses. The value of mi is

The π-functions, called support functions, are defined by duality from the

obtained from experiments performed on intact rock specimens. Values of this

strength conditions of the constituent materials ( F r (σ ) ≤ 0 for the rock

parameter for some typical rocks may be found in Hoek (1990).

material and F b (N ) ≤ 0 for the bolts):

failure condition is formulated in terms of the axial force N, the shear force V and the bending moment M per transversal length along x3-direction. The reinforcement action of bolts in the rock mass is probably a combination of axial, shear and flexural modes. However, in most conceptual or empirical models devised for assessing the effect of such reinforcement on rock slope stability, both the shear and bending strengths are neglected. In the subsequent analysis, it is assumed that the bolts act as bar-like inclusions that carry only tensile-compressive forces. Accordingly, the failure condition of the bolts only

(1) For the rock material:    r r π (ν ; [U ]) = sup{[U ]σ : ν | F (σ ) ≤ 0} σ 

π r (d ) = sup{σ : d | F r (σ ) ≤ 0} σ

(2) For the reinforcement layers:

   π b ([U (Pk )]) = sup{N [U (Pk )]e b | F b (N ) ≤ 0} N 

π b (d ) = sup{Nd | F b ( N ) ≤ 0} N

(7)

where N0 (resp. −N 0′ ) denotes the tensile (resp. compressive) strength of a

Hoek-Brown

strength

criterion,

compression (i.e. N 0 = N 0′ ), the aforementioned yield condition reduces to

(8)

From a macroscopic viewpoint, the contribution of reinforcement layers

may be represented by the average tensile strength σ + and compressive

(9)

π r (d ) =

1  n sσ c tr d + σ c  n 1−n − n 1−n  m 

n

  m M (d ) 1−n      t rd 

(tr d > 0)

M (d ) = [m ax(0, −d 1 ) + m ax(0, −d 2 ) + m ax(0, −d 3 )]1/n

(15) (16)

where d1, d2 and d3 are the eigenvalues of d . Similarly, it has been shown that the π r - function relative to a velocity

π r (ν ; [U ]) =

1  n sσ c [U ]ν + σ c  n 1−n − n 1−n  m 

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debonding along the reinforcement-rock mass interface. The assumption of perfect bonding between reinforcement and surrounding rock mass will thus

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The mechanical framework of kinematic approach of limit analysis (Salençon, 1983, 1990) shall be adopted to investigate the stability of

reinforced rock slopes. The upper bound theorem of limit analysis states that a necessary condition for the system to remain safe under the applied external

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(10)

(17)

M (ν ; [U ]) =

1 21/ n

(| [U ] | −[U ]ν )1/ n

(18)

It should be pointed out that conditions tr d > 0 in Eq. (15) and [U ]ν > 0 in Eq. (17) can be viewed as normality rule conditions, thus ensuring that

π r (d ) < +∞ and π r (ν ; [U ]) < +∞ , respectively. These conditions are necessary for the kinematic approach (Eq. (10)) to provide non-trivial upper bound solutions. As regards the support functions related to the strength criterion of the reinforcement layers, their expressions may be found for instance in Salençon (1983):

π b (d ) = m ax{N 0 d , − N 0′ d }

  ′ π ([U (Pk )]) = m ax{N 0 [U (Pk )]e b , −N 0 [U (Pk )]e b } b

Eq. (10) expresses in dual form the compatibility between the equilibrium

n

  m M (ν ; [U ])  1−n  ( [U ]ν > 0)   [U ]ν  

Finally, the reinforcement elements are considered to be fully encapsulated

in a stiff grout and their length is large enough to prevent any failure by

loading reads Pext (U ) ≤ Pm r (U )

closed-form

jump takes the following form:

It should be noted that σ + and σ − have dimensions of stress.

be adopted in the subsequent analysis.

(3D)

(2008, 2011, 2013):

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the bolts exhibit the same resistance per unit transversal length in tension and

three-dimensional

expressions of the associated π r - functions have been derived in Saada et al.

single reinforcement layer per unit transversal length. It is observed that when

2.3. Kinematic approach of limit analysis

(14)

The implementation of the kinematic approach requires the computation of

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F b (N ) = m ax{N − N 0 , − N − N 0′ } ≤ 0

strength −σ − defined by N n N′ n σ + = 0 = 0 N 0 , σ − = 0 = 0 N 0′ ∆H H ∆H H

(13)

the support functions defined by Eqs. (13) and (14). In the case of generalized

refers to the axial force N and can be conveniently expressed as

F b (N ) =| N | −N 0 ≤ 0

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Regarding the strength properties of reinforcement layers, the bolts are modeled in the context of plane strain framework as beams, so that the related

(19)

of the structure and the strength capacities of its constituent material. In the

where [U (Pk )] stands for the velocity jump across point Pk following the

above fundamental inequality, U is any virtual kinematically admissible

direction e b .

velocity field (termed failure mechanism in the sequel), and Pext (U ) stands for the rate of work performed by the external

loading defined by the

3. Application to reinforced rock slopes stability

gravitational and seismic forces: Pext (U ) = ∫Ω γ e 1Ud Ω + ∫Ω k h γ e 2Ud Ω

(11)

The general expression for the rate of maximum resisting work Pm r (U ) developed in the failure mechanism U is given by

bolts under loading (γ = γ e 1 , k h γ e 2 ) defined by gravity and earthquakeinduced forces. Simple dimensional arguments show that the stability of the

Pm r (U ) = ∫Ω π r ( d )d Ω + ∫Σ π r ( ν ,[U ])d Σ + ∫bolts π b ( d )d s + ∑ π b ([U (Pk )]) 14444442444444 3 144444 k 42444444 3 Pmr r (U )

The purpose of the following section is to apply the kinematic approach of limit analysis to assessing the stability of a rock slope reinforced by passive

reinforced rock slope is governed by means of the following dimensionless parameters: β , γ H / σ c , k h , m i , GSI , D , σ + / σ c , σ − / σ c . They are related to

Pmb r (U )

(12) and defined as the sum of the contribution of the rock mass Pmr r (U ) and that of reinforcement layers Pmb r (U ). In the expression of Pmr r (U ), d is the strain rate field in the rock mass associated with U , whereas [U ] is the jump of U

geometry, loading level, rock strength properties and reinforcement limits in tension and compression.

ACCEPTED MANUSCRIPT stability of homogeneous Mohr–Coulomb soil or rock slope. In such a mechanism, a volume TI1I2 of rock mass is rotating about a point Ω with an angular velocity ω.

The reinforcement layers are given the same virtual

motion as the surrounding rock mass (Fig. 3). The curve I1I2 separating this volume from the rest of the structure which is kept motionless is an arc of logspiral of angle ϕ and focus Ω. Accordingly, this curve can be defined in the polar coordinates (Ω, r, θ) as r (θ ) = r0 e (θ −θ1 ) t an ϕ

(20)

where the distance r0 = ΩI1 defines the radius of the log-spiral curve for θ =

θ1. The latter curve is emerging at point I2 located at the slope toe and defined by angular coordinate θ = θ2.

Pr (U ) = ∫Ω γ e 1Ud Ω =

r03 γω (f 1

Pk h (U ) = ∫Ω k h γ e 2Ud Ω =

+ f 2 + f3)

k h r03 γω (f 1′ + f 2′

(22) (23)

+ f 3′)

(24)

Expressions of the dimensionless functions f 1 , f 2 , f 3 , f 1′, f 2′ an d f 3′ are

given in the Appendix. Regarding the rate of maximum resisting work Pm r (U ) whose general expression is given by Eq. (12), the first and third integrals at the right-hand side are equal to zero, since there is no (virtual) deformation occurring inside TI1I2 (rigid-body motion). It then reduces to the sole contribution of the velocity jump along the log-spiral arc I1I2, expressed by the second and fourth integrals. Actually, the expression of the rate of maximum resisting work

The velocity field within the volume in motion is orthoradial: U = ωre θ = ωr (e 1 cos θ + e 2 sin θ ) (

The rate of work of the external forces comprises two contributions, corresponding to the two components (γ = γ e 1 , k h γ e 2 ) of the loading: Pext (U ) = Pr (U ) + Pk h (U )

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The kinematic approach developed herein is based on the rotational failure mechanism sketched in Fig. 2, which is usually employed for analyzing the

2

comprises two terms:

1

)

Pm r (U ) = Pmr r (U ) + Pmb r (U )

(25)

The first term at the right-hand side of Eq. (25) is related to the resisting

angle ϕ with respect to the tangent at the same point. Such a failure

contribution of the rock material developed in the velocity jump along the log-

mechanism involves three angular parameters, namely θ1, θ2 and ϕ.

spiral line I1I2:

θ

θ

Pmr r (U ) = ∫θ =2θ π r (ν ; [U ])

Ω

θ2 θ1

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It follows that the velocity jump at any point of curve I1I2 is inclined at

1

r (θ ) d θ = σ cωr02 f 4 cos ϕ

(26)

where the expression of the dimensionless function f4 is provided in the

ω

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Appendix.

The second term on the right-hand side of Eq. (25) represents the rate of

maximum resisting work developed by the reinforcement layers in the considered failure mechanism. Let us denote by Pk an intersecting point

between arc I1I2 and the reinforcement, and by θ = θk the associated angular

r0

coordinate. The velocity jump across the point Pk following the reinforcement

e ‒θ

e ‒r

direction e b is [U (Pk )] = U (θk ). It follows that k

I1

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T α

H n0

H

khγe2 ‒ φ e ‒3

e ‒2

V(θ)

velocity jump line I1I2. Expression of the dimensionless function f5 > 0, depending on parameters θ1, θ2, ϕ and N 0′ / N 0 , is provided in the Appendix. Note that the expression of summation in Eq. (27).

α = 0 o (configuration of horizontal reinforcement layers) or N 0′ / N 0 = 1 function f5 is in fact independent of the number of reinforcement layers n0.

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θ

Should the direction of reinforcement layers be not horizontal (i.e. α = 0 o ) and N 0′ ≠ N 0 , this function would a priori depends on n0. In all cases, the discrete summation defining f5 may be approximated by a continuous integral. The idea consists in replacing the discrete reinforcement distribution by an average density of reinforcements that are continuously distributed along the slope height (e.g. Michalowski, 1997; Siad, 2001). From a theoretical

ω

viewpoint, the accuracy of the latter approximation depends on the number of layers that is involved in the reinforcement scheme. Applying the fundamental kinematic inequality (Eq. (10)) yields the following upper bound condition:

Log-spiral

κ = γ H / σc ≤

e

−b

2

r0 f 4 + f 5σ + / σ c (f 1 + f 2 + f 3 ) + k h (f 1′ + f 2′ + f 3′)

(28)

where r0 = r0 / H . Eq. (28) holds for any admissible combination of angular

ϕ θk − ϕ

Fig. 3. Virtual motion of the reinforcement element induced by the rotational motion of volume TI1I2.

1 r0

Pk U −

α

Pmb r (U ) is computed keeping the discrete It can be readily established that when

(same resistance per unit transversal length in tension and compression), the

Fig. 2. Rotational log-spiral failure mechanism for reinforced rock slope.

Reinforcement

(27)

where the summation is extended to all reinforcement layers intersected by the

I2

β

e ‒1

i =1

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γ ‒

n0

Pmb r (U ) = ∑ π b ([U (Pk )]) = ∑ m ax{N 0U (θk )e b , −N 0′ U (θk )e b } = n 0 N 0 ωr0 f 5

parameters (θ1, θ2, ϕ). Alternatively, it may be convenient to express the stability condition of the reinforced rock slope by means of the following dimensionless parameter: X =

σ + n 0N 0 = γH γH 2

(29)

ACCEPTED MANUSCRIPT It was originally introduced by Michalowski (1997) to characterize the

The simulations presented in this section aim at capturing the influence of

required reinforcement strength. After rearrangement of the terms involved in

the rock strength parameters (σc/(γH), GSI, mi) on the reinforcement strength.

Eqs. (23), (24), (26) and (27) of Pext (U ) and Pext (U ) , the kinematic

The values of the remaining problem parameters are kept fixed to reference set

inequality (Eq. (10)) leads to a lower bound estimate for the required

data. In particular, the disturbance coefficient is maintained equal to D = 0

reinforcement strength:

(undisturbed in situ rock mass).

 2  1 r X ≥ X l = m ax r0 [(f 1 + f 2 + f 3 ) + k h (f 1′ + f 2′ + f 3′)] − 0 f 4  θ1 , θ2 , ϕ  κ  f 5 

Fig. 4 displays the variations of the dimensionless parameter Xl as a

(30)

function of the strength parameter σc/(γH) for different inclinations (α) of the

where Xl represents the best lower bound estimate for the required

reinforcement layers. As it might be expected, the required reinforcement

reinforcement strength that can be obtained from exploring the considered

strength is significantly affected by σc/(γH): the lower the value of σc/(γH) is,

class of failure mechanisms. It is computed from a maximization procedure

the higher the required reinforcement strength is. Recalling that the parameter

σc/(γH) characterizes the uniaxial compressive strength of the rock material,

the following constraints:

smaller values are therefore referring to much poorer rock quality and reduced

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with respect to the set of angular parameters (θ1, θ2, ϕ), which are subjected to

rock slope stability. Fig. 4 also shows that the reinforcement contribution to

0 < θ1 < θ2 < π − β , 0 < e (θ2 −θ1 ) t an ϕ sin θ 2 − sin θ1    cos θ − e (θ2 −θ1) t an ϕ cos θ 2 π − cot β  0 < ϕ < , 0 < (θ −θ1 ) tan ϕ 2 e 2 1 sin θ 2 − sin θ1 

(31)

rock slope stability is more efficient when the reinforcement layers are installed horizontally (α = 0°), since this orientation provides the smaller estimates for the required reinforcement strength. This result is likely

a lower bound to the required reinforcement strength is equivalent to seeking for an upper bound estimate for the safety factor κ = γ H / σ c (see Eq. (28)).

connected with the simplified assumption that the yield condition (Eq. (7)) associated with the reinforcement elements only refers to the axial force,

SC

It should be noted that the above approach described by Eq. (30) providing

disregarding the contributions of shear and bending strengths. 0.2

4. Numerical results

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This section provides some numerical results for the required reinforcement

α=0° α=10°

0.15

α=20°

strength derived from the rotational failure mechanism implemented in the

α=30°

seismic forces. For each selected model, the numerical value of lower bound

X1

context of reinforced rock slope under the combined action of gravity and

0.1

solution Xl is obtained from Eq. (30) through a constrained maximization

procedure based on the nonlinear sequential quadratic programming

algorithm. The subsequent numerical analysis consists in selected examples of

0.05

rocks and focuses on the effects of strength, loading and geometry parameters upon the required reinforcement strength.

0

A series of several numerical simulations was performed by varying the

TE D

value of each involved parameter with respect to the following reference set of data: β = 70°, σc/(γH) = 5, kh = 0.1, mi = 7, GSI = 10, D = 0, σ + / σ − = 1, n0 =

5

10

15

20 σc / (γH)

25

30

35

Fig. 4. Effect of rock uniaxial compressive strength on the required reinforcement strength.

6, whereas the inclination of reinforcement layers is lying within the range of 0°–30°. According to the classification reported in Hoek (1990), this set of

Figs. 5 and 6 show the variations of the lower bound estimates of

parameters characterizes a carbonate rock with well-developed crystal

dimensionless parameter Xl versus the geological strength index GSI and the

cleavage (dolomite, limestone, or slate) that is cut by many intersecting joints,

strength parameter mi, respectively. The same general trends observed for the dependence of Xl with respect to the uniaxial compressive strength parameter

the considered model data refer to reinforcement layers that exhibit the same

σc/(γH) still hold regarding the influence of parameters GSI and mi. Once

resistance per unit transversal length in tension and compression (i.e.

again, the horizontal inclination (α = 0°) appears to be the best orientation in

N 0 = N 0′ ). Although this assumption is not a limitation for the applicability of

terms of reinforcement efficiency. Qualitatively speaking, these illustrations

the present approach, its related consequences on the stability analysis will be

suggest a sharper increase of the required reinforcement strength for small

AC C

EP

resulting consequently in a relatively poor-quality blocky rock mass. Note that

discussed later in the paper. It is also recalled that in such a situation, the

values of σc/(γH) and mi when compared to that observed for the variations

lower bound estimate Xl is independent of the number of reinforcement layers

with respect to GSI.

n 0.

4.2. Effects of geometry and loading parameters

It should be remarked that the primary objective of the present study is not

Several numerical simulations were performed to investigate the effects of

to provide an exhaustive parametric analysis that would be necessary to build

rock slope inclination β and horizontal seismic coefficient kh on the lower

charts for practical use in rock engineering, but only to develop (under some

bound estimates of required reinforcement strength. Fig. 7 shows the results

restrictive assumptions) original yield design solutions for the problem of

of the first simulation aiming to capture the dependence of dimensionless

reinforced rock slope stability analyzed in the context of generalized Hoek-

parameter Xl on the slope inclination within the range of 55°–90°, while the

Brown strength criterion and involving seismic loading. Still, the parametric

remaining parameters are kept fixed to the reference data. As it could be

simulations presented in the paper are intended to give preliminary insights

expected, increasing the value of slope inclination induces a reduction in the

into the individual impact of some relevant parameters on the required

structure stability, which in turn leads to an increase in the amount of

reinforcement strength. The proposed solutions can notably provide useful

reinforcement required to prevent failure. For the particular case of horizontal

benchmark for more thorough stability analyses of reinforced rock slope with

reinforcement layers (i.e. α = 0°), the amount of reinforcement required for β

nonlinear failure criterion carried out in both static and seismic conditions.

= 90° is about twice as much as that required for β = 55°.

4.1. Assessing the effects of rock strength parameters

ACCEPTED MANUSCRIPT

0.2

0.2)/Xl(kh = 0) ≈ 4.5. As observed in Fig. 8, increasing the value of kh induces

α=0° α=10° α=20° α=30°

Xl

0.15

a moderate but effective increase in the required reinforcement for small values of horizontal seismic coefficient, followed by a sharp increase as soon as kh exceeds 0.1. 0.6

0.1

0.5 0.05

0 10

20

30

40

GSI Fig. 5. Lower bound estimate of required reinforcement strength versus geological strength index GSI.

0.2 0.1 0

0.3

0

α=0° α=10° α=20° α=30°

0.2

0.05

0.1

α=0° α=10° α=20° α=30°

0.15

0.2

kh

Fig. 8. Effects of earthquake intensity on required reinforcement strength.

SC

0.25

Xl

0.3

RI PT

Xl

0.4

4.3. Comparisons with finite element solutions In order to assess the accuracy of the lower bound predictions derived from

0.15 0.1

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the implementation of rotational failure mechanism described in Section 3, the results are compared for a selected model to FE solutions derived from the standard OptumG2 software (OptumCE, 2015), which implements a FE

0.05

formulation of static and kinematic approaches of limit analysis. This specific software has been selected for the analysis for the following reasons.

0 5

10

15

20

25

30

mi

35

FE analysis softwares proceed by means of incremental elastoplastic

Fig. 6. Effect of strength parameter mi on the lower bound estimate of required reinforcement strength.

calculations until the free plastic flow of the structure is reached. Actually, few softwares rely upon direct implementation of static and kinematic approaches of limit analysis theory. OptumG2 falls within this category and,

0.4 α=0°

TE D

as such, it appears more suitable for addressing the stability problem insofar as

α=10°

0.3

Xl

Concerning the numerical assessment of stability analysis, most of standard

α=20°

0.2

0 55

60

65

EP

0.1

70

75

80

procedure, OptumG2 takes advantage of the recent advances in the field of second-order

cone

programming,

together

with

an

adaptive

mesh

rearrangement procedure aiming at improving the quality of the obtained lower and upper bounds. In addition, the modified Hoek-Brown strength criterion is available as well as the possibility of including elements for bars, bending beams or interfaces. For comparison purposes, the following model data have been adopted: β = 70°, σc/(γH) = 5, mi = 7, GSI = 10, D = 0, σ + / σ − = 1, and the number and

85

90

AC C

β (°)

Fig. 7. Lower bound estimate of required reinforcement strength as a function of rock slope inclination.

the prediction of limit loads is the only objective. As regards the optimization

The second series of calculations is to investigate how earthquake sequences may affect the stability of rock slope, and consequently, the amount of reinforcement required for preventing failure. The obtained results are summarized in Fig. 8 in terms of dimensionless parameter Xl versus horizontal seismic coefficient kh, providing ample evidence of the destabilizing effects induced by seismic loading. As expected, the lower bound solution for the required reinforcement is very sensitive to variations of the horizontal seismic coefficient: the higher the kh is, the higher the Xl is, reflecting a reduction in rock slope stability. A quantitative description of the destabilizing effects associated with the occurrence of an earthquake can be defined by means of the ratio between the required reinforcement strength at a given seismic intensity kh and that required in static case (kh = 0). Referring to the considered reference data set, it is found that Xl(kh = 0.1)/Xl(kh = 0) ≈ 1.31 and Xl(kh =

inclination of the reinforcement layers are respectively fixed to n0 = 6 and α = 0°. The simulations have been performed considering two values for the horizontal seismic coefficient: kh = 0 (static case) and kh = 0.1. It should be first kept in mind that implementing the limit analysis lower bound theorem provides an upper bound estimate for the required reinforcement strength X, whereas that of limit analysis upper bound theorem leads to a lower bound estimate for the required reinforcement strength X. The FE discretization used for OptumG2 simulations is defined as follows: (1) In the lower bound static approach of limit analysis, the rock mass domain is discretized into 3-node triangular elements while the reinforcement layers are discretized into linear bar-like elements that take only tensile-compressive forces. The stress fields explored are piecewise linear with possible discontinuities between adjacent elements. (2) In the upper bound approach, 6-node triangular elements are used for the rock mass domain, together with 3-node bar elements for the

ACCEPTED MANUSCRIPT reinforcements, resulting in continuous piecewise quadratic velocity fields. Starting from initial geometry discretizations, both the static and kinematic analyses are performed using automatic adaptive refinement. For illustrative purpose, Fig. 9 shows the optimized FE mesh resulting from the simulation by means of OptumG2 software in the context of upper bound approach (kh = 0). Starting the kinematic analysis from a rather coarse geometry discretization, the optimized FE mesh yielding the best upper bound consists of 7117 quadratic triangular elements. The overall CPU time needed for achieving the

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upper bound kinematic analysis is about 90 s on a standard personal computer.

Fig. 10. Optimal velocity field obtained from OptumG2 FE simulations for kh = 0 (static

M AN U

SC

case).

Fig. 9. Optimized FE mesh obtained from upper bound kinematic analysis using OptumG2 software.

Fig. 11. Optimal velocity field obtained from OptumG2 FE simulations for kh = 0.1.

Table 1 summarizes the lower bound predictions derived from the present

yield design kinematic approach together with the FE solutions obtained from

TE D

OptumG2 software. It can be observed from this table that a good agreement is

achieved from the two distinct approaches, thus underlining the excellent performance of the log-spiral failure mechanism to predict the stability of

reinforced rock slopes under static and seismic loadings. The different

estimates in both cases of loading are very close to each other (relative discrepancy remains between 2.5% and 6.5%), which means that the exact

value of dimensionless parameter X is captured within a very narrow margin.

EP

Note that the lower bound estimates derived from the log-spiral failure mechanism are slightly below those computed from FE approach. Table 1. Estimates of required reinforcement strength from present approach and FE

kh = 0 kh = 0.1

Present work, lower bound 0.108 0.142

OptumG2, lower bound 0.112 0.151

AC C

simulations. Case

OptumG2, upper bound 0.115 0.155

The optimal velocity fields obtained from FE simulations corresponding to

Fig. 12. Optimal FE stress field: contours of generalized Hoek-Brown yield function for kh = 0.

kh = 0 (static case) and kh = 0.1 are plotted in Figs. 10 and 11, respectively. They clearly show a localization of the deformation pattern of the rock slope along a line separating the zone in motion from the rest of the structure, as well as in the vicinity of the slope toe. One may notice the remarkable similarity between the FE optimal velocity fields and the rotational log-spiral failure mechanism considered in the present approach.

The performance of the present approach can be further illustrated by plotting the contours of the generalized Hoek-Brown yield function corresponding to the optimal stress field obtained from the FE lower bound static analysis. The log-spiral curve defining the optimal failure mechanism is located within the zone where yield condition is reached (i.e. F r (σ ) = 0 ) by the optimal stress field (zone colored in red in Fig. 12).

4.4. Comments Coming back to the general framework for rock slope stability analysis, a series of comments regarding the basic assumptions and validity of the modeling deserves to be made herein. Some observations should also be

ACCEPTED MANUSCRIPT formulated in light of the parametric simulations undertaken and presented in

effect of reinforcement length on the rock stability has been disregarded.

the previous subsections:

From a theoretical viewpoint, this issue can be addressed by introducing a yield condition in terms of a limitation on the shearing component of

(2)

The closed-form expressions of the support functions (π-functions)

the stress vector acting at the interface of reinforcement/ rock (Anthoine,

computed for the generalized Hoek–Brown failure criterion allow for the

1989). Alternatively, the latter condition is classically expressed by

explicit calculation of the rate of maximum resisting work in any virtual

means of the pull-out strength of Nf per unit width and unit length of the

velocity field, thus ensuring that the kinematic approach preserves a

reinforcement layer (de Buhan, 1986). The effect of interface strength on

rigorous lower bound character for the required reinforcement strength.

the rock slope stability may therefore be assessed by resorting to the

In this respect, a major limitation of the method lies on the fact that it

concept

specifically yields lower bound estimates for the required reinforcement

mechanisms. Referring for instance to the rotational failure mechanism

of anchorage

with

specific

failure

strength, and as such, it proves unconservative.

implemented in Section 3, the contribution of reinforcements (Eq. (27)) to the rate of maximum resisting work

induced by seismic motions could be neglected because its magnitude is

following expression:

smaller compared with the static one. This is actually true as long as the

Pmb r (U ) = ∑ m in {π b ([U (Pk )]), N f Lf } k

Should the latter condition not be satisfied, the effect of inertia vertical forces shall be accounted for in design analyses by appropriate modification of the specific weight. Second, the destabilizing effects induced by the seepage forces associated with pore water pressure are disregarded in the present analysis. Previous investigations (e.g. Saada et rock slope stability due to the presence of seepage forces, which in turn

would imply an increase in the amount of reinforcement required to

(32)

where Lf stands for the anchorage length behind the log-spiral failure line I1I2 (Fig. 13), corresponding to the k-th reinforcement layer. It depends on whether the layer intersects the failure line or not (i.e. active or non-active reinforcement layer). In terms of virtual motion for the reinforcement layers, Eq. (32) stems from the considerations: (i)

of the virtual rotational motion that has been adopted in the case of

perfect bonding: ωr (s )e θ (s ) ( s < s k ) U (s ) =  (s > sk ) 0

M AN U

al., 2012) have provided ample evidence of the significant reduction in

should be replaced by the

SC

compared with gravity forces (less than 0.3g) (Chen and Liu, 1990).

(33)

(ii) of the virtual slipping motion:

prevent failure.

actually equal to the inclination of the velocity jump to a discontinuity

ωr (s )e θ (s ) U (s ) =  ωr (s k )(e θ (sk ) e b ) e b

surface. This property is derived from the normality rule of the strain

which involves slipping at the rock/reinforcement interface along the

The angle ϕ that defines the geometry of log-spiral failure surface is

(s < s k ) (s > s k )

(34)

rate tensor to the failure criterion. Consequently, the inclination ϕ has a

portion s k ≤ s ≤ L , but whose tangential component remains

clear interpretation in the Mohr plane as the inclination angle of a

continuous along the reinforcement, i.e. [U (s )]e b = 0. In such a failure

tangent to the failure envelope. The arbitrariness of the value of angle ϕ

TE D

is simply the consequence of operating with the nonlinear criterion. (4)

associated

As regards the loading mode, it is first assumed that the vertical force

magnitude of horizontal seismic acceleration remains moderate when

(3)

length

RI PT

(1)

mechanism, the rate of maximum resisting work due to the resistance to normal forces reduces to zero, i.e. π b ([U (Pk )]) = 0 .

Referring to practical situations of reinforced rock slope stability, the estimates derived for the required reinforcement strength in the context

It clearly appears from Eq. (32) giving the contribution of reinforcement

of present kinematic approach are expected:

strength to the rate of maximum resisting work that accounting for the yield

(i) to underestimate the amount of reinforcement when the

condition at the interface will induce a reduction in rock slope stability, and

compressive resistance of reinforcement layers is accounted for,

therefore an increase in required reinforcement. It is observed that other virtual motions involving slipping along the

enhanced. Actually, the ratio N 0′ / N 0 is likely to be smaller than

rock/reinforcement interface can be considered, such as that defined by the

EP

since the effect of strength reinforcement of the rock slope is

rotational motion of the rock mass block TI1I2, whereas the reinforcement

compressive strength is consistent with the idea that the primary

layers are given a virtual motion defined by the same normal velocity as the

purpose of incorporating reinforcing bolts is to provide initially

surrounding rock, while the tangential component of velocity is taken equal to

cohesionless (or low tensile strength) rocks with tensile stress

zero (i.e. U (s ) = ωr (s )(e θ (s ) n )n for s < sk). Such a failure mechanism exhibits

AC C

unity and probably near to zero (buckling phenomenon). A reduced

carrying capacities, thus implicitly neglecting their contribution to

slipping along the anchorage length 0 ≤ s < s k = L − Lf located in front of the

undergo compressive forces in the stability analysis.

log-spiral failure line.

(ii) to overestimate the amount of reinforcement when the shear and bending strengths of reinforcement are neglected, as has been assumed in the previous analysis. In the implementation of rotational log-spiral failure mechanism, only the contribution of mobilized resistance to normal forces has been considered for the reinforcements in the expression of maximum resisting work rate Pm r (U ) . However, this assumption reveals questionable in view of the virtual motion of reinforcement elements induced by rotational motion of the surrounding rock mass. In such a failure mechanism,

both

normal

and

shear

resistances

of

the

reinforcements are mainly mobilized (de Buhan and Salençon, 1993). (5)

The whole analysis relies upon the assumption of perfect bonding at the interface between the reinforcements and the rock. In particular, the

5. Conclusions In the present study, the kinematic approach of limit analysis implemented within the framework of the pseudo-static method has been developed to assess the stability conditions of a rock slope reinforced by a series of passive bolts under static and seismic loadings. Particular emphasis has been given to the evaluation of the amount of reinforcement required for preventing failure. At the material level, the strength properties of the rock are described by means of a generalized Hoek-Brown yield condition, which is known to reasonably well model the strength of isotropic rocks. The closed-form expressions formulated in previous works for the support functions ( πfunctions) associated with such a failure criterion allow for the analytical or semi-analytical derivation of rigorous upper bound limit analysis solutions for

ACCEPTED MANUSCRIPT the stability problem. Regarding the reinforcement elements, the passive bolts

horizontal seismic coefficient. Despite the limitation inherent to the pseudo-

are modeled as bar-like inclusions that are assumed to take only tensile-

static method, for the possible structure collapse caused by accumulated

compressive forces, perfect bonding being assumed at the interface with the

permanent displacement induced by the earthquake are disregarded, such a

surrounding rock mass. At the structural level, the effects of inertial forces

m

e

t

h

o

d

i

s

s

t

i

l

l

induced by earthquake events are addressed through the concept of average

I1

T

ω s=0

e = −t−

−b

Pk n

−

L

Lf s=L s = sk

I2

RI PT

t

Layer k

SC

Fig. 13. Geometry definitions for slipping failure mechanism at rock/reinforcement interface and associated anchorage length.

being widely used in geotechnical and rock engineering for its simplicity of

Referring to the notations introduced in Section 3, this appendix presents

From a practical viewpoint, stability conditions for the reinforced rock slope

the expressions of dimensionless functions f i and f i′ defining Pext (U ) and

M AN U

implementation together with its effectiveness to yield satisfactory predictions. are derived from implementation of the kinematic approach, making

Pm r (U ) developed in the rotational failure mechanism sketched in Fig. 2.

specifically use of the rotational failure mechanisms. The predictions thus

f1 =

computed are clearly formulated as lower bound estimates for the required

reinforcement strength. A parametric study has been undertaken to provide

preliminary insight into the influence of relevant parameters on the amount of reinforcement required to prevent failure. In light of this analysis, and within

the range of considered parameters, it is likely that the intensity of horizontal seismic coefficient is the parameter that most affects the stability of the reinforced rock slope. The accuracy of the proposed analysis has been

TE D

assessed by comparison of the lower bound predictions to FE solutions

mechanism to well capture the essential features of deformation localization pattern of the structure.

A main advantage of the approach lies on the fact that it requires a few

input parameters and can be operated with a low computational cost, allowing

EP

for the possibility of performing intensive parametric studies that might be useful to designers. Even though the applications have been restricted to the situation of perfect bonding between the reinforcement and the surrounding

AC C

Section 4.4, which provides a clear framework to assess the implication of slipping at the interface on the slope stability as well as on the influence of reinforcement length.

Finally, it should be mentioned that when a large number of reinforcement layers are involved, the limit analysis homogenization may offer an attractive and efficient approach to direct analyses (e.g. de Buhan, 1986; de Buhan et al., 1989). Taking advantage of both the density and the regularity of the reinforcement scheme, the homogenization approach stems from the heuristic idea that the reinforced rock medium can be perceived at the macroscopic

 cos(β + θT ) cos(β + θ2 )   sin β  −   sin (β + θT ) sin (β + θ 2 )  

f 1′ =

homogenized strength properties of the modified Hoek-Brown rock matrix reinforced by bolts as well as the associated π-functions, and then to develop the stability analysis at the level of the equivalent homogenized structure.

Appendix

(A3)

e 3(θ2 −θ1 ) t an ϕ (3 t an ϕ sin θ2 − cos θ2 ) − (3 tan ϕ sin θ1 − cos θ1 )

(A4)

3(9 tan 2 ϕ + 1)

TI 1 f 2′ = − sin 2 θ1 1 3 r0

(A5)

3

 cos(β + θ2 ) cos( β + θT )  1r   f 3′ = −  T  sin 3 (β + θT )  − cos β  − + 3  r0   sin (β + θ 2 ) sin ( β + θT )     sin β  1 1 −   2  sin 2 (β + θ2 ) sin 2 (β + θT )   f4 =

(A6) 1/(1−n ) 

 1 − sin ϕ  1  s n /(1−n ) − n 1/(1−n ) ]m n /(1−n )   + [n  2 m  2 sin ϕ  

 2(θ −θ ) t an ϕ − 1]  [e 2 1  (A7)

    (θk ≥ α )  sin (θk − α )   ak =  N 0′  − N sin (θk − α ) (θk ≤ α )   0  1 n0 (θ −θ ) t an ϕ f5 = ∑a e k 1 n 0 k =1 k

(A8)

The geometrical parameters involved in the above expressions are given by H r0 = (θ −θ ) t an ϕ (A9) e 2 1 sin θ2 − sin θ1

scale as a homogeneous but anisotropic continuum. In this context, an extension to be foreseen in the future will consist in formulating the

(A2)

3  cos β   1r  1 1 f 3 = −  T  sin 3 (β + θT )  −  2 + 2 3  r0  2  sin (β + θT ) sin (β + θ 2 )  

rock mass, a different yield condition at the reinforcement/rock interface can be easily included in the analysis following the methodology outlined in

(A1)

3(9 t an 2 ϕ + 1)

2  TI  TI   1 f 2 = − sin θ1  2 1 cos θ1 −  1   6  r0  r0   

derived from the OptumG2 software, demonstrating the ability of the analysis

to accurately predict the stability condition and the log-spiral failure

e 3(θ2 −θ1) t an ϕ (3 t an ϕ cos θ 2 + sin θ1 ) − (3 tan ϕ cos θ1 + sin θ1)

TI1 = r0

sin (θ1 + β ) − e (θ2 −θ1 ) t an ϕ sin (θ2 + β ) sin β

(A10)

The polar coordinates (rT, θT) of point T are rT = r0 1 − 2

 TI  TI1 cos θ1 +  1  r0  r0 

2

(A11)

ACCEPTED MANUSCRIPT  sin θ1   cos θ1 − TI1 / r0  

θT = arctan 

(A12)

The angle θk (k ∈ {1, 2, L , n 0 }) defines the angular coordinate of the intersecting point Pk between arc I1I2 and the k-th reinforcement layer. It is defined as the single root within the interval [θ1, θ2] of the following equation: rT sin (β + α )  1 H sin (θT − α ) − k −  =0 r0 sin β  2  r0 n 0

Geomechanics and Geoengineering 2010; 5(2): 99–108. Goodman RE. Introduction to rock mechanics. 2nd edition. New York: John Wiley & Sons Inc.; 1989. Guo D, Hamada M. Observed stability of natural and reinforced slopes during the 2008 Wenchuan earthquake. Journal of Japan Society of Civil Engineers, Ser. A1 (Structural Engineering & Earthquake Engineering (SE/EE)) 2012; 68(2): 481–94. Halabian AM, Sheikhbahaei AM, Hashemolhosseini SH. Three dimensional finite

(A13) When a reinforcement layer does not intersect the failure line (i.e. Eq. (A13) admits no solution within the interval [θ1, θ2]), the corresponding contribution ak is set to zero.

difference analysis of soil-nailed walls under static conditions. Geomechanics and Geoengineering 2012; 7(3): 183–96. He S, Ouyang C, Luo Y. Seismic stability analysis of soil nail reinforced slope using kinematic approach of limit analysis. Environmental Earth Science 2012; 66(1): 319–

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sin (θk − α )e (θk −θ1 ) t an ϕ −

Giri D, Sengupta A. Dynamic behavior of small-scale model of nailed steep slopes.

26.

Conflict of interest

Hoek E, Brown ET. Empirical strength criterion for rock masses. Journal of the Geotechnical Engineering Division, ASCE 1980; 106(GT9): 1013–36.

The authors wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.

Hoek E, Brown ET. Practical estimates of rock mass strength. International Journal of Rock Mechanics and Mining Sciences 1997; 34(8): 1165–86.

Hoek E, Carranza-Torres C, Corkum B. Hoek-Brown failure criterion – 2002 edition. In: 267–73.

SC

Proceedings of the North American Rock Mechanics Society Meeting, Toronto, 2002. p.

Acknowledgements

Hoek E. Estimating Mohr-Coulomb friction and cohesion values from the Hoek-Brown

The preparation of the paper had received financial support from Ecole des Ponts et Chaussées-ParisTech (France), the French Institute of Tunisia (French

Geomechanics Abstracts 1990; 27(3): 227–9. Hoek E. Strength of jointed rock masses. Géotechnique 1983; 33(3): 187–223.

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Embassy-Tunisia) and Laboratoire de Génie Civil (ENIT). The authors have

failure criterion. International Journal of Rock Mechanics and Mining Sciences and

greatly appreciated the financial support.

Hong YS, Chen RH, Wu CS, Chen JR. Shaking table tests and stability analysis of steep nailed slopes. Canadian Geotechnical Journal 2005; 42(5): 1264–79.

Jiang JC, Baker R, Yamagami T. The effect of strength envelope nonlinearity on slope

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Dr. Samir Maghous is a professor and doctoral student supervisor at Federal University of Rio Grande do Sul (UFRGS-Brazil), and a senior research

scholar of the Brazilian National Research Council (CNPq). He is the academic leader of the Center of Applied Computational Mechanics (CEMACOM) at UFRGS Civil Engineering Department. Dr. Maghous graduated in civil engineering from Ecole Nationale des Travaux Public de l´Etat (France), and received PhD in material and structural mechanics from Ecole Nationale des Ponts et Chaussées (France) in 1991, where he held a position of associate professor for 15 years. He coordinated several science and technology projects in both France and Brazil, and published more than 60 journal papers and 6 book chapters. The areas of expertise of Dr. Maghous include stability analysis of structures and geo-structures, theoretical and computational modeling in sedimentary basins, formulation of poromechanical constitutive modeling at large strains, modeling, material modeling by micromechanics, upscaling methods in geomechanics, constitutive modeling of cement based composites, micromechanical approaches to damage mechanics in rocks. He is also a corresponding member of Brazilian Tunnelling Committee (WG2).