International Journal of Rock Mechanics & Mining Sciences 89 (2016) 176–184
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International Journal of Rock Mechanics Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
Limit equilibrium method for rock slope stability analysis by using the Generalized Hoek–Brown criterion
crossmark
⁎
Deng Dong-pinga, , Li Lianga, Wang Jian-fengb, Zhao Lian-henga a b
School of Civil Engineering, Central South University, Changsha 410075, China Department of Architecture and Civil Engineering, City University of Hong Kong, Hong Kong 999077, China
A R T I C L E I N F O Keywords: Rock slope Stability analysis Limit equilibrium Generalized Hoek–Brown (GHB) criterion Factor of safety
1. Introduction Predicting the stability of rock slopes is a classic problem for geotechnical engineers and is crucial when designing dams, roads, tunnels, and other engineering structures.1 It has therefore attracted the attention of many researchers.2–12 The method used to describe the failure behavior of the rock mass is critical in evaluating the stability of rock slope. However, rock mass strength is a nonlinear stress function.13–16 Therefore, the linear Mohr–Coulomb (MC) criterion17–19 generally does not agree with the rock mass failure envelope, especially for slope stability problems where the rock mass is in a state of low confining stresses that make the nonlinearity more obvious.20,21 Using a large amount of experimental data from field tests conducted on rocks, Hoek and Brown13 proposed the nonlinear Hoek–Brown (HB) criterion in 1980. Thereafter, Hoek et al.22 improved the basic HB criterion with the latest version, that is, the generalized Hoek–Brown (GHB) criterion, in 2002, which has since been widely used for estimating the strength of rock and rock masses.23–28 Currently, the GHB criterion can reflect a rock's inherent nature and the effect of certain factors such as the strength of the rock, the number of structural planes, and the stress state, on the strength of rock mass; hence, the GHB criterion is essential in studying the deformation and failure characteristics of rock slopes. Compared with the linear MC criterion, which includes two parameter, cohesion c and internal friction angle φ, the nonlinear GHB criterion has a more complex expression and more strength parameters. Therefore, applying the nonlinear GHB criterion to the stability analysis of rock slopes is difficult. The limit analysis method
⁎
(LAM)1,29–36 has been adopted to study the stability of a rock slope with the nonlinear GHB criterion because it is ideally suited to modeling jointed and fissured materials with discontinuities that exist inherently throughout the mesh. For example, Li et al.1 used numerical limit analysis to produce stability charts for rock slopes. The numerical simulation method (NSM)37,38 can also be used to solve the aforementioned problem. However, in contrast to the linear MC criterion used in the limit equilibrium method (LEM), the nonlinear GHB criterion cannot be easily applied as the known condition in the limit equilibrium equations because of its complex expression; hence, obtaining the explicit solution for the slope factor of safety (FOS) with the nonlinear GHB criterion is difficult, and few studies have addressed this problem. Compared to the LAM and NSM, the LEM is based on a simpler theory and yields more reliable results, and thus this method is most preferred by engineers. Therefore, establishing the LEM for the stability analysis of a rock slope with the nonlinear GHB criterion would be a substantial development. At present, researchers usually convert the rock mass strength parameters into the equivalent MC parameters, as proposed by Hoek and Brown15 and Hoek et al.,22 following which the stability analysis of the rock slope is performed using the LEM and these equivalent MC parameters. However, studies1,29,32,39 have shown that this conversion produces inconsistent estimates of the slope stability FOS and that the difference in slope stability estimated using the equivalent parameters and the native yield criterion were as high as 64%. Subsequently, Li et al.32 argued that directly using the HB failure criterion in the calculations is the optimal method to solve rock and rock mass problems. For solving the limit equilibrium stability problem of slopes with
Corresponding author. E-mail addresses:
[email protected] (D. Dong-ping),
[email protected] (L. Liang),
[email protected] (W. Jian-feng),
[email protected] (Z. Lian-heng).
http://dx.doi.org/10.1016/j.ijrmms.2016.09.007 Received 29 February 2016; Received in revised form 22 June 2016; Accepted 17 September 2016 1365-1609/ © 2016 Elsevier Ltd. All rights reserved.
International Journal of Rock Mechanics & Mining Sciences 89 (2016) 176–184
D. Dong-ping et al.
the nonlinear strength criterion, Deng et al.40 proposed a Taylor series–based method for expanding the nonlinear MC criterion under the assumption of stresses on the slip surface to obtain linear equations for solving the slope FOS. In this paper, a new LEM to analyze rock slope stability by using the nonlinear GHB criterion is presented based on this same methodology. First, to obtain the approximate analytical formulas of shear strength expressed in the nonlinear GHB criterion, the GHB criterion is expanded using the Taylor series and under the assumption of stresses on the slip surface. Then, the static equilibrium conditions of the sliding body are combined to obtain the linear equations, solving which yields the stresses on the slip surface. Finally, the rock slope FOS is calculated using the known stresses on the slip surface. However, in contrast to the use of a single expression of shear strength in the nonlinear MC criterion, solving for shear strength in the nonlinear GHB criterion entails multiple nested formulas, which makes expanding the nonlinear GHB criterion using the Taylor series difficult. In addition, the instantaneous internal friction angle in the nonlinear GHB criterion must be iteratively solved to obtain the stresses on the slip surface. Therefore, the proposed method has some improvements to solve these aforementioned problems in relation to Deng's method. Further, the current method is suitable to analyze the stability of rock slopes with an arbitrarily shaped slip surface, and can be also easily programmed using a simple derivation process and yields reasonable and strict results. Meanwhile, a series of practical charts of the FOS for rock slope stability analysis is drawn by applying the proposed method.
α = 0.5 +
2.2. Linear equivalent MC strength parameters Eq. (1) shows the calculation of shear strength in the nonlinear GHB criterion. Because of its complexity, deriving a precise analytical expression of FOS for slope stability analysis with the nonlinear GHB criterion is not mathematically feasible. Therefore, to apply the nonlinear GHB criterion to the slope stability analysis by using existing methods, researchers have established the linear equivalent MC (EMC) strength parameters. For example, to obtain linear EMC strength parameters of the nonlinear GHB criterion, Hoek et al.22 established the σ and τf axes coordinate system and defined the tensile strength σt and the limit of the corresponding maximum confining pressure σ3max. For tensile stress ranging between σt and σ3max, the curve of the nonlinear GHB criterion was fitted adopting linear MC strength parameters. The principle of optimal fitting is that the areas covered by the curves corresponding to the two criterions within this specific range of σt and σ3max are the same. Accordingly, the value of the linear EMC strength parameters, c and φ, are given by Hoek et al.22 as
c=
σc fc [s (1 + 2α ) + (1 − α ) mb σ3n] fa 1 + fb fc / fa
⎡ fb fc ⎤ ⎥ φ = arcsin ⎢ ⎣ 2fa + fb fc ⎦
2.1. Nonlinear GHB criterion
σ3n =
ci = τf − σ tan φi
(3)
σ3 max σc
⎛ σ ⎞m σ3 max = k ⎜ cm ⎟ σcm ⎝ γH ⎠
σcm = σc
(4)
⎛ GSI − 100 ⎞ s = exp ⎜ ⎟ ⎝ 9 − 3D ⎠
(5)
(10)
[mb + 4s − α (mb − 8s )](mb /4 + s )α−1 2(1 + α )(2 + α )
(11)
Based on these above equations for calculating the linear EMC strength parameters, proposed in,22 other works have been also done to determine the equivalent shear strength parameters. For example, Li et al.1 found that the slope FOS was significantly overestimated when using k=0.72 and m=−0.91 in Eq. (10); hence, they used k=0.2 and m=−1.07 in Eq. (10) when slope angle β≥45°. Additionally, on the basis of the numerical method proposed by Kumar,42 Shen et al.21 proposed a new approximate analytical solution for estimating the linear EMC strength parameters from the GHB criterion for a highly fractured rock mass where 0 < GSI < 40. In this method, Shen et al.21 replaced the nonlinear function with a linear function through linear regression analysis and using s=0 when GSI < 40. The obtained shear stresses were consistent with those reported in the literature. In this work, for ensuring the uniformity of the comparative analysis, the linear EMC strength parameters are estimated using Hoek's method, as the other methods are necessary to be applied on the corresponding conditions.
where τf is the shear strength, σ is the normal stress, σc is the uniaxial compressive strength, mb is a material constant related to the intact material constant mi and reflects the hardness of the rock, s is a parameter that reflects the degree of fragmentation of the rock and ranges between 0.0 and 1.0, and α is a coefficient that reflects the characteristics of the rock mass. Unlike the linear MC criterion, ci and φi in Eq. (3) are not constants but variables exhibiting the nonlinear characteristics of rock strength. In Eq. (1), mb, s, and α can be calculated using the geological strength index (GSI) as
⎛ GSI − 100 ⎞ mb = exp ⎜ ⎟ mi ⎝ 28 − 14D ⎠
(9)
where k and m are empirical parameters (k=0.72 and m=−0.91 in slope engineering), γ is the unit weight of the rock mass, H is the slope height, and σcm is the rock mass strength, which is given by
(1)
(2)
(8)
where σ3max is the limit of maximum confining pressure. The relationship between σ3max and the rock mass strength σcm can be described as
σc cos φi
⎞⎛ ⎞(1− α ) ⎛ 1 sin φi ⎞(1− α ) 2 ⎛ σ =⎜ − 1⎟ ⎜1 + ⎟ ⎜mb + s⎟ ⎠ mb α ⎝ σc α ⎠ ⎝ sin φi ⎠⎝
(7)
where α, σc, mb, and s are as described in Eq. (1); fa = (1 + α )(2 + α ); fb = 6αmb ; and fc = (s + mb σ3n )α −1. The parameter σ 3nin Eq. (7) can be calculated as
The nonlinear HB criterion is used to calculate the strength of a joint geotechnical body and to assess the degree of interaction and conditions of the contact surfaces in the rock mass; hence, this criterion can be suitably applied to evaluating rock slope stability. The nonlinear HB criterion was initially proposed by Hoek and Brown13 in 1980 based on a combination of experimental studies and theoretical analysis. The latest version is the GHB criterion presented by Hoek et al.22 in 2002. The GHB criterion can also be expressed using instantaneous cohesion (ci) and internal friction angle (φi), similar to the linear MC criterion41,42:
⎛ σ ⎞α + s⎟ ⎜m sin φi α ⎝ b ⎠ σc 2(1 + α )
(6)
where D is a rock weakening factor related to the excavation mode and disturbance degree of the rock mass; it ranges from 0 (unperturbed state) to 1.
2. Nonlinear GHB and its linear equivalent MC strength parameters
τf =
1 [exp(−GSI/15) − exp(−20/3)] 6
177
International Journal of Rock Mechanics & Mining Sciences 89 (2016) 176–184
D. Dong-ping et al.
o(x c, y c)
³ 32 zone 2
dx d1 12
qx H
s(x) w c1
b1 A 0
x 31
Nonlinear GHB criterion
kVw
kHw
£ 0
linear Equivalent MC criterion
a1
g(x)
y
H zone 1
´f
B
qy
22
21
x
Fig. 2. Model for rock slope stability analysis with nonlinear GHB criterion.
11
normal, shear, and water forces on the slip surface, respectively, where δ is the horizontal inclination angle of the tangent to the slip surface.. Ignoring the interslice forces illustrated in Fig. 2, the force equilibrium conditions on the microunit slice a1b1c1d1 in the x and y directions are
11 — Shallow failure slip surface (SFSS); 21 — Intermediate failure slip surface (IFSS); 31 — Deep failure slip surface (DFSS); 12 — Normal stresses on SFSS; 22 — Normal stresses on IFSS; 32 — Normal stresses on DFSS; Fig. 1. Difference of rock slope stability analysis using nonlinear GHB criterion and linear EMC criterion. 11 — Shallow failure slip surface (SFSS); 21 — Intermediate failure slip surface (IFSS); 31 — Deep failure slip surface (DFSS); 12 — Normal stresses on SFSS; 22 — Normal stresses on IFSS; 32 — Normal stresses on DFSS.
kH w − τ0 − qx + (σ0 + u ) s′ = 0
(12)
(1 − kV ) w + qy − τ0 s′ − (σ0 + u ) = 0
(13)
where σ0 and τ0 are the normal and shear stresses on the slip surface, without considering the effect of interslice forces, respectively; u is the water pressure on the slip surface; s′ is the first derivative of slip surface equation s(x) with respect to the x-axis, and s′ = tan δ . Solving Eqs. (12) and (13), the normal stress σ0 can be obtained as
2.3. Comparison of the two criterions in slope stability analysis Fig. 1 shows the model for slope stability analysis with the nonlinear GHB and linear EMC criterions. The figure shows that τ fEMC > τ GHB in zone 1 and τ fEMC ≈ τ GHB in zone 2 for the curves of the f f two criterions. Failure slip surfaces on unstable slopes can be of three types: shallow failure slip surface (SFSS), intermediate failure slip surface (IFSS), and deep failure slip surface (DFSS). The normal stress distributions on these slip surfaces differ, indicating that slope stability obtained using the nonlinear GHB criterion and linear EMC criterion are not consistent with each other. For example, the normal stresses on an SFSS are small within zone 1, so the slope FOS determined using the linear EMC criterion is larger than that determined using the nonlinear GHB criterion. However, the normal stresses on an SFSS are large, with most of them being in zone 2, so the slope FOS determined by both the criterions are almost equal. Therefore, the linear EMC criterion overestimates the slope stability for an SFSS. Moreover, the nonlinear characteristics of rock strength cannot be studied using the EMC criterion..
σ0 =
[(1 − kV ) w + qy] + (−kH w + qx ) s′ 1 + (s′)2
−u
(14)
However, because of the interslice forces, σ0 obtained from Eq. (14) and the actual normal stress differ. Therefore, to obtain σ that is closer to the actual normal stress, σ0 is amended as (15)
σ = λ1 σ0
where λ1 is a variable, and σ is the normal stress on the slip surface. In slope stability analysis, the slope FOS (Fs) is generally defined as the ratio of failure shear force to the actual shear force on the slip surface (i.e., Fs=(τf dx)(τdx) in the microunit slice). Rearranging this expression yields τ=τf / Fs. Using Eq. (1), shear stress τ on the slip surface can be expressed as
τ=
⎞α 1 σc cos φi ⎛ σ + s⎟ ⎜m sin φi α ⎝ b ⎠ Fs 2(1 + σc ) α
(16)
Eq. (16) indicates that τ is a function of the normal stress σ. Using Eqs. (2) and (16), the first derivative of τ with respect to σ can be obtained as
3. LEM for rock slope stability analysis with nonlinear GHB criterion 3.1. Assumption of stresses on the slip surface
α cos φi
dτ 1 = dσ Fs
As shown in Fig. 2, the x-y coordinate system is established with its origin at the slope toe. A and B are the lower and upper sliding points of the slip surface, respectively; g(x) and s(x) are the equations for the slope outline and slip surface, respectively. The microunit slice a1b1c1d1 of width dx is selected for the force analysis. Under typical conditions, the forces acting on the microunit slice a1b1c1d1 are as follows: wdx is the gravity force; kHwdx and kVwdx are the seismic forces in the horizontal and vertical directions, respectively, kH being the horizontal and kV being the vertical seismic coefficient; qxdx and qydx are the external loads on the slope in the horizontal and vertical directions, respectively; σdx/cosδ, τdx/cosδ, and udx/cosδ are the
dφi = dσ
σ s + σc mb
⎛ − σci ⎜⎜sin φi + ⎝ 2(1 +
cos2 φi sin φi 1+ α
sin φi α ) α
2(1 − α ) ⎛ sin φ ασc cos φi ⎜sin φi − α i − ⎝
⎞ ⎟ dφi ⎟ dσ ⎞α ⎠ ⎛ σ ⎜mb + s⎟ ⎝ σc ⎠
1 sin φi
(17)
⎛ σ ⎞−α sin φi + s⎟ ⎜m ⎞ (1 + sin φi )−a ⎝ b σc ⎠ α − 1⎟ ⎠ (18)
Next, Eq. (2) is rearranged to determine the instantaneous internal friction angle φi of Eqs. (17) and (18). Accordingly, φi can be calculated as 178
International Journal of Rock Mechanics & Mining Sciences 89 (2016) 176–184
D. Dong-ping et al.
Table 1 Minimum FOS in rock slope example 1. case
Type of slip surface
1
Linear EMC criterion
Circular slip surface Arbitrary curved slip surface Circular slip surface Arbitrary curved slip surface Circular slip surface Arbitrary curved slip surface Circular slip surface Arbitrary curved slip surface Circular slip surface Arbitrary curved slip surface Circular slip surface Arbitrary curved slip surface
2 3 4 5 6
Nonlinear GHB criterion
Swedish method 1
Simplified Bishop method
Spencer method
Morgenstern-Price method
Swedish method 2
The current method
1.445 – 1.393 – 1.331 – 1.256 – 1.160 – 1.038 –
1.498 – 1.445 – 1.381 – 1.302 – 1.204 – 1.077 –
1.490 1.482 1.437 1.430 1.373 1.367 1.295 1.289 1.197 1.192 1.071 1.066
1.487 1.475 1.434 1.422 1.371 1.359 1.293 1.282 1.195 1.185 1.069 1.060
1.326 – 1.282 – 1.229 – 1.164 – 1.083 – 0.976 –
1.403 1.398 1.356 1.349 1.301 1.296 1.232 1.230 1.146 1.153 1.034 1.044
⎡ ⎞α ⎛ σ0 ⎢ ⎜mb σ + s⎟ ⎛ 2 c ⎠ ⎝ ⎢ α cos φ0 ⎜sin φ + cos φ0 τ02 = ⎢ σ 0 − σ c s × 0 α sin φ0 sin φ ⎜ + 2(1 + α ) 1+ α0 mb ⎝ ⎢ σc ⎣
7 5 6
/kPa
4
8
3
100
B1
B2
(1 − α )sin φ0 ⎛ sin φ ασc cos φ0 ⎜sin φ0 − α 0 − ⎝
R=30m 2 H =20m
75 50 25
1 0
5
10
15
20
25
30
35
⎤ ⎥ ⎥ σ ⎞⎥ 0 − 1⎟ ⎥ ⎠⎦
(22)
where H(σ–σ0) is the higher order error term and φ0 can be calculated using Eq. (12) (i.e., φi = φ0 when σ is replaced with σ0 in Eq. (19)). When σ0 is similar to σ, that is, λ1 tends to unity in Eq. (15), H(σ−σ0) in Eq. (20) contributes little to τ. In fact, as σ0 is approximately equal to σ, H(σ−σ0) can be neglected. Then, applying two variables, namely λ2 and λ3, to amend τ01 and τ02 in Eq. (20), respectively, τ can be calculated as
=45
A
1 sin φ0
⎞ ⎟ ⎟ ⎠
x/m
1 —— Circular slip surface 2 —— Arbitrary curved slip surface 3 —— Shear stress calculated using Eq. (20) on circular slip surface 4 —— Shear stress calculated using Eq. (20) with neglect of H(σ – σ0) on circular slip surface
(23)
5 —— Shear stress calculated using Eq. (23) on circular slip surface
τ = λ2 τ01 + λ3 τ02
6 —— Shear stress calculated using Eq. (20) on arbitrary curved slip surface
in which λ2 and λ3 also include the effect of H(σ−σ0) on τ. As shown in Fig. 2, the force equilibrium conditions in the x and y direction and the moment equilibrium condition of all forces about one point (xc, yc) in the sliding body can be determined as
7 —— Shear stress calculated using Eq. (20) with neglect of H(σ – σ0) on arbitrary curved slip surface 8 —— Shear stress calculated using Eq. (23) on arbitrary curved slip surface Fig. 3 Shear stresses on slip surfaces
Fig. 3. Shear stresses on slip surfaces. 1 —— Circular slip surface, 2 —— Arbitrary curved slip surface, 3 —— Shear stress calculated using Eq. (20) on circular slip surface, 4 —— Shear stress calculated using Eq. (20) with neglect of H(σ – σ0) on circular slip surface, 5 —— Shear stress calculated using Eq. (23) on circular slip surface, 6 —— Shear stress calculated using Eq. (20) on arbitrary curved slip surface, 7 —— Shear stress calculated using Eq. (20) with neglect of H(σ−σ0) on arbitrary curved slip surface, 8 —— Shear stress calculated using Eq. (23) on arbitrary curved slip surface.
sin φi =
(1
sin φ (1− α ) + α i) ⎞(1− α )
2 ⎛ σ + s⎟ ⎜m mb α ⎝ b σc ⎠
+ (1 +
∫ (−σs′ + τ ) dx − ∫ (kH w − qx + us′) dx = 0 ∫ (σ + τs′) dx − ∫ [(1 − kV ) w + qy − u] dx = 0 ∫ {[(1 − kV ) w + qy + u](x − xc ) + us′( yc − s) + kH w ( yc −
σc cos φ0 2(1 +
sin φ0 α ) α
⎛ σ0 ⎞α + s⎟ ⎜mb ⎝ σc ⎠
−
qx ( yc − g)} dx − ∫ [(−σs′ + τ )( yc − s ) + (σ + τs′)(x − xc )] dx = 0 a sin φi (1− α ) ) α
1 1 τ01 + (λ1 − 1) τ02 + H (σ − σ0 ) Fs Fs
τ01 =
(25) s+g ) 2
b
(26) (19)
In contrast to the traditional LEM, the solution Eqs., (24)–(26), are established according to the overall mechanical equilibrium conditions of the sliding body. For this reason, the interslice forces considered in the traditional LEM can be considered the internal forces in the current method, and they are not necessary to be appeared in these solution equations. Therefore, despite the use of the simple initial normal stress σ0 without considerations of the interslice forces, the current method satisfies all static equilibrium conditions of the sliding body. Further, other patterns of the initial normal stress with considerations of the interslice forces can be satisfied using the current method. In addition, the type of slip surface is not restricted when the mechanical equilibrium conditions of the sliding body is established by the current method, so the current method is suitable for arbitrary shaped slip surfaces. Substituting Eqs. (24) and (25) in Eq. (26) and simplifying, yields
Eq. (19) is an iterative formula; by assuming an initial value in Eq. (19), the final value can be obtained after several iterative cycles. Eq. (16) can be expressed using Taylor series expansion with the normal stress σ0 as the initial value and substituting it in Eqs. (17) and (18):
τ=
(24)
(20)
(21)
179
International Journal of Rock Mechanics & Mining Sciences 89 (2016) 176–184
D. Dong-ping et al.
Table 2 Minimum FOS in rock slope example 2. β
GSI
75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45 45
100 100 100 100 70 70 70 70 50 50 50 50 30 30 30 30 10 10 10 10 100 100 100 100 70 70 70 70 50 50 50 50 30 30 30 30 10 10 10 10
mi
5 15 25 35 5 15 25 35 5 15 25 35 5 15 25 35 5 15 25 35 5 15 25 35 5 15 25 35 5 15 25 35 5 15 25 35 5 15 25 35
Limit analysis-lower bound method (Li et al.1)
The current method
Nonlinear GHB criterion (σci/γH)crit 0.360 0.278 0.228 0.194 1.703 1.169 0.890 0.717 4.980 2.988 2.156 1.668 15.011 8.576 5.824 4.327 93.721 53.362 35.186 24.994 0.135 0.058 0.036 0.026 0.469 0.176 0.108 0.077 1.046 0.369 0.222 0.158 2.593 0.829 0.480 0.334 13.585 3.155 1.552 0.969
β
Limit analysis-lower bound method (Li et al.1)
The current method
Nonlinear GHB criterion
Nonlinear GHB criterion
Nonlinear GHB criterion
Fs
Fs1
%Diff
(σci/γH)crit
Fs
Fs1
%Diff
1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1.019 1.010 1.003 1.000 1.015 1.003 0.997 0.995 1.029 0.999 0.995 0.993 1.006 0.997 0.994 0.992 1.012 1.001 0.997 0.996 0.989 0.990 0.990 0.991 0.988 0.992 0.997 0.995 0.988 0.995 0.998 0.999 0.990 0.998 1.001 1.002 0.994 1.001 1.003 1.004
1.9% 1.0% 0.3% 0.0% 1.5% 0.3% −0.3% −0.5% 2.9% −0.1% −0.5% −0.7% 0.6% −0.3% −0.6% −0.8% 1.2% 0.1% −0.3% −0.4% −1.1% −1.0% −1.0% −0.9% −1.2% −0.8% −0.3% −0.5% −1.2% −0.5% −0.2% −0.1% −1.0% −0.2% 0.1% 0.2% −0.6% 0.1% 0.3% 0.4%
0.232 0.130 0.088 0.066 0.946 0.435 0.276 0.200 2.337 0.953 0.584 0.419 6.439 2.317 1.356 0.945 38.926 11.734 5.928 3.729 0.070 0.026 0.016 0.011 0.218 0.075 0.045 0.032 0.461 0.153 0.091 0.065 1.057 0.323 0.185 0.129 4.363 0.943 0.460 0.286
0.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
1.006 0.995 0.991 0.990 1.002 0.991 0.990 0.969 0.997 0.989 0.991 0.990 0.995 0.991 0.992 0.991 0.995 0.995 0.998 0.997 0.992 1.003 1.010 0.994 0.994 0.998 0.998 0.997 0.996 0.999 1.000 1.003 0.999 1.002 1.003 1.005 1.004 1.007 1.008 1.008
0.6% −0.5% −0.9% −1.0% 0.2% −0.9% −1.0% −3.1% −0.3% −1.1% −0.9% −1.0% −0.5% −0.9% −0.8% −0.9% −0.5% −0.5% −0.2% −0.3% −0.8% 0.3% 1.0% −0.6% −0.6% −0.2% −0.2% −0.3% −0.4% −0.1% 0.0% 0.3% −0.1% 0.2% 0.3% 0.5% 0.4% 0.7% 0.8% 0.8%
⎧
60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 60 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30
⎫
(27)
which shows that the position of point (xc, yc) has no influence on the overall moment equilibrium condition of the sliding body. Then, substituting Eqs. (15) and (23) into Eqs. (24), (25), and (27), the following linear equations for the variables λ1, λ2, and λ3 can be obtained: 3
∑ j =1 aij λj = bi where
i = 1, 2, 3
∫ Fs =
(28)
a12 = ∫ τ01dx ; a13 = ∫ τ02 dx ; a21 = ∫ σ 0 dx ; a22 = ∫ τ01s′dx ; ∫ σ 0 (x + s′s) dx ; a32 = ∫ τ01 (−s + s′x ) dx ; a33 = ∫ τ02 (−s + s′x ) dx ; b1 = ∫ (kH w − qx + us′) dx ; b 2 = ∫ [(1 − k V ) w + qy − u] dx ; b3 = ∫ {[(1 − k V ) w + qy + u] x +us′s −
σc cos φi α ⎛ sin φi ⎞ 2 ⎜1 + ⎟ α ⎠ ⎝
⎞α 1 ⎛ σ ⎜mb σ + s⎟ cos δ dx c ⎠ ⎝
∫ τ cos1 δ dx
(29)
where φi, mb, σc, s, and α are as defined in Eq. (1); σ, τ, and φi can be calculated using Eqs. (15), (23), and (19), respectively. In this work, rock dilatancy is not considered in the given nonlinear GHB criterion, so the current method should be used in the framework of associated flow law.
a11 = − ∫ σ 0 s′dx ;
a23 = ∫ τ 02 s′dx ;
5 15 25 35 5 15 25 35 5 15 25 35 5 15 25 35 5 15 25 35 5 15 25 35 5 15 25 35 5 15 25 35 5 15 25 35 5 15 25 35
According to the aforementioned definition of slope FOS (i.e., the ratio of the failure shear force to the actual shear force on the slip surface), the rock slope FOS (Fs) with the nonlinear GHB criterion can be calculated as
∫ [−(−σs′ + τ ) s + (σ + τs′) x] dx
=0
100 100 100 100 70 70 70 70 50 50 50 50 30 30 30 30 10 10 10 10 100 100 100 100 70 70 70 70 50 50 50 50 30 30 30 30 10 10 10 10
mi
3.2. Calculation of the rock slope FOS
∫ ⎨⎩ [(1 − kV ) w + qy + u] x + us′s − kH w s +2 g + qx g⎬⎭ dx −
GSI
a31 =
4. Comparative analyses of slopes examples
1 kH w (s + g) + qx g} dx . 2
Slope example 1: Rock slope height H=20 m and slope angle β=45°; rock destruction is subject to the nonlinear GHB criterion with the following parameters: γ=23.0 kN/m3, GSI=75, mi=15, and σc=0.4γH. Six cases, cases 1–6, are listed by varying D as 0.0, 0.2, 0.4, 0.6, 0.8,
After λ1, λ2, and λ3 are determined using Eq. (28), normal stress σ and shear stress τ can be obtained by substituting them in Eqs. (15) and (23), respectively.
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Fig. 4. FOS curve for stability analysis of the rock slope with nonlinear GHB criterion when D=0.0.
the xy axes coordinate system with the slope toe as the origin, the coordinates of the lower and upper sliding points A and B1 are (0 m, 0 m) and (24 m, 20 m), respectively, and R=30 m for the circular slip surface at the specific position. For the arbitrary curved slip surface, which consists of a plotline with n segments, the x coordinate of the end point in the i-th plotline is xi=ix0 −[(i−1)i]×[nx0−(xB−xA)]/[(n−1) n]+xA, and the corresponding y coordinate is yi=(xi−xA)×tan{2i(θ+δ0)/[n(n+1)]}+yA, where, δ0 is the horizontal inclination angle of the first segment of the plotline and is positive in the clockwise direction; x0 is the width of the first segment of the plotline, θ=arctan[(yB−yA)/(xB−xA)]; xA and xB are the x coordinates of the lower and upper sliding points of the slip surface, respectively; and yA and yB are the y coordinates of the lower and upper sliding points of the slip surface, respectively. When the coordinates of its lower and upper sliding points A and B2 are (0 m, 0 m) and (30 m, 20 m), respectively, the arbitrary curved slip surface at the specific position has the following parameters: δ0=−20°, n=100, and x0= L/n, where L is the horizontal distance between the lower and upper sliding points. Accordingly, three types of shear stresses on the circular and arbitrary curved slip surfaces at the specific positions are drawn (Fig. 3) by using the following parameters: (1) shear stress τ1 calculated using Eq. (20), (2) shear stress τ2 calculated using Eq. (20) after neglecting H(σ−σ0), and (3) shear stress τ3 calculated using Eq. (23). Then, Fig. 3 clarifies the following: (1) τ2 is nearly the same as τ1, thus affirming the feasibility of the H(σ−σ0) hypothesis, and (2) τ3, obtained according to the static equilibrium conditions of the sliding body, and τ1, obtained according to the shear strength reduced by FOS, differ slightly, which
and 1.0 in that order. Two types of slip surfaces, namely circular and arbitrary curved (proposed in43) slip surfaces, are adopted to analyze the slope stability by using LEM; the results obtained using different methods are listed in Table 1. The Swedish method has a simple implicit formula for normal stress on the slip surface, so it can apply the nonlinear GHB criterion to obtain the rock slope stability solution. The methods involving the EMC criterion and the nonlinear GHB criterion are termed Swedish method 1 and Swedish method 2, respectively. Table 1 clarifies the following points: (1) the current method provides smaller results than does the LEM by using the linear EMC criterion (i.e., the Swedish method 1, Simplified Bishop method, Spencer method, and Morgenstern–Price (M–P) method); the maximum difference between the results of the current method and those of the LEM is 6.77%, which shows the feasibility of the current method; (2) the current method gives larger and stricter results than does Swedish method 2 because the former satisfies all static equilibrium conditions of the slope sliding body, and the initial normal stress σ0, which is the same as the normal stress of Swedish method 2, is preferably amended using the current method to compensate for the influence of interslice forces on it; and (3) the current method is practical as it is suitable for an arbitrarily curved slip surface and uses a simple slope FOS calculation formula from the limit equilibrium theory. Moreover, circular and arbitrary curved slip surfaces at specific positions were used to analyze the effect of the neglected H(σ−σ0) for the shear stress hypothesis expressed as Eq. (20) on slope stability. In 181
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Fig. 5. FOS curve for stability analysis of the rock slope with nonlinear GHB criterion when D=0.5.
the same slope angle, the slope height has little effect on slope stability. Therefore, the drawn charts can be used for analyzing rock slope stability with arbitrary slope heights, as verified in the following section. When the slope FOS is less than 1.0, the slope is considered to be unstable; therefore, charts for slope FOS ranging from 1.0 to 5.0 are plotted in Figs. 4–6....
shows that the current method reflects the variation in stability at different points on the slip surface.. Slope example 2: The rock slope is as described by Li et al.,1 with slope height H=25 m, and the rock destruction is subject to the nonlinear GHB criterion with the following parameters: γ=23 kN/m3 and D = 0. Considering the slope FOS as unity, Li et al. 1 calculated various combinations of the slope angle β, GSI, mi, and the dimensionless parameter σc/(γH) by using the lower bound LAM under the nonlinear GHB criterion. To verify the feasibility of this work, the results of the current method were calculated using the parameters given in1 for a circular slip surface. Both sets of results are summarized in Table 2. Results of the lower bound LAM and the current method differed by less than 2%, evidencing the feasibility and reliability of the current method and that the work achieves the limit equilibrium analysis of rock slope with the nonlinear GHB criterion. The LEM is a simple method and is thus preferred by many designers; hence, the current method is expected to be widely applied.
5.2. Application and verification of charts Slope example 3: The rock slope height is H and slope angle is β, and the rock destruction is subject to the nonlinear GHB criterion with the following parameters: γ=23.0 kN/m3, GSI=20, D=0.0, mi=15, and σc=4 MPa. H and β are designed such that rock slope Fs=1.200. From the charts in Fig. 4(b), we determine that σc/γH=0.870, 1.811, 3.788, 8.528, 20.352,45.756, and 92.060 for β=30°, 40°, 50°, 60°, 70°, 80°, and 90°, respectively. Using the obtained σc/γH, we can solve for H and β (Table 3). The corresponding results obtained using different methods are also summarized in Table 3. Table clarifies the following points: (1) using these charts in Figs. 4–6, stability of a rock slope with arbitrary slope heights can be reliably and quickly determined; in addition, slope height and slope angle for a rock slope with certain security requirements can be also designed using these charts; (2) when the slope angle β≥60°, the results obtained using the nonlinear GHB and linear EMC criterions differ by 20% because the critical slip surfaces are SFSS; this is a limitation of the method using the linear EMC strength parameters.
5. Charts for rock slope stability analysis by using nonlinear GHB criterion 5.1. Slope FOS curve for stability analysis of rock slope (GSI = 20) To facilitate the engineering application, a series of charts for the stability analysis of rock slope is drawn using the current method (Figs. 4–6). In these charts, the rock slope height H is assumed to be 20 m, and representative rock parameters subject to the nonlinear GHB criterion are γ=23 kN/m3 and GSI=20. To analyze the rock slope stability, the circular slip surface is adopted. Using σc/(γH) as the xaxis, the slope FOS curves are drawn for the rock slopes with slope angle β=30°, 40°, 60°, 70°, 80°, and 90°; parameter mi=5, 15, and 25; and parameter D=0.0, 0.5, and 1.0. When γ, σc/(γH), and the other rock parameters have equal values for a homogeneous rock slope with
6. Conclusions Using the framework of the limit equilibrium theory, this work established a new method for analyzing rock slope stability with the nonlinear GHB criterion. In this method, three variables are used to assume the stresses on the slip surface, and GHB criterion is expanded using the Taylor series. Then, according to the static equilibrium 182
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Fig. 6. FOS curve for stability analysis of the rock slope with nonlinear GHB criterion when D=1.0. Table 3 Minimum FOS in rock slope example 3. Slope angle β (/°)
Slope height H (/m)
The designed FOS
The calculated minimum FOS Nonlinear GHB criterion
30 40 50 60 70 80 90
199.900 96.018 45.918 20.393 8.545 3.801 1.889
1.200 1.200 1.200 1.200 1.200 1.200 1.200
Linear EMC criterion
The current method
Swedish method 2
Swedish method 1
Simplified Bishop method
Spencer method
M-P method
1.215 1.210 1.203 1.200 1.201 1.201 1.201
1.140 1.141 1.138 1.140 1.148 1.169 1.201
1.146 1.205 1.318 1.497 1.732 1.953 2.097
1.213 1.262 1.357 1.505 1.676 1.818 1.870
1.209 1.255 1.350 1.504 1.735 1.916 2.070
1.209 1.254 1.347 1.503 1.741 1.913 2.069
the FOS derived by the current method is in close agreement with those derived using the traditional LEM and the linear EMC strength parameters instead of the nonlinear GHB criterion; this shows the feasibility of the current method. (2) Unlike the Swedish method using the nonlinear GHB criterion, the current method for the stability analysis of a rock slope with an arbitrarily shaped slip surface can satisfy all static limit equilibrium conditions of sliding body and amend the initial normal stress on slip surface without considering the effect of interslice
conditions of the sliding body, multiple linear equations for the three variables are derived. After the three variables are solved, the stress distributions on slip surface are obtained, and the slope FOS is calculated. Comparing the results of several examples and drawing a series of charts of the FOS curve for the rock slope stability analysis yielded the following conclusions: (1) The results of the current method differ from those of the lower bound LAM with the nonlinear GHB criterion by less than 2%, and
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