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Improved analytical solution for toppling stability analysis of rock slopes
ARTICLE IN PRESS
International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1361–1372 www.elsevier.com/locate/ijrmms
Improved analytical solution for toppling stability analysis of rock slopes C.H. Liua,, M.B. Jaksab, A.G. Meyersc a
Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China School of Civil and Environmental Engineering, University of Adelaide, SA 5005, Australia c Rocktest Consulting, SA 5001, Australia
b
Received 30 August 2007; received in revised form 11 December 2007; accepted 23 January 2008 Available online 7 March 2008
Abstract A solution is developed, based on a limit equilibrium approach, for analysing the toppling stability of rock slopes, which are characterized by blocks whose thickness is significantly smaller than the height of the block at the crest. The method is general in form and is applicable to a range of discontinuity geometries. In this study, the determination of the transition position from toppling to sliding is discussed in detail. It is clarified that the transition position from toppling to sliding should be determined by the frictional characteristics of the block base, for both cases of the friction angle along the dominant discontinuities being greater or less than the angle of the cut slope with the line normal to the dip of the dominant discontinuities. The effect of the angle of the block base with the normal to the dip of the dominant discontinuities, bbr, on the toppling stability is analysed. It is indicated that toppling stability greatly diminishes with increasing bbr, and that the ratio of the increment of the factor of safety to its initial value with bbr ¼ 0 will vary between 10% and 20% as bbr changes from 01 to 751, and 30% and 50% with bbr up to 7101. A large discrepancy occurs between the calculated factor of safety against its actual value when bbr is considered equal to zero, in the case of the block base being non-normal to the dip of the dominant discontinuities. A spreadsheet procedure is presented for facilitating the method and by which several cases of toppling are analysed. The results indicate that the proposed solution represents the asymptotic value of the support force necessary to stabilize the slope against toppling as the slenderness ratio tends to infinity and that, when the slenderness ratio is greater than approximately 15–25, the support force calculated by the proposed solution provides an accurate estimate of the actual value. r 2008 Elsevier Ltd. All rights reserved. Keywords: Rock slopes; Discontinuities; Toppling; Stability analysis; Analytical approach
1. Introduction Toppling is an important failure mode of rock slopes that has attracted much attention in the past several decades, and is particularly relevant to the construction of hydropower plants, expressways, open pits, and other civil engineering works. Rock slopes which are susceptible to toppling failure are those characterized by transverse fractures and a dominant discontinuity set such as bedding or foliation, steeply dipping inward, with a strike nearly parallel to the slope. Since toppling was observed in practice, many researchers [1–10] have made significant Corresponding author. Tel.: +86 27 87198931; fax: +86 27 87197386.
E-mail address: [email protected] (C.H. Liu). 1365-1609/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2008.01.009
contributions to its understanding. Two distinct and principal modes of toppling failure exist, namely flexural toppling and block toppling. This paper concentrates on the latter. Of the toppling research published in the literature, a mathematical solution proposed by Goodman and Bray [2] is widely adopted, which is based on the limit equilibrium method. This so-called ‘step-by-step’ approach has been further developed by others [11–13]. Other numerical techniques, such as the distinct element method (DEM), have been developed and applied in practice [5,14]. However, in some cases, where the thickness of blocks is small enough compared to the height of the block at the crest, the toppling slope can be assumed to be a continuous medium rather than a finite, discrete assemblage of
ARTICLE IN PRESS C.H. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1361–1372
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blocks [15]. Hence, it is possible to adopt a continuous approach to analyse toppling failure under such geometric conditions. Bobet [15] developed a solution to analyse a model dependent on both the block base and the base line normal to the dip of the dominant discontinuities, by establishing ordinary differential equations for rotational equilibrium. Sagaseta et al. [16] extended the continuum solution to incorporate the geometrical condition where the base line is not normal to the dip of the dominant discontinuities. These solutions represent useful advances in the techniques for evaluating block-toppling mechanisms. However, further work is required before they can be used confidently. The present study proposes an alternative solution for analysing toppling using a series of general expressions with consideration of various geometries of discontinuity sets.
of the blocks is greater than 20), the slope can be assumed to act as a continuum. Fig. 2 illustrates the geomechanical model for toppling failure of rock slopes considered as continua, where an orthogonal coordinate system is established with the x- and y-axes being, respectively, perpendicular and parallel to the dip of the dominant discontinuities. Expressions for the shapes of the slope surface and the base line can be, respectively, written as ( tan bgr x þ tan yr L; xpxc ; ys ¼ (1) tan bsr ðL xÞ; x4xc ; and yb ¼ tan yr ðL xÞ,
(2)
Y
2. Geomechanical model Almost all of the investigations of block-toppling failure documented in the literature are based on the limit equilibrium principle, and adopt the same two key assumptions; that is, on the side faces of the blocks, the conditions of limit friction equilibrium are met and the normal forces are applied at the tops of the blocks. Fig. 1 shows a typical geometrical model for the toppling failure of rock slopes, where the slope is bounded by the cut slope, the natural ground and the base line. When the thickness of the blocks, t, is small enough, compared to the height of the block at the crest (Sagaseta et al. [16] proposed that the slenderness ratio of the height of the slope to the thickness
g Natural ground
Cut slope
Base line 90°
H r
sr
L
s X
dx N
Cut slope
s
g
N+dN S
Y h
Natural ground
h b
S+dS hdx sin b
t
hdx cos b
Base line
H Block base
j
br
dx
b
dx
Base line Fig. 1. Typical geometrical model for toppling failure of rock slopes.
X Fig. 2. Geomechanical model for toppling failure: (a) geometrical conditions and (b) relationships of forces applied on a differential block.
ARTICLE IN PRESS C.H. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1361–1372
where bgr, yr and bsr are, respectively, the angles of the natural ground, the base line, and the cut slope with respect to the line normal to the dip of the dominant discontinuities with the forms bgr ¼ bg b, yr ¼ y b, and bsr ¼ bs b; bg is the dip angle of the natural ground; y is the plunge of the base line; bs is the dip angle of the cut slope; b is the plunge of the line normal to the dip of the dominant discontinuities with the form b ¼ 90 bj ; bj is the dip angle of the dominant discontinuities; L is the slope length along the line normal to the dip of the dominant discontinuities; and xc is the abscissa of the slope crest. The height of the blocks can be expressed as: h ¼ ðtan yr tan bgr Þx for blocks above the crest, and h ¼ ðtan bsr tan yr ÞðL xÞ for those blocks below the crest. For a slope whose geometry such as is shown in Fig. 2, the limit equilibrium equations for toppling failure are written as: dN 1 dys 1 þ þ tan fj N ¼ ghðsin bb cos bb sin bbr Þ, dx h dx 2 (3) dN ðcos bbr tan fj sin bbr Þ, dx
(4)
dN ðcos bbr þ tan fj sin bbr Þ, dx
(5)
s ¼ gh cos bb
t ¼ gh sin bb
where N is the normal force between block interfaces, s and t are the normal and shear stresses at the block base, fj is the friction angle at block interfaces, g is the unit weight of the rock mass, bb is the dip angle of the block base, and bbr is the angle of the block base with respect to the line normal to the dip of the dominant discontinuities with the form bbr ¼ bb b. The equilibrium equations for sliding failure are expressed as
3. Analysis 3.1. Assessment of failure modes To simplify the analyses that follow, the following auxiliary parameters are introduced: As ¼ tan bsr tan yr , Ag ¼ tan bgr tan yr , Aj ¼ tan fj tan yr .
(9)
From Eqs. (3)–(5), the solution for toppling above the crest can be integrated to yield: A2g 1 x2 , N ¼ gðsin bb cos bb sin bbr Þ 2 Aj 3Ag
(10)
s ¼ g cos bb Ag ðcos bbr tan fj sin bbr Þðtan bb sin bbr ÞAg x, 1þ Aj 3Ag (11) t ¼ g sin bb Ag ðcos bbr þ tan fj sin bbr Þðtan bb sin bbr ÞAg 1þ x. tan bb ðAj 3Ag Þ (12) When toppling occurs, the shear stress at the block base does not reach the limiting friction condition, defined as t ¼ s tan fb ; otherwise sliding would occur. Assuming this criterion to be applicable, an expression for the angle of the block base with the horizontal plane can be obtained from Eqs. (11) and (12): tan bb o tan fb þ ðtan bb sin bbr ÞAg cos bbr ðtan fj tan fb 1Þ sin bbr ðtan fj þ tan fb Þ . Aj 3Ag (13)
dN ¼ gh cos bb dx tan bb tan fb , sin bbr ðtan fb þ tan fj Þ þ cos bbr ð1 tan fb tan fj Þ
(6) s ¼ gh cos bb sin bbr ðtan bb þ tan fj Þ þ cos bbr ð1 tan bb tan fj Þ , sin bbr ðtan fb þ tan fj Þ þ cos bbr ð1 tan fb tan fj Þ (7) t ¼ gh cos bb tan fb sin bbr ðtan bb þ tan fj Þ þ cos bbr ð1 tan bb tan fj Þ , sin bbr ðtan fb þ tan fj Þ þ cos bbr ð1 tan fb tan fj Þ (8) where fb is the friction angle at the block base.
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The slope will tend to slide rather than topple when the inclination of the block base is larger than the angle defined by the right hand side of Eq. (13). The solution for the support force, P, to inhibit sliding is relatively straightforward and can be obtained from the following expressions: gAs LðL xc Þðsin bb cos bb tan fb Þ P¼ , (14) 2ðcos lbr sin lbr tan fb Þ " # cos bsr cos bgr sinðbsr yr Þ L¼ H, (15) sin bs sin bs sinðbgr yr Þ xc ¼
cos bgr sinðbsr yr Þ H, sin bs sinðbgr yr Þ
(16)
where P is the support force necessary to stabilize the slope; lbr is the angle of the support force with the block base, i.e. lbr ¼ l bb , where l is the plunge of the support force and H is the slope height.
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3.2. General solution
and
The support force necessary to stabilize the slope against toppling is of particular concern in stability analysis. When Eq. (13) is satisfied, the general solution for toppling below the slope crest can be obtained by integrating Eqs. (3)–(5), which yields:
dN T A2s ¼ gðsin bb cos bb sin bbr Þ ðL xÞ dx Aj 3As 3 þ gðsin bb cos bb sin bbr Þ 2 As ðAs Ag ÞðAj As Þ L x ðAj =As Þ3 ðAj 3As ÞðAj 3Ag Þ L xc
N¼
1 A2s gðsin bb cos bb sin bbr Þ ðL xc Þ2 2 Aj 3As " # As Ag L x ðAj =As Þ1 Lx 2 , (17) 3 L xc Aj 3Ag L xc
ðL xÞ.
(21)
The value of xm can be calculated from the following expression with dNS/dx=dNT/dx:
2ðAj 3Ag ÞAs L xm ðAj =As Þ3 ¼ L xc 3ðAs Ag ÞðAj As Þ
dN ðcos bbr tan fj sin bbr Þ, dx
(18)
dN ðcos bbr þ tan fj sin bbr Þ. dx
(19)
s ¼ g cos bb As ðL xÞ t ¼ g sin bb As ðL xÞ
! Aj 3As ðtan bb tan fb Þ 1þ , sin bbr ðtan fb þ tan fj Þ þ cos bbr ð1 tan fb tan fj Þ ðtan bb sin bbr ÞAs
If toppling occurs downslope to the slope toe, Eq. (17) indicates that the support force at the toe will be zero when AjXAs and infinite when AjoAs. In other words, the slope will always be maintained in a state of limit equilibrium when AjXAs but will be unstable when AjoAs. In practice, however, it is impossible for these situations to occur, and hence, there must exist a critical point of xm, where the failure mode changes from toppling to sliding. With respect to Eq. (17), three cases with Aj4As, AjoAs and Aj ¼ As are presented below.
(22)
where xm is the abscissa of the transition point It can be deduced from Eqs. (17)–(19) that Eq. (22) can also be obtained using the limit friction condition at the block base, t ¼ s tan fb , as shown in Fig. 3a. When bb ¼ b, Eq. (22) yields the same expression as that given previously in [16]. This indicates that, for the case of Aj4As, the conditions of dNS/dx ¼ dNT/dx and t ¼ s tan fb are
(+) x = x= tan b ,
3.2.1. Case 1: Aj4As (or fj4bsr) When Aj4As, failure will change from toppling to sliding at the appropriate point of xm, where the normal force, NS, for sliding is equal to that for toppling, NT. That is to say, for the differential block located at the transition position, on which a normal force, N, is exerted by the upper neighbouring toppling block, as shown in Fig. 2(b), the increment of the normal force in the case of sliding is of the same magnitude as that corresponding to toppling, i.e. dNS ¼ dNT. Therefore, the value of xm can be determined by setting the derivatives of Eqs. (6) and (17) equal to each other at the same point. Eqs. (6) and (17), for sliding and toppling, respectively, can be rewritten as: dN S ¼ gAs ðL xÞ cos bb dx
0 xm
xc
L
Abscissa
(+)
x = x=0
,
0
τ
x = x=0
xm
xc
L
x = x-= tan
b
tan bb tan fb sin bbr ðtan fb þ tan fj Þ þ cos bbr ð1 tan fb tan fj Þ
(20)
(−) Abscissa Fig. 3. Stresses of block bases below the crest: (a) Aj4As and (b) AjoAs.
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equivalent to each other and both can be used to determine the transition position of failure modes of the slope. Under the actions of the normal force, Nm, given by Eq. (17) with x ¼ xm, and its own weight, the part of the slope below the location of xm slides as an entire wedge, which is defined as the ‘toe wedge’ as shown in Fig. 4. The sliding equilibrium of the wedge along the direction of the block base is relatively straightforward to obtain and adopts the following form: P¼
1365
Y
Natural ground Cut slope Base line Toe wedge
1 ðW T ðsin bb cos bb tan fb Þ cos lbr sin lbr tan fb þ N m ðcos bbr þ sin bbr tan fb þ sin bbr tan fj cos bbr tan fb tan fj ,
xm
(23)
where WT is the weight of the wedge toe, W T ¼ gAs ðL xm Þ2 =2, and Nm is the normal force between the block interfaces located at the transition point. 3.2.2. Case 2: AjoAs (or fjobsr) When AjoAs, the normal stress at the block bases decreases downslope and will be zero at some point, while toppling continues from the crest to the slope toe. Since the block bases are unable to support a normal tensile stress, the solution is not applicable below this point. Sagaseta et al. [16] suggested that this point represents the position at which the failure mode changes. On first inspection, this appears to provide a method for determining the critical point of xm; that is, by setting the normal stress equal to zero. However, this conclusion is not valid. Fig. 3b presents the distributions of stresses at the block bases below the crest. It is shown that both the normal and shear stresses at the block bases decrease from the crest to the toe and become zero above the toe, but that the shear stress becomes zero before the normal stress does. In the zone from x ¼ xt ¼ 0 to x ¼ xs ¼ 0, there must be a critical point, xm, where the limit friction equilibrium of t ¼ s tan fb at the block base is reached. This indicates that failure mode changes at the point xt¼s tan fb rather than at xs ¼ 0. Therefore, the value of the critical point, xm, can be calculated from Eqs. (18) and (19) by assuming t ¼ s tan fb , with the following result:
X Nm tan j Nm
P
b
Horizontal direction
P
Block base
br
WT
L − xm
Fig. 4. Geomechanical conditions for the toe wedge: (a) location and (b) boundary conditions.
point, xm, where the normal force is still positive, the condition for toppling in Eq. (13) is not met and hence failure changes from toppling to sliding. The support force is again calculated using Eq. (23).
2ðAj 3Ag ÞAs L xm ðAj =As Þ3 ¼ L xc 3ðAs Ag ÞðAj As Þ (
) ðcos bb tan fb þ sin bb ÞðAj 3As Þ 1þ . ½cos bbr ð1 þ tan fj tan fb Þ þ sin bbr ðtan fj tan fb Þðsin bb cos bb sin bbr ÞAs
The corresponding force, Nm, is obtained by Eq. (17) with x ¼ xm. The toe wedge below xm will slide as a whole and it is impossible for it to topple because, at the critical
(24)
3.2.3. Case 3: Aj ¼ As (or fj ¼ bsr) The case where Aj ¼ As is the limit condition of Cases 1 and 2. In this case, the corresponding expression for
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C.H. Liu et al. / International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1361–1372
the normal force between the blocks is obtained from Eq. (17) as: 1 A2s ðL xc Þ2 N ¼ gðsin bb cos bb sin bbr Þ 2 Aj 3As " # As Ag Lx 2 3 . L xc Aj 3Ag
(25)
Assuming dNS/dx ¼ dNT/dx, it is a simple exercise to draw the conclusion that xm ¼ L. The following equation is then obtained: Aj Ag 3 N m ¼ gðsin bb cos bb sin bbr ÞAj ðL xc Þ2 . 4 Aj 3Ag (26) The support force can be calculated from Eq. (23) by setting WT ¼ 0. It should be noted that the sign convention adopted for all of the angles shown in Figs. 2 and 4 is such that positive implies an upward direction and vice versa. From the above analysis, it can be concluded that the failure modes change from toppling to sliding at a certain point when toppling occurs downslope, and that the transition between failure modes can be determined by limit friction equilibrium at the block base; that is, t ¼ s tan fb . It is noted that, in the case of bbro0, the kinematic mechanism of toppling is very different from that presented in this study because the rotation of a block may be restricted by the underlying rock, consequently, the proposed solution is not applicable in this case.
3.3. Spreadsheet implementation A spreadsheet, based on Microsoft Excel, has been developed to facilitate the relationships presented above. The spreadsheet is summarized in Fig. 5. Initial geometric and geotechnical parameters are input and some auxiliary parameters are output in the appropriate coordinates. The toppling or sliding failure modes are then determined by Eq. (13), which is modelled by Spreadsheet 1 (Fig. 5a). All relevant equations used to analyse toppling or sliding are included in Spreadsheet 2 (Fig. 5b). The support force for each of the cases mentioned above is then calculated separately. Depending on which of the four cases below satisfies the boundary conditions, the support force necessary to stabilize the slope against toppling or sliding is automatically determined from the four possible cases using a series of nested ‘‘IF’’ statements, and is equal to: If f1Xf2 (f1 and f2 being defined in Fig. 5(a)), then, P ¼ Psliding else if bsrofj, then P ¼ PCase 1 toppling else if bsr ¼ fj, then P ¼ PCase 3 toppling else P ¼ PCase 2 toppling
The spreadsheet procedure is easily adopted to model the solution presented above. For actual cases, one is concerned with not only the support force for a given factor of safety but also the factor of safety of the rock slope if no support is provided. For these purposes, modified friction coefficients are introduced into the
Input Initial Parameters Dip angle of the cut slope
s(°)
60
Dip angle of the natural ground
g(°)
10
Dip angle of the dominant discontinuities Dip angle of the block base
j(°) b(°)
80 15
Plunge of the base line
(°)
30
Friction angle along the block interfaces
j(°)
Friction angle along the block base
b(°)
25 Atan(tan j / Fs) 30 Atan(tan b / Fs)
Plunge of the support force
(°)
Unit weight of the rock mass
(kN/m3)
25
H(m)
20
Slope height
Angles with the normal to dominant discontinuities
(°)
10
sr(°)
50
gr(°)
0
br(°)
5 20 -25
r(°) rb(°) r(°) Auxiliary parameters
Slope length along the line normal to the dip Abscissa at the crest
L(m) xc(m)
48.629 33.772
25 30
-10
Output Parameters Plunge of the normal to dominant discontinuities
Geometrical Parameters Calculated
-20
As
0.827
Ag
-0.364
Aj
0.102
Determination of Failure Modes f1
0.268
f2
0.622
Failure mode: sliding as f1 ≥ f2; toppling as f1 < f2 Note: f1 = tan b f2 = tan b + (tan b - sin br) Ag
cosbr (tanj tanb − 1) − sinbr (tanj + tanb) Aj − 3Ag
Fig. 5. A spreadsheet procedure for toppling stability analysis. (a) Spreadsheet 1: Input and output parameters and determination of failure modes. (b) Spreadsheet 2: Calculations for the support force and the factor of safety.
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Sliding Failure for f1 ≥ f2 Support force
N /A
P(kN/m)
Toppling Failure for f1 ≥ f2 Case 1: sr < j Abscissa at the transition point
Normal force between block interfaces for x = xm
xm (m)
N/A
B1
-5.350
B2
-0.348
Nm (kN/m)
N/A
B3
N/A
B4
-0.876
P (kN/m)
N/A
Normal force between block interfaces for xm = L
Nm (kN/m)
N/A
Support force
P (kN/m)
N/A
Support force
B
xm = L − (L − xc)B1 2 Nm =
1 2
(sin b − cos b sin br) As − Ag
2
× (L − xc)2 B3 − 3
Aj − 3Ag
As2 Aj − 3As
B4
B3
Case 2: sr = j
B
xm = L − (L − xc)B5 2
Case 3: sr > j Abscissa at the transition point
Normal force between block interfaces for x = xm
xm (m)
41.225
B5
7.411
Nm (kN/m)
728.370 0.498
B6 Support force
P (kN/m)
371.613
Support force
P (kN/m)
371.613
Factor of safety
Fs
Nm =
1 2
(sin b − cos b sin br)
2
× (L − xc)2 B6 − 3
As − Ag Aj − 3Ag
As2 Aj − 3As
B
B6 4
Calculation Results
1.000
Fig. 5. (Continued)
spreadsheets by dividing the initial friction coefficients by the factor of safety. The support force can then be automatically calculated for any given factor of safety and, by adopting Excel’s Solver function, the factor of safety is determined by setting the support force equal to zero. 4. Sensitivity analysis of the angle bbr Previous research in relation to block toppling of rock slopes has generally assumed that the block bases are normal to the dip of the dominant discontinuities (i.e. bbr ¼ 01), whereas this may not always be the case. It is therefore prudent to identify the influence of different values of bbr on toppling stability.
A realistic case is analysed with bs ¼ 601, bg ¼ 101, y ¼ 301, l ¼ 101, g ¼ 25 kN/m3 and H ¼ 20 m, as shown in Fig. 2. The value of bbr varies between 7201. As mentioned previously, once toppling begins for the bbro0 case, the rotation of a block may be restricted by the underlying rock. This situation helps to stabilize the slope, thereby increasing the factor of safety. The result tends to be conservative, and hence the restriction is not considered in the discussion. Fig. 6 shows the relationship between the support force, P, and the angle bbr. It can be seen that P increases significantly with increasing angle bbr. To understand this further, the effect of bbr on the factor of safety is illustrated in Fig. 7. As shown in Figs. 7a and b, where the dip of the dominant discontinuities is equal to 701, it was found that Z
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1000 800 600 400 200 0 -200 -400 -600 -800
600
200
P (kN/m)
P (kN/m)
400
0 -200 -400 -600
-30
-20
-10
0 br (°)
10
20
30
-30
-20
-10
0 br (°)
10
20
30
-10
0 br (°)
10
20
30
j = 25°; b = 30°
Legend: Friction angles: j and b
j = 30°; b = 35° j = 35°; b = 40°
1200 1000 800 600 400 200 0 -200 -400 -600 -30
1500 1250 1000 750 500 250 0 -250 -500
P (kN/m)
P (kN/m)
j = 40°; b = 45°
-20
-10
0 10 br (°)
20
-30
30
-20
Fig. 6. Influence of bbr on support force: (a) bj ¼ 851, (b) bj ¼ 801, (c) bj ¼ 751, and (d) bj ¼ 701.
1.6
2
1.2
1.5
0.8
Fs
2.5
1
0.4
0.5
0.0
0
n
1368
-0.4 -30
3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -30
-20
-10
0 br (°)
10
20
30
-30
-20
Legend: Friction angles, j and b and Dip, j
-10
0 br (°)
10
20
30
j = 25°; b = 30° j = 30°; b = 35° j = 35°; b = 40° j = 40°; b = 45° j = 85° j = 80° j = 75°
-20
-10
0 br (°)
10
20
30
j = 70°
Fig. 7. Influence of bbr on the factor of safety: (a) and (b) for the case of bj ¼ 701, (c) for cases of bj ¼ 851, 801, 751, and 701.
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(defined as the ratio of the increment of the factor of safety, Fs, to its initial value, Fs,0, with bbr ¼ 0, i.e. Z ¼ ðF s F s;0 Þ=F s;0 ) remains relatively constant for different values of fj and fb, although the safety factor increases as fj and fb rise. Fig. 7c shows the relationship between Z and bbr for four cases of bj ¼ 851, 801, 751, and 701. It is observed that the increment of Z will vary by approximately 10–20% when bbr ranges between 01 and 751, and 30–50% when bbr varies by approximately 7101. In addition, the greater the change in bbr, the larger will be the increment of Z. It naturally follows that, when bbr is considered equal to zero, in the case of the block base being oblique to the dominant discontinuity dip, the factor of safety would be significantly different to its actual value. 5. Case studies In order to assess the validity of the proposed solution, the cases described by Sagaseta et al. [16] are analysed using the solution developed above. The results are given in Table 1, where the first four cases are characterized by Aj4As and the latter four cases have the geometrical condition of AjoAs. The results are compared with those obtained using the methods proposed by Goodman and Bray [2] and Sagaseta et al. [16]. It can be seen in Table 1 that, for the cases of Aj4As, the results agree well with those given by Sagaseta et al. approach. However, for the cases where AjoAs, there are large discrepancies between the methods. The discrepancies occur as a result of different approaches used to determine the location of the transition point for the case of AjoAs. As discussed above, the proposed solution determines the location of the transition point by evaluating the limit friction equilibrium at the block base, whereas the Sagaseta et al. method calculates the location by setting the normal stress of the block base equal to zero. It is demonstrated below that, in relation to this geometrical situation, the Sagaseta et al. method is inaccurate. In order to demonstrate this conclusion, the stress distributions at the block bases for the latter four cases in Table 1 are analysed using the Goodman and Bray
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method [2]. Fig. 8 shows the stresses at the block bases below the slope crest. It is assumed that toppling occurs downslope to the toe. Except for the case of G-B2, it is observed that the curves in Fig. 8 have similar shapes to those in Fig. 3b, which are obtained by the proposed method, and that the shear stress at the block bases reaches zero before the normal stress. For a slope with the geometric condition of AjoAs, the normal force between the blocks at the transition position between failure modes, Nm, increases as the abscissa of the transition position, xm , grows larger, but at the same time the effective resisting force of the toe wedge decreases. As a result, the support force must increase. Therefore, for the case of AjoAs, the support force calculated by the approach of Sagaseta et al. is greater than that given by the proposed solution. For the case of G-B2, the continuous method proposed above cannot be applied, as the slenderness ratio, w, defined as the ratio of the height of the block at the slope crest to its thickness, is 3.48 which is very small when compared to those ratios for the other three cases; i.e. 12.47–24.75. As shown in Table 1, this explains why the result given by the proposed solution is greater than that obtained from the Goodman and Bray method. However, the support forces associated with the other three cases, given by the two methods, demonstrate good agreement. Similar phenomena are also observed for the first four cases given in Table 1, where Aj4As. For the three cases of El Haya, G-B1a, and G-B1b, with slenderness ratios of 6.98, 3.96, and 3.96, respectively, the differences between the results given by the proposed solution and those from the Goodman and Bray method are large. Hence, the support force in these cases cannot be predicted by the proposed solution. For the case of Cereixal, with a larger slenderness ratio of 15.33, the results given by the two methods are almost identical. Fig. 9 shows the relationship between the support force calculated by the Goodman and Bray method and the slenderness ratio, for the cases listed in Table 1. It can be seen that the support force increases to an asymptotic value, which is equal to that calculated by the proposed solution and denoted by Plim, as the slenderness ratio increases, and that the proposed solution can predict
Table 1 Comparison between different approaches Case
El Haya G-B1a G-B1b Cereixal Derio G-B2 Paracuellos tunnel River Sil viaduct
Input parameters 3
Proposed solution
Sagaseta et al.
Goodman and Bray
d (m)
H (m)
g (kN/m )
bs (1)
bg (1)
bj (1)
y (1)
l (1)
fj (1)
fb (1)
w
Lxm Lxc
P (kN/m)
Lxm Lxc
P (kN/m)
Lxm Lxc
P (kN/m)
2 10 10 1 1 10 0.2 1
48 92.5 92.5 36 39 105.6 13 25
25 25 25 22 25 25 25 26
55 56.5 56.5 62 65 60 45 72
15 4 4 12 0 0 15 30
65 60 60 55 80 70 76 70
42 35.8 35.8 40 35 45 30 47
0 0 0 15 20 0 0 0
39 38.15 33 28 25 21.45 25 35
39 38.15 33 30 35 21.45 35 35
6.89 3.96 3.96 15.33 23.74 3.48 24.75 12.47
0.18 0.28 0.34 0.26 0.50 0.59 0.32 0.59
139.18 1826.74 5125.59 1480.51 361.61 8000.26 120.92 283.88
0.18 0.28 0.34 0.26 0.33 0.37 0.20 0.4
139.18 1826.74 5125.59 1480.51 1515.65 16175.3 188.17 557.56
0.20 0.32 0.42 0.32 0.47 0.63 0.28 0.55
22.04 21.89 1950.48 1370.30 268.35 279.70 101.33 272.23
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1000
0 -500
σ, τ (kPa)
σ, τ (kPa)
500
-1000 -1500
x m − xc
-2000 -2500 0
5
10 15 x − xc (m)
20
25
xm − xc
0 10 20 30 40 50 60 70 80 90 x − xc (m)
200
400
100 0
-100 -200
xm − x c
-300
200 σ, τ (kPa)
σ, τ (kPa)
900 800 700 600 500 400 300 200 100 0
0 -200
-400
xm − x c
-600
-400
-800 0
4
8 x − xc (m)
12
16
0
5
10
15
x − xc (m)
Fig. 8. Stress distributions along the block bases calculated by the Goodman and Bray method: (a) Debrio, (b) G-B2, (c) Paracuellos tunnel, and (d) River Sil viaduct.
relatively accurately the support force of rock slopes when the slenderness ratio reaches a value, which is shown in Fig. 9 to be approximately 15–25. It can also be observed in Fig. 9 that the Cereixal case results in a particular curve, which is different in nature to that of the other seven cases. That is, the support force in this case is always greater than 1044 kN/m, which is equal to the support force in the case of the slope sliding as a whole. As noted previously, when Eq. (13) is not met, the slope will slide rather than topple. If Eq. (13) and bbXfb are both satisfied, either toppling or sliding failure will occur and the support force against sliding is a minimum. Finally, when bb is less than fb, sliding is impossible, but toppling may occur. For the case of Cereixal, since bb ¼ 351 is greater than fb ¼ 301, and the right hand side of Eq. (13) is 0.7024, which is slightly larger than the left hand side of 0.6997, it is impossible for the slope to remain stable without an externally applied support force. However, for any of the other seven cases, because bb is less than fb, only toppling is possible. It is concluded from the above cases that the proposed solution can accurately predict the upper bound of the support force calculated by the Goodman and Bray method when the slenderness ratio tends to infinity. However, the approach by Sagaseta et al. provides a much higher value for the upper bound in the case of fjobsr, although it gives the same result as that by the proposed solution when fj4bsr. It is also indicated that, when the slenderness ratio is greater than about 15–25, the support
force calculated by the proposed solution can be taken as its actual value. 6. Conclusions This study presents an analytical approach for analysing block toppling of those slopes that can be considered as continua, which extends the continuum solution developed by Bobet and Sagaseta et al. to a wider range of situations, and clarifies that the transition point from toppling to sliding should be determined by the limit friction condition at the block base. For toppling failure to occur, the shear stress at the block base will not exceed the limit friction condition, otherwise sliding will occur. In other words, failure will change from toppling to sliding at the position where the limit friction equilibrium at the block base is met when toppling occurs downslope. This determination for the transition position gives a more precise result than that developed by Sagaseta et al. by setting the normal stress at the block base equal to zero. The likelihood for toppling increases greatly with increasing angle of bbr. The increment of Z (the ratio of the increment of the factor of safety to its initial value, with bbr ¼ 0) will vary between 10% and 20% when bbr increases from 01 to 751, and 30–50% when bbr varies between 7101. The factor of safety would be significantly different to its actual value when bbr is considered equal to zero, in the case of the block base being subvertical to the dip of the dominant discontinuities. It is indicated that it is
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2000
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60
6000
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80
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120
140
80
100
120
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80
100
120
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P = Plim
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G&B
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80
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G&B
80
20
40
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20
P = Plim
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P (kN/m)
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0
0
20
40
60
Fig. 9. Comparison between the proposed solution and the Goodman and Bray method: (a) El Haya, (b) G-B1a, (c) G-B1b, (d) Cereixal, (e) Debrio, (f) G-B2, (g) Paracuellos tunnel, and (h) River Sil viaduct.
significant to incorporate the angle, bbr, into the calculation procedure when analysing the toppling failure of rock slopes. A spreadsheet procedure has been developed for facilitating the proposed solution for the analysis of toppling stability of rock slopes. Several practical cases are analysed by the proposed solution as well as the methods by
Goodman and Bray and Sagaseta et al., and comparisons are made between them. The results indicate that the proposed solution represents an upper bound of that by the Goodman and Bray method as the slenderness ratio tends to infinity, but that the approach by Sagaseta et al. gives a much greater result than the upper bound in the case of fjobsr. The proposed solution can predict reasonably well
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the support force required to prevent toppling in those cases where the slenderness ratio is greater than approximately 15–25. Interested readers who wish to obtain a copy of the spreadsheet may do so by contacting any of the authors. References [1] de Freitas MH, Watters RJ. Some field examples of toppling failure. Ge´otechnique 1973;23(4):495–514. [2] Goodman RE, Bray JW. Toppling of rock slopes. In: Proceedings of the specialty conference rock engineering for foundations and slopes. Boulder, CO: American Society of Civil Engineers; 1977. p. 201–34. [3] Wyllie DC. Toppling rock slope failure examples of analysis and stabilization. Rock Mech 1980;13(2):89–98. [4] Hoek E, Bray JW. Rock slope engineering. London: Institute of Mining and Metallurgy; 1981. [5] Ishida T, Chigira M, Hibino S. Application of the distinct element method for analysis of toppling observed on a fissured slope. Rock Mech Rock Eng 1987;20(4):277–83. [6] Cruden DM. Limits to common toppling. Can Geotech J 1989;26: 737–42.
[7] Aydan O¨, Kawamoto T. The stability of slopes and underground openings against flexural toppling and their stabilisation. Rock Mech Rock Eng 1992;25(3):143–65. [8] Pang SK. Treatment of the tilting failure slope with pre-driving rock bolts. Chin J Rock Mech Eng 1993;12(4):30–44 [in Chinese]. [9] Adhikary DP, Dyskin AV, Jewell RJ, Stewart DP. A study of the mechanism of flexural toppling failure of rock slopes. Rock Mech Rock Eng 1997;30(2):75–93. [10] Nichol SL, Hungr O, Evans SG. Large-scale brittle and ductile toppling of rock slopes. Can Geotech J 2002;39:773–88. [11] Zanbak C. Design charts for rock slopes susceptible to toppling. J Geotech Eng ASCE 1983;109(8):1039–62. [12] Aydan O¨, Shimizu Y, Ichikawa Y. The effective failure modes and stability of slopes in rock mass with two discontinuity sets. Rock Mech Rock Eng 1989;22(3):163–88. [13] Kliche CA. Rock slope stability. Littleton, CO: Society for Mining, Metallurgy, and Exploration (SME); 1999. [14] Lanaro F, Jing L, Stephansson O, Barla G. DEM modelling of laboratory tests of block toppling. Int J Rock Mech Min Sci 1997;34:209–18. [15] Bobet A. Analytical solutions for toppling failure. Int J Rock Mech Min Sci 1999;36:971–80. [16] Sagaseta C, Sa´nchez JM, Can˜izal J. A general analytical solution for the required anchor force in rock slopes with toppling failure. Int J Rock Mech Min Sci 2001;38:421–35.
International Journal of Rock Mechanics & Mining Sciences 45 (2008) 1361–1372 www.elsevier.com/locate/ijrmms
Improved analytical solution for toppling stability analysis of rock slopes C.H. Liua,, M.B. Jaksab, A.G. Meyersc a
Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan 430071, China School of Civil and Environmental Engineering, University of Adelaide, SA 5005, Australia c Rocktest Consulting, SA 5001, Australia
b
Received 30 August 2007; received in revised form 11 December 2007; accepted 23 January 2008 Available online 7 March 2008
Abstract A solution is developed, based on a limit equilibrium approach, for analysing the toppling stability of rock slopes, which are characterized by blocks whose thickness is significantly smaller than the height of the block at the crest. The method is general in form and is applicable to a range of discontinuity geometries. In this study, the determination of the transition position from toppling to sliding is discussed in detail. It is clarified that the transition position from toppling to sliding should be determined by the frictional characteristics of the block base, for both cases of the friction angle along the dominant discontinuities being greater or less than the angle of the cut slope with the line normal to the dip of the dominant discontinuities. The effect of the angle of the block base with the normal to the dip of the dominant discontinuities, bbr, on the toppling stability is analysed. It is indicated that toppling stability greatly diminishes with increasing bbr, and that the ratio of the increment of the factor of safety to its initial value with bbr ¼ 0 will vary between 10% and 20% as bbr changes from 01 to 751, and 30% and 50% with bbr up to 7101. A large discrepancy occurs between the calculated factor of safety against its actual value when bbr is considered equal to zero, in the case of the block base being non-normal to the dip of the dominant discontinuities. A spreadsheet procedure is presented for facilitating the method and by which several cases of toppling are analysed. The results indicate that the proposed solution represents the asymptotic value of the support force necessary to stabilize the slope against toppling as the slenderness ratio tends to infinity and that, when the slenderness ratio is greater than approximately 15–25, the support force calculated by the proposed solution provides an accurate estimate of the actual value. r 2008 Elsevier Ltd. All rights reserved. Keywords: Rock slopes; Discontinuities; Toppling; Stability analysis; Analytical approach
1. Introduction Toppling is an important failure mode of rock slopes that has attracted much attention in the past several decades, and is particularly relevant to the construction of hydropower plants, expressways, open pits, and other civil engineering works. Rock slopes which are susceptible to toppling failure are those characterized by transverse fractures and a dominant discontinuity set such as bedding or foliation, steeply dipping inward, with a strike nearly parallel to the slope. Since toppling was observed in practice, many researchers [1–10] have made significant Corresponding author. Tel.: +86 27 87198931; fax: +86 27 87197386.
E-mail address: [email protected] (C.H. Liu). 1365-1609/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2008.01.009
contributions to its understanding. Two distinct and principal modes of toppling failure exist, namely flexural toppling and block toppling. This paper concentrates on the latter. Of the toppling research published in the literature, a mathematical solution proposed by Goodman and Bray [2] is widely adopted, which is based on the limit equilibrium method. This so-called ‘step-by-step’ approach has been further developed by others [11–13]. Other numerical techniques, such as the distinct element method (DEM), have been developed and applied in practice [5,14]. However, in some cases, where the thickness of blocks is small enough compared to the height of the block at the crest, the toppling slope can be assumed to be a continuous medium rather than a finite, discrete assemblage of
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blocks [15]. Hence, it is possible to adopt a continuous approach to analyse toppling failure under such geometric conditions. Bobet [15] developed a solution to analyse a model dependent on both the block base and the base line normal to the dip of the dominant discontinuities, by establishing ordinary differential equations for rotational equilibrium. Sagaseta et al. [16] extended the continuum solution to incorporate the geometrical condition where the base line is not normal to the dip of the dominant discontinuities. These solutions represent useful advances in the techniques for evaluating block-toppling mechanisms. However, further work is required before they can be used confidently. The present study proposes an alternative solution for analysing toppling using a series of general expressions with consideration of various geometries of discontinuity sets.
of the blocks is greater than 20), the slope can be assumed to act as a continuum. Fig. 2 illustrates the geomechanical model for toppling failure of rock slopes considered as continua, where an orthogonal coordinate system is established with the x- and y-axes being, respectively, perpendicular and parallel to the dip of the dominant discontinuities. Expressions for the shapes of the slope surface and the base line can be, respectively, written as ( tan bgr x þ tan yr L; xpxc ; ys ¼ (1) tan bsr ðL xÞ; x4xc ; and yb ¼ tan yr ðL xÞ,
(2)
Y
2. Geomechanical model Almost all of the investigations of block-toppling failure documented in the literature are based on the limit equilibrium principle, and adopt the same two key assumptions; that is, on the side faces of the blocks, the conditions of limit friction equilibrium are met and the normal forces are applied at the tops of the blocks. Fig. 1 shows a typical geometrical model for the toppling failure of rock slopes, where the slope is bounded by the cut slope, the natural ground and the base line. When the thickness of the blocks, t, is small enough, compared to the height of the block at the crest (Sagaseta et al. [16] proposed that the slenderness ratio of the height of the slope to the thickness
g Natural ground
Cut slope
Base line 90°
H r
sr
L
s X
dx N
Cut slope
s
g
N+dN S
Y h
Natural ground
h b
S+dS hdx sin b
t
hdx cos b
Base line
H Block base
j
br
dx
b
dx
Base line Fig. 1. Typical geometrical model for toppling failure of rock slopes.
X Fig. 2. Geomechanical model for toppling failure: (a) geometrical conditions and (b) relationships of forces applied on a differential block.
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where bgr, yr and bsr are, respectively, the angles of the natural ground, the base line, and the cut slope with respect to the line normal to the dip of the dominant discontinuities with the forms bgr ¼ bg b, yr ¼ y b, and bsr ¼ bs b; bg is the dip angle of the natural ground; y is the plunge of the base line; bs is the dip angle of the cut slope; b is the plunge of the line normal to the dip of the dominant discontinuities with the form b ¼ 90 bj ; bj is the dip angle of the dominant discontinuities; L is the slope length along the line normal to the dip of the dominant discontinuities; and xc is the abscissa of the slope crest. The height of the blocks can be expressed as: h ¼ ðtan yr tan bgr Þx for blocks above the crest, and h ¼ ðtan bsr tan yr ÞðL xÞ for those blocks below the crest. For a slope whose geometry such as is shown in Fig. 2, the limit equilibrium equations for toppling failure are written as: dN 1 dys 1 þ þ tan fj N ¼ ghðsin bb cos bb sin bbr Þ, dx h dx 2 (3) dN ðcos bbr tan fj sin bbr Þ, dx
(4)
dN ðcos bbr þ tan fj sin bbr Þ, dx
(5)
s ¼ gh cos bb
t ¼ gh sin bb
where N is the normal force between block interfaces, s and t are the normal and shear stresses at the block base, fj is the friction angle at block interfaces, g is the unit weight of the rock mass, bb is the dip angle of the block base, and bbr is the angle of the block base with respect to the line normal to the dip of the dominant discontinuities with the form bbr ¼ bb b. The equilibrium equations for sliding failure are expressed as
3. Analysis 3.1. Assessment of failure modes To simplify the analyses that follow, the following auxiliary parameters are introduced: As ¼ tan bsr tan yr , Ag ¼ tan bgr tan yr , Aj ¼ tan fj tan yr .
(9)
From Eqs. (3)–(5), the solution for toppling above the crest can be integrated to yield: A2g 1 x2 , N ¼ gðsin bb cos bb sin bbr Þ 2 Aj 3Ag
(10)
s ¼ g cos bb Ag ðcos bbr tan fj sin bbr Þðtan bb sin bbr ÞAg x, 1þ Aj 3Ag (11) t ¼ g sin bb Ag ðcos bbr þ tan fj sin bbr Þðtan bb sin bbr ÞAg 1þ x. tan bb ðAj 3Ag Þ (12) When toppling occurs, the shear stress at the block base does not reach the limiting friction condition, defined as t ¼ s tan fb ; otherwise sliding would occur. Assuming this criterion to be applicable, an expression for the angle of the block base with the horizontal plane can be obtained from Eqs. (11) and (12): tan bb o tan fb þ ðtan bb sin bbr ÞAg cos bbr ðtan fj tan fb 1Þ sin bbr ðtan fj þ tan fb Þ . Aj 3Ag (13)
dN ¼ gh cos bb dx tan bb tan fb , sin bbr ðtan fb þ tan fj Þ þ cos bbr ð1 tan fb tan fj Þ
(6) s ¼ gh cos bb sin bbr ðtan bb þ tan fj Þ þ cos bbr ð1 tan bb tan fj Þ , sin bbr ðtan fb þ tan fj Þ þ cos bbr ð1 tan fb tan fj Þ (7) t ¼ gh cos bb tan fb sin bbr ðtan bb þ tan fj Þ þ cos bbr ð1 tan bb tan fj Þ , sin bbr ðtan fb þ tan fj Þ þ cos bbr ð1 tan fb tan fj Þ (8) where fb is the friction angle at the block base.
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The slope will tend to slide rather than topple when the inclination of the block base is larger than the angle defined by the right hand side of Eq. (13). The solution for the support force, P, to inhibit sliding is relatively straightforward and can be obtained from the following expressions: gAs LðL xc Þðsin bb cos bb tan fb Þ P¼ , (14) 2ðcos lbr sin lbr tan fb Þ " # cos bsr cos bgr sinðbsr yr Þ L¼ H, (15) sin bs sin bs sinðbgr yr Þ xc ¼
cos bgr sinðbsr yr Þ H, sin bs sinðbgr yr Þ
(16)
where P is the support force necessary to stabilize the slope; lbr is the angle of the support force with the block base, i.e. lbr ¼ l bb , where l is the plunge of the support force and H is the slope height.
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3.2. General solution
and
The support force necessary to stabilize the slope against toppling is of particular concern in stability analysis. When Eq. (13) is satisfied, the general solution for toppling below the slope crest can be obtained by integrating Eqs. (3)–(5), which yields:
dN T A2s ¼ gðsin bb cos bb sin bbr Þ ðL xÞ dx Aj 3As 3 þ gðsin bb cos bb sin bbr Þ 2 As ðAs Ag ÞðAj As Þ L x ðAj =As Þ3 ðAj 3As ÞðAj 3Ag Þ L xc
N¼
1 A2s gðsin bb cos bb sin bbr Þ ðL xc Þ2 2 Aj 3As " # As Ag L x ðAj =As Þ1 Lx 2 , (17) 3 L xc Aj 3Ag L xc
ðL xÞ.
(21)
The value of xm can be calculated from the following expression with dNS/dx=dNT/dx:
2ðAj 3Ag ÞAs L xm ðAj =As Þ3 ¼ L xc 3ðAs Ag ÞðAj As Þ
dN ðcos bbr tan fj sin bbr Þ, dx
(18)
dN ðcos bbr þ tan fj sin bbr Þ. dx
(19)
s ¼ g cos bb As ðL xÞ t ¼ g sin bb As ðL xÞ
! Aj 3As ðtan bb tan fb Þ 1þ , sin bbr ðtan fb þ tan fj Þ þ cos bbr ð1 tan fb tan fj Þ ðtan bb sin bbr ÞAs
If toppling occurs downslope to the slope toe, Eq. (17) indicates that the support force at the toe will be zero when AjXAs and infinite when AjoAs. In other words, the slope will always be maintained in a state of limit equilibrium when AjXAs but will be unstable when AjoAs. In practice, however, it is impossible for these situations to occur, and hence, there must exist a critical point of xm, where the failure mode changes from toppling to sliding. With respect to Eq. (17), three cases with Aj4As, AjoAs and Aj ¼ As are presented below.
(22)
where xm is the abscissa of the transition point It can be deduced from Eqs. (17)–(19) that Eq. (22) can also be obtained using the limit friction condition at the block base, t ¼ s tan fb , as shown in Fig. 3a. When bb ¼ b, Eq. (22) yields the same expression as that given previously in [16]. This indicates that, for the case of Aj4As, the conditions of dNS/dx ¼ dNT/dx and t ¼ s tan fb are
(+) x = x= tan b ,
3.2.1. Case 1: Aj4As (or fj4bsr) When Aj4As, failure will change from toppling to sliding at the appropriate point of xm, where the normal force, NS, for sliding is equal to that for toppling, NT. That is to say, for the differential block located at the transition position, on which a normal force, N, is exerted by the upper neighbouring toppling block, as shown in Fig. 2(b), the increment of the normal force in the case of sliding is of the same magnitude as that corresponding to toppling, i.e. dNS ¼ dNT. Therefore, the value of xm can be determined by setting the derivatives of Eqs. (6) and (17) equal to each other at the same point. Eqs. (6) and (17), for sliding and toppling, respectively, can be rewritten as: dN S ¼ gAs ðL xÞ cos bb dx
0 xm
xc
L
Abscissa
(+)
x = x=0
,
0
τ
x = x=0
xm
xc
L
x = x-= tan
b
tan bb tan fb sin bbr ðtan fb þ tan fj Þ þ cos bbr ð1 tan fb tan fj Þ
(20)
(−) Abscissa Fig. 3. Stresses of block bases below the crest: (a) Aj4As and (b) AjoAs.
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equivalent to each other and both can be used to determine the transition position of failure modes of the slope. Under the actions of the normal force, Nm, given by Eq. (17) with x ¼ xm, and its own weight, the part of the slope below the location of xm slides as an entire wedge, which is defined as the ‘toe wedge’ as shown in Fig. 4. The sliding equilibrium of the wedge along the direction of the block base is relatively straightforward to obtain and adopts the following form: P¼
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Y
Natural ground Cut slope Base line Toe wedge
1 ðW T ðsin bb cos bb tan fb Þ cos lbr sin lbr tan fb þ N m ðcos bbr þ sin bbr tan fb þ sin bbr tan fj cos bbr tan fb tan fj ,
xm
(23)
where WT is the weight of the wedge toe, W T ¼ gAs ðL xm Þ2 =2, and Nm is the normal force between the block interfaces located at the transition point. 3.2.2. Case 2: AjoAs (or fjobsr) When AjoAs, the normal stress at the block bases decreases downslope and will be zero at some point, while toppling continues from the crest to the slope toe. Since the block bases are unable to support a normal tensile stress, the solution is not applicable below this point. Sagaseta et al. [16] suggested that this point represents the position at which the failure mode changes. On first inspection, this appears to provide a method for determining the critical point of xm; that is, by setting the normal stress equal to zero. However, this conclusion is not valid. Fig. 3b presents the distributions of stresses at the block bases below the crest. It is shown that both the normal and shear stresses at the block bases decrease from the crest to the toe and become zero above the toe, but that the shear stress becomes zero before the normal stress does. In the zone from x ¼ xt ¼ 0 to x ¼ xs ¼ 0, there must be a critical point, xm, where the limit friction equilibrium of t ¼ s tan fb at the block base is reached. This indicates that failure mode changes at the point xt¼s tan fb rather than at xs ¼ 0. Therefore, the value of the critical point, xm, can be calculated from Eqs. (18) and (19) by assuming t ¼ s tan fb , with the following result:
X Nm tan j Nm
P
b
Horizontal direction
P
Block base
br
WT
L − xm
Fig. 4. Geomechanical conditions for the toe wedge: (a) location and (b) boundary conditions.
point, xm, where the normal force is still positive, the condition for toppling in Eq. (13) is not met and hence failure changes from toppling to sliding. The support force is again calculated using Eq. (23).
2ðAj 3Ag ÞAs L xm ðAj =As Þ3 ¼ L xc 3ðAs Ag ÞðAj As Þ (
) ðcos bb tan fb þ sin bb ÞðAj 3As Þ 1þ . ½cos bbr ð1 þ tan fj tan fb Þ þ sin bbr ðtan fj tan fb Þðsin bb cos bb sin bbr ÞAs
The corresponding force, Nm, is obtained by Eq. (17) with x ¼ xm. The toe wedge below xm will slide as a whole and it is impossible for it to topple because, at the critical
(24)
3.2.3. Case 3: Aj ¼ As (or fj ¼ bsr) The case where Aj ¼ As is the limit condition of Cases 1 and 2. In this case, the corresponding expression for
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the normal force between the blocks is obtained from Eq. (17) as: 1 A2s ðL xc Þ2 N ¼ gðsin bb cos bb sin bbr Þ 2 Aj 3As " # As Ag Lx 2 3 . L xc Aj 3Ag
(25)
Assuming dNS/dx ¼ dNT/dx, it is a simple exercise to draw the conclusion that xm ¼ L. The following equation is then obtained: Aj Ag 3 N m ¼ gðsin bb cos bb sin bbr ÞAj ðL xc Þ2 . 4 Aj 3Ag (26) The support force can be calculated from Eq. (23) by setting WT ¼ 0. It should be noted that the sign convention adopted for all of the angles shown in Figs. 2 and 4 is such that positive implies an upward direction and vice versa. From the above analysis, it can be concluded that the failure modes change from toppling to sliding at a certain point when toppling occurs downslope, and that the transition between failure modes can be determined by limit friction equilibrium at the block base; that is, t ¼ s tan fb . It is noted that, in the case of bbro0, the kinematic mechanism of toppling is very different from that presented in this study because the rotation of a block may be restricted by the underlying rock, consequently, the proposed solution is not applicable in this case.
3.3. Spreadsheet implementation A spreadsheet, based on Microsoft Excel, has been developed to facilitate the relationships presented above. The spreadsheet is summarized in Fig. 5. Initial geometric and geotechnical parameters are input and some auxiliary parameters are output in the appropriate coordinates. The toppling or sliding failure modes are then determined by Eq. (13), which is modelled by Spreadsheet 1 (Fig. 5a). All relevant equations used to analyse toppling or sliding are included in Spreadsheet 2 (Fig. 5b). The support force for each of the cases mentioned above is then calculated separately. Depending on which of the four cases below satisfies the boundary conditions, the support force necessary to stabilize the slope against toppling or sliding is automatically determined from the four possible cases using a series of nested ‘‘IF’’ statements, and is equal to: If f1Xf2 (f1 and f2 being defined in Fig. 5(a)), then, P ¼ Psliding else if bsrofj, then P ¼ PCase 1 toppling else if bsr ¼ fj, then P ¼ PCase 3 toppling else P ¼ PCase 2 toppling
The spreadsheet procedure is easily adopted to model the solution presented above. For actual cases, one is concerned with not only the support force for a given factor of safety but also the factor of safety of the rock slope if no support is provided. For these purposes, modified friction coefficients are introduced into the
Input Initial Parameters Dip angle of the cut slope
s(°)
60
Dip angle of the natural ground
g(°)
10
Dip angle of the dominant discontinuities Dip angle of the block base
j(°) b(°)
80 15
Plunge of the base line
(°)
30
Friction angle along the block interfaces
j(°)
Friction angle along the block base
b(°)
25 Atan(tan j / Fs) 30 Atan(tan b / Fs)
Plunge of the support force
(°)
Unit weight of the rock mass
(kN/m3)
25
H(m)
20
Slope height
Angles with the normal to dominant discontinuities
(°)
10
sr(°)
50
gr(°)
0
br(°)
5 20 -25
r(°) rb(°) r(°) Auxiliary parameters
Slope length along the line normal to the dip Abscissa at the crest
L(m) xc(m)
48.629 33.772
25 30
-10
Output Parameters Plunge of the normal to dominant discontinuities
Geometrical Parameters Calculated
-20
As
0.827
Ag
-0.364
Aj
0.102
Determination of Failure Modes f1
0.268
f2
0.622
Failure mode: sliding as f1 ≥ f2; toppling as f1 < f2 Note: f1 = tan b f2 = tan b + (tan b - sin br) Ag
cosbr (tanj tanb − 1) − sinbr (tanj + tanb) Aj − 3Ag
Fig. 5. A spreadsheet procedure for toppling stability analysis. (a) Spreadsheet 1: Input and output parameters and determination of failure modes. (b) Spreadsheet 2: Calculations for the support force and the factor of safety.
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Sliding Failure for f1 ≥ f2 Support force
N /A
P(kN/m)
Toppling Failure for f1 ≥ f2 Case 1: sr < j Abscissa at the transition point
Normal force between block interfaces for x = xm
xm (m)
N/A
B1
-5.350
B2
-0.348
Nm (kN/m)
N/A
B3
N/A
B4
-0.876
P (kN/m)
N/A
Normal force between block interfaces for xm = L
Nm (kN/m)
N/A
Support force
P (kN/m)
N/A
Support force
B
xm = L − (L − xc)B1 2 Nm =
1 2
(sin b − cos b sin br) As − Ag
2
× (L − xc)2 B3 − 3
Aj − 3Ag
As2 Aj − 3As
B4
B3
Case 2: sr = j
B
xm = L − (L − xc)B5 2
Case 3: sr > j Abscissa at the transition point
Normal force between block interfaces for x = xm
xm (m)
41.225
B5
7.411
Nm (kN/m)
728.370 0.498
B6 Support force
P (kN/m)
371.613
Support force
P (kN/m)
371.613
Factor of safety
Fs
Nm =
1 2
(sin b − cos b sin br)
2
× (L − xc)2 B6 − 3
As − Ag Aj − 3Ag
As2 Aj − 3As
B
B6 4
Calculation Results
1.000
Fig. 5. (Continued)
spreadsheets by dividing the initial friction coefficients by the factor of safety. The support force can then be automatically calculated for any given factor of safety and, by adopting Excel’s Solver function, the factor of safety is determined by setting the support force equal to zero. 4. Sensitivity analysis of the angle bbr Previous research in relation to block toppling of rock slopes has generally assumed that the block bases are normal to the dip of the dominant discontinuities (i.e. bbr ¼ 01), whereas this may not always be the case. It is therefore prudent to identify the influence of different values of bbr on toppling stability.
A realistic case is analysed with bs ¼ 601, bg ¼ 101, y ¼ 301, l ¼ 101, g ¼ 25 kN/m3 and H ¼ 20 m, as shown in Fig. 2. The value of bbr varies between 7201. As mentioned previously, once toppling begins for the bbro0 case, the rotation of a block may be restricted by the underlying rock. This situation helps to stabilize the slope, thereby increasing the factor of safety. The result tends to be conservative, and hence the restriction is not considered in the discussion. Fig. 6 shows the relationship between the support force, P, and the angle bbr. It can be seen that P increases significantly with increasing angle bbr. To understand this further, the effect of bbr on the factor of safety is illustrated in Fig. 7. As shown in Figs. 7a and b, where the dip of the dominant discontinuities is equal to 701, it was found that Z
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1000 800 600 400 200 0 -200 -400 -600 -800
600
200
P (kN/m)
P (kN/m)
400
0 -200 -400 -600
-30
-20
-10
0 br (°)
10
20
30
-30
-20
-10
0 br (°)
10
20
30
-10
0 br (°)
10
20
30
j = 25°; b = 30°
Legend: Friction angles: j and b
j = 30°; b = 35° j = 35°; b = 40°
1200 1000 800 600 400 200 0 -200 -400 -600 -30
1500 1250 1000 750 500 250 0 -250 -500
P (kN/m)
P (kN/m)
j = 40°; b = 45°
-20
-10
0 10 br (°)
20
-30
30
-20
Fig. 6. Influence of bbr on support force: (a) bj ¼ 851, (b) bj ¼ 801, (c) bj ¼ 751, and (d) bj ¼ 701.
1.6
2
1.2
1.5
0.8
Fs
2.5
1
0.4
0.5
0.0
0
n
1368
-0.4 -30
3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -30
-20
-10
0 br (°)
10
20
30
-30
-20
Legend: Friction angles, j and b and Dip, j
-10
0 br (°)
10
20
30
j = 25°; b = 30° j = 30°; b = 35° j = 35°; b = 40° j = 40°; b = 45° j = 85° j = 80° j = 75°
-20
-10
0 br (°)
10
20
30
j = 70°
Fig. 7. Influence of bbr on the factor of safety: (a) and (b) for the case of bj ¼ 701, (c) for cases of bj ¼ 851, 801, 751, and 701.
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(defined as the ratio of the increment of the factor of safety, Fs, to its initial value, Fs,0, with bbr ¼ 0, i.e. Z ¼ ðF s F s;0 Þ=F s;0 ) remains relatively constant for different values of fj and fb, although the safety factor increases as fj and fb rise. Fig. 7c shows the relationship between Z and bbr for four cases of bj ¼ 851, 801, 751, and 701. It is observed that the increment of Z will vary by approximately 10–20% when bbr ranges between 01 and 751, and 30–50% when bbr varies by approximately 7101. In addition, the greater the change in bbr, the larger will be the increment of Z. It naturally follows that, when bbr is considered equal to zero, in the case of the block base being oblique to the dominant discontinuity dip, the factor of safety would be significantly different to its actual value. 5. Case studies In order to assess the validity of the proposed solution, the cases described by Sagaseta et al. [16] are analysed using the solution developed above. The results are given in Table 1, where the first four cases are characterized by Aj4As and the latter four cases have the geometrical condition of AjoAs. The results are compared with those obtained using the methods proposed by Goodman and Bray [2] and Sagaseta et al. [16]. It can be seen in Table 1 that, for the cases of Aj4As, the results agree well with those given by Sagaseta et al. approach. However, for the cases where AjoAs, there are large discrepancies between the methods. The discrepancies occur as a result of different approaches used to determine the location of the transition point for the case of AjoAs. As discussed above, the proposed solution determines the location of the transition point by evaluating the limit friction equilibrium at the block base, whereas the Sagaseta et al. method calculates the location by setting the normal stress of the block base equal to zero. It is demonstrated below that, in relation to this geometrical situation, the Sagaseta et al. method is inaccurate. In order to demonstrate this conclusion, the stress distributions at the block bases for the latter four cases in Table 1 are analysed using the Goodman and Bray
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method [2]. Fig. 8 shows the stresses at the block bases below the slope crest. It is assumed that toppling occurs downslope to the toe. Except for the case of G-B2, it is observed that the curves in Fig. 8 have similar shapes to those in Fig. 3b, which are obtained by the proposed method, and that the shear stress at the block bases reaches zero before the normal stress. For a slope with the geometric condition of AjoAs, the normal force between the blocks at the transition position between failure modes, Nm, increases as the abscissa of the transition position, xm , grows larger, but at the same time the effective resisting force of the toe wedge decreases. As a result, the support force must increase. Therefore, for the case of AjoAs, the support force calculated by the approach of Sagaseta et al. is greater than that given by the proposed solution. For the case of G-B2, the continuous method proposed above cannot be applied, as the slenderness ratio, w, defined as the ratio of the height of the block at the slope crest to its thickness, is 3.48 which is very small when compared to those ratios for the other three cases; i.e. 12.47–24.75. As shown in Table 1, this explains why the result given by the proposed solution is greater than that obtained from the Goodman and Bray method. However, the support forces associated with the other three cases, given by the two methods, demonstrate good agreement. Similar phenomena are also observed for the first four cases given in Table 1, where Aj4As. For the three cases of El Haya, G-B1a, and G-B1b, with slenderness ratios of 6.98, 3.96, and 3.96, respectively, the differences between the results given by the proposed solution and those from the Goodman and Bray method are large. Hence, the support force in these cases cannot be predicted by the proposed solution. For the case of Cereixal, with a larger slenderness ratio of 15.33, the results given by the two methods are almost identical. Fig. 9 shows the relationship between the support force calculated by the Goodman and Bray method and the slenderness ratio, for the cases listed in Table 1. It can be seen that the support force increases to an asymptotic value, which is equal to that calculated by the proposed solution and denoted by Plim, as the slenderness ratio increases, and that the proposed solution can predict
Table 1 Comparison between different approaches Case
El Haya G-B1a G-B1b Cereixal Derio G-B2 Paracuellos tunnel River Sil viaduct
Input parameters 3
Proposed solution
Sagaseta et al.
Goodman and Bray
d (m)
H (m)
g (kN/m )
bs (1)
bg (1)
bj (1)
y (1)
l (1)
fj (1)
fb (1)
w
Lxm Lxc
P (kN/m)
Lxm Lxc
P (kN/m)
Lxm Lxc
P (kN/m)
2 10 10 1 1 10 0.2 1
48 92.5 92.5 36 39 105.6 13 25
25 25 25 22 25 25 25 26
55 56.5 56.5 62 65 60 45 72
15 4 4 12 0 0 15 30
65 60 60 55 80 70 76 70
42 35.8 35.8 40 35 45 30 47
0 0 0 15 20 0 0 0
39 38.15 33 28 25 21.45 25 35
39 38.15 33 30 35 21.45 35 35
6.89 3.96 3.96 15.33 23.74 3.48 24.75 12.47
0.18 0.28 0.34 0.26 0.50 0.59 0.32 0.59
139.18 1826.74 5125.59 1480.51 361.61 8000.26 120.92 283.88
0.18 0.28 0.34 0.26 0.33 0.37 0.20 0.4
139.18 1826.74 5125.59 1480.51 1515.65 16175.3 188.17 557.56
0.20 0.32 0.42 0.32 0.47 0.63 0.28 0.55
22.04 21.89 1950.48 1370.30 268.35 279.70 101.33 272.23
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1000
0 -500
σ, τ (kPa)
σ, τ (kPa)
500
-1000 -1500
x m − xc
-2000 -2500 0
5
10 15 x − xc (m)
20
25
xm − xc
0 10 20 30 40 50 60 70 80 90 x − xc (m)
200
400
100 0
-100 -200
xm − x c
-300
200 σ, τ (kPa)
σ, τ (kPa)
900 800 700 600 500 400 300 200 100 0
0 -200
-400
xm − x c
-600
-400
-800 0
4
8 x − xc (m)
12
16
0
5
10
15
x − xc (m)
Fig. 8. Stress distributions along the block bases calculated by the Goodman and Bray method: (a) Debrio, (b) G-B2, (c) Paracuellos tunnel, and (d) River Sil viaduct.
relatively accurately the support force of rock slopes when the slenderness ratio reaches a value, which is shown in Fig. 9 to be approximately 15–25. It can also be observed in Fig. 9 that the Cereixal case results in a particular curve, which is different in nature to that of the other seven cases. That is, the support force in this case is always greater than 1044 kN/m, which is equal to the support force in the case of the slope sliding as a whole. As noted previously, when Eq. (13) is not met, the slope will slide rather than topple. If Eq. (13) and bbXfb are both satisfied, either toppling or sliding failure will occur and the support force against sliding is a minimum. Finally, when bb is less than fb, sliding is impossible, but toppling may occur. For the case of Cereixal, since bb ¼ 351 is greater than fb ¼ 301, and the right hand side of Eq. (13) is 0.7024, which is slightly larger than the left hand side of 0.6997, it is impossible for the slope to remain stable without an externally applied support force. However, for any of the other seven cases, because bb is less than fb, only toppling is possible. It is concluded from the above cases that the proposed solution can accurately predict the upper bound of the support force calculated by the Goodman and Bray method when the slenderness ratio tends to infinity. However, the approach by Sagaseta et al. provides a much higher value for the upper bound in the case of fjobsr, although it gives the same result as that by the proposed solution when fj4bsr. It is also indicated that, when the slenderness ratio is greater than about 15–25, the support
force calculated by the proposed solution can be taken as its actual value. 6. Conclusions This study presents an analytical approach for analysing block toppling of those slopes that can be considered as continua, which extends the continuum solution developed by Bobet and Sagaseta et al. to a wider range of situations, and clarifies that the transition point from toppling to sliding should be determined by the limit friction condition at the block base. For toppling failure to occur, the shear stress at the block base will not exceed the limit friction condition, otherwise sliding will occur. In other words, failure will change from toppling to sliding at the position where the limit friction equilibrium at the block base is met when toppling occurs downslope. This determination for the transition position gives a more precise result than that developed by Sagaseta et al. by setting the normal stress at the block base equal to zero. The likelihood for toppling increases greatly with increasing angle of bbr. The increment of Z (the ratio of the increment of the factor of safety to its initial value, with bbr ¼ 0) will vary between 10% and 20% when bbr increases from 01 to 751, and 30–50% when bbr varies between 7101. The factor of safety would be significantly different to its actual value when bbr is considered equal to zero, in the case of the block base being subvertical to the dip of the dominant discontinuities. It is indicated that it is
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G&B
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P = Plim
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P (kN/m)
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0
0
20
40
60
Fig. 9. Comparison between the proposed solution and the Goodman and Bray method: (a) El Haya, (b) G-B1a, (c) G-B1b, (d) Cereixal, (e) Debrio, (f) G-B2, (g) Paracuellos tunnel, and (h) River Sil viaduct.
significant to incorporate the angle, bbr, into the calculation procedure when analysing the toppling failure of rock slopes. A spreadsheet procedure has been developed for facilitating the proposed solution for the analysis of toppling stability of rock slopes. Several practical cases are analysed by the proposed solution as well as the methods by
Goodman and Bray and Sagaseta et al., and comparisons are made between them. The results indicate that the proposed solution represents an upper bound of that by the Goodman and Bray method as the slenderness ratio tends to infinity, but that the approach by Sagaseta et al. gives a much greater result than the upper bound in the case of fjobsr. The proposed solution can predict reasonably well
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the support force required to prevent toppling in those cases where the slenderness ratio is greater than approximately 15–25. Interested readers who wish to obtain a copy of the spreadsheet may do so by contacting any of the authors. References [1] de Freitas MH, Watters RJ. Some field examples of toppling failure. Ge´otechnique 1973;23(4):495–514. [2] Goodman RE, Bray JW. Toppling of rock slopes. In: Proceedings of the specialty conference rock engineering for foundations and slopes. Boulder, CO: American Society of Civil Engineers; 1977. p. 201–34. [3] Wyllie DC. Toppling rock slope failure examples of analysis and stabilization. Rock Mech 1980;13(2):89–98. [4] Hoek E, Bray JW. Rock slope engineering. London: Institute of Mining and Metallurgy; 1981. [5] Ishida T, Chigira M, Hibino S. Application of the distinct element method for analysis of toppling observed on a fissured slope. Rock Mech Rock Eng 1987;20(4):277–83. [6] Cruden DM. Limits to common toppling. Can Geotech J 1989;26: 737–42.
[7] Aydan O¨, Kawamoto T. The stability of slopes and underground openings against flexural toppling and their stabilisation. Rock Mech Rock Eng 1992;25(3):143–65. [8] Pang SK. Treatment of the tilting failure slope with pre-driving rock bolts. Chin J Rock Mech Eng 1993;12(4):30–44 [in Chinese]. [9] Adhikary DP, Dyskin AV, Jewell RJ, Stewart DP. A study of the mechanism of flexural toppling failure of rock slopes. Rock Mech Rock Eng 1997;30(2):75–93. [10] Nichol SL, Hungr O, Evans SG. Large-scale brittle and ductile toppling of rock slopes. Can Geotech J 2002;39:773–88. [11] Zanbak C. Design charts for rock slopes susceptible to toppling. J Geotech Eng ASCE 1983;109(8):1039–62. [12] Aydan O¨, Shimizu Y, Ichikawa Y. The effective failure modes and stability of slopes in rock mass with two discontinuity sets. Rock Mech Rock Eng 1989;22(3):163–88. [13] Kliche CA. Rock slope stability. Littleton, CO: Society for Mining, Metallurgy, and Exploration (SME); 1999. [14] Lanaro F, Jing L, Stephansson O, Barla G. DEM modelling of laboratory tests of block toppling. Int J Rock Mech Min Sci 1997;34:209–18. [15] Bobet A. Analytical solutions for toppling failure. Int J Rock Mech Min Sci 1999;36:971–80. [16] Sagaseta C, Sa´nchez JM, Can˜izal J. A general analytical solution for the required anchor force in rock slopes with toppling failure. Int J Rock Mech Min Sci 2001;38:421–35.