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A rigorous solution for the stability of polyhedral rock blocks
Computers and Geotechnics 90 (2017) 190–201
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Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Research Paper
A rigorous solution for the stability of polyhedral rock blocks Qinghui Jiang a,b,⇑, Chuangbing Zhou b a b
School of Civil Engineering, Wuhan University, Wuhan 430072, PR China School of Civil Engineering and Architecture, Nanchang University, Nanchang 330031, PR China
a r t i c l e
i n f o
Article history: Received 25 December 2016 Received in revised form 8 June 2017 Accepted 18 June 2017
Keywords: Polyhedral rock blocks Block theory Limit equilibrium methods Factor of safety Direction of sliding
a b s t r a c t Block theory has been widely applied to stability analysis of rock engineering due to its clear concept and elegant geometrical theory. For a general block with multiple discontinuity planes, it is assumed that contact is only maintained on a single plane (single-plane sliding) or two intersecting planes (double-plane sliding) in block theory analysis. Since the normal forces and shear resistances acting on the other discontinuity planes are omitted, it can cause unreasonable estimations of block failure modes and incorrect calculation of factors of safety. In this study, a new method is presented that permits to consider the contribution of the normal forces and shear resistances acting on each discontinuity plane to the block stability. The proposed method meets all of the force-equilibrium and moment-equilibrium conditions and provides a rigorous solution for stability of general blocks with any number of faces and any shape. Some typical polyhedral blocks in rock slopes are analyzed using block theory and the proposed method. The results indicate that the traditional block theory may give a misleading conclusion for the predictions of stability and sliding direction of rock blocks when contact occurs on more than two discontinuity planes. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction In a rock slope, there are usually numerous discontinuities of various attitudes and different scales. The spatial intersections of the discontinuities at the free surfaces of the slope may form a series of removable rock blocks with different shapes and sizes. It is necessary to evaluate the stability of these removable blocks to determine the excavation and reinforcement design of the slope. At present, stability analyses of a tetrahedral rock wedge formed by two intersecting discontinuity planes at the crest of a slope are well established in geotechnical literature. The early contributions to the issue arise from the work of Wittke [1], John [2], Londe et al. [3], Chan and Einstein [4], Hoek and Bray [5], and Priest [6]. Recent developments are given by Kumsar et al. [7], Wang and Yin [8], Yeung et al. [9], Chen [10], Jimenez and Sitar [11], Jiang et al. [12], among others. When a rock slope is intersected by more than two discontinuity sets, polyhedral rock blocks with general shapes can be created. Lin et al. [13], Ikegawa and Hudson [14], Jing [15], Lu [16] and Elmouttie et al. [17] developed geometrical identification methods for polyhedral rock blocks. For this type of complex blocks formed by any number of discontinuity planes, the ⇑ Corresponding author at: School of Civil Engineering, Wuhan University, Wuhan 430072, PR China. E-mail address: [email protected] (Q. Jiang). http://dx.doi.org/10.1016/j.compgeo.2017.06.012 0266-352X/Ó 2017 Elsevier Ltd. All rights reserved.
traditional wedge analysis methods are limited. An alternative method is to use block theory proposed by Goodman and Shi [18]. Block theory is a three-dimensional geometrical method with rigorous mathematical deduction that permits analysis of any shape of rock blocks. By means of the theorem of movability of block theory, one is able to analyze the system of joints and other rock discontinuities to find a list of removable blocks. Once the removable blocks are identified, the failure mode of the blocks can be judged according to kinematical admissibility conditions and the factor of safety of the blocks can be calculated by limit equilibrium methods. Since block theory was put forward, it has been widely applied to stability analysis and support design of rock engineering due to its clear concept and elegant geometrical theory [19–26]. However, there are limitations to the failure mode and stability analysis of the traditional block theory. Goodman and Shi [18] took the translational failure mode into account only and neglected the effect of the moments of external forces applied to a rock block. To study the rotational failure mechanism, Mauldon and Goodman [27], Tonon [28], and Tonon and Asadollahi [29] extended block theory to include the rotational failure modes and proposed vector analyses of keyblock rotations. Zhao and Wang [30] developed a lower bound limit method that can consider the sliding mode and rotation effect simultaneously. However, these analyses of incorporating rotational modes are limited to tetrahedral blocks bounded by
Q. Jiang, C. Zhou / Computers and Geotechnics 90 (2017) 190–201
joint planes. For general blocks with multiple discontinuity planes, at most two discontinuity planes are taken into account in stability analysis of block theory. This may lead to incorrect results for stability evaluation of the general blocks. Mauldon and Ureta [31] proposed an energy method for determination of the factor of safety against sliding failure for a special prismatic block with multiple sliding planes that form a cylindrical surface. Zhang et al. [32] presented a limit equilibrium analysis for a general block with more than two joint planes by dividing the block into multiple sub-blocks along the line of intersection of joint planes. In both Zhang’s and Mauldon-Ureta approaches, only force-equilibrium conditions can be satisfied. Sun et al. [33] proposed further an optimization model based on limit equilibrium analysis to evaluate the stability of rock blocks with multiple sliding planes, which meets all force and moment equilibrium conditions. Since a symmetric plane, or sliding direction, needs to be pre-determined in the analysis, the method proposed by Sun et al. is only appropriate for polyhedral blocks with a symmetrical geometry. The behavior of polyhedral blocks with arbitrary joint planes can also be simulated by numerical methods based on the discontinuous media models, such as 3d DDA and 3d DEM [34–36]. However, the determination of the safety factor by the numerical methods involves complex interface contact treatment and time-consuming trial and error calculations using the reduction of the strength parameters. The purpose of this study is to develop a rigorous method for stability assessment of general blocks with any number of faces and any shape. To overcome the limitations described above, a new method is proposed that considers the effect of the normal forces and shear resistances acting on multiple discontinuity planes on the block stability. The proposed method meets all of the force-equilibrium and moment-equilibrium conditions and provides a rigorous solution for stability analysis of general blocks. Finally, some typical polyhedral blocks in rock slopes are analyzed using the block theory and proposed method, and some interesting findings are illustrated by this study. 2. Block theory 2.1. Kinematic analysis Suppose that a removable block is formed by multiple discontinuity planes and the face and upper surface of a slope (Fig. 1). Let ni be the upward unit normal to discontinuity plane i, v i be the unit normal vector of discontinuity plane i pointing to the interior of the block, r represent the active resultant acting on the block, and s represent the direction of movement of the block. There are usually three types of failure modes for the removable block: lifting, single-plane sliding and double-plane sliding. From Goodman and Shi [18], the kinematic admissibility for the block failure can be described as follows: 2.1.1. Lifting When a removable block is lifting (Fig. 1a), the movement direction of the block is parallel to the direction of the active resultant, that is,
s ¼ r=jrj
ð1Þ
For the failure mode of lifting, the kinematic conditions for the block instability are
s v l > 0 for all l
s ¼ si ¼ ðni rÞ ni =jni rj
191
ð3Þ
where si is the orthographic projection of r on plane i. For the mode of single-plane sliding, the kinematic conditions for the block failure are
r vi 6 0
ð4Þ
and
s v l P 0 for all l; l–i
ð5Þ
2.1.3. Double-plane sliding When a removable block slides along discontinuity planes i and j (Fig. 1c), the movement direction of the block is parallel to the intersection line of the two planes, that is,
ni nj sign½ðni nj Þ r jni nj j
s ¼ sij ¼
ð6Þ
For the mode of double-plane sliding, the kinematic conditions for the block failure are
si v j 6 0;
sj v i 6 0
ð7Þ
and
s v l P 0 for all l; l–i or j
ð8Þ
Here, si and sj are the orthographic projections of r on planes i and j, respectively. 2.2. Factor of safety If the failure mode of a removable block is identified by the kinematic admissibility, the factor of safety of the block can be calculated using the limit equilibrium method. This method was originally proposed for tetrahedral blocks by Wittke [1], John [2] and Londe et al. [3], and further extended to the blocks with any number of faces by Goodman and Shi [18]. As shown in Fig. 2, Goodman and Shi introduced a fictitious force Fs to bring a removable block to a limit equilibrium state. The resultant of shear forces acting on joint planes can be written as
Ts ¼
X
N l tan /l s Fs
ð9Þ
l
where N l is the normal force acting on the discontinuity plane l, and /l is the friction angle of discontinuity plane l. For evaluation of the block stability, the conventional definition of safety factor is introduced to bring the block to a limiting state. Further considering the cohesion of joint planes, the resultant of shear forces on discontinuity planes can be rewritten as
Ts ¼
X 1 l
Fs
ðN l tan /l þ cl Al Þs
ð10Þ
where F s is the factor of safety, cl and Al are the cohesion and area of discontinuity plane l. Therefore, the force equilibrium equations for the block can be expressed as
rþ
X Nl v l Ts ¼ 0
ð11Þ
l
ð2Þ
2.1.2. Single-plane sliding When a removable block slides along plane i only (Fig. 1b), the movement direction of the block is parallel to si , that is,
2.2.1. Lifting In this case, all of the discontinuity planes will open and the normal reaction forces N l ¼ 0 for all the discontinuity planes. The factor of safety of the removable block is defined as zero.
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Upper surface
n1 v1
Plane 1
Slope face
n2
v2 r(s)
Plane 2
(a)
Upper surface
Plane 1
v1(n1)
Movement direction of the block
Plane 2
v2(n2)
r
n3
Plane 3
Slope face
n3 r
Plane 3
s
r
v3(n3)
(b) Movement direction of the block
Upper surface
Plane 1
v1(n1)
r
n1
n2
Plane 2
v2(n2) r
Slope face
s
Plane 2
Plane 1
(c) Fig. 1. Failure mode for a removable block: (a) lifting; (b) single-plane sliding; and (c) double-plane sliding.
2.2.2. Single-plane sliding In this case, the removable block maintains contact with joint plane i only and all the other discontinuity planes will open. Since N l ¼ 0 for all l–i, Eq. (11) reduces to
r þ Ni vi
1 ðNi tan /i þ ci Ai Þsi ¼ 0 Fs
ð12Þ
By using the vector operation to Eq. (12), the safety factor of the block can be derived as follows:
Ni ¼ jr ni j T i ¼ jr ni j Fs ¼
Ni tan /i þ ci Ai Ti
ð13Þ
2.2.3. Double-plane sliding In this case, the removable block maintains contact with joint planes i and j and all the other discontinuity planes will open. Since N l ¼ 0 for all l–i or j, Eq. (11) reduces to
r þ Ni v i þ Nj vj
ð16Þ
By using the vector operation to Eq. (16), the safety factor of the block can be derived as follows:
Ni ¼
ð14Þ ð15Þ
1 ðN i tan /i þ ci Ai þ Nj tan /j þ cj Aj Þsij ¼ 0 Fs
Nj ¼
jðr nj Þ ðni nj Þj jni nj j2 jðr ni Þ ðni nj Þj jni nj j2
ð17Þ
ð18Þ
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From Eqs. (15) and (20), the contributions of at most two discontinuity planes to the factor of safety are taken into account. For a removable block with multiple discontinuities, it is assumed that contact is only maintained on a single plane (single-plane sliding) or two intersecting planes (double-plane sliding), and all the other discontinuity planes will open in block theory analysis. In fact, the block is to keep contact with the other discontinuities in a limiting state. With three or more contact planes, if the normal forces and shear resistances acting on the contact planes are omitted, it can cause unreasonable calculation of factors of safety.
r v
3. The proposed stability method for polyhedral blocks 3.1. Basic assumptions l
N
s
lv l
N Fs
an lt
ls
Ts
Fig. 2. Explanation of fictitious force F and movement direction vector s (modified after [13]).
T ij ¼
jr ðni nj Þj jni nj j
ð19Þ
Fs ¼
Ni tan /i þ ci Ai þ Nj tan /j þ cj Aj T ij
ð20Þ
Without loss of generality, Fig. 4a shows the geometry of a typical removable block with multiple discontinuity planes. Illustrated in Fig. 4b is the plan view of the polyhedral block, illustrated in Fig. 4c is a vertical differential column with a base area of dS in the block. Different with the conventional block theory, we assume that the projection of movement direction of the removable block on the x-y plane is unique. The direction of sliding, denoted by b, is defined as an angle calculated anticlockwise from the positive xaxis to the opposite direction of the projection (Fig. 4b). Based on this assumption, the unit vector tl of the shear force acting on the discontinuity plane l can be described as:
tlx ¼ sin c cos b;
t ly ¼ sin c sin b;
tlz ¼ cos c
ð21Þ
where c is the angle between the shear force and the upward vertical direction. Since the direction of the shear force is perpendicular to the unit normal vector v l , we have
v lx sin c cos b þ v ly sin c sin b þ v lz cos c ¼ 0
ð22Þ
By solving Eq. (22), the unit vector of the shear force on the discontinuity plane l can be readily determined as follows:
2.3. Limitations of block theory analysis According to the above-mentioned analyses, the traditional block theory considers translational failure modes only and the factor of safety of the block is obtain using force-equilibrium Eq. (11). This means that the only force-equilibrium conditions are satisfied, but the moment-equilibrium conditions are ignored in the stability analysis of the block. From Eqs. (9) and (10), it has been found that the block theory actually involves an assumption that the direction of shear forces on all the contact planes is unique, which is in the opposite direction to the movement of the block. For a removable block with multiple discontinuities, the directions of shear forces acting on the different discontinuity planes are obviously different, as shown in Fig. 3a. Furthermore, if there is an anti-dip bottom face among the discontinuity planes forming a polyhedral block (Fig. 3b), the block is deemed to be immovable according to the kinematic analysis of block theory. However, a sliding failure for the block may still occur when the sliding force along joint plane 1 is greater than the resisting force on joint plane 2.
T tl ¼ ½ tlx tly tlz ¼ v lz cosb=J v lz sinb=J
where J ¼
ðv lx cosb þ v ly sinbÞ=J
T
ð23Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v 2lz þ ðv lx cos b þ v ly sin bÞ2 .
Now we select an arbitrary differential column in the block for analysis (Fig. 4c). rl v l dS and sl tl dS are the normal force and shear force acting on the column base, respectively. kdw is the gravity of the column. dp is the resultant of inter-column forces acting on the row-interfaces ABFE and DCGH, and dq is the resultant of intercolumn forces acting on the column-interfaces ADHE and BCGF. The force equilibrium equations for the column can be expressed as follows:
rl v l dS þ sl tl dS kdw þ dp þ dq ¼ 0 According to the dot product of the Eq. (24) and
rl
dw dp dq ¼ v lz vl vl dS dS dS
ð24Þ
v l , we obtain ð25Þ
Considering
hv lz dS dw ¼ c Free face
Free face Joint plane 1 Joint plane 1 Joint plane 2
Joint plane 2
(a) Fig. 3. Types of blocks, in two dimensions.
(b)
ð26Þ
and h are the unit weight and height of the differential colwhere c umn, respectively. Substituting Eq. (26) into Eq. (25), we obtain
rl ¼ rext þ rint
ð27Þ
where rext ¼ v c represents the contribution of the gravity acting on the column to rl , rint represents the contribution of the internal 2 lz h
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Upper surface Plane 1 Plane 3 Plane 2 Plane m
Slope face
z
Line of intersection
y x
(a)
y
G
H
Column (plan view)
F
E
β
dq
Failure surface (plan view)
kdw C
dp t dS
D
Sliding direction (Plan view)
l l
l
v l dS
B
A
x
(c)
(b)
Fig. 4. (a) Geometry of a rock block with multiple discontinuity planes; (b) plan view of the failure surface and sliding direction; and (c) forces acting on a differential column in the block.
forces in the block to to
rint
dp þ dq ¼ v l dS
rl and is statically indeterminate, and equals
K ¼ ð1
ð28Þ
According to Eq. (27), we can construct an approximation to in the following way
rl ¼ rext þ nðx; y; kÞ
There are many choices of n. In this study, we use a polynomial function to construct n. Letting the polynomial basis
rl
ð29Þ
where k is an unknown vector of 4-order, k ¼ ð k1 k2 k3 k4 ÞT . n is a function in the horizontal coordinate pair of (x, y) with k. The reason for introducing four unknowns lies in that the six equilibrium conditions for the removable block are able to solve at most six unknowns and the safety factor and sliding direction have occupied two unknowns.
x
y
xy ÞT , we have
nðx; y; kÞ ¼ KT k
ð30Þ
Therefore, the total normal stress on the column base can be approximated by
rl ¼ rext þ KT k
ð31Þ
By the Mohr-Coulomb criterion, the shear stress on the column base is determined as:
sl ¼
ðrl uÞ tan /l þ cl Fs
where u is pore pressure on the base of the column.
ð32Þ
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3.2. Global equilibrium equations for removable blocks Assuming that the sliding surface of the removable block is composed of m discontinuity planes (see Figs. 4 and 5). The global force equilibrium equations can be written in the vector form as:
l
l¼1
rl v l dS þ
m ZZ X l
l¼1
sl tl dS þ f ¼ 0
J3
ð33Þ
J1
where f represents the resultant of external forces acting on the block, including weight, seismic forces and support forces from rock bolts or cables. The global moment equilibrium equations can be written in the vector form as: m ZZ X l
l¼1
ðx v l Þrl dS þ
m ZZ X l
l¼1
ðx tl Þsl dS þ m ¼ 0
ð34Þ
where x represents the coordinates of the center of the column base. m represents the resultant moment of the external forces acting on the block about the coordinate axes. Substituting Eq. (32) into Eqs. (33) and (34), we obtain m ZZ X l
l¼1
m ZZ X
ðF s v l þ tan /l tl Þrl dS þ
tl ðcl u tan /l ÞdS þ F s f ¼ 0 l
l¼1
ð35Þ m ZZ X l
l¼1
m ZZ X
½F s ðx v l Þ þ tan /l ðx tl Þrl dS þ
l¼1
ðx tl Þ ð36Þ
Eqs. (35) and (36) can be rewritten in a more compact form as follows:
gðk; b; F s Þ ¼
l¼1
l
ðF s v 0l þ tan /l t0l Þrl dS þ
m ZZ X l¼1
l
t0l ðcl u
0
tan /l ÞdS þ F s f ¼ 0
ð37Þ
where g denotes the vector of unbalanced forces and moments. v , t 0 and f are three 6-order vectors, defined by
v
xv
;
t0 ¼
t xt
;
0
f ¼
f
0
ð38Þ
m
There are six unknowns, namely k, b, and F s involved in the system (37), which can be efficiently solved by the Newton method. For application of the Newton method, the Jacobian of function gðk; b; F s Þ is needed and defined as:
J¼
h
@g @k1
@g @k2
@g @k3
@g @k4
@g @b
@g @F s
Fig. 6. Geometries of a symmetric rock wedge and pentahedral block considered in Example 1.
Since the six equations in the system (37) are functionally independent, the Jacobian of gðk; b; F s Þ is invertible for any ðk; b; F s Þ. Assuming that values of k, b and F s at the kth step are obtained as kk , bk and F s;k , then their values at the (k + 1)th step are modified as
0
kkþ1
i
ð39Þ
Pl an e1
z
Plane m
x
0
kk
1 ð40Þ
The convergence criterion for k, b and F s can be defined as
f ¼ kðkTkþ1 ; bkþ1 ; F s;kþ1 Þ ðkTk ; bk ; F s;k Þk 6 e where
tdS
vdS
y Fig. 5. A 2d schematic plot for the removable block in the yz plane.
ð41Þ
e is user-specified tolerances.
Based on the above formulations, an iterative algorithm for stability analysis of polyhedral blocks is summarized in the following steps: (1) Divide the polyhedral block into a number of columns. (2) Calculate the unit normal vector v and the area of the column base for all columns. (3) Specify the tolerance e and initial values of F s , b and k. In general, it can be assumed that F s ¼ 10, b ¼ 1 and
(4) (5)
(7)
2 ne Pla
1
B C B C 1 @ bkþ1 A ¼ @ bk A ½Jðkk ; bk ; F s;k Þ gðkk ; bk ; F s;k Þ F s;kþ1 F s;k
(6)
f
x
3.3. Numerical implementation
0
v0 ¼
z
Slope face
l
ðcl u tan /l ÞdS þ F s m ¼ 0
m ZZ X
y
J2
H=50 m
m ZZ X
Upper surface
(8) (9) (10)
k ¼ ½ 0 0 0 0 T for the first iteration. The tolerance parameters is usually specified as 0.001. Set the iteration step k 1. Calculate the unit vector t of the shear stress acting on the column base using Eq. (23). Calculate the normal stress r and shear stress s acting on the column base using Eqs. (31) and (32). Calculate the vector g representing the three unbalanced forces and three unbalanced moments using Eq. (37) for all of columns. Calculate the Jacobian of g using Eq. (39). Obtain the updated values of F s , b and k using Eq. (40). If the convergence criterion (41) is satisfied, then end calculation; otherwise, let k k þ 1 and turn to step (5).
4. Examples Three typical examples involving seven polyhedral rock blocks are analyzed using block theory and the proposed method. The size of columns for discretization of the blocks is 1 m 1 m. The initial values of the unknowns are: F s ¼ 10, b ¼ 1, and k ¼ ½ 0
0
0
0 T .
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Table 1 Orientation of discontinuity planes forming rock blocks in Example 1.
Joint J1 Joint J2 Joint J3 Slope face Upper surface
Dip (degrees)
Dip direction
Friction angle
Cohesion (kPa)
45 45 40 45 10
115 245 180 180 180
20 20 20
30 30 30
Note: Height of slope = 50 m and unit weight of rock = 26 kN/m3.
Table 2 Results for rock blocks in Example 1. Block no.
1 2
Joint planes
J1, J2 J1, J2, J3
Block theory
Proposed method
Factor of safety
Direction of sliding
Factor of safety
Direction of sliding
1.457 1.388
1.571 1.571
1.435 1.188
1.578 1.576
The results indicate that the proposed method has rapid convergence and all examples are convergent within, at most, six iteration steps.
4.1. Example 1 In this example a rock slope with a strike of EW is considered as shown in Fig. 6. The dips of face and upper surface of the slope are 45° and 10°, respectively. The height of the slope is 50 m, and the unit weight of rock is 26 kN/m3. There are three joint planes, J1, J2 and J3, developed in this slope. The attitudes and strength properties of the joint planes are listed in Table 1. Two removable blocks were analyzed in the slope. One is a wedge block formed by J1 and J2, the other is a pentahedral block formed by J1, J2 and J3. For comparison, both the traditional block theory and proposed method are used to evaluate the stability of the two rock blocks. The calculation results are summarized in Table 2. It was found that, for the wedge block involving two sliding surfaces only, the safety factor obtained using two different approaches are in reasonably close agreement. For the pentahedral block with three contact planes, the factor of safety obtained by the block theory is 1.388, whereas that obtained by the proposed method is 1.188. According to kinematic conditions (7) and (8) of block theory, the pentahedral block was identified as doubleplane sliding along J1 and J2. In this case, the removable block maintains only contact with J1 and J2 and joint plane J3 will open. From Eq. (20), the contributions from J1 and J2 to the safety factor are considered, and the effects of J3 are neglected. Since the dip of J3 is greater than the plunge of the line of intersection of J1 and J2, J3 has unfavorable effects on the stability of the pentahedral block. This would lead to overestimation of the stability of the block if J3 acting as the rear sliding surface is neglected. Although the two removable blocks in this example have different geometric configurations, the sliding surfaces forming the two blocks are symmetrical about the y-axis. It can be seen from Table 2 that the sliding directions for both cases are correctly predicted by the block theory and proposed method. The results would be anticipated for a problem with symmetrical failure surfaces. Once Eq. (37) is solved, the normal stress acting on the contact planes could be computed using Eq. (31). Shown in Fig. 7 is the distribution of normal stresses over the contact planes of the pentahedral block. It can be observed that in a limiting state, the normal stresses on J3 are positive and cannot be neglected. For the symmetrical pentahedral block, the distribution of normal stresses takes on a good symmetry, which indicates the results are reasonable.
Fig. 7. Contours of normal stresses over the failure surfaces of the pentahedral block formed by J1, J2 and J3.
4.2. Example 2 The face of a rock slope shown in Fig. 8 has a dip of 60° and a dip direction of 180°. The upper surface of the slope is horizontal. The height of the slope is 76 m, and the unit weight of rock is 26 kN/m3. There are three joint planes, J1, J2 and J3, developed in this slope. The attitudes and strength properties of the joint planes are listed in Table 3. Two removable blocks were analyzed in the slope. One is an asymmetric wedge block formed by J1 and J2, the other is an asymmetric pentahedral block formed by J1, J2 and J3. Both block theory and the proposed method are employed to evaluate the stability of the two polyhedral blocks. The calculation results are summarized in Table 4. It can be observed that, for the asymmetric wedge block, the safety factor and sliding direction obtained using two different approaches are in reasonably close agreement. For the asymmetric pentahedral block, the value of safety factor from block theory is significantly smaller than that from the proposed method. Furthermore, the sliding direction obtained by block theory is obviously incorrect because the failure surfaces forming the pentahedral block is laterally asymmetrical about the y-axis. The cause for the block theory yielding a lower safety factor and an incorrect sliding direction lies in that, the block theory placed the failure mode of the pentahedral block as single-plane sliding along J3 and neglected the contributions of J1 and J2 to the stability of the block. In fact, the areas of J1, J2 and J3 forming the block are 2118 m2, 1868 m2 and 374 m2, respectively. According to the calculation results from the proposed rigorous method, the stability and sliding direction of the pentahedral block are mainly controlled by joint planes J1 and J2.
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4.3. Example 3
J2
J1
J3
J3
z y
x
Fig. 8. Geometries of an asymmetric rock wedge and two pentahedral blocks considered in Example 2.
Table 3 Orientation of discontinuity planes forming rock blocks in Example 2. Surface
Dip (degrees)
Dip direction
Friction angle
Cohesion (kPa)
Joint J1 Joint J2 Joint J3/J30 Slope face Upper surface
50.1 63.0 25.2/5 60 0
125 233 180/0 180 0
30 30 20
50 50 30
The face of a rock slope shown in Fig. 10 has a dip of 60° and a dip direction of 180°. The upper surface of the slope is horizontal. There are four joint planes, J1, J2, J3 and J4, developed in this slope. The attitudes and strength properties of the joint planes are listed in Table 5. It can be observed from Fig. 10 that the four joint planes in this slope intersect to form a symmetrical hexahedral block. For this specified block, the intersection line s13 of planes J1 and J3 is parallel to the intersection line s23 of planes J2 and J3. Assuming that gravity is the only contributor to the active resultant r acting on the block, the projection s3 of r on plane J3 is also parallel to s13 and s23 . According to the kinematic analysis of block theory, the failure mode of the block can be identified as single-plane sliding along J3, or double-plane sliding along J1 and J3 or J2 and J3. Obviously, the block is singular in mathematics from the analysis of block theory. To avoid the singularity, we slightly adjust J1 (124.7°\45.4°) and J2 (235.3°\45.4°) to J10 (127.3°\44.6°) and J20 (232.7°\44.6°), respectively. Using block theory, the block formed by J10 , J20 , J3 and J4 is identified as single-plane along plane J3. The factor of safety of the block can be determined as 0.71 by Eq. (15). For comparison, the two hexahedral blocks involved in this example are also analyzed using the proposed method. The calculation results are listed in Table 6. For the two symmetrical blocks, the calculated sliding directions agree well with the direction of
Note: Height of slope = 76 m and unit weight of rock = 26 kN/m3.
Table 4 Results for rock blocks in Example 2. Block no.
Joint planes
Block theory
Proposed method
Factor of safety
Direction of sliding
Factor of safety
Direction of sliding
1 2 3
J1, J2 J1, J2, J3 J1, J2, J30
1.325 0.805 –
1.761 1.571 –
1.309 1.280 1.349
1.772 1.744 1.764
Now we specify the orientation of J3 as J30 (0°\5°), as shown in Fig. 8. Due to joint plane J30 dipping into the face of the slope, the pentahedral block formed by J1, J2 and J30 is immovable according to the traditional block theory. From a practical point of view, the block still has the risk of failure if the sliding force along J1 and J2 is greater than the resisting force on J30 . Using the proposed method, the safety factor and sliding direction of the block can be determined as 1.349 and 1.764. The results demonstrate that for the block with an anti-dip joint plane, J1 and J2 are still main factors contributing to the stability and sliding direction of the failure mass. Obviously, the block theory highly overestimates the stability of the block due to neglecting the contributions of J1 and J2. Illustrated in Fig. 9a and b are the distributions of normal and shear stresses on the joint planes of the block, respectively. It can be observed that the normal stresses are positive and satisfy the physical admissible condition on failure surfaces. Although the maximum normal stress occurs on the joint plane J30 , the shear stress on J30 is not the maximum due to the effect of strength properties. This indicates that the slide-resistance action of J30 is not remarkable although J30 dips into the slope face.
Fig. 9. Contours of stresses over the failure surfaces of the pentahedral block formed by J1, J2 and J30 : (a) normal stresses; (b) shear stresses.
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5. Discussion
J2
J1
5.1. Effects of discretized column size
J4
J2
J1 J3
z y
x
Fig. 10. Geometries of two symmetric hexahedral blocks considered in Example 3.
Table 5 Orientation of discontinuity planes forming rock blocks in Example 3.
Joint J1/J10 Joint J2/J20 Joint J3 Joint J4 Slope face Upper surface
Dip (degrees)
Dip direction
Friction angle
Cohesion (kPa)
45.4/44.6 45.4/44.6 30 45 60 0
124.7/127.3 235.3/232.7 180 180 180 0
30 30 20 30
100 100 50 100
Note: Height of slope = 50 m and unit weight of rock = 26 kN/m3.
In order to examine the effects of the size of columns on solution accuracy, various column sizes are adopted for analysis. For Example 1, seven different column sizes are employed to discretize the polyhedral blocks. Fig. 12 shows the factor of safety and sliding direction versus the number of columns relationships. It can be observed that the safety factor and sliding direction approach the corresponding steady values with increasing numbers of columns, or equivalently, decreasing sizes of columns. This indicates that there is a reasonable convergence trend for the solutions of the proposed method with decreasing column size. For Examples 2 and 3, the evolution trends of solutions with the size of columns are similar to that of Example 1. For illustration, Fig. 13 shows variation of the calculated factors of safety and sliding directions with the number of columns for Example 2. Assuming the convergence value is the exact solution, Fig. 14 shows variation of relative error with the discretized column numbers for the three examples involving seven polyhedral blocks. It can be observed that, for these examples, the proposed method has an enough precision if the total number of columns for discretization of the polyhedral blocks is greater than 1000. 5.2. Effects of approximation functions to the normal stress To investigate the effects of different approximation functions to the normal stress on solution precision, three approximation
Table 6 Results for rock blocks in Example 3. Block no.
Joint planes
Block theory Factor of safety
Direction of sliding
Factor of safety
Direction of sliding
1
J1, J2, J3, J4 J10 , J20 , J3, J4
0.711a/ 0.871b 0.710
1.571
1.297
1.570
1.571
1.292
1.571
2
Proposed method
Note: a- single-plane sliding; b- double-plane sliding.
the symmetrical axis. Compared to the first block, the second block has only a very slight change in the geometry and volume. Consequently, the safety factors of the two blocks obtained using the proposed method are in reasonably close agreement. By comparing the results listed in Table 6, it was found that the stability of the two polyhedral blocks are significantly underestimated by block theory. The main reasons lie in that the traditional block theory can take into account the contribution of two sliding planes to the block stability at most. Furthermore, it is not necessary to determine the failure mode of the failure mass in advance in the proposed method. The singular problem in the analysis of block theory can be effectively overcome. Shown in Fig. 11a and b are the distributions of normal and shear stresses over the failure surfaces of the hexahedral block formed by J10 , J20 , J3 and J4, respectively. Since the block is characterized by the symmetry of geometry of discontinuity planes, the distributions of normal and shear stresses show also a good symmetry. Although the maximum normal stresses occur on plane J3, the maximum shear stresses exist in planes J10 , J20 and J4. This demonstrates that the resisting forces acting on J10 , J20 and J4 help stabilize the removable block. Because the shear resistances on J10 , J20 and J4 are omitted in block theory analysis, the stability of the block is significantly underestimated by block theory.
Fig. 11. Contours of stresses over the failure surfaces of the hexahedral block formed by J10 , J20 , J3 and J4: (a) normal stress; (b) shear stress.
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1.38
1.60
1.50
1.52
1.45 1.40
3
7
1.35
1
4 5
1.30
2
1.48
Factor of safety Size of columns
Direction of sliding
1.44
8
1.75
1.34 1.32
1.70
1.30
2 3 4 5 Size of columns 6
1.28 1.26 1.24
1.65 Factor of safety Direction of sliding
1.60 1.55
2000
1.40 0
0.5
0.6
1.22 0
1.25
0.7
1
Direction of sliding
1.56
Factor of safety
Factor of safety
1.55
1.80
1.36
Direction of sliding
1.60
1000 2000 3000 4000 5000 6000 7000 8000 9000
4000 6000 8000 Number of columns
Number of columns
10000
(a)
(a) 1.35
1.18 3 7
1.14
Fcator of safety
Size of columns
Direction of sliding
8
1.10 0
1.52 1.48
4 5
1.12
1
2
1000
2000
3000
4000
5000
6000
1.44
1.71
1.29 1.26 4 5 6
1.23
0.6
0.7
1 2 3
0.5
Factor of safety Size of columns
1.68
Direction of sliding
Direction of sliding
1.56
1.20
Factor of safety
Factor of safety
1.22
1.32
Direction of sliding
1.24
1.16
1.74
1.60
1.65
1.20 0
2000
1.40 7000
4000
6000
8000
10000
Number of columns
(b)
Number of columns
(b)
ð42Þ
nðx; y; kÞ ¼ ðk1 þ k2 x þ k3 y þ k4 xyÞrext
ð43Þ
Taking Example 1 in the following analysis as an instance. For the two cases in Example 1, the size of columns for discretisation of the blocks is set to 1 m 1 m. The results from the three approximation functions are listed in Table 7. Although we construct different functions to approximate rl , the calculated factors of safety and direction of sliding are in reasonably close agreement because the normal and shear forces obtained using different approximation functions must meet all of force and moment equilibrium conditions in the present method. It can be also seen from Table 7 that the solutions are convergent within five to seven iterations corresponding to different approximation functions. This indicates that the present method has rapid convergence and is suitable to use in practice.
1.38
1.75
1.35
1 2 3 4 5 Size of columns 6
1.32 1.29 1.26
0
2000
To examine the numerical performance of the proposed method, the initial iteration value of the factor of safety are set to be very far from the real solution and the precision of convergence is set to be enough high. Letting F s;0 ¼ 100, b0 ¼ 1, k0 ¼ ½ 0 0 0 0 T and e ¼ 109 , all seven blocks in the abovementioned three examples are used for re-analysis. Fig. 15a–c shows the iteration process of the calculated factor of safety, direction of sliding and convergence precision, respectively. Although the initial iteration values are far from the real solutions, only six to seven iterations are required to reach a convergence precision less than 109 . This indicates that the proposed method has a high
1.74
Factor of safety Direction of sliding
1.73 1.72
6000
8000
10000
Number of columns (c) Fig. 13. Variation of calculated factors of safety and directions of sliding with column size: (a) Case 1, Example 2; (b) Case 2, Example 2; and (c) Case 3, Example 2.
Table 7 Results from different approximation functions for blocks in Example 1.
Wedge block
Pentahedral block
5.3. Numerical performance of the proposed method
4000
0.5
0.6
0.7
Direction of sliding
nðx; y; kÞ ¼ k1 x þ k2 y þ k3 x2 þ k4 y2
1.76
1.41
Factor of safety
functions are adopted for analysis. Except Eq. (30), the other two approximation functions are
1.77
1.44
Fig. 12. Variation of calculated factors of safety and directions of sliding with column size: (a) Case 1, Example 1; (b) Case 2, Example 1.
Safety factor Sliding direction Iteration times Safety factor Sliding direction Iteration times
Eq. (30)
Eq. (42)
Eq. (43)
1.435 1.578 6 1.188 1.576 5
1.435 1.578 5 1.187 1.576 7
1.431 1.582 5 1.183 1.578 5
convergence precision and a fast convergence speed. It can be also observed from Fig. 15c that when the tolerance is specified as 103 , the calculation results for the seven cases are convergent within, at most, six iteration steps. If the tolerance is set to be lower than 103 , the increase of iteration steps to reach convergence is not significant for the present method. However, a tolerance of 103 is generally acceptable and a higher precision seems not necessary for practical engineering problems.
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0.10
100
Relative error
0.06
Factor of safety
Example 1(Case 1) Example 1(Case 2) Example 2(Case 1) Example 2(Case 2) Example 2(Case 3) Example 3(Case 1) Example 3(Case 2)
0.08
0.04 0.02
Example 1 (Case 1) Example 1 (Case 2) Example 2 (Case 1) Example 2 (Case 2) Example 2 (Case 3) Example 3 (Case 1) Example 3 (Case 2)
10
0.00 100
1000
1
10000
0
1
2
3
Number of columns
4
5
6
7
8
Iteration step
(a)
(a) 2.2
0.05
0.03
Direction of sliding
0.04
Relative error
2.0
Example 1(Case 1) Example 1(Case 2) Example 2(Case 1) Example 2(Case 2) Example 2(Case 3) Example 3(Case 1) Example 3(Case 2)
0.02 0.01
1.8 1.6 1.4
Example 1 (case 1) Example 1 (case 2) Example 2 (case 1) Example 2 (case 2) Example 2 (case 3) Example 3 (case 1) Example 3 (case 2)
1.2 1.0
0.00
0.8 100
1000
0
10000
1
2
3
Number of columns
4
5
6
7
8
Iteration step
(b)
(b)
Fig. 14. Variation of relative errors in safety factor and sliding direction with the number of columns: (a) safety factor; (b) sliding direction.
1000
The computational efficiency is an important part for this proposed method to find eventually wide application. The computational costs are measured by the time of evaluation before convergence. The iteration algorithm was run on an Apple laptop with Inter(R) Core(TM) i7-4870HQ CPU @ 2.50 GHz 2.50 GHz. The computational time for all seven blocks in this study is ranging from 6 ms to 25 ms, which demonstrates the proposed method has a high computation efficiency. 6. Conclusion
Convergence precision
10 0.1 1E-3 1E-5 Example 1 (case 1) Example 1 (case 2) Example 2 (case 1) Example 2 (case 2) Example 2 (case 3) Example 3 (case 1) Example 3 (case 2)
1E-7 1E-9 1E-11 1E-13 1E-15
For general polyhedral blocks formed by multiple discontinuity planes, at most two planes can be considered in the stability analysis of block theory. The study presents a rigorous solution to stability of the polyhedral blocks that remove the limitations of the block theory method. Based on this work, the following conclusions can be drawn: (1) By introducing an assumed normal stress distribution on contact planes, the proposed method can effectively consider the effects of the normal force and shear resistances on each discontinuity plane on the block stability. Furthermore, the proposed method satisfies all of the force-
1
2
3
4
5
6
7
8
Iteration step
(c) Fig. 15. (a) Calculated factor of safety versus iteration step number; (b) Calculated direction of sliding versus iteration step number; and (c) Convergence parameter versus iteration step number.
equilibrium and moment-equilibrium conditions, which provides a rigorous solution for stability assessment of general removable blocks.
Q. Jiang, C. Zhou / Computers and Geotechnics 90 (2017) 190–201
(2) The proposed method can not only calculate the factor of safety, but also accurately predict the direction of sliding of the removable block. This provides an insight for the design optimization of block reinforcement where the optimal trend of bolting forces is in a direction opposite to that of the sliding direction of the block. (3) The results from the three examples involving seven polyhedral blocks indicate that this proposed method is applicable to stability assessment of polyhedral blocks in rock slopes. For tetrahedral rock wedges, the calculated factors of safety and sliding directions from the proposed method and block theory are in reasonably close agreement. However, for polyhedral rock blocks with three or more contact planes, the traditional block theory may give misleading conclusions for the predictions of stability and sliding direction. (4) It is not necessary to determine the failure mode of polyhedral blocks in advance in the proposed method. Therefore, the singular problem caused by failure mode change in the analysis of block theory can be effectively overcome.
Acknowledgements The work reported in this paper has received financial support from the National Natural Science Foundation of China (Nos. 51679173) and Natural Science Foundation of Hubei Province, China (No. 2016CFA083). This support is gratefully acknowledged. References [1] Wittke WW. Method to analyze the stability of rock slopes with and without additional loading (in German). Felsmechanick und Ingenieurgeolgie 1965;30 (Supp. II):52–79. [2] John KW. Graphical stability analysis of slopes in jointed rock. J Soil Mech Found Div (ASCE) 1968;94(2):497–526. [3] Londe P, Vigier G, Vormeringer R. Stability of rock slopes, a three-dimensional study. J Soil Mech Found Div (ASCE) 1969;95(1):235–62. [4] Chan HC, Einstein HH. Approach to complete limit equilibrium analysis for rock wedges – the method of ‘‘artificial supports”. Rock Mech 1981;14 (2):59–86. [5] Hoek ET, Bray JW. Rock slope engineering. 3rd ed. London, England: Inst of Min and Metallurgy; 1981. [6] Priest SD. Hemispherical projection methods in rock mechanics. London: George Allan and Unwin; 1985. [7] Kumsar H, Aydan Ö, Ulusay R. Dynamic and static stability assessment of rock slopes against wedge failures. Rock Mech Rock Eng 2000;33(1):31–51. [8] Wang YJ, Yin JH. Wedge stability analysis considering dilatancy of discontinuities. Rock Mech Rock Eng 2002;35(2):127–37. [9] Yeung MR, Jiang QH, Sun N. Validation of block theory and threedimensional discontinuous deformation analysis as wedge stability analysis methods. Int J Rock Mech Min Sci 2003;40:265–75.
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[10] Chen ZY. A generalized solution for tetrahedral rock wedge stability analysis. Int J Rock Mech Min Sci 2004;41:613–28. [11] Jimenez-Rodriguez R, Sitar N. Rock wedge stability analysis using system reliability methods. Rock Mech Rock Eng 2007;40(4):419–27. [12] Jiang QH, Liu XH, Wei W, Zhou CB. A new method for analyzing the stability of rock wedges. Int J Rock Mech Min Sci 2013;60:413–22. [13] Lin D, Fairhurst C, Starfield AM. Geometrical identification of three dimensional rock block systems using topological techniques. Int J Rock Mech Min Sci Geomech Abstr 1987;24(6):331–8. [14] Ikegawa Y, Hudson JA. A novel automatic identification system for threedimensional multi-block systems. Eng Comp 1992;9:169–79. [15] Jing L. Block system construction for three-dimensional discrete element models of fractured rocks. Int J Rock Mech Min Sci 2000;37:645–59. [16] Lu J. Systematic identification of polyhedral rock blocks with arbitrary joints and faults. Comput Geotech 2002;29:49–72. [17] Elmouttie M, Poropat G, Krähenbühl G. Polyhedral modelling of rock mass structure. Int J Rock Mech Min Sci Geomech Abstr 2010;47(4):544–52. [18] Goodman RE, Shi GH. Block theory and its application to rock engineering. London: Prentice-Hall; 1985. [19] Mauldon M. Keyblock probabilities and size distributions: a first model for impersistent 2-D Fractures. Int J Rock Mech Min Sci Geomech Abstr 1995;32 (6):575–83. [20] Lee IM, Park JK. Stability analysis of tunnel keyblock: a case study. Tunnelling Underground Space Technol 2000;15(4):453–62. [21] Hatzor YH. Keyblock stability in seismically active rock slopes—snake path cliff, Masada. J Geotech Geoenviron Eng (ASCE) 2003;129(8):697–710. [22] Goodman RE, Powell C. Investigations of blocks in foundations and abutments of concrete dams. J Geotech Geoenviron Eng (ASCE) 2003;129(2):105–16. [23] Jimenez-Rodriguez R, Sitar N. Influence of stochastic discontinuity network parameters on the formation of removable blocks in rock slopes. Rock Mech Rock Eng 2008;41(4):563–85. [24] Yarahmadi-Bafghi AR, Verdel T. The probabilistic key-group method. Int J Numer Anal Methods Geomech 2004;28(11):899–917. [25] Menéndez-Díaz A, González-Palacio C, Álvarez-Vigil AE, González-Nicieza C, Ramírez-Oyanguren P. Analysis of tetrahedral and pentahedral key blocks in underground excavations. Comput Geotech 2009;36(6):1009–23. [26] Kulatilake PHSW, Wang LQ, Tang HM, Liang Y. Evaluation of rock slope stability for Yujian River dam site by kinematic and block theory analyses. Comput Geotech 2011; 38(?):-. [27] Mauldon M, Goodman RE. Vector analysis of keyblock rotation. J Geotech Geoenviron Eng (ASCE) 1996;122(12):976–87. [28] Tonon F. Generalization of Mauldon’s and Goodman’s analysis of keyblock rotations. J Geotech Geoenviron Eng (ASCE) 1998;124(10):913–22. [29] Tonon F, Asadollahi P. Validation of general single rock block stability analysis (BS3D) for wedge failure. Int J Rock Mech Min Sci 2008;45:627–37. [30] Zhao J, Wang J. Lower bound limit analysis of wedge stability using block element method. Comput Geotech 2017;86:120–8. [31] Mauldon M, Ureta J. Stability analysis of rock wedges with multiple sliding surfaces. Geotech Geol Eng 1996;14:51–66. [32] Zhang QH. Block-dividing limit equilibrium analysis method with multiple sliding planes. Chin J Rock Mech Eng 2007;26:1625–32. [33] Sun GH, Zheng H, Huang YY. Stability analysis of statically indeterminate blocks in key block theory and application to rock slope in Jinping-I Hydropower Station. Eng Geol 2015;186:57–67. [34] Nezami EG, Hashash YMA, Zhao D, Ghaboussi J. A fast contact detection algorithm for 3-D discrete element method. Comput Geotech 2004;31:575–87. [35] Keneti AR, Jafari A, Wu JH. A new algorithm to identify contact patterns between convex blocks for three-dimensional discontinuous deformation analysis. Comput Geotech 2008;35:746–59. [36] Wu JH, Lin WK, Hu HT. Assessing the impacts of a large slope failure using 3DEC: The Chiu-fen-erh-shan residual slope. Comput Geotech 2017;88:32–45.
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Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo
Research Paper
A rigorous solution for the stability of polyhedral rock blocks Qinghui Jiang a,b,⇑, Chuangbing Zhou b a b
School of Civil Engineering, Wuhan University, Wuhan 430072, PR China School of Civil Engineering and Architecture, Nanchang University, Nanchang 330031, PR China
a r t i c l e
i n f o
Article history: Received 25 December 2016 Received in revised form 8 June 2017 Accepted 18 June 2017
Keywords: Polyhedral rock blocks Block theory Limit equilibrium methods Factor of safety Direction of sliding
a b s t r a c t Block theory has been widely applied to stability analysis of rock engineering due to its clear concept and elegant geometrical theory. For a general block with multiple discontinuity planes, it is assumed that contact is only maintained on a single plane (single-plane sliding) or two intersecting planes (double-plane sliding) in block theory analysis. Since the normal forces and shear resistances acting on the other discontinuity planes are omitted, it can cause unreasonable estimations of block failure modes and incorrect calculation of factors of safety. In this study, a new method is presented that permits to consider the contribution of the normal forces and shear resistances acting on each discontinuity plane to the block stability. The proposed method meets all of the force-equilibrium and moment-equilibrium conditions and provides a rigorous solution for stability of general blocks with any number of faces and any shape. Some typical polyhedral blocks in rock slopes are analyzed using block theory and the proposed method. The results indicate that the traditional block theory may give a misleading conclusion for the predictions of stability and sliding direction of rock blocks when contact occurs on more than two discontinuity planes. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction In a rock slope, there are usually numerous discontinuities of various attitudes and different scales. The spatial intersections of the discontinuities at the free surfaces of the slope may form a series of removable rock blocks with different shapes and sizes. It is necessary to evaluate the stability of these removable blocks to determine the excavation and reinforcement design of the slope. At present, stability analyses of a tetrahedral rock wedge formed by two intersecting discontinuity planes at the crest of a slope are well established in geotechnical literature. The early contributions to the issue arise from the work of Wittke [1], John [2], Londe et al. [3], Chan and Einstein [4], Hoek and Bray [5], and Priest [6]. Recent developments are given by Kumsar et al. [7], Wang and Yin [8], Yeung et al. [9], Chen [10], Jimenez and Sitar [11], Jiang et al. [12], among others. When a rock slope is intersected by more than two discontinuity sets, polyhedral rock blocks with general shapes can be created. Lin et al. [13], Ikegawa and Hudson [14], Jing [15], Lu [16] and Elmouttie et al. [17] developed geometrical identification methods for polyhedral rock blocks. For this type of complex blocks formed by any number of discontinuity planes, the ⇑ Corresponding author at: School of Civil Engineering, Wuhan University, Wuhan 430072, PR China. E-mail address: [email protected] (Q. Jiang). http://dx.doi.org/10.1016/j.compgeo.2017.06.012 0266-352X/Ó 2017 Elsevier Ltd. All rights reserved.
traditional wedge analysis methods are limited. An alternative method is to use block theory proposed by Goodman and Shi [18]. Block theory is a three-dimensional geometrical method with rigorous mathematical deduction that permits analysis of any shape of rock blocks. By means of the theorem of movability of block theory, one is able to analyze the system of joints and other rock discontinuities to find a list of removable blocks. Once the removable blocks are identified, the failure mode of the blocks can be judged according to kinematical admissibility conditions and the factor of safety of the blocks can be calculated by limit equilibrium methods. Since block theory was put forward, it has been widely applied to stability analysis and support design of rock engineering due to its clear concept and elegant geometrical theory [19–26]. However, there are limitations to the failure mode and stability analysis of the traditional block theory. Goodman and Shi [18] took the translational failure mode into account only and neglected the effect of the moments of external forces applied to a rock block. To study the rotational failure mechanism, Mauldon and Goodman [27], Tonon [28], and Tonon and Asadollahi [29] extended block theory to include the rotational failure modes and proposed vector analyses of keyblock rotations. Zhao and Wang [30] developed a lower bound limit method that can consider the sliding mode and rotation effect simultaneously. However, these analyses of incorporating rotational modes are limited to tetrahedral blocks bounded by
Q. Jiang, C. Zhou / Computers and Geotechnics 90 (2017) 190–201
joint planes. For general blocks with multiple discontinuity planes, at most two discontinuity planes are taken into account in stability analysis of block theory. This may lead to incorrect results for stability evaluation of the general blocks. Mauldon and Ureta [31] proposed an energy method for determination of the factor of safety against sliding failure for a special prismatic block with multiple sliding planes that form a cylindrical surface. Zhang et al. [32] presented a limit equilibrium analysis for a general block with more than two joint planes by dividing the block into multiple sub-blocks along the line of intersection of joint planes. In both Zhang’s and Mauldon-Ureta approaches, only force-equilibrium conditions can be satisfied. Sun et al. [33] proposed further an optimization model based on limit equilibrium analysis to evaluate the stability of rock blocks with multiple sliding planes, which meets all force and moment equilibrium conditions. Since a symmetric plane, or sliding direction, needs to be pre-determined in the analysis, the method proposed by Sun et al. is only appropriate for polyhedral blocks with a symmetrical geometry. The behavior of polyhedral blocks with arbitrary joint planes can also be simulated by numerical methods based on the discontinuous media models, such as 3d DDA and 3d DEM [34–36]. However, the determination of the safety factor by the numerical methods involves complex interface contact treatment and time-consuming trial and error calculations using the reduction of the strength parameters. The purpose of this study is to develop a rigorous method for stability assessment of general blocks with any number of faces and any shape. To overcome the limitations described above, a new method is proposed that considers the effect of the normal forces and shear resistances acting on multiple discontinuity planes on the block stability. The proposed method meets all of the force-equilibrium and moment-equilibrium conditions and provides a rigorous solution for stability analysis of general blocks. Finally, some typical polyhedral blocks in rock slopes are analyzed using the block theory and proposed method, and some interesting findings are illustrated by this study. 2. Block theory 2.1. Kinematic analysis Suppose that a removable block is formed by multiple discontinuity planes and the face and upper surface of a slope (Fig. 1). Let ni be the upward unit normal to discontinuity plane i, v i be the unit normal vector of discontinuity plane i pointing to the interior of the block, r represent the active resultant acting on the block, and s represent the direction of movement of the block. There are usually three types of failure modes for the removable block: lifting, single-plane sliding and double-plane sliding. From Goodman and Shi [18], the kinematic admissibility for the block failure can be described as follows: 2.1.1. Lifting When a removable block is lifting (Fig. 1a), the movement direction of the block is parallel to the direction of the active resultant, that is,
s ¼ r=jrj
ð1Þ
For the failure mode of lifting, the kinematic conditions for the block instability are
s v l > 0 for all l
s ¼ si ¼ ðni rÞ ni =jni rj
191
ð3Þ
where si is the orthographic projection of r on plane i. For the mode of single-plane sliding, the kinematic conditions for the block failure are
r vi 6 0
ð4Þ
and
s v l P 0 for all l; l–i
ð5Þ
2.1.3. Double-plane sliding When a removable block slides along discontinuity planes i and j (Fig. 1c), the movement direction of the block is parallel to the intersection line of the two planes, that is,
ni nj sign½ðni nj Þ r jni nj j
s ¼ sij ¼
ð6Þ
For the mode of double-plane sliding, the kinematic conditions for the block failure are
si v j 6 0;
sj v i 6 0
ð7Þ
and
s v l P 0 for all l; l–i or j
ð8Þ
Here, si and sj are the orthographic projections of r on planes i and j, respectively. 2.2. Factor of safety If the failure mode of a removable block is identified by the kinematic admissibility, the factor of safety of the block can be calculated using the limit equilibrium method. This method was originally proposed for tetrahedral blocks by Wittke [1], John [2] and Londe et al. [3], and further extended to the blocks with any number of faces by Goodman and Shi [18]. As shown in Fig. 2, Goodman and Shi introduced a fictitious force Fs to bring a removable block to a limit equilibrium state. The resultant of shear forces acting on joint planes can be written as
Ts ¼
X
N l tan /l s Fs
ð9Þ
l
where N l is the normal force acting on the discontinuity plane l, and /l is the friction angle of discontinuity plane l. For evaluation of the block stability, the conventional definition of safety factor is introduced to bring the block to a limiting state. Further considering the cohesion of joint planes, the resultant of shear forces on discontinuity planes can be rewritten as
Ts ¼
X 1 l
Fs
ðN l tan /l þ cl Al Þs
ð10Þ
where F s is the factor of safety, cl and Al are the cohesion and area of discontinuity plane l. Therefore, the force equilibrium equations for the block can be expressed as
rþ
X Nl v l Ts ¼ 0
ð11Þ
l
ð2Þ
2.1.2. Single-plane sliding When a removable block slides along plane i only (Fig. 1b), the movement direction of the block is parallel to si , that is,
2.2.1. Lifting In this case, all of the discontinuity planes will open and the normal reaction forces N l ¼ 0 for all the discontinuity planes. The factor of safety of the removable block is defined as zero.
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Upper surface
n1 v1
Plane 1
Slope face
n2
v2 r(s)
Plane 2
(a)
Upper surface
Plane 1
v1(n1)
Movement direction of the block
Plane 2
v2(n2)
r
n3
Plane 3
Slope face
n3 r
Plane 3
s
r
v3(n3)
(b) Movement direction of the block
Upper surface
Plane 1
v1(n1)
r
n1
n2
Plane 2
v2(n2) r
Slope face
s
Plane 2
Plane 1
(c) Fig. 1. Failure mode for a removable block: (a) lifting; (b) single-plane sliding; and (c) double-plane sliding.
2.2.2. Single-plane sliding In this case, the removable block maintains contact with joint plane i only and all the other discontinuity planes will open. Since N l ¼ 0 for all l–i, Eq. (11) reduces to
r þ Ni vi
1 ðNi tan /i þ ci Ai Þsi ¼ 0 Fs
ð12Þ
By using the vector operation to Eq. (12), the safety factor of the block can be derived as follows:
Ni ¼ jr ni j T i ¼ jr ni j Fs ¼
Ni tan /i þ ci Ai Ti
ð13Þ
2.2.3. Double-plane sliding In this case, the removable block maintains contact with joint planes i and j and all the other discontinuity planes will open. Since N l ¼ 0 for all l–i or j, Eq. (11) reduces to
r þ Ni v i þ Nj vj
ð16Þ
By using the vector operation to Eq. (16), the safety factor of the block can be derived as follows:
Ni ¼
ð14Þ ð15Þ
1 ðN i tan /i þ ci Ai þ Nj tan /j þ cj Aj Þsij ¼ 0 Fs
Nj ¼
jðr nj Þ ðni nj Þj jni nj j2 jðr ni Þ ðni nj Þj jni nj j2
ð17Þ
ð18Þ
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From Eqs. (15) and (20), the contributions of at most two discontinuity planes to the factor of safety are taken into account. For a removable block with multiple discontinuities, it is assumed that contact is only maintained on a single plane (single-plane sliding) or two intersecting planes (double-plane sliding), and all the other discontinuity planes will open in block theory analysis. In fact, the block is to keep contact with the other discontinuities in a limiting state. With three or more contact planes, if the normal forces and shear resistances acting on the contact planes are omitted, it can cause unreasonable calculation of factors of safety.
r v
3. The proposed stability method for polyhedral blocks 3.1. Basic assumptions l
N
s
lv l
N Fs
an lt
ls
Ts
Fig. 2. Explanation of fictitious force F and movement direction vector s (modified after [13]).
T ij ¼
jr ðni nj Þj jni nj j
ð19Þ
Fs ¼
Ni tan /i þ ci Ai þ Nj tan /j þ cj Aj T ij
ð20Þ
Without loss of generality, Fig. 4a shows the geometry of a typical removable block with multiple discontinuity planes. Illustrated in Fig. 4b is the plan view of the polyhedral block, illustrated in Fig. 4c is a vertical differential column with a base area of dS in the block. Different with the conventional block theory, we assume that the projection of movement direction of the removable block on the x-y plane is unique. The direction of sliding, denoted by b, is defined as an angle calculated anticlockwise from the positive xaxis to the opposite direction of the projection (Fig. 4b). Based on this assumption, the unit vector tl of the shear force acting on the discontinuity plane l can be described as:
tlx ¼ sin c cos b;
t ly ¼ sin c sin b;
tlz ¼ cos c
ð21Þ
where c is the angle between the shear force and the upward vertical direction. Since the direction of the shear force is perpendicular to the unit normal vector v l , we have
v lx sin c cos b þ v ly sin c sin b þ v lz cos c ¼ 0
ð22Þ
By solving Eq. (22), the unit vector of the shear force on the discontinuity plane l can be readily determined as follows:
2.3. Limitations of block theory analysis According to the above-mentioned analyses, the traditional block theory considers translational failure modes only and the factor of safety of the block is obtain using force-equilibrium Eq. (11). This means that the only force-equilibrium conditions are satisfied, but the moment-equilibrium conditions are ignored in the stability analysis of the block. From Eqs. (9) and (10), it has been found that the block theory actually involves an assumption that the direction of shear forces on all the contact planes is unique, which is in the opposite direction to the movement of the block. For a removable block with multiple discontinuities, the directions of shear forces acting on the different discontinuity planes are obviously different, as shown in Fig. 3a. Furthermore, if there is an anti-dip bottom face among the discontinuity planes forming a polyhedral block (Fig. 3b), the block is deemed to be immovable according to the kinematic analysis of block theory. However, a sliding failure for the block may still occur when the sliding force along joint plane 1 is greater than the resisting force on joint plane 2.
T tl ¼ ½ tlx tly tlz ¼ v lz cosb=J v lz sinb=J
where J ¼
ðv lx cosb þ v ly sinbÞ=J
T
ð23Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v 2lz þ ðv lx cos b þ v ly sin bÞ2 .
Now we select an arbitrary differential column in the block for analysis (Fig. 4c). rl v l dS and sl tl dS are the normal force and shear force acting on the column base, respectively. kdw is the gravity of the column. dp is the resultant of inter-column forces acting on the row-interfaces ABFE and DCGH, and dq is the resultant of intercolumn forces acting on the column-interfaces ADHE and BCGF. The force equilibrium equations for the column can be expressed as follows:
rl v l dS þ sl tl dS kdw þ dp þ dq ¼ 0 According to the dot product of the Eq. (24) and
rl
dw dp dq ¼ v lz vl vl dS dS dS
ð24Þ
v l , we obtain ð25Þ
Considering
hv lz dS dw ¼ c Free face
Free face Joint plane 1 Joint plane 1 Joint plane 2
Joint plane 2
(a) Fig. 3. Types of blocks, in two dimensions.
(b)
ð26Þ
and h are the unit weight and height of the differential colwhere c umn, respectively. Substituting Eq. (26) into Eq. (25), we obtain
rl ¼ rext þ rint
ð27Þ
where rext ¼ v c represents the contribution of the gravity acting on the column to rl , rint represents the contribution of the internal 2 lz h
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Upper surface Plane 1 Plane 3 Plane 2 Plane m
Slope face
z
Line of intersection
y x
(a)
y
G
H
Column (plan view)
F
E
β
dq
Failure surface (plan view)
kdw C
dp t dS
D
Sliding direction (Plan view)
l l
l
v l dS
B
A
x
(c)
(b)
Fig. 4. (a) Geometry of a rock block with multiple discontinuity planes; (b) plan view of the failure surface and sliding direction; and (c) forces acting on a differential column in the block.
forces in the block to to
rint
dp þ dq ¼ v l dS
rl and is statically indeterminate, and equals
K ¼ ð1
ð28Þ
According to Eq. (27), we can construct an approximation to in the following way
rl ¼ rext þ nðx; y; kÞ
There are many choices of n. In this study, we use a polynomial function to construct n. Letting the polynomial basis
rl
ð29Þ
where k is an unknown vector of 4-order, k ¼ ð k1 k2 k3 k4 ÞT . n is a function in the horizontal coordinate pair of (x, y) with k. The reason for introducing four unknowns lies in that the six equilibrium conditions for the removable block are able to solve at most six unknowns and the safety factor and sliding direction have occupied two unknowns.
x
y
xy ÞT , we have
nðx; y; kÞ ¼ KT k
ð30Þ
Therefore, the total normal stress on the column base can be approximated by
rl ¼ rext þ KT k
ð31Þ
By the Mohr-Coulomb criterion, the shear stress on the column base is determined as:
sl ¼
ðrl uÞ tan /l þ cl Fs
where u is pore pressure on the base of the column.
ð32Þ
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3.2. Global equilibrium equations for removable blocks Assuming that the sliding surface of the removable block is composed of m discontinuity planes (see Figs. 4 and 5). The global force equilibrium equations can be written in the vector form as:
l
l¼1
rl v l dS þ
m ZZ X l
l¼1
sl tl dS þ f ¼ 0
J3
ð33Þ
J1
where f represents the resultant of external forces acting on the block, including weight, seismic forces and support forces from rock bolts or cables. The global moment equilibrium equations can be written in the vector form as: m ZZ X l
l¼1
ðx v l Þrl dS þ
m ZZ X l
l¼1
ðx tl Þsl dS þ m ¼ 0
ð34Þ
where x represents the coordinates of the center of the column base. m represents the resultant moment of the external forces acting on the block about the coordinate axes. Substituting Eq. (32) into Eqs. (33) and (34), we obtain m ZZ X l
l¼1
m ZZ X
ðF s v l þ tan /l tl Þrl dS þ
tl ðcl u tan /l ÞdS þ F s f ¼ 0 l
l¼1
ð35Þ m ZZ X l
l¼1
m ZZ X
½F s ðx v l Þ þ tan /l ðx tl Þrl dS þ
l¼1
ðx tl Þ ð36Þ
Eqs. (35) and (36) can be rewritten in a more compact form as follows:
gðk; b; F s Þ ¼
l¼1
l
ðF s v 0l þ tan /l t0l Þrl dS þ
m ZZ X l¼1
l
t0l ðcl u
0
tan /l ÞdS þ F s f ¼ 0
ð37Þ
where g denotes the vector of unbalanced forces and moments. v , t 0 and f are three 6-order vectors, defined by
v
xv
;
t0 ¼
t xt
;
0
f ¼
f
0
ð38Þ
m
There are six unknowns, namely k, b, and F s involved in the system (37), which can be efficiently solved by the Newton method. For application of the Newton method, the Jacobian of function gðk; b; F s Þ is needed and defined as:
J¼
h
@g @k1
@g @k2
@g @k3
@g @k4
@g @b
@g @F s
Fig. 6. Geometries of a symmetric rock wedge and pentahedral block considered in Example 1.
Since the six equations in the system (37) are functionally independent, the Jacobian of gðk; b; F s Þ is invertible for any ðk; b; F s Þ. Assuming that values of k, b and F s at the kth step are obtained as kk , bk and F s;k , then their values at the (k + 1)th step are modified as
0
kkþ1
i
ð39Þ
Pl an e1
z
Plane m
x
0
kk
1 ð40Þ
The convergence criterion for k, b and F s can be defined as
f ¼ kðkTkþ1 ; bkþ1 ; F s;kþ1 Þ ðkTk ; bk ; F s;k Þk 6 e where
tdS
vdS
y Fig. 5. A 2d schematic plot for the removable block in the yz plane.
ð41Þ
e is user-specified tolerances.
Based on the above formulations, an iterative algorithm for stability analysis of polyhedral blocks is summarized in the following steps: (1) Divide the polyhedral block into a number of columns. (2) Calculate the unit normal vector v and the area of the column base for all columns. (3) Specify the tolerance e and initial values of F s , b and k. In general, it can be assumed that F s ¼ 10, b ¼ 1 and
(4) (5)
(7)
2 ne Pla
1
B C B C 1 @ bkþ1 A ¼ @ bk A ½Jðkk ; bk ; F s;k Þ gðkk ; bk ; F s;k Þ F s;kþ1 F s;k
(6)
f
x
3.3. Numerical implementation
0
v0 ¼
z
Slope face
l
ðcl u tan /l ÞdS þ F s m ¼ 0
m ZZ X
y
J2
H=50 m
m ZZ X
Upper surface
(8) (9) (10)
k ¼ ½ 0 0 0 0 T for the first iteration. The tolerance parameters is usually specified as 0.001. Set the iteration step k 1. Calculate the unit vector t of the shear stress acting on the column base using Eq. (23). Calculate the normal stress r and shear stress s acting on the column base using Eqs. (31) and (32). Calculate the vector g representing the three unbalanced forces and three unbalanced moments using Eq. (37) for all of columns. Calculate the Jacobian of g using Eq. (39). Obtain the updated values of F s , b and k using Eq. (40). If the convergence criterion (41) is satisfied, then end calculation; otherwise, let k k þ 1 and turn to step (5).
4. Examples Three typical examples involving seven polyhedral rock blocks are analyzed using block theory and the proposed method. The size of columns for discretization of the blocks is 1 m 1 m. The initial values of the unknowns are: F s ¼ 10, b ¼ 1, and k ¼ ½ 0
0
0
0 T .
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Table 1 Orientation of discontinuity planes forming rock blocks in Example 1.
Joint J1 Joint J2 Joint J3 Slope face Upper surface
Dip (degrees)
Dip direction
Friction angle
Cohesion (kPa)
45 45 40 45 10
115 245 180 180 180
20 20 20
30 30 30
Note: Height of slope = 50 m and unit weight of rock = 26 kN/m3.
Table 2 Results for rock blocks in Example 1. Block no.
1 2
Joint planes
J1, J2 J1, J2, J3
Block theory
Proposed method
Factor of safety
Direction of sliding
Factor of safety
Direction of sliding
1.457 1.388
1.571 1.571
1.435 1.188
1.578 1.576
The results indicate that the proposed method has rapid convergence and all examples are convergent within, at most, six iteration steps.
4.1. Example 1 In this example a rock slope with a strike of EW is considered as shown in Fig. 6. The dips of face and upper surface of the slope are 45° and 10°, respectively. The height of the slope is 50 m, and the unit weight of rock is 26 kN/m3. There are three joint planes, J1, J2 and J3, developed in this slope. The attitudes and strength properties of the joint planes are listed in Table 1. Two removable blocks were analyzed in the slope. One is a wedge block formed by J1 and J2, the other is a pentahedral block formed by J1, J2 and J3. For comparison, both the traditional block theory and proposed method are used to evaluate the stability of the two rock blocks. The calculation results are summarized in Table 2. It was found that, for the wedge block involving two sliding surfaces only, the safety factor obtained using two different approaches are in reasonably close agreement. For the pentahedral block with three contact planes, the factor of safety obtained by the block theory is 1.388, whereas that obtained by the proposed method is 1.188. According to kinematic conditions (7) and (8) of block theory, the pentahedral block was identified as doubleplane sliding along J1 and J2. In this case, the removable block maintains only contact with J1 and J2 and joint plane J3 will open. From Eq. (20), the contributions from J1 and J2 to the safety factor are considered, and the effects of J3 are neglected. Since the dip of J3 is greater than the plunge of the line of intersection of J1 and J2, J3 has unfavorable effects on the stability of the pentahedral block. This would lead to overestimation of the stability of the block if J3 acting as the rear sliding surface is neglected. Although the two removable blocks in this example have different geometric configurations, the sliding surfaces forming the two blocks are symmetrical about the y-axis. It can be seen from Table 2 that the sliding directions for both cases are correctly predicted by the block theory and proposed method. The results would be anticipated for a problem with symmetrical failure surfaces. Once Eq. (37) is solved, the normal stress acting on the contact planes could be computed using Eq. (31). Shown in Fig. 7 is the distribution of normal stresses over the contact planes of the pentahedral block. It can be observed that in a limiting state, the normal stresses on J3 are positive and cannot be neglected. For the symmetrical pentahedral block, the distribution of normal stresses takes on a good symmetry, which indicates the results are reasonable.
Fig. 7. Contours of normal stresses over the failure surfaces of the pentahedral block formed by J1, J2 and J3.
4.2. Example 2 The face of a rock slope shown in Fig. 8 has a dip of 60° and a dip direction of 180°. The upper surface of the slope is horizontal. The height of the slope is 76 m, and the unit weight of rock is 26 kN/m3. There are three joint planes, J1, J2 and J3, developed in this slope. The attitudes and strength properties of the joint planes are listed in Table 3. Two removable blocks were analyzed in the slope. One is an asymmetric wedge block formed by J1 and J2, the other is an asymmetric pentahedral block formed by J1, J2 and J3. Both block theory and the proposed method are employed to evaluate the stability of the two polyhedral blocks. The calculation results are summarized in Table 4. It can be observed that, for the asymmetric wedge block, the safety factor and sliding direction obtained using two different approaches are in reasonably close agreement. For the asymmetric pentahedral block, the value of safety factor from block theory is significantly smaller than that from the proposed method. Furthermore, the sliding direction obtained by block theory is obviously incorrect because the failure surfaces forming the pentahedral block is laterally asymmetrical about the y-axis. The cause for the block theory yielding a lower safety factor and an incorrect sliding direction lies in that, the block theory placed the failure mode of the pentahedral block as single-plane sliding along J3 and neglected the contributions of J1 and J2 to the stability of the block. In fact, the areas of J1, J2 and J3 forming the block are 2118 m2, 1868 m2 and 374 m2, respectively. According to the calculation results from the proposed rigorous method, the stability and sliding direction of the pentahedral block are mainly controlled by joint planes J1 and J2.
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4.3. Example 3
J2
J1
J3
J3
z y
x
Fig. 8. Geometries of an asymmetric rock wedge and two pentahedral blocks considered in Example 2.
Table 3 Orientation of discontinuity planes forming rock blocks in Example 2. Surface
Dip (degrees)
Dip direction
Friction angle
Cohesion (kPa)
Joint J1 Joint J2 Joint J3/J30 Slope face Upper surface
50.1 63.0 25.2/5 60 0
125 233 180/0 180 0
30 30 20
50 50 30
The face of a rock slope shown in Fig. 10 has a dip of 60° and a dip direction of 180°. The upper surface of the slope is horizontal. There are four joint planes, J1, J2, J3 and J4, developed in this slope. The attitudes and strength properties of the joint planes are listed in Table 5. It can be observed from Fig. 10 that the four joint planes in this slope intersect to form a symmetrical hexahedral block. For this specified block, the intersection line s13 of planes J1 and J3 is parallel to the intersection line s23 of planes J2 and J3. Assuming that gravity is the only contributor to the active resultant r acting on the block, the projection s3 of r on plane J3 is also parallel to s13 and s23 . According to the kinematic analysis of block theory, the failure mode of the block can be identified as single-plane sliding along J3, or double-plane sliding along J1 and J3 or J2 and J3. Obviously, the block is singular in mathematics from the analysis of block theory. To avoid the singularity, we slightly adjust J1 (124.7°\45.4°) and J2 (235.3°\45.4°) to J10 (127.3°\44.6°) and J20 (232.7°\44.6°), respectively. Using block theory, the block formed by J10 , J20 , J3 and J4 is identified as single-plane along plane J3. The factor of safety of the block can be determined as 0.71 by Eq. (15). For comparison, the two hexahedral blocks involved in this example are also analyzed using the proposed method. The calculation results are listed in Table 6. For the two symmetrical blocks, the calculated sliding directions agree well with the direction of
Note: Height of slope = 76 m and unit weight of rock = 26 kN/m3.
Table 4 Results for rock blocks in Example 2. Block no.
Joint planes
Block theory
Proposed method
Factor of safety
Direction of sliding
Factor of safety
Direction of sliding
1 2 3
J1, J2 J1, J2, J3 J1, J2, J30
1.325 0.805 –
1.761 1.571 –
1.309 1.280 1.349
1.772 1.744 1.764
Now we specify the orientation of J3 as J30 (0°\5°), as shown in Fig. 8. Due to joint plane J30 dipping into the face of the slope, the pentahedral block formed by J1, J2 and J30 is immovable according to the traditional block theory. From a practical point of view, the block still has the risk of failure if the sliding force along J1 and J2 is greater than the resisting force on J30 . Using the proposed method, the safety factor and sliding direction of the block can be determined as 1.349 and 1.764. The results demonstrate that for the block with an anti-dip joint plane, J1 and J2 are still main factors contributing to the stability and sliding direction of the failure mass. Obviously, the block theory highly overestimates the stability of the block due to neglecting the contributions of J1 and J2. Illustrated in Fig. 9a and b are the distributions of normal and shear stresses on the joint planes of the block, respectively. It can be observed that the normal stresses are positive and satisfy the physical admissible condition on failure surfaces. Although the maximum normal stress occurs on the joint plane J30 , the shear stress on J30 is not the maximum due to the effect of strength properties. This indicates that the slide-resistance action of J30 is not remarkable although J30 dips into the slope face.
Fig. 9. Contours of stresses over the failure surfaces of the pentahedral block formed by J1, J2 and J30 : (a) normal stresses; (b) shear stresses.
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5. Discussion
J2
J1
5.1. Effects of discretized column size
J4
J2
J1 J3
z y
x
Fig. 10. Geometries of two symmetric hexahedral blocks considered in Example 3.
Table 5 Orientation of discontinuity planes forming rock blocks in Example 3.
Joint J1/J10 Joint J2/J20 Joint J3 Joint J4 Slope face Upper surface
Dip (degrees)
Dip direction
Friction angle
Cohesion (kPa)
45.4/44.6 45.4/44.6 30 45 60 0
124.7/127.3 235.3/232.7 180 180 180 0
30 30 20 30
100 100 50 100
Note: Height of slope = 50 m and unit weight of rock = 26 kN/m3.
In order to examine the effects of the size of columns on solution accuracy, various column sizes are adopted for analysis. For Example 1, seven different column sizes are employed to discretize the polyhedral blocks. Fig. 12 shows the factor of safety and sliding direction versus the number of columns relationships. It can be observed that the safety factor and sliding direction approach the corresponding steady values with increasing numbers of columns, or equivalently, decreasing sizes of columns. This indicates that there is a reasonable convergence trend for the solutions of the proposed method with decreasing column size. For Examples 2 and 3, the evolution trends of solutions with the size of columns are similar to that of Example 1. For illustration, Fig. 13 shows variation of the calculated factors of safety and sliding directions with the number of columns for Example 2. Assuming the convergence value is the exact solution, Fig. 14 shows variation of relative error with the discretized column numbers for the three examples involving seven polyhedral blocks. It can be observed that, for these examples, the proposed method has an enough precision if the total number of columns for discretization of the polyhedral blocks is greater than 1000. 5.2. Effects of approximation functions to the normal stress To investigate the effects of different approximation functions to the normal stress on solution precision, three approximation
Table 6 Results for rock blocks in Example 3. Block no.
Joint planes
Block theory Factor of safety
Direction of sliding
Factor of safety
Direction of sliding
1
J1, J2, J3, J4 J10 , J20 , J3, J4
0.711a/ 0.871b 0.710
1.571
1.297
1.570
1.571
1.292
1.571
2
Proposed method
Note: a- single-plane sliding; b- double-plane sliding.
the symmetrical axis. Compared to the first block, the second block has only a very slight change in the geometry and volume. Consequently, the safety factors of the two blocks obtained using the proposed method are in reasonably close agreement. By comparing the results listed in Table 6, it was found that the stability of the two polyhedral blocks are significantly underestimated by block theory. The main reasons lie in that the traditional block theory can take into account the contribution of two sliding planes to the block stability at most. Furthermore, it is not necessary to determine the failure mode of the failure mass in advance in the proposed method. The singular problem in the analysis of block theory can be effectively overcome. Shown in Fig. 11a and b are the distributions of normal and shear stresses over the failure surfaces of the hexahedral block formed by J10 , J20 , J3 and J4, respectively. Since the block is characterized by the symmetry of geometry of discontinuity planes, the distributions of normal and shear stresses show also a good symmetry. Although the maximum normal stresses occur on plane J3, the maximum shear stresses exist in planes J10 , J20 and J4. This demonstrates that the resisting forces acting on J10 , J20 and J4 help stabilize the removable block. Because the shear resistances on J10 , J20 and J4 are omitted in block theory analysis, the stability of the block is significantly underestimated by block theory.
Fig. 11. Contours of stresses over the failure surfaces of the hexahedral block formed by J10 , J20 , J3 and J4: (a) normal stress; (b) shear stress.
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1.38
1.60
1.50
1.52
1.45 1.40
3
7
1.35
1
4 5
1.30
2
1.48
Factor of safety Size of columns
Direction of sliding
1.44
8
1.75
1.34 1.32
1.70
1.30
2 3 4 5 Size of columns 6
1.28 1.26 1.24
1.65 Factor of safety Direction of sliding
1.60 1.55
2000
1.40 0
0.5
0.6
1.22 0
1.25
0.7
1
Direction of sliding
1.56
Factor of safety
Factor of safety
1.55
1.80
1.36
Direction of sliding
1.60
1000 2000 3000 4000 5000 6000 7000 8000 9000
4000 6000 8000 Number of columns
Number of columns
10000
(a)
(a) 1.35
1.18 3 7
1.14
Fcator of safety
Size of columns
Direction of sliding
8
1.10 0
1.52 1.48
4 5
1.12
1
2
1000
2000
3000
4000
5000
6000
1.44
1.71
1.29 1.26 4 5 6
1.23
0.6
0.7
1 2 3
0.5
Factor of safety Size of columns
1.68
Direction of sliding
Direction of sliding
1.56
1.20
Factor of safety
Factor of safety
1.22
1.32
Direction of sliding
1.24
1.16
1.74
1.60
1.65
1.20 0
2000
1.40 7000
4000
6000
8000
10000
Number of columns
(b)
Number of columns
(b)
ð42Þ
nðx; y; kÞ ¼ ðk1 þ k2 x þ k3 y þ k4 xyÞrext
ð43Þ
Taking Example 1 in the following analysis as an instance. For the two cases in Example 1, the size of columns for discretisation of the blocks is set to 1 m 1 m. The results from the three approximation functions are listed in Table 7. Although we construct different functions to approximate rl , the calculated factors of safety and direction of sliding are in reasonably close agreement because the normal and shear forces obtained using different approximation functions must meet all of force and moment equilibrium conditions in the present method. It can be also seen from Table 7 that the solutions are convergent within five to seven iterations corresponding to different approximation functions. This indicates that the present method has rapid convergence and is suitable to use in practice.
1.38
1.75
1.35
1 2 3 4 5 Size of columns 6
1.32 1.29 1.26
0
2000
To examine the numerical performance of the proposed method, the initial iteration value of the factor of safety are set to be very far from the real solution and the precision of convergence is set to be enough high. Letting F s;0 ¼ 100, b0 ¼ 1, k0 ¼ ½ 0 0 0 0 T and e ¼ 109 , all seven blocks in the abovementioned three examples are used for re-analysis. Fig. 15a–c shows the iteration process of the calculated factor of safety, direction of sliding and convergence precision, respectively. Although the initial iteration values are far from the real solutions, only six to seven iterations are required to reach a convergence precision less than 109 . This indicates that the proposed method has a high
1.74
Factor of safety Direction of sliding
1.73 1.72
6000
8000
10000
Number of columns (c) Fig. 13. Variation of calculated factors of safety and directions of sliding with column size: (a) Case 1, Example 2; (b) Case 2, Example 2; and (c) Case 3, Example 2.
Table 7 Results from different approximation functions for blocks in Example 1.
Wedge block
Pentahedral block
5.3. Numerical performance of the proposed method
4000
0.5
0.6
0.7
Direction of sliding
nðx; y; kÞ ¼ k1 x þ k2 y þ k3 x2 þ k4 y2
1.76
1.41
Factor of safety
functions are adopted for analysis. Except Eq. (30), the other two approximation functions are
1.77
1.44
Fig. 12. Variation of calculated factors of safety and directions of sliding with column size: (a) Case 1, Example 1; (b) Case 2, Example 1.
Safety factor Sliding direction Iteration times Safety factor Sliding direction Iteration times
Eq. (30)
Eq. (42)
Eq. (43)
1.435 1.578 6 1.188 1.576 5
1.435 1.578 5 1.187 1.576 7
1.431 1.582 5 1.183 1.578 5
convergence precision and a fast convergence speed. It can be also observed from Fig. 15c that when the tolerance is specified as 103 , the calculation results for the seven cases are convergent within, at most, six iteration steps. If the tolerance is set to be lower than 103 , the increase of iteration steps to reach convergence is not significant for the present method. However, a tolerance of 103 is generally acceptable and a higher precision seems not necessary for practical engineering problems.
200
Q. Jiang, C. Zhou / Computers and Geotechnics 90 (2017) 190–201
0.10
100
Relative error
0.06
Factor of safety
Example 1(Case 1) Example 1(Case 2) Example 2(Case 1) Example 2(Case 2) Example 2(Case 3) Example 3(Case 1) Example 3(Case 2)
0.08
0.04 0.02
Example 1 (Case 1) Example 1 (Case 2) Example 2 (Case 1) Example 2 (Case 2) Example 2 (Case 3) Example 3 (Case 1) Example 3 (Case 2)
10
0.00 100
1000
1
10000
0
1
2
3
Number of columns
4
5
6
7
8
Iteration step
(a)
(a) 2.2
0.05
0.03
Direction of sliding
0.04
Relative error
2.0
Example 1(Case 1) Example 1(Case 2) Example 2(Case 1) Example 2(Case 2) Example 2(Case 3) Example 3(Case 1) Example 3(Case 2)
0.02 0.01
1.8 1.6 1.4
Example 1 (case 1) Example 1 (case 2) Example 2 (case 1) Example 2 (case 2) Example 2 (case 3) Example 3 (case 1) Example 3 (case 2)
1.2 1.0
0.00
0.8 100
1000
0
10000
1
2
3
Number of columns
4
5
6
7
8
Iteration step
(b)
(b)
Fig. 14. Variation of relative errors in safety factor and sliding direction with the number of columns: (a) safety factor; (b) sliding direction.
1000
The computational efficiency is an important part for this proposed method to find eventually wide application. The computational costs are measured by the time of evaluation before convergence. The iteration algorithm was run on an Apple laptop with Inter(R) Core(TM) i7-4870HQ CPU @ 2.50 GHz 2.50 GHz. The computational time for all seven blocks in this study is ranging from 6 ms to 25 ms, which demonstrates the proposed method has a high computation efficiency. 6. Conclusion
Convergence precision
10 0.1 1E-3 1E-5 Example 1 (case 1) Example 1 (case 2) Example 2 (case 1) Example 2 (case 2) Example 2 (case 3) Example 3 (case 1) Example 3 (case 2)
1E-7 1E-9 1E-11 1E-13 1E-15
For general polyhedral blocks formed by multiple discontinuity planes, at most two planes can be considered in the stability analysis of block theory. The study presents a rigorous solution to stability of the polyhedral blocks that remove the limitations of the block theory method. Based on this work, the following conclusions can be drawn: (1) By introducing an assumed normal stress distribution on contact planes, the proposed method can effectively consider the effects of the normal force and shear resistances on each discontinuity plane on the block stability. Furthermore, the proposed method satisfies all of the force-
1
2
3
4
5
6
7
8
Iteration step
(c) Fig. 15. (a) Calculated factor of safety versus iteration step number; (b) Calculated direction of sliding versus iteration step number; and (c) Convergence parameter versus iteration step number.
equilibrium and moment-equilibrium conditions, which provides a rigorous solution for stability assessment of general removable blocks.
Q. Jiang, C. Zhou / Computers and Geotechnics 90 (2017) 190–201
(2) The proposed method can not only calculate the factor of safety, but also accurately predict the direction of sliding of the removable block. This provides an insight for the design optimization of block reinforcement where the optimal trend of bolting forces is in a direction opposite to that of the sliding direction of the block. (3) The results from the three examples involving seven polyhedral blocks indicate that this proposed method is applicable to stability assessment of polyhedral blocks in rock slopes. For tetrahedral rock wedges, the calculated factors of safety and sliding directions from the proposed method and block theory are in reasonably close agreement. However, for polyhedral rock blocks with three or more contact planes, the traditional block theory may give misleading conclusions for the predictions of stability and sliding direction. (4) It is not necessary to determine the failure mode of polyhedral blocks in advance in the proposed method. Therefore, the singular problem caused by failure mode change in the analysis of block theory can be effectively overcome.
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