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Probabilistic treatment of the sliding wedge with EzSlide
Engineering Geology 50 (1998) 153–163
Probabilistic treatment of the sliding wedge with EzSlide Ping Feng *, Emery Z. Lajtai Department of Civil and Geological Engineering, University of Manitoba, Winnipeg, Manitoba, Canada R3T 5V6 Received 10 September 1997; received in revised form 14 November 1997; accepted 14 November 1997
Abstract Many of the variables that govern rock slope stability, the orientation of discontinuities, the strength parameters belonging to them and the loading conditions, are random variables. Therefore, the safety factor itself is a statistic. This paper analyzes the effect of parameter variability through the use of the Monte Carlo simulation technique. To meet this goal, a Windows based program, EzSlide written in Visual Basic has been developed. EzSlide treats both the single and the multiple-wedge problem and it is equipped with all the tools necessary to conduct a probabilistic analysis of the sliding wedge. The capabilities of the Program are introduced through an application. The field data for the demonstration comes from highway rock cuts along the TransCanada Highway 17a near Kenora, Ontario. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Computer simulation; Probability; Rock; Slope; Stability
1. Introduction Rock slope designs, particularly in hard rock, are based on the limiting equilibrium method that considers the equilibrium of forces acting on a potentially unstable wedge. The sliding wedge is formed by the slope face, the top slope surface and by two intersecting discontinuities. The limiting equilibrium method has been used for slope stability in both deterministic and probabilistic analyses. In the deterministic method, the orientation of the two discontinuities and their shear strength as well as external forces are pre-determined and treated as constant parameters. Most of the parameters, however, are characterized by uncertainty; they are not constants. The orienta* Corresponding author. Tel: (204) 474-8701; Fax: (204) 275-7514; e-mail: [email protected]
tions of rock joints surveyed from field investigation always vary, even within the same set. The strength properties of the rock mass are inherently uncertain; in material testing unique values are seldom measured. External forces such as water forces and earthquake loading are changing all the time. Therefore, a probabilistic approach to rock slope stability is more desirable. The Monte Carlo simulation technique is a numerical method that solves mathematical problems through random sampling. A computer program generates random variates to model the variation in controlling parameters. Since 1985, several computer programs have been written using the Monte Carlo simulation technique (Lajtai and Carter, 1989; Singh et al., 1985; Tamimi et al., 1989; Piteau et al., 1985; Quek and Leung, 1995). Some of them consider only plane (two-dimensional ) slides (e.g. Singh et al., 1985), others
0013-7952/98/$ – see front matter © 1998 Elsevier Science B.V. All rights reserved. PII: S0 0 1 3 -7 9 5 2 ( 9 8 ) 0 0 00 7 - 6
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Fig. 1. A typical screen output from EzSlide. An editable spreadsheet contains all the safety factors. The same data are graphed in the cumulative distribution form and the Weibull distribution is fitted to it automatically.
include wedge slides as well (Quek and Leung, 1995; Lajtai and Carter, 1989; Piteau et al., 1985). Lajtai and Carter developed two DOS-based programs, GEOSLIDE and PROSLIDE, to conduct a deterministic and a probabilistic analysis, respectively (Lajtai and Carter, 1989). The analysis uses three-dimensional-vector algebra, in essence following the routine described in Hoek and Bray (1977). This paper is a continuation of this work. The new Windows program EzSlide has been developed using the object-oriented programming language—Visual Basic (professional edition version 5.0) that runs under either Windows 95 or Windows NT. EzSlide has the standard Windows interface complete with multiple windows, menus, toolbar, push buttons, dialog boxes and various other controls. There is an editable data input/output spreadsheet component and a twodimensional/three-dimensional graphics package for data presentation ( Fig. 1). The flow of the analysis is managed with mouse clicks or through input from the keyboard. EzSlide comprises three major parts: deterministic analysis for a single wedge; probabilistic analysis for a multi wedge system; and a collection of utilities for the post-
analysis of the results. It uses GEOSLIDE/ PROSLIDE by making use of transferable functions and subroutines. However, it is a stand-alone program with several significant additions. In GEOSLIDE/PROSLIDE, only the geological data are statistics; the external forces are entered as constants. In EzSlide, all the parameters are distributed. Another important addition is the sensitivity analysis for slope geometry; slope strike, slope dip and the slope height can be varied within the same simulation run. GEOSLIDE/PROSLIDE uses the Coulomb–Mohr shear strength parameters exclusively. EzSlide gives a choice. Either the Coulomb–Mohr or the non-linear Barton specification can be selected to represent the form of shear resistance along the sliding plane(s). EzSlide is distributed free of charge and it is available through the Internet. Download it from Lajtai’s homepage at the University of Manitoba (http://home.cc.umanitoba.ca/~lajtai). 2. The sliding wedge in deterministic analysis The stability analysis of the single wedge is the heart of both the single- and the multi-wedge
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problem. It is essentially a rigid-body analysis of a mass of rock delimited by four plane surfaces: the slope face, the top slope and the two discontinuities (planes A and B). The analysis in EzSlide uses three-dimensional-vector algebra, in essence following the routine described in Hoek and Bray (1977) and as coded by Carter and Lajtai (1992) in the DOS-based program, GEOSLIDE. In the single wedge analysis, the kinematic freedom of the wedge to slide is evaluated first. It is undertaken using the techniques outlined by Hoek and Bray (Hoek and Bray, 1977) and Goodman (1980). Three types of sliding modes are included in EzSlide: (1) wedge slide along the intersection of the two discontinuities; (2) wedge slide along the dip direction of one of the two planes; and (3) plane sliding along a single plane (Hoek and Bray, 1977). A wedge sliding along the line of intersection of two discontinuities is possible when the plunge of this line is less than the apparent dip of the slope, measured in a vertical plane containing the slide direction. Plane sliding occurs when the strike of one of the discontinuities is within ±20° of the strike of the slope and the dip of the plane is less than the dip of the slope. If the wedge is found free to slide, the geometry of the wedge is developed; the edges, the angles, the areas and the volume are computed. The external forces (water, earthquake, anchor and other external forces) are vector-summed and the resultant is resolved into components normal and parallel with the slidedirection. Following this, the safety factor, defined as a ratio of the resisting force that opposes sliding to the driving force, is computed. In EzSlide, two shear strength criteria are included. The first is the linear Mohr–Coulomb criterion representing a linear relationship between the shear strength and the normal stress:
(1976):
t=c+s tan w n
Here R , R are the resisting forces mobilized A B along plane A and B, t , t are the unit shear A B resistance of the plane A and B respectively as calculated from Eq. (2). The safety factor equaling 1.0 represents the limiting equilibrium condition. This exists when the resisting forces and driving
(1)
Here t is the peak strength, s is the effective n normal stress on the slide surface, c is unit cohesion and w the angle of friction. The second criterion is the non-linear shear strength proposed by Barton
C
A B D JCS
+w (2) B s n JRC is the joint roughness coefficient with its values varying from 0 for the smoothest to 20 for the roughest surface. JCS is the joint wall compressive strength. Barton suggested that JCS is equal to the uniaxial compressive strength s of the rock c if the joint surface is unweathered or the normal stress level is very high, and may reduce to 1/4s c if the joint walls are weathered or the normal stress is moderate to low. w is the basic friction B angle that has a range from 25° to 35° for unweathered rock surfaces. According to Barton, when the value of the term in the outside bracket of Eq. (2) exceeds 70°, Eq. (2) is no longer applicable. In EzSlide, when the value of s +JRC log B 10 (JCS/s )>70°, 70° is substituted. n When the Mohr–Coulomb criterion is selected for the shear strength of discontinuities, the safety factor for a typical wedge is computed as: t=s tan JRC log 10 n
c A +c A +N tan w +N tan w B B A A B B SF= A A D
(3)
In Eq. (3), c , c are the unit cohesion along plane A B A and B. A , A are the area of the plane A and A B B; N , N are the components of the resultant A B normal to plane A and B respectively. w , w are A B the friction angle of plane A and B. D is the driving force along the slide direction. It is calculated by resolving the resultant force parallel with the slide direction. D is treated as a vector. It is positive when it points downslope, negative when it points upslope. If Barton’s specification is used, the safety factor is given by: R +R t A +t A B= A A B B SF= A D D
(4)
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Fig. 2. A partially failed single wedge on the north side of TransCanada Highway 17A, near Kenora, Ontario. Note the block of rock at the top, which is still in position. On the assumption that the safety factor is one, this wedge was back-analyzed to find the non-linear shear strength parameters of the Barton specification.
forces are equal. A safety factor between 0 and 1 signals failure by gravitational sliding. A safety factor between 0 and −1 would also represent failure, but by sliding upwards (as it could under the effect of an oversized anchor). Single-wedge analysis concludes with the threedimensional display of the examined rock wedge, the results of intermediate calculations and the safety factor. The rock wedge can be rotated, stretched along coordinate axes, enlarged or minimized. As an example, the case of a partially failed rock wedge (Fig. 2) is examined using the single wedge analysis routine in EzSlide on the assumption that SF=1. The geometry data and a set of parameters producing SF=1 are listed in Table 1. Since, none of the strength parameters are known, there are several possible combinations that could
Fig. 3. The effect of JCS on the safety factor (w =26°). B
produce a safety factor at unity. Table 1 gives only one possible solution. To find this set of parameters, the dependence of the safety factor on the three strength parameters of the non-linear shear strength criterion were evaluated in detail. Fig. 3 shows the influence of JCS ( joint wall compressive strength) on the safety factor for three different JRCs when the basic friction angle is 26°. The results show that the safety factor increases with increasing JCS in a nonlinear form. The rate of increase is greater when JCS is <50 MPa. In general, JCS has a small influence on the safety factor when the ratio of JCS/s is large. The n normal stresses on the two planes of the Kenora site are only ca 20 kPa, which gives a large JCS/s ratio, >2000. Fig. 3 also indicates that n JCS has more of an impact on the safety factor when the JRC is large. This is perhaps how it should be. Rough surfaces would be more likely to involve the crushing of asperities, a process that is controlled by the strength of the intact rock (JCS ).
Table 1 The geometry and the estimated strength parameters of the partially failed wedge located on the north side of TransCanada Highway 17A ( Fig. 2)
Slope Top slope Plane A Plane B Height=4 m.
Azimuth (°)
Plunge (°)
JCS (MPa)
w (°) B
JRC
10 10 296 46
0 85 36 20
— — 50 50
— — 26 26
— — 5 5
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Fig. 4. The effect of the basic friction angle on the safety factor.
Fig. 5. The effect of JRC on the safety factor (w =26°). B
The effect of the basic friction angle on the safety factor is shown in Fig. 4. The safety factor increases with the basic friction angle in a linear form, at the same rate irrespective the value of JCS; the safety factor increases by ca 0.04 with every one-degree increase in the friction angle. The influence of JRC on the safety factor under different JCS values is shown in Fig. 5. The safety factor increases nonlinearly with JRC till it reaches a value of ca 14; after this the safety factor remains constant. With increasing JRC, the shear strength increases as Eq. (2). However, when the value arctan(t/s )>70°, that is, w +JRC log n B 10 (JCS/s )>70°, Barton’s Eq. (2) is no longer valid. n In this case, the assumption of w +JRC log B 10 (JCS/s )=70° is made. Therefore the safety factor n appears to become constant when JRC approaches 20. The non-linear shear strength parameters of the
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Fig. 6. The size effect on the safety factor using both the Mohr–Coulomb and the Barton specification for the shear strength of the sliding planes.
Barton specification are relatively easy to estimate following the instructions in Barton’s paper (Barton, 1976). Except for the back-analysis of a failing sliding wedge, there is no reliable way to estimate the unit cohesion of the Mohr–Coulomb law. The selection of one or the other shear strength criterion should not, however, be regarded simply as a matter of convenience. A good illustration of the difference in outcome comes from the examination of the size effect. Starting with slope height of 4 m and adjusting the parameters to yield a safety factor of ca 1.23 for both criteria, the safety factor was recalculated for several slope heights to 28 m ( Fig. 6). The safety factor based on the non-linear parameters is relatively insensitive to the size of the sliding wedge. This contrasts with the much higher sensitivity obtained while using the Mohr–Coulomb parameters.
3. Multi wedge analysis with EzSlide The multi wedge analysis uses the Monte Carlo simulation. The process starts with data entry for the slope geometry, the controlling discontinuities and the selection of options for the random variable processing of the strength parameters and/or the external forces. The simulation finishes with the display of summary results. Graphical displays for the safety factor distribution are accessed through the Post-Analysis menu. For the standard multi-wedge run, the slope
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geometry (the dip, the dip direction of the slope and the top slope and the height of the slope) is constant. The joint data are entered as a Discontinuity List, which includes the orientation and the strength parameters for each individual plane of weakness and a numeric domain (set) designation. If the joint data have not been separated into domains, EzSlide will do this on request. The discontinuity data may be entered directly from the keyboard or retrieved from a text file. Once the Discontinuity List is ready, the next step is to enter the external loading condition. The loading condition in rock slope stability problems includes the water force, the anchor force, other external forces, the earthquake loading and the surcharge. The loading condition is entered in one of two ways. The first handles forces by assuming that they are constant; the second that some or all are variables. Variability is accepted for magnitude only; the orientation of an external force is treated as constant. Since the user is rarely in position to provide the exact distribution for a parameter, it was decided to ask for only three representative values, a low, a high and a most likely value. The term most likely value is not intended to be a statistical parameter (mean, mode etc.). The user may even choose to enter the same value for all three. This would treat the parameter as a constant rather than a variable. The user is asked to identify the type of the distribution (normal, Weibull, triangular etc.), which would be the most applicable for the given parameter. In case of doubt, the best choices are the normal, Weibull and triangular distributions. In view of uncertainties about the value of parameters, the effect of selecting the ‘‘wrong’’ distribution is minor. Using the three values, EzSlide constructs the selected probability distribution using the maximum-likelihood method (MLE ) for finding the statistical parameters. For example, if the normal distribution is selected, the three representative values input by the user are used to find the two required parameters, the mean and the standard deviation. For the other distributions, the computations are somewhat more involved. Details can be found in Feng (1997), which is available in a compressed format (PingThes.Zip) and can be downloaded from the ftp server of the University of Manitoba:
ftp://ftp.cc.umanitoba.ca/pub/Rock_Mechanics. During simulation, the variates are produced form the theoretical distribution, subject to the setting of another control. The user may or may not want to produce variates that are under or above the entered low and maximum values. Therefore, the user is given the option to select either the truncated or the non-truncated version of the theoretical distribution. After all the parameters have been entered, the user is asked to select among various options that will control the run of the Monte Carlo simulation. The simulation is undertaken in one of two ways in EzSlide. The major difference lies in the treatment of the variability of joint strength parameters and the loading conditions. The first method follows the procedure used in PROSLIDE using only the Discontinuity List with its joint orientation and the strength parameters listed for each joint. In this process, both the orientation and the strength parameters of the joints are used, as they have originally been measured (estimated ). The second method ignores the strength parameters listed in the Discontinuity List and generates their value from the theoretical distributions. For selecting plane A and B of the sliding wedge, there are again two options. One governs the reuse of discontinuities to serve as either plane A or plane B. This is the ‘‘replacement’’ option. The second option instructs the program to either use or ignore the separation of joints into sets. This control is called ‘‘domain control’’. In the ‘‘replacement’’ procedure a particular combination of plane A and B is used only once if no-replacement is selected. With replacement allowed the same wedge can be reused several times. If the user selects the first method discussed above, that is, the strength parameters of the Discontinuity List are used and all the external forces are treated deterministically, there would be little point in selecting the same wedge twice. In case of ‘‘no-replacement’’, the maximum number of discontinuity combinations to form a wedge from the discontinuity population N is N*(N−1)/2. With domain control selected, the program will take planes A and B from different sets. With no domain control, planes A and B could come from the same set. The second method for the Monte Carlo simula-
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P. Feng, E.Z. Lajtai / Engineering Geology 50 (1998) 153–163 Table 2 Input for the random variables Parameter
Low
High
Most likely
Probability distribution
JCS (MPa) w (°) B JRC (No. 1) JRC (No. 2) JRC (No. 3)
25 23 4 3 3
50 26 8 9 6
37.5 24 5 5 4
Normal Triangular Normal Normal Normal
tion uses theoretical distributions to produce variates for loading and/or strength. However, the orientation of joints is still taken from the Discontinuity List, but the strength parameters in the listing are ignored, and the strength parameters are derived from a theoretical distribution. In this procedure of simulation, the same wedge can be reused several times, since the strength parameters are now random variables. There is no limit on the number of simulations. When the simulation is complete, a set of safety factors for all the kinematically free wedges is produced. The safety factors are classified under the three failure types: (1) plane sliding along a single plane; (2) wedge sliding along the intersection of two planes; (3) wedge sliding along the dip direction of one of the two planes. The safety factors are separated into two groups: those signaling failure; and those representing stable conditions. The probability of failure is defined on basis of the relative size of theses two groups. The relative frequency of occurrence ( p ) for a i failed wedge in failure mode i is: n (5) p= i i N T where n is the number of kinematically free wedges i in failure mode i and N is the total number of T wedges analyzed (total number of simulations including both the kinematically free and the notdaylighting wedges). The probability of failure for each mode p is f,i the ratio of number of failed wedges to the number
of kinematically free wedges in that mode: n p = f,i (6) f,i n i where n is the number of wedges with safety f,i factor <1 in mode i. The probability of system failure p is calcuf,sys lated according to Quek and Leung (1995): N 3 =∑ p p = F (7) f,sys f,i i N i=1 T where N is the total number of failed wedges. F N refers to the total number of wedges anaT lyzed. One can, however, interpret this in two ways: N is either the total number of discontinuity T combinations that were looked at or only those combinations that form a kinematically free wedge (N ). Depending on this definition, the computed K probability of failure for the system will be quite different. Lajtai and Carter (1989) used the number of the kinematically free wedges (N ) as the divisor: K N p = F (8) f,sys N K In EzSlide, the probability of failure is calculated and displayed both ways leaving the user in the position of the arbitrator. p
4. The Kenora example To demonstrate the use of EzSlide, the joint system measured on the south side of TransCanada highway 17A at Kenora is entered for the probabi-
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listic analysis. In the field, a representative number of joints were measured for orientation and their non-linear strength parameters were estimated at the same time. JRC was estimated through the visual comparison of the joint surface with standard profiles. Low, high and most likely values for JCS and w were estimated using references Hoek B and Bray (1977) and Barton (1976). For the simulation, the loading conditions and the strength parameters are treated as random variables. The probability distribution for JCS and JRC is assumed to be the normal distribution. The probability distribution of the basic friction angle w is taken as the triangular distribution. The three B representative values and the probability distribution for the random variables are listed in Table 2. The Monte Carlo simulation has been conducted using the ‘‘no domain control’’ and the ‘‘replacement’’ options. The only external force used is the water force for three different conditions ( Table 3). The first simulation is for dry conditions, that is, only the weight induces sliding. The other two
simulations add a water effect in the 20–40% and 30–50% range. Under dry conditions, two sliding modes are possible, but failure occurs only by the wedge sliding along the intersection. The probability of failure ( p ) in this failure mode is 14.3%. f,i The system probability of failure (P ) is 6.9% f,sys when based on N and 4.0% using N . When K T water pressure acts on the rock slope, the probability of system failure (based on N ) increases K to 8.3% (20–40% water level ) and 9.3% (30–50% water level ), respectively.
5. Optimization routines in EzSlide During the design stage, it may be possible to alter the strike or dip of the slope to reduce the probability of failure. EzSlide is an effective tool for this purpose. There are three routines for optimized computation in EzSlide. One allows the horizontal rotation of the slope strike. The second routine examines the influence of the slope angle
Table 3 Monte Carlo simulation Water level
Wedge slide along intersection
Plane slide
Total
(A) Number of kinematically free wedges in the three failure modes Dry 1410 0 20–40% 1403 0 30–50% 1397 0
1513 1539 1546
2923 2942 2925
Water level
Plane slide
Total
Wedge slide along intersection
Wedge slide along single plane
Wedge slide along single plane
(B) Number of failed wedges in the three failure modes Dry 202 0 20–40% 243 0 30–50% 273 0
Water level
Wedge slide along intersection
(C) Probability of failure (%) Dry 14.3 20–40% 17.2 30–50% 19.8
Plane slide
0 0 0
Note: N , Number of kinematically free wedges; N , total number of wedges. K T
0 0 0
System based on N K
6.9 8.3 9.3
202 243 273
System based on N T
4.0 4.9 5.5
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Fig. 7. The probability of failure as a function of the slope angle.
on the safety factor of the slope. The third one finds the effect of changing the slope height. These routines are part of the probabilistic analysis of the multi-wedge system. The optimization process involves the running of the multi wedge routine over and over again using 19 stations within a specified range of the slope strike, dip or height. At each station, a full Monte Carlo simulation is conducted. The percentage of the kinematically free wedges and the system failure probability are calculated and displayed in a graphical format. For the Kenora joint system, the percentage of kinematically free wedges and the probability of failure vary with increasing slope dip as shown in Fig. 7. When the slope dip is changed from 0 to 35°, the number of kinematically free wedges increases from 0 to 60% of the total observed wedges. However, the number of failed wedges increases only from 0 to 4.5% of the total wedges. The slope angle for this site does not have a significant influence on creating kinematically free wedges when the slope angle is above 35°. The effect of changing the slope strike on the probability of failure is shown in Fig. 8. There is an optimum slope strike in the 240–380° range. The north and south walls of Highway 17A are actually at 100 and 280°, respectively. The worst possible orientation for the highway would be at 200°.
6. Distribution of the safety factor The analysis for the distribution of the safety factor starts with the plotting of the safety factor
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Fig. 8. The probability of failure as a function of the slope strike. It changes dramatically as the slope strike is rotated.
data in a cumulative distribution form. The theoretical Weibull distribution is fitted to the cumulative data automatically. Alternatively, the safety factor data can be displayed in the form of a histogram. This in turn can be edited for range and bin size. Many researchers have analyzed the safety factor distribution based on the assumption tha it is normal (Piteau et al., 1985; Call et al., 1976, etc.). In fact the distribution of the safety factors is often skewed and almost always multi-modal. The advantage of the Weibull distribution is that it can accommodate skewed distributions. None of the distributions can, however, work with multimodal data. The most useful values of the safety factor are ca 1 and less. Therefore for the initial display, EzSlide truncates the data to the range of −1.99
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Fig. 9. The cumulative distribution and the histogram displaying safety factors in the plane sliding mode. Note multi-modality resulting from the fact that joints come in sets.
multi-modal with at least five peaks observed. Each sub-distribution corresponds to a specific set of joints.
7. Summary and Conclusions The primary purpose of this paper is to introduce the computer program EzSlide, which has been constructed to help the rock engineer in designing rock slopes in discontinuous hard rocks. In competent rocks the most common failure mode is by the gravitational sliding of blocks (wedges) of rock. The slope and the top slope surface and at least two natural discontinuities ( joints) of the rock mass delimit a potentially sliding block. For a given slope geometry, the controlling structures are the two natural discontinuities. At a particular site, the rock mass may contain many (hundreds?)
discontinuities. Any two of these may combine with the slope to form the potentially sliding wedge(s). EzSlide is designed to make the selection without any input from the user beyond supplying the joint listing. A single visit to a site to collect joint data and some educated guessing at the strength parameters is all that is required to complete a full probabilistic analysis. EzSlide is a Windows-based program with a user-friendly interface. No separate manual comes with the Program, but there is a comprehensive help system to explain and guide all aspects of the procedure. EzSlide provides functions for both the deterministic and the probabilistic analysis of sliding failure in rock slopes. In addition it offers procedures to facilitate back and sensitivity analysis and has subroutines to optimize the slope strike and the slope angle. Using the rock cuts along the TransCanada Highway 17a near Kenora, Ontario, EzSlide was put to work to demonstrate its capabilities. Using the example of a partially failed rock wedge, a back-analysis was conducted to define the nonlinear strength parameters of the Barton specification. Next a sensitivity analysis was conducted to find the major controlling parameter for the stability of the vertical rock cuts. The value of JCS is found to have an insignificant impact on the safety factor and the probability of failure at this site. This would probably be the case for all shallow slopes where the fracture of asperities is unlikely. For the Kenora rock cuts, JRC, controls stability. The probability analysis through EzSlide produces a distribution of the safety factor. The nature of the distribution, however, is complex. In almost all cases the distribution is multi-modal, presenting several peaks. The multi-modality comes from the fact that joint orientations do not form a continuous distribution; joints come in sets. The purpose of constructing EzSlide was to provide the geotechnical engineer with a computer tool to get the best estimate of slope stability based on information that is available on rock and loading conditions. When the information is inadequate or even faulty, as the case often is in rock engineering, EzSlide is no more reliable than a stability analysis conducted by other means. The
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power of EzSlide lies in its capacity to identify and analyze all the sliding wedges that can be produced based on the information provided on discontinuities. The actual values included in the safety factor distribution are accurate only if both the geometry and the strength data are accurate. Since this is rarely the case, the actual value of the safety factor for a given wedge should be viewed with extreme caution. Emphasis should rather be placed on comparative studies. One wedge may be safer than another, based on relative position within the distribution. Since the discontinuity condition (orientation and strength) does not change with slope strike and dip, comparisons of the failure probabilities produced by EzSlide for different slope geometries are always meaningful. There is always some reluctance among engineers to express stability in probabilistic terms. In communications with the public, the term probability of failure certainly sounds more threatening than safety factor. On the technical level, however, the stability of a rock slope controlled by discontinuities of variable geometry and strength cannot be described through a single value of the safety factor. Given a distribution of safety factors, there is no choice but to characterize the stability of a slope in terms of the ratio of failing and total wedges, that is, by using the term probability of failure.
Acknowledgment The authors would like to thank Laura Wytrykush for coding the stereonet procedures of EzSlide. Appreciation goes to Dr Brian Stimpson for his valuable discussions. Undergraduate students of the Geological engineering Program at
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the University of Manitoba have already used the program EzSlide. Many corrections and improvement have already been made following student feedback.
References Barton, N., 1976. The shear strength of rock and rock joints. Int. J. Rock Mech. Mining Sci. Geomech. Abstr. 13, 255–279. Call, R.D., Savely, J.P., Nicholas, D.E., 1976. Estimation of joint set characteristics from surface mapping data. In: Proceedings 17th U.S. Symposium on Rock Mechanics, Snowbird, Utah, Society of Mining Engineers of the American Institute of Mining, Metallurgical and Petroleum Engineers, New York, pp. 65–73. Carter, B.J., Lajtai, E.Z., 1992. Rock slope stability and distributed joint systems. Canada Geotech. J. 29, 53–60. Feng, P., 1997. Probabilistic treatment of the sliding wedge. Unpublished MSc. thesis, Department of Civil and Geological Engineering, University of Manitoba, Canada. Goodman, R.E., 1980. Introduction to Rock Mechanics, 2nd edn. Wiley, New York. Hoek, E., Bray, J.W., 1977. Rock Slope Engineering, 2nd edn. The Institution of Mining and Metallurgy, London. Lajtai, E.Z., Carter, B.J., 1989. GEOSLIDE—A computer code on the IBM PC for the analysis of rock slopes. Department of Civil and Geological Engineering, University of Manitoba, Winnipeg, Canada. Piteau, D.R., Stewart, A.F., Martin, D.C., Trenholme, B.S., 1985. A combined limit equilibrium statistical analysis of wedges for design of high rock slopes. In: Dowding, C.H. ( Ed.), Rock Masses: Proceedings of the Geotechnical Engineering Division. American Society of Civil Engineers, Denver, CO, pp. 93–121. Quek, S.T., Leung, C.F., 1995. Reliability-based stability analysis of rock excavations. Int. J. Rock Mech. Mining Sci. Geomech. Abstr. 32 (6), 617–620. Singh, R.N., Denby, B., Brown, D.J., 1985. The characteristics of coal measures instability in British surface mines. In: 26th US Symposium on Rock Mechanics, Rapid City, slope SD, A.A. Balkema, Rotterdam, pp. 41–48. Tamimi, S., Amadei, B., Frangopol, D.M., 1989. Monte Carlo simulation of rock slope reliability. Computer and Structures 33 (6), 1495–1505.
Probabilistic treatment of the sliding wedge with EzSlide Ping Feng *, Emery Z. Lajtai Department of Civil and Geological Engineering, University of Manitoba, Winnipeg, Manitoba, Canada R3T 5V6 Received 10 September 1997; received in revised form 14 November 1997; accepted 14 November 1997
Abstract Many of the variables that govern rock slope stability, the orientation of discontinuities, the strength parameters belonging to them and the loading conditions, are random variables. Therefore, the safety factor itself is a statistic. This paper analyzes the effect of parameter variability through the use of the Monte Carlo simulation technique. To meet this goal, a Windows based program, EzSlide written in Visual Basic has been developed. EzSlide treats both the single and the multiple-wedge problem and it is equipped with all the tools necessary to conduct a probabilistic analysis of the sliding wedge. The capabilities of the Program are introduced through an application. The field data for the demonstration comes from highway rock cuts along the TransCanada Highway 17a near Kenora, Ontario. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Computer simulation; Probability; Rock; Slope; Stability
1. Introduction Rock slope designs, particularly in hard rock, are based on the limiting equilibrium method that considers the equilibrium of forces acting on a potentially unstable wedge. The sliding wedge is formed by the slope face, the top slope surface and by two intersecting discontinuities. The limiting equilibrium method has been used for slope stability in both deterministic and probabilistic analyses. In the deterministic method, the orientation of the two discontinuities and their shear strength as well as external forces are pre-determined and treated as constant parameters. Most of the parameters, however, are characterized by uncertainty; they are not constants. The orienta* Corresponding author. Tel: (204) 474-8701; Fax: (204) 275-7514; e-mail: [email protected]
tions of rock joints surveyed from field investigation always vary, even within the same set. The strength properties of the rock mass are inherently uncertain; in material testing unique values are seldom measured. External forces such as water forces and earthquake loading are changing all the time. Therefore, a probabilistic approach to rock slope stability is more desirable. The Monte Carlo simulation technique is a numerical method that solves mathematical problems through random sampling. A computer program generates random variates to model the variation in controlling parameters. Since 1985, several computer programs have been written using the Monte Carlo simulation technique (Lajtai and Carter, 1989; Singh et al., 1985; Tamimi et al., 1989; Piteau et al., 1985; Quek and Leung, 1995). Some of them consider only plane (two-dimensional ) slides (e.g. Singh et al., 1985), others
0013-7952/98/$ – see front matter © 1998 Elsevier Science B.V. All rights reserved. PII: S0 0 1 3 -7 9 5 2 ( 9 8 ) 0 0 00 7 - 6
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Fig. 1. A typical screen output from EzSlide. An editable spreadsheet contains all the safety factors. The same data are graphed in the cumulative distribution form and the Weibull distribution is fitted to it automatically.
include wedge slides as well (Quek and Leung, 1995; Lajtai and Carter, 1989; Piteau et al., 1985). Lajtai and Carter developed two DOS-based programs, GEOSLIDE and PROSLIDE, to conduct a deterministic and a probabilistic analysis, respectively (Lajtai and Carter, 1989). The analysis uses three-dimensional-vector algebra, in essence following the routine described in Hoek and Bray (1977). This paper is a continuation of this work. The new Windows program EzSlide has been developed using the object-oriented programming language—Visual Basic (professional edition version 5.0) that runs under either Windows 95 or Windows NT. EzSlide has the standard Windows interface complete with multiple windows, menus, toolbar, push buttons, dialog boxes and various other controls. There is an editable data input/output spreadsheet component and a twodimensional/three-dimensional graphics package for data presentation ( Fig. 1). The flow of the analysis is managed with mouse clicks or through input from the keyboard. EzSlide comprises three major parts: deterministic analysis for a single wedge; probabilistic analysis for a multi wedge system; and a collection of utilities for the post-
analysis of the results. It uses GEOSLIDE/ PROSLIDE by making use of transferable functions and subroutines. However, it is a stand-alone program with several significant additions. In GEOSLIDE/PROSLIDE, only the geological data are statistics; the external forces are entered as constants. In EzSlide, all the parameters are distributed. Another important addition is the sensitivity analysis for slope geometry; slope strike, slope dip and the slope height can be varied within the same simulation run. GEOSLIDE/PROSLIDE uses the Coulomb–Mohr shear strength parameters exclusively. EzSlide gives a choice. Either the Coulomb–Mohr or the non-linear Barton specification can be selected to represent the form of shear resistance along the sliding plane(s). EzSlide is distributed free of charge and it is available through the Internet. Download it from Lajtai’s homepage at the University of Manitoba (http://home.cc.umanitoba.ca/~lajtai). 2. The sliding wedge in deterministic analysis The stability analysis of the single wedge is the heart of both the single- and the multi-wedge
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problem. It is essentially a rigid-body analysis of a mass of rock delimited by four plane surfaces: the slope face, the top slope and the two discontinuities (planes A and B). The analysis in EzSlide uses three-dimensional-vector algebra, in essence following the routine described in Hoek and Bray (1977) and as coded by Carter and Lajtai (1992) in the DOS-based program, GEOSLIDE. In the single wedge analysis, the kinematic freedom of the wedge to slide is evaluated first. It is undertaken using the techniques outlined by Hoek and Bray (Hoek and Bray, 1977) and Goodman (1980). Three types of sliding modes are included in EzSlide: (1) wedge slide along the intersection of the two discontinuities; (2) wedge slide along the dip direction of one of the two planes; and (3) plane sliding along a single plane (Hoek and Bray, 1977). A wedge sliding along the line of intersection of two discontinuities is possible when the plunge of this line is less than the apparent dip of the slope, measured in a vertical plane containing the slide direction. Plane sliding occurs when the strike of one of the discontinuities is within ±20° of the strike of the slope and the dip of the plane is less than the dip of the slope. If the wedge is found free to slide, the geometry of the wedge is developed; the edges, the angles, the areas and the volume are computed. The external forces (water, earthquake, anchor and other external forces) are vector-summed and the resultant is resolved into components normal and parallel with the slidedirection. Following this, the safety factor, defined as a ratio of the resisting force that opposes sliding to the driving force, is computed. In EzSlide, two shear strength criteria are included. The first is the linear Mohr–Coulomb criterion representing a linear relationship between the shear strength and the normal stress:
(1976):
t=c+s tan w n
Here R , R are the resisting forces mobilized A B along plane A and B, t , t are the unit shear A B resistance of the plane A and B respectively as calculated from Eq. (2). The safety factor equaling 1.0 represents the limiting equilibrium condition. This exists when the resisting forces and driving
(1)
Here t is the peak strength, s is the effective n normal stress on the slide surface, c is unit cohesion and w the angle of friction. The second criterion is the non-linear shear strength proposed by Barton
C
A B D JCS
+w (2) B s n JRC is the joint roughness coefficient with its values varying from 0 for the smoothest to 20 for the roughest surface. JCS is the joint wall compressive strength. Barton suggested that JCS is equal to the uniaxial compressive strength s of the rock c if the joint surface is unweathered or the normal stress level is very high, and may reduce to 1/4s c if the joint walls are weathered or the normal stress is moderate to low. w is the basic friction B angle that has a range from 25° to 35° for unweathered rock surfaces. According to Barton, when the value of the term in the outside bracket of Eq. (2) exceeds 70°, Eq. (2) is no longer applicable. In EzSlide, when the value of s +JRC log B 10 (JCS/s )>70°, 70° is substituted. n When the Mohr–Coulomb criterion is selected for the shear strength of discontinuities, the safety factor for a typical wedge is computed as: t=s tan JRC log 10 n
c A +c A +N tan w +N tan w B B A A B B SF= A A D
(3)
In Eq. (3), c , c are the unit cohesion along plane A B A and B. A , A are the area of the plane A and A B B; N , N are the components of the resultant A B normal to plane A and B respectively. w , w are A B the friction angle of plane A and B. D is the driving force along the slide direction. It is calculated by resolving the resultant force parallel with the slide direction. D is treated as a vector. It is positive when it points downslope, negative when it points upslope. If Barton’s specification is used, the safety factor is given by: R +R t A +t A B= A A B B SF= A D D
(4)
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Fig. 2. A partially failed single wedge on the north side of TransCanada Highway 17A, near Kenora, Ontario. Note the block of rock at the top, which is still in position. On the assumption that the safety factor is one, this wedge was back-analyzed to find the non-linear shear strength parameters of the Barton specification.
forces are equal. A safety factor between 0 and 1 signals failure by gravitational sliding. A safety factor between 0 and −1 would also represent failure, but by sliding upwards (as it could under the effect of an oversized anchor). Single-wedge analysis concludes with the threedimensional display of the examined rock wedge, the results of intermediate calculations and the safety factor. The rock wedge can be rotated, stretched along coordinate axes, enlarged or minimized. As an example, the case of a partially failed rock wedge (Fig. 2) is examined using the single wedge analysis routine in EzSlide on the assumption that SF=1. The geometry data and a set of parameters producing SF=1 are listed in Table 1. Since, none of the strength parameters are known, there are several possible combinations that could
Fig. 3. The effect of JCS on the safety factor (w =26°). B
produce a safety factor at unity. Table 1 gives only one possible solution. To find this set of parameters, the dependence of the safety factor on the three strength parameters of the non-linear shear strength criterion were evaluated in detail. Fig. 3 shows the influence of JCS ( joint wall compressive strength) on the safety factor for three different JRCs when the basic friction angle is 26°. The results show that the safety factor increases with increasing JCS in a nonlinear form. The rate of increase is greater when JCS is <50 MPa. In general, JCS has a small influence on the safety factor when the ratio of JCS/s is large. The n normal stresses on the two planes of the Kenora site are only ca 20 kPa, which gives a large JCS/s ratio, >2000. Fig. 3 also indicates that n JCS has more of an impact on the safety factor when the JRC is large. This is perhaps how it should be. Rough surfaces would be more likely to involve the crushing of asperities, a process that is controlled by the strength of the intact rock (JCS ).
Table 1 The geometry and the estimated strength parameters of the partially failed wedge located on the north side of TransCanada Highway 17A ( Fig. 2)
Slope Top slope Plane A Plane B Height=4 m.
Azimuth (°)
Plunge (°)
JCS (MPa)
w (°) B
JRC
10 10 296 46
0 85 36 20
— — 50 50
— — 26 26
— — 5 5
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Fig. 4. The effect of the basic friction angle on the safety factor.
Fig. 5. The effect of JRC on the safety factor (w =26°). B
The effect of the basic friction angle on the safety factor is shown in Fig. 4. The safety factor increases with the basic friction angle in a linear form, at the same rate irrespective the value of JCS; the safety factor increases by ca 0.04 with every one-degree increase in the friction angle. The influence of JRC on the safety factor under different JCS values is shown in Fig. 5. The safety factor increases nonlinearly with JRC till it reaches a value of ca 14; after this the safety factor remains constant. With increasing JRC, the shear strength increases as Eq. (2). However, when the value arctan(t/s )>70°, that is, w +JRC log n B 10 (JCS/s )>70°, Barton’s Eq. (2) is no longer valid. n In this case, the assumption of w +JRC log B 10 (JCS/s )=70° is made. Therefore the safety factor n appears to become constant when JRC approaches 20. The non-linear shear strength parameters of the
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Fig. 6. The size effect on the safety factor using both the Mohr–Coulomb and the Barton specification for the shear strength of the sliding planes.
Barton specification are relatively easy to estimate following the instructions in Barton’s paper (Barton, 1976). Except for the back-analysis of a failing sliding wedge, there is no reliable way to estimate the unit cohesion of the Mohr–Coulomb law. The selection of one or the other shear strength criterion should not, however, be regarded simply as a matter of convenience. A good illustration of the difference in outcome comes from the examination of the size effect. Starting with slope height of 4 m and adjusting the parameters to yield a safety factor of ca 1.23 for both criteria, the safety factor was recalculated for several slope heights to 28 m ( Fig. 6). The safety factor based on the non-linear parameters is relatively insensitive to the size of the sliding wedge. This contrasts with the much higher sensitivity obtained while using the Mohr–Coulomb parameters.
3. Multi wedge analysis with EzSlide The multi wedge analysis uses the Monte Carlo simulation. The process starts with data entry for the slope geometry, the controlling discontinuities and the selection of options for the random variable processing of the strength parameters and/or the external forces. The simulation finishes with the display of summary results. Graphical displays for the safety factor distribution are accessed through the Post-Analysis menu. For the standard multi-wedge run, the slope
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geometry (the dip, the dip direction of the slope and the top slope and the height of the slope) is constant. The joint data are entered as a Discontinuity List, which includes the orientation and the strength parameters for each individual plane of weakness and a numeric domain (set) designation. If the joint data have not been separated into domains, EzSlide will do this on request. The discontinuity data may be entered directly from the keyboard or retrieved from a text file. Once the Discontinuity List is ready, the next step is to enter the external loading condition. The loading condition in rock slope stability problems includes the water force, the anchor force, other external forces, the earthquake loading and the surcharge. The loading condition is entered in one of two ways. The first handles forces by assuming that they are constant; the second that some or all are variables. Variability is accepted for magnitude only; the orientation of an external force is treated as constant. Since the user is rarely in position to provide the exact distribution for a parameter, it was decided to ask for only three representative values, a low, a high and a most likely value. The term most likely value is not intended to be a statistical parameter (mean, mode etc.). The user may even choose to enter the same value for all three. This would treat the parameter as a constant rather than a variable. The user is asked to identify the type of the distribution (normal, Weibull, triangular etc.), which would be the most applicable for the given parameter. In case of doubt, the best choices are the normal, Weibull and triangular distributions. In view of uncertainties about the value of parameters, the effect of selecting the ‘‘wrong’’ distribution is minor. Using the three values, EzSlide constructs the selected probability distribution using the maximum-likelihood method (MLE ) for finding the statistical parameters. For example, if the normal distribution is selected, the three representative values input by the user are used to find the two required parameters, the mean and the standard deviation. For the other distributions, the computations are somewhat more involved. Details can be found in Feng (1997), which is available in a compressed format (PingThes.Zip) and can be downloaded from the ftp server of the University of Manitoba:
ftp://ftp.cc.umanitoba.ca/pub/Rock_Mechanics. During simulation, the variates are produced form the theoretical distribution, subject to the setting of another control. The user may or may not want to produce variates that are under or above the entered low and maximum values. Therefore, the user is given the option to select either the truncated or the non-truncated version of the theoretical distribution. After all the parameters have been entered, the user is asked to select among various options that will control the run of the Monte Carlo simulation. The simulation is undertaken in one of two ways in EzSlide. The major difference lies in the treatment of the variability of joint strength parameters and the loading conditions. The first method follows the procedure used in PROSLIDE using only the Discontinuity List with its joint orientation and the strength parameters listed for each joint. In this process, both the orientation and the strength parameters of the joints are used, as they have originally been measured (estimated ). The second method ignores the strength parameters listed in the Discontinuity List and generates their value from the theoretical distributions. For selecting plane A and B of the sliding wedge, there are again two options. One governs the reuse of discontinuities to serve as either plane A or plane B. This is the ‘‘replacement’’ option. The second option instructs the program to either use or ignore the separation of joints into sets. This control is called ‘‘domain control’’. In the ‘‘replacement’’ procedure a particular combination of plane A and B is used only once if no-replacement is selected. With replacement allowed the same wedge can be reused several times. If the user selects the first method discussed above, that is, the strength parameters of the Discontinuity List are used and all the external forces are treated deterministically, there would be little point in selecting the same wedge twice. In case of ‘‘no-replacement’’, the maximum number of discontinuity combinations to form a wedge from the discontinuity population N is N*(N−1)/2. With domain control selected, the program will take planes A and B from different sets. With no domain control, planes A and B could come from the same set. The second method for the Monte Carlo simula-
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P. Feng, E.Z. Lajtai / Engineering Geology 50 (1998) 153–163 Table 2 Input for the random variables Parameter
Low
High
Most likely
Probability distribution
JCS (MPa) w (°) B JRC (No. 1) JRC (No. 2) JRC (No. 3)
25 23 4 3 3
50 26 8 9 6
37.5 24 5 5 4
Normal Triangular Normal Normal Normal
tion uses theoretical distributions to produce variates for loading and/or strength. However, the orientation of joints is still taken from the Discontinuity List, but the strength parameters in the listing are ignored, and the strength parameters are derived from a theoretical distribution. In this procedure of simulation, the same wedge can be reused several times, since the strength parameters are now random variables. There is no limit on the number of simulations. When the simulation is complete, a set of safety factors for all the kinematically free wedges is produced. The safety factors are classified under the three failure types: (1) plane sliding along a single plane; (2) wedge sliding along the intersection of two planes; (3) wedge sliding along the dip direction of one of the two planes. The safety factors are separated into two groups: those signaling failure; and those representing stable conditions. The probability of failure is defined on basis of the relative size of theses two groups. The relative frequency of occurrence ( p ) for a i failed wedge in failure mode i is: n (5) p= i i N T where n is the number of kinematically free wedges i in failure mode i and N is the total number of T wedges analyzed (total number of simulations including both the kinematically free and the notdaylighting wedges). The probability of failure for each mode p is f,i the ratio of number of failed wedges to the number
of kinematically free wedges in that mode: n p = f,i (6) f,i n i where n is the number of wedges with safety f,i factor <1 in mode i. The probability of system failure p is calcuf,sys lated according to Quek and Leung (1995): N 3 =∑ p p = F (7) f,sys f,i i N i=1 T where N is the total number of failed wedges. F N refers to the total number of wedges anaT lyzed. One can, however, interpret this in two ways: N is either the total number of discontinuity T combinations that were looked at or only those combinations that form a kinematically free wedge (N ). Depending on this definition, the computed K probability of failure for the system will be quite different. Lajtai and Carter (1989) used the number of the kinematically free wedges (N ) as the divisor: K N p = F (8) f,sys N K In EzSlide, the probability of failure is calculated and displayed both ways leaving the user in the position of the arbitrator. p
4. The Kenora example To demonstrate the use of EzSlide, the joint system measured on the south side of TransCanada highway 17A at Kenora is entered for the probabi-
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listic analysis. In the field, a representative number of joints were measured for orientation and their non-linear strength parameters were estimated at the same time. JRC was estimated through the visual comparison of the joint surface with standard profiles. Low, high and most likely values for JCS and w were estimated using references Hoek B and Bray (1977) and Barton (1976). For the simulation, the loading conditions and the strength parameters are treated as random variables. The probability distribution for JCS and JRC is assumed to be the normal distribution. The probability distribution of the basic friction angle w is taken as the triangular distribution. The three B representative values and the probability distribution for the random variables are listed in Table 2. The Monte Carlo simulation has been conducted using the ‘‘no domain control’’ and the ‘‘replacement’’ options. The only external force used is the water force for three different conditions ( Table 3). The first simulation is for dry conditions, that is, only the weight induces sliding. The other two
simulations add a water effect in the 20–40% and 30–50% range. Under dry conditions, two sliding modes are possible, but failure occurs only by the wedge sliding along the intersection. The probability of failure ( p ) in this failure mode is 14.3%. f,i The system probability of failure (P ) is 6.9% f,sys when based on N and 4.0% using N . When K T water pressure acts on the rock slope, the probability of system failure (based on N ) increases K to 8.3% (20–40% water level ) and 9.3% (30–50% water level ), respectively.
5. Optimization routines in EzSlide During the design stage, it may be possible to alter the strike or dip of the slope to reduce the probability of failure. EzSlide is an effective tool for this purpose. There are three routines for optimized computation in EzSlide. One allows the horizontal rotation of the slope strike. The second routine examines the influence of the slope angle
Table 3 Monte Carlo simulation Water level
Wedge slide along intersection
Plane slide
Total
(A) Number of kinematically free wedges in the three failure modes Dry 1410 0 20–40% 1403 0 30–50% 1397 0
1513 1539 1546
2923 2942 2925
Water level
Plane slide
Total
Wedge slide along intersection
Wedge slide along single plane
Wedge slide along single plane
(B) Number of failed wedges in the three failure modes Dry 202 0 20–40% 243 0 30–50% 273 0
Water level
Wedge slide along intersection
(C) Probability of failure (%) Dry 14.3 20–40% 17.2 30–50% 19.8
Plane slide
0 0 0
Note: N , Number of kinematically free wedges; N , total number of wedges. K T
0 0 0
System based on N K
6.9 8.3 9.3
202 243 273
System based on N T
4.0 4.9 5.5
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Fig. 7. The probability of failure as a function of the slope angle.
on the safety factor of the slope. The third one finds the effect of changing the slope height. These routines are part of the probabilistic analysis of the multi-wedge system. The optimization process involves the running of the multi wedge routine over and over again using 19 stations within a specified range of the slope strike, dip or height. At each station, a full Monte Carlo simulation is conducted. The percentage of the kinematically free wedges and the system failure probability are calculated and displayed in a graphical format. For the Kenora joint system, the percentage of kinematically free wedges and the probability of failure vary with increasing slope dip as shown in Fig. 7. When the slope dip is changed from 0 to 35°, the number of kinematically free wedges increases from 0 to 60% of the total observed wedges. However, the number of failed wedges increases only from 0 to 4.5% of the total wedges. The slope angle for this site does not have a significant influence on creating kinematically free wedges when the slope angle is above 35°. The effect of changing the slope strike on the probability of failure is shown in Fig. 8. There is an optimum slope strike in the 240–380° range. The north and south walls of Highway 17A are actually at 100 and 280°, respectively. The worst possible orientation for the highway would be at 200°.
6. Distribution of the safety factor The analysis for the distribution of the safety factor starts with the plotting of the safety factor
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Fig. 8. The probability of failure as a function of the slope strike. It changes dramatically as the slope strike is rotated.
data in a cumulative distribution form. The theoretical Weibull distribution is fitted to the cumulative data automatically. Alternatively, the safety factor data can be displayed in the form of a histogram. This in turn can be edited for range and bin size. Many researchers have analyzed the safety factor distribution based on the assumption tha it is normal (Piteau et al., 1985; Call et al., 1976, etc.). In fact the distribution of the safety factors is often skewed and almost always multi-modal. The advantage of the Weibull distribution is that it can accommodate skewed distributions. None of the distributions can, however, work with multimodal data. The most useful values of the safety factor are ca 1 and less. Therefore for the initial display, EzSlide truncates the data to the range of −1.99
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Fig. 9. The cumulative distribution and the histogram displaying safety factors in the plane sliding mode. Note multi-modality resulting from the fact that joints come in sets.
multi-modal with at least five peaks observed. Each sub-distribution corresponds to a specific set of joints.
7. Summary and Conclusions The primary purpose of this paper is to introduce the computer program EzSlide, which has been constructed to help the rock engineer in designing rock slopes in discontinuous hard rocks. In competent rocks the most common failure mode is by the gravitational sliding of blocks (wedges) of rock. The slope and the top slope surface and at least two natural discontinuities ( joints) of the rock mass delimit a potentially sliding block. For a given slope geometry, the controlling structures are the two natural discontinuities. At a particular site, the rock mass may contain many (hundreds?)
discontinuities. Any two of these may combine with the slope to form the potentially sliding wedge(s). EzSlide is designed to make the selection without any input from the user beyond supplying the joint listing. A single visit to a site to collect joint data and some educated guessing at the strength parameters is all that is required to complete a full probabilistic analysis. EzSlide is a Windows-based program with a user-friendly interface. No separate manual comes with the Program, but there is a comprehensive help system to explain and guide all aspects of the procedure. EzSlide provides functions for both the deterministic and the probabilistic analysis of sliding failure in rock slopes. In addition it offers procedures to facilitate back and sensitivity analysis and has subroutines to optimize the slope strike and the slope angle. Using the rock cuts along the TransCanada Highway 17a near Kenora, Ontario, EzSlide was put to work to demonstrate its capabilities. Using the example of a partially failed rock wedge, a back-analysis was conducted to define the nonlinear strength parameters of the Barton specification. Next a sensitivity analysis was conducted to find the major controlling parameter for the stability of the vertical rock cuts. The value of JCS is found to have an insignificant impact on the safety factor and the probability of failure at this site. This would probably be the case for all shallow slopes where the fracture of asperities is unlikely. For the Kenora rock cuts, JRC, controls stability. The probability analysis through EzSlide produces a distribution of the safety factor. The nature of the distribution, however, is complex. In almost all cases the distribution is multi-modal, presenting several peaks. The multi-modality comes from the fact that joint orientations do not form a continuous distribution; joints come in sets. The purpose of constructing EzSlide was to provide the geotechnical engineer with a computer tool to get the best estimate of slope stability based on information that is available on rock and loading conditions. When the information is inadequate or even faulty, as the case often is in rock engineering, EzSlide is no more reliable than a stability analysis conducted by other means. The
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power of EzSlide lies in its capacity to identify and analyze all the sliding wedges that can be produced based on the information provided on discontinuities. The actual values included in the safety factor distribution are accurate only if both the geometry and the strength data are accurate. Since this is rarely the case, the actual value of the safety factor for a given wedge should be viewed with extreme caution. Emphasis should rather be placed on comparative studies. One wedge may be safer than another, based on relative position within the distribution. Since the discontinuity condition (orientation and strength) does not change with slope strike and dip, comparisons of the failure probabilities produced by EzSlide for different slope geometries are always meaningful. There is always some reluctance among engineers to express stability in probabilistic terms. In communications with the public, the term probability of failure certainly sounds more threatening than safety factor. On the technical level, however, the stability of a rock slope controlled by discontinuities of variable geometry and strength cannot be described through a single value of the safety factor. Given a distribution of safety factors, there is no choice but to characterize the stability of a slope in terms of the ratio of failing and total wedges, that is, by using the term probability of failure.
Acknowledgment The authors would like to thank Laura Wytrykush for coding the stereonet procedures of EzSlide. Appreciation goes to Dr Brian Stimpson for his valuable discussions. Undergraduate students of the Geological engineering Program at
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the University of Manitoba have already used the program EzSlide. Many corrections and improvement have already been made following student feedback.
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