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A new approach for application of rock mass classification on rock slope stability assessment
Engineering Geology 89 (2007) 129 – 143 www.elsevier.com/locate/enggeo
A new approach for application of rock mass classification on rock slope stability assessment Ya-Ching Liu ⁎, Chao-Shi Chen Department of Resources Engineering, National Cheng Kung University, Tainan, 701, Taiwan Received 24 May 2005; received in revised form 21 March 2006; accepted 15 September 2006 Available online 21 November 2006
Abstract The objective of this paper is to present a new rock mass classification system which can be appropriate for rock slope stability assessment. In this paper an evaluation model based on combining the Analytic Hierarchy Process (AHP) and the Fuzzy Delphi method (FDM) was presented for assessing slope rock mass quality estimates. This research treats the slope rock mass classification as a group decision problem, and applies the fuzzy logic theory as the criterion to calculate the weighting factors. In addition, several rock slopes of the Southern Cross-Island Highway in Taiwan were selected as the case study examples. After determining the slope rock mass quality estimates for each cases, the Linear Discriminant Analysis (LDA) model was used to classify those that are stable or not, and the discriminant functions which can determine failure probability of rock slopes were carried out by the LDA procedure. Afterward, the results may be compared with slope unstable hazards occurring actually, and then the relation and difference between them were discussed. Results show that the proposed method can be used to assess the stability of rock slopes according to the rock mass classification procedure and the failure probability in the early stage. © 2006 Elsevier B.V. All rights reserved. Keywords: Rock mass classification; Rock slope stability; AHP; LDA
1. Introduction Due to the complexity and uncertainty of geomechanical factors affecting underground construction, the empirical design method is still widely used in current engineering practices. In the 1970s, several rock mass classification systems were proposed for tunneling and underground excavation, which belonged to the empirical design methods with rudiments of the expert system. In the last decades, the rock mass classification concept
⁎ Corresponding author. Tel.: +886 6 275 7575x62840. E-mail address: [email protected] (Y.-C. Liu). 0013-7952/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2006.09.017
has been applied extensively in engineering design and construction such as tunnels, slopes and foundations for a long time. The main objective of rock mass classification is used to provide quantitative data and guidelines for engineering purposes that can improve originally abstract descriptions of geological formation. Until now for rock engineering, the most commonly used rock mass classification systems are the Rock Structure Rating, RSR (Wickham et al., 1972), the Rock Mass Rating, RMR (Bieniawski, 1973, 1975, 1979, 1989), and the NGI Q-system (Barton et al., 1974). However, these traditional classification systems, which ignored the regional and local geological features as well as rock properties, were constructed with the fixed weight for each rating factor.
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Moreover, initial rating systems such as the RMR were formulated for tunnel engineering but more recent methods (Selby, 1980; Haines and Terbrugge, 1991; Romana, 1991; Hack, 1998) incorporate a systemic procedure to indicate slope stability in terms of a slope rock mass quality estimate. In addition, the main island of Taiwan is relatively young in geological terms and is situated at plate borders, thus rock property, rock strength, overburden, excavation span and groundwater and differ greatly from those in the area where the well-known rock mass classifications originated. Therefore, it is necessary to start building up a new rock mass classification system, which can be suitable for determining the slope stability in Taiwan.
The main objective of this paper is to present a systemic procedure combined the Analytic Hierarchy Process (AHP, Saaty, 1980) and the Fuzzy Delphi method (FDM, Kaufmann and Gupta, 1988) for assessing slope rock mass quality estimates. This research treats rock classification as a group decision problem, and applies the fuzzy logic theory on the criterion of weighting calculations. In this paper, several rock slopes of the Southern Cross-Island Highway in Taiwan were selected as the case study examples. The proposed procedure was applied to determine the rating of rock slope with the hierarchy and weighting factors that are modified for rock slopes. After determining slope rock mass quality
Fig. 1. A flow chart of the proposed method.
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estimates for each case, the Linear Discriminant Analysis (LDA) model was used to classify those are stable or not, and the discriminant functions which can determine failure probability of rock slopes were carried out by the LDA procedure. That is the most important reason for LDA being utilized in this study. Afterward, the results may be compared with unstable slope hazards occurring actually, and then the relation and difference between them were discussed. Finally, we summarize the results to derive a slope rock mass classification system with the failure probability. Results show that the proposed method can be used to assess the stability of rock slopes in the early stage. 2. Methodology The objective of this paper is to introduce a different viewpoint to establish a rock mass quality evaluation model for slopes. In developing the analytical framework, two issues are addressed, which are expressed briefly as follows: many decisions involve criteria and goals, many of which are conflicting with some quantitative and some qualitative. We called this type of decision-making as Multiple Criteria DecisionMaking (MCDM). One of the methods employed to support MCDM is the AHP. In addition to MCDM, another key point is that groups must make decisions. It is known that group decision-making is a very important and powerful tool to accelerate the consensus of various opinions from experts, which are experienced in practices. In this section, the FDM was taken to
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synthesize their responses for the questionnaires. The FDM is a methodology in which subjective data of experts are transformed into quasi-objective data using the statistical analysis and fuzzy operations. The main advantages of FDM (Kaufmann and Gupta, 1988) are that it can reduce the numbers of surveys to save time and cost and it also includes the individual attributes of all experts. Thus that can effectively determine the weighting of each parameter with the variation of geological conditions based on only required two rounds of investigations and comprehensive discussions by a group of experts. From the above-mentioned point, the same method can be applied to the problem of slope rock mass quality evaluation. A new and simplified approach is presented for the slope engineering. Fig. 1 illustrates the flow chart of this study. 2.1. Rock mass quality evaluation analysis of slopes The major steps for evaluating slope rock mass quality are organized as follows: 1. Define the problem and determine its goal (slope rock mass quality estimates) 2. Select and determine the rock mass parameters for different types of engineering projects (such as slopes). 3. Structure the hierarchy from the top (the objectives from a decision-maker's viewpoint) through the intermediate levels (criteria on which subsequent levels depend) to the lowest level, which usually contains the list of alternatives.
Table 1 The fundamental scale of AHP (Saaty, 1980) Intensity of importance
Definition
Explanation
1 2 3
Equal importance Weak Moderate importance
4 5
Moderate plus Strong importance
6 7
Strong plus Very strong or demonstrated importance
8 9
Very, very strong Extreme importance
Reciprocals of above
If activity i has one of the above nonzero numbers assigned to it when compared with activity j, then j has the reciprocal value when compared with i Ratios arising from the scale
Two activities contribute equally to the objective – Experience and judgment slightly favor one activity over another – Experience and judgment strongly favor one activity over another – An activity is favored very strongly over another; its dominance demonstrated in practice – The evidence favoring one activity over another is of the highest possible order of affirmation A reasonable assumption
Rational
If consistency were to be forced by obtaining n numerical values to span the matrix
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4. Design the format of questionnaire items as to process according to the hierarchy in step 2. And then collect the input by a pairwise comparison of decision elements. 5. On the basis of the data obtained from the respondents through the questionnaires, construct a set of pairwise comparison matrices (size n × n) for each of the lower levels with one matrix for each element in the level immediately above by using the relative scale measurement which is the same as Saaty's scale (Table 1). The pair-wise comparisons are done in terms of which element dominates the other. 6. Use the eigenvalue method to estimate the consistence index. 7. Determine whether the input data satisfies a “consistence check”. If it does not, go back to step 1 and redo the pairwise comparisons. In this step, the inconsistency of judgments through the matrix can be captured using the largest eigenvalue, λmax. Given an n × n square matrix, a number, (λmax − n), measures the deviation of the judgments from the consistent approximation. The closer λmax is to n, the more consistent is the result. The deviation of consistency is represented by the Consistency Index (CI), which is defined as, CI ¼ ðkmax −nÞ=ðn−1Þ
ð1Þ
8. Calculate the relative fuzzy weights of the decision elements using the following three steps based on the FDM and aggregate the relative fuzzy weights to obtain scores for the decision alternation. (1) Compute the triangular fuzzy numbers (TFNs) ãij as defined in Eq. (2). In this work, the TFNs (shown as Fig. 2) that represent the pessimistic, moderate and
Fig. 2. The membership function of the Fuzzy Delphi Method.
optimistic estimate are used to represent the opinions of experts for each activity time. f ð2Þ a ¼ ða ; d ; g Þ ij
ij
ij
ij
aij ¼ Minðbijk Þ; k ¼ 1; N ; n dij ¼
n
j bijk
k¼1
ð3Þ
1=n ; k ¼ 1; N ; n
gij ¼ Maxðbijk Þ; k ¼ 1; N ; n
ð4Þ ð5Þ
Where, αij ≤ δij ≤ γij, αij,δij,γij ∈ [1/9, 1] ∪ [1, 9] and αij,δij,γij are obtained from Eq. (3) to Eq. (5). αij indicates the lower bound and γij indicates the upper bound. βijk indicates the relative intensity of importance of expert k between activities i and j. n is the number of experts in consisting of a group. (2) Following outlined above, we obtained a fuzzy ∼ positive reciprocal matrix A f f f f A ¼ ½ aij ; aij aji c1; 8i; j ¼ 1; 2; N ; n Or 2
ð1; 1; 1Þ f A ¼ 4 ð1=g12 ; 1=d12 ; 1=a12 Þ ð1=g13 ; 1=d13 ; 1=a13 Þ
3 ða13 ; d13 ; g13 Þ ða12 ; d12 ; g12 Þ ð1; 1; 1Þ ða23 ; d23 ; g23 Þ 5 ð1=g23 ; 1=d23 ; 1=a23 Þ ð1; 1; 1Þ
ð6Þ (3) Calculate the relative fuzzy weights of the evaluation factors. f f f f f Z i ¼ ½f aij N f ain 1=n ; Wi ¼ Z i ð Z i P N P Z n Þ−1 ð7Þ Where ∼ a 1 ⊗∼ a 2 ≅ (α1 × α2, δ1 × δ2, γ1 × γ2); the symbol ⊗ here denotes the multiplication of fuzzy numbers and the symbol ⊕ here denotes the addition of fuzzy ∼ numbers. Wi is a row vector in consist of a fuzzy weight ∼ of the ith factor. Wi = (ω1, ω2, …ωn), i = 1, 2, …n, and Wi is a fuzzy weight of the ith factor. The defuzzification is based on geometric average method. Appendix A illustrates in detail. Among all stages of the above description, the stage for factors and hierarchy decision is technically the most important one. Based on the findings of the field investigation, literature review and collected assistant data, 16 parameters were found relevant to slope instability. For our purposes, there are three main aspects for rock mass quality evaluation of slopes, i.e. geological, geometric, and environmental factors. Then, we can define that the slope rock mass quality estimate is equal to the summation of total weights of the three main aspects. The
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total weights were determined by the procedure described above. The proposed rock mass quality evaluating system for slopes contains four layers of hierarchy as shown in Fig. 3. In this paper, through one round of investigation papers responded to by 33 experts
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and through a series of comprehensive discussions, the assessment factors and fuzzy weights can be composed into a multidimensional questionnaire resulting in 16 variable groups as shown in Fig. 3. The numbers listed in the brackets are the total weights of each factor. The
Fig. 3. The hierarchy and the weights for evaluating slope rock mass.
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Fig. 4. The geological map of the study area (from Mei-Shan to Ya-Kou in Taiwan).
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next step was to assess the evaluation of the rock mass quality. For this purpose, we have to collect the relevant information given in Wang (2001), and then we can compute slope rock mass quality estimates combining the slope case records through an onsite survey and the algorithm introduced in this paper. By aggregating the relative weights and rating value interval for all parameters, results of the calculation for these slope ratings have been done completely using the weighted method. Finally, slope rock mass quality estimates were calculated for each slope, higher values of the rating indicate higher degrees of slope instability. 2.2. Discriminant analysis (DA) Discrimination and classification are multivariate techniques concerned with separating distinct sets of objects (or observations) and with allocating new objects (or observations) to previously defined groups. Discriminant Analysis is rather exploratory in nature. As a separative procedure, it is often employed on a onetime basis in order to investigate observed differences when causal relationships are not well understood. Classification procedures are less exploratory in the sense that they lead to well-defined rules, which can be adopted for assigning new objects.
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There are several purposes for the DA as shown in the following. (Tao, 2003) – To classify cases into groups using a discriminant prediction equation. – To investigate independent variable mean differences between groups formed by the dependent variable. – To determine the percent of variance in the dependent variable explained by the independents. – To determine the percent of variance in the dependent variable explained by the independents over and above the variance accounted for by control variables, using sequential discriminant analysis. – To assess the relative importance of the independent variables in classifying the dependent variable. – To discard variables which are little related to group distinctions. – To test theory by observing whether cases are classified as predicted. There are many possible techniques for classification of data. The Linear Discriminant Analysis (LDA) is a classical statistics approach for classifying samples of unknown classes, based on training samples with known classes. The LDA has been previously applied to sample classification of experiments data. Therefore, we adopt
Fig. 5. Photos showing the failed slopes located in this study area.
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Table 2 The rating value interval for slope rock mass quality estimate 1-1 Geological factor — geological structure: A (10), B (8), C (6), D (4), E (2) Criteria
The rating value interval
Discontinuity sets
Very few Non-conspicuous fracture A Very low N2 m A Close None A Strike Dip angle
Condition of fracture
Joint filling
Discontinuity orientation
Sliding surface roughness
None None A
Very choppy A
Few One B Lower 60 cm–2 m B Open Harder filling B N46–90W b30 B Choppy irregular B
Medium Two sets C Medium 20–60 cm C Open Hard filling C N0–45W 31–45 C Choppy regular C
More Three sets D High 6–20 cm D Open Soft filling D N0–45E 46–60 D Slightly rough D
Much more Above four E Very high b6 cm E Open Very soft E N46–90E N61 E Smooth E
1-2 Geological factor — formation Criteria
The rating value interval
Weathering
Very low Fresh
Rock strength Schmidt hardness, N (kg/cm2)
A Very high N500
Rock type
A Volcanic metamorphic rock A
Lower Slightly weathered B High 350–500 B Sandstone B
Medium Moderately weathered C Medium 200–350 C Combined layer C
High Highly weathered D Lower 75–200 D Slate D
Very high Decomposed E Very low b75 E Shale E
Lower 21–40 B
Medium 41–60 C
High 61–80 D
Very high N81 E
Gradual 31–45 B
Medium 46–60 C
Steep 61–75 D
Very steep 75–90 E
Narrow 21–40 B
Medium 41–60 C
Wide 61–80 D
Very wide N81 E
2-1 Geometric factor — height of slope Criteria
The rating value interval
Height (m)
Very low b20 A
2-2 Geometric factor — gradient Criteria
The rating value interval
Gradient (°)
Very gradual 30 A
2-3 Geometric factor — width of slope Criteria
The rating value interval
Width (m)
Very narrow b20 A
2-4 Geometric factor — inclined direction: A (10), B (7.5), C (5), D (2.5) Criteria
The rating value interval
Inclined direction
NW A
North — north, E — east, W — west, S — south
NE B
SE C
SW D
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Table 2 (continued ) 2-5 Geometric factor — orientation of the road cut: A (10), B (7.5), C (5), D (2.5) Criteria
The rating value interval
Orientation of the road cut
Dip slope Slope N dip angle A
Slope b dip angle B
Escarpment slope C
Slanting slope D
3-1 Environmental factor — groundwater Criteria
The rating value interval
Surface with or without runoff
N No trace A Completely dry A
Surface with or without seepage
N Doubtful trace B Damp B
N With C Wet C
Yes Dripping D Dripping D
Yes Flowing E Flowing E
Good 50–75% B
Medium 25–50% C
Poor 10–25% D
Very poor b10% E
3-2 Environmental factor — vegetation Criteria
The rating value interval
Vegetation (condition and density)
Perfect N75% A
3-3 Environmental factor — failure record Criteria
The rating value interval
Based on weather the slope failure happened
Never A
Ever E
A (10), E (2.5).
the LDA to classify which slope is stable or not and derive the discriminant functions. The problem we are concerned about in this paper is with separation and classification for two populations. This concept is introduced as follows: We consider the two classes labeled as Ω1 and Ω2. The objects are ordinarily separated or classified on the basis of measurements on. For instance, p associated random variables X′=[X1, X2…Xp]. The observed values of X differ to some extent from one class to the other. We can think of the totality of values from the fist class as being the population of X values for Ω1 and those from the second class as the population of X values for Ω2. These two populations can then be described by probability density functions f1(x) and f2(x), and consequently, we can talk of assigning observations to populations or objects to class interchangeably (Johnson and Wichern, 1998). 3. Cases study
method from the previous discussion. These slopes under study in this paper are located at road cuts marked by A1–A80 (the failed sites) and B1–B81 (the stable sites) along the Southern Cross-Island Highway between Mei-shan and Ya-kou in Taiwan. The geological map of this area is shown in Fig. 4. There are several prominent and recurring discontinuity sets in these road cuts. For example, the two slopes of phyllite originally with a gradient of about 64° to 80° (see Fig. 5) are both unstable. Other features that may bear upon the slope stability are as follows: 1. The common rock types are sedimentary rock and metamorphic rock. 2. There are prominent foliations in the argillite. 3. There are multiple discontinuity sets, many of which intersect each other. 4. These are presumed to have low shear strength under the forceful erosion. 5. Clay-rich weathering products are present in some well-developed discontinuity sets and shear zones.
3.1. Description of field conditions In this section a total of 161 cases of rock slopes were carried out using the rock mass quality estimation
According to the records of the onsite investigation, it can be found that the most probable mode of failure is wedge unstable. Planar unstable and toppling are the
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other two possible modes of failure. These unstable slopes present a potential hazard to passing motorists. The maximum safe angles range from 25° to 60°, which are smaller than the slope angle of most cut slopes in this region marked by 110 K to 146 K along the highway. 3.2. Data acquirement and questionnaire investigation A total of 161 slopes are examined for the rock mass quality evaluating and the results are used to assess the stability of existing road cut slopes located in the research area. More details of the geological properties of slopes are illustrated in the Master's thesis of Wang (2001). In general, many of the classification parameters for the different rock mass classifications considered are similar. Among them, the most important point, however, is the information of the intact rock strength, discontinuity conditions, and groundwater, etc. In this paper we take various parameters generated from the opinion of experts and the previous literature to determine the slope rock mass quality estimates. After ascertaining the parameters in use, the main proposed procedure was carried out. Through one round of investigation papers, the weighting hierarchy structure for slope rock mass quality evaluating is obtained as shown in Fig. 3. The total weights of parameters are listed in the brackets. 3.3. Result and discussion of slope rock mass evaluation By aggregating the relative weights and the rating value interval (see Table 2) for all parameters, results of the calculation for these slope ratings have been done completely using the weighted method. To clarify the description, an example is given as follows: Take case number A-27 and B-57 (see Table 3) as example. The slope rock mass quality estimate (shown as Table 4) for A27 and B57 are 50.1 and 73.9. These could be computed on the basis of Eqs. (8), (9): EstimateP A27 ¼ 0:3525 ½0:6024 ð0:1598 2 þ 0:216 2 þ 0:1364 2 þ 0:265 10 þ 0:2228 6Þ þ 0:3796 ð0:2403 4 þ 0:2966 4 þ 0:4631 2Þ þ 0:3403 ½0:0999 8 þ 0:2914 8 þ 0:0625 6 þ 0:3335 5 þ 0:2127 2:5 þ 0:3072 ½0:3844 6 þ 0:265 8 þ 0:3506 2 ¼ 50:1 ð8Þ
EstimateP B57 ¼ 0:3525 ½0:6024 ð0:1598 4 þ 0:216 8 þ 0:3164 4 þ 0:265 10 þ 0:2228 10Þ þ 0:3796 ð0:2403 4 þ 0:2966 4 þ 0:4631 8Þ þ 0:3403 ½0:0999 6 þ 0:2914 6 þ 0:0625 6 þ 0:3335 7:5 þ 0:2127 2:5 þ 0:3072 ½0:3844 9 þ 0:265 10 þ 0:3506 10 ¼ 73:9 ð9Þ All case records are partially listed in Table 3 and plotted in scattering Fig. 6. Fig. 6 shows the correlation between the observed behaviors (failed or stable) and the slope rock mass quality estimates. To proceed further into an analysis of the results, the statistical analysis tool is used and the related results are summarized as follows. From the results as shown in Table 4, the statistical analysis presents the spread and variability of the ratings for the failed and the stable group. The distribution can be seen in Fig. 7. The stable group exhibits a wide range (78.9–58.4) of values, with a mean of 68.2 and a welldefined peak (standard deviation 5.2). In contrast, the failed group presents a narrower range (66.9–39.1) of values, with a mean of 51.4 and a flat peak (standard deviation 5.81). Initial rough analysis shows that the stable slope rock mass must be rated over 65 points. Moreover in order to avoid any slope instability risk, the slope rated value below 55 points should be taken to support or to prevent a possible instability in the foreseeable future. The empirically found limit value range Table 3 Partial case records of slope site investigation in this study (numbers in the table present the score for the parameters) Case no.
A27
A28
A29
A30
B57
B58
Discontinuity sets Condition of fracture Joint filling Discontinuity orientation Sliding surface roughness Rock type Rock strength Weathering Height of slope Gradient of slope Width of slope Inclined direction of slope Orientation of the road cut Ground water Vegetation Failure record
2 2 2 10 6 4 4 2 8 8 6 5 2.5 6 8 2
2 4 2 3 8 6 6 6 6 6 4 2.5 7.5 7 8 2
2 6 8 10 6 8 8 8 8 6 10 5 2.5 8 8 2
2 8 2 10 4 8 10 8 8 6 8 5 2.5 6 8 2
4 8 4 10 10 4 4 8 6 6 6 7.5 2.5 9 10 10
4 8 2 5 10 4 4 6 6 8 2 5 7.5 6 10 10
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Table 4 The slope rock mass estimates partly listed Case no.
A1
A2
A3
A13
A27
A30
B57
B81
Site (km) Estimate
110.2 74.1
110.5 49.37
110.8 59.96
112.95 66.69
122.3 50.1
117.05 77.52
133.2 73.9
146.3 46.66
of slope rating for whether unstable or not is 55 to 65 points. No slope has been rated as slope rock mass quality estimated below 39 points in this study. To proceed further into a definition of the best criterion to classify the two slope groups in this study, we will make use of the DA. 4. Approach to classify rock slopes stability
discriminant function. In the next stage, we present the LDA model and the corresponding results. As mentioned above, we have to construct a model for classifying the failed slope group and the stable one. These two groups and the data corresponding required to the factor were analyzed by means of the DA to obtain a linear model. In this section, the software used in LDA was SPSS Standard Version 12.0 for the relevant calculations. The SPSS uses the statistical decision theory method for
4.1. The linear discriminant analysis (LDA) model The data given in Table 4 are used for discussing the analytical approach to the DA. The ratings of two slope groups were determined by the proposed procedure. Fig. 7 indicates a histogram for the both groups. Table 5 gives means and standard deviations for the two groups. Using an independent sample t-test (see Table 6) can assess the differences in the means of the two groups. The t-value for testing equality of the means of the two groups is − 19.45 for slope rock mass quality estimates. The t-test suggests that the two groups are significantly different with respect to slope rock mass quality estimates at a significance level of 0.05. That is, slope rock mass quality estimate does discriminate between the two groups and consequently will be used to create the
Fig. 6. The correlation between the observed behaviors (failed or stable) and the slope rock mass quality estimate.
Fig. 7. The histogram for two group slopes: (a) failed slopes, (b) stable slopes.
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Table 5 The descriptive statistics of the slope rock mass quality estimates for two groups
Table 7 The summarized results of LDA by SPSS (a) Classification processing summary
Failed or not Group Valid N Mean
Statistic Std. error
Median Minimum Maximum Std. deviation Interquartile range
1 (failed)
2 (stable)
80 51.35 0.65 50.68 39.08 66.86 5.81 7.85
81 68.24 0.58 67.76 58.40 78.92 5.20 8.76
Processed Excluded Missing or out-of-range group codes At least one missing discriminating variable
0 0 161
Used in output (b) Prior probabilities for groups
classifying sample observations into various groups. This method minimizes misclassification errors that are taking into account prior probabilities and misclassification costs. From results of SPSS shown in Table 7, it can be seen that this model is a linear combination of the slope rock mass quality estimates and the constant item, which separate the two groups in the best way. SPSS computes the classification functions of Fisher's linear discriminant function mode as follows: The function equation for the failed slope is d1 : Y ðX Þ ¼ −44:11 þ 1:6909X1
161
Group
Prior
1 2 Total
.497 .503 1.000
Case used in analysis Unweighted
Weighted
80 81 161
80.000 81.000 161.000
(c) Classification function coefficients Group 1
2
Rock mass estimate 1.691 Constant −44.104 Fisher's linear discriminant functions
2.247 − 77.354
(d) Classification matrix (p = . meant the prior probability for the group) Group
Classification matrix (total score) Rows: observed classifications
ð10Þ
Columns: predicted classifications
The function equation for the stable slope is d2 : ZðX Þ ¼ −77:3622 þ 2:2472X1
Percent
ð11Þ G_1:1 G_2:2 Total
Y(X ) and Z(X ) are referred to as Fisher's linear discriminant function and X1 is the slope rock mass quality estimate. Observations are assigned to the group with the largest classification score. Assign the observation to group d1 if Y(X ) N Z(X ) and to group d2 if Y(X ) b Z(X ). Table 8 shows the output generated by the DA procedure in SPSS. We can see the correct classification rate is 91.93%. How good is this classification rate? Huberty (1984) has proposed approximate test statistics
G_1:1
G_2:2
Correct
p=.
p=.
90.00000 93.82716 91.92547
72 5 77
49,689
50,311
8 76 84
that can be used to evaluate the statistical and the practical significance of the overall classification rate given. Our method achieves a statistically significant result with respect to the current situation.
Table 6 The independent sample t-test result for distinguishing between the differences in the means of the two groups T-tests: grouping: category (total score) Group 1:1 Variable
Group 2:2 Mean 1
Mean 2
t-value
df
p
Valid N1
Valid N2
St.d 1
St.d 2
F-ratio
P
Rock mass estimate
51.35
68.24
− 19.449
159
0.0
80
81
5.81
5.20
1.249
0.324
St.d: Standard deviation; df: degrees of freedom.
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Table 8 Result of the LDA model
X0 X1
Constant Slope rock mass quality estimate
Failed (d1)
Stable (d2)
−44.1100 1.6909
− 77.3622 2.2472
4.2. The probability of slope failure (Lin, 2001) In this paper an equation, Eq. (12) to calculate the failure probability P(%) of rock slopes based on linear discriminant functions obtained in previous analysis is introduced in the following. eY P¼ Y ðe þ eZ Þ
ð12Þ
Where Y is the linear discriminant function of the failed slopes and Z is the linear discriminant function of the stable slopes. Substituting Eq. (10) and Eq. (11) into Eq. (12), we can obtain P¼
Fig. 8. The failure probability distribution.
e−44:11þ1:6909X1 þ e−77:3622þ2:2472X1 Þ
ð13Þ
ðe−44:11þ1:6909X1
Now that the equation is derived, we can take the ratings of the two slope groups into account the failure probability for each slope. The results are obtained from the failure probability analysis. Table 9 partially shows the failure probability of slopes and the category. The failure probability distribution as shown in Fig. 8 indicates that the cases considered in this study are extreme examples (failed or stable). 4.3. The stability classification of rock slope As previously discussed, a probabilistic approach using the DA functions was applied in the analysis to quantify the slope stability. To summarize our interpretation of the results, a proposed classification as shown in Table 10 can be categorized according to their slope rock mass quality estimates, the failure probability and the procedure to deal with during the monitoring period.
There are three types of class I, class II (IIA, IIB, IIC, IID) and class III, which are used to classify the stability of rock slopes. The calculation of the failure probability of a slope with the new method developed in this study gives a more distinctive differentiation between failed and stable conditions than with existing rock mass classification systems for slopes (Kentli and Topal, 2004). 5. Conclusions The main results of this paper are twofold. The first is that the proposed method has provided a useful basis for introducing quantitative rock mass assessment into rock slope engineering, where there are numerous potential failure locations and several different failure cases at the Southern Cross-Island Highway in Taiwan. The proposed method can be successfully applied to determine the rock mass quality estimates for rock slopes. Results roughly show that it can be used to determine the rock mass quality estimates of slopes as a simple safety assessment method. A second result of this paper presents that the LDA model can be applied to distinguish between the failed and stable slope group. And this analysis also assesses to establish the discriminant functions to evaluate the failure probability of the slopes. On the basis of our earlier discussion we could conclude that this new proposed method can be used to judge the stability problems
Table 9 Partially showing the probability of slope failure and the category Case number
1
2
3
4
5
6
7
8
9
Marked (km) Category Failure probability
110.2 2 0.03%
110.5 1 99.69%
110.8 2 47.41%
111 1 86.53%
111.2 2 15.16%
111.3 1 99.98%
111.5 1 99.12%
111.6 2 6.87%
111.8 2 0.03%
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Table 10 The slope stability classified based on slope quality estimate and failure probability
1. Compute the triangular fuzzy numbers (TFNs) ãij = (αij, δij, γij) According Eq. (3)–Eq. (5)
Class FDAHP rock mass estimate
Failure Stability probability class
Recommended treatment
aij ¼ MinðBijk Þ; k ¼ 1; N ; n
I
78–100
0%
None
II A II B
61–77 59–60
II C
56–58
II D
42–55
b15% 60.6%– 46.9% 89.1 %– 72.9% N93%
III
41–0
100%
Fully stable Stable Tend to be unstable More unstable Unstable
Fully unstable
Usual monitoring Up frequent measure Add support to wall or concrete Protection technique needed and very often revise support used Re-excavation and design
of rock slopes according to the proposed procedure and the failure probability. Acknowledgments
In all acceptable responses for matrix 1, n is equal to 15 (the other 8 responses were under the mark and rejected. The ij subscript notation here represents that i is 1 to 3 and j is 1 to 3. Thus from Appendix Table C-1, we could know B121 = 1/6, B124 = 4, B134 = 6, B235 = 1/3, B211 = 6, B214 = 1/4, B314 = 1/6, B325 = 3,…etc. Derived from above description, we can obtain the triangular fuzzy numbers (TFNs) f a ¼ ða ; d ; g Þk ¼ 1; N ; 15 ij
ij
ij
ij
a12 ¼ MinðB12k Þ ¼ 1=6; a13 ¼ MinðB13k Þ ¼ 1=7; a23 ¼ MinðB23k Þ ¼ 1=5 15 1=15 ¼ 1:333534; d13 ¼ 1:510459; d12 ¼ j B12k k¼1 15 1=15 ¼ 0:969667 d23 ¼ j B23k k¼1
This study is supported by the research project of Taiwan National Science Council through contract number NSC-91-2211-E-006-049. Comments by two anonymous reviewers are greatly acknowledged. Appendix A. The weighting factors determination Among the 33 survey respondents, 4 were from university professor in geological engineering area and 29 were from government, engineering consultants companies and the other organizations. Based on the survey responses, the proposed method was applied and evaluation model was constructed. The weighting factors for each criterion listed in this study hierarchy were presented in the following explanation. Referred to the article described in Section 2.1, take matrix 1 as an exemplification. Table A-1 All acceptable responses for the matrix 1 Geological Geometric factor factor
Environmental factor
Geological factor
1
(1/7,1,1,6,3,7,1, 3,3,1/5,6,5,3,1/6,1)
Geometric factor Environmental factor
Positive reciprocal Positive reciprocal
(1/6,1,1,4,5, 1/3,1,3,5,1,5,3, 1/2,1/5,3) 1 Positive reciprocal
(1/4,1,1,3,1/3,7,1,3, 1/3,1/5,3,3,3,1/3,1/5) 1
g12 ¼ MaxðB12k Þ ¼ 5; g13 ¼ MaxðB13k Þ ¼ 7; g23 ¼ MaxðB23k Þ ¼ 7 According the positive reciprocal rule, aij Vdij Vgij ; aij ; dij ; gij ϵ 19 ; 1 [ ½1; 9 f a21 ¼ ð1=a12 ; 1=d12 ; 1=g12 Þ ¼ ð6; 1=1:333534; 1=5Þ ¼ ð6; 0:749887; 0:2Þ Because 0.2b 0.749887 b 6 thus ã21 = (0.2, 0.749887, 6) In the same way, ã31 = (1/7, 0.66205, 7) f a21 ¼ ð0:2; 0:749887; 6Þ; f a32 ¼ ð1=7; 1:031282; 5Þ 2. Following outlined above, we obtained a fuzzy positive reciprocal matrix à 2 ð1; 1; 1Þ f 4 A ¼ ð0:2; 0:749887; 6Þ ð1=7; 0:66205; 7Þ
3 ð1=6; 1:333534; 5Þ ð1=7; 1:510459; 7Þ ð1; 1; 1Þ ð1=5; 0:969667; 7Þ 5 ð1=7; 1:031282; 5Þ ð1; 1; 1Þ
3. Calculate the relative fuzzy weights of the evaluation factors. f Z 1 ¼ ½f a11 f a12 f a13 1=3 ¼ ½0:2877; 1:2629; 3:2711; f Z 2 ¼ ½f a21 f a22 f a23 1=3 ¼ ½0:3420; 0:8992; 3:4760; f a31 f a32 f a33 1=3 ¼ ½0:2733; 0:8806; 3:2711; Z 3 ¼ ½f Xf Z i ¼ ½0:9030; 3:0427; 10:0182
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Reciprocal ∑Z˜i = [0.998, 0.3287, 1.1075] f f f f f W 1 ¼ Z 1 ð Z 1 P Z 2 P Z 3 Þ−1 ¼ ½0:0287; 0:4151; 3:6226 f W 2 ¼ ½0:0341; 0:2955; 3:8496; f W 3 ¼ ½0:0273; 0:2894; 3:6226 1=3 Therefore, W1 ¼ j3i¼1 xj ¼ 0:3525; W2 ¼ 0:3403; W3 ¼ 0:3072 The weighting factors for three aspects are: geological factor (0.3525), geometrical factor (0.3403) and environmental factor (0.3072). References Barton, N., Lien, R., Lunde, J., 1974. Engineering classification of rock masses for the design of tunnel support. Rock Mechanics 6 (4), 189–236. Bieniawski, Z.T., 1973. Engineering classification of jointed rock masses. Transactions of the South African Institution of Civil Engineers 15 (12), 335–344. Bieniawski, Z.T., 1975. Case studies: prediction of rock mass behavior by the geomechanics classification. Proc. 2nd Australia–New Zealand Conference Geomechanics, Brisbane, pp. 36–41. Bieniawski, Z.T., 1979. The geomechanics classification in rock engineering application. Proceedings Fourth Congress of the International Society for Rock Mechanics, vol. 2, pp. 41–48. Bieniawski, Z.T., 1989. Engineering Rock Mass Classifications. Wiley, New York. 251 pp. Hack, H.R.G.K., 1998. Slope Stability Probability Classification, vol. 43. ITC Delft Publication, Netherlands, Enschede. 273 pp.
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Haines, A., Terbrugge, P.J., 1991. Preliminary estimation of rock slope stability using rock mass classification systems. In: Wittke, W. (Ed.), Proc 7th Congr on Rock Mechanics 2, ISRM, Aachen, Germany. Balkema, Rotterdam, pp. 887–892. Huberty, C.J., 1984. Issues in the use and interpretation of discriminant analysis. Psychological Bulletin 95 (1), 156–171. Johnson, R.A., Wichern, D.W., 1998. Applied Multivariate Statistical Analysis. Prentice-Hall, Englewood Cliffs, N.J. Kaufmann, A., Gupta, M.M., 1988. Fuzzy Mathematical Models in Engineering and Management Science. North-Holland, Amsterdam. Kentli, B., Topal, T., 2004. Assessment of rock slope stability for a segment of the Ankara–Pozanti motorway, Turkey. Engineering Geology 74, 73–90. Lin, Y.C., 2001. Application of Discriminate Functions on Evaluating Failure Probability of Slopes, Master's thesis in Taiwan (in Chinese), 179 pp. Romana, M., 1991. SMR Classification. In: Wittke, W. (Ed.), Proc 7th Congr on Rock Mechanics 2, ISRM, Aachenm, Germany. Balkema, Rotterdam, pp. 955–960. Saaty, T.L., 1980. The Analytic Hierarchy Process. McGraw-Hill International Book Company, New York. Selby, M.J., 1980. A rock mass strength classification for geomorphic purposes: with tests from Antarctica and New Zealand. Zeitschrift für Geomorphologie 23, 31–51. Tao, Li, 2003. Using discriminant analysis for multi-class classification. Third IEEE International Conference on Data Mining, vol. 11, p. 589. Wickham, G.E., Tiedemann, H.R., Skinner, E.H., 1972. Support determination based on geologic predictions. Proc. Conf. Rapid Excavation and Tunneling, pp. 43–64. Wang, J.R., 2001. Analysis of the engineering geological parameters in highway rock slope stability assessment by field investigation, Master's thesis in Taiwan (in Chinese), 74 pp.
A new approach for application of rock mass classification on rock slope stability assessment Ya-Ching Liu ⁎, Chao-Shi Chen Department of Resources Engineering, National Cheng Kung University, Tainan, 701, Taiwan Received 24 May 2005; received in revised form 21 March 2006; accepted 15 September 2006 Available online 21 November 2006
Abstract The objective of this paper is to present a new rock mass classification system which can be appropriate for rock slope stability assessment. In this paper an evaluation model based on combining the Analytic Hierarchy Process (AHP) and the Fuzzy Delphi method (FDM) was presented for assessing slope rock mass quality estimates. This research treats the slope rock mass classification as a group decision problem, and applies the fuzzy logic theory as the criterion to calculate the weighting factors. In addition, several rock slopes of the Southern Cross-Island Highway in Taiwan were selected as the case study examples. After determining the slope rock mass quality estimates for each cases, the Linear Discriminant Analysis (LDA) model was used to classify those that are stable or not, and the discriminant functions which can determine failure probability of rock slopes were carried out by the LDA procedure. Afterward, the results may be compared with slope unstable hazards occurring actually, and then the relation and difference between them were discussed. Results show that the proposed method can be used to assess the stability of rock slopes according to the rock mass classification procedure and the failure probability in the early stage. © 2006 Elsevier B.V. All rights reserved. Keywords: Rock mass classification; Rock slope stability; AHP; LDA
1. Introduction Due to the complexity and uncertainty of geomechanical factors affecting underground construction, the empirical design method is still widely used in current engineering practices. In the 1970s, several rock mass classification systems were proposed for tunneling and underground excavation, which belonged to the empirical design methods with rudiments of the expert system. In the last decades, the rock mass classification concept
⁎ Corresponding author. Tel.: +886 6 275 7575x62840. E-mail address: [email protected] (Y.-C. Liu). 0013-7952/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2006.09.017
has been applied extensively in engineering design and construction such as tunnels, slopes and foundations for a long time. The main objective of rock mass classification is used to provide quantitative data and guidelines for engineering purposes that can improve originally abstract descriptions of geological formation. Until now for rock engineering, the most commonly used rock mass classification systems are the Rock Structure Rating, RSR (Wickham et al., 1972), the Rock Mass Rating, RMR (Bieniawski, 1973, 1975, 1979, 1989), and the NGI Q-system (Barton et al., 1974). However, these traditional classification systems, which ignored the regional and local geological features as well as rock properties, were constructed with the fixed weight for each rating factor.
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Moreover, initial rating systems such as the RMR were formulated for tunnel engineering but more recent methods (Selby, 1980; Haines and Terbrugge, 1991; Romana, 1991; Hack, 1998) incorporate a systemic procedure to indicate slope stability in terms of a slope rock mass quality estimate. In addition, the main island of Taiwan is relatively young in geological terms and is situated at plate borders, thus rock property, rock strength, overburden, excavation span and groundwater and differ greatly from those in the area where the well-known rock mass classifications originated. Therefore, it is necessary to start building up a new rock mass classification system, which can be suitable for determining the slope stability in Taiwan.
The main objective of this paper is to present a systemic procedure combined the Analytic Hierarchy Process (AHP, Saaty, 1980) and the Fuzzy Delphi method (FDM, Kaufmann and Gupta, 1988) for assessing slope rock mass quality estimates. This research treats rock classification as a group decision problem, and applies the fuzzy logic theory on the criterion of weighting calculations. In this paper, several rock slopes of the Southern Cross-Island Highway in Taiwan were selected as the case study examples. The proposed procedure was applied to determine the rating of rock slope with the hierarchy and weighting factors that are modified for rock slopes. After determining slope rock mass quality
Fig. 1. A flow chart of the proposed method.
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estimates for each case, the Linear Discriminant Analysis (LDA) model was used to classify those are stable or not, and the discriminant functions which can determine failure probability of rock slopes were carried out by the LDA procedure. That is the most important reason for LDA being utilized in this study. Afterward, the results may be compared with unstable slope hazards occurring actually, and then the relation and difference between them were discussed. Finally, we summarize the results to derive a slope rock mass classification system with the failure probability. Results show that the proposed method can be used to assess the stability of rock slopes in the early stage. 2. Methodology The objective of this paper is to introduce a different viewpoint to establish a rock mass quality evaluation model for slopes. In developing the analytical framework, two issues are addressed, which are expressed briefly as follows: many decisions involve criteria and goals, many of which are conflicting with some quantitative and some qualitative. We called this type of decision-making as Multiple Criteria DecisionMaking (MCDM). One of the methods employed to support MCDM is the AHP. In addition to MCDM, another key point is that groups must make decisions. It is known that group decision-making is a very important and powerful tool to accelerate the consensus of various opinions from experts, which are experienced in practices. In this section, the FDM was taken to
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synthesize their responses for the questionnaires. The FDM is a methodology in which subjective data of experts are transformed into quasi-objective data using the statistical analysis and fuzzy operations. The main advantages of FDM (Kaufmann and Gupta, 1988) are that it can reduce the numbers of surveys to save time and cost and it also includes the individual attributes of all experts. Thus that can effectively determine the weighting of each parameter with the variation of geological conditions based on only required two rounds of investigations and comprehensive discussions by a group of experts. From the above-mentioned point, the same method can be applied to the problem of slope rock mass quality evaluation. A new and simplified approach is presented for the slope engineering. Fig. 1 illustrates the flow chart of this study. 2.1. Rock mass quality evaluation analysis of slopes The major steps for evaluating slope rock mass quality are organized as follows: 1. Define the problem and determine its goal (slope rock mass quality estimates) 2. Select and determine the rock mass parameters for different types of engineering projects (such as slopes). 3. Structure the hierarchy from the top (the objectives from a decision-maker's viewpoint) through the intermediate levels (criteria on which subsequent levels depend) to the lowest level, which usually contains the list of alternatives.
Table 1 The fundamental scale of AHP (Saaty, 1980) Intensity of importance
Definition
Explanation
1 2 3
Equal importance Weak Moderate importance
4 5
Moderate plus Strong importance
6 7
Strong plus Very strong or demonstrated importance
8 9
Very, very strong Extreme importance
Reciprocals of above
If activity i has one of the above nonzero numbers assigned to it when compared with activity j, then j has the reciprocal value when compared with i Ratios arising from the scale
Two activities contribute equally to the objective – Experience and judgment slightly favor one activity over another – Experience and judgment strongly favor one activity over another – An activity is favored very strongly over another; its dominance demonstrated in practice – The evidence favoring one activity over another is of the highest possible order of affirmation A reasonable assumption
Rational
If consistency were to be forced by obtaining n numerical values to span the matrix
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4. Design the format of questionnaire items as to process according to the hierarchy in step 2. And then collect the input by a pairwise comparison of decision elements. 5. On the basis of the data obtained from the respondents through the questionnaires, construct a set of pairwise comparison matrices (size n × n) for each of the lower levels with one matrix for each element in the level immediately above by using the relative scale measurement which is the same as Saaty's scale (Table 1). The pair-wise comparisons are done in terms of which element dominates the other. 6. Use the eigenvalue method to estimate the consistence index. 7. Determine whether the input data satisfies a “consistence check”. If it does not, go back to step 1 and redo the pairwise comparisons. In this step, the inconsistency of judgments through the matrix can be captured using the largest eigenvalue, λmax. Given an n × n square matrix, a number, (λmax − n), measures the deviation of the judgments from the consistent approximation. The closer λmax is to n, the more consistent is the result. The deviation of consistency is represented by the Consistency Index (CI), which is defined as, CI ¼ ðkmax −nÞ=ðn−1Þ
ð1Þ
8. Calculate the relative fuzzy weights of the decision elements using the following three steps based on the FDM and aggregate the relative fuzzy weights to obtain scores for the decision alternation. (1) Compute the triangular fuzzy numbers (TFNs) ãij as defined in Eq. (2). In this work, the TFNs (shown as Fig. 2) that represent the pessimistic, moderate and
Fig. 2. The membership function of the Fuzzy Delphi Method.
optimistic estimate are used to represent the opinions of experts for each activity time. f ð2Þ a ¼ ða ; d ; g Þ ij
ij
ij
ij
aij ¼ Minðbijk Þ; k ¼ 1; N ; n dij ¼
n
j bijk
k¼1
ð3Þ
1=n ; k ¼ 1; N ; n
gij ¼ Maxðbijk Þ; k ¼ 1; N ; n
ð4Þ ð5Þ
Where, αij ≤ δij ≤ γij, αij,δij,γij ∈ [1/9, 1] ∪ [1, 9] and αij,δij,γij are obtained from Eq. (3) to Eq. (5). αij indicates the lower bound and γij indicates the upper bound. βijk indicates the relative intensity of importance of expert k between activities i and j. n is the number of experts in consisting of a group. (2) Following outlined above, we obtained a fuzzy ∼ positive reciprocal matrix A f f f f A ¼ ½ aij ; aij aji c1; 8i; j ¼ 1; 2; N ; n Or 2
ð1; 1; 1Þ f A ¼ 4 ð1=g12 ; 1=d12 ; 1=a12 Þ ð1=g13 ; 1=d13 ; 1=a13 Þ
3 ða13 ; d13 ; g13 Þ ða12 ; d12 ; g12 Þ ð1; 1; 1Þ ða23 ; d23 ; g23 Þ 5 ð1=g23 ; 1=d23 ; 1=a23 Þ ð1; 1; 1Þ
ð6Þ (3) Calculate the relative fuzzy weights of the evaluation factors. f f f f f Z i ¼ ½f aij N f ain 1=n ; Wi ¼ Z i ð Z i P N P Z n Þ−1 ð7Þ Where ∼ a 1 ⊗∼ a 2 ≅ (α1 × α2, δ1 × δ2, γ1 × γ2); the symbol ⊗ here denotes the multiplication of fuzzy numbers and the symbol ⊕ here denotes the addition of fuzzy ∼ numbers. Wi is a row vector in consist of a fuzzy weight ∼ of the ith factor. Wi = (ω1, ω2, …ωn), i = 1, 2, …n, and Wi is a fuzzy weight of the ith factor. The defuzzification is based on geometric average method. Appendix A illustrates in detail. Among all stages of the above description, the stage for factors and hierarchy decision is technically the most important one. Based on the findings of the field investigation, literature review and collected assistant data, 16 parameters were found relevant to slope instability. For our purposes, there are three main aspects for rock mass quality evaluation of slopes, i.e. geological, geometric, and environmental factors. Then, we can define that the slope rock mass quality estimate is equal to the summation of total weights of the three main aspects. The
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total weights were determined by the procedure described above. The proposed rock mass quality evaluating system for slopes contains four layers of hierarchy as shown in Fig. 3. In this paper, through one round of investigation papers responded to by 33 experts
133
and through a series of comprehensive discussions, the assessment factors and fuzzy weights can be composed into a multidimensional questionnaire resulting in 16 variable groups as shown in Fig. 3. The numbers listed in the brackets are the total weights of each factor. The
Fig. 3. The hierarchy and the weights for evaluating slope rock mass.
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Fig. 4. The geological map of the study area (from Mei-Shan to Ya-Kou in Taiwan).
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next step was to assess the evaluation of the rock mass quality. For this purpose, we have to collect the relevant information given in Wang (2001), and then we can compute slope rock mass quality estimates combining the slope case records through an onsite survey and the algorithm introduced in this paper. By aggregating the relative weights and rating value interval for all parameters, results of the calculation for these slope ratings have been done completely using the weighted method. Finally, slope rock mass quality estimates were calculated for each slope, higher values of the rating indicate higher degrees of slope instability. 2.2. Discriminant analysis (DA) Discrimination and classification are multivariate techniques concerned with separating distinct sets of objects (or observations) and with allocating new objects (or observations) to previously defined groups. Discriminant Analysis is rather exploratory in nature. As a separative procedure, it is often employed on a onetime basis in order to investigate observed differences when causal relationships are not well understood. Classification procedures are less exploratory in the sense that they lead to well-defined rules, which can be adopted for assigning new objects.
135
There are several purposes for the DA as shown in the following. (Tao, 2003) – To classify cases into groups using a discriminant prediction equation. – To investigate independent variable mean differences between groups formed by the dependent variable. – To determine the percent of variance in the dependent variable explained by the independents. – To determine the percent of variance in the dependent variable explained by the independents over and above the variance accounted for by control variables, using sequential discriminant analysis. – To assess the relative importance of the independent variables in classifying the dependent variable. – To discard variables which are little related to group distinctions. – To test theory by observing whether cases are classified as predicted. There are many possible techniques for classification of data. The Linear Discriminant Analysis (LDA) is a classical statistics approach for classifying samples of unknown classes, based on training samples with known classes. The LDA has been previously applied to sample classification of experiments data. Therefore, we adopt
Fig. 5. Photos showing the failed slopes located in this study area.
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Table 2 The rating value interval for slope rock mass quality estimate 1-1 Geological factor — geological structure: A (10), B (8), C (6), D (4), E (2) Criteria
The rating value interval
Discontinuity sets
Very few Non-conspicuous fracture A Very low N2 m A Close None A Strike Dip angle
Condition of fracture
Joint filling
Discontinuity orientation
Sliding surface roughness
None None A
Very choppy A
Few One B Lower 60 cm–2 m B Open Harder filling B N46–90W b30 B Choppy irregular B
Medium Two sets C Medium 20–60 cm C Open Hard filling C N0–45W 31–45 C Choppy regular C
More Three sets D High 6–20 cm D Open Soft filling D N0–45E 46–60 D Slightly rough D
Much more Above four E Very high b6 cm E Open Very soft E N46–90E N61 E Smooth E
1-2 Geological factor — formation Criteria
The rating value interval
Weathering
Very low Fresh
Rock strength Schmidt hardness, N (kg/cm2)
A Very high N500
Rock type
A Volcanic metamorphic rock A
Lower Slightly weathered B High 350–500 B Sandstone B
Medium Moderately weathered C Medium 200–350 C Combined layer C
High Highly weathered D Lower 75–200 D Slate D
Very high Decomposed E Very low b75 E Shale E
Lower 21–40 B
Medium 41–60 C
High 61–80 D
Very high N81 E
Gradual 31–45 B
Medium 46–60 C
Steep 61–75 D
Very steep 75–90 E
Narrow 21–40 B
Medium 41–60 C
Wide 61–80 D
Very wide N81 E
2-1 Geometric factor — height of slope Criteria
The rating value interval
Height (m)
Very low b20 A
2-2 Geometric factor — gradient Criteria
The rating value interval
Gradient (°)
Very gradual 30 A
2-3 Geometric factor — width of slope Criteria
The rating value interval
Width (m)
Very narrow b20 A
2-4 Geometric factor — inclined direction: A (10), B (7.5), C (5), D (2.5) Criteria
The rating value interval
Inclined direction
NW A
North — north, E — east, W — west, S — south
NE B
SE C
SW D
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Table 2 (continued ) 2-5 Geometric factor — orientation of the road cut: A (10), B (7.5), C (5), D (2.5) Criteria
The rating value interval
Orientation of the road cut
Dip slope Slope N dip angle A
Slope b dip angle B
Escarpment slope C
Slanting slope D
3-1 Environmental factor — groundwater Criteria
The rating value interval
Surface with or without runoff
N No trace A Completely dry A
Surface with or without seepage
N Doubtful trace B Damp B
N With C Wet C
Yes Dripping D Dripping D
Yes Flowing E Flowing E
Good 50–75% B
Medium 25–50% C
Poor 10–25% D
Very poor b10% E
3-2 Environmental factor — vegetation Criteria
The rating value interval
Vegetation (condition and density)
Perfect N75% A
3-3 Environmental factor — failure record Criteria
The rating value interval
Based on weather the slope failure happened
Never A
Ever E
A (10), E (2.5).
the LDA to classify which slope is stable or not and derive the discriminant functions. The problem we are concerned about in this paper is with separation and classification for two populations. This concept is introduced as follows: We consider the two classes labeled as Ω1 and Ω2. The objects are ordinarily separated or classified on the basis of measurements on. For instance, p associated random variables X′=[X1, X2…Xp]. The observed values of X differ to some extent from one class to the other. We can think of the totality of values from the fist class as being the population of X values for Ω1 and those from the second class as the population of X values for Ω2. These two populations can then be described by probability density functions f1(x) and f2(x), and consequently, we can talk of assigning observations to populations or objects to class interchangeably (Johnson and Wichern, 1998). 3. Cases study
method from the previous discussion. These slopes under study in this paper are located at road cuts marked by A1–A80 (the failed sites) and B1–B81 (the stable sites) along the Southern Cross-Island Highway between Mei-shan and Ya-kou in Taiwan. The geological map of this area is shown in Fig. 4. There are several prominent and recurring discontinuity sets in these road cuts. For example, the two slopes of phyllite originally with a gradient of about 64° to 80° (see Fig. 5) are both unstable. Other features that may bear upon the slope stability are as follows: 1. The common rock types are sedimentary rock and metamorphic rock. 2. There are prominent foliations in the argillite. 3. There are multiple discontinuity sets, many of which intersect each other. 4. These are presumed to have low shear strength under the forceful erosion. 5. Clay-rich weathering products are present in some well-developed discontinuity sets and shear zones.
3.1. Description of field conditions In this section a total of 161 cases of rock slopes were carried out using the rock mass quality estimation
According to the records of the onsite investigation, it can be found that the most probable mode of failure is wedge unstable. Planar unstable and toppling are the
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other two possible modes of failure. These unstable slopes present a potential hazard to passing motorists. The maximum safe angles range from 25° to 60°, which are smaller than the slope angle of most cut slopes in this region marked by 110 K to 146 K along the highway. 3.2. Data acquirement and questionnaire investigation A total of 161 slopes are examined for the rock mass quality evaluating and the results are used to assess the stability of existing road cut slopes located in the research area. More details of the geological properties of slopes are illustrated in the Master's thesis of Wang (2001). In general, many of the classification parameters for the different rock mass classifications considered are similar. Among them, the most important point, however, is the information of the intact rock strength, discontinuity conditions, and groundwater, etc. In this paper we take various parameters generated from the opinion of experts and the previous literature to determine the slope rock mass quality estimates. After ascertaining the parameters in use, the main proposed procedure was carried out. Through one round of investigation papers, the weighting hierarchy structure for slope rock mass quality evaluating is obtained as shown in Fig. 3. The total weights of parameters are listed in the brackets. 3.3. Result and discussion of slope rock mass evaluation By aggregating the relative weights and the rating value interval (see Table 2) for all parameters, results of the calculation for these slope ratings have been done completely using the weighted method. To clarify the description, an example is given as follows: Take case number A-27 and B-57 (see Table 3) as example. The slope rock mass quality estimate (shown as Table 4) for A27 and B57 are 50.1 and 73.9. These could be computed on the basis of Eqs. (8), (9): EstimateP A27 ¼ 0:3525 ½0:6024 ð0:1598 2 þ 0:216 2 þ 0:1364 2 þ 0:265 10 þ 0:2228 6Þ þ 0:3796 ð0:2403 4 þ 0:2966 4 þ 0:4631 2Þ þ 0:3403 ½0:0999 8 þ 0:2914 8 þ 0:0625 6 þ 0:3335 5 þ 0:2127 2:5 þ 0:3072 ½0:3844 6 þ 0:265 8 þ 0:3506 2 ¼ 50:1 ð8Þ
EstimateP B57 ¼ 0:3525 ½0:6024 ð0:1598 4 þ 0:216 8 þ 0:3164 4 þ 0:265 10 þ 0:2228 10Þ þ 0:3796 ð0:2403 4 þ 0:2966 4 þ 0:4631 8Þ þ 0:3403 ½0:0999 6 þ 0:2914 6 þ 0:0625 6 þ 0:3335 7:5 þ 0:2127 2:5 þ 0:3072 ½0:3844 9 þ 0:265 10 þ 0:3506 10 ¼ 73:9 ð9Þ All case records are partially listed in Table 3 and plotted in scattering Fig. 6. Fig. 6 shows the correlation between the observed behaviors (failed or stable) and the slope rock mass quality estimates. To proceed further into an analysis of the results, the statistical analysis tool is used and the related results are summarized as follows. From the results as shown in Table 4, the statistical analysis presents the spread and variability of the ratings for the failed and the stable group. The distribution can be seen in Fig. 7. The stable group exhibits a wide range (78.9–58.4) of values, with a mean of 68.2 and a welldefined peak (standard deviation 5.2). In contrast, the failed group presents a narrower range (66.9–39.1) of values, with a mean of 51.4 and a flat peak (standard deviation 5.81). Initial rough analysis shows that the stable slope rock mass must be rated over 65 points. Moreover in order to avoid any slope instability risk, the slope rated value below 55 points should be taken to support or to prevent a possible instability in the foreseeable future. The empirically found limit value range Table 3 Partial case records of slope site investigation in this study (numbers in the table present the score for the parameters) Case no.
A27
A28
A29
A30
B57
B58
Discontinuity sets Condition of fracture Joint filling Discontinuity orientation Sliding surface roughness Rock type Rock strength Weathering Height of slope Gradient of slope Width of slope Inclined direction of slope Orientation of the road cut Ground water Vegetation Failure record
2 2 2 10 6 4 4 2 8 8 6 5 2.5 6 8 2
2 4 2 3 8 6 6 6 6 6 4 2.5 7.5 7 8 2
2 6 8 10 6 8 8 8 8 6 10 5 2.5 8 8 2
2 8 2 10 4 8 10 8 8 6 8 5 2.5 6 8 2
4 8 4 10 10 4 4 8 6 6 6 7.5 2.5 9 10 10
4 8 2 5 10 4 4 6 6 8 2 5 7.5 6 10 10
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Table 4 The slope rock mass estimates partly listed Case no.
A1
A2
A3
A13
A27
A30
B57
B81
Site (km) Estimate
110.2 74.1
110.5 49.37
110.8 59.96
112.95 66.69
122.3 50.1
117.05 77.52
133.2 73.9
146.3 46.66
of slope rating for whether unstable or not is 55 to 65 points. No slope has been rated as slope rock mass quality estimated below 39 points in this study. To proceed further into a definition of the best criterion to classify the two slope groups in this study, we will make use of the DA. 4. Approach to classify rock slopes stability
discriminant function. In the next stage, we present the LDA model and the corresponding results. As mentioned above, we have to construct a model for classifying the failed slope group and the stable one. These two groups and the data corresponding required to the factor were analyzed by means of the DA to obtain a linear model. In this section, the software used in LDA was SPSS Standard Version 12.0 for the relevant calculations. The SPSS uses the statistical decision theory method for
4.1. The linear discriminant analysis (LDA) model The data given in Table 4 are used for discussing the analytical approach to the DA. The ratings of two slope groups were determined by the proposed procedure. Fig. 7 indicates a histogram for the both groups. Table 5 gives means and standard deviations for the two groups. Using an independent sample t-test (see Table 6) can assess the differences in the means of the two groups. The t-value for testing equality of the means of the two groups is − 19.45 for slope rock mass quality estimates. The t-test suggests that the two groups are significantly different with respect to slope rock mass quality estimates at a significance level of 0.05. That is, slope rock mass quality estimate does discriminate between the two groups and consequently will be used to create the
Fig. 6. The correlation between the observed behaviors (failed or stable) and the slope rock mass quality estimate.
Fig. 7. The histogram for two group slopes: (a) failed slopes, (b) stable slopes.
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Table 5 The descriptive statistics of the slope rock mass quality estimates for two groups
Table 7 The summarized results of LDA by SPSS (a) Classification processing summary
Failed or not Group Valid N Mean
Statistic Std. error
Median Minimum Maximum Std. deviation Interquartile range
1 (failed)
2 (stable)
80 51.35 0.65 50.68 39.08 66.86 5.81 7.85
81 68.24 0.58 67.76 58.40 78.92 5.20 8.76
Processed Excluded Missing or out-of-range group codes At least one missing discriminating variable
0 0 161
Used in output (b) Prior probabilities for groups
classifying sample observations into various groups. This method minimizes misclassification errors that are taking into account prior probabilities and misclassification costs. From results of SPSS shown in Table 7, it can be seen that this model is a linear combination of the slope rock mass quality estimates and the constant item, which separate the two groups in the best way. SPSS computes the classification functions of Fisher's linear discriminant function mode as follows: The function equation for the failed slope is d1 : Y ðX Þ ¼ −44:11 þ 1:6909X1
161
Group
Prior
1 2 Total
.497 .503 1.000
Case used in analysis Unweighted
Weighted
80 81 161
80.000 81.000 161.000
(c) Classification function coefficients Group 1
2
Rock mass estimate 1.691 Constant −44.104 Fisher's linear discriminant functions
2.247 − 77.354
(d) Classification matrix (p = . meant the prior probability for the group) Group
Classification matrix (total score) Rows: observed classifications
ð10Þ
Columns: predicted classifications
The function equation for the stable slope is d2 : ZðX Þ ¼ −77:3622 þ 2:2472X1
Percent
ð11Þ G_1:1 G_2:2 Total
Y(X ) and Z(X ) are referred to as Fisher's linear discriminant function and X1 is the slope rock mass quality estimate. Observations are assigned to the group with the largest classification score. Assign the observation to group d1 if Y(X ) N Z(X ) and to group d2 if Y(X ) b Z(X ). Table 8 shows the output generated by the DA procedure in SPSS. We can see the correct classification rate is 91.93%. How good is this classification rate? Huberty (1984) has proposed approximate test statistics
G_1:1
G_2:2
Correct
p=.
p=.
90.00000 93.82716 91.92547
72 5 77
49,689
50,311
8 76 84
that can be used to evaluate the statistical and the practical significance of the overall classification rate given. Our method achieves a statistically significant result with respect to the current situation.
Table 6 The independent sample t-test result for distinguishing between the differences in the means of the two groups T-tests: grouping: category (total score) Group 1:1 Variable
Group 2:2 Mean 1
Mean 2
t-value
df
p
Valid N1
Valid N2
St.d 1
St.d 2
F-ratio
P
Rock mass estimate
51.35
68.24
− 19.449
159
0.0
80
81
5.81
5.20
1.249
0.324
St.d: Standard deviation; df: degrees of freedom.
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Table 8 Result of the LDA model
X0 X1
Constant Slope rock mass quality estimate
Failed (d1)
Stable (d2)
−44.1100 1.6909
− 77.3622 2.2472
4.2. The probability of slope failure (Lin, 2001) In this paper an equation, Eq. (12) to calculate the failure probability P(%) of rock slopes based on linear discriminant functions obtained in previous analysis is introduced in the following. eY P¼ Y ðe þ eZ Þ
ð12Þ
Where Y is the linear discriminant function of the failed slopes and Z is the linear discriminant function of the stable slopes. Substituting Eq. (10) and Eq. (11) into Eq. (12), we can obtain P¼
Fig. 8. The failure probability distribution.
e−44:11þ1:6909X1 þ e−77:3622þ2:2472X1 Þ
ð13Þ
ðe−44:11þ1:6909X1
Now that the equation is derived, we can take the ratings of the two slope groups into account the failure probability for each slope. The results are obtained from the failure probability analysis. Table 9 partially shows the failure probability of slopes and the category. The failure probability distribution as shown in Fig. 8 indicates that the cases considered in this study are extreme examples (failed or stable). 4.3. The stability classification of rock slope As previously discussed, a probabilistic approach using the DA functions was applied in the analysis to quantify the slope stability. To summarize our interpretation of the results, a proposed classification as shown in Table 10 can be categorized according to their slope rock mass quality estimates, the failure probability and the procedure to deal with during the monitoring period.
There are three types of class I, class II (IIA, IIB, IIC, IID) and class III, which are used to classify the stability of rock slopes. The calculation of the failure probability of a slope with the new method developed in this study gives a more distinctive differentiation between failed and stable conditions than with existing rock mass classification systems for slopes (Kentli and Topal, 2004). 5. Conclusions The main results of this paper are twofold. The first is that the proposed method has provided a useful basis for introducing quantitative rock mass assessment into rock slope engineering, where there are numerous potential failure locations and several different failure cases at the Southern Cross-Island Highway in Taiwan. The proposed method can be successfully applied to determine the rock mass quality estimates for rock slopes. Results roughly show that it can be used to determine the rock mass quality estimates of slopes as a simple safety assessment method. A second result of this paper presents that the LDA model can be applied to distinguish between the failed and stable slope group. And this analysis also assesses to establish the discriminant functions to evaluate the failure probability of the slopes. On the basis of our earlier discussion we could conclude that this new proposed method can be used to judge the stability problems
Table 9 Partially showing the probability of slope failure and the category Case number
1
2
3
4
5
6
7
8
9
Marked (km) Category Failure probability
110.2 2 0.03%
110.5 1 99.69%
110.8 2 47.41%
111 1 86.53%
111.2 2 15.16%
111.3 1 99.98%
111.5 1 99.12%
111.6 2 6.87%
111.8 2 0.03%
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Table 10 The slope stability classified based on slope quality estimate and failure probability
1. Compute the triangular fuzzy numbers (TFNs) ãij = (αij, δij, γij) According Eq. (3)–Eq. (5)
Class FDAHP rock mass estimate
Failure Stability probability class
Recommended treatment
aij ¼ MinðBijk Þ; k ¼ 1; N ; n
I
78–100
0%
None
II A II B
61–77 59–60
II C
56–58
II D
42–55
b15% 60.6%– 46.9% 89.1 %– 72.9% N93%
III
41–0
100%
Fully stable Stable Tend to be unstable More unstable Unstable
Fully unstable
Usual monitoring Up frequent measure Add support to wall or concrete Protection technique needed and very often revise support used Re-excavation and design
of rock slopes according to the proposed procedure and the failure probability. Acknowledgments
In all acceptable responses for matrix 1, n is equal to 15 (the other 8 responses were under the mark and rejected. The ij subscript notation here represents that i is 1 to 3 and j is 1 to 3. Thus from Appendix Table C-1, we could know B121 = 1/6, B124 = 4, B134 = 6, B235 = 1/3, B211 = 6, B214 = 1/4, B314 = 1/6, B325 = 3,…etc. Derived from above description, we can obtain the triangular fuzzy numbers (TFNs) f a ¼ ða ; d ; g Þk ¼ 1; N ; 15 ij
ij
ij
ij
a12 ¼ MinðB12k Þ ¼ 1=6; a13 ¼ MinðB13k Þ ¼ 1=7; a23 ¼ MinðB23k Þ ¼ 1=5 15 1=15 ¼ 1:333534; d13 ¼ 1:510459; d12 ¼ j B12k k¼1 15 1=15 ¼ 0:969667 d23 ¼ j B23k k¼1
This study is supported by the research project of Taiwan National Science Council through contract number NSC-91-2211-E-006-049. Comments by two anonymous reviewers are greatly acknowledged. Appendix A. The weighting factors determination Among the 33 survey respondents, 4 were from university professor in geological engineering area and 29 were from government, engineering consultants companies and the other organizations. Based on the survey responses, the proposed method was applied and evaluation model was constructed. The weighting factors for each criterion listed in this study hierarchy were presented in the following explanation. Referred to the article described in Section 2.1, take matrix 1 as an exemplification. Table A-1 All acceptable responses for the matrix 1 Geological Geometric factor factor
Environmental factor
Geological factor
1
(1/7,1,1,6,3,7,1, 3,3,1/5,6,5,3,1/6,1)
Geometric factor Environmental factor
Positive reciprocal Positive reciprocal
(1/6,1,1,4,5, 1/3,1,3,5,1,5,3, 1/2,1/5,3) 1 Positive reciprocal
(1/4,1,1,3,1/3,7,1,3, 1/3,1/5,3,3,3,1/3,1/5) 1
g12 ¼ MaxðB12k Þ ¼ 5; g13 ¼ MaxðB13k Þ ¼ 7; g23 ¼ MaxðB23k Þ ¼ 7 According the positive reciprocal rule, aij Vdij Vgij ; aij ; dij ; gij ϵ 19 ; 1 [ ½1; 9 f a21 ¼ ð1=a12 ; 1=d12 ; 1=g12 Þ ¼ ð6; 1=1:333534; 1=5Þ ¼ ð6; 0:749887; 0:2Þ Because 0.2b 0.749887 b 6 thus ã21 = (0.2, 0.749887, 6) In the same way, ã31 = (1/7, 0.66205, 7) f a21 ¼ ð0:2; 0:749887; 6Þ; f a32 ¼ ð1=7; 1:031282; 5Þ 2. Following outlined above, we obtained a fuzzy positive reciprocal matrix à 2 ð1; 1; 1Þ f 4 A ¼ ð0:2; 0:749887; 6Þ ð1=7; 0:66205; 7Þ
3 ð1=6; 1:333534; 5Þ ð1=7; 1:510459; 7Þ ð1; 1; 1Þ ð1=5; 0:969667; 7Þ 5 ð1=7; 1:031282; 5Þ ð1; 1; 1Þ
3. Calculate the relative fuzzy weights of the evaluation factors. f Z 1 ¼ ½f a11 f a12 f a13 1=3 ¼ ½0:2877; 1:2629; 3:2711; f Z 2 ¼ ½f a21 f a22 f a23 1=3 ¼ ½0:3420; 0:8992; 3:4760; f a31 f a32 f a33 1=3 ¼ ½0:2733; 0:8806; 3:2711; Z 3 ¼ ½f Xf Z i ¼ ½0:9030; 3:0427; 10:0182
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Reciprocal ∑Z˜i = [0.998, 0.3287, 1.1075] f f f f f W 1 ¼ Z 1 ð Z 1 P Z 2 P Z 3 Þ−1 ¼ ½0:0287; 0:4151; 3:6226 f W 2 ¼ ½0:0341; 0:2955; 3:8496; f W 3 ¼ ½0:0273; 0:2894; 3:6226 1=3 Therefore, W1 ¼ j3i¼1 xj ¼ 0:3525; W2 ¼ 0:3403; W3 ¼ 0:3072 The weighting factors for three aspects are: geological factor (0.3525), geometrical factor (0.3403) and environmental factor (0.3072). References Barton, N., Lien, R., Lunde, J., 1974. Engineering classification of rock masses for the design of tunnel support. Rock Mechanics 6 (4), 189–236. Bieniawski, Z.T., 1973. Engineering classification of jointed rock masses. Transactions of the South African Institution of Civil Engineers 15 (12), 335–344. Bieniawski, Z.T., 1975. Case studies: prediction of rock mass behavior by the geomechanics classification. Proc. 2nd Australia–New Zealand Conference Geomechanics, Brisbane, pp. 36–41. Bieniawski, Z.T., 1979. The geomechanics classification in rock engineering application. Proceedings Fourth Congress of the International Society for Rock Mechanics, vol. 2, pp. 41–48. Bieniawski, Z.T., 1989. Engineering Rock Mass Classifications. Wiley, New York. 251 pp. Hack, H.R.G.K., 1998. Slope Stability Probability Classification, vol. 43. ITC Delft Publication, Netherlands, Enschede. 273 pp.
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