a new approach for application of rock mass classification

(2007) 129–143www.elsevier.com/locate/enggeo

Engineering Geology 89

A new approach for application of rock mass classificationon 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 2006Available 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 stabilityassessment. In this paper an evaluation model based on combining the Analytic Hierarchy Process (AHP) and the Fuzzy Delphimethod (FDM)was presented for assessing slope rockmass quality estimates. This research treats the slope rockmass classification asa group decision problem, and applies the fuzzy logic theory as the criterion to calculate theweighting factors. In addition, several rockslopes of the Southern Cross-Island Highway in Taiwan were selected as the case study examples. After determining the slope rockmass 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 betweenthem were discussed. Results show that the proposed method can be used to assess the stability of rock slopes according to the rockmass 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 geomecha-nical factors affecting underground construction, theempirical design method is still widely used in currentengineering practices. In the 1970s, several rock massclassification systems were proposed for tunneling andunderground excavation, which belonged to the empir-ical 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 andconstruction such as tunnels, slopes and foundations for along time. The main objective of rock mass classificationis used to provide quantitative data and guidelines forengineering purposes that can improve originally abstractdescriptions of geological formation. Until now for rockengineering, the most commonly used rock massclassification systems are the Rock Structure Rating,RSR (Wickham et al., 1972), the Rock Mass Rating,RMR (Bieniawski, 1973, 1975, 1979, 1989), and the NGIQ-system (Barton et al., 1974). However, these traditionalclassification systems, which ignored the regional andlocal geological features as well as rock properties, wereconstructed with the fixed weight for each rating factor.

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130 Y.-C. Liu, C.-S. Chen / Engineering Geology 89 (2007) 129–143

Moreover, initial rating systems such as the RMR wereformulated for tunnel engineering but more recent meth-ods (Selby, 1980; Haines and Terbrugge, 1991; Romana,1991; Hack, 1998) incorporate a systemic procedure toindicate slope stability in terms of a slope rock massquality estimate. In addition, the main island of Taiwan isrelatively young in geological terms and is situated atplate borders, thus rock property, rock strength, overbur-den, excavation span and groundwater and differ greatlyfrom those in the area where the well-known rock massclassifications originated. Therefore, it is necessary tostart building up a new rock mass classification system,which can be suitable for determining the slope stability inTaiwan.

Fig. 1. A flow chart of th

The main objective of this paper is to present a sys-temic procedure combined the Analytic Hierarchy Pro-cess (AHP, Saaty, 1980) and the Fuzzy Delphi method(FDM, Kaufmann and Gupta, 1988) for assessing sloperock mass quality estimates. This research treats rockclassification as a group decision problem, and appliesthe fuzzy logic theory on the criterion of weightingcalculations.

In this paper, several rock slopes of the SouthernCross-Island Highway in Taiwan were selected as thecase study examples. The proposed procedure wasapplied to determine the rating of rock slope with thehierarchy and weighting factors that are modified forrock slopes. After determining slope rock mass quality

e proposed method.

131Y.-C. Liu, C.-S. Chen / Engineering Geology 89 (2007) 129–143

estimates for each case, the Linear Discriminant Anal-ysis (LDA) model was used to classify those are stable ornot, and the discriminant functions which can determinefailure probability of rock slopes were carried out by theLDA procedure. That is the most important reason forLDA being utilized in this study. Afterward, the resultsmay be compared with unstable slope hazards occurringactually, and then the relation and difference betweenthem were discussed. Finally, we summarize the resultsto derive a slope rock mass classification system with thefailure probability. Results show that the proposed meth-od can be used to assess the stability of rock slopes in theearly stage.

2. Methodology

The objective of this paper is to introduce a differentviewpoint to establish a rock mass quality evaluationmodel for slopes. In developing the analytical frame-work, two issues are addressed, which are expressedbriefly as follows: many decisions involve criteria andgoals, many of which are conflicting with somequantitative and some qualitative. We called this typeof decision-making as Multiple Criteria Decision-Making (MCDM). One of the methods employed tosupport MCDM is the AHP. In addition to MCDM,another key point is that groups must make decisions. Itis known that group decision-making is a very importantand powerful tool to accelerate the consensus of variousopinions from experts, which are experienced inpractices. In this section, the FDM was taken to

Table 1The fundamental scale of AHP (Saaty, 1980)

Intensity of importance Definition

1 Equal importance2 Weak3 Moderate importance

4 Moderate plus5 Strong importance

6 Strong plus7 Very strong or demonstrated importance

8 Very, very strong9 Extreme importance

Reciprocals of above If activity i has one of the above nonzero numbewhen compared with activity j, then j has the recwhen compared with i

Rational Ratios arising from the scale

synthesize their responses for the questionnaires. TheFDM is a methodology in which subjective data ofexperts are transformed into quasi-objective data usingthe statistical analysis and fuzzy operations. The mainadvantages of FDM (Kaufmann and Gupta, 1988) arethat it can reduce the numbers of surveys to save time andcost and it also includes the individual attributes of allexperts. Thus that can effectively determine theweightingof each parameter with the variation of geological con-ditions based on only required two rounds of inves-tigations and comprehensive discussions by a group ofexperts. From the above-mentioned point, the samemeth-od can be applied to the problem of slope rock massquality evaluation. A new and simplified approach ispresented for the slope engineering. Fig. 1 illustrates theflow chart of this study.

2.1. Rock mass quality evaluation analysis of slopes

The major steps for evaluating slope rock mass qual-ity are organized as follows:

1. Define the problem and determine its goal (slope rockmass quality estimates)

2. Select and determine the rock mass parameters fordifferent types of engineering projects (such as slopes).

3. Structure the hierarchy from the top (the objectivesfrom a decision-maker's viewpoint) through the in-termediate levels (criteria on which subsequent levelsdepend) to the lowest level, which usually containsthe list of alternatives.

Explanation

Two activities contribute equally to the objective–Experience and judgment slightly favor oneactivity over another–Experience and judgment strongly favor oneactivity over another–An activity is favored very strongly over another;its dominance demonstrated in practice–The evidence favoring one activity over another isof the highest possible order of affirmation

rs assigned to itiprocal value

A reasonable assumption

If consistency were to be forced by obtaining nnumerical values to span the matrix


4. Design the format of questionnaire items as to pro-cess according to the hierarchy in step 2. And thencollect the input by a pairwise comparison of de-cision elements.

5. On the basis of the data obtained from the respondentsthrough the questionnaires, construct a set of pair-wise comparison matrices (size n×n) for each of thelower levels with one matrix for each element in thelevel immediately above by using the relative scalemeasurement which is the same as Saaty's scale(Table 1). The pair-wise comparisons are done interms of which element dominates the other.

6. Use the eigenvalue method to estimate the consis-tence index.

7. Determine whether the input data satisfies a “consis-tence check”. If it does not, go back to step 1 and redothe pairwise comparisons. In this step, the inconsis-tency of judgments through the matrix can be cap-tured using the largest eigenvalue, λmax. Given ann×n square matrix, a number, (λmax−n), measuresthe deviation of the judgments from the consistentapproximation. The closer λmax is to n, the moreconsistent is the result. The deviation of consistency isrepresented by the Consistency Index (CI), which isdefined as,

CI ¼ ðkmax−nÞ=ðn−1Þ ð1Þ8. Calculate the relative fuzzy weights of the decision

elements using the following three steps based on theFDM and aggregate the relative fuzzy weights toobtain scores for the decision alternation.

(1) Compute the triangular fuzzy numbers (TFNs)ãij as defined in Eq. (2). In this work, the TFNs (shownas 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 ofexperts for each activity time.

faij ¼ ðaij; dij; gijÞ ð2Þ

aij ¼ MinðbijkÞ; k ¼ 1; N ; n ð3Þ

dij ¼ jn

k¼1bijk

� �1=n

; k ¼ 1; N ; n ð4Þ

gij ¼ MaxðbijkÞ; k ¼ 1; N ; n ð5Þ

Where, αij≤δij≤γij, αij,δij,γij∈ [1/9, 1]∪ [1, 9] andαij,δij,γij are obtained from Eq. (3) to Eq. (5). αijindicates the lower bound and γij indicates the upperbound. βijk indicates the relative intensity of importanceof expert k between activities i and j. n is the number ofexperts in consisting of a group.

(2) Following outlined above, we obtained a fuzzypositive reciprocal matrix A

∼

fA ¼ ½faij�;faij �fajic1; 8i; j ¼ 1; 2; N ; n

Or

fA ¼

ð1; 1; 1Þ ða12; d12;g12Þ ða13; d13;g13Þð1=g12; 1=d12; 1=a12Þ ð1; 1; 1Þ ða23; d23;g23Þð1=g13; 1=d13; 1=a13Þ ð1=g23; 1=d23; 1=a23Þ ð1; 1; 1Þ

24

35

ð6Þ(3) Calculate the relative fuzzy weights of the eval-

uation factors.

fZi ¼ ½faij � N �fain�1=n;fWi ¼ fZi � ðfZiP N PfZnÞ−1

ð7Þ

Where a∼1⊗a∼2≅ (α1×α2, δ1×δ2, γ1×γ2); the sym-bol ⊗ here denotes the multiplication of fuzzy numbersand the symbol ⊕ here denotes the addition of fuzzynumbers. W

∼i is a row vector in consist of a fuzzy weight

of the ith factor. W∼i=(ω1, ω2, …ωn), i=1, 2, …n, and Wi

is a fuzzy weight of the ith factor. The defuzzificationis based on geometric average method. Appendix Aillustrates in detail.

Among all stages of the above description, the stagefor factors and hierarchy decision is technically the mostimportant one. Based on the findings of the field inves-tigation, literature review and collected assistant data, 16parameters were found relevant to slope instability. Forour purposes, there are three main aspects for rock massquality evaluation of slopes, i.e. geological, geometric,and environmental factors. Then, we can define that theslope rock mass quality estimate is equal to the sum-mation of total weights of the three main aspects. The


total weights were determined by the proceduredescribed above. The proposed rock mass qualityevaluating system for slopes contains four layers ofhierarchy as shown in Fig. 3. In this paper, through oneround of investigation papers responded to by 33 experts

Fig. 3. The hierarchy and the weights

and through a series of comprehensive discussions, theassessment factors and fuzzy weights can be composedinto a multidimensional questionnaire resulting in 16variable groups as shown in Fig. 3. The numbers listedin the brackets are the total weights of each factor. The

for evaluating slope rock mass.

Fig. 4. The geological map of the study area (from Mei-Shan to Ya-Kou in Taiwan).

134Y.-C

.Liu,

C.-S.

Chen

/Engineering

Geology

89(2007)

129–143


next step was to assess the evaluation of the rock massquality. For this purpose, we have to collect the relevantinformation given in Wang (2001), and then we cancompute slope rock mass quality estimates combiningthe slope case records through an onsite survey and thealgorithm introduced in this paper. By aggregating therelative weights and rating value interval for allparameters, results of the calculation for these sloperatings have been done completely using the weightedmethod. Finally, slope rock mass quality estimates werecalculated for each slope, higher values of the ratingindicate higher degrees of slope instability.

2.2. Discriminant analysis (DA)

Discrimination and classification are multivariatetechniques concerned with separating distinct sets ofobjects (or observations) and with allocating newobjects (or observations) to previously defined groups.Discriminant Analysis is rather exploratory in nature. Asa separative procedure, it is often employed on a one-time basis in order to investigate observed differenceswhen causal relationships are not well understood.Classification procedures are less exploratory in thesense that they lead to well-defined rules, which can beadopted for assigning new objects.

Fig. 5. Photos showing the failed slo

There are several purposes for the DA as shown inthe following. (Tao, 2003)

– To classify cases into groups using a discriminantprediction equation.

– To investigate independent variable mean differencesbetween groups formed by the dependent variable.

– To determine the percent of variance in the dependentvariable explained by the independents.

– To determine the percent of variance in the dependentvariable explained by the independents over and abovethe variance accounted for by control variables, usingsequential discriminant analysis.

– To assess the relative importance of the independentvariables in classifying the dependent variable.

– To discard variables which are little related to groupdistinctions.

– To test theory by observing whether cases are clas-sified as predicted.

There are many possible techniques for classificationof data. The Linear Discriminant Analysis (LDA) is aclassical statistics approach for classifying samples ofunknown classes, based on training samples with knownclasses. The LDA has been previously applied to sampleclassification of experiments data. Therefore, we adopt

pes located in this study area.

Table 2The 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 Few Medium More Much moreNon-conspicuous fracture One Two sets Three sets Above fourA B C D E

Condition of fracture Very low Lower Medium High Very highN2 m 60 cm–2 m 20–60 cm 6–20 cm b6 cmA B C D E

Joint filling Close Open Open Open OpenNone Harder filling Hard filling Soft filling Very softA B C D E

Discontinuity orientation Strike None N46–90W N0–45W N0–45E N46–90EDip angle None b30 31–45 46–60 N61

A B C D ESliding surface roughness Very choppy Choppy irregular Choppy regular Slightly rough Smooth

A B C D E

1-2 Geological factor — formation


Weathering Very low Lower Medium High Very highFresh Slightly

weatheredModeratelyweathered

Highlyweathered

Decomposed

A B C D ERock strength Schmidt hardness, N (kg/cm2) Very high N500 High

350–500Medium200–350

Lower75–200

Very lowb75

A B C D ERock type Volcanic metamorphic rock Sandstone Combined layer Slate Shale

A B C D E

2-1 Geometric factor — height of slope


Height (m) Very low Lower Medium High Very highb20 21–40 41–60 61–80 N81A B C D E

2-2 Geometric factor — gradient


Gradient (°) Very gradual Gradual Medium Steep Very steep30 31–45 46–60 61–75 75–90A B C D E

2-3 Geometric factor — width of slope


Width (m) Very narrow Narrow Medium Wide Very wideb20 21–40 41–60 61–80 N81A B C D E

2-4 Geometric factor — inclined direction: A (10), B (7.5), C (5), D (2.5)


Inclined direction NW NE SE SWA B C D

North — north, E — east,W — west, S — south


Table 2 (continued )

2-5 Geometric factor — orientation of the road cut: A (10), B (7.5), C (5), D (2.5)


Dip slopeOrientation of the road cut SlopeNdip angle Slopebdip angle Escarpment slope Slanting slope

A B C D

3-1 Environmental factor — groundwater


Surface with or without runoff N N N Yes YesNo trace Doubtful trace With Dripping FlowingA B C D E

Surface with or without seepage Completely dry Damp Wet Dripping FlowingA B C D E

3-2 Environmental factor — vegetation


Vegetation (condition and density) Perfect Good Medium Poor Very poorN75% 50–75% 25–50% 10–25% b10%A B C D E

3-3 Environmental factor — failure record


Based on weather the slope failure happened Never EverA E

A (10), E (2.5).


the LDA to classify which slope is stable or not andderive the discriminant functions. The problem we areconcerned about in this paper is with separation andclassification for two populations. This concept isintroduced as follows:

We consider the two classes labeled as Ω1 and Ω2.The objects are ordinarily separated or classified on thebasis of measurements on. For instance, p associatedrandom variables X′=[X1, X2…Xp]. The observed valuesof X differ to some extent from one class to the other. Wecan think of the totality of values from the fist class asbeing the population of X values for Ω1 and those fromthe second class as the population of X values for Ω2.These two populations can then be described byprobability density functions f1(x) and f2(x), andconsequently, we can talk of assigning observations topopulations or objects to class interchangeably (Johnsonand Wichern, 1998).

3. Cases study

3.1. Description of field conditions

In this section a total of 161 cases of rock slopes werecarried out using the rock mass quality estimation

method from the previous discussion. These slopesunder study in this paper are located at road cuts markedby A1–A80 (the failed sites) and B1–B81 (the stablesites) along the Southern Cross-Island Highway be-tween Mei-shan and Ya-kou in Taiwan. The geologicalmap of this area is shown in Fig. 4. There are severalprominent and recurring discontinuity sets in these roadcuts. For example, the two slopes of phyllite originallywith a gradient of about 64° to 80° (see Fig. 5) are bothunstable. Other features that may bear upon the slopestability are as follows:

1. The common rock types are sedimentary rock andmetamorphic 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.

According to the records of the onsite investigation,it can be found that the most probable mode of failure iswedge unstable. Planar unstable and toppling are the

Table 3Partial case records of slope site investigation in this study (numbers inthe table present the score for the parameters)

Case no. A27 A28 A29 A30 B57 B58

Discontinuity sets 2 2 2 2 4 4Condition of fracture 2 4 6 8 8 8Joint filling 2 2 8 2 4 2Discontinuity orientation 10 3 10 10 10 5Sliding surface roughness 6 8 6 4 10 10Rock type 4 6 8 8 4 4Rock strength 4 6 8 10 4 4Weathering 2 6 8 8 8 6Height of slope 8 6 8 8 6 6Gradient of slope 8 6 6 6 6 8Width of slope 6 4 10 8 6 2Inclined direction of slope 5 2.5 5 5 7.5 5Orientation of the road cut 2.5 7.5 2.5 2.5 2.5 7.5Ground water 6 7 8 6 9 6Vegetation 8 8 8 8 10 10Failure record 2 2 2 2 10 10


other two possible modes of failure. These unstableslopes present a potential hazard to passing motorists.The maximum safe angles range from 25° to 60°, whichare smaller than the slope angle of most cut slopes in thisregion 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 massquality evaluating and the results are used to assess thestability of existing road cut slopes located in the re-search area. More details of the geological properties ofslopes are illustrated in the Master's thesis of Wang(2001). In general, many of the classification parametersfor the different rock mass classifications considered aresimilar. Among them, the most important point, how-ever, is the information of the intact rock strength,discontinuity conditions, and groundwater, etc. In thispaper we take various parameters generated from theopinion of experts and the previous literature to deter-mine the slope rock mass quality estimates. After as-certaining the parameters in use, the main proposedprocedure was carried out. Through one round of in-vestigation papers, the weighting hierarchy structure forslope rock mass quality evaluating is obtained as shownin Fig. 3. The total weights of parameters are listed in thebrackets.

3.3. Result and discussion of slope rock mass evaluation

By aggregating the relative weights and the ratingvalue interval (see Table 2) for all parameters, results ofthe calculation for these slope ratings have been donecompletely using the weighted method. To clarify thedescription, an example is given as follows:

Take case number A-27 and B-57 (see Table 3) asexample. The slope rock mass quality estimate (shownas Table 4) for A27 and B57 are 50.1 and 73.9. Thesecould be computed on the basis of Eqs. (8), (9):

EstimatePA27 ¼ 0:3525� ½0:6024� ð0:1598� 2þ 0:216� 2þ 0:1364� 2þ 0:265� 10þ 0:2228� 6Þþ 0:3796� ð0:2403� 4þ 0:2966� 4þ 0:4631� 2Þ�þ 0:3403� ½0:0999� 8þ 0:2914� 8þ 0:0625� 6þ 0:3335� 5þ 0:2127� 2:5�þ 0:3072� ½0:3844� 6þ 0:265� 8þ 0:3506� 2� ¼ 50:1

ð8Þ

EstimatePB57 ¼ 0:3525� ½0:6024� ð0:1598� 4þ 0:216� 8þ 0:3164� 4þ 0:265� 10þ 0:2228� 10Þþ 0:3796� ð0:2403� 4þ 0:2966� 4þ 0:4631� 8Þ�þ 0:3403� ½0:0999� 6þ 0:2914� 6þ 0:0625� 6þ 0:3335� 7:5þ 0:2127� 2:5�þ 0:3072� ½0:3844� 9þ 0:265� 10þ 0:3506� 10� ¼ 73:9

ð9Þ

All case records are partially listed in Table 3 andplotted in scattering Fig. 6. Fig. 6 shows the correlationbetween the observed behaviors (failed or stable) and theslope rock mass quality estimates. To proceed furtherinto an analysis of the results, the statistical analysis toolis used and the related results are summarized as follows.From the results as shown in Table 4, the statisticalanalysis presents the spread and variability of the ratingsfor the failed and the stable group. The distribution canbe 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 well-defined peak (standard deviation 5.2). In contrast, thefailed group presents a narrower range (66.9–39.1) ofvalues, with a mean of 51.4 and a flat peak (standarddeviation 5.81). Initial rough analysis shows that thestable slope rock mass must be rated over 65 points.Moreover in order to avoid any slope instability risk, theslope rated value below 55 points should be taken tosupport or to prevent a possible instability in the fore-seeable future. The empirically found limit value range

Table 4The slope rock mass estimates partly listed

Case no. A1 A2 A3 A13 A27 A30 B57 B81

Site (km) 110.2 110.5 110.8 112.95 122.3 117.05 133.2 146.3Estimate 74.1 49.37 59.96 66.69 50.1 77.52 73.9 46.66


of slope rating for whether unstable or not is 55 to 65points. No slope has been rated as slope rock massquality estimated below 39 points in this study. Toproceed further into a definition of the best criterion toclassify the two slope groups in this study, we will makeuse of the DA.

4. Approach to classify rock slopes stability

4.1. The linear discriminant analysis (LDA) model

The data given in Table 4 are used for discussing theanalytical approach to the DA. The ratings of two slopegroups were determined by the proposed procedure.Fig. 7 indicates a histogram for the both groups. Table 5gives means and standard deviations for the two groups.Using an independent sample t-test (see Table 6) canassess the differences in the means of the two groups.The t-value for testing equality of the means of the twogroups is −19.45 for slope rock mass quality estimates.The t-test suggests that the two groups are significantlydifferent with respect to slope rock mass quality es-timates at a significance level of 0.05. That is, slope rockmass quality estimate does discriminate between the twogroups and consequently will be used to create the

Fig. 6. The correlation between the observed behaviors (failed orstable) and the slope rock mass quality estimate.

discriminant function. In the next stage, we present theLDAmodel and the corresponding results. Asmentionedabove, we have to construct a model for classifying thefailed slope group and the stable one. These two groupsand the data corresponding required to the factor wereanalyzed by means of the DA to obtain a linear model. Inthis section, the software used in LDA was SPSSStandard Version 12.0 for the relevant calculations. TheSPSS uses the statistical decision theory method for

Fig. 7. The histogram for two group slopes: (a) failed slopes, (b) stableslopes.

Table 5The descriptive statistics of the slope rock mass quality estimates fortwo groups

Failed or not

Group 1 (failed) 2 (stable)

Valid N 80 81Mean Statistic 51.35 68.24

Std. error 0.65 0.58Median 50.68 67.76Minimum 39.08 58.40Maximum 66.86 78.92Std. deviation 5.81 5.20Interquartile range 7.85 8.76

Table 7The summarized results of LDA by SPSS

(a) Classification processing summary

Processed 161Excluded

Missing or out-of-range group codes 0At least one missing discriminating variable 0

Used in output 161

(b) Prior probabilities for groups

Group Prior Case used in analysis

Unweighted Weighted

1 .497 80 80.0002 .503 81 81.000Total 1.000 161 161.000

(c) Classification function coefficients

Group

1 2

Rock mass estimate 1.691 2.247Constant −44.104 −77.354Fisher's linear discriminant functions

(d) Classification matrix (p= . meant the prior probability for thegroup)

Group Classification matrix (total score)

Rows: observed classifications

Columns: predicted classifications

Percent G_1:1 G_2:2

Correct p= . 49,689 p= . 50,311

G_1:1 90.00000 72 8G_2:2 93.82716 5 76Total 91.92547 77 84


classifying sample observations into various groups.This method minimizes misclassification errors that aretaking into account prior probabilities and misclassifi-cation costs. From results of SPSS shown in Table 7, itcan be seen that this model is a linear combination of theslope rock mass quality estimates and the constant item,which separate the two groups in the best way. SPSScomputes the classification functions of Fisher's lineardiscriminant function mode as follows:

The function equation for the failed slope is

d1 : Y ðX Þ ¼ −44:11þ 1:6909X1 ð10Þ

The function equation for the stable slope is

d2 : ZðX Þ ¼ −77:3622þ 2:2472X1 ð11ÞY(X ) and Z(X ) are referred to as Fisher's linear

discriminant function and X1 is the slope rock massquality estimate. Observations are assigned to the groupwith the largest classification score. Assign theobservation to group d1 if Y(X )NZ(X ) and to group d2if Y(X )bZ(X ).

Table 8 shows the output generated by the DAprocedure in SPSS. We can see the correct classificationrate is 91.93%. How good is this classification rate?Huberty (1984) has proposed approximate test statistics

Table 6The independent sample t-test result for distinguishing between the differen

T-tests: grouping: category (total score)

Group 1:1

Variable Group 2:2

Mean 1 Mean 2 t-value df p

Rock mass estimate 51.35 68.24 −19.449 159 0.0

St.d: Standard deviation; df: degrees of freedom.

that can be used to evaluate the statistical and thepractical significance of the overall classification rategiven. Our method achieves a statistically significantresult with respect to the current situation.

ces in the means of the two groups

Valid N1 Valid N2 St.d 1 St.d 2 F-ratio P

80 81 5.81 5.20 1.249 0.324

Table 8Result of the LDA model

Failed (d1) Stable (d2)

X0 Constant −44.1100 −77.3622X1 Slope rock mass quality estimate 1.6909 2.2472

Fig. 8. The failure probability distribution.


4.2. The probability of slope failure (Lin, 2001)

In this paper an equation, Eq. (12) to calculate thefailure probability P(%) of rock slopes based on lineardiscriminant functions obtained in previous analysis isintroduced in the following.

P ¼ eY

ðeY þ eZÞ ð12Þ

Where Y is the linear discriminant function of thefailed slopes and Z is the linear discriminant function ofthe stable slopes.

Substituting Eq. (10) and Eq. (11) into Eq. (12), wecan obtain

P ¼ e−44:11þ1:6909X1

ðe−44:11þ1:6909X1 þ e−77:3622þ2:2472X1Þ ð13Þ

Now that the equation is derived, we can take theratings of the two slope groups into account the failureprobability for each slope. The results are obtained fromthe failure probability analysis. Table 9 partially showsthe failure probability of slopes and the category. Thefailure probability distribution as shown in Fig. 8 indi-cates that the cases considered in this study are extremeexamples (failed or stable).

4.3. The stability classification of rock slope

As previously discussed, a probabilistic approachusing the DA functions was applied in the analysis toquantify the slope stability. To summarize our interpre-tation of the results, a proposed classification as shownin Table 10 can be categorized according to their sloperock mass quality estimates, the failure probability andthe procedure to deal with during the monitoring period.

Table 9Partially showing the probability of slope failure and the category

Case number 1 2 3 4

Marked (km) 110.2 110.5 110.8 111Category 2 1 2 1Failure probability 0.03% 99.69% 47.41% 86.53%

There are three types of class I, class II (IIA, IIB, IIC,IID) and class III, which are used to classify the stabilityof rock slopes. The calculation of the failure probabilityof a slope with the new method developed in this studygives a more distinctive differentiation between failedand stable conditions than with existing rock massclassification systems for slopes (Kentli and Topal,2004).

5. Conclusions

The main results of this paper are twofold. The first isthat the proposed method has provided a useful basis forintroducing quantitative rock mass assessment into rockslope engineering, where there are numerous potentialfailure locations and several different failure cases at theSouthern Cross-Island Highway in Taiwan. The pro-posed method can be successfully applied to determinethe rock mass quality estimates for rock slopes. Resultsroughly show that it can be used to determine the rockmass quality estimates of slopes as a simple safety as-sessment method. A second result of this paper presentsthat the LDAmodel can be applied to distinguish betweenthe failed and stable slope group. And this analysis alsoassesses to establish the discriminant functions to evaluatethe failure probability of the slopes. On the basis of ourearlier discussion we could conclude that this new pro-posed method can be used to judge the stability problems

5 6 7 8 9

111.2 111.3 111.5 111.6 111.82 1 1 2 215.16% 99.98% 99.12% 6.87% 0.03%

Table 10The slope stability classified based on slope quality estimate andfailure probability

Class FDAHProck massestimate

Failureprobability

Stabilityclass

Recommendedtreatment

I 78–100 0% Fullystable

None

II A 61–77 b15% Stable Usual monitoringII B 59–60 60.6%–

46.9%Tend tobe unstable

Up frequent measure

II C 56–58 89.1 %–72.9%

Moreunstable

Add support to wallor concrete

II D 42–55 N93% Unstable Protection techniqueneeded and very oftenrevise support used

III 41–0 100% Fullyunstable

Re-excavation anddesign


of rock slopes according to the proposed procedure andthe failure probability.

Acknowledgments

This study is supported by the research project ofTaiwan National Science Council through contractnumber NSC-91-2211-E-006-049. Comments by twoanonymous reviewers are greatly acknowledged.

Appendix A. The weighting factors determination

Among the 33 survey respondents, 4 were fromuniversity professor in geological engineering area and29 were from government, engineering consultantscompanies and the other organizations. Based on thesurvey responses, the proposed method was applied andevaluation model was constructed. The weightingfactors for each criterion listed in this study hierarchywere presented in the following explanation.

Referred to the article described in Section 2.1, takematrix 1 as an exemplification.

Table A-1All acceptable responses for the matrix 1

Geologicalfactor

Geometricfactor

Environmentalfactor

Geologicalfactor

1
(1/6,1,1,4,5,1/3,1,3,5,1,5,3,1/2,1/5,3)
(1/7,1,1,6,3,7,1,3,3,1/5,6,5,3,1/6,1)

Geometricfactor

Positivereciprocal

1
(1/4,1,1,3,1/3,7,1,3,1/3,1/5,3,3,3,1/3,1/5)
Environmentalfactor

Positivereciprocal

Positivereciprocal

1

1. Compute the triangular fuzzy numbers (TFNs)ãij=(αij, δij, γij)

According Eq. (3)–Eq. (5)

aij ¼ MinðBijkÞ; k ¼ 1; N ; n

In all acceptable responses for matrix 1, n is equal to15 (the other 8 responses were under the mark andrejected. The ij subscript notation here represents that i is1 to 3 and j is 1 to 3. Thus from Appendix Table C-1, wecould 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 thetriangular fuzzy numbers (TFNs)

faij ¼ ðaij; dij; gijÞk ¼ 1; N ; 15

a12 ¼ MinðB12kÞ ¼ 1=6; a13 ¼ MinðB13kÞ ¼ 1=7;

a23 ¼ MinðB23kÞ ¼ 1=5

d12 ¼ j15

k¼1B12k

� �1=15

¼ 1:333534; d13 ¼ 1:510459;

d23 ¼ j15

k¼1B23k

� �1=15

¼ 0:969667

g12 ¼ MaxðB12kÞ ¼ 5; g13 ¼ MaxðB13kÞ ¼ 7;

g23 ¼ MaxðB23kÞ ¼ 7

According the positive reciprocal rule, aijVdijVgij;aij; dij; gijϵ

19 ; 1� � [ ½1; 9�

fa21 ¼ ð1=a12; 1=d12; 1=g12Þ ¼ ð6; 1=1:333534; 1=5Þ¼ ð6; 0:749887; 0:2Þ

Because 0.2b0.749887b6 thus ã21=(0.2, 0.749887, 6)In the same way, ã31= (1/7, 0.66205, 7)

fa21 ¼ ð0:2; 0:749887; 6Þ;fa32 ¼ ð1=7; 1:031282; 5Þ

2. Following outlined above, we obtained a fuzzypositive reciprocal matrix Ã

fA ¼ð1; 1; 1Þ ð1=6; 1:333534; 5Þ ð1=7; 1:510459; 7Þ

ð0:2; 0:749887; 6Þ ð1; 1; 1Þ ð1=5; 0:969667; 7Þð1=7; 0:66205; 7Þ ð1=7; 1:031282; 5Þ ð1; 1; 1Þ

24

35

3. Calculate the relative fuzzy weights of the eval-uation factors.

fZ1 ¼ ½fa11 �fa12 �fa13�1=3 ¼ ½0:2877; 1:2629; 3:2711�;fZ2 ¼ ½fa21 �fa22 �fa23�1=3 ¼ ½0:3420; 0:8992; 3:4760�;fZ3 ¼ ½fa31 �fa32 �fa33�1=3 ¼ ½0:2733; 0:8806; 3:2711�;Xf

Zi ¼ ½0:9030; 3:0427; 10:0182�


Reciprocal ∑Z̃i=[0.998, 0.3287, 1.1075]

fW 1 ¼ f

Z1 � ðfZ1PfZ2P

fZ3Þ−1

¼ ½0:0287; 0:4151; 3:6226�fW 2 ¼ ½0:0341; 0:2955; 3:8496�;fW 3 ¼ ½0:0273; 0:2894; 3:6226�

Therefore, W1¼ j3i¼1xj

� �1=3¼ 0:3525;W2 ¼ 0:3403;W3 ¼ 0:3072

The weighting factors for three aspects are: geolog-ical factor (0.3525), geometrical factor (0.3403) andenvironmental factor (0.3072).

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a new approach for application of rock mass classification

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