A graphical approach for slope mass rating (SMR)

Engineering Geology 124 (2012) 67–76

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A graphical approach for slope mass rating (SMR) R. Tomás ⁎, A. Cuenca, M. Cano, J. García-Barba Departamento de Ingeniería de la Construcción, Obras Públicas e Infraestructura Urbana. Escuela Politécnica Superior, Universidad de Alicante, P.O. Box 99, E-03080 Alicante, Spain

a r t i c l e

i n f o

Article history: Received 3 January 2011 Received in revised form 4 October 2011 Accepted 8 October 2011 Available online 18 October 2011 Keywords: Geomechanical classification SMR Basic RMR Stereographic projection

a b s t r a c t Slope mass rating (SMR) is a commonly used geomechanical classification for the characterization of rock slopes. SMR is computed adding to basic rock mass rating (RMR) index, calculated by characteristic values of the rock mass, several correction factors depending of the discontinuity–slope parallelism, the discontinuity dip, the relative dip between discontinuity and slope and the employed excavation method. In this work a graphical method based on the stereographic representation of the discontinuities and the slope to obtain correction parameters of the SMR (F1, F2 and F3) is presented. This method allows the SMR correction factors to be easily obtained for a simple slope or for several practical applications as linear infrastructures slopes, open pit mining or trench excavations. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Rock mass classification systems are a worldwide communication system for explorers, designers and constructors that facilitate characterization, classification and knowledge of rock mass properties. They provide quantitative data and guidelines for engineering purposes that can improve originally abstract descriptions of rock mass from inherent and structural parameters (Liu and Chen, 2007; Pantelidis, 2009) by a simple arithmetic algorithm (Romana, 1997). The main advantage of using a rock mass classification scheme is that it is a simple and effective way of representing rock mass quality and of encapsulating precedent practice (Harrison and Hudson, 2000). Nevertheless, rock mass classifications present some well-known limitations. Hack (2002) stated that generally rock mass classifications consider parameters related with slope geometry, intact rock strength, discontinuity spacing or block size and shear strength along discontinuities, some of which are difficult or impossible to measure (e.g. water pressure) or have a limited influence on slope stability (e.g. intact rock strength). Pantelidis (2009) referred to these parameters as “questionable,” including those that: (a) are unsuitable for use in slope stability problems, (b) are attributed into the systems in an erroneous manner, (c) although, in practice, they play significant role regarding stability of slopes, they exert a minor influence on the system, or, (d) present several major disadvantages related to their definition. All the previous mentioned causes can introduce some uncertainties during the rock mass characterization process that can affect the final computed indexes and the inferred geomechanical quality and parameters. As a consequence, rock mass classifications on their own should only be used for preliminary planning

⁎ Corresponding author. Tel.:+34 9659034003093. E-mail address: [email protected] (R. Tomás). 0013-7952/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2011.10.004

purposes or within the overall engineering design process (Bieniawski, 1997). Some of the existing geomechanical classifications for slopes are rock mass rating (RMR, Bieniawski, 1976, 1989), rock mass strength (RMS, Selby, 1980), Slope mass rating (SMR, Romana, 1985), slope rock mass rating (SRMR, Robertson, 1988), rock mass rating, mining rock mass rating (MRMR, Laubscher, 1990), mining rock mass rating modified (MRMR modified, Haines and Terbrugge, 1991), Chinese slope mass rating (CSMR, Chen, 1995), natural slope methodology (NSM, Shuk, 1994), modified rock mass rating (M-RMR, Ünal, 1996), slope stability probability classification (SSPC, Hack, 1998; Hack et al., 2003), modified slope stability probability classification (SSPC modified, Lindsay et al., 2001), continuous rock mass rating (Sen and Sadagah, 2003), continuous slope mass rating (Tomás et al., 2007) and an alternative rock mass classification system proposed by Pantelidis (2010). Among all geomechanical classifications listed above, SMR is universally used (Romana et al., 2001, 2003, 2005). It is derived from the basic RMR (Bieniawski, 1989), initially created for tunneling applications, although its author also included proposals for slope correction factors in order to take into account the influence of the discontinuities orientation on slope stability. In practice, RMR is difficult to apply to slopes as there is no exhaustive definition for the selection of correction factors. The detailed quantitative definition of the correction factors (Irigaray et al., 2003) is one of the most important advantages of SMR classification. Both RMR and SMR are discrete classifications, computed by assigning a specific rating to each parameter included, depending on the value adopted by the variable that controls the parameter under consideration. The aim of this study is to propose a graphical method for the determination of slope mass rating correction factors. The present work

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Table 1 Correction parameters for SMR (modified from Romana (1985) by Anbalagan et al. (1992)). Type of failure P T W P/T/W P/W P/W T P W T P/T/W

Very favorable

Favorable Normal Unfavorable Very unfavorable

>30°

30–20°

20–10° 10–5°

b5°

F1

0.15

0.40

0.70

1.00

B |βj| ó |βi| F2

b 20° 0.15 1.00 >10°

20–30° 0.40

30–35° 35–45° 0.70 0.85

>45° 1.00

10–0°

0°

b(− 10°)

b 110° 0

110–120° >120° – −6 − 25 − 50

– − 60

+ 15 + 10 +8

Blasting or mechanical Deficient blasting

0 −8

A

C

| α j − α s| |αj−αs−180| |αi − αs|

βj − βs βi − βs βj + βs

F3

Excavation method (F4) Natural slope Presplitting Smooth blasting

0.85

0–(−10°)

P: planar failure; T: toppling failure; W: wedge failure. αj: dip direction of the discontinuity; αs: dip direction of the slope; αi: dip direction of the intersection line of two sets of discontinuities; βj: discontinuity dip; βi: angle of plunge of the intersection line of two sets of discontinuities; βs: slope dip.

is devoted to define stereoplots that can be used in rock mass slopes studies in order to easily interpret and compute SMR correction factors.

2. Slope mass rating (SMR) classification The slope mass rating (SMR) index, proposed by Romana (1985), is calculated by determining four correction factors to the basic RMR (Bieniawski, 1989). These factors depend on the existing relationship between discontinuities affecting the rock mass and the slope, and the slope excavation method. It is obtained using Eq. (1).

SMR ¼ RMRb þ ðF 1  F 2  F 3 Þ þ F 4

ð1Þ

where: – RMRb is the basic RMR index resulting from Bieniawski's rock mass classification without any correction. Therefore, it is calculated according to RMR classification parameters (Bieniawski, 1989). – F1 depends on the parallelism (A in Table 1) between discontinuity dip direction, αj, (or the trend of the intersection line, αi, in the case of wedge failure) and slope dip, αs (Table 1). – F2 depends on the discontinuity dip, βj, in the case of planar failure and the plunge of the intersection line, βi, in wedge failure (B in Table 1). For toppling failure, this parameter adopts the value 1.0. This parameter is related to the probability of discontinuity shear strength (Romana, 1993). – F3 depends on the relationship (C in Table 1) between slope, βs, and discontinuity, βj, dips (toppling or planar failure cases) or the plunge of the intersection line (wedge failure case) (Table 1). This parameter retains the Bieniawski adjustment factors that vary from 0 to − 60 points and express the probability of discontinuity outcropping on the slope face (Romana, 1993) for planar and wedge failure.

Fig. 1. Proposed diagram for ψ determination in planar failure case.

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Fig. 2. Proposed diagram for ψ determination in wedge failure case.

Fig. 3. Proposed diagram for ψ determination in toppling failure case.

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Fig. 4. Graphical construction of the stereoplot used for the determination F3 parameter for planar failure case. PS is the slope pole.

Fig. 5. Graphical construction of the stereoplot used for the determination F3 parameter for wedge failure case. PS is the slope pole and βS is the slope dip. L.m.d.: line of maximum dip.

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Fig. 6. Graphical construction of the stereoplot used for the determination F3 parameter for toppling failure case. PS is the slope pole.

– F4 is a correction factor that depends on the excavation method used (Table 1). Eq. (1) can be rewritten as: SMR ¼ RMRb þ ðψ  F 3 Þ þ F 4

ð2Þ

where F1 × F2 has been grouped in the same term (ψ) that varies from 0 to 1. This term can be considered as the percentage of factor F3 mobilized. As F4 parameter has an irreplaceable descriptive character and depends on the excavation method, the method proposed in this work is only focused in the determination of the geometrical terms ψ and F3 in Eq. (2) using the stereographic projection of rock mass slope discontinuities. 3. Stereographic diagrams proposed for slope mass rating One of the most important aspects of rock slope analysis is the systematic collection and representation of geological data (Hoek and Bray, 1981). Field collected discontinuity orientation data are analyzed by the use of stereographic projection techniques that allow graphical representation and interpretation of the discontinuity data by means of the great circles or the poles to the planes. When huge amounts of discontinuity data are available the presentation of a large number of great circles onto a single stereoplot makes the data difficult to interpret. In these cases it is preferable to represent the inclination and azimuth of a plane by means of the pole of the plane. The poles are presented in two dimensions by projecting them onto the horizontal or equatorial reference plane. When a

large number of measurements have been plotted a recognition of pole concentrations and discontinuity patterns becomes evident; this may be assisted by the use of density contouring of the pole plots. This allows a quantitative assessment of the influence of discontinuities on the behavior of the rock mass and will provide the necessary information for determining rock mass classification values and failure mechanisms. In the case of SMR, once the different discontinuity sets have been identified and rock mass has been fully characterized, Table 1 can be used to compute F1, F2 and F3 correction parameters. An alternative graphical method for F1, F2 and F3 parameters calculus is proposed using stereographical projection stereoplots.

3.1. Graphical determination of ψ parameter Figs. 1, 2 and 3 show the stereographic diagrams proposed for ψ determination associated to each type of failure, planar, wedge and toppling respectively. These diagrams are obtained considering the combined values of F1 and F2 proposed by Romana (1985) that depend on the parallelism between discontinuity dip direction (or the trend of the intersection line in the case of wedge failure) and slope dip direction and the discontinuity dip that conditions the radial lines and the concentric areas respectively drawn onto the stereoplots. For the use of these diagrams the slope and the discontinuities (or the intersection line for wedge failure) have to be represented in equiangular projection over the lower hemisphere using tracing paper. The tracing paper is superimposed to the Figs. 1, 2 or 3, depending of the mode of failure, and rotated around the point O to

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Fig. 7. Poles diagram (Px) and great circles of the discontinuities (Jx) affecting the slope (S).

match the slope dip direction to the direction indicated in the figures. The pole position of each discontinuity set indicates the type of failure and the corresponding ψ values. 3.2. Graphical determination of F3 parameter F3 depends on the relationship between slope and discontinuity dips or the intersection line plunge (Table 1). Because this parameter depends on slope dip, a specific stereoplot has to be used for each slope dip value. The construction of the stereoplot is easy and also depends on the type of failure mechanism. Fig. 4 shows the construction of the F3 parameter for planar failure. As it can be seen, once the slope plane has been represented onto stereographic projection, a semicircle with a radius OPS from the center O has to be drawn in the upper part of the projection diagram, where PS is the slope plane pole. Posteriorly, two concentric semicircles to OPS one at a ±10° has to be drawn using the graduated diameter of the graphic. The four obtained areas provide the F3 values that vary from 0 to −60 points. Fig. 4 shows the location of the areas with the correction values that grow towards the center (O) of the diagram. Fig. 5 corresponds to the stereoplot used for F3 determination for wedge failure cases. In order to define the punctuation sectors a semicircle from the center O and with a radius OQ must be drawn. Q is the pole projection of the line of maximum dip (l.m.d. in Fig. 5) of the slope, S. Finally, two new concentric semicircles have to be drawn at ±10° using the axis graduation. Notice that the punctuation decreases outward of the diagram varying between 0 and −60 points. When toppling is the compatible failure, the stereoplot shown in Fig. 6 has to be used. This stereoplot can be easily drawn scribing an arc, radius OPS, from the center (O) of the diagram to intersect the

graduated diameter at point M. From this point we have to measure 110° and 120° over the graduated diameter in the direction of the center O obtaining S and T points. Taking the distances OS and OT and scribing an arc from center O the different punctuation sectors are defined. F3 values increase towards the center of the diagram varying from 0 to − 25 points. As it was previously mentioned, the stereoplots for F3 determination depend on the slope dip and as a consequence different stereoplots have to be drawn for each slope dip value. However, once the stereoplot has been built for a particular dip it can be used for the determination of F3 correction parameter of other slopes with the same dip. For the exploitation of the above defined stereoplots, we proceed the same way as with the stereoplots proposed for ψ correction parameter. 4. Application example The proposed graphical methodology is applied for the determination of the SMR correction factors of a rocky slope and compared with the original values obtained from Romana's (1985) discrete classification. The studied slope (S) has a 210° and 60° dip direction and dip respectively. This slope is affected by four discontinuities sets that are represented in Fig. 7 by means of their great circles (J1 to J4) and their respective poles (P1 to P4) whose orientations (dip direction and dip) are: J1 (60/70), J2 (235/40), J3 (150/80) and J4 (16/10). When the discontinuity sets and the slope have been represented in equiangular projection the obtained stereographic diagram has to be superimposed to the stereoplots and oriented in order to match the slope (S) dip direction with the dip direction of the stereoplot. As it is well known, for this purpose tracing paper can be used in

R. Tomás et al. / Engineering Geology 124 (2012) 67–76

order to represent and to overlay the discontinuities representation to the stereoplots. The analysis of the position of the poles of the discontinuities or the intersection lines of the wedges allows identifying the feasible type of failure mechanism and the punctuation corresponding to every case (Figs. 8 and 9). Fig. 8a corresponds to planar failure case. The overlay of the discontinuity sets shows that only P2 and P3 poles are compatible with planar failure (they are contained in shadow area). ψ values are deduced from the position of the P2 and P3 poles respecting the defined sectors and its values are ψ2 = 0.34 and ψ3 = 0.15 respectively. Analyzing Fig. 8b we can infer that only J1–J2 and J2–J3 discontinuity combinations are feasible wedges cinematically compatible with slope geometry (which are contained in shadow area). For these intersection lines ψ12 = 0.02 and ψ23 = 0.34 values are computed. Toppling failure correction factors are obtained from Fig. 8c. As it can be seen in Fig. 8c only J1 and J4 are cinematically compatible with toppling failure (which are contained in shadow area). The ψ values for these discontinuity sets are determined by the location of the poles J1 and J4 in the stereoplot and adopt the values ψ1 = 0.15 and ψ4 = 0.70 respectively. Once ψ values have been determined for all discontinuity sets and for the wedges intersection lines, next step is the determination of F3 correction parameter. Although ψ stereoplots are independent of the slope geometry the stereoplots proposed for F3 determination depends on the slope dip and as a consequence an ad hoc construction has to be performed following the process explained in previous section. When stereoplots have been drawn for the 60° dip slope of the studied example and discontinuities have been overlapped and rotated to match the slope dip direction with the direction indicated in the stereoplot we only have to determine the position of the discontinuity poles and the intersection lines poles of the wedges to know the value of the correction parameters (Fig. 9a to c). F3 values for planar failure cases (J2 and J3) are − 60 and 0 points respectively (Fig. 9a). For wedges J1–J2 and J2–J3 the F3 correction parameter adopts values of − 60 points in both cases (Fig. 9b). When the considered compatible failure mechanism is toppling, F3 acquires the values of − 25 and 0 points for J1 and J4 discontinuity sets respectively (Fig. 9c). Graphically calculated ψ and F3 correction parameters are summarized in Table 2. Conventional parameters of F1, F2 and F3 computed numerically from Romana's (1985) SMR classification (Table 1) are also included in Table 2. Notice that the resulting corrections are equal except for the J1–J2 wedge where the values obtained by graphical and analytical methods differ 0.1 points due to the rounding of the assigned scores for each sector in Figs. 1 and 2 that are obtained by multiplying F1 by F2 original parameters proposed by Romana (1985).

5. Some interesting applications of graphical slope mass rating approach Some practical applications of proposed approach for the SMR calculus are presented. The proposed stereoplots for ψ determination (Figs. 1 to 3) are independent of the slope geometry and as a consequence they are valid for all slopes cases. In contrast, F3 stereoplots (Figs. 4 to 6) have to be drawn for a specific slope because they depend on the slope dip. One of the advantages of the proposed graphical method is the possibility to easily compute SMR correction factors for different slope orientations affected by the same sets of discontinuities. This fact is usually presented in linear infrastructures (roads, railways,

Fig. 8. Determination of ψ (F1 × F2) for a 60° dip slope affected by Fig. 7 discontinuities sets: (a) planar failure, (b) wedge failure and (c) toppling failure.

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2channels, etc.) and open pit mining where the slopes excavated over wide rock masses present the same dip but different strikes that change along the infrastructure or the exploitation sector. An example of the firstly referred case is showed in Fig. 10a and b that corresponds to a linear infrastructure excavated through a rock mass affected by a family of discontinuities with a dip direction and a dip equals to 235° and 40° respectively. As seen, once the stereoplots have been drawn, SMR correction parameters can be easily determined by rotating the stereoplot to match the dip direction of the stereoplot with the dip direction of the dip direction of the stereoplot. Fig. 10c shows the position of the discontinuity poles for the different slope strikes of the linear infrastructure. Notice that the punctuations for each slope orientation and the changes of punctuations are immediately determined by rotating the stereoplot in order to match the slope and the stereoplot dip directions. The different positions of the discontinuity pole (PJ) for the whole slope strikes describe its geometric path that allows to easily understand how correction parameters (and consequently SMR) change with the slope orientation. The knowledge of the joint pole geometric path can be very useful for identifying the critical slope orientation that corresponds to the higher combination of ψ and F3 correction parameters. For trench excavation, where the slopes have the same direction and dip but opposite dip direction, the graphical method can be effortlessly applied by simply rotating the stereoplot 180° to determine ψ and F3 correction parameters that are immediately determined by simple measurement of the position of the discontinuity (or the line of intersection for wedge failure) (Fig. 10b and c). The previously presented applications of this methodology use a representative dip and dip direction of each discontinuity set for the SMR calculus. However, this graphical methodology can provide another interesting application that consists of representing the poles of all the discontinuities measured at field (Fig. 11) in order to take into account the dispersion of the discontinuities orientation to determine the maximum, minimum, mode and other simple statistics for ψ and F3 values. Notice that Fig. 11 only includes planar cases simply for clearness of the figure. Nevertheless toppling and wedge failure cases can be also taken into account by representing the discontinuity poles and the great circles of all discontinuities field measurements respectively using the corresponding stereoplots.

6. Conclusions A graphical method for slope mass rating F1, F2 and F3 correction parameters determination is proposed based on stereographical projection. F4 parameter has an irreplaceable descriptive character and as a consequence it keeps its original way of being computed. For obtaining the SMR correction parameters the discontinuity sets represented on equiangular stereographical projection have to be superimposed to the proposed stereoplots and rotated to match the slope dip direction with the stereoplot dip direction. Subsequently, the numerical values of the correction parameters are directly obtained from stereoplots determining the position of the discontinuity pole (for planar and toppling failure modes) or the intersection lines poles (for wedge failure mode). F1 and F2 parameters are grouped into a parameter named ψ that represents the percentage of F3 mobilized and that is easily computed by representing the discontinuity sets into the proposed stereoplots for each type of failure mode, valid for all slopes regardless of the orientation of the slope and discontinuity studied. Moreover, the proposed stereoplots for F3 parameter depend of the type of failure. However, these stereoplots also depend on the Fig. 9. Determination of F3 for a 60° dip slope (S) affected by Fig. 7 discontinuities sets (J1, J2, J3 and J4): (a) planar failure, (b) wedge failure and (c) toppling failure.

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Table 2 SMR correction factors computed using original discrete (D) function proposed by Romana (1985) and graphical method (G) proposed in this paper. A: parallelism between discontinuity dip direction (or the trend of the intersection line in the case of wedge failure) and slope dip direction; B: discontinuity dip or angle of plunge of the intersection line of two sets of discontinuities; C: discontinuity (or intersection line) dip and slope dip relationship. T: toppling; P: planar; W: wedge; NFW: non feasible wedge. Set

Type of failure

αj

βj

αs

βs

A

B

C

Method

F1

F2

Ψ = F1 × F2

F3

F1 × F2 × F3

J1

T

60

70

210

60

30

70

130

J2

P

235

40

210

60

25

40

− 20

J3

P

150

80

210

60

60

80

20

J4

T

16

10

210

60

14

10

70

J1–J2

W

149

3

210

60

61

3

− 57

J1–J3

NFW

86

68

210

60

56

68

–

J1–J4

NFW

333

7

210

60

57

7

–

J2–J3

W

232

40

210

60

22

40

− 20

J2–J4

NFW

319

5

210

60

71

5

–

J3–J4

NFW

61

7

210

60

31

7

–

G D G D G D G D G D G D G D G D G D G D

– 0.15 – 0.40 – 0.15 – 0.70 – 0.15 – – – – – 0.40 – – – –

– 1.00 – 0.85 – 1.00 – 1.00 – 0.15 – – – – – 0.85 – – – –

0.15 – 0.34 – 0.15 – 0.70 – 0.02 – – – – – 0.34 – – – – –

− 25 − 25 − 60 − 60 0 0 0 0 − 60 − 60 – – – – − 60 − 60 – – – –

− 3.7 − 3.7 − 20.4 − 20.4 0 0 0 0 − 1.2 − 1.3 – – – – − 20.4 − 20.4 – – – –

Fig. 10. Practical application of SMR graphical approach for the determination of correction parameters in a linear infrastructure.

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the projects vigrob-157 uausti10–18, uausti11–11 and gre09–40 and by the Generalitat Valenciana within project gv/2011/044. References

Fig. 11. Application of SMR graphical methodology using field discontinuity measures for the determination of ψ parameter for planar failure mode.

slope dip and as a consequence a different stereoplot should be used for each slope dip. The main advantage of this methodology is the possibility to be used for easily calculating the correction parameters of SMR in cases where all the slopes have the same dip with different strike as in linear infrastructures and open pit mining. Another significant improvement of this methodology is the possibility of working with the field measurements of all discontinuities (of the poles of the intersection lines for wedge failure cases) in order to determine the distribution of the correction parameters values.

Acknowledgements Authors thank anonymous reviewers and A. Singleton (University of Glasgow) for their useful comments and the review of the paper. This work was partially funded by the University of Alicante under

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