Empirical methods to estimate the strength of jointed rock masses

Engineering Geology 77 (2005) 127 – 137 www.elsevier.com/locate/enggeo

Empirical methods to estimate the strength of jointed rock masses Mahendra Singha,*, K. Seshagiri Raob a

Department of Civil Engineering, Indian Institute of Technology, Roorkee, Roorkee UA 247667, India b Department of Civil Engineering, Indian Institute of Technology, Delhi, New Delhi 110016, India Received 3 June 2003; accepted 1 September 2004 Available online 26 October 2004

Abstract The ultimate strength and deformation of jointed rock mass are important parameters that designers look for in selecting sites for foundations of civil and mining engineering structures in rocks. In the field, it is extremely difficult to stress a rock mass to its ultimate failure stress. Consequently, the field tests are performed up to a certain predefined stress level. It is shown in this study that the results of these tests can be used to arrive at the ultimate strength of the rock mass. A large number of uniaxial compressive strength (UCS) tests were conducted on the specimens of jointed block mass having various combinations of orientations and different levels of interlocking of joints. Four dominating modes of failure were observed. The strength and the tangent (elastic) modulus values of the mass for a specific failure mode, when plotted on the Deere– Miller classification chart, are found to follow an empirical straight line, indicating strong correlation between them. Using the gradient of this empirical line, correlations have thus been suggested to assess the ultimate strength of the jointed rock mass. The findings of the study have been verified by applying it to estimate the ultimate rock mass strength of nine rock types from few dam sites in the lower Himalayas. The ultimate strength obtained by the present methodology is compared with that obtained through the Q classification system. It is concluded that reasonably good estimates on field strength of jointed rocks are possible by using the correlations suggested in this study. D 2004 Elsevier B.V. All rights reserved. Keywords: Ultimate strength; Tangent modulus; Field modulus; Jointed rock mass; Failure mode

1. Introduction While dealing with rocks as a foundation material, civil and mining engineers are confronted with discontinua. A jointed rock mass is an assemblage of intact rock pieces separated by discontinuities such * Corresponding author. E-mail addresses: [email protected] (M. Singh)8 [email protected] (K. Seshagiri Rao). 0013-7952/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2004.09.001

as bedding planes, joints, shear planes, and faults. At a shallow depth where stresses are low, the failure of the intact rock material is minimal, and the behaviour of the rock mass is controlled by sliding along the discontinuities (Hoek, 2000). As a result, the design parameters for jointed rock masses depend both on the intact rock material and the discontinuities present in the mass. Strength and modulus of deformation of the rock mass are the two most important parameters the designers have to assess. The best estimate of the

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Table 1 Properties of the model material used (Singh, 1997) Serial No. Property 1 2 3 4 5 6 7

Value

Uniaxial compressive strength, rci Failure strain in uniaxial compression, e ai Brazilian strength, r tb Tangent modulus at 50% of failure stress, E i Friction angle of intact material, / i Friction angle along the cut joints of the elemental blocks, / j Deere and Miller (1966) classification

17.13 MPa 0.50% 2.49 MPa 5.344 GPa 338 378 EM

design parameters can only be made through large size field testing of the mass and loading it up to failure. It is, however, extremely difficult, if not

impossible, to stress a large volume of jointed mass in the field up to ultimate failure. A better alternative is to get the deformability characteristics by stressing a limited area of the mass up to a certain stress level and then relate the ultimate strength of the mass to the laboratory uniaxial compressive strength (UCS) of the rock material through a strength reduction factor (SRF, ratio of the rock mass strength to the intact rock strength). It is shown in this paper that the strength reduction factor and the modulus reduction factor (MRF, ratio of the rock mass modulus to the intact rock modulus) are correlated with each other. Using the correlations suggested in this paper, the field strength may be assessed through the observed field modulus, laboratory strength and modulus of

Fig. 1. Configuration of joints tested.

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intact rock. The applicability of the approach has been demonstrated by applying it to the uniaxial jacking test results collected from few important hydroelectric projects situated in the lower Himalayas. The findings have been deduced from an extensive experimental programme, as discussed in the next section.

2. Experimental programme Physical model tests are one of the best ways to understand the mechanism of failure of jointed rock masses. Due to ease in working and reproducibility of results, model materials have been used to simulate the rock material. Major physical model studies reported in the literature are those by Goldstein et al. (1966), Hayashi (1966), Brown (1970a,b), Brown and Trollope (1970), Walker (1971), Ladanyi and Archambault (1972), Einstein and Hirschfeld (1973), Lama (1974), Baoshu et al. (1986) and Yang and Huang (1995). Most of these studies were directed towards explaining the behaviour of jointed mass in a confined state. Singh (1997) carried out an extensive experimental study and tested more than 80 block specimens of a jointed mass of model material

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under uniaxial compression (Singh et al., 2002). The specimens were formed with cut blocks of lime silica model material having properties indicated in Table 1. The various configurations of the joints adopted in the laboratory study are shown in Fig. 1. The size of the specimen was 15
15
15 cm, and it consisted of three sets of joints. Joint Set-I was continuous and was inclined at variable angle dhT with the horizontal; Set-II was staggered at variable stepping dsT for each h (except for Type-B specimens); and Set-III was always kept vertical. By varying inclination h and stepping s, various modes of failure were obtained. Besides Type-A, three more configurations of joints, namely, Types-B, -C and -D, were also used (Fig. 1). The tests were performed under uniaxial loading with a strain controlled loading arrangement. During testing, the deformations were continued beyond the failure of the specimen until the load reduced to about half the peak load, and the deformations during loading were measured in X, Y, and Z directions. Axial stress and the corresponding strains in the Z and X directions were computed and presented in the form of stress–strain curves for each specimen. Some typical stress–strain curves (specimen Type-A, h = 808, s = 0, 1/8, 2/8, and 3/8) are presented in

Fig. 2. Some typical stress–strain curves for Type-A specimens.

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Fig. 2. The tangent modulus of the mass was computed by drawing a tangent to the stress–axial curve at 50% of the failure stress.

sketches of the failure modes observed are presented in Fig. 3. The failure modes obtained are:

3.1. Failure modes

(i) splitting of intact material of the elemental blocks, (ii) shearing of intact block material, (iii) rotation of the blocks, and (iv) sliding along the critical joints.

The failure modes were identified based on the visual observations at the time of failure. Although there were complex and mixed modes, it was possible to identify the most dominating one. Schematic

These modes were observed to depend on the combination of orientation h and the stepping s (Table 2). The angle h in this study represents the angle between the normal to the joint plane and the loading

3. Results and discussions

Fig. 3. Modes of failure of jointed mass (Singh et al., 2002).

M. Singh, K. Seshagiri Rao / Engineering Geology 77 (2005) 127–137 Table 2 Modes of failure in jointed mass

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3.2.3. Sliding The specimen fails due to sliding on the continuous joints. The mode is associated with large deformations, stick–slip phenomenon, and poorly defined peak in stress–strain curves. This mode occurs in the specimen with joints inclined between hc208–308 if the interlocking is nil or low (see Table 2 for details). For orientations, hc358– 658 sliding occurs invariably for all the interlocking conditions. 3.2.4. Rotation The mass fails due to rotation of the elemental blocks. It occurs for all interlocking conditions if the continuous joints have h N 708, except for h equal to 908 when splitting is the most probable failure mode.

direction, whereas the stepping s represents the level/ extent of interlocking of the mass. The following observations were made on the effect of the orientation of the joints and their interlocking on the failure modes. These observations may be used as rough guidelines to assess the probable modes of failure under a uniaxial loading condition in the field. 3.2. Failure mode vis-a`-vis orientation and stepping of the joints 3.2.1. Splitting Material fails due to tensile stresses developed inside the elemental blocks. The cracks are roughly vertical with no sign of shearing. The specimen fails in this mode when joints are either horizontal or vertical and are tightly interlocked due to stepping. 3.2.2. Shearing In this category, the specimen fails due to shearing of the elemental block material. Failure planes are inclined and are marked with signs of displacements and formation of fractured material along the sheared zones. This failure mode occurs when the continuous joints are close to horizontal (i.e., h V 108) and the mass is moderately interlocked (Table 2). As the angle h increases, the tendency to fail in shearing reduces, and sliding takes place. For hc308, shearing occurs only if the mass is highly interlocked due to stepping.

3.3. Strength and tangent modulus Deere and Miller (1966) proposed a classification for intact rocks. In this classification, the rocks are represented by a two-lettered symbol; the first letter represents the range of the uniaxial compressive strength (UCS) and the second letter represents the modulus ratio (ratio of tangent modulus to the UCS). This classification, therefore, gives an idea of the two most important properties used in design, i.e., the strength and the deformability of the rock. Ramamurthy and Arora (1993) extended the use of this classification to jointed rocks. They however suggested different ranges of the strength and modulus ratio for different categories of the jointed rocks (Tables 3a,b). The laboratory strength and the tangent moduli of the specimens tested in this study are presented in the form of the classification charts (Figs. 4–8). The results have been plotted according to different Table 3(a) Strength classification of intact and jointed rocks (Ramamurthy and Arora, 1993) Class

Description

UCS, MPa

A B C D E F

Very high strength High strength Moderate strength Medium strength Low strength Very low strength

N250 100–250 50–100 25–50 5–25 b5

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Table 3(b) Modulus ratio classification of intact and jointed rocks (Ramamurthy and Arora, 1993) Class

Description

Modulus ratio, M rj

A B C D E

Very high modulus ratio High modulus ratio Medium modulus ratio Low modulus ratio Very low modulus ratio

N500 200–500 100–200 50–100 b50

modes of failure. In Fig. 8, the results belonging to all the modes are combined. The position of the intact material on the classification chart is designated by point dIT. It is interesting to note that the points representing the jointed mass on these charts lie very near to a best fitting straight line, starting from the point dIT. The coefficients of correlation of the best fitting line in Figs. 4–8 are 0.91, 0.94, 0.87, 0.95, and 0.94, respectively. An empirical line can therefore be found to represent the positions of the points belonging to the jointed mass. It is inferred that if an intact rock is intersected by joints, its strength and tangent modulus reduce in such a manner that its position on the Deere–Miller classification chart moves on an empirical line that has a specific gradient. How much the position moves from the intact rock position depends on

Fig. 4. Strength and tangent modulus values for splitting mode of failure (chart as per Ramamurthy and Arora, 1993).

Fig. 5. Strength and tangent modulus values for shearing mode of failure (chart as per Ramamurthy and Arora, 1993).

frequency, orientation, and interlocking of the joints. The gradients of this empirical line for splitting, shearing, sliding and rotational modes are observed as 1.8, 1.8, 1.5 and 1.4, respectively (Figs. 4–7). Sometimes the details on the orientation of the joints and their interlocking conditions may not be available. For these situations, where no assessment

Fig. 6. Strength and tangent modulus values for sliding mode of failure (chart as per Ramamurthy and Arora, 1993).

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is possible about the failure mode, an average value of the gradient of the empirical line may be taken as 1.6 (Fig. 8). The empirical line discussed above may be used to assess the ultimate strength of the jointed rock mass as given below:

 Gradient of the line ¼ log Ej =Ei =log rcj =rci  1=Gradient r E ð1Þ Z rcjci ¼ Eji or SRF ¼ ðMRFÞn ;

ð2Þ

where SRF is the strength reduction factor, r cj/r ci; MRF is the modulus reduction factor, E j/E i; n = 1/ Gradient, which is 0.56 for splitting and shearing, 0.66 for sliding, 0.72 for rotation, 0.63 for average of all modes; E i and E j are the tangent moduli (elastic) of the intact and jointed rock, respectively; and r ci and r cj are the uniaxial compressive strength of intact and jointed rock, respectively. Eq. (1) indicates that the ultimate strength r cj, and the tangent modulus E j of a jointed mass are correlated with each other through the intact rock properties r ci and E i. It is assumed that r ci and E i will be available from laboratory tests, and the value of the

Fig. 8. Strength and tangent modulus values for all modes of failure (chart as per Ramamurthy and Arora, 1993).

E j will be obtained from the field tests. The probable mode of failure may be assigned depending on the orientation and the interlocking of the joints by using Table 2. The ultimate strength of the jointed mass can thus be assessed by using the expression given above (Eq. (1)). In the absence of information about the orientation of joints and interlocking, the average value for all the modes may be used to assess the strength value. Alternatively, one can work out the range of values in which the strength of the mass is likely to lie for different modes of failure.

4. Application to the field 4.1. Jointed rocks

Fig. 7. Strength and tangent modulus values for rotational mode of failure (chart as per Ramamurthy and Arora, 1993).

Mehrotra (1992) has discussed the results of laboratory tests and the field uniaxial jacking tests performed at several dam sites in India. The details of the field investigations of the rocks are presented in Table 4. The tests were performed on 10 rock types, which were classified as poor to fair rock masses based on RMR. Uniaxial jacking tests were performed as per the Indian Standard IS:7317 (1974) by stressing two parallel flat rock faces on the opposite walls of a drift by means of a hydraulic jack (Mehrotra, 1992). Two

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Table 4 Details of field investigation on rocks (after Mehrotra, 1992) Dam site Jamrani, Nainital Kotlibehl, Rishikesh, Lakhwar Dam

Srinagar Tehri Utyasu

Rock type Sandstone Claystone Limestone Shale Slate Xenolith Trap rock Metabasic Quartzite Phyllite Quartzite

presented in Table 5. The field strength is estimated as:

Number of uniaxial jacking tests 9 4 10 3 10 8 29 7 5 22 13

circular plates of 60-cm diameter were used at the top and bottom of the loading jack to stress the rock faces. The stress was applied in two cycles as shown in Fig. 9. The displacements were measured at the loading surface through dial gauges. The second cycle of stress deformation curve was used to compute the field modulus. The following expression was used to compute the field modulus: Ee ¼

mð1  m2 ÞP pffiffiffi ; A de

Field strength ¼ Intact rock strength   observed field modulus; E e n :
intact rock modulus; E i It is also observed from Table 5 that instead of reporting a single value, Mehrotra (1992) has reported the ranges of r ci, E i, and E e. The minimum and maximum values of rock mass strength r cj are therefore computed using Eq. (2) and reported in Table 6. It may be noted that, for claystone, the UCS of the intact rock was not available. Singh et al. (1997), based on the back analysis of several tunnels, have suggested the following expressions for rock mass strength based on quality index Q. rcj ¼ 7cQ1=3 MPa:

Barton (2002) modified Eq. (4) and included in it the uniaxial compressive strength r ci and suggested the following expression: 

ð3Þ

where, E e is the field modulus of elasticity of the rock mass, d e is the elastic deformation of the rock face in cm, m = 0.96 for circular plate of 25 mm thickness (used), m is the Poisson’s ratio (assumed as 0.2), P is the load on the test plate in kg, and A is the area of the plate in cm2 (2828 cm2 for 60 cm diameter plate). The summary of the in situ test results (Mehrotra, 1992) is presented in Table 5.

ð4Þ

rcj ¼ 5c

Qrci 100

1=3 MPa;

ð5Þ

where c is the unit weight of the rock in gm/cm3 and r cj and r ci are in MPa. The values of the rock mass strength through expressions given by Singh et al. (1997) and Barton (2002) are also presented in the Table 6. A comparison of the estimated values of the rock mass strength r cj,

4.1.1. Estimation of the rock mass strength To estimate the field strength of a rock mass, as per Eq. (2), the following inputs are required: ! observed laboratory strength of intact rock, r ci, ! observed tangent modulus of intact rock, E i, ! observed field modulus, E e, and ! data on mapping of joints to assess failure mode. It should be noted that, from now on, E j in Eq. (1) is replaced by E e. The laboratory strength and modulus values of the intact rocks along with the field modulus values reported by Mehrotra (1992) are

Fig. 9. Definitions of the field modulus of elasticity (Mehrotra, 1992).

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Table 5 Summary of laboratory and in situ test results for various rock types (after Mehrotra, 1992) Serial No.

Rock type

Intact rock strength, r ci (MPa)

Natural moisture content (%)

Average density (g/cm3)

Intact rock modulus, E i (GPa)

Observed field modulus, E e (GPa)

Rock mass quality, Q

Rock mass rating, RMR

1 2 3 4 5 6 7 8 9 10

Sandstone Claystone Slate Xenolith Trap rock Shale Limestone Metabasic Quartzite Phyllite

32.00–75.00 – 1.00–38.00 21.00 98.00–196.50 16.80–37.00 21.00–49.00 70.90–104.00 67.00–128.00 38.00–133.00

1.25–1.50 2.25–4.50 0.30–0.94 0.25 0.28–0.30 1.0–1.95 0.30–2.50 0.30–0.60 0.40–0.45 0.35–0.90

2.42 2.48 2.73 2.73 2.74 2.27 2.29 2.81 2.70 2.73

6.76 8.41 20.00 14.74 12.35–36.43 10.80 11.90 21.00–22.40 28.25–49.80 6.68–7.07

1.75–2.90 1.43–3.67 0.98–7.80 2.95 1.98–13.00 2.22–2.95 0.55–4.80 4.38–7.11 0.98–14.37 0.73–4.13

0.7–2.0 0.7–1.8 0.3–3.9 1.5–1.6 1.7–11.7 0.9–1.5 0.1–4.0 3.3–4.7 0.3–19 0.3–4.1

20–45 24–38 18–45 28–33 30–61 25–30 11–53 37–60 27–71 18–50

through the present method, and the two other approaches is presented in Fig. 10. The following observations can be made from this comparison: (1)

(2)

Almost all the lower bound values of the ranges of r cj predicted by the present method are in close proximity of the values predicted through Q system. It is therefore concluded that a reasonable assessment of the ultimate strength of the rock mass is possible through the observed field modulus without stressing the rock mass up to failure. The upper bound values of r cj are found to have more deviations compared to those predicted through Q system. The difference in values predicted through Singh et al. (1997) and Barton (2002) expressions is low, which is obvious, as Barton (2002) had modified the Singh et al. (1997) expression.

5. Conclusions Experiments have been conducted on block-jointed mass under uniaxial compression condition. Several geometries of the joint configurations were used to obtain various modes of failure. Four dominating modes under uniaxial stress conditions have been observed. The results obtained on strength and tangent modulus have been used to arrive at the relations linking the properties of the jointed mass with those of the intact rock. These properties in general depend on the failure mode, which in turn depends on the joint configuration. The validity of the expressions suggested has been verified by applying them to the field data obtained from several dam sites from northern India. The following conclusions are drawn: (1)

The ultimate strength and the tangent modulus of jointed block mass tested in the laboratory are

Table 6 Computation of rock mass strength Serial No.

Rock type

rci (MPa)

MRF=E e /E i

SRF=(MRF)0.63

rcj from SRF (MPa)

1 2 3 4 5 6 7 8 9 10

Sandstone Claystone Slate Xenolith Trap rock Shale Limestone Metabasic Quartzite Phyllite

32.00–75.00 – 1.00–38.00 21.00 98.00–196.50 16.8–37.00 21.00–49.00 70.9–104.00 67.00–128.00 38.00–133.00

0.259–0.429 – 0.049–0.39 0.200 0.160–0.357 0.206–0.273 0.047–0.407 0.209–0.317 0.035–0.289 0.109–0.584

0.427–0.587 – 0.149–0.553 0.363 0.316–0.522 0.369–0.442 0.145–0.567 0.373–0.485 0.120–0.457 0.248–0.713

13.66–44.00 – 0.15–20.99 7.63 30.93–102.57 6.20–16.34 3.03–27.66 25.38–50.47 8.06–58.50 9.42–94.83

r cj from Q (MPa) Barton (2002)

Singh et al. (1997)

7.34–13.85 – 1.97–15.50 9.28–9.48 16.24–38.95 6.05–9.33 3.16–14.33 18.65–23.84 7.91–39.11 6.62–24.03

15.04–21.34 – 12.79–30.08 21.86–22.35 22.89–43.54 15.34–18.18 7.44–55.27 29.28–32.94 12.65–50.43 12.79–30.56

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r cj

Fig. 10. Comparison of the rock mass strength predicted through different approaches.

(2)

(3)

(4)

correlated with each other through intact rock properties. On the Deere–Miller classification chart, this relationship may be represented by an empirical line starting from the intact rock position. Due to presence of joints, the strength and modulus of the rock mass are reduced, and the point representing the jointed rock on Deere– Miller classification chart moves along this empirical line. The gradient of the empirical line specifies the failure mode. For splitting, shearing, sliding, and rotation, the gradient on log–log scale is 1.8, 1.8, 1.5, and 1.4, respectively. The strength reduction factor (SRF) may be estimated as SRF = (MRF)n , where index n may be taken as 0.56, 0.56, 0.66, and 0.72 for splitting, shearing, sliding and rotation modes, respectively. In case no assessment of failure mode is possible, an average value of 0.63 may be used for a rough estimate of the strength. A comparison of the ultimate rock mass strength predicted through the present method indicates close prediction with the results from Singh et al. (1997) and Barton (2002) for lower bound values of Q. For upper bound values, the suggested method in its present form predicts higher results compared with the other two methods.

Appendix A. List of symbols and abbreviations h r ci

Inclination of continuous joints with horizontal Uniaxial compressive strength of intact rock

Uniaxial compressive strength of the jointed rock/rock mass /i Friction angle of intact material /j Friction angle along the cut joints of the elemental blocks r tb Brazilian strength Ee Observed field modulus of elasticity Ej Laboratory tangent modulus of jointed rock Ei Laboratory tangent modulus of intact rock MRF Modulus reduction factor n Index RMR Rock mass rating s Stepping of joints SRF Strength reduction factor Q Rock mass quality index

Acknowledgements Some part of the work presented in this paper has been taken from the PhD thesis of the first author, which was completed under the supervision of Prof. T. Ramamurthy of I.I.T. Delhi and the coauthor of this paper. The authors gratefully acknowledge the contribution of Prof. Ramamurthy in completion of the work. The authors also thank the reviewers Dr. N. Barton and Prof. H.H. Einstein for their valuable suggestions to make the paper in the present form.

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