Rock Mechanics, Nelson & Laubach (eds) O 1994 Balkema, Rotterdam, ISBN 90 !i4 10 380 8

A new empirical peak shear strength criterion for rock joints

Pinnaduwa H.S.W. Kulatilake, Guohua Shou & Robert M.Morgan University of Arizona, Tucson, A r k , USA

Tsan-Hwei Huang National Taiwan UniversifX Taipei, Taiwan

ABSTRACT: The paper presents findings of an experimental as well as theoret ica l /analyt ica l research i n i t i a t e d (a ) to quantify jo in t surface roughness on a plane and then (b) to develop a new peak shear s t rength c r i t e r i o n f o r rock j o i n t s which has the capabi l i ty of capturing the anisot ropic behavior of peak shear strength due to the anisotropy of j o i n t surface roughness. The resu l t s show c lea r ly tha t a t l e a s t two parameters a r e required to quantify j o i n t roughness i n one dimension. By l imi t ing t o only s t a t i s t i c a l parameters, an average slope of the p ro f i l e (average 1 angle) is suggested to quantify the large scale undulations of the p r o f i l e (non-stationary p a r t of the p ro f i l e ) and the root mean square of the slope (2,3 is suggested t o quantify the small scale roughness of the p r o f i l e (s ta t ionary p a r t of the p ro f i l e ) . In order t o capture the anis t ropic roughness, i t was found necessary to calcula te both averages I and 2,' i n d i f fe ren t d i rec t ions on the j o i n t surface. The shear strength r e s u l t s of a model material j o i n t show tha t the suggested new equation, which i s based on very l imited experimental data, has a good capabi l i ty of predic t ing the anisot ropic peak shear strength of the jo in t s .

1 INTRODUCTION

In the mining, c i v i l , and petroleum engineering disc ipl ines , the engi- neers confront problems associated with geotechnical systems which are i n o r on jo inted rock. Due t o the complex nature of natura l j o i n t pat terns i n rock, and the uncertainty involved i n estimating geomechanical proper- t i e s of j o i n t s and i n t a c t rock, the prediction of mass mechanical and hydraulic proper t ies of rock a t a large scale using available techniques presents a great challenge fo r the geo-engineering profession.

Strength and deformation behavior of jo in t s and flow properties of j o i n t s depend very much on the surface roughness of jo in t s . These e f f e c t s a r i s e from the f a c t tha t the surfaces composing a j o i n t a re rough and mismatched a t some sca le . The shape, s i ze , number, and s t rength of contacts between the surfaces control the mechanical proper t ies . The separation between the surfaces or the "aperture" determines the hydrau- l i c proper t ies . The " Jo in t Roughness Coefficient," JRC (Barton and Chou- bey, 19771, various s t a t i s t i c a l parameters and the f r a c t a l dimension, D, (Mandelbrot, 1983) have been suggested as parameters to quantify roughness of rock j o i n t p r o f i l e s along l ines (or in one dimension).

Barton and Choubey (1977) have suggested two methods to choose an appropriate value for JRC. In the first method, the roughness profile of the natural joint is visually compared with ten standard profiles pub­lished by Barton and Choubey (1977). For these standard profiles JRC values have been assigned between 0 and 20 in steps of 2 starting from the smoothest to the roughest. This visual comparison method has been found to be subjective and unreliable by some investigators (Maerz et al. 1990; Miller et al. 1990; among others). In the other method, tilt and pull tests should be performed on natural discontinuities to estimate JRC by back analyzing the test results using the empirical equation suggested for peak shear strength by Barton and Choubey (1977) along with the esti­mations for normal stress and basic friction angle on the joint plane, and joint wall compressive strength. Some researchers (Miller et al., 1990, among others) feel that the second method has little practical merit because roughness preferably should be used to predict shear strength not vice-versa.

Researchers have tried statistical parameters such as centerline aver­age value, mean square value, root mean square (RMS) value, mean square of the first derivative, (Zz) RMS of the first derivative, RMS of the second derivative, probability density function, auto-correlation func­tion, spectral density function, structure function (SF), roughness pro­file index (Rp) and micro-average i angle (Ai) [Wu and Ali, 1978; Tse and Cruden, 1979; Krahn and Morgenstern, 1979; Maerz et al. 1990] to quantify roughness. Some of these parameters are conceptually the same and hence they are dependent. At present, no firm conclusions are available in the literature evaluating how good these statistical parameters are in quan­tifying roughness. It is very important to note that only a single sta­tistical parameter has been used to quantify rock joint roughness. It is highly questionable whether a single statistical parameter is adequate in capturing the roughness arising from both stationary a~d non-stationary parts of a profile.

Some researchers (Brown & Scholz, 1985; Andrle and Abrahams, 1989; Chesters et al., 1989; Miller et al., 1990; Power et al., 1991; Huang et al., 1992) have investigated the possibility of usi.ng fractal dimension to provide a measure for roughness of rock joints. These investigations have led to controversial findings. The main reason for this is our limited understanding of the application of fractal theory in quantifying the roughness of natural rock joints.

The brief literature review given above shows clearly that further research on roughness characterization of natural, rock joints is badly needed. A part of the paper presents findings of a research initiated on roughness quantification of natural rock joints.

The main contributors for the development of peak shear strength crite­ria in the low normal effective stress range are Ladanyi and Archambault (1969) and Barton and Choubey (1977). The joint peak shear strength shows anisotropic properties due to roughness variation with the shearing direction in direct shear tests (Huang and Doong. 1990). Even along one particular direction, the shear strength of a natural joint can be dif­ferent between the forward and the backward directions. To capture the above observations, either the existing shear strength criteria should be improved or a new peak shear strength criterion should be developed. A part of the paper presents findings of a research initiated to develop a new empirical peak shear strength criterion.

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(a) (b) (c)

Figure 1 Photographs illustrating the surfaces of (a) natural joint, (b) silicone rubber cast, and (c) model joint.

2 EXPERIMENTAL STUDY

Silicone rubber was used to obtain a pair of perfectly mated silicone rubber casts to represent the topographic features of a sandstone rock joint of about 100 mm diameter. These silicone rubber casts were then used to prepare about 65 model material joints (a mixture of plaster of paris, sand and water) having the same cross section and topographic fea­tures of the natural rock joint . The surfaces of the natural joint, silicone rubber cast and the model joint are shown in Fig . 1.

Since all the model material joints were made from the same silicone rubber cast, it was sufficient to measure the surface roughness of only one model material joint surface. A stylus profilograph was used to perform the roughness measurements. Roughness profiles were obtained in six directions (at every 30 degrees) on the model joint surface. In each direction, roughness profiles were obtained along nine parallel lines which are spaced at equal distance. Along each line the surface height was recorded at every 0.2 mm up to an accuracy of 0.01 mm.

Each pair of perfectly mated model material casts was used to prepare a model material shear sample to perform a direct shear test in a large Wykeham Farrance shear machine having a square cross section shear box of size 12 inches. Each shear sample was subjected to a selected normal stress in the range 0.5 - 10 kgj cm2 and was sheared either in the forward or backward direction along one of the six directions used for roughness measurements . Using this testing scheme , a total of twelve direct shear test results were obtained for a selected normal stress. Five different normal stresses were used to obtain a total of 60 direct shear test results. All shear tests were performed until failure. The shear strength data obtained made it possible to study the anisotropic behavior of the peak shear strength of the joint resulting from anisotropic sur­face roughness. Four shear tests were performed under different normal stresses on smooth horizontal joint surfaces made out of the model material to estimate its basic friction angle. Two tests were performeci to determine the joint compressive strength of the model material.

3 QUANTIFICATION OF JOINT ROUGHNESS

In this investigation, only the statistical parameters were considered in quantifying joint roughness. Future research will focus on investigating the possibility of using fractal theory in quantifying joint roughness .

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All the statistical parameters mentioned in the Introduction were included in the investigation to select the best parameter(s) to quantify roughness. Out of these parameters, the statistical parameters asso­ciated with the slope of the profile (Z2.A,.Rp) and SF were found to have strong correlations with JRC. Out of the slope parameters, (Z2) was found to be the best one because it changed significantly from the smoothest profile to the roughest of Barton's profiles and also showed the capabil­ity of reflecting the influence of anisotropic roughness on anisotropic peak shear strength. It is quite clear from the definitions for Z2, SF, Ai and R p , that SF, A, and Rp are all highly correlated to Z2' In addi­tion, Z2 is unitless, and is better than SF, which has units. Therefore, out of the statistical parameters investigated, Z2 was found to be the best parameter.

If the joint surface is inclined (not horizontal) with respect to large-scale undulations, then the shear strength of the joint in the upward inclined direction will be different than the shear strength of the joint in the opposite direction. However, Z2 will be the same for both directions. Therefore, Z2 alone is not sufficient to quantify roughness. It seems that 22 is a good parameter to quantify small-scale roughness. To quantify large-scale undulations, it is necessary to define another statistical parameter.

It may be important to note that all 10 of Barton and Choubey's surface profiles can be considered as stationary profiles. The stationary pro­files satisfy the following second-order stationary properties: (a) The mean surface is horizontal with respect to spatial location; (b) The variance of the surface height around the mean surface is a constant with respect to spatial location; and (c) The covariance function of the sur­face height depends only on the lag distance irrespective of the spatial location.

However, many rock joint surface profiles can exhibit non-stationary properties. Neither JRC nor 22 can capture the non-stationarity. At the simplest level, the non-stationarity of the joint surface can be repre­sented by a linear function with a positive or a negative slope. This slope can be denoted by I and can be estimated through regression analysis. For most joint surfaces, this linear function will be suffi­cient to model the non-stationary portion of the joint surface profile. Therefore, in this investigation, to model the non-stationary part or the large-scale undulations of the surface profile, the parameter I was selected. To model the stationary part or the small-scale roughness of the surface profile, the statistical parameter Z2 was chosen.

To estimate the I of the surface in a certain direction, each I of the nine profiles in the selected direction was calculated; then, using the following formula, the weighted average value of I was estimated and used as one of the roughness parameters.

( 1 ) A verage I 1 9

-9-l..Ii l L l i·1 I . I I-I

where I j = I angle estimated for jth surface height profile in the direc­tion considered Ii = length of the jih surface height profile in the direction con­sidered.

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:s ..... ::r: <.!> jjj ::r: w u it ~ ::> (f)

XI X2 X:s

YHI IY;ly. ... ,

, AJ:I AJ:. .)(;-1 )(; X .. ,

y.

Xn

I ... " I L

DISTANCE ALONG A MEASUREMENT LINE (X)

Figure 2 The diagram used to define statistical parameters, for a joint surface profile.

For each profile, in a certain direction, a modified Z2 (z'J was calcu­lated using the following equation:

(2) Z' 2 ~ 1 n-I _

.-=-1 I «(Yi;l Yi+l) - (Yi-n i-I

IlI.X)2 ]

1/2

where Y,.l - measured surface height corresponding to X •• l (See Fig. 2) :V ... - calculated surface height from linear trend equation at x ...

y, - measured surface height corresponding to Xi (See Fig. 2)

Yi= calculated surface height from linear trend equation at Xi

D.x = Xi .. - x. for any i

n - number of y measurements made on the profile.

From equation (2) it is clear that the linear trend was removed from the measured values of surface height before calculating z'z values. There­fore, it provides a measure of small-scale roughness or roughness arising due to the stationary part of the surface profile. For each of the six directions considered. an average Z'2 value was computed according to the following equation:

(3) Average Z'2 1 9

9-I(z' I l/-l 1- I

If

where (z 2) j = calculated z' 2 value for j th profile in the direction

considered lj = length of the j th profile in the direction considered.

Calculated average I and average Z'2 values clearly showed that the sur­face roughness of the natural joint considered is anisotropic.

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4 SUGGESTED EMPIRICAL PEAK SHEAR STRENGTH CRITERION FOR ROCK JOINTS

A part of direct shear test results obtained for the model joint surface in different directions under different normal stresses are shown in Fig. 3. These results show very clearly that the peak shear strength is ani­sotropic and it depends on the normal stress. Also, the results show the possibility of getting different peak shear strength values in the forward and the backward directions along a particular direction. The basic friction angle (,.) and the joint compressive strength (JCS) of the model material were found to be 24 degrees and 40 kg/cmZ, respectively.

The following equation is suggested to estimate peak shear strength of a rock joint under low normal effective stresses:

(4) -C On tan ($0 + A(z' 2)8 {Jog(JCS /0 n)} C + I)

where A, Band C are coefficients to be determined by fitting a curve to the experimental data.

The peak shear strength data for o. ~ 0.5, 3 2 and 10.0 kg/cmz were used along with the z·2.1.~. and JCS values to estimate the coefficients A, B and C through multiple linear regression analysis. The following regres­sion equation was obtained at a multiple R value of 0.9674.

(5) -c=on tan + 25.28(Z· 2)°·0646 {log (J CS /0 n)} 0.969\ + I)

Equation (5) was then used to predict the peak shear strength at an - 1.0 and 5.0 kg/cmz in different directions. These predicted peak shear strengths are compared with the measured peak shear strengths in Fig. 3. Three researchers who work in rock engineering were asked to estimate JRC values for all the roughness profiles considered in this research. The mean JRC value obtained for each profile from the three researchers were used in equation (6) to estimate an average JRC value for each of the six directions considered.

(6) Average JRC 1 9

-9-I(JRC)i l i '" i-I L lj f- I

where JRC = mean JRC value for jth profile in the direction considered

li = length of the jth surface height profile in the direction considered.

The average JRC value estimated for each direction was then used in Bar­ton's shear strength equation to predict the peak shear strength of the model joint in the direction considered for On = 1.0 and 5.0 kg/cm2. These shear strengths were predicted in all the twelve directions and are shown in Fig. 3. It is clear from Fig. 3 that the shear strengths pre­dicted by Barton's equation underestimates the actual peak shear strength significantly. Also, note that the Barton's equation is incapable of capturing the anisotropic shear strength occurring between the forward and the backward directions during shearing in any particular direction. Figure 3 shows clearly that the new equation has better capability for predicting joint peak shear strength compared to the Barton's equation.

570

'so , ,

180 - -.-EI

'J"f¥'

~ o

\ \

\

o 01

'\ ' /

'0/ /\"1-"7"" /' 'r-:..., .\1-/"'.)\

~

: "-{. /' \ '.

f- .j Sf?-, -) .... )­\ -k·~1~ . :

'. '.1 \ )", '<' 1"..J._.'v .' .... /". 'T '\ /' .... " -,,\' /.- .. t- ..

I ~

<\f

/

/ \ l' I

, o I' N

\

'b o

/'

.."

SCALE ALONG Mf( RADIAL DIRECTION

o 4 8 kg/em2

'!>o

_ 0

J'J'o

••••• MEASURED VALUE FROM EXPERIMENT FOR c;, "1.0 kg/em2

•• -.- PREDICTED VALUE FROM NEW EOUATION FOR c;, .. 1.0 kg/em2

- - - - PREDICTED VALUE FROM BARTON'S EOUATION FOR (7. "'1.0 kg/em2

00000 MEASURED VALUE FROM EXPERIMENT FORq; =5.0kg/em2 2

-- PREDICTED VALUE FROM NEW EOUATION FOR c;, .. 5.0 kg/em

- .. - .. PREDICTED VALUE FROM BARTON'S EOUATION FOR 0. "'5.0 kg/em2

Figure 3 Comparison between measured and predicted peak shear strengths, based on the new equation and on Barton's equation.

Figure 3 also shows that the new equation has the capability of capturing the anisotropic behavior of the peak shear strength arising due to' the anisotropic roughness.

5 CONCLUSIONS

The results of this investigation show clearly that at least two parame­ters are required to quantify joint surface roughness. By limiting only to statistical parameters, an average I angle is suggested to capture the large-scale undulations and zz' is suggested to capture the small-scale

571

roughness. Both zz' and I should be calculated in different directions in order to capture the anisotropic roughness which is present in most of the natural rock joint surfaces. Based on very limited experimental data, a new equation is suggested to predict the peak shear strength of joint surfaces. The validation exercise performed shows clearly that the new equation has a good capability of predicting anisotropic peak shear strength of joints. In practice, the new equation should be used along with a factor of safety of about 1.5. This investigation was only a research initiation on this topic. Further experimental as well as theo­retical/analytical work is very much needed in order to develop new tools to quantify joint surface roughness and to predict the peak shear strength of rock joints.

ACKNOWLEDGEMENT

The Arizona Mining and Mineral Resources Research Institute under Grant No. Gll14l04 provided partial financial support for this study. It is gratefully acknowledged.

REFERENCES

Andrle, R. & A.D. Abrahams 1989. Fractal techniques and the surface roughness of talus slopes. Earth Surface Processes and Landforms 14:197-209.

Barton, N.R. & V. Choubey 1977. The shear strength of rock joints in theory and practice. Rock Mechanics (Springer-Verlag) 10:1-54.

Brown, S.R. & C.H. Scholz 1985. Broad band width study of the topography of natural rock surfaces. J. Geophys. Res. 90:12575-12582.

Chesters, S., H.Y. Wen, M. Lundin, & G. Kasper 1989. Fractal based char­acterization of surface texture. Applied Surface Science, 40:185-192.

Huang, S.L., S.M. Oelfke, & R.C. Speck 1992. Applicability of fractal characterization and modelling to rock joint profiles, Int. J. Rock Mech. Min. Sci. 29:89-98.

Huang, T.H. & Y.S. Doong 1990, Anisotropic shear strength of rock joints. Proc. of the Int. Symp. on Rock Joints, Loen, Norway: 211-218.

Krahn, J. & N.R. Morgenstern 1979. The ultimate frictional resistance of rock discontinuities. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 16:127-133.

Ladanyi, B. & G. Archambault 1969. Simulation of the shear behavior of a jointed rock mass. 11th Symp. On Rock Mech. Cal. 7:105-125.

Mandelbrot, B. 1983. The fractal geometry of nature. W. Freeman, San Francisco, 468 pp.

Maerz, N.H., J.A. Franklin, & C.P. Bennett 1990. Joint Roughness measure­ment using shadow profilometry. Int. Jour. of Rock Mech. Min. Sci., 27/5:329-343.

Miller, S.M., P.C. McWilliams, & J.C. Kerkering 1990. Ambiguities in estimating fractal dimensions of rock fracture surfaces. Proc. 31st u.s. Symp. on Rock Mech. 471-478.

Power, W.L. & T.E. Tullis 1991. Euclidean and fractal models for the description of rock surface roughness. J. Geophys. Res. 96:415-424.

Tse, R. & D.M. Cruden 1979. Estimating joint roughness coefficients. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 16:303-307.

Wu, T.H. & E.K. Ali 1978. Statistical representation of joint roughness. Int. J. Rock. Mech. Min. Sci. & Geomech. Abstr. 15:259-262.

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