Microsoft Word - cet-01.docx CHEMICAL ENGINEERING TRANSACTIONS VOL. 46, 2015 A publication of The Italian Association of Chemical Engineering Online at www.aidic.it/cet Guest Editors: Peiyu Ren, Yancang Li, Huiping Song Copyright © 2015, AIDIC Servizi S.r.l., ISBN 978-88-95608-37-2; ISSN 2283-9216 Slice Method for Rock Slope Stability Analysis Based on the Hoek–Brown Criterion Xiaohu Huanga, Changming Wang*a, Shaohua Kuangb, Tianzuo Wanga a College of Construction Engineering, Jilin University, Changchun 130026, China, b Liaoning Jiaotong Investigation and Design Institute, Shenyang 110005, China. wangcm@jlu.edu.cn Hoek-Brown criterion is widely used to predict rock strength and analyze stability. Convert the H-B formula to find the relationship between the normal stress σn and shear stress τ of the sliding surface. Combined with the H-B criterion, gives a formula Fs to solve the safety factor. The research shows that the Fs is associated with H-B constants mi, geological strength index GSI, disturbance factor D, slice parameter αi , ration of normal stress and saturated single axis compressive strength Sri. Through building rock slope model, to analyze the safety factor Fs at different ranges of parameters. Compared with the results of Swedish slice method, Janbu method, Bishop method and M-P method, it displayed the new algorithm’s rationality and limitations .Finally verifying the applicability of the new algorithm’s through the engineering example. 1. Introduction The stability analysis of rock slope has been the key and hot point in engineering research. The traditional methods such as Swedish slice, Jan-bu, Bishop and M-P based on M-C criterion. The cohesion c and the internal friction angle φ are used to calculate the safety factor value. They have been matured in theory, and widely used in project practice. But the M-C criterion can’t reflect the influence of the nonlinear failure and engineering geology for rock masses. In 1980, E. Hoek and E. T. Brown proposed the H-B criterion to supplement the deficiency of the M-C criterion (HOEK E, BROWN E T (1980)). The H–B strength criterion can describe the destruction and deformation characteristics of rock mass with good accuracy, reflect the nonlinear characteristics of rock destroyed, and well present the surface stress state, blasting damage and stress relieving effect on strength. The relevant parameters concerning H–B criterion are difficult to be determined and the corresponding applicability for jointed rock mass is poor, as GSI involved is difficult to be determined. Because of this, through it is widely used in rock mass strength research, but it can’t be used in the stability analysis of Rock Slope directly. Many scholars have done a lot of research on this subject, strength reduction method based on the Hoek–Brown Criterion was used to analysis of slope stability(Lin Hang et al. (2007); Wu Shunchuan et al. (2006)). But most of scholars explore the relationship between the cohesion c and the internal friction angle φ with the H–B parameters mi, a, s and the geological strength index GSI. The H-B criterion is not used to solve slope stability analysis question directly. The H–B strength criterion was used to analysis of slope stability combined with computer technology(Carranza-Torres C (2004); Shen J et al. (2012)). This article will use the H-B criterion to analysis slope stability combined with Matlab and offer a reference of application for the H-B criterion directly. 2. Formula of Fs under the Hoek-Brown criterion E. Hoek and E. T. Brown (HOEK E, BROWN E T (1980)). proposed The H-B strength criterion in 1980. The equations are expressed as follows:       0.51 3 3= + ( 1) b c c m (1) Where σ1 is the maximum principal stresses, σ3 is the minimum principal stresses, σc is the uniaxial compressive strength of the intact rock, mb is the H–B constant of the intact rock. DOI: 10.3303/CET1546120 Please cite this article as: Huang X.H., Wang C.M., Kuang S.H., Wang T.Z., 2015, Slice method for rock slope stability analysis based on the hoek–brown criterion, Chemical Engineering Transactions, 46, 715-720 DOI:10.3303/CET1546120 715 E. Hoek and E. T. Brown(HOEK E (1992); HOEK E et al. (1998)) improved the H–B strength criterion in 1992. The equations are expressed as follows:      1 3 3= + ( ) ab c c m s (2) Where mb, s and a are the H–B parameters that depend on the degree of fracturing of the rock mass and can be estimated from the Geological Strength Index (GSI) (HOEK E et al. (2002)), given by    100 exp( ) 28 14 b i GSI m m D (3)    100 exp( ) 9 3 GSI s D (4)      20 15 3 1 1 ( ) 2 6 GSI a e e (5) Derivate of the formula (2), the new equations are expressed as follows:       131 3 1 ( )ab b ci d am m s d (6) It should be noted that the expressions (7) and (8) are valid for any non-linear failure envelope. In the case of the generalized Hoek-Brown failure criterion (2), after replacing the functions σ1 and dσ1/dσ3 in equations (7) and (8), we get             3 3 ci 13ci ci ci ( ) 2 ( ) a b b n a b b am m s am m s (9)             （ ） 13 ci3 13ci ci ci 1 ( ) +s 2 ( ) a b b ab a b b am m s m am m s (10) In order to calculate shear stress τ, for the given values of the input parameters mb, s, a, σci and σn, Eq. (9) is solved iteratively to calculate the σ3 value. Having obtained σ3, Eq. (10) can be used to calculate shear stress τ. τ/σci is a function of σn/σci, mb, GSI and D. Therefore, τ/σci can be expressed as follows:     4 ci ci ( , , , )n if m GSI D (11) The factor of safety (FS) will be defined as the ratio of the moment Ms=∑WiRsinαi, associated with forces that tend to stabilize the slope, and the moment Mw=∑TiR, associated with forces that tend to destabilize the slope.      s sin R i s i i M T R F M W R (12) Where Ti and Wi are the forces acting at an arbitrary slice (i), and R and R sinαi are the corresponding distances between the forces and the center of the failure surface —see Fig.1 and Fig.2. For the arbitrary slice (i) represented in Fig.1 and Fig.2, the forces Ti and Wi can be expressed in terms of the shear stress τ(i) and normal stresses σ(i) n acting on the base of the slice (i, on the failure surface) as follows, o A B R θ -2 -1 0 1 2 3 4 5 Fig.1 Slip surface of rock slope Fig.2 Stresses on a given slice bi hi Wi li αi Ti Ni 716  *i i iT l and    ( )cos ii i i n iN W l The factor of safety (FS) can be equally written as               ( ) ( ) s ( ) ( ) cos sin sin cos i i i i i i n i n i i i l F l (13) Replacing equations (11) into equation (13), the factor of safety (FS) can be equally written as          ( ) 4 ci s ( ) ( , , , )cos sin i n i i i n i ci f m GSI D F (14) The equations (14) show that the factor of safety (FS) is a function associated with σ (i) n /σci, αi, mi, GSI and D. At the same time, σ(i) n is associated with γ, hi and αi as follows,      ( ) ci ci/ cos / i n i i ih Sr (15) Thus, replacing equations (15) into equation (14), the factor of safety (FS) can be equally written as         4 s 5 ( cos , , , )cos ( , , , , ) sin i i i i i i i i i f Sr m GSI D F f Sr GSI D m SR (16) Figure 3: Geological strength index (GSI) in Hoek-Brown criterion 3. Example and comparison 3.1 The value of rock parameter: The equations (16) show that the values of GSI and mi are the key elements that must be considered when analysis slope stability with the H-B criterion. It is clear that the rock mass strength parameters are sensitive to the GSI value by E. Hoek and E. T. Brown. The lack of parameters to describe surface conditions of the discontinuities and the rock mass structure prevents to obtain a more precise value of GSI. Sonmez introduced rock structural hierarchy and rock surface condition rating to achieve quantitative GSI system (SONMEZ H, ULUSAY R (1999)). Su Yong-hua introduced rock mass block index and weathering index to Rock Structure Blocky: very well interlocked undisturbed rock mass consisting of cubical blocks formed by three orthogonal discontinuity sets Very Blocky: interlocked, partially disturbed rock mass with multifaceted angular blocks formed by four or more Blocky/disturbed: folded and or faulted with angular blocks formed by many intersecting discontinuity sets Disintegrated: poorly interlocked, heavily broken rock mass with a mixture of angular and rounded rock F ra ct a l d im e n si o n D r 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 026 5 4 378911 10121314151617 1 3.00 2.95 2.90 2.85 2.80 2.75 2.70 2.65 2.60 2.55 2.50 2.40 2.30 2.20 2.10 Surfaces Conditions Very good: very rough and fresh unweathered surfaces Good: rough, maybe slightly weathered or iron stained Fait: smooth and moderately weathered and altered surfaces Poor: Slickensided or highly weathered surfaces or compact coatings with fillings or Very poor: Slickensided or very weathered surfaces with soft clay 717 depict characteristics of rock mass, realizing the quantification of GSI (Su Yong-hua et al. (2009)). In the aspects of quantitative GSI system, Cai and Kaiser et al proposed a method based on block size and joint condition factor JC, while Hu Sheng-ming developed an approach relying on joint count Jv and joint condition factor JC (Hu Sheng-ming, Hu Xiu-wen (2011)). These GSI quantitative methods have enriched the theory of GSI concerning rock classification and improved the precision of GSI value. However, the abovementioned methods have serious limitations, as they cannot widen the application of H–B strength criterion in the jointed rock mass with apparent anisotropy, and cannot reflect the influence of rock mass discontinuity on the rock mass strength. Meanwhile, GSI cannot present three-dimensional characteristics of jointed rock mass, and is not capable of reflecting the spatial distribution of rock mass discontinuity, which affects the estimation of GSI. For these reasons, the authors suggest two terms namely, ‘fractal dimension, Dr’ based on distribution of rock masses discontinuities and ‘surface condition rating, SCR’, estimated from the input parameters. It is a more precise GSI value from intersection point of fractal dimension and SCR ratings when the modified GSI chart is used in Fig.3. 3.2 Values of GSI impact on the safety factor Fs : 3.2.1 Verification model According to the model index, the distance from slope toe to left side is 1.5 times long than the height of slope, the distance from slope top to right side is 2.5 times long than slope height, the slope angle is 45°, slope height is 40 meter in Fig 4. The slope consisting of highly fractured rock masses with the following input parameters. Table 1:The values of Fs under different GSI and mi GSI 10 30 50 70 100 Fs 1.162 1.435 2.130 2.951 9.091 Table 2: The values of Fs under different mi mi 5 10 15 20 25 30 Fs 1.237 1.718 2.103 2.337 2.619 2.817 The equation (16) shows that the values of σci, mi and GSI are necessary factor for the safety factor Fs. Table 1 shows that the values of the safety factor Fs change as the values of GSI changing with the following input parameters: D = 0, σci = 10MPa, mi = 10. It shows that the variation of the safety factor Fs increases gradually 100m 60m 200m 8 0 m 4 0 m Fig.4 Model size of rock slope (unit: m) Fig.7 Simplified CAD drawing of Bridge 20m 165.60m 20m 36.47° Fig.5 Relationships between GSI and Fs 0 20 40 60 80 100 0.000 2.000 4.000 6.000 8.000 10.000 Fig.6 Relationships between mi and Fs 3.000 2.500 2.000 1.500 1.000 0 5 10 15 20 25 30 718 in a small range when GSI<60 and increases rapidly when GSI>60 in Fig.5. Actually, the better interlocked undisturbed rock masses are, the higher values of GSI are. At the same time, the less development structure and surfaces conditions of rock are, the higher the values of safety factor Fs are. Table 2 shows that the values of the safety factor Fs change as the values of mi changing with the following input parameters: D = 0, σci = 10MPa, GSI = 30. It shows that the variation of the safety factor Fs increases gradually as mi increased in Fig.6. The values of mi have the great influence to the stability of rock in a certain range, but beyond this range, the influence tends to stagnation. 4. Engineering example A slope from Wujiang river build a bridge located at Guiyang province. The necessary data were collected by authors from this bridge. The slope with the height of 165.60m and the angle of 36.47° has a sequence consisting of compact limestone. There are two penetrating joints of the slope which have dip angle with the following parameters: J1: 0°–20°∠80–85°; J2:70°–90°∠80–85°. Two joints are "X" shape and cut rock with the angle of 70°. And second fault activity had occurred in the region before. Many joints and cracks were developed nearby the "X" shape and formed many discontinuities for rock masses. It shows that the rock slope of the bridge draw by CAD in Fig.7. Table 3: Calculation parameters Slice number Calculate parameter hi li αi Sri σ3/σci τ/σci 1 7.5912 10.3170 -4.8788 0.0068 0.0040 0.0175 2 19.0600 7.3835 16.4730 0.0163 0.0103 0.0301 3 26.8180 7.3835 16.4730 0.0230 0.0149 0.0379 4 35.7900 12.4910 27.4900 0.0283 0.0187 0.0403 5 43.5810 7.9583 32.8890 0.0327 0.0218 0.0419 6 47.7710 5.9337 32.8890 0.0358 0.0241 0.0446 7 54.4550 15.6050 37.8560 0.0384 0.0259 0.0438 8 63.0130 10.5740 42.0400 0.0418 0.0284 0.0436 9 65.5760 8.2310 44.6210 0.0417 0.0284 0.0418 10 63.7750 8.6263 46.3820 0.0393 0.0266 0.0389 11 61.7760 8.6263 46.3820 0.0380 0.0257 0.0381 12 58.4200 15.7960 49.5590 0.0338 0.0227 0.0332 13 50.8770 17.5610 52.0600 0.0279 0.0184 0.0277 14 41.8500 12.3100 54.2570 0.0218 0.0141 0.0223 15 35.1350 8.9323 55.6460 0.0177 0.0113 0.0188 16 29.1080 9.4411 57.2890 0.0140 0.0088 0.0154 17 22.7910 9.0233 59.7570 0.0102 0.0062 0.0116 18 15.2390 10.5440 59.7560 0.0069 0.0041 0.0090 19 6.5650 10.0700 61.8360 0.0028 0.0015 0.0048 20 1.1385 2.7461 61.8370 0.0005 0.0001 0.0020 The value of GSI is estimate to 30 by table 1 and the value of mi is 10. The value of D is equal 0 because of no excavation. The value of GSI, mi and D were used to calculate the H–B parameters mb, s and a. The results were mb =0.8208, s = 0.0004 and a = 0.5223. The rock slope consisting of highly fractured rock masses with the following input parameters: σci =28MPa, γ = 27 KN/m3, φ = 32°and c = 330 Kpa. The model with the height of 230 m and the width of 310 m and the slope height of 170 m was used to analysis slope stability with the help of Slope/W module under SIGMA/W in Geostudio 2004. Number of slices is 20, water pressure and disturbance is ignored. The most dangerous slip surface from the rock slope model is calculated by the Geostudio2004. Table 3 shows parameters for slices in the model which used by the Geostudio 2004. The value of Sri is calculated by the value of γ, hi and σci. The value of σ3/σci is calculated by 719 equations (9) for each slice. The value of τ/σci is calculated by equations (10) for each slice. Finally, the value of the safety factor Fs was calculated by equations (13) and equations (14):          s ( / )cos 0.4205 1.3833 sin 0.3072 ci i i i F SR At the same time, the values of the safety factor Fs were 1.408, 1.396, 1.430 and 1.436 by the method of slice, Jan-bu, Bishop and M-P. Compared with all results, the value calculated by the new method is smaller 5. Conclusions The H-B strength criterion is widely used in underground and slope engineering. But it is seldom used to research slope stability because of the H–B parameters are difficult to determine and the applicability is poor. The exact value of the GSI is very difficultly obtained by E. Hoek and E. T. Brown. It is a more precise GSI value from intersection point of fractal dimension based on distribution of rock masses discontinuities and SCR ratings when the modified GSI chart is used in Fig.3. It is found that the factor of safety (FS) can be equally written as equations (16). The fractal dimension of rock discontinuities is a function which combined with rock strength, the formation of environmental and engineering geological characteristics. It is feasible that the three-dimensional fractal dimension of rock discontinuities is used to describe rock structures. The result of the safety factor Fs from the H-B criterion compared with the result from the method of M-C, Jan- bu, Bishop and M-P shows that the error is existed in different methods. The value of the safety factor Fs is 1.3833and is smaller than the results of the other methods is 1.408, 1.396, 1.430 and 1.436. References Cai M., Kaisera P.K., Unob H. (2004). Estimation of rock mass deformation modulus and strength of jointed hard rock masses using the GSI system [J]. International Journal of Rock Mechanics & Mining Sciences, 41: 3－19. doi:10.1016/S1365-1609(03)00025-X. Hoek E., Brown E.T. (1980). Empirical strength criterion for rock masses [J]. J. Geotech. Engng. 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