HOEKBROWN FAILURE CRITERION   EDITION Evert Hoek Consulting Engineer V ncouver C nada Carlos Carranz aTorres Itasca Consulting Group Inc
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HOEKBROWN FAILURE CRITERION EDITION Evert Hoek Consulting Engineer V ncouver C nada Carlos Carranz aTorres Itasca Consulting Group Inc

Minneapolis USA Brent Corkum Rocscience Inc Toronto Canada ABST RACT The HoekBrown failure criterio n for rock m sse s is wid ly accepted a nd has been applied in a large num ber of projects around the world While in gene ral it has been found to b

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HOEKBROWN FAILURE CRITERION EDITION Evert Hoek Consulting Engineer V ncouver C nada Carlos Carranz aTorres Itasca Consulting Group Inc

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HOEK-BROWN FAILURE CRITERION 2002 EDITION Evert Hoek Consulting Engineer, V ncouver, C nada Carlos Carranz a-Torres Itasca Consulting Group Inc., Minneapolis, USA Brent Corkum Rocscience Inc., Toronto, Canada ABST RACT : The Hoek-Brown failure criterio n for rock m sse s is wid ly accepted a nd has been applied in a large num ber of projects around the world. While, in gene ral, it has been found to be satisfactory, there are som uncertainties and inaccuracies that h ve made th e criterion inconv enient to ap ply and to incorpora e into num erical m odels and lim it eq uilib

rium program s. In particula , the dif culty of f nding an acc ptab le equivalent friction angle and cohe sive strength for a given rock ss has been a problem since the publication of the criterion in 1980. This paper resolves all these issues and sets out a recomm ended sequence of calculations for applyi ng the criterion. An associated W ndows program called R ocLab has been developed to provide a convenient m eans of solvi ng and plotting the equations presented in this paper. Hoek and Brown [1, 2] introduced their failure criterion in an attem t to provide input data for the analyses

required for the design of underground excavations in hard rock . The criterion was derived om the results of resea ch into the brittle f ilu of intac rock by Hoek [ and on model studies of jointed rock m ss behavi our by Brown [4]. The criterion started from t e properties of intact rock and then introdu ced factors to reduce these properties on the basis of the characteristics of joints in a rock m ss. The authors sought to link the em pirical criterion to geological observations by eans of one of the available rock m ss classification schem s and, for this purpose, they chose the Rock Mass

Rating proposed by Bieniawski [ ]. Because of the lack o suitable alternatives, the criterion was soon adopted by the rock m echanics community and its use quickly spread beyond the original lim its used in deriving the strength reduction relationships. C onsequently, it became necessary to re-exam ne these relationships and to introdu ce new elem ents f om time to tim e to account for the wide range of practical problems to which the criterion was being applied. Typical of these enhan cem ents were th e in tro duction of the idea of undisturbed and disturbed rock m es Hoek and Brown [6],

a nd the introduction of a modified criterion to for ce the ro ck m ss tensile strength to zero for very poor quality rock m sses (Hoek, W od and Shah, [7]). One of the early difficu lties aros e becaus m ny geotechnical problem s, part icularly slope stability issues, are more convenientl y dealt with in term of shear and n rm al stress es rath er th an the principal stress relationships of the original Hoek-Brown (1) where and are the m jor and m nor effective princ pal s sses a f ilu re is the uniax ial com ressive streng th of the intac rock m terial and and are m terial constan s, where =

1 for intac rock. An exact relationship between equation 1 and the norm l and shear stresses at failure was derived by J. W Bray (reported by Ho ek [8]) and later by Ucar [9] and Lon [10]. Hoek [12] discussed the derivation of equivalent friction ang es and coh si ve strengths for various practical situations. Thes e derivations were based Lo nde s e ons were l r f t cont ai n er ro rs alth ugh th e co ep ts in tro ced by Londe were extrem ely tan in th e app licatio n o th e Ho ek-Brown criteri on to tun elling em s (Carra nza To rre s a Fa irh rst, [ 11] )
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upon tangents to

the Mohr envelope derived by Bray. Hoek [13] suggested that the cohesive streng th determ ined by f itting a tangent to the curvilinear Mohr envelope is an upper bound value and m y give optim istic resu lts in stability calcu lations. Consequently, an average v lue, determ ined by f itting a linear Mohr-Coulom b relationship by least squares m thods, m y be more appropriate. In this pape r Hoek also introdu ced the concept of the Generalized Hoek-Brown criterio in which the shape of the principal stress plot or the Mohr envelope could be adjusted by m eans of a variab le coefficient in place

of the square root term in equation 1. 100 exp (4) (5) 20 15 is a factor which depends upon the degree of disturbance to which the rock m ss has been subjected by blast dam ge a nd stress relaxation. It varies from 0 for undisturbe d in situ rock m sses to 1 for very disturbed rock m sses. Guidelines for the selection of are discussed in a later section. The uniaxial com ressive strength is obtained by setting in equation 2, giving: (6) Hoek and Brown [14] a tte ted to consolid ate all the previou enhancem ents into a com rehensive presentation of the failure criterion and they gave a num er

of worked exam pl es to illustrate its practical application. and, the tensile strength is: (7) Equation 7 is obtain d by settin g in equation 2. This represents a condition of biaxial tension. Hoek [8] showed that, for brittle m terials, the uniaxial tensile str ngth is equ l to the biaxial tensile strength. In addition to the changes in the equations, it was also recogn ised th at the Rock Mass Ratin of Bieniawski was no longer ad equ te as a v hicle for rela ting th e f ilure crite rion to geologica l observations in the field, pa rticularly for very we ak rock m sses. This resulted in the

introduction of the Geological Strength Inde x (GSI) by Hoek, Wood and Shah [7], Hoek [13] and Hoek, Kaiser and Bawden [15]. This index was subsequently extended for weak rock m sses in a series of papers by Hoek, Marinos and Benissi [ 16], Hoek and Marinos [17, 18] and Marinos and Hoek [19]. Note tha the switch at GSI = 25 for the coefficients and (Hoek and Brown, [14]) has been elim inated in equations 4 and 5 which give sm ooth continuous trans itions for th e entire rang e of GSI values. The num rical values of and , given by these equ tions, are v ry close to those given by the previous

equations and it is no t nec ssar f r readers to revis it and m ke corrections to old calcu lations. The Geological Strength Inde x will not be discussed in the f llo wing text, which will co ncentr ate on the sequence of calculations now proposed for the application of the Generalized Hoek Br own criter ion to jointed rock m sses. Nor al and shear s res es are related to prin cipal stresses by the equations published by Balm er [20]. (8) This is expressed as (9) (2) where (10) where is a reduced value of the material constant and is given by 14 28 100 exp (3) The rock m ss m odulus of defor

ation is given by: and are constan s f r the rock mass given by the following relationships: 40 10 (( 10 100 (11a)
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The equivalent plot, in term s of the m jor and minor principal stresses, is defined by: Equation 11a applies for 100 MPa. For 100 MPa, use equation 11b. sin sin sin cos (15) 40 10 (( 10 (11b) Note that th e origina equation proposed by Hoek and Brown [14] has been m odified, by the inclusion of the factor , to allow for the effects of bl ast dam ge and stress relaxation. Since m st geotechn cal sof ware is still written in term s of the Mohr-Coulom failure

criterion, it is necessary to determ ine equiva lent angles of friction and cohesiv streng ths for each rock m ss and stres range. This is done by fitting an average linear rela tionsh to the cu rv e generated by solving equation 2 for a range of nor principal stress values defined by max , as illus ted in Figure 1. The fitting pro cess involv s balancing the areas above and below the Mohr-Coulom b plot. This results in the f llo wing equations f r the angle of friction and cohesive strength : )( sin (12) )( )( (13) Figure 1: R lationships between major and m nor principal stresses for H

ek-Brown and equivalent Mohr-Coulomb criteria. The uniaxia l com ressive streng th o the rock mass is given by equation 6. Failu re initiates at the boundary of an excavation when is exceeded by the stress induced on that boundary. The failure propagates f om this initia tion po int into a b ax ia l stres field and it even tually stab ilizes when the local strength, defined by equation 2, is higher than the indu ced stresses and . Most num rical models ca n f llow this p oces s of f acture propagation and this level of detailed analysis is very im portant when cons ider ing the stability of

excavations in rock an d when designing support system s. where max Note that the value of max , the upper lim it of confining stress over which the relationship between the Hoek-Brown and the Mohr-Coulom b criteria is co nsidered, has to be determ ined for each individu al case. Guid eline f r selec ting th ese values for slopes as well as sh allow and deep tunnels are presented later. The Mohr-Coulom b shear strength , for a given norm l stress , is found by subs titution of these values of and in to th e e quation tan However, there are tim s when it is usef ul to consider the overall

behaviour of a rock m ss rather than the detailed failure propagation process described above. For exam ple, when considering the strength of a pillar, it is usef ul to have an estim ate of the overall streng th of the pillar rather
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than a detailed knowledge of the extent of fracture propagation in the pilla r. This le ad s to the con cept of a global rock m ss strength and Hoek and Brown [14] proposed that this could be estim ated from the Mohr-Coulom b relationship: sin cos (16) with and determ ined for the stress range giving )( )) (17) The issue o determ ining the app

riate value of for use in equations 12 and 13 depends upon the specific application. Two cases will be investigated: max 1. Tunnels where th e v lue of is that which gives equivalent characteristic curves for the two failure criteria for deep tunnels or equivalent subsidence profiles for shallow tunnels. max 2. Slopes here the calcula ted factor of safety and the sha e and loc tion of th e f ilu re surface have to be equiv lent. For the case of deep tunnels, closed for solutions for both the Generalized Hoek-Brown and the Mohr-Coulomb criteria have been used to generate hundreds of solutions

and to find the value of that g ves eq uivalen characteristic curves. max For shallow tunnels, where the depth below surface is less than 3 tunnel diam eters, com arative num rical studies of the extent of failure and the gnitude of surface subsidence g ve an id en tical rela tionsh to tha o tained f deep tun els, provided that caving to s rface is avo ded. The results of the studies for deep tunnels are plotted in Figure 2 and the f itted eq uation f r both cases is: 94 max 47 (18) where is the rock m ss strength, defined by equation 17, is the unit weight of the rock m ss and is the depth of

the tunnel below surface. In cases where the ho rizon al stress is higher than the vertical stress, the horizont al stress value should be used in place of . Figure 2: Relationship for the calculation of Vc 3m for equivalent Mohr-Coulom b a nd Hoek-Brown param ters for tunnels. Equation 18 applies to a ll underground excavations, which are surrounded by a z one of failure that does not extend to surface. F r stud ies of problem s such as block caving in m nes it is recomm ended that no attem t sho ld be m de to re late the Hoek-Br wn and Mohr-Coulom b param ters and that the determ inatio n of

m teria properties and subsequent analysis should be base d on only one of these criteria. Sim ilar studies for slopes, using Bishops circular failure analy is for a wid range of slope geom etries and rock m ss properties, gave: 91 max 72 (19) where is the height of the slope. Experience in the design of slopes in very large open pit mines has shown that the Hoek-Brown criterion for undisturbed in situ rock m sses ( = 0) results in rock m ss properties that are too optim istic [ 21, 22]. The effects of heavy blast
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dam ge as well as str relief due to rem oval of the overburden

result in disturba nce of the rock m ss. It is consid er ed that th e distu rbed rock m ss properties [6], = 1 in equations 3 and 4, are more appropriate for these rock m sses. Lorig and Varona [23] show ed that factors su ch as the la tera l c onf inem ent pr oduced by different radii of curvature of slopes (in plan) as com ared with their height also have an influence on the degree of disturbance. Sonm ez and Ulusay [24] back-analysed five slope failures in open pit coal m nes in Turkey and attem ted to assign d stu bance facto s to each ro ck ss based upon their assessm ent of the rock

mass properties predicted by th e Hoek-Brown criterion. Unfortunately, one of the slope failures appears to be stru cturally c ontrolled while anot r cons ists o a transpo ted waste pile. The authors consider tha the Hoek-Brown crite rion is not applic able to the two cases. Cheng and Liu [25] report the results of very careful back analysis of deform ation m urem ents, from extensom eters placed before the commencem ent of excavation, in the Mingtan power cavern in Taiwan. It was found that a zone of blast dam ge extended for a distance of approxim tely 2 m around all large excavations. The

back-calculated streng th and defor ation properties of the dam ged rock mass give an equivalent disturbance factor = 0.7. From these references it is clear that a large number of factors can influence the degree of disturbance in the rock m ss surrounding an excavation and that it y never be possible to quantify these factors precisely. H wever, based on their experience and on an ana is of a ll th e details co ntained in these papers, the authors have attem ted to draw up a set of guidelines for estim ting the factor and these are summ arised in Table 1. The influence of this disturbance

factor can be large. This is illu str te d by a typical exam ple in which = 50 MPa, = 10 and GSI = 45. For an undisturbed in situ rock m ss surrounding a tunnel at a depth of 100 m with a disturbance factor = 0, the equivalent friction angle is 47.16 while the cohesive strength is 0.58 MPa. A rock m ss with the sam basic param ters but in highly disturbed slope of 100 m he ight, with a disturbance factor of = 1, has an equivalent friction angle of 27.61 and a cohesive strength of 0.35 MPa. Note tha th ese a e guid lines only and the reader would be w ll advised to apply the values given with

caution. However, they can be used to provide a realistic starting point for any design and, if the observed or m easured perform nce of the excavation turns out to be better th predic ted, the disturb nce factors can b adjus ed d wnwards. A num er of uncertain ties and practical problems in using the Hoek-Brown failure criterion have been addressed in this pap r. W erever poss ble, an attem t has been m de to provide a rigorous and unam iguous m thod for calc ulating or estim ating the input param ters required for the analysis. These thods ha ve all been im ple ented in a W ndows program

called RocLab that can be downloaded (free) from This progra m include s ta bles and charts f estim ating the uniaxia l co mpressive s reng th of the inta ct r ck elem ents ( ), the m terial cons tant and the Geological Strength Index ( ). The authors wish to ack now ledge the contributions of Professor E.T. Brown in reviewing a draft of this paper and in participating in the developm ent of the Hoek-Brown criterion for the past 25 years.
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ating disturbance factor Appearance of rock ss Descri ption of rock ss Suggested value of Excellent quality controlled blasting or

excavation by Tunnel Boring Machine results in m ni l disturbance to the confined rock ss surrounding a tunnel. = 0 Mechanical or hand excavat ion in poor quality rock sses (no blasting) results in m nim l disturbance to ss. Where squeezing problem s result in significan floor heave, disturbance can be severe unless a temporary invert, as sh own in the p hotograph, is placed. = 0 = 0.5 No invert Very poor quality blasting in a hard rock tunnel results in severe local dam ge, extending 2 or 3 m i the surrounding rock m ss. = 0.8 Sm all scale blasting in civ il engineering slopes results in m

odest rock m ss dam ge, particularly if controlled blasting is used as shown on the left hand side of the photograph. However, stress relief results in som disturbance. = 0.7 Good blasting = 1.0 Poor blasting Very large open pit m ne slopes suffer significant disturbance due to heavy pr oduction blasting and also due to stress relief from overburden rem oval. In som softer ro cks ex cavation can be carried out by ripping and dozing and the degree of da mage to the slopes is less. = 1.0 Production blasting = 0.7 Mechanical excavation
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1. Hoek, E. and Brown, E.T. 1980. Empirical

strength criterion for rock masses. J. Geotech. Engng Div., ASCE 106 (GT9), 1013-1035. 2. Hoek, E. and Brow n, E.T. 1980. , London, Instn Min. Metall. 3. Hoek, E. 1968 . Brittle failure of rock. . (eds K.G. Stagg and O.C. Zienkiewicz), 99-124. London: Wiley 4. Brown, E.T. 1970. Strength of models of rock with intermittent joints. 96 , SM6, 1935-1949. 5. Bieniawski Z.T. 1976. Rock mass classification in rock engineering. In ., (ed. Z.T. Bieniawski) , 97-106. Cape Town, Balkema. 6. Hoek, E. and Brown, E.T. 1988. The Hoek-Brown failure criterion - a 1988 update. (ed. J.C. Curran), 31-38. Toronto,

Dept. Civil Engineering, University of Toronto. 7. Hoek, E., Wood D. and Shah S. 1992. A modified Hoek- Brown criterion for jointed rock masses. , (ed. J.A. Hudson), 209-214. London, Brit. Geotech. Soc. 8. Hoek, E. 1983. Strength of jointed rock masses, 23rd. Rankine Lecture. 33 (3), 187-223. 9. Ucar, R. (1986) Determination of shear failure envelope in rock masses. 112 , (3), 303-315. 10. Londe, P. 1988. Discussion on the determination of the shear stress failure in rock masses. ASCE J Geotech Eng Div, 14 , (3), 374-6. 11. Carranza-Torres, C., and Fairhurst, C. 1999. General formulation of

the elasto-plas tic response of openings in rock using the Hoek-Brown failure criterion. 36 (6), 777-809. 12. Hoek, E. 1990. Estimating Mohr-Coulomb friction and cohesion values from the Hoek-Brown failure criterion. . 12 (3), 227-229. 13. Hoek, E. 1994. Strength of rock and rock masses, 2 (2), 4-16. 14. Hoek, E. and Brown, E.T. 1997. Practical estimates of rock mass strength. . 34 (8), 1165-1186. 15. Hoek, E., Kaiser P.K. and Bawden W.F. 1995. . Rotterdam, Balkema. 16. Hoek, E., Marinos, P. and Be nissi, M. 1998. Applicability of the Geological Strength In dex (GSI) classification for very

weak and sheared rock masses. The case of the Athens Schist Formation. 57 (2), 151-160. 17. Marinos, P and Hoek, E. 2000. GSI A geologically friendly tool for rock mass strength estimation. 18. Hoek, E. and Marinos, P. 2000. Predicting Tunnel Squeezing. Part 1 November 2000, Part 2 December, 2000 19. Marinos. P, and Hoek, E. 2001. Estimating the geotechnical properties of heterogeneous rock masses such as flysch. Accepted for publication in the 20. Balmer, G. 1952. A general analytical solution for Mohr's envelope. 52, 1260-1271. 21. Sjberg, J., Sharp, J.C., and Malorey, D.J. 2001

Slope stability at Aznalcllar. In (eds. W.A. Hustrulid, M.J. McCarter and D.J.A. Van Zyl). Littleton: Society for Mining, Metallurgy and Exploration, Inc., 183-202. 22. Pierce, M., Brandshaug, T., and Ward, M. 2001 Slope stability assessment at the Main Cresson Mine. In (eds. W.A. Hustrulid, M.J. McCarter and D.J.A. Van Zyl). Littleton: Society for Mining, Metallurgy and Exploration, Inc., 239-250. 23. Lorig, L., and Varona, P. 20 01 Practical slope-stability analysis using finite-difference codes. In (eds. W.A. Hustrulid, M.J. McCarter and D.J.A. Van Zyl). Littleton: Society for Mining,

Metallurgy and Exploration, Inc., 115-124. 24. Sonmez, H., and Ulusay, R. 1999. Modifications to the geological strength in dex (GSI) and thei r applicability to the stability of slopes. 36 (6), 743-760. 25. Cheng, Y., and Liu, S. 1990. Power caverns of the Mingtan Pumped Storage Project, Taiwan. In (ed. J.A. Hudson), Oxford: Pergamon, , 111-132.