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- 1. HOEK-BROWN FAILURE CRITERION – 2002 EDITIONEvert HoekConsulting Engineer, Vancouver, CanadaCarlos Carranza-TorresItasca Consulting Group Inc., Minneapolis, USABrent CorkumRocscience Inc., Toronto, CanadaABSTRACT: The Hoek-Brown failure criterion for rock masses is widely accepted and has been applied ina large number of projects around the world. While, in general, it has been found to be satisfactory, there aresome uncertainties and inaccuracies that have made the criterion inconvenient to apply and to incorporateinto numerical models and limit equilibrium programs. In particular, the difficulty of finding an acceptableequivalent friction angle and cohesive strength for a given rock mass has been a problem since thepublication of the criterion in 1980. This paper resolves all these issues and sets out a recommendedsequence of calculations for applying the criterion. An associated Windows program called “RocLab” hasbeen developed to provide a convenient means of solving and plotting the equations presented in this paper.1. INTRODUCTION strength to zero for very poor quality rock masses (Hoek, Wood and Shah, [7]).Hoek and Brown [1, 2] introduced their failurecriterion in an attempt to provide input data for the One of the early difficulties arose because manyanalyses required for the design of underground geotechnical problems, particularly slope stabilityexcavations in hard rock. The criterion was derived issues, are more conveniently dealt with in terms offrom the results of research into the brittle failure of shear and normal stresses rather than the principalintact rock by Hoek [3] and on model studies of stress relationships of the original Hoek-Brownjointed rock mass behaviour by Brown [4]. The criterion, defined by the equation:criterion started from the properties of intact rock 0.5and then introduced factors to reduce these σ σ1 =σ3 + σ ci m 3 + s (1)properties on the basis of the characteristics of σ ci joints in a rock mass. The authors sought to link theempirical criterion to geological observations by where σ 1 and σ 3 are the major and minor effective means of one of the available rock mass principal stresses at failureclassification schemes and, for this purpose, they σ ci is the uniaxial compressive strength of the intactchose the Rock Mass Rating proposed by rock material andBieniawski [5]. m and s are material constants, where s = 1 for intact rock.Because of the lack of suitable alternatives, thecriterion was soon adopted by the rock mechanics An exact relationship between equation 1 and thecommunity and its use quickly spread beyond the normal and shear stresses at failure was derived byoriginal limits used in deriving the strength J. W. Bray (reported by Hoek [8]) and later by Ucarreduction relationships. Consequently, it became [9] and Londe1 [10].necessary to re-examine these relationships and tointroduce new elements from time to time to Hoek [12] discussed the derivation of equivalentaccount for the wide range of practical problems to friction angles and cohesive strengths for variouswhich the criterion was being applied. Typical of practical situations. These derivations were basedthese enhancements were the introduction of theidea of “undisturbed” and “disturbed” rock masses 1 Londe’s equations were later found to contain errorsHoek and Brown [6], and the introduction of a although the concepts introduced by Londe were extremelymodified criterion to force the rock mass tensile important in the application of the Hoek-Brown criterion to tunnelling problems (Carranza-Torres and Fairhurst, [11])
- 2. upon tangents to the Mohr envelope derived by GSI − 100 s = exp (4)Bray. Hoek [13] suggested that the cohesive 9 − 3D strength determined by fitting a tangent to thecurvilinear Mohr envelope is an upper bound value a= + e 2 6 ( 1 1 −GSI /15 −20/ 3 −e ) (5)and may give optimistic results in stabilitycalculations. Consequently, an average value, D is a factor which depends upon the degree ofdetermined by fitting a linear Mohr-Coulomb disturbance to which the rock mass has beenrelationship by least squares methods, may be more subjected by blast damage and stress relaxation. Itappropriate. In this paper Hoek also introduced the varies from 0 for undisturbed in situ rock masses toconcept of the Generalized Hoek-Brown criterion in 1 for very disturbed rock masses. Guidelines for thewhich the shape of the principal stress plot or the selection of D are discussed in a later section.Mohr envelope could be adjusted by means of avariable coefficient a in place of the square root The uniaxial compressive strength is obtained byterm in equation 1. setting σ 3 = 0 in equation 2, giving: Hoek and Brown [14] attempted to consolidate all σ c = σ ci .s a (6)the previous enhancements into a comprehensivepresentation of the failure criterion and they gave a and, the tensile strength is: sσ cinumber of worked examples to illustrate its σt = − (7) mbpractical application. Equation 7 is obtained by setting σ 1 = σ 3 = σ t in In addition to the changes in the equations, it was equation 2. This represents a condition of biaxialalso recognised that the Rock Mass Rating of tension. Hoek [8] showed that, for brittle materials,Bieniawski was no longer adequate as a vehicle for the uniaxial tensile strength is equal to the biaxialrelating the failure criterion to geological tensile strength.observations in the field, particularly for very weakrock masses. This resulted in the introduction of the Note that the “switch” at GSI = 25 for theGeological Strength Index (GSI) by Hoek, Wood coefficients s and a (Hoek and Brown, [14]) hasand Shah [7], Hoek [13] and Hoek, Kaiser and been eliminated in equations 4 and 5 which giveBawden [15]. This index was subsequently smooth continuous transitions for the entire range ofextended for weak rock masses in a series of papers GSI values. The numerical values of a and s, givenby Hoek, Marinos and Benissi [16], Hoek and by these equations, are very close to those given byMarinos [17, 18] and Marinos and Hoek [19]. the previous equations and it is not necessary for readers to revisit and make corrections to oldThe Geological Strength Index will not be discussed calculations.in the following text, which will concentrate on thesequence of calculations now proposed for the Normal and shear stresses are related to principalapplication of the Generalized Hoek Brown stresses by the equations published by Balmer [20].criterion to jointed rock masses.2. GENERALIZED HOEK-BROWN CRITERION σ 1 + σ 3 σ 1 − σ 3 dσ 1 dσ 3 − 1 σn = − ⋅ (8) 2 2 dσ 1 dσ 3 + 1 This is expressed as ( ) d σd σ d σd σ + 1 a σ τ = σ1 − σ 1 3 (9) σ1 = σ3 + σ ci mb 3 + s (2) 3 σ ci 1 3 wherewhere mb is a reduced value of the material constant ( dσ 1 dσ 3 = 1 + amb mbσ 3 σ ci + s ) a −1 (10)mi and is given by GSI − 100 3. MODULUS OF DEFORMATION mb = mi exp (3) 28 − 14D The rock mass modulus of deformation is given by:s and a are constants for the rock mass given by the D σfollowing relationships: E m (GPa) = 1 − ci ⋅ 10 ((GSI −10) / 40) (11a) 2 100
- 3. The equivalent plot, in terms of the major and minorEquation 11a applies for σ ci ≤ 100 MPa. For σ ci > principal stresses, is defined by:100 MPa, use equation 11b. 2c cos φ 1 + sin φ σ1 = + σ 3 (15) D 1 − sin φ 1 − sin φ E m (GPa) = 1 − ⋅ 10 ((GSI −10) / 40) (11b) 2Note that the original equation proposed by Hoekand Brown [14] has been modified, by the inclusionof the factor D, to allow for the effects of blastdamage and stress relaxation.4. MOHR-COULOMB CRITERIONSince most geotechnical software is still written interms of the Mohr-Coulomb failure criterion, it isnecessary to determine equivalent angles of frictionand cohesive strengths for each rock mass and stressrange. This is done by fitting an average linearrelationship to the curve generated by solvingequation 2 for a range of minor principal stressvalues defined by σ t < σ 3 < σ 3 max , as illustrated inFigure 1. The fitting process involves balancing theareas above and below the Mohr-Coulomb plot.This results in the following equations for the angleof friction φ and cohesive strength c : 6amb ( s + mb σ 3n ) a −1 φ = sin −1 (12) Figure 1: Relationships between major and minor 2(1 + a)(2 + a) + 6amb ( s + mb σ ) a −1 3n principal stresses for Hoek-Brown and equivalent Mohr-Coulomb criteria.c = [ ] σ ci (1 + 2a)s + (1 − a)mbσ 3n (s + mbσ 3n ) a −1 ( (1 + a)(2 + a) 1 + 6amb (s + mbσ 3n ) a −1 ) ((1 + a)(2 + a)) 5. ROCK MASS STRENGTH (13) The uniaxial compressive strength of the rock masswhere σ 3n = σ 3 max σ ci σ c is given by equation 6. Failure initiates at the boundary of an excavation when σ c is exceeded by the stress induced on that boundary. The failureNote that the value of σ 3 max , the upper limit of propagates from this initiation point into a biaxialconfining stress over which the relationship stress field and it eventually stabilizes when thebetween the Hoek-Brown and the Mohr-Coulomb local strength, defined by equation 2, is higher thancriteria is considered, has to be determined for each the induced stresses σ 1 and σ 3 . Most numerical individual case. Guidelines for selecting these models can follow this process of fracturevalues for slopes as well as shallow and deep propagation and this level of detailed analysis istunnels are presented later. very important when considering the stability of excavations in rock and when designing supportThe Mohr-Coulomb shear strength τ , for a given systems.normal stress σ , is found by substitution of thesevalues of c and φ in to the equation: However, there are times when it is useful to consider the overall behaviour of a rock mass rather τ = c + σ tan φ (14) than the detailed failure propagation process described above. For example, when considering the strength of a pillar, it is useful to have an estimate of the overall strength of the pillar rather
- 4. than a detailed knowledge of the extent of fracture and H is the depth of the tunnel below surface. Inpropagation in the pillar. This leads to the concept cases where the horizontal stress is higher than theof a global “rock mass strength” and Hoek and vertical stress, the horizontal stress value should beBrown [14] proposed that this could be estimated used in place of γH .from the Mohr-Coulomb relationship: 2c cos φ σ cm = (16) 1 − sin φ with c and φ determined for the stress rangeσ t < σ 3 < σ ci / 4 giving (mb + 4s − a(mb − 8s ))(mb 4 + s )a −1 σ cm = σ ci ⋅ (17) 2(1 + a)(2 + a)6. DETERMINATION OF σ′3MAXThe issue of determining the appropriate value ofσ 3 max for use in equations 12 and 13 depends upon the specific application. Two cases will beinvestigated: 1. Tunnels − where the value of σ 3 max is that Figure 2: Relationship for the calculation of σ′3max which gives equivalent characteristic curves for equivalent Mohr-Coulomb and Hoek-Brown for the two failure criteria for deep tunnels parameters for tunnels. or equivalent subsidence profiles for shallow tunnels. Equation 18 applies to all underground excavations, which are surrounded by a zone of failure that does 2. Slopes – here the calculated factor of safety not extend to surface. For studies of problems such and the shape and location of the failure as block caving in mines it is recommended that no surface have to be equivalent. attempt should be made to relate the Hoek-BrownFor the case of deep tunnels, closed form solutions and Mohr-Coulomb parameters and that thefor both the Generalized Hoek-Brown and the determination of material properties and subsequentMohr-Coulomb criteria have been used to generate analysis should be based on only one of these criteria.hundreds of solutions and to find the value of σ 3 max that gives equivalent characteristic curves. Similar studies for slopes, using Bishop’s circularFor shallow tunnels, where the depth below surface failure analysis for a wide range of slope geometriesis less than 3 tunnel diameters, comparative and rock mass properties, gave:numerical studies of the extent of failure and themagnitude of surface subsidence gave an identical −0.91 σ 3 max σ relationship to that obtained for deep tunnels, = 0.72 cm (19) σ cm γH provided that caving to surface is avoided. The results of the studies for deep tunnels are where H is the height of the slope.plotted in Figure 2 and the fitted equation for bothcases is: 7. ESTIMATION OF DISTURBANCE FACTOR D −0.94 σ 3 max σ = 0.47 cm (18) Experience in the design of slopes in very large σ cm γH open pit mines has shown that the Hoek-Brown criterion for undisturbed in situ rock masses (D = 0)where σ cm is the rock mass strength, defined by results in rock mass properties that are tooequation 17, γ is the unit weight of the rock mass optimistic [21, 22]. The effects of heavy blast
- 5. damage as well as stress relief due to removal of the φ = 27.61° and a cohesive strength of c = 0.35overburden result in disturbance of the rock mass. It MPa.is considered that the “disturbed” rock massproperties [6], D = 1 in equations 3 and 4, are more Note that these are guidelines only and the readerappropriate for these rock masses. would be well advised to apply the values given with caution. However, they can be used to provideLorig and Varona [23] showed that factors such as a realistic starting point for any design and, if thethe lateral confinement produced by different radii observed or measured performance of theof curvature of slopes (in plan) as compared with excavation turns out to be better than predicted, thetheir height also have an influence on the degree of disturbance factors can be adjusted downwards.disturbance. 8. CONCLUSIONSonmez and Ulusay [24] back-analysed five slopefailures in open pit coal mines in Turkey and A number of uncertainties and practical problems inattempted to assign disturbance factors to each rock using the Hoek-Brown failure criterion have beenmass based upon their assessment of the rock mass addressed in this paper. Wherever possible, anproperties predicted by the Hoek-Brown criterion. attempt has been made to provide a rigorous andUnfortunately, one of the slope failures appears to unambiguous method for calculating or estimatingbe structurally controlled while another consists of a the input parameters required for the analysis. Thesetransported waste pile. The authors consider that the methods have all been implemented in a WindowsHoek-Brown criterion is not applicable to these two program called “RocLab” that can be downloadedcases. (free) from www.rocscience.com. This program includes tables and charts for estimating theCheng and Liu [25] report the results of very careful uniaxial compressive strength of the intact rockback analysis of deformation measurements, from elements ( σ ci ), the material constant mi and theextensometers placed before the commencement of Geological Strength Index (GSI).excavation, in the Mingtan power cavern in Taiwan.It was found that a zone of blast damage extended 9. ACKNOWLEDGEMENTSfor a distance of approximately 2 m around all largeexcavations. The back-calculated strength and The authors wish to acknowledge the contributionsdeformation properties of the damaged rock mass of Professor E.T. Brown in reviewing a draft of thisgive an equivalent disturbance factor D = 0.7. paper and in participating in the development of the Hoek-Brown criterion for the past 25 years.From these references it is clear that a large numberof factors can influence the degree of disturbance inthe rock mass surrounding an excavation and that itmay never be possible to quantify these factorsprecisely. However, based on their experience andon an analysis of all the details contained in thesepapers, the authors have attempted to draw up a setof guidelines for estimating the factor D and theseare summarised in Table 1.The influence of this disturbance factor can belarge. This is illustrated by a typical example inwhich σ ci = 50 MPa, mi = 10 and GSI = 45. For anundisturbed in situ rock mass surrounding a tunnelat a depth of 100 m, with a disturbance factor D = 0,the equivalent friction angle is φ = 47.16° while thecohesive strength is c = 0.58 MPa. A rock masswith the same basic parameters but in highlydisturbed slope of 100 m height, with a disturbancefactor of D = 1, has an equivalent friction angle of
- 6. Table 1: Guidelines for estimating disturbance factor D Appearance of rock mass Description of rock mass Suggested value of D Excellent quality controlled blasting or excavation by Tunnel Boring Machine results in minimal disturbance D=0 to the confined rock mass surrounding a tunnel. Mechanical or hand excavation in poor quality rock masses (no blasting) results in minimal disturbance to D=0 the surrounding rock mass. Where squeezing problems result in significant floor D = 0.5 heave, disturbance can be severe unless a temporary No invert invert, as shown in the photograph, is placed. Very poor quality blasting in a hard rock tunnel results in severe local damage, extending 2 or 3 m, in the D = 0.8 surrounding rock mass. Small scale blasting in civil engineering slopes results D = 0.7 in modest rock mass damage, particularly if controlled Good blasting blasting is used as shown on the left hand side of the photograph. However, stress relief results in some D = 1.0 disturbance. Poor blasting Very large open pit mine slopes suffer significant D = 1.0 disturbance due to heavy production blasting and also Production due to stress relief from overburden removal. blasting In some softer rocks excavation can be carried out by D = 0.7 ripping and dozing and the degree of damage to the Mechanical slopes is less. excavation
- 7. 10. REFERENCES 17. Marinos, P and Hoek, E. 2000. GSI – A geologically friendly tool for rock mass strength estimation. Proc.1. Hoek, E. and Brown, E.T. 1980. Empirical strength GeoEng2000 Conference, Melbourne. criterion for rock masses. J. Geotech. Engng Div., ASCE 18. Hoek, E. and Marinos, P. 2000. Predicting Tunnel 106 (GT9), 1013-1035. Squeezing. Tunnels and Tunnelling International. Part 12. Hoek, E. and Brown, E.T. 1980. Underground – November 2000, Part 2 – December, 2000 Excavations in Rock, London, Instn Min. Metall. 19. Marinos. P, and Hoek, E. 2001. – Estimating the3. Hoek, E. 1968. Brittle failure of rock. In Rock Mechanics geotechnical properties of heterogeneous rock masses in Engineering Practice . (eds K.G. Stagg and O.C. such as flysch. Accepted for publication in the Bulletin of Zienkiewicz), 99-124. London: Wiley the International Association of Engineering Geologists4. Brown, E.T. 1970. Strength of models of rock with 20. Balmer, G. 1952. A general analytical solution for Mohrs intermittent joints. J. Soil Mech. Foundn Div., ASCE 96, envelope. Am. Soc. Test. Mat. 52, 1260-1271. SM6, 1935-1949. 21. Sjöberg, J., Sharp, J.C., and Malorey, D.J. 2001 Slope5. Bieniawski Z.T. 1976. Rock mass classification in rock stability at Aznalcóllar. In Slope stability in surface engineering. In Exploration for Rock Engineering, Proc. mining. (eds. W.A. Hustrulid, M.J. McCarter and D.J.A. of the Symp., (ed. Z.T. Bieniawski) 1, 97-106. Cape Van Zyl). Littleton: Society for Mining, Metallurgy and Town, Balkema. Exploration, Inc., 183-202.6. Hoek, E. and Brown, E.T. 1988. The Hoek-Brown failure 22. Pierce, M., Brandshaug, T., and Ward, M. 2001 Slope criterion - a 1988 update. Proc. 15th Canadian Rock stability assessment at the Main Cresson Mine. In Slope Mech. Symp. (ed. J.C. Curran), 31-38. Toronto, Dept. stability in surface mining. (eds. W.A. Hustrulid, M.J. Civil Engineering, University of Toronto. McCarter and D.J.A. Van Zyl). Littleton: Society for Mining, Metallurgy and Exploration, Inc., 239-250.7. Hoek, E., Wood D. and Shah S. 1992. A modified Hoek- Brown criterion for jointed rock masses. Proc. Rock 23. Lorig, L., and Varona, P. 2001 Practical slope-stability Characterization, Symp. Int. Soc. Rock Mech.: Eurock analysis using finite-difference codes. In Slope stability in ‘92, (ed. J.A. Hudson), 209-214. London, Brit. Geotech. surface mining. (eds. W.A. Hustrulid, M.J. McCarter and Soc. D.J.A. Van Zyl). Littleton: Society for Mining, Metallurgy and Exploration, Inc., 115-124.8. Hoek, E. 1983. Strength of jointed rock masses, 23rd. Rankine Lecture. Géotechnique 33 (3), 187-223. 24. Sonmez, H., and Ulusay, R. 1999. Modifications to the geological strength index (GSI) and their applicability to9. Ucar, R. (1986) Determination of shear failure envelope the stability of slopes. Int. J. Rock Mech. Min. Sci., 36 (6), in rock masses. J. Geotech. Engg. Div. ASCE. 112, (3), 743-760. 303-315. 25. Cheng, Y., and Liu, S. 1990. Power caverns of the10. Londe, P. 1988. Discussion on the determination of the Mingtan Pumped Storage Project, Taiwan. In shear stress failure in rock masses. ASCE J Geotech Eng Comprehensive Rock Engineering. (ed. J.A. Hudson), Div, 14, (3), 374-6. Oxford: Pergamon, 5, 111-132.11. Carranza-Torres, C., and Fairhurst, C. 1999. General formulation of the elasto-plastic response of openings in rock using the Hoek-Brown failure criterion. Int. J. Rock Mech. Min. Sci., 36 (6), 777-809.12. Hoek, E. 1990. Estimating Mohr-Coulomb friction and cohesion values from the Hoek-Brown failure criterion. Intnl. J. Rock Mech. & Mining Sci. & Geomechanics Abstracts. 12 (3), 227-229.13. Hoek, E. 1994. Strength of rock and rock masses, ISRM News Journal, 2 (2), 4-16.14. Hoek, E. and Brown, E.T. 1997. Practical estimates of rock mass strength. Intnl. J. Rock Mech. & Mining Sci. & Geomechanics Abstracts. 34 (8), 1165-1186.15. Hoek, E., Kaiser P.K. and Bawden W.F. 1995. Support of underground excavations in hard rock. Rotterdam, Balkema.16. Hoek, E., Marinos, P. and Benissi, M. 1998. Applicability of the Geological Strength Index (GSI) classification for very weak and sheared rock masses. The case of the Athens Schist Formation. Bull. Engg. Geol. Env. 57(2), 151-160.

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