HOEK-BROWN FAILURE CRITERION ��� 2002 EDITION Evert Hoek Consulting Engineer, Vancouver, Canada Carlos Carranza-Torres Itasca Consulting Group Inc., Minneapolis, USA Brent Corkum Rocscience Inc., Toronto, Canada ABSTRACT: The Hoek-Brown failure criterion for rock masses is widely accepted and has been applied in a large number of projects around the world. While, in general, it has been found to be satisfactory, there are some uncertainties and inaccuracies that have made the criterion inconvenient to apply and to incorporate into numerical models and limit equilibrium programs. In particular, the difficulty of finding an acceptable equivalent friction angle and cohesive strength for a given rock mass has been a problem since the publication of the criterion in 1980. This paper resolves all these issues and sets out a recommended sequence of calculations for applying the criterion. An associated Windows program called ���RocLab��� has been developed to provide a convenient means of solving and plotting the equations presented in this paper. 1. INTRODUCTION Hoek and Brown [1, 2] introduced their failure criterion in an attempt to provide input data for the analyses required for the design of underground excavations in hard rock. The criterion was derived from the results of research into the brittle failure of intact rock by Hoek [3] and on model studies of jointed rock mass behaviour by Brown [4]. The criterion started from the properties of intact rock and then introduced factors to reduce these properties on the basis of the characteristics of joints in a rock mass. The authors sought to link the empirical criterion to geological observations by means of one of the available rock mass classification schemes and, for this purpose, they chose the Rock Mass Rating proposed by Bieniawski [5]. Because of the lack of suitable alternatives, the criterion was soon adopted by the rock mechanics community and its use quickly spread beyond the original limits used in deriving the strength reduction relationships. Consequently, it became necessary to re-examine these relationships and to introduce new elements from time to time to account for the wide range of practical problems to which the criterion was being applied. Typical of these enhancements were the introduction of the idea of ���undisturbed��� and ���disturbed��� rock masses Hoek and Brown [6], and the introduction of a modified criterion to force the rock mass tensile strength to zero for very poor quality rock masses (Hoek, Wood and Shah, [7]). One of the early difficulties arose because many geotechnical problems, particularly slope stability issues, are more conveniently dealt with in terms of shear and normal stresses rather than the principal stress relationships of the original Hoek-Brown riterion, defined by the equation: c ' 3 ' 3 ' 1 ��� ��� ���0.5 ��� ��� ��� ��� + +�� = s��� m ci ci �� �� �� �� (1) where ' 1 �� and ' 3 �� are the major and minor effective principal stresses at failure ci �� is the uniaxial compressive strength of the intact rock material and m and s are material constants, where s = 1 for intact rock. An exact relationship between equation 1 and the normal and shear stresses at failure was derived by J. W. Bray (reported by Hoek [8]) and later by Ucar [9] and Londe1 [10]. Hoek [12] discussed the derivation of equivalent friction angles and cohesive strengths for various practical situations. These derivations were based 1 Londe���s equations were later found to contain errors although the concepts introduced by Londe were extremely important in the application of the Hoek-Brown criterion to tunnelling problems (Carranza-Torres and Fairhurst, [11])

upon tangents to the Mohr envelope derived by Bray. Hoek [13] suggested that the cohesive strength determined by fitting a tangent to the curvilinear Mohr envelope is an upper bound value and may give optimistic results in stability calculations. Consequently, an average value, determined by fitting a linear Mohr-Coulomb relationship by least squares methods, may be more appropriate. In this paper Hoek also introduced the concept of the Generalized Hoek-Brown criterion in which the shape of the principal stress plot or the Mohr envelope could be adjusted by means of a variable coefficient a in place of the square root term in equation 1. ��� ��� ��� ��� ��� ��� ���100��� = GSI s 3D 9 exp (4) (e���GSI/15 ) (5) 6 1 2 1 ��� + = e���20/3 a D is a factor which depends upon the degree of disturbance to which the rock mass has been subjected by blast damage and stress relaxation. It varies from 0 for undisturbed in situ rock masses to 1 for very disturbed rock masses. Guidelines for the selection of D are discussed in a later section. The uniaxial compressive strength is obtained by setting in equation 2, giving: 0 ' 3 = �� ci c .sa �� �� = (6) Hoek and Brown [14] attempted to consolidate all the previous enhancements into a comprehensive presentation of the failure criterion and they gave a number of worked examples to illustrate its practical application. and, the tensile strength is: ci t mb s�� �� ��� = (7) Equation 7 is obtained by setting in equation 2. This represents a condition of biaxial tension. Hoek [8] showed that, for brittle materials, the uniaxial tensile strength is equal to the biaxial tensile strength. t �� �� ��1 = = ' 3 ' In addition to the changes in the equations, it was also recognised that the Rock Mass Rating of Bieniawski was no longer adequate as a vehicle for relating the failure criterion to geological observations in the field, particularly for very weak rock masses. This resulted in the introduction of the Geological Strength Index (GSI) by Hoek, Wood and Shah [7], Hoek [13] and Hoek, Kaiser and Bawden [15]. This index was subsequently extended for weak rock masses in a series of papers by Hoek, Marinos and Benissi [16], Hoek and Marinos [17, 18] and Marinos and Hoek [19]. Note that the ���switch��� at GSI = 25 for the coefficients s and a (Hoek and Brown, [14]) has been eliminated in equations 4 and 5 which give smooth continuous transitions for the entire range of GSI values. The numerical values of a and s, given by these equations, are very close to those given by the previous equations and it is not necessary for readers to revisit and make corrections to old calculations. The Geological Strength Index will not be discussed in the following text, which will concentrate on the sequence of calculations now proposed for the application of the Generalized Hoek Brown criterion to jointed rock masses. Normal and shear stresses are related to principal stresses by the equations published by Balmer [20]. 2 2 ' ' ' ' ' ' ' ' ' 3 1 3 1 3 1 3 +1 ���1 ��� ��� ��� + = �� �� �� ��1 �� d�� d�� d�� d�� n (8) 2. GENERALIZED HOEK-BROWN CRITERION This is expressed as ( ) 1 ' ' ' ' ' ' 1 3 1 3 1 3 + ��� = �� �� �� �� d�� d�� d�� d (9) ci ci���mb s��� ��� ��� ���a ��� ��� ��� + + = �� ��3 �� ��3 ��1 ' ' ' (2) where (mb�� ' ' ' 3 3 1 1 + + = ci s)a���1 amb d�� d�� �� (10) where mb is a reduced value of the material constant mi and is given by ��� ��� ��� ��� ��� ���100��� = GSI mi mb 28���14D exp (3) 3. MODULUS OF DEFORMATION The rock mass modulus of deformation is given by: s and a are constants for the rock mass given by the following relationships: 40) / 100 2 (GPa) ���10) ���10((GSI ��� ��� ��� ���1 ��� ��� ��� = ci D Em �� (11a)