1 Introduction

The Hoek–Brown (HB) failure criterion is widely used in rock mass characterization of tunnels (e.g. Zhao 2000; Deák et al. 2013), road cuts and cliff face stability analyses (e.g. Lindsay et al. 2001). It was developed to characterize the behaviour of rock mass by using laboratory strength parameters and material constants. The criterion was introduced by Hoek and Brown (1980) and the following form is used for intact rock:

$$\sigma_{1} = \sigma_{3} + \sigma_{ci} \left( {m_{i} \frac{{\sigma_{3} }}{{\sigma_{ci} }} + 1} \right)^{0.5}$$
(1)

where σ1: the major principal stress at failure, σ3: the minor principal stress or confining pressure, mi: Hoek–Brown material constant, σci: the uniaxial compressive strength of the intact rock.

According to Eq. (1), two independent parameters are necessary, namely the:

  • uniaxial compressive strength (σci), and the

  • Hoek–Brown material constant of the intact rock (mi)

Hoek and Brown (1997) suggested that these values should be determined by numerous triaxial tests (minimum five sets of test), applying different confining pressures (σ3) between 0 and 0.5 × σci. These laboratory tests are time consuming, expensive and in many cases there are not enough (or suitable) samples to obtain the necessary amount of data. Hoek (2007) provided values of mi constants for different types of rocks. The values of mi are in between 7 and 35, however several factors influence these values. Mineral composition, grain size and cementation are among others that control the mi (Table 1).

Table 1 Suggested values of the Hoek–Brown constant ‘mi’ for intact rock (Hoek 2007) (values in parenthesis are estimates)
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Nevertheless, the detremination of the exact mi value is very important for the rock engineering design, due to the sensitivity of the Hoek–Brown equations (Ván and Vásárhelyi 2014). Shen and Karakus (2014) also emphasized the difficulties in determining the m i values of rocks. They suggested to normalize the Hoek–Brown constant (m i) by using strength of the rock (σci). The modified version of the Hoek–Brown Eq. (1) is as follows:

$$\sigma_{1} = \sigma_{3} + \sigma_{ci} \left( {m_{in} \sigma_{3} + 1} \right)^{0.5}$$
(2)

where m in = m i/σci.

In their paper the Hoek–Brown parameter (mi) as the function of the uniaxial strength of the rock (σci) (Fig. 1a) was illustrated. The strength dependence of the normalized Hoek–Brown constant (min) (Fig. 1b) were also calculated rock types.

Fig. 1
figure 1

a Relations between m i and σci; b relations between min and σci both for limestone (Shen and Karakus 2014)

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By analyzing the data of five different rock types they found a correlation between the normalized material constant (min) and the uniaxial compressive strength of the rocks (σci). The mi value of limestone was found to be in between 7 and 14. By fitting a regression curve, According to Shen and Karakus (2014) the Eq. (3) can be used to estimate min values from uniaxial compressive strength (σci) values for different rock types:

$$m_{in} = A\sigma_{ci}^{B}$$
(3)

where A and B are the material parameters, of various rock types (Table 2).

Table 2 The modified Hoek–Brown parameters (A and B) by Shen and Karakus (2014)
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2 Analyzing existing suggested methods for Hoek–Brown constant

Sheorey (1997) collected and published 187 independent triaxial test results and calculated the Hoek–Brown parameter (mi) of intact rocks. By using this data set and calculating with Eq. (3) a strong relationship between the normalized Hoek–Brown constant (mi) and the uniaxial compressive strength of the rock (σci) was found (Fig. 2).

Fig. 2
figure 2

Normalized Hoek–Brown parameter as a function of uniaxial compressive strength–data from Sheorey (1997)

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According to these results, a new equation can be formulated:

$$m_{in} = 10.94\sigma_{ci}^{ - 0.998} \approx 10\sigma_{ci}^{ - 1}$$
(4)

It is well known, that when the Brinke number (\(R = \sigma _{{\text{c}}} /\left| {\sigma _{{\text{t}}} } \right|\)) is higher than 8, the R-value is equal to Hoek–Brown constant, i.e. equals to m i (Cai 2010):

$$m_{i} \approx \frac{{\sigma_{c} }}{{\left| {\sigma_{t} } \right|}} = R$$
(5)

Statistically, the Brinke number is around 10 (Andreev 1995).

The A and B values given by Shen and Karakus (2014) (cf. Eq. 3) were also recalculated for different rock types, using the data set published by Sheorey (1997) (Table 3).

Table 3 The calculated modified Hoek–Brown parameters, data from Sheorey (1997)
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This equation (Eq. 3) was also used to evaluate the data set of Hungarian granitic rocks of Bátaapáti site (Hungary). A radioactive waste repository (for L/LW) was designed and had been built during the past 10 years with a tunnel system of more than 6 km in length at that site. The engineering geology and rock engineering behaviour of this area was described by Deák et al. (2014).

The design of the tunnel systems was based mainly on the Hoek–Brown theory, which included the determination of Hoek–Brown constants of different granitic rock types. Multiple Failure State tests have been used, according to the ISRM method (Kovári et al. 1983). 44 samples were tested representing three slightly different lithologies: monzonitic (15), monzogranitic (14) and hybrid (contaminated monzonite and monzogranite) (15) rocks. All the samples were tested under air-dry conditions. The data was presented by Vásárhelyi et al. (2013). For triaxial tests right circular cylinders were prepared (according to ISRM, Kovári et al. 1983) with diameter of 38 mm, and with height to diameter ratio of 2:1. No direct relationship was found between the Hoek–Brown constant (m i) and the uniaxial strength of the rock (σci) (Fig. 3a). However, when the normalized Hoek–Brown constants are plotted against the uniaxial compressive strength a good correlation was recorded (Fig. 3b). It is in good agreement with the findings of Shen and Karakus (2014). A and B values were also calculated for each tested lithotypes according to Eq. 3 (Table 4).

Fig. 3
figure 3

a Relations between mi and σci; b relations between min and σci for granitic rocks of Bátaapáti (Hungary)

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Table 4 The modified Hoek–Brown parameters of the granitic rocks from Bátaapáti
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3 Correlation between the constants

The correlation between A and B constants (according to Eq. 3) was also studied. Using all calculated and published data (Sheorey 1997; Shen and Karakus 2014; Vásárhelyi et al. 2013), the following relationship was found (Fig. 4):

$$B = - 0.212\;\ln \;(A){-}0.468 \, \left( {R^{2} = \, 0.763} \right)$$
(6)
Fig. 4
figure 4

Relations between A and B constants using published and calculated data (data from: Sheorey 1997; Shen and Karakus 2014; Vásárhelyi et al. 2013)

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Plotting the data set of measured 44 granitic samples (Fig. 5) and applying the same formula for granitic rocks of Bátaapáti (Hungary), the following equation was found:

$$B = - 0.182\;\ln (A) - 0.515\;(R^{2} = 0.848)$$
(7)
Fig. 5
figure 5

Relations between A and B constants; granitic rocks from Bátaapáti (Hungary)

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The above listed equations provide a good estimation of rock mass strength of various lithologies and can be used for quick calculations when limited amount of laboratory results are available.

4 Applying the theory for rock mass

According to Hoek et al. (2002), the Hoek–Brown equation for rock mass is:

$$\sigma_{1} = \sigma_{3} + \sigma_{ci} \left( {m_{b} \frac{{\sigma_{3} }}{{\sigma_{ci} }} + s} \right)^{a}$$
(8)

where m b, s and a Hoek–Brown parameters depend on both the Geological Strength Index (GSI) and the damage factor (D). Both s and a parameters are independent of the mi value, thus the original formula can be used. For determining the Hoek–Brown parameter m b, originally the following form has been used:

$$m_{b} = m_{i} \exp \left( {\frac{GSI - 100}{28 - 14D}} \right)$$
(9)

According to Shen and Karakus (2014), this equation can be rewritten in the following form, using Eq. (3)

$$m_{b} = A\sigma_{ci}^{B + 1} \exp \left( {\frac{GSI - 100}{28 - 14D}} \right)$$
(10)

where GSI (Geological Strength Index) and D (damage value) following the definition of Hoek et al. (2002).

Accordingly, the Hoek–Brown equation for rock mass can be modified applying the theory of Shen and Karakus (2014), using Eqs. (8) and (10):

$$\sigma_{1} = \sigma_{3} + \sigma_{ci} \left( {A\sigma_{ci}^{B} \sigma_{3} + s} \right)^{a}$$
(11)

Using the Eq. (11) the following relationship can be calculated for the granitic rocks of Bátaapáti:

$$\sigma_{1} = \sigma_{3} + \sigma_{ci} \left( {216\sigma_{ci}^{ - 1.53} \sigma_{3} + s} \right)^{a}$$
(11)

5 Conclusions

The modified Hoek–Brown failure envelope (Shen and Karakus 2014) was recalculated by using the data set published by Sheorey (1997). New material constants are suggested for coal, limestone, quartzite, slate and sandstone. For the tested 44 granitic samples of Bátaapáti (Hungary), representing three lithotypes the application of the modified Hoek–Brown failure envelope proved to be more reliable than the classical one. It was shown that by using this method the failure envelope can be determined more exactly. A logarithmic correlation between the empirical constants (A and B) was also found, which strongly suggest that the assumption that these constants were independent could not be confirmed. It was also denoted that the failure envelope of the intact rock can be determined more exactly with the help of the new suggested equation. It also should be noted that the values of A and B depend on various factors including rock state, rock types, and specimen geometry.