Abstract
The rock mass deformation modulus, E_{rm}, is an input parameter for most numerical modeling to verify the deformation behavior of rocks due to rock engineering activities within/on it. Among the most common methodologies used for estimating this parameter, empirical correlations based on rock mass classification schemes (e.g., RQD, RMR, GSI, and Q) stand out the most, principally because of their low cost when compared to the other methods. Herein, the main correlations used in practice are evaluated and comparted for 48 different rock quality scenarios, previously characterized and classified according to rock mass classification systems. The results obtained by each of the empirical correlations demonstrated that normalized correlations, that is, based on the ratio of the rock mass and intact rock modulus, E_{rm}/E_{i}, underestimate the E_{rm} values when compared to those results obtained from notnormalized correlations in the scenarios of better quality rock masses. For poor quality rock mass scenarios, both nonnormalized and normalized correlation presented similar results. The correlations proposed by Hoek and Diederichs (2006) and Galera et al. (2007) estimated more central Erm values when compared to the other correlations, for all quality scenarios, while the Mitri et al. (1994) and Sonmez et al. (2006) methods estimated most high and low values of E_{rm}, respectively.
Keywords:
deformation modulus; empirical methods; classification schemes; rock masses
1. Introduction
The deformation modulus of the rock mass, E_{rm}, is an essential input parameter for many analyses of jointed rock mass behavior that includes deformations (Hoek and Diederichs, 2006HOEK, E.; DIEDERICHS, M. S. Empirical estimation of rock mass modulus. International Journal of Rock mechanics and Mining Sciences , v. 43, n. 2, p. 203215, 2006.). It can be estimated directly through field tests or indirectly by using empirical relationships based on classification schemes, e.g., Rock Quality Designation (Deere et al., 1967), Rock Mass Rating (Bieniawski, 1973BIENIAWSKI, Z. T. Engineering classification of jointed rock masses. Civil Engineer in South Africa , v. 15, n. 12, 1973., 1989), QSystem (Barton et al., 1974BARTON, N.; LIEN, R.; LUNDE, J. Engineering classification of rock masses for the design of tunnel support. Rock Mechanics ,v. 6, n. 4, p. 189236, 1974.) and Geological Strength Index (Hoek, 1994). Considering that field tests are timeconsuming, expensive, and often difficult to be conducted, as reported by Zhang (2016), the second approach is often used by the rock mechanics community to estimate the rock mass deformation modulus.
From the various empirical correlations proposed for estimating E_{rm}, standing out are those using as input parameters the RQD index (Coon and Merritt, 1970COON, R. F.; MERRITT, A. H. Predicting in situ modulus of deformation using rock quality indexes. In: DETERMINATION of the in situ modulus of deformation of rock. West Conshohocken, PA: ASTM International, 1970.; Gardner, 1987GARDNER, W. S. Design of drilled piers in the Atlantic Piedmont. In: SMITH, R. E. (ed.). Foundations and excavations in decomposed rock of the Piedmont province. Reston, Virginia: ASCE, 1987. p. 6286.; Zhang and Einstein, 2004ZHANG, L. Evaluation of rock mass deformability using empirical methodsA review. Underground Space, v. 2, n. 1, p. 115, 2017.), the RMR number (Bieniawski, 1978BIENIAWSKI, Z. T. Determining rock mass deformability: experience from case histories. International Journal of Rock mechanics and Mining Sciences & Geomechanics Abstracts, v. 15, n.5, p. 237247, 1978.; Serafim and Pereira, 1983SERAFIM, J. L; PEREIRA, J. P. Consideration of the geomechanical classification of Bieniawski. In: INTERNATIONAL SYMPOSIUM ON ENGINEERING GEOLOGY AND UNDERGROUND CONSTRUCTION, 1983, Lisboa, Portugal. Proceedings […] . [S. l.] : SPG  Sociedade Portuguesa de Geotecnia, 1983. p. 3344.; Nicholson and Bieniawski, 1990NICHOLSON, G. A.; BIENIAWSKI, Z. T. A nonlinear deformation modulus based on rock mass classification. International journal of Mining and geological engineering , v. 8, n. 3, p. 181202, 1990.; Mitri et al., 1994MITRI, H. S.; EDRISSI, R.; HENNING, J. G. Finiteelement modeling of cablebolted stopes in hardrock underground mines. TransactionsSociety for Mining Metallurgy and Exploration Incorporated , v. 298, p. 18971902, 1994.; Read et al., 1999READ, S. A. L.; PERRIN, N. D.; RICHARDS, L. R. Applicability of the HoekBrown failure criterion to New Zealand greywacke rocks. In: ISRM CONGRESS, 9th, 1999, Paris, France. Proceedings […] . [S. l.] : International Society for Rock Mechanics and Rock Engineering, 1999.; Gokceoglu et al., 2003GOKCEOGLU, C.; SONMEZ, H.; KAYABASI, A. Predicting the deformation moduli of rock masses. International Journal of Rock mechanics and Mining Sciences , v. 40, n. 5, p. 701710, 2003.; Ramamurthy, 2004RAMAMURTHY, T. A geoengineering classification for rocks and rock masses. International Journal of Rock mechanics and Mining Sciences , v. 41, n. 1, p. 89101, 2004.; Sonmez et al., 2006SONMEZ, H. et al. Estimation of rock modulus: for intact rocks with an artificial neural network and for rock masses with a new empirical equation. International Journal of Rock mechanics and Mining Sciences , v. 43, n. 2, p. 224235, 2006.; Galera et al., 2007GALERA, J. M.; ÁLVAREZ, M.; BIENIAWSKI, Z. T. Evaluation of the deformation modulus of rock masses using RMR: comparison with dilatometer tests. In: ISRM WORKSHOP, 2007, Madrid, Spain. Proceedings […]. [S. l.] : ISRM, 2007. p. 7177.; Lowson and Bieniawski, 2013LOWSON, A. R.; BIENIAWSKI, Z. T. Critical assessment of RMR based tunnel design practices: a practical engineer’s approach. In: RAPID EXCAVATION AND TUNNELLING CONFERENCE, 2013, Washington, DC, USA. Proceedings […] . Englewood, Colorado: SME, 2013. p. 2326.), the Qvalue (Barton et al., 1980; Grimstad and Barton, 1993GRIMSTAD, E.; BARTON, N. Updating of the Qsystem for NMT. In: INTERNATIONAL SYMPOSIUM ON SPRAYED CONCRETE: MODERN USE OF WET MIX SPRAYED CONCRETE FOR UNDERGROUND SUPPORT, 1993, Fagernes, Oslo, Norway. Proceedings […]. Oslo: Norwegian Concrete Association, 1993.; Barton, 1995; Pamström and Singh, 2001; Barton, 2002; Ramamurthy, 2004), and the GSI (Hoek and Brown, 1997HOEK, E.; BROWN, E. T. Practical estimates of rock mass strength. International Journal of Rock mechanics and Mining Sciences , v. 34, n. 8, p. 11651186, 1997.; Hoek et al., 2002; Gokceoglu et al., 2003; Hoek and Diederichs, 2006).
Because of the number of correlations in literature, this article will evaluate their behavior in determining the deformation modulus of the rock mass for 48 different rock quality scenarios, previously characterized and classified according to rock mass classification systems.
2. Rock mass classification systems (RMCS) and their indices
A reliable prediction of rock mass behavior under different stress conditions is critical for the design of most rock engineering projects. However, this is not a trivial task due to the heterogeneous and anisotropic characteristics of the rock material and the scale effect existing in jointed rock masses. These features combined make conventional laboratory and field tests, which are necessary to estimate the mechanical parameters of this geomaterial for this purpose, an extremely costeffective solution.
Dealing with this limitation, different rock mass classification systems (RMCS) were proposed to be used as a guide for different rock engineering applications, taking into consideration the individual characteristics of the intact rock and the discontinuities within the rock masses. Among these RMCS, the classification schemes designed by Deere et al. (1967), Bieniawski (1973BIENIAWSKI, Z. T. Engineering classification of jointed rock masses. Civil Engineer in South Africa , v. 15, n. 12, 1973., ^{1978)}BIENIAWSKI, Z. T. Determining rock mass deformability: experience from case histories. International Journal of Rock mechanics and Mining Sciences & Geomechanics Abstracts, v. 15, n.5, p. 237247, 1978., Barton et al. (1974)BARTON, N.; LIEN, R.; LUNDE, J. Engineering classification of rock masses for the design of tunnel support. Rock Mechanics ,v. 6, n. 4, p. 189236, 1974., and Hoek (1994)HOEK, E. Strength of rock and rock masses. ISRM News Journal , v. 2, n. 2, p. 416, 1994. have been widely used to quantify the rock mass quality based on their indices, i.e., RQD, RMR, Q e GSI, respectively.
First, the Rock Quality Designation (RQD) index was idealized by Dr. Don U. Deere in 1964 and later presented for the first time in published form by Deere et al. (1967), as reported by Deere and Deere (1988)DEERE, D. U.; DEERE, D. W. The rock quality designation (RQD) index in practice. In: KIRKALDIE, L. (ed.). Rock classification systems for engineering purposes.West Conshohocken, PA: ASTM International, 1988. p. 91101.. It represents a modified borehole core recovery percentage in which all the pieces of sound core over 10 cm long are summed and divided by the length of the core run, allowing a measurement of the rock mass quality. The RQD index ranges from 0 to 100%, where high RQD values will identify a rock mass with good quality, while an RQD ranging from 0 to 50% are indicative of poor quality rock mass (Deere et al., 1967DEERE, D. U. et al. Design of surface and nearsurface construction in rock. In: US SYMPOSIUM ON ROCK MECHANICS (USRMS), 8th, 1966, Minneapolis, Minnesota. Proceedings […]. [S. l.] : American Rock Mechanics Association, 1967.). Although the RQD is a singlebased parameter, only considering the discontinuity frequency within the rock mass, it is still widely used for estimating the rock’s mechanical properties, e.g., the deformation modulus (Zhang and Einstein, 2004ZHANG, L.; EINSTEIN, H. H. Using RQD to estimate the deformation modulus of rock masses. International Journal of Rock mechanics and Mining Sciences , v. 41, n. 2, p. 337341, 2004.).
The Rock Mass Rating (RMR) system, also known as the Geomechanics Classification, was introduced first by Bieniawski (1973)BIENIAWSKI, Z. T. Engineering classification of jointed rock masses. Civil Engineer in South Africa , v. 15, n. 12, 1973. and later modified over the years by the author until its last version presented by Bieniawski (1989). This system allows classification of the rock mass quality based on six different parameters: (1) the uniaxial compressive strength of the intact rock; (2) the RQD; (3) the spacing of discontinuities; (4) the condition of discontinuities; (5) the groundwater conditions; and (6) the orientation of discontinuities. Each of these parameters will attribute a rating for the rock mass, which combined, generate the overall RMR value of it, varying from 0 to 100. In a direct comparison to Deere’s index, the RMR provides a much more realistic quality condition of the jointed geomaterial due to the numbers of variables analyzed.
Almost one year after the first publication of Bieniawski’s classification scheme, Barton et al. (1974)BARTON, N.; LIEN, R.; LUNDE, J. Engineering classification of rock masses for the design of tunnel support. Rock Mechanics ,v. 6, n. 4, p. 189236, 1974. introduced a new system to classify rock masses for engineering purposes, the Qsystem. Similar to the Geomechanics Classification, this system also provides a numerical assessment of the rock mass quality using six parameters: (1) the RQD; (2) the number of joint sets; (3) the roughness of the discontinuities; (4) the degree of alteration of the discontinuities; (5) the water inflow; and (6) the stress conditions. The numerical value of Q ranges on a logarithmic sale from 0.001, for exceptionally poor rock mass quality, up to 1000, for exceptionally good rock mass quality. The main difference between the RMR and Q indices lies in the fact that the first uses the compressive strength of the intact rock as an input parameter, while the second takes into account the stress condition of the rock mass. Besides that, for similar conditions of discontinuities within the rock mass, i.e., spacing, aperture, roughness, infilling and weathering, they attribute different weightings to compose their indices.
Lastly, the Geological Strength Index (GSI) was designed by Hoek (1994)HOEK, E. Strength of rock and rock masses. ISRM News Journal , v. 2, n. 2, p. 416, 1994. as a tool to describe a rock mass qualitatively based upon geological observations. According to the author, the GSI use as input parameters the two major features that most influence the mechanical properties of a rock mass, i.e., the overall structure conditions, also described as the blockiness (Hoek and Brown, 2019HOEK, E.; BROWN, E. T. The HoekBrown failure criterion and GSI  2018 edition. Journal of Rock Mechanics and Geotechnical Engineering , v. 11, n. 3, p. 445463, 2019.), and the surface condition of the discontinuities. Besides describing the rock mass quality, Hoek’s index is frequently used for numerical modeling in jointed rock masses, since it is used to determine the empirical constants of the generalized HoekBrown failure criterion (Hoek et al., 2002).
Although the RQD, RMR, Q, and GSI systems were introduced as an alternative methodology to evaluate the rock mass quality for rock engineering practices, especially to overcome the limitation of testing these jointed geomaterials as a conventional engineering material, e.g., humanmade materials and soils, they were designed initially for different applications. Under these circumstances, these systems have both their strengths and weaknesses that need to be assessed individually according to some variables, such as the insitu site accessibility conditions, the experience of the engineering and geological staff, and the previously recorded local geological features.
3. Empirical methods
This study selected the most relevant empirical methods presented in literature to estimate the deformation modulus of jointed rock masses based on their qualitative index values derived from Deere’s (1967), Bieniawski’s (1973BIENIAWSKI, Z. T. Engineering classification of jointed rock masses. Civil Engineer in South Africa , v. 15, n. 12, 1973.,^{1989)}BIENIAWSKI, Z. T. Engineering rock mass classifications New York: John Wiley & Sons, 1989. 251 p., Barton’s (1974BARTON, N.; LIEN, R.; LUNDE, J. Engineering classification of rock masses for the design of tunnel support. Rock Mechanics ,v. 6, n. 4, p. 189236, 1974., ^{2002)}BARTON, N. Some new Qvalue correlations to assist in site characterisation and tunnel design. International Journal of Rock mechanics and Mining Sciences, v. 39, n. 2, p. 185216, 2002., and Hoek’s (1994)HOEK, E. Strength of rock and rock masses. ISRM News Journal , v. 2, n. 2, p. 416, 1994. classification schemes. Table 1 presents the RQD based correlations, where all of them are normalized, i.e., expressed in terms of the deformation ratio, E_{rm}/E_{i}. Coon and Merritt’s (1970)COON, R. F.; MERRITT, A. H. Predicting in situ modulus of deformation using rock quality indexes. In: DETERMINATION of the in situ modulus of deformation of rock. West Conshohocken, PA: ASTM International, 1970. and Gardner’s (1987)GARDNER, W. S. Design of drilled piers in the Atlantic Piedmont. In: SMITH, R. E. (ed.). Foundations and excavations in decomposed rock of the Piedmont province. Reston, Virginia: ASCE, 1987. p. 6286. correlations are very similar, the only difference is that the first is only applied for RQD > 57%, while the second gives an arbitrary value of E_{rm}/E_{i} for RQD < 57%. The correlation proposed by Zhang and Einstein (2004)ZHANG, L.; EINSTEIN, H. H. Using RQD to estimate the deformation modulus of rock masses. International Journal of Rock mechanics and Mining Sciences , v. 41, n. 2, p. 337341, 2004. can be applied for the full range for RQD.
Since the Rock Mass Rating system, alias Geomechanics Classification, initially presented by Bieniawski (1973)BIENIAWSKI, Z. T. Engineering classification of jointed rock masses. Civil Engineer in South Africa , v. 15, n. 12, 1973., was probably the most used classification scheme during the '80s and '90s, many authors suggested correlations to estimate deformation modulus of rock masses using this system. These methods can be divided into two major groups, the nonnormalized equations, and the normalized equations, and are summarized in Tables 2 and 3, respectively. Among the nonnormalized correlations, the ones proposed by Bieniawski (1978), Equation 5, for RMR > 50, and Serafim and Pereira (1983)SERAFIM, J. L; PEREIRA, J. P. Consideration of the geomechanical classification of Bieniawski. In: INTERNATIONAL SYMPOSIUM ON ENGINEERING GEOLOGY AND UNDERGROUND CONSTRUCTION, 1983, Lisboa, Portugal. Proceedings […] . [S. l.] : SPG  Sociedade Portuguesa de Geotecnia, 1983. p. 3344., Equation 6, for RMR ≤ 56, stand out in literature. While for the normalized correlations, Nicholson and Bieniawski’s (1990)NICHOLSON, G. A.; BIENIAWSKI, Z. T. A nonlinear deformation modulus based on rock mass classification. International journal of Mining and geological engineering , v. 8, n. 3, p. 181202, 1990., Equation 11, and Sonmez et al.’s (2006)SONMEZ, H. et al. Estimation of rock modulus: for intact rocks with an artificial neural network and for rock masses with a new empirical equation. International Journal of Rock mechanics and Mining Sciences , v. 43, n. 2, p. 224235, 2006., Equation 12, are commonly used for estimating the deformation modulus of rock masses using an RMR number for numerical modeling.
Regarding the empirical methods developed based on the Qvalue, most of the proposed correlation is indicated for fair, good and very good rock qualities, e.g., Equations 16 and 17, proposed by Grimstad and Barton (1993)GRIMSTAD, E.; BARTON, N. Updating of the Qsystem for NMT. In: INTERNATIONAL SYMPOSIUM ON SPRAYED CONCRETE: MODERN USE OF WET MIX SPRAYED CONCRETE FOR UNDERGROUND SUPPORT, 1993, Fagernes, Oslo, Norway. Proceedings […]. Oslo: Norwegian Concrete Association, 1993. and Barton (1995)BARTON, N. The influence of joint properties in modelling jointed rock masses. In: ISRM CONGRESS, 8th, 1995, Tokyo, Japan. Proceedings […]. [S. l.]: International Society for Rock Mechanics and Rock Engineering, 1995., respectively. For weak rocks, Barton's (2002) correlation, Equation 19, is recommended, which takes the strength of intact rock into account. Table 4 summarizes these Q based methods, which all are notnormalized equations.
Lastly, because of its direct link to the HoekBrown empirical constants and consequently to other engineering parameters, such as the MohrCoulomb strength parameters, i.e., cohesive strength and the angle of friction, several empirical methods for estimating rock mass deformation using GSI have been proposed. Among these methods, Table 5 presents several GSIbased methods to estimate the deformation modulus of rock masses, in which the generalized Hoek and Diederichs correlation, Equation 23, stands as the only normalized equation.
4. Scenarios database
To assess the behavior of the empirical correlations presented above, they were applied to estimate the deformability modulus of rock masses from a database of different lithologies, previously characterized and classified according to the rock mass classifications systems, i.e., RQD, RMR, Q, and GSI. Table 6 presents the database of 48 scenarios selected.
Respecting the chosen database described in Table 6, the authors took into consideration grouping them according to the following criteria: (i) lithology variability; (ii) wide range of jointed rock mass quality; (iii) availability of mechanical properties of the intact rock previously estimated through direct or indirect tests; (iv) multiple classifications for the same site using the RMCS discussed in this study; and (v) sites characterized for different civil engineering purposes, e.g., underground excavations and dam foundations.
For the comparative study between the correlations, the 48 scenarios were divided into two groups, taking into consideration the similarity between the quality of the rock mass. The first group, SGI, consists of the poor quality rock mass scenarios, while the second group, SGII, comprises the rock mass with better quality. The scenarios presented in SGI are: S1, S2, S4, S5, S6, S13, S15, S18, S19, S20, S21, S22, S27, S29, S32, S34, S35, S36, S38, S42, S43, S44, S45, S46, and S47. The SGII group covers the following scenarios: S3, S7, S8, S9, S10, S11, S12, S14, S16, S17, S23, S24, S25, S26, S28, S30, S31, S33, S37, S39, S40, S41, and S48. Table 7 gives the descriptive statistics of both groups.
5. Comparative analysis and results
The first comparative study considered the variation of estimated values for each scenario, based on the application of the possible correlations available. Regarding the results of the SGI, Figure 1 shows the boxwhisker plots displaying the distribution of E_{rm} estimated of the scenarios in this group. In summary, Equations (12) and (19) were responsible for estimating the maximum deformation modulus in 28% (S1, S15, S19, S20, S36, and S38), 20% (S4, S18, S29, S34, and S42) and 28% (S2, S5, S21, S22, S27, S46, and S47) of the scenarios analyzed, respectively.
Boxwhisker plots displaying the distribution of E_{rm} estimated for the scenarios in SGI.
On the other hand, the correlation suggested by Equation (13) estimated in 80% (S2, S4, S13, S15, S18, S19, S20, S21, S22, S27, S29, S32, S34, S35, S42, S43, S44, and S45) the minimum values for the SGI. Equation (3), which is a RQDbased method proposed by Zhang and Einstein (2004)ZHANG, L.; EINSTEIN, H. H. Using RQD to estimate the deformation modulus of rock masses. International Journal of Rock mechanics and Mining Sciences , v. 41, n. 2, p. 337341, 2004., was responsible for estimating the other 20% minimum values in this group. Equations (14), (20), (22), (23) and (24) gave the closest result to the mean in each study scenario, especially the generalized equation of Hoek and Diederichs (2006)HOEK, E.; DIEDERICHS, M. S. Empirical estimation of rock mass modulus. International Journal of Rock mechanics and Mining Sciences , v. 43, n. 2, p. 203215, 2006..
Regarding the group with better quality rock masses, SGII, Figure 2 shows the distribution behavior of E_{rm} calculated for each scenario in this group. Equations (7) and (12) estimated almost 61% (S3, S11, S12, S17, S23, S24, S25, S26, S28, S30, S31, S33, S39, and S40) of the maximum scenarios’ values of rock mass deformation modulus. It was Equation (23), however, that estimated the highest values (S8, S9, S16, and S48). In common, these are the highest quality rock mass scenarios in SGII.
In contrast, Equations (3), (8) and (13) were responsible for estimating the minimum deformation modulus in 17% (S12, S14, S26, S48), 26% (S3, S23, S25, S37, S39 and S40) and 22% (S7, S10, S11, S16 and S24) of the scenarios analyzed, respectively. For the rock quality range in SGII, Equations (11) and (24) estimated values of deformation modulus with closest result to the mean in each study scenario.
Based on the results presented above, it is noticed that nonnormalized correlations tend to estimate higher values of deformation modulus as rock quality increases in most cases. In order to show this behavior, Figure 3 presents a comparison between the arithmetic mean for each scenario of the E_{rm} estimated using the normalized and nonnormalized correlations.
Estimated average values of rock mass deformation modulus for nonnormalized (NN) and normalized (N) empirical correlations.
For some scenarios with good rock mass quality, e.g., S8, S9, S16, and S48, there is noticeably a higher peak of the deformation modulus calculated by the nonnormalized equations when compared with the normalized ones. In these cases, Equations (4), (5), (7), and (9) estimated E_{rm} values up to 3 times more than the values obtained using Equations (11) to (15) and (24).
For the weak rock mass scenarios, the difference between the notnormalized and the normalized correlations is much smaller. There are even situations where normalized equations give higher values, e.g., S3 and S40, which are justified by the mathematical function of the correlations used. In both scenarios cited, the correlation proposed by Mitri et al. (1994)MITRI, H. S.; EDRISSI, R.; HENNING, J. G. Finiteelement modeling of cablebolted stopes in hardrock underground mines. TransactionsSociety for Mining Metallurgy and Exploration Incorporated , v. 298, p. 18971902, 1994., which is a function of cosine, estimates higher values for poor to fair quality rock masses when compared with other normalized and notnormalized equations.
For the third and final analysis of this study, the behavior of the estimated deformation modulus values was considered, based on the same type of classification system for each scenario. Regarding the RQDbased methods, Figure 4 presents the range of rock mass deformation modulus calculated using Equations (1) to (3). For the scenarios with rock masses with RQD > 64%, Equations (1) and (2) estimated higher values of E_{rm} compared to the values estimated by Equation (3). In total, Equation (3) only overestimated E_{rm} values in less than 7% of the total scenarios (S14, S28, and S31). Consequently, the correlation proposed by Zhang and Einstein (2004)ZHANG, L.; EINSTEIN, H. H. Using RQD to estimate the deformation modulus of rock masses. International Journal of Rock mechanics and Mining Sciences , v. 41, n. 2, p. 337341, 2004. predicted more conservative values for the scenarios analyzed in this study.
In relation to the values estimated using the RMRbased correlations for the all the 48 scenarios (Figure 5), Equations (9) and (14) presented the mean values for poorer quality rocks, while Equations (11) and (14) presented average values for rocks with better quality. On the other hand, Equations (12) and (13) resulted in overestimated and underestimated values of E_{rm}, respectively, for the full range of rock mass quality.
Concerning the empirical methods that use Barton’s (1974)BARTON, N.; LIEN, R.; LUNDE, J. Engineering classification of rock masses for the design of tunnel support. Rock Mechanics ,v. 6, n. 4, p. 189236, 1974. Qvalue as an input parameter for estimating the deformation modulus of rock masses, Figure 6 gives the behavior of the E_{rm} values estimated using Equations (16) to (19). For the lowquality rocks scenarios, i.e., from the SGI group, Equation (19) were the only that could be used, and yet resulted in overestimated values compared to other classification scheme methods, as pointed out before. For the scenarios with better rock mass quality, in general, Equations (16) to (17), when applicable, estimated similar E_{rm} values.
To conclude, the GSIbased methods, among the other systems, were responsible for estimating the greatest variation of rock mass deformation modulus, especially for the scenarios from the SGII group. Taking into consideration only the estimated values using Equations (20) to (24), as illustrated in Figure 7, it is remarkable that these equations estimated similar E_{rm} values for scenarios formed by rock masses with poor qualities, putting in evidence the generalized equation of Hoek and Diederichs (2006)HOEK, E.; DIEDERICHS, M. S. Empirical estimation of rock mass modulus. International Journal of Rock mechanics and Mining Sciences , v. 43, n. 2, p. 203215, 2006..
6. Conclusions
Herein, analyzed and compared were the main empirical correlations derived from the most used rock mass classification schemes (RQD, RMR, Q, and GSI) for estimating the rock mass deformation modulus. A database with 48 different scenarios of rock masses from different lithologies, previously characterized and classified according to rock mass classification systems, was used to achieve these goals. From the results discussed earlier, the conclusions obtained are as follow:

For both poor and good quality rock mass scenarios, the correlations proposed by Hoek and Diederichs (2006)HOEK, E.; DIEDERICHS, M. S. Empirical estimation of rock mass modulus. International Journal of Rock mechanics and Mining Sciences , v. 43, n. 2, p. 203215, 2006. and Galera et al. (2007)GALERA, J. M.; ÁLVAREZ, M.; BIENIAWSKI, Z. T. Evaluation of the deformation modulus of rock masses using RMR: comparison with dilatometer tests. In: ISRM WORKSHOP, 2007, Madrid, Spain. Proceedings […]. [S. l.] : ISRM, 2007. p. 7177., Equations (14) and (24), respectively, estimated more central Erm values when compared to the other correlations.

Concerning the nonnormalized and normalized equations, it is noticed that for higher quality rock masses, the former tends to estimate higher values of Erm when compared to the latter. However, for conditions of rock masses with lower quality, they behave more similarly.

Sonmez et al.’s (2006)SONMEZ, H. et al. Estimation of rock modulus: for intact rocks with an artificial neural network and for rock masses with a new empirical equation. International Journal of Rock mechanics and Mining Sciences , v. 43, n. 2, p. 224235, 2006. method estimated the lower value of Erm among all 24 correlations for most scenarios, mainly for those in the SGI group. On the other hand, Mitri et al.’s (1994)MITRI, H. S.; EDRISSI, R.; HENNING, J. G. Finiteelement modeling of cablebolted stopes in hardrock underground mines. TransactionsSociety for Mining Metallurgy and Exploration Incorporated , v. 298, p. 18971902, 1994. method estimated the highest values of Erm, mainly for those in the SGII.

Due to the number of different methods proposed for estimating deformation modulus, the authors of this article recommend, when possible, that three or more methods should be used to give a good confidence range of values.
Acknowledgments
The authors gratefully acknowledge the National Council for Scientific and Technological Development (CNPq) for their financial support.
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Publication Dates

Publication in this collection
15 Jan 2021 
Date of issue
JanMar 2021
History

Received
11 Nov 2019 
Accepted
05 Oct 2020