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O R I G I N A L P A P E R
Estimating the Strength of Jointed Rock Masses
Lianyang Zhang
Received: 19 January 2009/ Accepted: 14 July 2009
Springer-Verlag 2009
Abstract Determination of the strength of jointed rock
masses is an important and challenging task in rockmechanics and rock engineering. In this article, the existing
empirical methods for estimating the unconfined com-
pressive strength of jointed rock masses are reviewed and
evaluated, including the jointing index methods, the joint
factor methods, and the methods based on rock mass
classification. The review shows that different empirical
methods may produce very different estimates. Since in
many cases, rock quality designation (RQD) is the only
information available for describing rock discontinuities, a
new empirical relation is developed for estimating rock
mass strength based on RQD. The newly developed
empirical relation is applied to estimate the unconfined
compressive strength of rock masses at six sites and the
results are compared with those from the empirical meth-
ods based on rock mass classification. The estimated
unconfined compressive strength values from the new
empirical relation are essentially in the middle of the
estimated values from the different empirical methods
based on rock mass classification. Similar to the existing
empirical methods, the newly developed relation is only
approximate and should be used, with care, only for a first
estimate of the unconfined compressive strength of rock
masses. Recommendations are provided on how to apply
the newly developed relation in combination with the
existing empirical methods for estimating rock mass
strength in practice.
Keywords Rock mass strength
Rock mass classification RQD Empirical methods
1 Introduction
Reliable estimation of the strength and deformation prop-
erties of jointed rock masses is very important for safe and
economical design of civil structures such as houses, dams,
bridges, and tunnels founded on or in rock. As it is well
known, natural rock masses consist of intact rock blocks
separated by discontinuities such as joints, bedding planes,
folds, sheared zones, and faults. Because of the discontin-
uous nature of rock masses, it is important to choose the
right domain that is representative of the rock mass affected
by the structure analyzed (see Fig. 1). The behavior of the
rock mass is dependent on the relative scale between
the problem domain and the rock blocks formed by the
discontinuities. For example, when the structure being
analyzed is much larger than the rock blocks formed by the
discontinuities, the rock mass may be simply treated as an
equivalent continuum for the analysis (Brady and Brown
1985; Brown 1993; Hoek et al. 1995; Zhang 2005).
Treating the jointed rock mass as an equivalent continuum
(i.e., the equivalent continuum approach) has been widely
used in rock engineering. To apply the equivalent contin-
uum approach in analysis and design, the equivalent
strength and deformation properties need be determined.
Although the properties of the intact rock between the
discontinuities and the properties of the discontinuities
themselves can be determined in the laboratory, the direct
physical measurements of the properties of the jointed rock
mass are very expensive and time consuming, if not
impossible (Zhang and Einstein2004; Zhang2005; Edelbro
et al. 2006). Moreover, the interaction between the intact
L. Zhang (&)
Department of Civil Engineering and Engineering Mechanics,
University of Arizona, Tucson, AZ, USA
e-mail: [email protected]
1 3
Rock Mech Rock Eng
DOI 10.1007/s00603-009-0065-x
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rocks and the discontinuities is often complex and less well
understood than the behavior of the individual units,
making it difficult to predict the properties of the jointed
rock mass solely from the data on the intact rock and the
discontinuities. Researchers have extensively studied the
deformability of jointed rock masses and different empiri-
cal methods have been proposed for estimating the defor-
mation modulus of jointed rock masses, including Deere
et al. (1967), Coon and Merritt (1970), Bieniawski (1978),
Barton et al. (1980), Barton (1983), Serafim and Pereira
(1983), Hoek and Brown (1997), Zhang and Einstein
(2004), and Hoek and Diederichs (2006). For the strength
of jointed rock masses, however, further work is required
to develop more precise, practical, and easy-to-use methods
for determining the rock mass strength (Edelbro et al.
2006).
In this article, the existing empirical methods for esti-
mating the unconfined compressive strength of jointed rock
masses are first reviewed and evaluated in Sect.2. The
review shows that different empirical methods may provide
very different estimates. Since in many cases, rock quality
designation (RQD) is the only information available for
describing rock discontinuities, a new empirical relation is
developed for estimating rock mass strength based on RQD
in Sect. 3. Then in Sect. 4, the newly developed empirical
relation is applied to estimate the unconfined compressive
strength of rock masses at six sites and the results are
compared with those from the existing empirical methods.
A discussion and recommendations about applying the
newly developed relation in combination with existing
empirical methods in practice are provided in Sect.5.
Finally, the conclusions are presented in Sect. 6.
2 Existing Empirical Methods for Estimating
the Strength of Jointed Rock Masses
There are at present several types of empirical methods for
estimating the strength of jointed rock masses. The fol-
lowing provides a brief review and evaluation of some of
these methods.
2.1 Jointing Index Methods
Jointing index methods are based on an index defined as
the ratio of sample length to discontinuity spacing or
number of blocks contained in the sample. Several
researchers, including Protodyakonov and Koifman (1964),
Goldstein et al. (1966), Vardar (1977), and Aydan et al.
(1997), have proposed empirical relations between the
strength ratio (rcm/rc) and the jointing index (L/l) based on
experimental studies on jointed rock samples, where rcmand rc are the unconfined compressive strength, respec-
tively, of the rock mass and the intact rock, Lis the samplelength, and l is the discontinuity spacing. Since these
empirical relations are in similar format, the following only
describes the empirical relationship of Goldstein et al.
(1966).
Goldstein et al. (1966) conducted uniaxial compression
tests on composite specimens made from cubes of plaster
of Paris and suggested the following relationship based on
the test results:
rcm
rca 1a
L
l
e1
wherercm, rc, L, and l are as defined earlier; and a and eare constants with e\1. Figure2 shows the variation of
rcm/rcwithL/lbased on Eq.1for different values ofa and
e. As L/l increases (i.e., more discontinuities are included
in a rock mass sample of length L), the unconfined com-
pressive strength of the rock mass decreases. How fast
rcm/rc decreases with L/l depends on the magnitude of
constants a and e. The decrease ofrcm/rc with L/l will be
faster for smaller a or larger e. The values of a and e
depend on the strength and orientation of the discontinu-
ities (Aydan et al.1997; Jade and Sitharam2003). Specific
studies should be conducted to determine the values ofa
andebefore applying the relation 1to estimate the strength
of a specific jointed rock mass.
2.2 Joint Factor Methods
The joint factor methods relate the strength ratiorcm/rcto a
joint factor that is related to joint frequency, joint orienta-
tion, and joint strength (Arora 1987; Ramamurthy 1993;
Jade and Sitharam2003). Arora (1987) conducted tests on
intact and jointed specimens of plaster of Paris, sandstones,
Intact rock
One discontinuit set
Two discontinuity sets
Many discontinuities
Heavily jointed rock mass
Fig. 1 Simplified representation of the influence of scale on the type
of rock mass behavior (after Hoek et al. 1995)
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and granite in uniaxial and triaxial compression. The resultsindicate that the important factors that influence the strength
and deformation modulus values of jointed rock masses are
joint frequency, joint orientation, and joint strength. Based
on the results, he defined a joint factor Jf to consider the
combined effect of these three factors as
JfJn
nr 2
where Jn is the joint frequency (number of joints per
meter), which is simply obtained by dividing the number of
joints by the specimen length in meters;n is an inclination
parameter depending on the orientation of the joint, b (theangle between the loading direction and the joint plane);
and r is the joint strength parameter depending on the
joint condition. The value of n is obtained by taking the
ratio of log(strength reduction) at b = 90 to log(strength
reduction) at the desired value of b. The parameter n is
found to be essentially independent of joint frequency Jn.
The joint strength parameterris obtained from a shear test
along the joint and is given by
r sj
rnj3
where sj is the shear strength along the joint; and rnj the
normal stress on the joint. The variation ofnwithb and the
values ofrfor both intact (unfilled, fresh, and not weathered)
and gouge filled joints are provided by Ramamurthy (1993)
and Ramamurthy and Arora (1994) (see Tables 1,2,3).
Based on the results of uniaxial and triaxial tests of
intact and jointed specimens, Arora (1987) and Rama-
murthy (1993) proposed the following empirical relation
between unconfined compressive strength ratio rcm/rc and
joint factor Jf:
rcm
rcexp0:008Jf 4
Jade and Sitharam (2003) expanded the database used
by Arora (1987) and Ramamurthy (1993) and conducted
detailed statistical analyses of all the data. Based on the
statistical analysis, the best empirical relationship between
rcm/rc and Jfwas found as follows:
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1 3 5 7 9 11
cm
/c
L/l
a = 0.75
a = 0.25
e = 0.3
0.5
0.7
0.5
0.7
e = 0.3
Fig. 2 Variation of rcm/rc with L/l based on Eq.1 for different
values ofa and e Table 2 Suggested values of joint strength parameter rfor different
values ofrc (after Ramamurthy1993)
Unconfined compressive
strength of intact
rockrc (MPa)
Joint strength
parameterr
Remarks
2.5 0.30 Fine grained
micaceous to
coarse grained5.0 0.45
15.0 0.60
25.0 0.70
45.0 0.80
65.0 0.90
100.0 1.00
Table 3 Suggested values of joint strength parameter r for filled
joints (after Ramamurthy1993)
Gouge material Friction angle/j () Joint strength parameter r
Gravelly sand 45 1.00
Coarse sand 40 0.84
Fine sand 35 0.70
Silty sand 32 0.62
Clayey sand 30 0.58
Clayey silt
Clay25% 25 0.47
Clay50% 15 0.27
Clay75% 10 0.18
Table 1 Variation of inclina-
tion parameter n with joint
orientationb(after Ramamurthy
1993)
Joint
orientationb
()
Inclination
parametern
0 0.82
10 0.46
20 0.11
30 0.05
40 0.09
50 0.30
60 0.46
70 0.64
80 0.82
90 0.95
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rcm
rcab exp
Jfc
5
where a, b, and c are constants equal to 0.039, 0.893, and
160.99, respectively, for the database analyzed (see Fig. 3).
It can be seen that Eq.4 is a special form of Eq. 5 with
a = 0, b = 1, andc = 125. It is worth noting that there is a
large scatter for the test data and it is very possible that an
estimation value from Eq.5 is more than two times or less
than half of the measured value ofrcm.
2.3 Methods Based on Rock Mass Classification
Methods based on rock mass classification are the most
widely used empirical methods for estimating rock mass
strength. Over the years, many rock mass classification
systems have been proposed and used in engineering
practice, including the RQD (Deere 1967), the rock mass
rating (RMR) (Bieniawski 1976, 1989), tunneling quality
index (Q) (Barton et al. 1974; Barton 2002), geological
strength index (GSI) (Hoek et al. 1995, 1998), and rock
mass index (RMi) (Palmstrom1996a,b). Some systems are
developed by modification of existing ones to suit specific
applications. For example, the mining rock mass rating
(MRMR) system was developed by modifying the RMR
system for mining applications (Laubscher1990) and the
rock mass number (N) system is a modified Q-system (Goel
et al. 1995). A review of the different rock mass classifi-
cation systems can be found in Edelbro (2003). Table4
lists the parameters considered in different classification
systems.
Rock mass classification systems have been used to
estimate the strength of jointed rock masses by different
0
Best fitting curve:
+=
99.160exp893.0039.0 f
c
cm J
200 400 600 8000
0.1
0.2
0.9
0.8
0.7
0.6
0.4
0.5
1.0
0.3
Joint factor Jf
Unconfinedcompressive
strengthratiocm
/c
Fig. 3 Unconfined compressive test data and fitted relation between
rcm/rc and Jf (from Jade and Sitharam 2003)
Table4
Parametersconsideredin
differentclassificationsystems(afterEdelbro2003)
Classificationsystem
RMR
MRMR
RM
S
Q
N
RMi
GSI
Parameters
UCS
UCS
UC
S
JointsetnumberJn
JointsetnumberJn
UCS
Surfacecondition
RQD
RQD
RQ
D
RQD
RQD
BlockvolumeVb
Structure/interlocking
ofrockblocks
Jointspacing
Jointspacing
Jointspacing
JointroughnessJr
JointroughnessJr
JointroughnessjR
Jointcondition
Jointcondition
Jointcondition
JointalternationJa
JointalternationJa
JointalternationjA
Groundwate
rcondition
Groundwatercondition
Groundwatercondition
Jointwaterreduction
factorJw
Jointwaterreduction
factorJw
Jointsizeand
terminationjL
Stressreduction
factorSRF
Adjustment
parameters
Jointorientation
Jointorientation,
blastingandweathering
Jointsets
UCSunconfinedcompressivestrengthofintactrockmateria,RQDrockqualitydesignation
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researchers (e.g., Yudhbir and Prinzl 1983; Laubscher
1984; Ramamurthy et al.1985; Ramamurthy1996; Kulhawy
and Goodman 1987; Trueman 1988; Kalamaras and
Bieniawski 1993; AASHTO 1996; Bhasin and Grimstad
1996; Sheorey1997; Singh et al. 1997; Aydan and Dalgic
1998; Singh and Goel 1999; Asef et al. 2000; Hoek 2004,
personal communication; Edelbro et al. 2006). Kulhawy
and Goodman (1987) suggested that, as a first approxi-
mation, the unconfined compressive strength rcm of rock
masses be taken as 0.33rc when RQD is less than about70% and then linearly increasing to 0.8rc when RQD
increases from 70 to 100% (see Fig. 4), where rc is the
unconfined compressive strength of the intact rock. The
Standard Specifications for Highway Bridges adopted by
the American Association of State Highway and Trans-
portation Officials (AASHTO 1996) suggest that rcm be
estimated using the following expression
rcm aErc 6a
aE 0:0231RQD1:32 0:15 6b
in which RQD is expressed as a percent. The variation of
the unconfined compressive strength ratio rcm/rc with
RQD based on Eq. 6a,6bis also shown in Fig. 4. It can be
seen that the general trend of these two relations between
rcm/rcand RQD is about the same:rcm/rcis constant when
RQD is smaller than a certain value and then linearly
increases when RQD increases. Obviously, it is inappro-
priate to assume that rcm/rc is constant when RQD varies
from 0 to a certain value (70% for the relation of Kulhawy
and Goodman and 64% for the relation of AASHTO). For
example, for a very poor quality rock mass (RQD\ 25%)
and a fair quality rock mass (RQD = 5075%), different
rcm/rc values should be expected.
While the basis for the suggestion by Kulhawy and
Goodman (1987) is not clear, the reduction factor aE (note
the subscript E) in Eq.6a, 6b is the reduction factor
originally proposed by Gardner (1987) for estimating the
rock mass deformation modulus Em from the intact rock
deformation modulusEr:
Em aEEr: 7Gardner (1987) derived the reduction factoraEbased on
the Em/Er versus RQD data of Coon and Merritt (1970),
which are shown in Fig.5. It can be seen that the data for
RQD\ 64% is very limited, which is probably the reason
whyaE was assumed to be constant for RQD\ 64%.
Table5 lists the empirical relations based on the three
widely used rock mass classification systems, RMR,Q, and
GSI, for estimating the unconfined compressive strength
rcmof jointed rock masses. It should be noted that when a
rock mass classification system is used for estimating rock
mass strength (and deformation properties), only the
inherent parameters of intact rock and discontinuities need
be considered for evaluation of the classification index.
Other parameters such as groundwater and in situ stress
should not be considered to modify the classification index
because they are considered in the analysis of rock struc-
tures. For example, when RMR is used for rock mass
strength estimation, the rock mass should be assumed
completely dry and a very favorable discontinuity orien-
tation should be assumed (Hoek et al. 1995, 2002).
Depending on the engineering problem analyzed, pore
Unconfinedcompressivestrengthratiocm
/c
RQD (%)
Kulhawy and Goodman (1987)
AASHTO (1996)
Fig. 4 Variation of unconfined compressive strength ratio rcm/rcwith RQD suggested respectively by Kulhawy and Goodman (1987)
and AASHTO (1996)
0
0.2
0.4
0.6
0.8
1
0 20 40 60 80 100
Em
/Er
RQD (%)
Em/Er = 0.0231RQD-1.32
Dworshak Dam, Granite Gneiss, Su rface Gages
Dworshak Dam, Granite Gneiss, Buried Gages
Two Forks Dam, Gneiss
Yellowtail Dam, Limestone
Glen Canyon Dam, Sandstone
Em/Er = 0.15
Fig. 5 Data of deformation modulus ratio Em/Er versus RQD (after
Coon and Merritt 1970)
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pressures and discontinuity orientation can be considered
together with strength as input parameters of the analysis.
Excluding these parameters from classification index
evaluation will ensure no double accounting for a
parameter.
Figure6 shows a comparison of some of the empirical
relations with the in situ test data from Aydan and Dalgic
(1998), Palmstrom (1995) and Cai et al. (2004). It can be
seen that (1) there is a large scatter for the in situ test data,
reflecting the difficulty to conduct accurate measurements
of in situ rock mass strength; (2) different empirical rela-
tions may provide very different estimation values; and (3)
the average trend of the different empirical relations are in
good agreement with the measuredrcm.
2.4 Discussion
Both the jointing index methods and the joint factor
methods are developed based on laboratory test data of
intact and jointed specimens. These methods consider the
effect of joint frequency, joint orientation, and joint
strength on the strength of jointed rock masses: The
jointing index methods use L/l for joint frequency and
factors a and e reflect the effect of joint orientation and
strength; while the joint factor methods combine the effect
of joint frequency, orientation, and strength in a single
factor Jf. Application of the jointing index methods and
the joint factor methods to estimate the strength of field
jointed rock masses require extensive work to obtain the
information on joint frequency, joint orientation, and joint
strength.
The empirical methods based on rock mass classification
treat the rock mass as an equivalent continuum and may or
may not consider the effect of joint orientations. It need be
noted that some of the empirical relations based on rock
mass classification are simply derived from their corre-
sponding strength criteria. For example, the empirical
relation of Hoek et al. (2002) (Eq. 13 in Table5) can be
derived from the empirical HoekBrown strength criterion:
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0 20 40 60 80 100
RMR or GSI
Unconfinedcompressiv
estrengthratiocm
/c
In situ test data (Aydan & Dalgic, 1998)
cm/c= exp(7.65((RMR-100)/100))(Yudhbir & Prinzl, 1983)
cm/c= exp((RMR-100)/24)(Kalamaras & Bieniawski, 1993)
cm/c= exp((RMR-100)/18)
(Hoek et al., 2004)cm/c= exp((RMR-100)/20)(Sheorey, 1997)
cm/c= RMR/(RMR+6(100-RMR))(Aydan & Dalgic, 1998)
cm/c= 0.036exp(GSI/30)(Hoek, 2004
1
1
2
2
3
3
4
4
5
5
6
6
In situ test data (Palmstrom, 1995)
In situ test data (Cai et al., 2004)
Fig. 6 Variation of unconfined compressive strength ratio rcm /rcwith RMR or GSI
Table 5 Empirical relations based on rock mass classification for estimating unconfined compressive strength rcm of rock masses (modified
from Zhang2005)
Authors Relation Equation #
Yudhbir and Prinzl (1983) rcmrc
e7:65RMR100
100 (8)
Laubscher (1984) and Singh and Goel (1999) rcmrc
RMR Rating forrc106
(9)
Ramamurthy et al. (1985) and Ramamurthy (1996) rcmrc
eRMR100
18:75 (10)
Trueman (1988) and Asef et al. (2000) rcm 0:5e0:06RMR
(MPa) (11)Kalamaras and Bieniawski (1993) rcm
rce
RMR10024 (12)
Hoek et al. (2002) rcmrc
eGSI100
93D121
6 e
GSI15e
203
(13)
Bhasin and Grimstad (1996) and Singh and Goel (1999) rcm 7cfcQ1=3 (MPa) where fc = rc/100 forQ[ 10
andrc[ 100 MPa, otherwise fc = 1; and c is the unit
weight of the rock mass in g/cm3.
(14)
Sheorey (1997) rcmrc
eRMR100
20 (15)
Aydan and Dalgic (1998) rcmrc
RMRRMR6100RMR (16)
Barton (2002) rcm 5cQrc=1001=3
(MPa) where c is the unit weight
of the rock mass in g/cm3.
(17)
Hoek (2004, personal communication) rcmrc
0:036eGSI30 (18)
Singh et al. (1997) rcm 7cQ1=3 (MPa) wherec is the unit weight of the rock mass in g/cm3. (19)
rcunconfined compressive strength of intact rock materia, RMR rock mass rating,GSIgeological strength index, Q tunneling quality index, andD factor indicating the degree of disturbance due to blast damage and stress relaxation
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r01 r03 rc mb
r03rc
s
a20
where rc is the unconfined compressive strength of the
intact rock; r01 and r03, respectively, the major and minor
effective principal stresses; and mb, and s and a the
constants that depend on the characteristics of the rock
mass and can be estimated from GSI as follows (Hoek et al.2002):
mb exp GSI100
2814D
mi 21
s exp GSI100
93D
22
a1
2
1
6expGSI=15 exp20=3 23
where mi is a material constant for the intact rock and
depends on the rock type (texture and mineralogy); andD a
factor that depends on the degree of disturbance due toblast damage and stress relaxation. Values ofD range from
0 for undisturbed in situ rock masses to 1 for very disturbed
rock masses.
From Eq.20, the unconfined compressive strength of
the rock mass can be derived as
rcm sarc: 24
Substitution of s and a in Eq.24, respectively, with
Eqs.22 and 23will result in Eq. 13 in Table 5.
As shown in Table 4, many factors need be considered
for evaluating the classification indices. In many cases,
however, the available information may not be sufficientfor evaluating the classification index. For example, in
routine subsurface investigations, it is often that the only
information available about discontinuities is RQD.
Therefore, it is practically important to develop an empir-
ical method based on RQD for estimating the strength of
rock masses.
3 New Relation Between Unconfined Compressive
Strength and RQD
As seen in Sect. 2, different empirical relations may pro-vide very different estimation values of the unconfined
compressive strength of jointed rock masses. It is also
known that, in many cases, RQD may be the only infor-
mation available about discontinuities. So, a new empirical
relation between the unconfined compressive strength and
RQD will be derived here.
Zhang and Einstein (2004) expanded the database
shown in Fig. 5 by collecting the data from the published
literature (see Fig. 7). The expanded database covers the
entire range 0 B RQD B 100% and shows a nonlinear
variation ofEm/Er with RQD. The rocks for the expanded
database include mudstone, siltstone, sandstone, shale,
dolerite, granite, limestone, greywacke, gneiss, and granite
gneiss. Again, one can see the large scatter of the data,
especially when RQD[ 65%. Zhang and Einstein (2004)
discussed the possible causes for the large scatter, includ-
ing test methods, directional effect, discontinuity condi-
tions, and insensitivity of RQD to discontinuity frequency
(or spacing). Using the expanded database, Zhang and
Einstein (2004) derived the following RQD - Em/Er rela-
tion for the average trend (RQD in %):
aE Em=Er100:0186RQD1:91: 25
The average RQD - Em/Er relation (Eq.25) gives
aE = 0.95 at RQD = 100%, which makes sense because
there may be discontinuities in rock masses at RQD =
100% and thus Em may be smaller than Er even when
RQD = 100%.
Researchers in rock mechanics and rock engineering
have studied the relation between the unconfined com-
pressive strength ratiorcm/rcand the deformation modulus
ratio Em/Er and found that they can be related approxi-mately by the following equation (Ramamurthy 1993;
Singh et al. 1998; Singh and Rao2005):
rcm
rc
Em
Er
q aE
q 26
in which the power q varies from 0.5 to 1.0 and is most
likely in the range of 0.61 to 0.74 with an average of 0.7. It
can be seen that the AASHTO method (Eq. 6a,6b) uses the
upper bound value ofq = 1.0.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 50 60 70 80 90 100
Em
/Er
RQD (%)
Coon and Merritt (1970)
Em/Er = 100.0186RQD-1.91
r2 = 0.76
Ebisu et al. (1992)
Bieniawski (1978)
Fig. 7 Expanded data and derived new relation between deformation
modulus ratio Em/Er and RQD (after Zhang and Einstein2004)
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It needs to be noted that the relation betweenrcm/rcand
Em/Er(Eq.26) was derived based only on triaxial test data
on jointed rock mass specimens with different joint fre-
quencies, orientations, and conditions (Ramamurthy1993;
Singh et al. 1998; Singh and Rao 2005) and has not been
tested against field cases. The power q in Eq.26 may vary
significantly for different rock types and discontinuity
conditions. Nevertheless, using the average value ofq = 0.7, the unconfined compressive strength of rock mass
can be related to the unconfined compressive strength of
intact rock approximately by
rcm
rc aE
0:7: 27
Combining Eqs.25 and 27, the following empirical
relation can be derived for estimating the unconfined
compressive strength of rock masses from RQD:
rcm=rc 100:013RQD1:34: 28
Due to the reasons stated above, using the rcm/rcversusEm/Errelationship of Eq.26 may or may not be appropriate
for deriving thercm/rcversus RQD relation. It is taken as a
first step and applying the derived rcm/rc versus RQD
relation to 15 cases in Sect.4 will indicate to what extent it
can be practically used.
Figure8 shows the comparison of the newly developed
empirical relation28with the suggestions respectively by
Kulhawy and Goodman (1987) and AASHTO (1996). The
newly developed rcm/rc versus RQD relation covers
the entire range 0 B RQD B 100% continuously. For
RQD[ 70%, the new rcm/rc versus RQD relation is in
good agreement with the suggestions of Kulhawy and
Goodman (1987) and AASHTO (1996). For RQD\ 70%,
however, the new rcm/rc versus RQD relation is different
from the suggestions of Kulhawy and Goodman (1987) and
AASHTO (1996), with the new rcm/rc versus RQD rela-
tion considering the continuous variation of rcm/rc with
RQD while the suggestions of Kulhawy and Goodman
(1987) and AASHTO (1996) assume constant rcm/rc
values.
4 Applications
In this section, the newly developed rcm/rc versus RQD
relation is used to estimate the unconfined compressive
strength of rock masses at six sites with detailed geotech-
nical information available: the Sulakyurt dam site in
central Turkey (Ozsan et al.2007), the Tannur Dam site in
south Jordan (El-Naqa and Kuisi2002), the Urus Dam site
also in central Turkey (Ozsan and Akin2002), a high tower
site at Tenerife Island (Justo et al. 2006), an open pit minesite in the vicinity of Berlin, Germany (Alber and Heiland
2001), and a site with jointed basaltic rocks on the
Columbia Plateau in Washington State (Schultz1996). The
results are compared with those from the empirical meth-
ods based on rock mass classification to indirectly check
the accuracy of the developedrcm/rcversus RQD relation.
In other words, the rcm is first obtained with the RQD
based relation and then compared to the rcmobtained with
the related rock mass classifications. Table6 lists the
properties of rocks at the six sites. As can be seen in
Table6, the cases cover a reasonable but clearly limited
range of rock types.
According to Ozsan et al. (2007), the site consists of
moderately to highly weathered granite and diorite of
Paleocene age. Detailed site investigation was carried out,
including field observations, discontinuity surveying, core
drilling, laboratory tests, and rock mass classification. The
unconfined compressive strength and the RQD, RMR, Q,
and GSI values for both granite and diorite were obtained
as shown in Table6. Using the developed relation
between rcm/rc and RQD (Eq.28), the unconfined com-
pressive strength of the granite and diorite are estimated
respectively as 4.36 and 2.87 MPa as shown in Table 7.
Using the empirical methods based on rock mass classi-
fication listed in Table5, the unconfined compressive
strength of the granite and diorite can also be estimated as
shown in Table7. The estimated rock mass strength
values from the different empirical methods based on rock
mass classification cover a large range: from 0.22 to
8.14 MPa for granite and from 0.14 to 6.91 MPa for
diorite, respectively. For the other five sites, the rock
mass unconfined compressive strength can also be esti-
mated using the rcm/rc versus RQD relation (Eq. 28) and
Unconfinedcom
pressivestrengthratiocm
/c
RQD (%)
Kulhawy and Goodman (1987)
AASHTO (1996)
Developed:34.1RQD013.0
ccm 10/ =
Fig. 8 Comparison of the developed rcm/rc versus RQD relation
with suggestions respectively by Kulhawy and Goodman (1987) and
AASHTO (1996)
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some of the empirical methods based on rock mass
classification listed in Table5. These results are also
presented in Table7. The estimated values from the
developed rcm/rc versus RQD relation are within the
range of the estimated values from the different empirical
methods based on rock mass classification, except for
andesite (case #8) whose estimated value from the
developed rcm/rc versus RQD relation is outside the
range but very close the highest of the estimated values
from the different empirical methods based on rock mass
classification. So, the developed rcm/rc versus RQD
relation can estimate rock mass strength values that are in
Table 6 Summary of rock properties at six sites (after Ozsan et al.2007; El-Naqa and Kuisi2002; Ozsan and Akin2002; Justo et al.2006; Alber
and Heiland2001; Schultz1996)
# Rock rc (MPa) RQD (%) RMR Q GSI References
1 Granite 74.0 017 (8.5) 2128 (24) 0.040.13 (0.08) 1624 (19) Ozsan et al. (2007)
2 Diorite 60.0 12 (1.5) 1723 (21) 0.0250.1 (0.05) 1218 (16)
3 Limestone (L1) 31.0 54 57 4.23 52 El-Naqa and Kuisi (2002)
4 Limestone (L2) 13.0 50 59 5.29 545 Limestone (R1) 37.0 48 59 5.29 54
6 Limestone (R2) 27.0 45 54 3.04 59
7 Marly Limestone 28.0 44 55 3.39 50
8 Andesite 93.0 41 34 0.56 41 Ozsan and Akin (2002)
9 Basalt 142.0 15 38 0.63 42.5
10 Tuff 24.0 10 21 0.11 31
11 Basalt (d1) 69.0 77 59 6.6 52 Justo et al. (2006)
12 Basalt (d2) 15.0 42.5 38 3.4 39
13 Basalt (d3) 13.0 0 25 0 28
14 Limestone 40.0 50 58 53 Alber and Heiland (2001)
15 Basalt 66.0 60 76 71 Schultz (1996)
Values in the parentheses are the average
Table 7 Estimated rock mass strength (rcm) values for the rocks listed in Table 6 using the developed empirical relation (Eq. 28) and the
empirical methods based rock mass classification (Eqs. 819)
Eq. # rcm (MPa)
1a 2a 3a 4a 5a 6a 7a 8a 9a 10a 11a 12a 13a 14a 15a
(28) 4.36 2.87 7.13 2.65 7.12 4.75 4.78 14.5 10.2 1.48 31.6 2.45 0.59 8.17 18.2
(8)b 0.22 0.14 1.16 0.56 1.61 0.80 0.90 0.60 1.24 0.06 3.00 0.13 0.04 1.61 10.5
(10)b 1.28 0.89 3.13 1.46 4.15 2.32 2.54 2.75 5.20 0.36 7.75 0.55 0.24 4.26 18.4
(11)b 2.11 1.76 15.3 17.2 17.2 12.8 13.6 3.85 4.89 1.76 17.2 4.89 2.24 16.2 47.8
(12)b 3.12 223 5.17 2.36 6.70 3.97 4.29 5.95 10.7 0.89 12.5 1.13 0.57 6.95 24.3
(13)b 0.54 0.33 2.10 0.99 2.81 1.53 1.68 3.27 5.47 0.44 4.66 0.47 0.19 2.86 13.2
(14)b 6.03 4.15 8.42 3.81 10.8 6.57 7.07 13.0 20.9 1.63 25.3 3.67
(15)b 1.66 1.16 3.61 1.67 4.76 2.71 2.95 3.43 6.40 0.46 8.88 0.68 0.31 4.90 19.9
(16)b 3.70 2.55 5.61 2.51 7.16 4.42 4.74 7.35 13.2 1.02 13.4 1.39 0.68 7.48 22.8
(17)b 5.26 4.16 13.1 10.6 15.0 11.2 11.8 9.72 11.8 3.02 23.2 9.28
(18)b 5.05 3.68 6.32 2.83 8.06 4.98 5.34 13.1 21.1 2.43 14.1 1.98 1.19 8.43 25.3
(19)b 8.14 6.91 27.2 29.3 29.3 24.3 25.2 13.9 14.7 6.80 36.7 24.5
Rangec 0.22
8.14
0.14
6.91
1.16
27.2
0.56
29.3
1.61
29.3
0.80
24.3
0.90
25.2
0.60
13.9
1.24
21.1
0.06
6.80
3.00
36.7
0.13
24.5
0.04
2.24
1.61
16.2
10.5
47.8
a The numbers refer to the case numbers shown in Table 6
b See Table5 for the specific equationsc The range are for the empirical methods based rock mass classification (Eqs. 819)
Estimating the Strength of Jointed Rock Masses
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reasonable agreement with those from the empirical
methods based on rock mass classification.
Figure9 summarizes the results for all 15 cases at the
six sites using the developed rcm/rc versus RQD relation
and the empirical methods based on rock mass classifica-
tion. It can be seen clearly that the estimated values from
the developedrcm/rcversus RQD relation are essentially in
the middle of the estimated values from the different
empirical methods. The relations of Singh et al. (1997),
Trueman (1988), and Asef et al. (2000) tend to estimate
high rcm values (upper bound), whereas the relation pro-
posed by Yudhbir and Prinzl (1983) estimates low rcmvalues (lower bound). Some relations, such as those pro-
posed by Kalamaras and Bieniawski (1993), Sheorey
(1997), and Aydan and Dalgic (1998) tend to give average
(medium)rcm values.
5 Discussion and Recommendations
Determination of the strength of jointed rock masses is an
important and challenging task in rock mechanics and rock
engineering. The newly developed rcm/rc versus RQD
relation provides a convenient way for estimating theunconfined compressive strength of rock masses because,
in many cases, RQD is the only available information
about discontinuities in routine site investigations. How-
ever, care should be taken when applying the developed
empirical relation for determining the unconfined com-
pressive strength of jointed rock masses, because of the
following reasons:
The relation between the unconfined compressive
strength ratio rcm/rc and the deformation modulus
ratio Em/Er (Eq.26) is based on limited laboratory test
data and has not been tested against field cases. For
different rock types and discontinuity conditions, the
powerqin Eq.26 may vary significantly from the value
of 0.7 used in the derivation.
The reduction factor aE is based on the Em/Er versus
RQD data shown in Fig. 7, which have a large scatter,
especially when RQD[ 65%. It is expected that thercm/rc versus RQD data should also have a large
scatter.
RQD is only one of the many factors that affect the
strength of jointed rock masses. Other factors such as
the discontinuity surface conditions can have a great
effect on the strength of jointed rock masses.
To apply the developed rcm/rc versus RQD relation for
estimation of rock mass strength, the following guidance
should be followed:
1. When RQD is the only information available about
rock discontinuities, the rcm/rc versus RQD relationcan be used to estimate the rock mass strength but care
should be taken when applying the estimated values.
The rcm/rc versus RQD relation should be used only
for a first estimation.
2. When RQD and other information are available for
evaluating the rock mass classification indices, the
rcm/rc versus RQD relation should be used together
with the empirical methods based on rock mass
classification to evaluate the rock mass strength. The
estimated value from thercm/rc versus RQD relation
can be compared with the range of the estimated
values from the empirical methods based on rock mass
classification to get an idea on the effect of RQD on
rock mass strength.
6 Conclusions
Different empirical methods are available for estimating
the strength of jointed rock masses. The empirical methods
may provide very different estimation values of the
unconfined compressive strength of jointed rock masses.
The newly developed rcm/rc versus RQD relation in this
article provides a convenient way for estimating the
unconfined compressive strength of jointed rock masses
because, in many cases, RQD is the only available infor-
mation about rock discontinuities. The developed rcm/rcversus RQD relation can provide estimated rock mass
strength values that are often in reasonable agreement with
those from the empirical methods based on rock mass
classification. To apply the developed rcm/rc versus RQD
relation for estimation of rock mass strength in practice, the
0.0
10.0
20.0
30.0
40.0
50.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Yudhbir & Prinzl (1983)Ramamurthy et al. (985) & Ramamurthy (1986, 1993)Trueman (1998) & Asef et al. (2000)Kalamaras & Bieniawski (1993)Hoek et al. ( 2002)Bhasin & Grimstad (1996) and Singh & Goel (1999)Sheorey (1997)Aydan & Dalgic (1998)Barton (2002)Hoek (2004)Singh et al. (1997)
Estimatedcm
(MPa)
Case No.
Developed cmvs. RQD
Fig. 9 Estimated rock mass strength values from the existing
empirical methods and the developed rcm/rc versus RQD relation
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limitations need to be considered and the recommendations
in Sect. 5 should be followed.
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