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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)

    L. Zhang

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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

    Estimating the Strength of Jointed Rock Masses

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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

    L. Zhang

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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)

    Estimating the Strength of Jointed Rock Masses

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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

    L. Zhang

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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)

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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

    L. Zhang

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    limitations need to be considered and the recommendations

    in Sect. 5 should be followed.

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