Rock Mech. Rock Engng. (1995) 28 (3), 135-144

Rock Mechanics and Rock Engineering

9 Springer-Verlag 1995 Printed in Austria

Elastic Moduli for Fractured Rock Mass By

T. H. Huang 1, C. S. Chang 2, and Z. Y. Yang 3 1 Department of Civil Engineering, National Taiwan University, Taipei, Taiwan 2 Department of Civil Engineering, University of Massachusetts, Amherst, U.S.A. 3 Department of Civil Engineering, Tamkang University, Taipei Hsien, Taiwan

Summary The presence of joint discontinuities has long been recognized as an important factor influencing the mechanical behavior of rock masses. This paper proposes a stress-strain model for an assemblage of intact rock blocks separated by joint planes. The stress-strain relationship accounts for spacings and orientations of the joint sets. Closed-form expressions of elastic moduli for a rock mass with three intersecting sets of joints are derived explicitly in terms of properties of joints and intact rock. Applicability of the derived expressions are evaluated by comparing the predicted results with experimental results from physical model tests.

1. Introduction In the analysis of engineering problems dealing with rock, it is essential to model the discontinuities in the rock mass. Modelling of joints as interface elements in finite element analyses can be found in the work by G o o d m a n (1976) and G o o d m a n et al. (1968). Modelling of discrete block systems can be found in the work by Hart and Cundall (1988) and Shi (1988). For highly fractured jointed rock masses, where the joint spacing is small compared to the scale of the structure being analyzed, it is more convenient to treat the jointed rock mass as an "equivalent" continuum material. Along this line, work can be found by many investigators, e.g., Stephansson (1981), Gerrard (1982 a,b), Yoshinaka and Yamabe (1986), Amadei and G o o d m a n (1981), Morland (1974), Chen (1989, 1990), Zienkiewicz and Pande (1977), Chang and Huang (1989), etc. However, closed-form expressions for elastic moduli are available only for rock masses with three sets of orthogonal intersecting joints. This paper intends to derive closedform expressions of elastic moduli for rock masses with three sets of nonorthogonal intersecting joints. The derivation is based on the work by Chang and Huang (1989), which modelled discontinuities in the jointed rock mass using a slip concept (Chang et al. 1989 a,b). Applicability of the derived formulations of

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elastic moduli are evaluated using experimental results measured from physical model tests.

2. Stress-Strain Modelling for Rock Masses This section briefly describes the stress-strain model (Chang and Huang, 1989) for a rock mass consisting of M sets of joints. Subjected to a small increment of stress Acrkl, the rock mass has a corresponding incremental strain Aeij. The incremental strain consists of two components: one induced from the movements of the M sets of joints, Ae/~, and the other from intact rock deformation, Ae~j. Thus

The strain of the intact rock can be expressed as ae/5 = C,.Sk,Acrk,,

(2)

where Aei5 is the strain of intact rock due to a change of stresses A~Ykl. C~kl is the flexibility tensor. The strain from joint movements can be expressed as a function of stress Ae~- = Ciy~lA~rkl, s

(3)

where C~kl is the flexibility tensor of the joint sets. Combining Eqs. (1), (2), and (3), the overall constitutive tensor of the rock mass becomes

aeij = Ci;e,a~k,; Cqk, = C~k, + C~k,. (4) y To derive the flexibility tensor of joint sets, Cijkl , the following relationships are considered:

1. Behavior of a single joint The general constitutive relation for a joint can be expressed as

A61 = DIjA'I-j;

(5)

or

[ A6t J

Dtn

Dts

D~t

A%

Du

A%

(6)

where the subscript 'n' represents the direction normal (i.e. perpendicular to the joint plane; 's' and 't' represent two orthogonal directions on the joint plane. Thus AT~ is the normal stress; A% and A N are the shear stresses; A6~ is the closure displacement of the joint; and A~s and A6t are the shear displacements. The tensor Dxy (I, J = n, s, t) represents nine components of the joint flexibility matrix.

2. Stress in the rock mass and traction on a joint plane Considering that an element of rock mass consists of a number of joints in various

Elastic Moduli for Fractured Rock Mass

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orientations, the traction on the n-th joint plane can be obtained directly from the stress in the rock mass based on Cauchy's stress formula:

= A

S,

(7)

where n~ is the unit vector outwardly perpendicular to the joint plane.

3. Strain of rock mass and joint movements Based on the principle of energy conservation in a representative rock mass with volume V, the work done by stress is equal to the summation of work done by traction at each joint due to joint movements. Thus

~ijAeij:~)__~

6~A,

(8)

n

where A n is the total area of the n-th set of parallel joint planes in the representative rock mass which can be estimated from the field geological report. The area for this set of joints is equal to the unit volume divided by the average spacing between joints, S n, i.e., A n = (V/S~). Substituting Eq. (7) into Eq. (8), the strain of rock mass induced from joint movements is given by: M

n=l

4. Local and global coordinate transformation Traction and displacement of a joint are vectors which can be transformed between local and global coordinate systems, for example, A~y = LjjA~j;

(10)

/',~-f = L~iA~-i. The transformation tensor Ljj is composed of direction cosines between the local systems n, s, t and the global system i, j, k. Using Eqs. (9), (5) and (7), the following constitutive tensor can be obtained m

n

n

n

t7

n

1

C~k, = ~-'~ ni L}jDjLLLlnl~ ~ .

(11)

n=l

3. Joint Properties The following two assumptions are generally made for the behavior of a single joint: 1. Deformation behavior is the same in all directions on the joint plane. Thus

Dss = Dtt; and Dsn = Dtn, D~f = Dns, Dst =Dts; and 2. The coupling effect of shear displacement caused by the normal stress is neglected, thus D~,, = 0. However, it is noted that these assumptions may not be true in some cases as experimentally observed by Huang and Doong (1990) and Jing et al. (1992).

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With these two assumptions, the nine coefficients in the joint flexibility matrix (see Eq. (6)) can be simplified to three independent coefficients. They represent the following three different types of behavior: 1. Closure Behavior: Dnn relates the stress and the displacement normal to a joint plane. 2. Shear Behavior: D~s relates shear stress and shear displacement. 3. Dilatancy Behavior: Dns relates normal displacement and shear stress. The joint flexibilities, D,~n,Dss, are the inverse of the conventional joint stiffnesses, K~, Ks, measured in shear tests. For convenience, a dilatancy factor, A is defined as: A6n = XASs.

(12)

Therefore D,~s = D~A. The joint stiffnesses and dilatancy are influenced by many factors, such as: surface properties, joint type, stress applied and shearing displacement (Ladanyi and Archambault, 1970; Barton, 1974; Bandis et al., 1983). The joint stiffnesses and dilatancy may be treated as constants in a small increment of load. For a large increment of load, constant joint stiffness and dilatancy can only be used as approximate estimation.

4. Elastic Moduli of Rock Mass with Intersecting Joint Sets

Based on the formulation described in previous sections, the constitutive constants

J

/

J

/

/

/ /

/

J J

J

J J

J

J

j

J J

j J

z

J

J

>y X Fig. 1. Rock mass with three intersecting sets of joints

Elastic Moduli for Fractured Rock Mass

139

are now explicitly derived for a rock mass with three intersecting sets of joints as shown in Fig. 1. The angle between first two sets of joints is 0. The first two sets of joints are assumed to have the same joint stiffnesses, Kn, K~,, which are different from the joint stiffnesses, K,,3, K~3, of the third set of joints. The spacings are same for the first two sets of joints denoted by S; the spacings are denoted by S3 for the third set of joints. We first derive the closed-form expressions of moduli based on Eq. (11), neglecting the effect of joint dilatancy. The stress-strain relationship in Eq. (4) for rock masses with three intersecting sets of joints is given by

ex yj

/kGxx /kCzzI A%y A%,z A%x r

1 1 Eo+Ex uo Vxy

Vo Eo 1

uyx Ey 1

Vo Eo Vo

~x Ez Uyz

Eo

G

Eo E,,

Eo

Ez

vo Eo

u~; Ex

vo Eo

1 1 E o 'E~

t

Vzy Ey

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Go

~

1

Gxy 1

GO

+

1

0

Gyz 1

mO-yy a z. I m~

0

1

Go +---.G~x

(13)

Ary.

/~"F'X.T

where the properties of intact rock are denoted by: Young's modulus Eo, shear modulus Go, and Poisson's ratio u0; the properties of joints are denoted by: joint Young's moduli Ex, Ey, E~ and joint shear moduli Gxy, Gy> Gzx, which are functions of the stiffness, orientations and spacings of the joint sets, given as follows: 1

1

Ex-- Kn3S3,

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2sin 2 0 2 ['c~

1

2cos2(O/2)(sin2(O/2)Kn +cos2(O/2)Ks~

gzz =

\

1

1

_

7v, 7 s Ks S

(14)

'

2 sin 20(K~ + Ks)

Gyz

KnKsS

'

1

1

2 sin 0 sin 2(0/2)

G~x

Ks3S ~

KsS

~'yz __ ~zy

g

l"xy

)'

2 sin 0 cos 2(0/2)

Gxy Ks3S3-~ 1

+ sin2(O/2)K]~

Yyx

Ex - - uy

-

K~, - K s

'

sin2 o,

--07

Uzx _ Uxz _ O.

E~

Ex

When the angle between the two cross sets of joints 0 = 90 ~ Eq. (14) reduces to the expression proposed by Amadei and Goodman (1981) for orthogonal joint sets. Eq. (14) shows that, for non-orthogonal joint sets, the orientations of joints have a significant effect on the moduli of the rock mass. A rock mass with isotropic intact rock material becomes highly anisotropic with the presence of joints. It is noted that Eq. (13) is limited to conditions of zero joint dilation. When joint dilatancy is considered, the joint flexibility tensor DIj in Eq. (11) is no longer symmetric. Consequently, the symmetric condition of Cijkl does not hold and the derived stress-strain matrix does not resemble orthotropic or any other type of conventional elastic continuum. Complete expressions of the stress-strain relationship including effects of dilation are cumbersome. However, the elastic modulus, Ez, for the rock mass with three intersecting joint sets (Fig. 1) under uni-axial tests can be given by

1 _ 2cos2(0/2) sI/n-i2(O2/)Kn+c~ \

~E _z

/ A 1 / _ sin(0) 2 ~ sS q E0 .

KnK, S

(15)

Neglecting the effects of dilatancy, Eq. (15) becomes

1 ez =

2cos2(O/2)(sin2(O/2)Kn+cos2(O/2)Ks~ \

which reduces to the form in Eq. (14).

KnK,s

1 / +go

(16)

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5. Predicted and Measured Results

The large scale physical model test for jointed rock mass performed by Yang (1992) is used here to validate the present formulation. Instead of natural rock, the specimens were made from plaster blocks. Using plaster blocks, it was easier to make specimens with the identical joint properties and desired configuration of joint sets. Prismatic rock mass specimens are examined with configurations of joint sets similar to Fig. 1. Three different types of configurations were selected with angles 0 equal to 30, 80, and 90 degrees. The spacings were 5 cm for all joint sets. The model material was a mixture of plaster, sand, and water in the proportions of 1 : 0.25 : 0.92 by weight. All model specimens were cured at a temperature of 25~ and at a relative humidity of 55% for four to seven days. The physical characteristics of the intact rock model material are: 1. modulus of elasticity, E 0 = 4380 MPa to 4730 MPa; and 2. Poisson's ratio = 0.17 to 0.21. The material shows moderately brittle behavior with a compressive to tensile strength ratio of 4.8. (Huang and Yang, 1991a). The parallel artificial joints with identical surface characteristics were produced by tensile fractures generated in a fashion similar to the Brazilian test. Using a large controllable double-blade guillotine, the model material can be broken into small blocks with joint sets in specific orientations and spacings. Based on surface roughness and direct shear tests on specimens (Huang and Yang, 1991b), properties for the artificial joints tested under a range of normal stress from 0.23 to 1.03 MPa are as follows: 1. Closure Behavior: K n = 8 - 16MPa/mm. 2. Shear Behavior: K s = 0.31 - 0.413 MPa/mm. 3. Dilatancy Behavior: A = 0.2 - 0.4 These blocks were assembled to form specific configurations of prismatic specimens of a rock mass as shown in Fig. 1. From uniaxial compression tests on these specimens, the measured Young's modulus are respectively 260 MPa, 77.5MPa, 6 5 M P a for specimens with three different types of configurations, 0 = 30 ~ 80 ~ and 90 ~ The moduli of specimens with different configurations of joint sets are first predicted using Eq. (16) neglecting the effects of dilatancy. The upper solid curve in Fig. 2 is predicted using the stiffer estimates: Kn = 16 MPa/mm, K s = 0.413 MPa/ mm, and E0 = 4730 MPa; the lower curve in Fig. 2 is predicted using the softer estimates: Kn = 8 MPa/mm, Ks = 0.31 MPa/mm, and E0 = 4380MPa. The range of predicted moduli are compared in Fig. 2 with the moduli measured from specimens in uniaxial compression tests. The comparisons show that the predicted range of moduli is much smaller than the measured moduli of the rock mass specimens. The discrepancies may be primarily attributed to the fact that dilatancy is neglected in Eq. (16). Moduli of the rock mass are then predicted using Eq. (15) considering the effect of dilatancy. In Fig. 3, the moduli of specimens are predicted for dilatancy factors

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T.H. Huang et al. BOO

I

i

1

n

I

700 9=

Experiments

600 500 IZ, =Z N

400 300 200 ~

100 0

0

MPa/mm I

I

15

30

8

60

45

75

90

(degree)

Fig. 2. Predicted moduli of rock mass without consideration of joint dilatancy

ranging from 0 to 0.4 using the stiffer estimates: Kn = 16MPa/mm, K s = 0.413MPa/mm, and E0 = 4730MPa. The predicted moduli show better agreement with the measured ones. Figure 3 shows that rock mass moduli increase significantly with joint dilatancy. Hence, joint dilatancy plays an important role which cannot be neglected in estimating moduli of a rock mass. It is noted that, in the derivation of rock mass moduli, Eq. (7) is used for 800

i

i

i

700

i

i

O= Experiments

600 500

~= 0.4

400 300 2OO IO0 I

I

I

I

I

15

30

45

60

75

O

90

(degree)

Fig. 3. Effects o f joint dilatancy on the predicted moduli of rock mass

Elastic Moduli for Fractured Rock Mass determining the joint traction from the applied stress. The underlying assumption of uniform stress over the rock corresponds to a statistically admissible stress field, thus solution. It is therefore expected that, even with the dilatancy, the overall range of predicted moduli is still moduli.

143

use of Eq. (7) implies an mass. This assumption leads to a lower bound consideration of joint less than the measured

6. Conclusions A stress-strain model for jointed rock masses is described which accounts for the spacing, orientations, and stiffness of joint sets. Explicit expressions of moduli are derived for a model rock mass with three non-orthogonal intersecting sets of joints. The derived expressions indicate that joint dilatancy and orientations significantly affect the rock mass moduli. Due to the uniform stress assumption of the theoretical model, the derived expressions are expected to predict moduli lower than the true moduli of rock mass. Comparisons show reasonable agreement of the predicted moduli with those measured on artifically made rock mass specimens. With proper accounting for the range of joint stiffness and dilatancy, the derived expressions are useful in estimating the range of elastic moduli of jointed rock masses.

Acknowledgements The second author wishes to express his gratitude to the National Science Council, Republic of China for providing research support during his sabbatical leave at the Dept. of Civil Engineering, National Taiwan University, from January to June 1993. The assistance of Mr. C. H. Chan and Ms. T. L. Wang in preparation of this manuscript are also acknowledged.

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Chen, E. P. (1989): A constitutive model for jointed rock mass with orthogonal sets of joints. J. Appl. Mech. 56, 25-32. Chen, E. P. (1990): A constitutive model for jointed rock mass with two intersecting sets of joints. In: Rossmanith(ed), Mechanics of jointed and fractured rock, Balkema, Rotterdam, 519-526. Gerrard, C. M. (1982a): Elastic models of rock masses having one, two and three sets of joints, Int. J. Rock Mech. Min. Sci. Geomech. Abstr., 19, pp. 15-23. Gerrard, C. M. (1982b): Joint compliances a basis for rock mass properties and the design of supports. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 19, 285-305. Goodman, R. E. (1976): Methods of geological engineering in discontinuous rocks, West Publ., St. Paul, Minn. Goodman, R. E., Taylor, R. L., Brekke, T. L. (1968): A model for the mechanics of jointed rock. ASCE J. Soil Mech. Foundation Div. SM 3, 637-659. Hart, R. D., Cundall, P. A. (1988): Formulation of a three-dimensional distinct element model - Part II. Mechanical calculations for motion and interaction of a system composed of many polyhedral blocks. Int, J. Rock Mech. Min. Sci. Geomech. Abstr. 25 (3), 117-125. Huang, T. H., Doong, Y. S. (1990): Anisotropic shear strength of rock joints. In: Proc., Int. Symp. on Rock Joints, Loen, Norway, 211-218. Huang, T. H., Yang, Z. Y. (1991 a): Stress-strain behavior of three dimensional jointed rock masses. In: Proc., Seventh Int. Congress on Rock Mechanics, Aachen, Germany, 261 264. Huang, T. H., Yang, Z. Y. (1991b): Generation of artificial extension joints and their mechanical properties. In: Proc., 32nd U.S. Symp. on Rock Mechanics, Oklahoma, 1105-1114. Jing, L,, Nordlund, E., Stephansson, O. (1992): An experimental study on the anisotropy and stress-dependency of the strength and deformability of rock joints. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 29 (6), 535-542. Ladanyi, B., Archambault, G. (1970): Simulation of shear behavior of a jointed rock mass. In: Proc., 1lth U.S. Symp. Rock Mechanics, AIME, 105-125. Morland, L. W. (1974): Continuum model of regularly jointed medium. J. Geophys. Res. 79 (2) 357 362. Shi, G. H. (1988): Discontinuous deformation analysis - A new model for the static and dynamics of block systems. Ph.D. Thesis, University of California, Berkeley. Stephansson, O. (1981): The Nasliden Project-rock mass investigations. In: Stephansson, O., Jones, M. J.. (eds.), Applications of rock mechanics to cut and fill mining, I. M. M., London, 145-161. Yang, Z. Y. (1992): Strength and deformation of physical models of rock mass with regular joint sets. Ph.D. Dissertation, National Taiwan University. Yoshinaka, R., Yamabe, T. (1986): Jointed stiffness and the deformation behavior of discontinuous rock. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 23 (1), 295-303. Zienkiewicz, O. C., Pande, G. N. (1977): Time-dependent multilaminate model of rocks - a numerical study of deformation and failure of rock masses. Int. J. Numer. Analyt. Meth. Geomech. 1,219-247. Authors' address: Dr. Tsan H. Huang, Department of Civil Engineering, National Taiwan University, Taipei, Taiwan.