The effect of joint constitutive laws on the modelling of an underground excavation and comparison with in situ measurements

lnt. J. Rock Mech. Min. Sci. Vol. 34, No. 1, pp. 97-115, 1997

Pergamon PII: S0148-9062(96)00014-9

© 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0148-9062/97 $17.00 + 0.00

The Effect of Joint Constitutive Laws on the Modelling of an Underground Excavation and Comparison with in situ Measurements M. SOULEY# V. H O M A N D t $ A. THORAVAL§ The paper discusses the results of a numerical study which examines the influence of joint constitutive models on the response of a specific jointed rock mass made up of an anisotropic rock (slate). It presents a comparison between predicted convergences and displacements of rock mass surrounding a pilot gallery, and those measured during field investigations. To perform this analysis, the two-dimensional Distinct Element Method code, UDEC, is used and three joint laws are compared. Based on field investigations for site characterisation, two models are constructed according to fracture density (MODEL A and MODEL B). For each of them, the influence of joint constitutive law on the stability of the gallery is examined, and comparisons between the investigations and predictions are made. Examination of the results shows that there is no noticeable change in stress magnitudes between laws. The displacement magnitudes depend on (1) the constitutive law, (2) the input model parameters and (3) the fracture density. A parametric study in the case of MODEL A indicates that a relatively good match between predicted and measured displacements around the gallery can be observed in certain areas. © 1997 Elsevier Science Ltd. All rights reserved.

1. INTRODUCTION

element codes are not adapted (e.g. problems where multiple bodies are in contact or problems where new contact surfaces develop). In other words, discontinuum analyses, through their ability to take into account the behaviour of discontinuities, are now suitable for design calculations to evaluate the effect of rock joints on the response of excavations in jointed rock masses. The various analyses described in this paper were conducted using the two-dimensional UDEC code first introduced by Cundall [9] and further developed by Cundall and other authors [10-12]. UDEC enables discontinuities to be modelled explicitly, and block caving and joint slip and/or separation induced by excavation to be simulated. In other words, the UDEC algorithm includes not only continuum theory representation for the blocks, but also force~lisplacement laws which specify forces between blocks. The main objectives of the study discussed in this paper are to illustrate:

The presence of discontinuities affects, significantly, the mechanical behaviour of rock masses by reducing their capability to support loading and unloading in both normal and shear directions. Therefore, joint-controlled mechanisms are important considerations in the design of deep underground excavations for mining, oil or gas production, and civil construction. With the new advanced techniques of numerical analysis such as DEM (Distinct Element Method) which have evolved during the last decade, several codes such as UDEC have been developed in geomechanics and applied to engineering practices [1-6]. In fact, the development of general purpose discrete element codes (UDEC: Universal Distinct Element Code, 3DEC: 3D Distinct Element Code [7], DDA: Discontinuous Deformation Analysis [8] ...) provides an effective tool for analysing certain engineering problems for which traditional continuum finite

(i) The influence of joint constitutive law on the response of a given underground excavation as a function of fracture density. The constitutive laws include:

tLaboratoire de G6om6canique E.N.S.G., P.O. Box 40, 54501 Vandoeuvre-Es-Nancy, France. ++Author to whom correspondence should be addressed. §Laboratoire de M6canique des Terrains INERIS-I~cole des Mines-Parc de Saurupt, 54042 Nancy Cedex, France. 97

98

SOULEY et al.: JOINT CONSTITUTIVE LAWS AND SITE MEASUREMENTS • Law 1: the elastic-linear model, involving the Mohr-Coulomb criterion in shear direction. In this model, the normal and shear stress increments are related to the normal and shear displacement increments, respectively, through two constants k, (normal stiffness) and ks (shear stiffness). The current shear stress is limited by the Mohr-Coulomb friction criterion. • Law 2: the continuously yielding model, proposed by Cundall and Hart [12], and later revised by Lemos [10], is intended to simulate empirically, the intrinsic mechanism of progressive damage of joints under shear loading. In order to take into account the non-linear response of rock joints, a revised version of the continuously yielding joint model was proposed by Cundall and Hart. In this formulation the stiffness parameters k, and ks increase with normal stress, in a given range of values specified by kjmin and kjmax (.]" = n,s). It is interesting to note that, outside this range of values k, and ks remain constant. Furthermore, joints were assigned to the same variation of stiffness in opening or re-closure which occurred during subsequent unloading and re-loading. However, this formulation may produce unacceptable results when large variations of normal stress accompany reversals in both normal and shear directions [13-15]. • Law 3: the model of Souley et al. [14], which extends the non-linear Saeb and Amadei model [17] for rock joints to include cyclic loading paths. In this model, the normal and shear stress increments are related to the normal and

shear displacement increments through a (2 x 2) non-symmetric stiffness matrix whose terms depend on the intrinsic parameters describing the mechanical behaviour of joints. It takes into account the effect of loading and unloading in both normal and shear directions which cannot be neglected particularly for structures in jointed rock mass [3]. Note that this model was recently implemented in UDEC by Souley and Homand [18]. (ii) A comparison between the numerical predictions of convergences and expansions (displacements of rock mass surrounding the excavation) with those measured in the gallery. (iii) A sensitivity analysis was performed by varying the intrinsic properties of prominent discontinuities, in order to assess their effect. 2. GEOMETRY, I N S I T U AND BOUNDARY CONDITIONS, AND MECHANICAL PROPERTIES The site is constituted of anisotropic rock materials (slate) in which a pilot gallery was excavated. The first part of the tunnel (case 1) is perpendicular to the plane of transverse isotropy (e.g. schistosity plane) and the second one is parallel to the schistosity (case 2) [Fig. 1]. A statistical characterisation of the different families of fractures enabled the site fracturing to be generated in three-dimensions using RESOBLOK software [5, 19]. It was, therefore, possible to obtain the vertical sections in both planes parallel and perpendicular to the plane of isotropy. In this paper, the analysis is limited to the part of the gallery located perpendicular to the schistosity, where almost all of the instruments were set

Case 2

A | y=-11.5m

X

-t ............... I

i s

A

B

B

Case 1 x =30m Schistosity plane

-schistosity ~ - -

R

I

5m

I

.....................

7y

Fig. 1. Schematicmap of the experimental gallery excavated in slates.

SOULEY et al.:

JOINT CONSTITUTIVE LAWS AND SITE M E A S U R E M E N T S

(case I). Site fracture characterisation demonstrated the existence of five faults located near the gallery and their persistence was observed along the total length of the gallery during excavation. Moreover, there are other discontinuities such as diaclases which constitute a secondary fracturing. In the first approach, we have not considered this secondary fracturing, and the corresponding geometrical model (MODEL A) is represented in Fig. 2(a). The secondary fracturing is modelled according to two joint sets with an average spacing ranging between 1 and 2 meters (m), and an orientation of about 87 and 160 degrees (°). The corresponding geometrical model which contains both major faults and secondary fractures is shown in Fig. 2(b) and called M O D E L B. Both Fig. 2(a) and (b) are two-dimensional strain representations of a 2.65 m high and 3 m wide excavation located 500 m below the surface. These models are assumed to be parallel to the schistosity plane. The rock mass is modelled using the fully deformable blocks assumption. Therefore, blocks are subdivided into finite difference zones for the calculation of internal stresses and Strains. Also, rock blocks are assumed to be isotropic and linearly elastic materials. The values of the elastic constants used for all calculations are derived from laboratory investigations and according to the following equations [20]: Ell_ + 2vt + nv~

E

(1 + v,) 2

vl + nv] where n El v (I + v,) = ~"

Px

I

I I I I I I t I I I I

i

2.5

7.5

I

Young's modulus, E[GPa] Poisson's ratio, v

For each geometry (MODEL A and M O D E L B), the modelling sequences are performed as follows: (i) The model without excavation is consolidated under in situ stresses. During consolidation, the model is loaded using a combination of gravity ( g = 9.81 m/s2), initial in situ stresses and boundary stresses in order to produce the in situ stress field. (ii) The excavation is introduced using the boundary element conditions applied to the boundaries domain, and the model is cycled to an equilibrium state for stress redistribution. Note that the boundary element approach provides a more accurate representation of the far field stress [4, 11, 21, 22]. The stress field in the rock mass was measured by field investigations using hydraulic fracturing. These investigations show that the virgin stress field in the rock mass is represented by the following relation: a H = 1.33"ov

(2)

where OH is the /n situ horizontal stress and Ov is the overburden pressure, ov is assumed to vary with depth according to av = ?*h

(3) 35 (m) 30

25

25

20

20 X

~.

---22

15

~

10

10

5

5

,..,j

~

I

I I ~''.,,jk'', J j

0 I

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0

Py

I

12.5 17.5 22.5 27.5 32.5 (m)

I

(m)2.5 -

Major faults

115 0.2.

35 (m) 30

15

Py I

response in the plane of transverse isotropy to a stress 0, (0, = o - = 0) acting parallel or normal to it, respectively. Values of E~, E2, v~ and v2 derived from laboratory tests are: E~ = 111.9 GPa, E2 = 40.9 GPa, v~ = 0.17 and v2 = 0.16. Therefore, equation (1) provides the values of E and v:

(1)

Previous constants in equation (1) have the following definitions: E~ and E2 are Young's moduli in the plane of transverse isotropy and the direction normal to it, respectively; v, and v2 are Poisson's ratio characterising the lateral strain

99

-

--

I

7.5

I

I

I

I

I

12.5 17.5 22.5 27.5 32.5 37.5

Major faults Secondary fracturing

Fig. 2. (a) Geometric model: M O D E L A P,.: horizontal profile; p,.: vertical profile. (b) Geometrical model: M O D E L B P,.: horizontal profile; p,.: vertical profile.

100

SOULEY

et al.:

JOINT

CONSTITUTIVE

where ~ --= 0.024 MN/m 3 which is the unit weight of the rock and h is the depth beneath the surface. In particular at the centre of the gallery (h = 500 m), the initial horizontal and vertical stresses are, respectively, 16 and 12 MPa. All of the parameters needed for secondary fractures in the case of law 3 (MODEL B) are derived from laboratory analysis tests and summarised in Table 1. As reported in Ref. [15], normal deformability is linear in the case of laws 1 and 2, whereas the experimental normal stress vs the normal closure curve provides a non-linear variation of normal stiffness as a function of normal stress, particularly at high normal stress levels. In order to make a more rigorous comparison between these laws, it is necessary to back-calculate the constant k., which characterises laws 1 and 2, according to the normal stress level, and also the hyperbolic model. Therefore, assuming that the normal stress acting on joints is equal to the horizontal stress magnitude at the centre of the gallery, normal stiffness k, which is used for laws 1 and 2, was calculated according to equation (4) [see Section 3.3]. The value of k, is approximately equal to - 17.4 MPa/mm as reported in Tables 1 and 2 remains constant. Unlike the rock mass properties and the secondary fracture properties in MODEL B [Fig. 2(b)], it was not possible to deduct in advance the mechanical parameters of the faults shown in Fig. 2(a) and (b). The initial normal stiffness k,~ and the maximum normal closure Vm were calculated based on a recent study reported by Arif [21] and the empirical relationships suggested by Bandis et al. [23]. Note that, the fault deformation properties used in the models (A and B) and summarised in Table 2 are consistent with the literature. The rock strength parameters needed in peak strength (law 3) are equal to: ac: uniaxial compressive strength = 120 MPa, a,: tensile strength = 5 MPa. The key parameters used in the analysis are listed in Tables 1 and 2. These are defined as follows: • • • • • •

k,~: initial normal stiffness (law 3), k,: current normal stiffness (laws 1 and 2), ks: shear stiffness, Vm: normal maximum closure, aT: normal transitional stress, B0: ratio of residual to peak strengths at very low or zero normal stress, • Ur: residual displacement, T a b l e 1. J o i n t i n p u t p a r a m e t e r s f o r e a c h l a w used in M O D E L Law 1

Law 2

Law 3

k,=-17.4MPa/mm k~ = 2.5 M P a / m m ~b = 2Y" i,, = 0 "~

k.=-17.4MPa/mm ks = 2.5 M P a / m m q~h = 25 '~ ¢0 = 25 '~ jr ----0.001 m m

k.~= -10MPa/mm k~ = 2.5 M P a / m m av = 100 M P a ~b,, = 25 'j i,, = 0 ~ V,, = 5 m m B.= lur=3.25mm

B

LAWS AND

SITE MEASUREMENTS

*Jr: joint roughness, • ~b: internal friction angle, • ~bb: basic friction angle of the rock surfaces, • i0: peak angle of dilatancy at very low or zero normal stress, .q~,: angle of friction for sliding along the asperities, and • ¢~ = Cb + i0. For instance, the initial dilatancy i0 is set to zero. Therefore, the parameters B0 and Ur are formal in this case. In other words, under constant normal stress and without dilatancy, law 3 is similar to law 1 in the shear direction, except for the peak strength criterion, whereas in the normal direction, law 3 is reduced to the hyperbolic model proposed by Bandis et al. [23]. The influence of dilatancy will be discussed in the parametric study (Section 3.3). 3. RESULTS AND DISCUSSION For each law applied in this study, two simulations, corresponding to MODEL A and MODEL B, were performed. We analysed: • the normal and shear displacements along the specific fault FA located at the left side of the cavity and represented in Fig. 2(a) [MODEL A], • the normal and shear stresses along FA (MODEL A), • the horizontal and vertical stresses along the horizontal and vertical profiles P~. and p, [Fig. 2(a) and (b)], • the displacement magnitudes along P,. and P~. (MODEL A and MODEL B). As discussed in Refs [14, 18] joint normal behaviour, joint shear behaviour and joint peak shear strength in the case of law 3, were based on experimental observations. In comparison to laws 1 and 2, law 3 integrates more realistically the properties of joint behaviour and joint strengths which can be determined from conventional normal compression tests and direct shear tests under constant normal stress. For that reason we consider law 3 as a reference law when interpreting data. 3.1. Effect of joint laws: M O D E L A Figure 3(a) illustrates the direction of block movements and shear displacement magnitudes through major discontinuities for law 1. The excavating of the pilot gallery induces a decompression of the rock mass surrounding the cavity leading then, to a displacement of blocks and an increase in shear displacement along the discontinuities close to the excavation. In particular, Fig. 3(b) shows the normal displacement along the specific fault, FA, situated on the left side of the gallery. Note that the fault (FA) opening is located at the level of the excavation whereas, above and below this level, the fault remains entirely closed and the normal closure decreases from the immediate roof-wall to the model

SOULEY et al.:

JOINT CONSTITUTIVE LAWS AND SITE MEASUREMENTS

101

(a)

A * Values of shear displacements are in millimetres

(b) Region 1 closure

Region 2 opening

Region 3 closure

i

A

B 32.5 m

closure--0.011 Inm

closure=0.01mm opening=0.57mm

Fig. 3. MODEL A, case of laws 1 and 2: (a) shear displacement magnitudes along discontinuous; (b) normal behaviour of fault FA.

boundaries. Therefore, the fault opening at the cavity level will provoke an amount of block displacement (block delimited by FA and excavation) towards the excavation and will prevent the block located on the left side of FA, from moving closer to the cavity. The values of normal displacement along FA and shear displacement magnitudes through discontinuities reported in Fig. 3(a) and (b) are in mm and correspond to laws 1 and 2. The same evolution of block movement, shear displacement along discontinuities and normal behaviour of the fault FA were observed according to laws 2 and 3. It is interesting to notice that no difference between laws 1 and 2 in terms of shear displacement and closure/opening magnitudes was noted. 3.1.1. Displacements analysis along fault FA. Figure 4 shows the evolution of the normal displacement along FA as a function of law. The abscissa axis represents the profile of FA where the origin is located at the base of the model (e.g. point A in Fig. 3). This indicates that the curves of normal displacement according to laws 1 and 2 are identical, whereas, the comparison according to law 3 shows a slight difference in the magnitude of

the fault opening. However, the curve forms match qualitatively. In fact, for the geometry of the discontinuities reported in Fig. 2(a), the normal behaviour plays an important role in comparison to shear behaviour particularly at the level of the excavation. Shear displacement along FA is equal to zero at the centre of the cavity and reaches a maximum at the immediate roof-wall. In other words, there is no significant change in shear displacement magnitudes along the discontinuities at cavity level (Fig. 5). The average of normal stress acting along FA near the cavity is estimated to be 16 MPa. Then, based on back-calculation (according to Mohr-Coulomb's criterion), the theoretical peak shear displacement is approximately equal to 12.9 mm. The maximum shear displacement (located at the immediate roof) resulting from these simulations is less than 3% of the previous theoretical peak displacement. Therefore the magnitudes of shear displacement are too small and as a result, the shear behaviour of these faults is elastic and entirely governed by the shear stiffness. Also, no significant change in normal displacement between laws 1 and 2 on the one hand, and laws 1 and

102

SOULEY et al.:

JOINT CONSTITUTIVE LAWS AND SITE MEASUREMENTS

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Fig. 4. MODEL A'. Normal displacement along the fault FA.

3 on the other hand, was noted except for the maximum opening magnitude (Fig. 4). According to: (1) hyperbolic model [23], (2) law 3 parameters (maximum closure and initial normal stiffness reported in Table 2), and (3) assuming that the normal stress acting on FA is equal to the normal stress magnitude at the centre of 0.4

I

the cavity (e.g. 16 MPa), the calculated normal stiffness in the linear case (laws 1 and 2) kn according to the hyperbolic model (equation 4), is equal to - 3 . 4 M P a / m m , whereas the input value used is - 2 MPa/mm (Table 2). Notice that the input value of k, ( - 2 MPa/mm) used is greater than the value

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SOULEY et al.:

J O I N T CONSTITUTIVE LAWS A N D SITE M E A S U R E M E N T S

Table 2. Fault input parameters for each law used in M O D E L A and MODEL B

Law 1

Law 2

Law 3

k. =

k° =

k°~ =

- 2 MPa/mm

k~ = 0 . 5 M P a / m m ~b = 2 2 °

- 2 MPa/mm

- 2 MPa/mm

k~ = 0 . 5 M P a / m m (~b = 2 2 °

k~ = 0 . 5 M P a / m m (7T = 1 0 0 M P a

~bp = 2 2 °

q~,, = 2 2 ° i0 = 0 °

I"o = 0 °

j~ = 0 . 0 0 1 m m

V~ = 2 5 m m Bo=lu~=3.25mm

obtained from a hyperbolic response ( - 3 . 4 MPa/mm). Consequently, laws 1 and 2 overestimate the joint opening at the cavity level and underestimate the normal closure far from the cavity. The shear displacement along FA [profile AB in Fig. 3(a)] as a function of law is represented in Fig. 5. As mentioned earlier, the shear displacement magnitudes are independent of the law, at cavity level, whereas far from the excavation, laws 1 and 2 slightly underestimate the shear displacement. Consequently, based on the displacement analysis along the discontinuities in MODEL A, the previous study leads to the following conclusion: no significant difference was obtained to allow a distinction to be made between laws. Note that (1) when compared to law 3, block movements on both sides of the cavity are slightly greater in the case of laws 1 and 2 due to a large amount of joint opening, and (2) block movements at the roof-wall close to the cavity, are slightly more important according to law 3.

3.1.2. Displacement analysis along profiles Px and Py. Figure 6(a) illustrates the evolution of horizontal displacement along the horizontal profile p,. As previously mentioned, concerning the normal displacement along FA, no significant difference between 0.8-

.......

T . . . . . . . . . . . . . . . .

0.7- . . . . . . .

~

.._

0.6-

'-"

/

~ ........

,

,,

F~ultFA

:..o

.--

,........

-I

II

I

laws can be observed. However, for all laws, the change in block displacement on both sides of FA is clearly shown in Fig. 6(a). The block situated to the left of FA is constrained to move away from the cavity due to fault opening, and maximum horizontal displacement at the left of FA is estimated to be 0.05ram. Consequently, the block situated to the right of FA is constrained to move "freely" towards the cavity due to fault opening and excavating. It is interesting to note that near the right side of FA, the horizontal displacement corresponds to the maximum fault opening (e.g. 0.55 mm) predicted according to each law. Finally, the slight difference in horizontal displacement between laws is closely connected to the normal behaviour of the fault (FA) shown in Fig. 4, and the change in values of the horizontal displacement along P, corresponds to the discrepancy in normal opening between laws. Figure 6(b) shows the variation of the vertical displacement along the vertical profile P~. As reported in shear displacement analysis along FA far from the cavity (Fig. 5), laws 1 and 2 underestimate the vertical displacement in comparison to law 3. Also, the vertical displacement is closely related to shear displacement along FA. Recalling that discontinuities represented in MODEL A are approximately sub-vertical it then becomes easy to correlate the vertical displacement along p, to the shear displacement along FA. 3.1.3. Stress analysis along profiles Px and Py. Stress analysis was carried out using the ratios a.~.~/a.~,~.,,and a.~./a.i..,.; where the subscripts xx and yy denote, respectively, horizontal and vertical stress, " f " and "i" denote final and initial stress obtained after equilibrium during excavation and consolidation. The initial horizontal and 0.35

i

J 0.3

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:;-L. , °Y

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0.4-

:........

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=

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(a)

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i / i

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0.1- . . . . . . . . ÷. . . . . . . . :. . . . . . . . -:. . . . . . . . : . . . . . . . 4 0 ' 0

II It

~0.25

i i........

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i i i

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............

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i

0.15

t~ .

Lawl

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. . . . . . . . . . . . . j. . i J i i i i

/ 4 ,

*

I I

~ 0.5-

103

i 15

> ....

0.1

i

0.05 0

i ........

.......

i

3 6 9 12 Vertical displacement Py (m)

(b)

Fig. 6. MODELA: Displacementsalong the profiles.(a) Horizontaldisplacementalong P,; (b) Vertical displacementalong p,.

15

104

SOULEY et

JOINT CONSTITUTIVE LAWS AND SITE MEASUREMENTS

al.:

Table 3. Mean values of initial compressive stress obtained after consolidation along profiles P~ and p,. a.i., (MPa) aj.,. (MPa) Profile P,. 16 12 Profile p,. 16.5 12.5

vertical stresses after consolidation are approximately uniform along the profiles P, and p,.. Mean values of these stresses are listed in Table 3. Variations of horizontal and vertical stresses along the horizontal profile P,. as a function of law are illustrated in Fig. 7(a) and (b). Far from the fracturing and excavation (x = 0 m), the three laws are identical and the stress ratio remains closely equal to one. This is because, far from the set of discontinuities, the effects of contact forces between blocks vanish and in this case, the medium behaves as a continuum medium. Furthermore, in this area (far from excavating) there is no change in stress distribution in comparison to the initial stress field. However, in the vicinity of fault FA and the excavation (x near 15 m), the horizontal stress decreases progressively towards the centre of the gallery along p,. The stress ratio is reduced in value from 1 to 0.2 near the cavity. The slight change in horizontal stress between laws observed in Fig. 7(a), is in

accordance with the examination of normal stress evolution. Recall that laws 1 and 2 underestimate the normal stress magnitude at the gallery level due to the difference in current normal stiffness between hyperbolic and linear models in the direction normal to the fault FA. The vertical stress [Fig. 7(b)] is practically constant along the horizontal profile Px, except in the block mass delimited by FA and the gallery. In this case, the vertical stress ratio, first increases suddenly and approaches a value of 2 near FA, and second decreases very fast and becomes negative (tension) near the excavation (at the immediate left side) and the tensile stress reaches a value of - 6 MPa. It is also interesting to note that the examination of the horizontal and vertical streses along the vertical profile P~. not presented here, showed that the stress curves as a function of law are practically similar, except near the immediate wall (y ,~ 15 m) where laws 1 and 2 tend to slightly overestimate the horizontal stress. In conclusion, all three laws are reduced to a similar behaviour as the difference in results is negligible. Finally, it should be noted that the global behaviour of the rock mass may sometimes be captured with sufficient accuracy using the simpler joint model (e.g. Mohr-Coulomb model).

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Hodzontal profile Px (m)

:

0

.

15

3

; i

- 0

i

i

i

6

9

12

: i

i

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

i

i

. . . . . . . .

(b)

I

FA •

Lawl Law2 Law3

x

Fig. 7. MODEL A: Stress ratio along the horizontal profile P,. (a) Horizontal stress ratio along P,; (b) Vertical stress ratio along P,.

i

SOULEY et al.:

JOINT CONSTITUTIVE LAWS AND SITE MEASUREMENTS

105

ROOF -

T3. / I'8 T2

RIGHT SIDE

¢

:

:

,2.65m

4m 3m 3m ~ - - 3 m ~ C1

¢q

WALL 3m (a)

(b)

Fig. 8. Expansion monitoring: (a) Position of extensometers; (b) Location of displacements measured.

3.2. Comparison between calculations and measurements

Field measurements were conducted during the construction of the pilot gallery and the following instruments were used: • eight extensometers were installed at the roofwall and both sides of the cavity to monitor, respectively, vertical and horizontal displacements in the rock mass surrounding the gallery (Fig. 8). In particular, one of the locations where extensometers were installed, is shown in Fig. 8(b). • 12 convergencemeters (convergence monitoring of the gallery) distributed at two stations (six convergencemeters per station) were sited as illustrated in Fig. 9(b). Each element of the monitoring system (instruments in the monitored section of the gallery) was installed as close as possible to the gallery faces during the excavation, then the displacements measured were assumed to be the total displacements induced by excavating. Therefore, we will directly compare the simulated displacements to those observed.

The numerical predictions of displacements in the rock mass surrounding the excavation provided by MODEL A according to the laws (1-3) were compared with the field measurements. Table 4 shows details of the predicted and measured displacements. A large disparity in terms of measured displacements between STATION 1 (T~/T2/T3/T4) and STATION 2 (Ts/T6/TT/ Ts) can be observed, except for extensometers T2 and T6. It is interesting to note that significant discrepancies exist between extensometers T3 and TT. Significant increases in rock mass movements were measured near the level of the tunnel roof (case T3). This scatter of data could result from specific local conditions in which these extensometers were installed (cracks, secondary fractures ...). Consequently, given the scatter of measured data, only global patterns and orders of magnitude can be discussed between numerical predictions and field measurements. Then it may be seen that the numerical predictions and measured data are reasonably close for extensometers TI/T5 (immediate wall) and T2/T6 (left side of the excavation). In comparison, the data measured by T3/T7 show a larger value of displacements. When compared

/ D10~D!1 D 1 2 / D4,"D~ D6 ° y " ~ , " ~

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9

~D8

ROOF-WALL

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iLeft

~

Right

67

r~ .66 ¢q

CAylTY l ~ , j ~

3m

~ .66 m .66

~- - CAVIITYI- - tMediu m ~ L o w e r .75 .75 .75 .75 m

(a)

(b)

Fig. 9. Convergence monitoring: (a) Position of convergencemeters; (b) Location of convergences measured.

106

SOULEY et al.:

J O I N T C O N S T I T U T I V E LAWS A N D SITE M E A S U R E M E N T S

Table 4. Comparison between observed and calculated displacements in rock mass surrounding the gallery

T2

~-

-i

3m Expansions (mm)

In situ measures

Upper (T3/TT)

CI C2 C3 CI C2 C3 C1 C2 C3 CI C2 C3

Lower (TI/Ts)

Left (T2/T 0

Right (T4/Ts)

Law 1

Law 2

Law 3

Tt/T2/T3/T4

Ts/T6/TT/T8

0.22 0.18 0.16 0.17 0.17 0.15 0.76 0.75 0.74 1.86 1.85 1.57

0.22 0.18 0.16 0.17 0.17 0.15 0.76 0.75 0.74 1.86 1.85 1.57

0.23 0.19 0.16 0.18 0.17 0.15 0.72 0.70 0.68 1.69 1.68 1.40

6.90 2.50 -0.52 0.20 0.10 0.40 0.38 0.31 0.10 0.10 0.06

0.93 ---0.50 -0.60 0.47 0.38 -0.30 --

to the case of the right side, rock mass movements decrease. To illustrate this, the measured displacements recorded by T4/T8 are 15 times less than the computed values. Table 4 illustrates the comparison of displacements around the cavity as a function of law. No significant difference can be observed. Once more, laws 1 and 2 are similar, whereas slight changes between law 3, and laws 1 and 2 are in accordance with the changes observed in normal and shear displacements along discontinuities, as explained previously. The main comparisons between predicted and measured convergences of the gallery faces are

summarised in Table 5. A similar trend in terms of convergences between laws can be observed once again, and no significant difference was obtained between laws. Obviously, there are discrepancies between the in situ measurement results and numerical predictions. The measured convergences are higher than the calculated ones, up to a factor of 4 at certain areas. The reasons for these discrepancies are probably: (1) the local conditions acting on the instruments as earlier mentioned: in this analysis, the techniques for installing devices must be taken into account properly, and

Table 5. Comparison between observed and calculated gallery convergences

7

I i " " D3~ " v

3m Convergence (mm) Both sides D~/D7 D.,/D, D3/D9 Roof-wall D4/DJl) Ds/D, D6/D~2

In situ measures

Law 1

Law 2

Law 3

upper medium lower

2.23 2.71 2.17

2.23 2.71 2.17

2.09 2.53 2.03

1.79 0.73 0.29

0,16 1.94 0.19*

left centre right

0.55 0.65 0.59

0.55 0.65 0.59

0.57 0.67 0.62

2.47 1.36" 2.12"

-4.53 3.76

*Suspect values.

1, 2 . . . .

6

7, 8 . . . .

12

SOULEY et al.:

JOINT CONSTITUTIVE LAWS A N D SITE M E A S U R E M E N T S

Table 6. Parameters used for parametric study

Basic case (reference) New values

k.i (MPa/mm)

k~ (MPa/mm)

i0 (°)

Vm (mm)

- 2

0.5

0

25

-- 0.2 - 20

0.05 50

10

10 5

(ks) controls the shear displacement for a given normal stress level. A similar trend can be observed in Fig. 10 where ks is now equal to 0.05, 0.5 and 5. However, compared to the basic case, it is interesting to note that the change in ks does not affect significantly the curve of normal displacement along FA. Figure 11 illustrates the variations in normal displacement along FA according to the different values of k,~ ( - 0 . 2 , - 2 and - 2 0 M P a / m m ) . An increase in kni indicates that the fault becomes, in the normal direction, less deformable at the level of the excavation and the immediate roof-wall, whereas an increase in kni decreases the normal closure far from the gallery. In fact the gallery level corresponds to the region where the fault opens (reduction in normal stress), whereas far from the gallery, FA remains entirely closed. The value - 0 . 2 MPa/mm of kni, compared to the basic case, indicates that a decrease in k.~ does not affect the change in normal behaviour. In order to explain this, the evolution of the current normal stiffness (k.) as a function of the initial normal stiffness (k.0 is performed for a given Vm (maximum closure) and a,0 (normal stress). According to the hyperbolic model when the initial dilatancy is set to zero (to = 0), k, is written as follows:

(2) the fact that it was not possible to deduct in advance the mechanical properties of the major discontinuities from field investigations. In order to take into account the latter point, a parametric study on the properties of prominent discontinuities was performed according to each law. As the parametric study concerns in particular the main parameters shared by the three laws which characterise the intrinsic behaviour of rock discontinuities, only the results corresponding to law 3 (e.g. reference law) will be presented here.

3.3. Parametric study (MODEL A and law 3) In order to approach more realistically the behaviour of MODEL A, a parametric study was carried out to assess the effect of the intrinsic properties of predominant joint set (faults). In this parametric study, the geometry of Fig. 2(a) was adopted. The study is based on the parameters summarised in Table 6 (k,~ = initial normal stiffness; k~ = initial shear stiffness, I'm = maximum normal closure, and i0 = initial dilatancy). In such a simulation, only a single parameter is modified. 3.3.1. Displacement analysis along F~. Figure 10 shows the variations of shear displacement along FA for ks = 0.05, 0.5 (basic case) and 5 MPa/mm. An increase in ks indicates that the fault FA becomes less deformable in the shear direction, because shear behaviour depends greatly on the value of k~. Recall that the shear stiffness 0.35 . . . . . . . . . . . . . . . . . . .

107

where a,0 > 0 compression, k.~ < 0. From equation (4) and assuming that the normal stress acting along FA is equal to the initial horizontal stress (tr,0 = trH = 16 MPa), and according to the basic maximum closure (Vm = 25 mm), it easily follows that k, increases under very low initial normal stiffness kni. In particular, considering these assumptions: tr.0 = 16 MPa and Vm = 25 mm, k, approaches the value of - 3 . 5 M P a / m m for both initial normal stiffness

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, --

..

108

SOULEY et al.:

JOINT CONSTITUTIVE LAWS AND SITE MEASUREMENTS

0.6 -I-. . . . . . . . .

I

I

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~

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Basic case (kni=-2) kni=-0.2 kni=-20

0.3 0.2

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i

i0.1

,

t

Fault FA ~rofile (m): 0

5

~

1

15

~

,

25

30

,

,

35 ,

Fig. 11. Normal displacement along FA as a function of the initial normal stiffness: MODEL A.

k,~ = - 0 . 2 and - 2 MPa/mm. It appears that there is no significant difference in normal behaviour for these two values of kn~. In other respects, the examination of shear displacement variations along FA for different k,~ shows that an increase in k,~ (1) reduces the maximum shear displacement (near the level of the gallery), and (2) increases the shear displacement far from the excavation due to the change in normal stress. Also, the influence of kn~ on the shear displacement along FA extends to the model boundaries. 0.8

As previously discussed, the effect of an increase in ko~ on the normal and shear displacements along FA is similar to the trends which were observed for Vm when equal to 10 or 5 mm. However, the range of the decrease in Vmis lower than that obtained from an increase of ko~. To illustrate this, the value 5 mm of Vmpresents a 50% decrease in the maximum fault opening of FA at the centre of the gallery.

3.3.2. Displacement and stress analysis along profiles Px and Py. Figure 12(a) and (b) shows, respectively, the variations in horizontal and vertical displacements along 0.4

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(b)

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15

SOULEY et al.:

JOINT CONSTITUTIVE LAWS AND SITE MEASUREMENTS

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Fig. 13. Parametric study: MODEL A: (a) Horizontal stress ratio along P,; (b) Vertical stress ratio along Px.

P, and p, for the basic case, and kni = - 2 0 MPa/mm, I'm = 5 mm, ks = 5 MPa/mm. Recall that the basic case corresponds to k,~ = - 2 MPa/mm, I'm = 25 mm and ks = 0.5 MPa/mm. The effects of these parameters are briefly discussed below. An increase in k,~ (or decrease in Vm), leads systematically to a reduction in the horizontal 1.8

i

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Vertical profile Py (m) (b)

Fig. 14. Parametric study: MODEL A: (A) Horizontal stress ratio along Py; (b) Vertical stress ratio along Py.

12

15

110

SOULEY et al.:

JOINT CONSTITUTIVE LAWS A N D SITE M E A S U R E M E N T S

Table 7. Parametric study: predicted and measured convergences Convergence (mm) Both sides

Roof-wall

upper medium lower left centre right

Basic

k,~ = - 2 0 MPa/mm

V~ = 5 mm

k~ = 5 MPa/mm

2.09 2.53 2.03 0.57 0.67 0.62

1.41 1.68 1.39 0.67 0.78 0.73

1.65 1.99 1.62 0.64 0.75 0.70

2.01 2.44 1.96 0.48 0.56 0.50

In situ measures

1, 2 . . . .

6

1.79 0.73 0.29 2.47 1.36" 2.12"

7, 8 . . . .

12

0.16 1.94 0.19' -4.53 3.76

*Suspect values.

the current normal stiffness and then reduces fault deformability• It is therefore not surprising that block 1 is subjected to an upward thrust towards the gallery, whereas block 2 is less confined in comparison to the basic case. As illustrated in Fig. 12(a), horizontal displacements along P,. depend, greatly, on kn~ and Vm in comparison to ks. On the other hand, the horizontal displacement along P,. is not affected much by the value of k~ in opposition to the variation in vertical displacement along P). as a function of ks given in Fig. 12(b). Also, from Fig. 12(a) and (b), it can be observed how the changes of k,~, ks and Vm affect the displacement variations at the model boundaries (far from the gallery). Figure 13(a) and (b) and Fig. 14(a) and (b) show respectively, the variations in stress ratios a.,..~/ax.,, r i and a.~,./a.i,:~, along P,- and p,. for k n i = - 2 0 M P a / m m , Vm = 5 mm and ks = 5 MPa/mm. Recall that the average of the initial horizontal and vertical stresses (a!,.,.and aj.).) are, respectively, 16 MPa and 12 MPa along p,., and 16.5 MPa and 12.5 MPa along p~.. As explained in FA analysis according to an increase in kn~ (or decrease in Vm), the difference between results obtained from the basic case and those obtained from the parametric study in terms of stress change is strongly correlated to the previous discussion concerning fracture behaviour as a function of kni, Vm and k~. In particular, an increase in k.~ reflects a reduction in horizontal stress along p, due to low horizontal confinement (normal stress acting along FA) induced by FA. 3.3.3. Comparison with in situ measurements. Table 7 summarises the effect of k,~, Vm and ks on the gallery convergence and leads to the following conclusions: - - a n increase in k,~ reduces the convergence on both sides of the gallery and increases the roof-wall convergence,

- - a n increase in k~ reduces the gallery convergence, - - a decrease in V~ has a similar trend to an increase in k,~. It may be seen that the numerical convergence predicted at both sides of the gallery using the value - 2 0 MPa/ mm of k,~ (or Vm = 5 mm) and the measured data are in reasonably close agreement at certain areas, although an increase in k,~ tends to increase the roof-wall convergence. However, the measured convergence remains higher than the one which was calculated• Examination of Table 8 shows that an increase in kni or a decrease in Vm reduces the horizontal displacement on both sides (left and right)• Nevertheless, an increase in ks produces an opposite effect, except on the right side of the gallery. From the results shown in Table 8, it follows that the displacement magnitudes in rock mass surrounding the gallery on the right side are significantly higher than values for the left side. This can be justified in terms of fault density and the interdependence between fractures on the right side. In order to capture the role of dilatancy, simulation using i0 = 10° was performed, then the parameter B0 was reduced to 0.9. Results are similar to those obtained earlier for the zero value of dilatancy. From laboratory tests, it is now widely accepted that dilatancy takes place when a rough joint is subjected to shear loading under constant normal stress or under low normal stiffness• However, joints in a given rock mass are not only subjected to these boundary conditions, but also to normal stress depending on rock mass deformability• In particular, rock mass surrounding the fracture tends to reduce more or less completely the joint dilatancy phenomenon• This is observed when (1) fracture sets do not intersect a free surface (excavation for example), (2) fracture set is subjected to external forces (bars, tubes •..) [24], and (3) during laboratory shear tests under high

Table 8. Parametric study: predicted and measured expansions Expansions (mm) Upper (T3/TT)

Lower (TI/Ts)

Left (T2/T6)

Right (T4/Tg)

CI C2 C3 C1 C2 C3 C1 C2 C3 C1 C2 C3

k,~ = - 2 0 MPa/mm

Vm = 5

k~ = 5

Basic

mm

MPa/mm

Tt/T2/T3/T4

In situ measures

T~/T6/T7/T8

0.23 0.19 0.16 0.18 0.17 0.15 0.72 0.70 0.68 1.69 1.68 1.40

0.27 0.22 0.18 0.21 0.20 0.15 0.54 0.50 0.46 0.98 0.94 0.88

0.26 0.21 0.17 0.20 0.19 0.15 0.59 0.56 0.53 1.24 1.22 1.17

0.22 0.19 0.17 0.18 0.18 0.15 0.73 0.72 0.72 1.62 1.61 1.40

6.90 2.50 -0.52 0.20 0.10 0.40 0.38 0.31 0.10 0.10 0.06

0.93 ---0.50 -0.60 0.47 0.38 -0.30 --

SOULEY et al.: JOINT CONSTITUTIVE LAWS AND SITE MEASUREMENTS

II1

normal stiffness levels. Recall that in a recent paper, 3.4. Effect of joint laws: M O D E L B Souley and H o m a n d [14] showed the role of dilatancy The geometrical model including both the five when the fracture sets intersect the excavation. major fractures and the two joint sets is represented in It is also interesting to note that similar parametric Fig. 2(b) [MODEL B]. The mechanical properties of studies not presented here and based on normal stiffness, major and secondary fractures according to each law are shear stiffness and dilatancy were performed according reported in Tables 2 and 1, respectively. It must be to laws 1 and 2. The results obtained behave similarly to emphasised that M O D E L A and M O D E L B represent those discussed in the parametric study according to two different physical situations, and so only a law 3. qualitative comparison between these models can be Definitively, the main conclusions of this parametric made. In other words, to compare quantitatively the two study can be summarised as follows: models, it is necessary to determine a block elastic • An increase in kn~ reduces joint deformability and moduli in the case of M O D E L A which should be therefore decreases the displacement field. Also, a reduced to represent the equivalent continuum formed decrease in kni does not imply systematically an by the rock and the secondary fractures. For a complete discussion of equivalent moduli determination for increase in joint deformability. • An increase in kn~ is very similar to a decrease in fractured rock mass according to joint orientations, we refer the reader to Ref. [16]. Vm. The evolution of normal and shear displacement along • In any event, the results of this parametric study FA for M O D E L B has globally the same trend as that certainly indicate that a relatively good match of observed using M O D E L A. However, the magnitudes of computed and measured convergences, and these displacements increase strongly. This has been expansions is more likely to be obtained for an attributed to the deformability of secondary fractures. increase in kni and/or a decrease in gm in certain As an illustration, far from the excavation, the fault places. closure (FA) is near the value of - 0 . 0 2 5 mm in both Recall that the previous study concerns the model models, whereas the maximum opening in M O D E L B which contains only the major faults. However, as reaches 2.75mm. Also, the normal closure at the mentioned above, the medium is made up of some other roof-wall is almost near the value of 0.5 mm. discontinuities (diaclases). In order to take this fact into Figure 15(a) and (b) gives the variation of the account, we present in the following section, the results horizontal and vertical displacement along P,- and P~., obtained when secondary fractures are introduced, in respectively. Some of the general features of these curves are as follows: addition to the previous major faults.

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112

SOULEY et al.:

JOINT CONSTITUTIVE LAWS A N D SITE M E A S U R E M E N T S

(a) Far from the gallery, there is no noticeable difference between laws. A plausible explanation is that, fractures at these locations are not significantly subjected to the excavation conditions but only subjected to in situ conditions. In other words, displacement and/or stress variations induced by gallery excavating vanish at infinity. This fact is also illustrated on the stress curves reported in Fig. 16(a) and (b). (b) It appears from Fig. 15(a) and (b) that laws 1 and 3 differ only slightly except for the horizontal displacement magnitudes near the left side (x = 15 m). (c) Law 2 overestimates the displacement from 0% at y = 0 to 85% when y = 15 m, and from 26% ( x = 0 m ) to 61% ( x = 15m) along p~. and P,-, respectively. The maximum displacement values are reached at the gallery faces. It is interesting to note that the displacement curve forms match qualitatively for both profiles. (d) From results obtained, it follows that pronounced horizontal and vertical displacements occur along P,- and p~., respectively, when the secondary fractures are taken into account.

Also, in these figures, an increase in displacement magnitudes can be observed from the model boundaries (in situ conditions effects) to the centre of gallery (excavating effects). In conclusion, the effect of the laws on the displacement magnitudes after exacavating becomes dominant with the secondary fractures. Figure 16(a) and (b) shows the variation of the horizontal and vertical stress ratios along the vertical profile p,. As observed in M O D E L A, results indicate again that the stress ratio has to fall between a very narrow range of values as a function of law. For a fixed law, the horizontal stress try,.approaches the initial stress o-!,,. (ratio close t o one) determined by hydraulic fracturing for y ranged from 0 to 9 m. At the immediate wall location, horizontal stress increases rapidly, reaches a maximum near the average value of 26.2MPa (e.g. ratio = 1.5) and decreases very fast to the value of 0.6 MPa. The resulting vertical stress state decreases from a value of 11.6MPa to a negative value of - 0 . 3 M P a , which indicates a tensile stress, and is attained at the wall side. The vertical stress ratio less than one, results from the vertical decompression of rock mass along p, due to excavating. Note that this decompression is accompanied by a development of shear stresses and, therefore, the changes of principal stresses

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12

15

SOULEY et al.:

J O I N T CONSTITUTIVE LAWS A N D SITE M E A S U R E M E N T S

113

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-,- Law 3 Fig. 17. Influence of joint laws: M O D E L B: (a) Horizontal stress ratio along P,.; (b) Shear stress along P,..

Table 9. M O D E L B: calculated convergences and expansions as a function of law Convergence (mm)

Expansions Law 1

Both sides Dr/D7 D2/D8 D3/D9 Roof-wall D4/Dt0 Ds/Dll D6/D]_~

Law 2

Law 3

upper medium lower

13.57 21.77 20.23

14.87 26.15 24.06

9.98 15.23 14.26

left centre right

11.13 12.94 12.67

13.61 15.72 15.52

10.01 10.93 10.54

orientations and magnitudes. The maximum shear stress is reached near the value of 3.61 MPa and corresponds at the same location to the horizontal stress reduction. Figure 17(a) gives the variation in the horizontal stress ratio along the horizontal profile P, as a function of law and indicates that the horizontal stress is always compressive. It can be seen that the stress magnitudes for the three laws differ only slightly. As mentioned above for vertical stress along P,, horizontal stress

Upper Ci C2

C3 Lower Ci C2 C3 Left Ct C2 C3 Right Ci C2 C3

Law 1

Law 2

Law 3

2.84 2.37 1.73

3.56 3.00 2.19

2.79 2.28 1.57

8.02 7.91 7.16

9.68 9.55 8.61

6.15 6.07 5.27

5.32 4.80 4.15

6.95 6.32 5.45

4.25 3.76 3.10

14.67 14.00 13.39

17.21 16.44 15.72

9.67 9.02 8.40

decompression can be observed along Px with a ratio smaller than one. Also, this decompression is accompanied by substantial shear stresses which depend on the law particularly at the immediate left side location. The shear stress curve along P, is illustrated in Fig. 17(b). Also, examination of the vertical stress along P, (not reported herein) has shown that the maximum vertical stress component is greater than twice the initial stress magnitude at x = 14 m, whereas in other locations, the

114

SOULEY et al.: JOINT CONSTITUTIVE LAWS AND SITE MEASUREMENTS

vertical stress is approximately equal to the initial value (stress ratio equal to one). This is in accordance with the previous variation of the shear stress reported in Fig. 17(b). Table 9 summarises the effect of laws on the gallery convergences and displacement magnitudes in rock mass surrounding the gallery (expansions) with respect to the density of the fracture system. As is made clear by Table 9 above, the effect of laws becomes more considerable in comparison to the previous analysis ( M O D E L A). In fact, the convergences for law 2, effectively show an increase of their amplitude at the locations of measurements. For example, convergence at points D2/D3 or D3/D9 (lower) is increased by 70%, and for D6/D~2 (right) by 47% as compared to the convergence amplitude at the same location according to the reference law (e.g. law 3). In the case of law 1, the convergence is increased by approximately 40% and 16% at the gallery sides and the wall-roof, respectively, as can be seen in Table 9. R o o f expansions (upper) showed a slight difference between laws ( 2 4 % for C~ and C2), whereas the maximum difference between laws 1 and 3, or laws 2 and 3 can attain 50-87% at the right side. Clearly, Table 9 also shows that the change in the variation o f gallery boundaries or expansions in rock mass surrounding the gallery depend on the constitutive law. This was not correctly elucidated in the previous section. Therefore, the local comparisons presented herein show the effect of laws. 5. CONCLUSIONS The effect of joint constitutive laws on the modelling of a given jointed rock mass as a function of joint density was examined for a specific pilot gallery using the numerical code, UDEC. The two existing laws in U D E C were compared to the non-linear model of Saeb and Amadei modified by Souley et al. [14] for rock joints undergoing monotonic and cyclic loading sequences, and which was recently implemented in U D E C [15]. Also, some comparisons between calculations and observations were made. Based on the analysis reported here, the main conclusions of this study can be summarised as follows:

cannot be made easily. Nevertheless, a parametric study was carried out to assess the effect of law 3 properties on the magnitude and distribution of stresses and displacements around the cavity. In this parametric study, it is shown that an increase in kni or a decrease in Vm leads to a reduction in fault deformability, and therefore a reduction in the displacement values around the cavity. In particular, an increase in kni or decrease in Vm indicates that a relatively good match between computed and measured displacements in certain areas can be obtained. Comparisons between calculations and measurements were only carried out on the measured displacements. If we had had the stress measurements after excavating, the analysis would have been even more complete. Finally, this study shows that when the medium is slightly fractured, the three laws are qualitatively and quantitatively similar. MODEL

B

When secondary fracturing is taken into account, the results of simulations change significantly according to the law, particularly for displacement analysis. Indeed, joint density reduces the final stresses and increases the displacement magnitudes. In particular, the values of displacements are 10-13 times greater than the computed values in the case of M O D E L A. Contrary to M O D E L A where the global behaviour of the rock mass can be captured with sufficient accuracy using a simpler joint model, the results obtained according to M O D E L B indicate that the difference between laws can significantly increase with joint density. The study does show that, in particular situations, a simpler model (e.g. Mohr-Coulomb), with the correct parameters, matches the performance of more elaborate models (e.g. Cundall's C-Y model, Saeb and Amadei, or Souley et al.). Also, it illustrates the necessity of taking into account more realistic joint models and a good choice of the mechanical properties of joints for better predictions of displacements/stresses in complex practical problems. Acknowledgement--The authors wish to acknowledge the Agence

Nationale Pour la Gestation des D6chets Radioactifs for the financial support for this project. Accepted for publication 14 March 1996.

MODEL

A

It appeared that the values of stresses and displacements are identical for laws 1 and 2. These values do not change much with law 3. The principal difference between laws 1 and 2 on one hand, and law 3 on the other hand, was recorded in the opening of the faults at the level of the cavity. Consequently, laws 1 and 2 overestimate slightly the horizontal displacements at both sides of the gallery in comparison with law 3. A significant discrepancy in terms of observed measurements (convergences and expansions) between two adjoining stations was noticed. Consequently, comparisons between calculations and measurements

REFERENCES

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SOULEY et al.:

JOINT CONSTITUTIVE LAWS AND SITE MEASUREMENTS

5. Baroudi H., Piguet J. P., Arif I., Josien J. P. and Lebon P. Mod6lisation des cavit6s de stockage et prise en compte des discontinuit6s des massifs rocheux. Int. Congress on Rock Mechanics, Aachen, 16-20 Sept., Balkema, Vol. 1, pp. 675-678 (1991). 6. Kaneshiro J. Y., Madianos M. N., Rosidi D., White R. K. and McManus R. A. Discrete element modeling for the Beldon Siphon Rock slope remediation project--verification and reliability. Rock Mechanics (Edited by Nelson and Laubach), pp. 1137-1144. Balkema, Rotterdam (1994). 7. Hart R. D., Cundall P. A. and Lemos J. V. Formulation of three-dimensional distinct element model--Part II: mechanical calculations for motion and interaction of a system of many polyhedral blocks. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 25, 117-126 (1988). 8. Shi G. Discontinuous deformation analysis--a new numerical model for the statics and dynamics of block systems. Ph.D. thesis, University of California, Berkeley, 295pp. (1988). 9. Cundall P. A. A computer model for simulating progressive large scale movements in blocky rock system. Int. Proe. Symp. o f Int. Society for Rock Mech., Vol. 1, paper II-8 pp. 128-132. Nancy, France (1971). 10. Lemos J. V. A distinct element model for dynamic analysis of jointed rock with application to dam foundations and fault motion. Ph.D. thesis, University of Minnesota, Minneapolis 295pp. (1987). 11. Lorig L. J., Brady B. H. G. and Cundall P. A. Hybrid distinct element--boundary element analysis of jointed rock. Int. J. Rock Mech.Min. Sci. & Geomech. Abstr. 23, 303-312 (1986). 12. Cundall P. A. and Hart R. D. Analysis of block test No. 1 inelastic rock mass behavior: phase 2--a generalization of joint behavior. ltasca Consulting Group, Rockwell Hanford Operations, Subcontract SA-957, final Report, March (1984). 13. Souley M. Mod61isation des massifs rocheux fractur6s par la m6thode des 616ments distincts; influence de la Ioi de

14.

15. 16. 17. 18.

19. 20. 21. 22. 23. 24.

115

comportement des discontinuit6s sur la stabilit6 des ouvrages. Th~se de Doctorat de I'INPL, Nancy, France, 143pp. (1993). Souley M., Homand F. and Amadei B. An extension to the Saeb and Amadei constitutive model for rock joints to include cyclic loading paths. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 32, 101-109 (1995). UDEC. Universal Distinct Element Code (version 2.0). User's manual, ITASCA Consulting Group (1993). Huang T. H., Chang C. S. and Yang Z. Y. Elastic moduli for fractured rock mass. Rock Mech. Rock Engng 28, 135-144 (1995). Saeb S. and Amadei B. Modelling rock joints under shear and normal loading, lnt. J. Rock Mech. Min. Sci. & Geornech. Abstr. 29, 267-278 (1992). Souley M. and Homand F. Stability of jointed rock masses evaluated by UDEC with an extended Saeb-Amadei constitutive law. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 33, 233-244 (1996). Heliot D. Generating of blocky rock mass. Int. J. Rock Mech. Min. Sci. & Geornech. Abstr. 25, 127-138 (1988). Fine J. et Vouille G. La m6thode des 616ments finis appliqu6e ~ la m~canique des roches, l~cole Nationale Sup~rieure des Mines de Paris, Intern Report (1978). Arif, I. Mod61isation des milieux anisotropes et fractures. Th6se de Doctorat de I'INPL, Nancy, France, 196pp. (1991). Beer G. and Poulsen B. A. Efficient numerical modelling of faulted rock using the boundary element method. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 31, 485-506 (1994). Bandis S. C., Lumsden A. C. and Barton N. Fundamentals of rock joint deformation. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 20, 249-268 (1983). Rochet L. Propri6t6s m6caniques des discontinuit~s des massifs rocheux. In La m~canique des roches appliquke aux ouvrages due Gbnie Civil (Edited by Panet), 235pp. ENPC, Paris (1976).