Effect of intermediate principal stress on strength of anisotropic rock mass

PII:S0886-7798(98)00023-6

Effect of Intermediate Principal Stress on Strength of Anisotropic Rock Mass Bhawani Singh, R. K. Goel, V. K. Mehrotra, S. K. Garg, and M. R. Allu Abstract - - The Mohr-Coulomb criterion needs to be modified for highly anisotropic rock material and jointed rock masses. Taking % into account, a new strength criterion is suggested because both a2 and a3 would contribute to the ¢ormal stress on the existing plane of weakness. This criterion explains the enhancement of strength (%- a3) in the underground openings because a~along the tunnel axis is not relaxed significantly. Another caLtse of strength enhancement is less reduction t~nthe mass modulus in tunnels due to constrained dilatancy. Empirical correlations obtained from data from block shear tests and uniaxial jacking tests have been suggested to estimate new strength parameters. A correlation for the tensile strength of the rock mass is presented. Finally, Hock and Brown theory is extended to account for az A common strength criterion for both supported underground openings and rock slopes is suggested.

© 1998ElsevierScienceLtd

Introduction C

omputer modelling of rock structures has reached an advanced stage of development, in which analytical studies are used in the practical design of support systems for tunnels, caverns, slopes and dam foundations, etc. In mining engineering as well, computer modelling has gained wide popularity. Experience suggests that the reliability of computer modelling is increased if realistic constitutive equations and parameters are used in the analysis. Experience has led to the following strategy of refinement in the design of support systems. 1. In feasibility studies, empirical correlations may be used for estimating rock parameters. 2. For major projects, in-situ tests should be conducted at the design stage to determine the actual rock parameters. It is suggested that exLensive in-situ triaxial testing (with ~ , ~2 and ~3 applied on sides of the cube of rock mass) should be conducted because cr2is found to affect both strength and modulus of deformation of the rock mass. This is the motivation for the research, which points to the urgent need for in-situ triaxial tests. (Significant developments have already taken place on polyaxial tests on concrete, and have been reported in the Lecture Notes o f a Workshop on Behaviour o f Concrete Under Multiaxial States o f Stress

held at New Delhi, India, in 1987. For example, sophisticated equipment may be adopted for in-situ triaxial tests on rock masses; and the fziction between flat jacks and sides of the rock mass may be reduced significantly by inserting sets of three thin Teflon-coated steel plates in between them).

Present addresses: Bha.wani Singh, Professor, Dept. of Civil Engineering, University ofRoorkee, Roorkee - 247 667, India; R. K. Goel, Scientist, CMRI l~gional Centre, CBRI, Roorkee - 247 667, India; V. K. Mehrotra, Supdt. Engineer, Upper Ganga Canal Modernization, U.P. Irrigation, Roorkee - 247 667, India; S. K. Garg, SRF, Dept. of Civil Engineering, University of Roorkee, Roorkee - 247 667, India; M. R. Allu, Student, Dept. of Civil Engineering, University of Roorkee, Roorkee - 247 667, India.

Tunnelling and Underground Space Technology,Vol. 13, No. 1, pp. 71-79, 1998 C 1998 Elsevier Science Ltd Printed in Great Britain, All rig]~ts reserved 0886-7798/98 $19.00 +0.00

3. At the initial construction stage, instrumentation should be carried out in the drifts, caverns and other important locations in order to obtain field data on displacements, both on the supported excavated surfaces and within the rock mass. Instrumentation is also essential for monitoring construction quality. Experience has confirmed that in complex geological environments, instrumentation is the key to achieving a safe and steady tunnelling rate. These data should be utilized in the computer modelling for back analysis of parameters (Sakurai 1993). At present, non-linear back analysis is difficult and prone to mistakes and misinterpretations. However, experience on m a n y future projects should make advances possible. 4. At the construction stage, forward analysis of rock structures should be carried out using back-analyzed parameters of rock masses. Repeated cycles of back analysis and forward analysis will eliminate m a n y inherent uncertainties in the geological mapping and engineering behaviour of the rock mass. Stresses and strains vary considerably within the ground and with excavation phases. Failure will develop where stresses exceed the strength. The zone of failure will propagate with phases of excavation. Thus, the sequence of excavation would govern the shape of final failure zone and needs to be simulated. 5. The aim of computer modelling should be to design site-specific support systems, not simply to analyse the strains and stresses in the idealized geological environment. In the case of a non-homogeneous and complex geological environment, which is difficult to predict, slightly conservative values of rock parameters may be assumed for the purpose of designing site-specific remedial measures and accounting for inherent uncertainties in geological and geotechnical investigations. Stresses in the shotcrete may be reduced significantly if the spray of the shotcrete is slightly delayed in non-squeezing ground conditions. The delay, however, should be within the stand-up time. This basic "design as you go~ strategy of computer modelling has been suggested by Hoek (1997).

Pergamon

Effect of Intermediate Principal Stress on Tangential Stress at Failure in Tunnels

Strength Enhancement

The intermediate principal stress (o2) along the tunnel axis (Fig. 1) m a y be of the order of about half of the tangential stress (0 I)in deep tunnels. According to W a n g and K e m e n y (1995), o 2 has a strong effect on o~ at failure even if o 3 equals zero. Their polyaxial laboratory tests on hollow cylinders suggest the following strength criterion: 02 q¢

(1)

(~1 = q¢ + (.4 + f) (% + %)

for % < < %

where f = material constant (0.10-0.20) A = material constant (0.75-2.00) q~ = average uniaxial compressive strength of rock material (°2 = °3 = 0) for various orientations of planes of weakness In the case of unsupported tunnels, (~3 = 0 on its periphery. Thus, Eq. 1 simplifies to °-!1= 1 + A I~2](1-f) qc [ q~J

(2)

It may be inferred from Eq. (2) that o 2 will enhance (~3 at failure to the extent of 75-200 percent when °2 = qc' In fact, strength enhancement may be much greater, since propagation of fractures will occur only on the excavated face (Bazant et al. 1993). Murrell (1963) suggested a 100 percent increase in o 2 = 0.5~ and o3 = 0.

Uniaxial Compressive Strength of Rock Mass

(qcm.$,) The strain is a better measure of failure than the stress because the rock mass damage is directly dependent upon the strain. The strain at failure is bound to increase with more fractures in the rock mass. Therefore, the strength reduction factor (q=o/q¢) should be greater than the mass modulus reduction ~ctor (E ° ]Er). In dynamic conditions as well, the dynamic strength l:e~uction factor should be greater than the dynamic modulus reduction factor (Ed /Er). Ramamurthy (1993) and his co-workers (Roy 1993) have conducted extensive triaxial tests on the dry models of jointed rock mass using plaster of paris (qc= 9.46 MPa). They have simulated a wide range of rock mass conditions by varying joint frequency, inclination of joints, thickness of joint fillings, etc. Their extensive test data suggest the following approximate correlation, irrespective of the rock mass classification. qcmass _ Emass 0.7

where average uniaxial compressive strength (UCS) of model of jointed rockmass; q¢ -- average UCS of model material (plaster of paris); average mass modulus of deformation of model Emass of jointed rock mass; and Sr = average modulus of deformation of model material. The power in Eq. (3) varies from 0.5 to 1.0. Griffith's theory (Jaeger and Cook 1969) of tensile failure suggests that the power is 0.5, whereas Sakurai (1994), on the basis of back-calculation of several case histories, has found that the above power is about 1.0 for jointed rock masses. Therefore, the power of 0.7 in Eq. (3) appears to be realistic. Thus, Eq. (3) may be used reliably to estimate the uniaxial compressive strength of rock mass (q . . . . ) from the values of E .... obtained from uniaxial jacking tests. q0~,, =

----

7 2 TUNNELLING AND UNDERGROUND SPACE TECHNOLOGY

It is observed that as one goes farther inside an approach drii~, the joints become tighter and the length of joints decreases significantly because of tectonic movement. This observation confirms the authors' experience that both E and q¢.... are signifcantly higher in deep tunnels than thosa~ near ground and slope for the same value of rock mass quality except near faults and thrusts. Further, the dilatancy in the rock mass is constrained in tunnels. Therefore, failure may not occur along rough joints and the strength of rock mass may be nearly equal to the strength of the rock material (Pande 1997). Considerable strength enhancement has been observed by Singh et al. (1997). Therefore, on the basis of careful back analysis of data collected from 60 supported tunnels, they recommended that: (i) The mobilised UCS is about 77Q1~3MPa for Q < 10, 100 > q¢ > 2MPa, Jw = 1 and Jr/Ja < 1.5. Eq. (3) also predicts the same order of UCS. (ii) The mobilised UCS for good and massive rock mass is about 77.f Qm MPa. Where f = %/100 for Q > 10 and q¢ > 100 MPa; otherwise, f¢ = 1. Laubscher (1984) found UCS equal to qc (RMRrating for q~)/10~ for hard rock mines, which is nearly same as the above UCS. (iii) The mobilised residual cohesion c is approximately equal to 0.1 MPa and is not negligible. The mobilised residual angle of internal friction c is about 10 ° less than ~p but greater than 14 °. (iv) Hoek and Brown (1982) criterion is applicable to the rock slopes and open cast mines with weathered and saturated rock mass. Block shear test results suggest qcmass to be 0.387Q ~ MPa, as joint orientation becomes a very important factor due to unconstrained dilatancy and negligible intermediate principal stress unlike in tunnels. Therefore, the block shear test is recommended only for slopes and not for supported deep openings. The strength is very low because dilatancy is unconstrained in rock slopes as normal stresses on joints are governed by weight of the wedge (Pande 1997). The above strength parameters are intended only for 2D stress analysis of underground openings.

Proposed Strength Criteria for Anisotropic Rock Mass The object of continuum characterization is to find overall constitutive equations between average stresses and average strains within a unit cube of a jointed rock mass. The loaded area must be sufficiently large compared to the joint spacing (Singh 1973a). Coulomb's theory for an anisotropic material predicts that the average shear strength ((~i - o3) will depend upon the average confining pressure (a. + o3)/2 because failure will occur only along pre-existing planes of weaknesses and not along critical plane, as in isotropic material like soil. This hypothesis needs to be examined because it has important applications in rock mechanics. Consider a cube of rock mass with two or more joint sets as shown in Figure 1. If high intermediate principal stress is applied on the two opposite faces of the cube, then the chances of wedge failure are generally greater than the chances of planar failure as found in the triaxial tests. The shear stress along the line of intersection of joint planes will be proportional to (~1- (~3because (~s will act to reduce shear stress. The normal stress on both the joint planes will be proportional to (o° + o3)/2. Hence, the criterion for peak failure may be as fo|lows:

V o l u m e 13, Number 1, 1 9 9 8

1

?

Mo,d¢ of failure in rock mass with 2 Joint sets

(b)

~--

o-~ :pO

max Cross- section Direction

L o ngit udi n el- s ¢ction of

cq-, o~ and

o~ in t,h¢

tunnel

(d) Figure 1. Mode of faEure of anisotropic rock mass under polyaxial stress conditions.

Volume 13, N u m b e r 1, 1998

r

TUNNELLINGAND UNDERGROUNDSPACE TECHNOLOGY73

U 1 - (~3 = q c m a ~

+

A (a2 + ~3)

(4)

where qcmass

= average uniaxial compressive strength of

block of rock mass for various orientations of principal stresses; al, ¢~2,~3 = final effective principal stresses which are equal to in-situ stress plus induced stress minus seepage pressure; A = average constant for various orientations of principal stresses (1 - 6) 2 sin Cp ( 1 - sin Cp)

the residual strength criterion, the rock mass is governed by Mohr's theory as follows: al - a3 = %r + a3 Ar (5) where qcr = average residual uniaxial compressive strength of rockmass 2crcos¢, ( 1 - sin ~, ) c = 0.1MPa = 0 if e 1 - e3 > critical strain _

Ar=

(

2 sin Cr 1-sinCr)

(6a)

(6b)

~,= ¢~p-10 o > 1 4 ° ~p = peak angle of internal friction of rock mass Obviously, Eq. (5) is also applicable for peak shear strength of a brecciated, highly fractured and weathered The proposed strength criteria reduces to the Mohrrock mass because such a rock mass is almost like isotropic Coulomb criterion for triaxial conditions. soil mass. Thus, significant strength enhancement in the supAt very high confining pressures, a 2 and cys, the rock ported underground opening is expected to result from a 2or mass behaves like a ductile material (Roy 1993). Its the in-situ stress along tunnels and caverns axes (Fig. ld). residual strength is also equal to the peak strength. It It is understood t h a t ~2 pre-stresses rock wedges and preseems t h a t at such high confining pressures Mohr's theory vents their failure in both the roof and the walls. However, should be valid. a s is released because of the stress-free excavation boundFor more accurate analysis, an anisotropic strength aries. In the case of rock slopes, on the other hand, both a= criterion (Eq. 4) m a y be developed. The uniaxial compresand a 3 are negligible, and so there is insignificant or no sive strength (c~ ,) and mass modulus of deformation enhancement of the strength in the case of slopes. There( E ,) will depen~"~upon orientation of principal stresses fore, a block shear test on the rock mass gives realistic wit~ respect to joint planes. Moreover, constant A will also results for rock slopes and dam abutments only, because ~2 change with the orientation of the joint planes. However, is zero in this test. Thus, Eq. (4) m a y give a general failure anisotropic strength criteria m a y be too complicated and criterion of jointed rock masses for underground openings, impractical for the purpose of designs. rock slopes and foundations. A n o t h e r cause of s t r e n g t h e n h a n c e m e n t is higher uniaxial compressive strength of the rock mass ( q ~ , . ) due Evaluation of Parameter A from Distressed to higher E . , caused by constrained dilatancy and reUndergroundOpenings strained fracture propagation near the face of the excavation only in the u n d e r g r o u n d structures. In rock slopes, F a i l u r e o f t h e r o c k mass m a y t a k e place a t some locations E is found to be much lower because of complete relaxwithin deep tunnels and caverns. The stress analysis will a t ~ n ofin-situ stress and low confining pressures ~2 and a 3 give m a x i m u m tangential stress (a. ) in the distressed 1 . . . . ~max and excessive weathering. Therefore, q ..... will also be low ocatlons (Fig. ld). In such conditions, the parameter A in near rock slopes. Eq. 4 may be estimated as follows: In the back analysis, A, E and q . . . . should be derived very carefully from Eq. 3 and 4 and the feedback of instrumentation data at the beginning of the construction stage. With these values, forward analysis 0-672 / should be attempted taking into account - 2.8 ( ~ + 0 , 7 ) T n r n c sequence of excavation, as mentioned above. 16 The proposed strength criteria is different from Mohr's strength theory, which tl 4 works very well for soils and isotropic mateu rials. There is a basic difference in the ~12structure of soil and a rock mass. Soils generally have no pre-existing planes of weaknesses, and therefore planar failure ~10 -[sat = can occur on a typical plane with dip direction towards ~3- However, rock has preexisting planes ofweaknesses such as joints ~ 8 and bedding planes. Failure occurs mostly along these planes of weaknesses. In the = 6 triaxial test on arock mass, planar failure takes place along the weakest joint plane. In a polyaxial stress field, the wedge type of failure may be the dominant mode of failure, if o 2 >> ~3" Therefore, Mohr's theory 2 I' t • may not be valid for anisotropic and jointed I rock masses. I i I 0 2 4 6 8 10 12 14 16 18 20 However, in the event of residual failure, 2 Normal stress (Or), k g / /c m 2 - - - - ~ considerable slip has taken place along joints and m a n y new fracture planes have developed, reducing the rock mass to almost an Figure 2. Estimation of tensile strength of rock mass from Mohr 's isotropic soil-like structure. Therefore, in Envelope.

74 TUNNELLINGAND UNDERGROUNDSPACE TECHNOLOGY

Volume 13, N u m b e r 1, 1998

A _-> 2 ( o ~ . - qCm., )

o.l,o

Po (7)

qtj : 0.029.1(. O, 0.31

where Po = horizontal in-.situ stress along the

X

X Dry ('Mchrotra, 1992)

Sat (Mehrotra 1992) axis of the tuztnel cavern and 0.3o shaft. +__~ Tensile stress in wall of cavern of BQSlmnII Prdj¢ct (Singh I~ Agorwo11995) It is unlikely that failure of the rock mass in compression will take place where the maximum in-situ horizontal stress acts along the axis of tunnels and caverns. Eq. (7) ~" 0.20 predicts that squeezing conditions may occur in the tunnels Where parameter A is relatively small because the in-situ stress along the tunnel axis will not cause much strength enhanement. A survey of reported case histories of squeezing ground in the O.lC Himalayas and in other countries indicates that Jr/Ja was less than 0.5 (Goel 1994). In other words, squeezing had taken place where Jr/Ja was less than 0.5 and A was small. The o-oo 0"0 1.0 2.0 3.0 40 5.0 6.0 theory also predicts that squeezing is unX'. Q0.31 likely to occur where J:r/Ja is higher than 1.0 even if overburden H :> 350 Q0.33,because of Figure 3. Plot between % and QO.3£ the enormous strength enhancement due to 02 along the tunnel axis. It appears that rock burst may occur where A is found to be high. This is because strain energy released per unit area of excavation could be higher than the permismine pillars may be eliminated. The distinct element model sible limit. will then also enable the designers to predict the size of selfsupporting tunnels more realistically. The tensile stress in the roofofa tunnel of span B will be of the order ofyB in the Tensile Strength of Discontinuous Joints and vertical direction. Equating this with %., the span of selfRock Mass supporting tunnels obtained from Eq. (8~ would be 2.9 Q031 The joints inside tunnels are generally discontinuous m. Barton et al. (1974) found the self-supporting span to be joints, i.e., the joints exposed to a length less than, say, 5 m, 2 Q0.4 m. This comparison is very encouraging. Wedge except for bedding planes. Such discontinuous joints have analysis considering qt" and in-situ stress along the tunnel tensile strength. Mehrotra (1996) has performed 44 shear axis will give the spa~ of a self-supporting tunnel more block tests on both nearly dry and saturated rock masses. accurately. He has also obtained non-linear strength envelopes for various rock conditions. These strength envelopes were Tensile Strength of Rock Mass extrapolated carefully in the tensile stress region so that it It should be noted that the tensile strength of the rock is tangential to Mohr's circle for uniaxial tensile strength, mass under compressive polyaxial stress conditions will be as shown in Figure 2. It was noted that the non-linear higher than %. Figure 4 shows the relationship between o~ strength envelopes for both dry and saturated rock masses and G3 at failure using Eq. (4) both in compression and in converged to nearly the same uniaxial tensile strength (qt,) tension. It is felt that the value of A in tension is much across discontinuous joints within the blocks of rock masse~. higher than in the compression. Thus, it follows that It is related to Barto;a's rock mass quality Q (Fig. 3) as follows: ~1 + qtma, = q. . . . . + O2 ( A ) - qt . . . . (~__At) % =0.029 7 QO.31fc MPa (8) where qtmass = qtj + D [ a2 ( ~ - ) - (31 ] (9) fc = correction factor for UCS; = 1if Q< 10 and qc < 100 MPa; and D= qtj = 1 =(0.1-0.5) qcmass (0.5At + 1) = 1-~ if Q > 10 and qc > 100 MPa.

The tensile strength across discontinuous joints is, therefore, not zero as generally assumed; it is found to be significant, especially in hard rocks. In the Himalayan region, thin bands of weak rocks are found within good rock masses, and sometimes these bands lie just above the roof' of tunnels and caverns. Separation between stronger rock above and weak bands occurs where the tensile stress is grcater than % of the weak band. Longer rock bolts are needed soon after excavation to halt this separation, as well as to stabilise the roofs. Thus, tensile strength q~ should be estimated for the minimum value of Q in the bd_nd and adjoining rock mass. By considering the ~ensile strength across discontinuous joints in the distinct element model of rock masses, artificial buckling ofjointed rock columns in the walls of caverns and

Volume 13, Number I, 1998

It may be seen that the tensile strength of rock mass qtm,, is enhanced by compressive stress o2. Thus, in the conventional hydraulic fracturing test, the maximum horizontal stress may be underestimated. Past experience shows that the hydraulic fracturing test slightly underestimates the stress gradient for Pm~. and Ph,,, in comparison to estimates by the overcoring method (Stephansson 1993). The limited data of India's National Institute of Rock Mechanics (NIRM) (Sengupta 1997) indicate that hydraulic fracturing appears to give enhanced tensile strength of the rock mass (qm,,) which is equal to the difference between fracture initiation pressure (Pf) and fracture reopening pressure (Pr)" This tensile strength is found to increase with Pf and decrease with overburden pressure. This trend is the same as that suggested by Eq. (9). Further work is in

TUNNELLING ANDUNDERGROUND SPACETECHNOLOGY 75

progress at NIRM on the polyaxial criterion of tensile strength of the rock mass.

Strength of Mine Pillars Eq. (4) also suggests that the crushing strength of pillar (a~) is likely to be increased significantly by in-situ stress ((~2) along the axis of mine road ways• On excavation, ((~3)is released but ~2 is present. Thus, the strength of pillar in hard rock mines may be obtained by ~i = qc=,, + A ( ~ )

(10)

The philosophy of mine planning should be to take advantage of in-situ horizontal stress by preserving it as much as possible. The conventional crushing strength should be replaced by new in-situ biaxial tests on a block of rock mass with ~2 acting on opposite sides.

Validation of Proposed Strength Criterion Wang and Kemeny (1995) have conducted extensive polyaxial tests on hollow cylinders of Apache Leap tuff. The tuff is an anisotropic rock material and therefore is likely to show the dominant role of~ 2 on strength. Their data for t~sts are plotted in Figure 5 between ~ - ~3 at failure (Allu 1997). Figure 5a shows plot for ~ and a~ when a 8 = 0. Figure 5 shows a very good fit for the proposed strength criterion (Eq. 4) with A = 4.9. Prior to this, other combinations such as ~. and ~ , ( ~ 1 and (~2+ ~3, ~. - ~ , and a~+ (~ + (~3were also p]~otted, ~ u t these plots ~id not give better correlations. Further, their data shows that triaxial tests give same value of A as that from polyaxial tests. It should be noted that in hard rocks the correlation is likely to be non-linear. Their linear approximation is accepted for the practical design of rock structures. The study shows that the time is ripe to move to for insiSu polyaxial tests for the future advancement of rock mechanics.

Correlations for Rock Mass Deformation Modulus (Emas$) Extensive uniaxial jacking tests have been conducted by the U.P. Irrigation Research Institute in various dam projects

in Himalaya (Mehrotra 1993). The analysis of 30 sets of test data by Singh (1997) gave the following approximate correlation for the rock mass modulus of deformation (E .... ). E .... = 0.07 E r MR°.~e QO.31 GPa (11) where MR = modUlus ratio of rock material/1000 = Er/qc;, Er = modulus of elasticity of rock material in GPa; q~ = UCS of rock material in MPa. It may be mentioned that total displacement of the plate for all cycles of loading and unloading was considered in estimating the value of E ma s s . Some engineers believe that • . . . E d obtained from the last cycle of umaxial jacking test should be used in Eq. (11) in place ofE because the first cycle is not reliable. The modulus of deformation of a jointed rock mass has also been obtained from tunnel closure data in about 35 instrumented and supported tunnels, with a correlation coefficient of 0.85 for nearly dry and weak rock masses as follows (Singh 1997): Era, ' = Q°36H°~ GPa for Q < 10 (12) E=... = 0.3 H ~ i 0 (RMR- 20/38) a

GPa

(13) (Verman 1993, 1997)

= 0.16 to 0.3

where H = overburden in meters > 50 m. It may be noted that'the modulus of deformation (E ~,,) depends upon the confining pressure and thus upon the overburden (H). It is therefore suggested that the mass modulus of deformation may be related to mean confining pressure as follows. E

~. r~2+a~ ]~ ~ " = - ° ~--~Y/-,

(14)

where Eo = modulus of deformation at mean confining pressure Pa; Pa = atmospheric pressure (0.1 MPa) for normalization; a = average constant (0.16 to 0.3). The modulus of elasticity of rock mass (E) was also obtained from field data from 30 uniaxial jacking tests by Mehrotra (1993), with a correlation coefficient of 0.96 for both dry and saturated rock mass as fellows (Singh 1997): E = 1.5 Q0.SEr0.14 GPa (15)

Emass vs E °

A

C~crnasS

~n~ss ~n~le

0

O-3 compressive

Figure 4. Proposed bilinear strength criterion of anisotropic jointed rock mass (Allu 1997).

76 TUNNELLINGANDUNDERGROUNDSPACETECHNOLOGY

In the stress analysis, E should be used where stress relaxation occurs. In the case of a tunnel at shallow depth below ground level, the vertical stress is reduced in comparison to that in the original state before the excavation, whereas the horizontal stress increases. This explains the stress induced anisotropy. Thus, the horizontal modulus of deformation is E m and the vertical modulus of deformation'~s E . It is suggested tIiat ~B should be used in seismic analyses of concrete dams because of the fast loading and unloading of the rock foundation• Similarly, for machine foundations E should be used to estimate the coefficieert ofuniferm compression since the block resonance test is not suitable for rock masses. It may be noted that in sophisticated machines, dynamic amplitude is not allowed t~. exceed even 0.1 mm for perfect machining.

Volume 13, Number 1, 1998

Low Shear Modulus of Jointed Rock Mass

350.00

A jointed rock mass is highly anisotropic because the shear stiff~.Less of joints is much less than its normal stiffness. Therefore, the shear modulus of the rock mass is much lower than its modulus of deformation. This special nature of anisotropy explains the different shapes of the pressure bulb below a strip foundation. It also predicts that subsidence above an underl~ound opening, particularly a mine opening, is restricted to a narrow width above the opening. The value of the shear modulus (G) has been estimated from a few case histories as follows (Singh 1973b).

o~=o

/

.?

300.00

250.00 L~

/

2(?0-00

X

G = Em~ 10

150.00

For mine subsidence, Bahuguna (1993) has back-analysed mo~"ethan 100 cases of coal mine subsidence and has found G/Ed to be dependent on the type of rock mass. Obviously, the axis of anisotropy should be assumed along a bedding plane or critical joint set.

where Phmax = average effective maximum horizontal in-situ stress in the rock mass on ground surface. Phmi, = average effi.~ctiveminimum horizontal in-situ stress in the rock mass on ground surface.

oT

+ s

1°

m

mr

=

S0.33

Volume 13, Number i, 1998

-F 2 . 4 7

125.00

(o"~2-l-o- ~)

x

x xx

x

y,

x

30O

x

xX x xX x~ ( x

x

x x

R x

x

200

100

1

J

20

/-,0

(b)

I

60 o.-~--t.-o-~

I

80

I

lOO

120

F i g u r e 5(a): P l o t b e t w e e n cT1 a n d Gs (G3 = O ) f o r p o l y a x i a l tests. F i g u r e 5(b): P l o t b e t w e e n cz2 a n d G3 a n d G1 - c[3 f o r p o l y a x i a l tests.

(18)

= 0.16MR°'SQ°'22 fromEq.(ll)

2 = a3 = 0

= strength reduction factor

:qc

I

where = LE - - - r - - J r

o'~

x

Hoek's (1994) critex:ton may be generalised for anisotropic and jointed rock mass by replacing os by average confining pressure as follows:

s n IN.... 10.7

-

x x

Generalised Hoek and Brown's Criterion

[

100"00

400

Thus, P h ~ may exceed q ~ , , at surface in the rock mass; and Mohr's theory, which suggests that P ~ cannot exceed q~,,, (= 0.387Q '~ MPa), appears to be invalid. The insitu stress measurements (Sheorey 1994) suggest that q is about 1 MPa and A is 2.9 in Eq. (17). This is an encouraging observation.

((~2+ (~3) 2 qc

t

I 7SO0

(17)

Ph~ < - q ..... + Ph~i~(~/,, A

O'i ----($3 + qc m

I 50.00

SO0,

In-Situ Stress at Ground Surface Tectonic movement across plate boundaries leads to the development of high horizontal compressive stresses even near the surface of the upper plate. One may assume that the stress near the plate boundary may be in a state of failure in the upper plate. Insitu stresses at the ground surface may be related to Eq. (4) with o s = 0; thus, it appears that

215 "00

0.00 (o)

(16)

(19) qcmass q¢ (20)

n

= 0.5 = 0.65

RMR < 0.6 for RMR < 25

- 2--~0-

(21) The above strength criterion reduces to the original equation of Hoek (1994) for triaxial stress conditions (°2 = o3). Thus, the generalised non-linear criterion of strength takes into account the strength enhancement due to a high value of ~2 and a higher value of E m g underground openings. In-situ triaxial tests will greatl~V~elp in developing the non-linear polyaxial strength criterion.

Dynamic Strength of Rock Mass There is likely to be dynamic strength enhancement due to the impulsive seismic loading by a high intensity earthquake with nearby epicentre. It appears logical to assume that dynamic strain at failure should be of the same order as the static strain at failure for a given confining stress. Dynamic strain at failure should be proportional to the modulus of elasticity of the rock mass (Ee), and static strain at failure should be proportional to E . ,. Eq. (3) tentatively suggests the following correla"t'~on for dynamic strength (qcmdy~)enhancement,

TUNNELLINGANDUNDERGROUNDSPACETECHNOLOGY77

Rapid Sympathetic Failure of Rock Mass within the Entire Failure Zone in Tunnels In ductile materials, failure is dependent only on the state of stress or strain at a particular point. The failure behaviour does not depend upon the state of stresses at nearby points. In the case of brittle materials like rocks, there may be unstable and widespread fracture propagation in the entire shear failure zone, starting from the point of maximum shear strain. In conQ. tinuum analysis, complex and unstable fracture propagation cannot be simulated easily. In work softening analysis as well, this type of simulation is unsatisfactory at very large strains because local loss of strength cr=o does not automatically lead to sudden loss of cohesive strength throughout the shear failure zone. A continuum characterization of a zone of unstable fracture propagation may be poscr~O sible by considering sympathetic failure proOnset of sympathetic failure cess. In such cases, complete loss of residual 6 elo cohesion of rock mass may be assumed to occur at a very fast rate throughout the shear Tunnel closure failure zone where the deviatoric strain % £o) exceeds a critical limit at any point within Figure 6. Effect of sympathetic failure of rock mass on ground responce tl~is failure zone. It appears that unstable curve. fracture propagation will be governed by the strain limit rather than the stress limit. The sympathetic failure is like the weakest link faihire in which the failure of the weakest link qcmdyn_( Ee ) 0'7 results in complete loss of}trcngth of the entire chain. This sympathetic failure process is observed in tunnelling through - ~E--~ / (22) highly squeezing ground conditions with an inadequate and where delayed support system. Figure 6 shows a typical ground E e = elastic or instantqneous modulus of elasticity of rock response (reaction) curve-for such a tunnelling condition mass from Eq. (15). (Singh et al. 1997). It may be noted that there is a sudden In seismic analysis, dynamic strength enhancement may rise in the support pressures after about 6 percent of tunnel be quite high, particularly for weatheredrock mass, as the wall closure. Of course, the critical sympathetic strain instantaneous modulus o f elasticity will be much higher failure limit will be different for hard rocks, medium hard than the long-term mass modulus E . rocks, etc. According to U.S.B.R. (1977) deslgn ~ practlce, " the coheIn the non-linear analysis, this rise could not be simusive resistance associated with intact bridges across disconlated satisfactorily even after considering the work softentinuous joints m a y b e quite high at low normal stresses, ing constitutive equations of the rock mass. This would be such as in the case of wedges in tunnels. possible if the critical strain of sudden sympath.etic failure In the case of joints filledwith overconsolidated siltor is assumed in the analysis. If the deviatoric strain % - ~3) clay with PI > 5, there m a y be some cohesive resistance. or (£e - £r) is less than the critical strain, analysis may be Under seismic loading the dynamic cohesion m a y increase perormed assuming appropriate residual cohesion of the enormously, rock mass. At the event of overstraining at any point Cdyn ~- C . . . . lidatedundrained (23) beyond the critical strain, a rapid loss of residual cohesion Particles of soil and rock take some time to slip with to zero may be assumed at all points simultaneously within respect to each other because of the inertial forces of parthe failure zone whenever the deviatoric strain exceeds the critical limit at any stage of excavation. ticles during seismic loading and creep movement. Therefore, m u c h higher dynamic stress is needed to develop The proposed sympathetic failure criterion may thus failure strain. Consequently, dynamic strength enhanceenable designers to advise caution on the permissible limit ment in cohesion is likely to be very high along filled of closffres/displacements to field engineers when thick discontinuities in the rock mass. plastic shear zones are encountered or when highly squeezing Clayey gouge in shear zones is generally highly overground conditions are encountered in tunnels and caverns. consolidated. The over consolidated (PI > 5) clayey soils generate more and more negative pore water pressure with Conclusions successive cycles ofimpulsive seismic loading. The negative There is an urgent need to conduct the in-situ triaxial pore water pressure shows up as enormous increase in the tests in the field, particularly in major projects. The triaxial apparent dynamic cohesion. (C¢o~l~,~d~d~..d)" test data are likely to give realistic constitutive equations According to Jaeger and Cook (1969), enhancement in for highly anisotropic rock masses. The expectation is that dynamic angle of friction of smooth rock joints m a y be the field data should show the dominant role of o 2 on the about 2 °. strength of rock mass (o~-on). The following conclusions Extensive research is urgently needed to obtain more presented in support of the need for further research have realistic correlations for dynamic strength enhancement. been justified by laboratory polyaxial tests on anisotropic Hope is high because concrete dams have performed well rock material. despite the effects of earthquakes of very high intensity.

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Volume 13, Number i, 1998

(i)

Mohr's t h e o r y needs to be modified for highly anisotropic m a t e r i a l such as a j o i n t e d rock mass, b e c a u s e t h e failure will be governed b y the w e a k e s t p l a n e in t h e rock mass. In such cases, t h e role o f o 2 should be n e a r l y equal to t h e role of. In o t h e r words t h e s t r e n g t h ( o ~ - o 3) will be governed by t h e m e a n confining p r e s s u r e [(o s + o2)/2] as follows, a n d not by o s alone: o l - o~, = qcma. + A [ ~

q. . . . . qb

]

[Ema~l °'7 --L Er Jo2=o3=0

The proposed extension of Mohr-Coulomb's theory for h i g h l y anisotropic rock m a s s m a y g e n e r a t e confidence a m o n g t u n n e l / m i n e engineers a n d geologists. (ii) H o w e v e r , Molar's t h e o r y will be applicable to t h e r e s i d u a l s t a t e of failure since t h e rock m a s s is likely to be r e d u c e d to a n isotropic soil m a s s (Eq.5). (iii) The tensile stxength is found to be significant according to Eq. (9). I n t h e d i s t i n c t e l e m e n t model Eq. (8) is r e c o m m e n d e d for e s t i m a t i n g tensile s t r e n g t h across discontinuous joints: q~ = 0.029 Q0Sl fc MPa. (iv) The modulus of deformation of rock mass is found to be pressure dependent, particularly for poor rock masses, according to Eqs. (12), (13) and (14). (v) Generalised Hoek and Brown's criterion is also suggested to account for o s (Eq. 18). Approximate correlations are also suggested for parameters m, s ands. (vi) Sympathetic failure of the rock mass m a y occur within a broken zone in squeezing ground conditions wherever the strain exceeds a criticalstrain. In this event, widespread fracture propagation will take place throughout the broken zone with zero cohesion, followed by a sudden rise in the support pressure. Hence, efforts should be m a d e to predict and avoid such catastrophic failure.

Acknowledgment The a u t h o r s a r e d e e p l y g r a t e f u l to Prof. J. N e d o m a , Prof. E. Hoek, Prof. G. N. Pande, Prof. V. S. V u t u k u r i , Dr. V. I. Rechitski, Dr. V. M. S h a r m a , Dr. B. D a s g u p t a , Prof. M. N. V i l a d k a r , Sri N. K. S a m a d h i y a , Dr. U. N. S i n h a , Dr. R. A n b a l a g a n , Prof. P. K. J a i n , Prof. R a m e s h C h a n d r a , Dr. G. C. N a y a k , Dr. A. K, D h a w a n , Dr. S u b h a s h M i t r a , a n d m a n y r e s e a r c h e r s for t h e i r m o r a l s u p p o r t a n d v a l u a b l e suggestions.

References Allu, M. R. 1997. Strength Characteristics of Jointed Rock Mass, 60. M.E. thesis,Civil Engineering Department, University of Roorkee, India. Bahuguna, P. P.; Singh, Bhawani; Srivastava, A. M. C. and Saxena,N. C. 1993. Semi empirical method for calculation of maximum subsidence in coal mines. Geotechnical and Geological Engineering 11,249-2q31. Barton, N.; Lien, R.; and Lunde, J. 1974. Engineering classification of rock masses for design of tunnel support. Rock Mechanics 64, 189-236. Bazant, Z. P.; Lin, F. B.; and Lippmann, H. 1993. Fracture energy release and size effect in borehole breakout. Int. Jr. Num. & Analytical Methods in C-eomech. 17, 1-14. Goel, R. K. 1994.Correlations for Predicting Support Pressures and Closures in Tunnels, 347. Ph.D. thesis, Visvesvaraya Regional Engineering College, Nagpur, India.

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Grimstad, E. and Bhasin, R. 1996. Stress-strength relationships and stabilityin hard rock. Proc. Conference on Recent Advances in Tunnelling Technology, New Delhi, India, Vol. I, 3-8. Hoek, E. and Brown, E. T. 1982. Underground Excavations in Rock (rev. ed.). London: Institution of Mining and Metallurgy. Hoek, E. 1994. Strength of Rock and Rock Masses. ISRM News Journal 2, 4-16. Hoek, E. 1997. Tunnelling through rocks. Lecture, University of Roorkee, Roorkee, India, Feb. 7, 1997. Jaeger, J. C. and Cook, N. G. W. 1969. Fundamentals of Rock Mechanics, Art. 3.4. London: Methew and Co. Ltd. Laubscher, D. H. 1984. Design aspects and effectiveness of support system in different mining conditions. Trans. Inst. Mining and Metallurgy 93, A70-81. Lecture Notes of a Workshop on Behaviour of Concrete under Multiaxial States of Stress, 1.1 to 5.75. Organised by Central Board of Irrigation and Power and Central Soils and Material Research Station, New Delhi, India, 1987. Mehrotra, V. I~ 1993. Estimation of Engineering Parameters ofRock Mass, 267. Ph.D. Thesis, University of Roorkee, Roorkee, India. Mehrotra,V. K. 1996. Failure envelopes for jointed rocks in Lesser Himalaya. Jrnl of Rock Mechanics and Tunnelling Technology (Indian Society of Rock Mechanics and Tunnelling Technology) 2(1), 59-74. Murrell, S. A. K. 1963. A criterion for brittle fracture of rocks and concrete under triaxial stress and the effect of pore pressure on the criteria. Fifth Syrup. on Rock Mech., Univ. of Minnesota (C. Fairhurst, ed.), 563-577. Pergamon Press: Oxford, Pande, G. N. 1997. SQCC Lecture on Applications of the Homogenisation Techniques in Soil Mechanics and Structure Masonry, University of Roorkee, India, Sept. 26, 1997. Ramamurthy, T. 1993. Strength and modulus responses of anisotropic rocks. Comprehensive Rock Engineering, Vol. 1, 313-329. ROy, Nagendra. 1993. Engineering Behaviour of Rock Masses through Study of Jointed Models, 365. Ph.D.Thesis, Civil Engineering Department, I.I.T., New Delhi, India. Sakurai, S. 1993. Back Analysis in Rock Engineering. ISRMNews Journal 2(2), 4-16. Sengupta, S. 1997. Personal communications. Sheorey, P. R. 1994. A theory for insitu stresses in isotropic and transversely isotropic rock. Int. J. Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 31(1), 23-34. Singh, Bhawani. 1973a. Continuum characterization of jointed rock mass--Part I. Int. J. Rock Mechanics and Mining Sciences and Geomech Abstracts 10, 311-335. Singh, Bhawani. 1973b. Continuum characterization of jointed rock mass--Part II: Significance of low shear modulus. Int. J. Rock Mechanics and Mining Sciences and Geomech Abstracts 10, 337-349. Singh, Narendra and Agarwal, Sanjay. 1995. FEM Analysis and design of roof support system for power house and transfomer hall cavities of Baspa - II project. Conf. on Design and Construction of Underground Structures, New Delhi, India, 323-334. Singh, Bhawani ; Viladkar, M. N.; Samadhiya, N. K.; and Mehrotra, V. K. 1997. Rock mass strength parameters mobilized in tunnels. Tunnelling and Underground Space Technology 12(3). Singh, Suneel. 1997. Time-Dependent Deformation Modulus of Rocks in Tunnels, 65. M.E. Thesis, Civil Engineering Department, University of Roorkee, India. Stephansson, O. 1993. Rock stress in the Fennoscandian Shield. Comprehensive Rock Engineering, Vol. 2, 445-459. U.S.B.R. 1997. Design of Arch Dams. Design Manual for Concrete Arch Dams. A Water Resources Technical Publication. Denver, Colorado. Verman, Manoj. 1993. Rock Mass - Tunnel support Interaction Analysis, 267. Ph.D.Thesis, University of Roorkee, Roorkee, India. Verman, Manoj; Singh, Bhawani; Viladkar, M. N.; and Jethwa, J. L. 1997. Effect of tunnel depth on modulus of deformation of rock mass. Rock Mechanics and Rock Engineering 30(3), 121-127. Wang, R. and Kemeny J. M. 1995. A New Empirical Failure Criterion Under Polyaxial Compressive Stresses. XXXV U.S. Symposium on Rock Mechanics, Reno, Nevada (J J. K. Daemen and R. A. Schultz, eds.), 453-459.

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