Design for grouted rock bolts based on the convergence control method

Int. J. RockMech. Min.Sci. &Geomech.Abstr.Vol. 27. No. 4. pp. 269-281, 1990

0148-9062/90$3.00 + 0.00 Copyright ~ 1990 PergamonPress plc

Printed in Great Britain.All rights reserved

Design for Grouted Rock Bolts Based on the Convergence Control Method B. I N D R A R A T N A f P. K. KAISER**

A design approach for the application offully grouted bolts as a passive support system in underground openings is introduced in this paper. The analytical solution was developed based on elasto-plastic concepts and was verified by laboratory simulations with physical models. The application o f the theory to one case history and a comparison with an empirical design method ( R M R ) are presented. Computational examples and general design guides are provided to facilitate the use of the proposed method in practice.

NOMENCLATURE a= r= K,, = a,, =

tunnel radius; distance from tunnel centre to point of interest; ratio of horizontal to vertical in situ stress; far field stress horizontal stress; tr, = radial stress; tro= tangential stress; ~, = uniaxial compressive strength; a , = s.a c = post peak compressive (residual) strength; s = post-peak strength reduction factor; 0 < s < I; at = tensile strength; ay = yield strength of bolts; m = tan2(n/4 + 4U2); ,P= plastic radial strain increment; 0p = plastic tangential strain increment; ~r= failure strain in uniaxial compression test; ~, = a,/E = critical strain; q~ = friction angle of Mohr--Coulomb material; c = cohesion intercept; = dilation coefficient; E = Young's modulus; G = shear modulus; v = Poisson's ratio; L = length of bolt; d = diameter of bolt; Sr = tangential (circumferential) bolt spacing; St = longitudinal bolt spacing; p = radial distance to the neutral point on a bolt; ~. = friction factor for bolt/ground interaction; /~ -~ bolt density parameter; R = radius of the plastic zone; R* = radius of equivalent plastic zone; u, = wall displacement of the unsupported opening; u,* = wall displacement of the reinforced opening; u*/u, = normalized covergence ratio; u, = elastic wall displacement; ~* = radial strain at the reinforced tunnel wall; i = bolt effectiveness; t/= scale factor;

m = model; p = prototype; b = bolt; g = ground; j =joint;

Superscript * = material with properties equivalent to those of a reinforced rock mass. INTRODUCTION

D i s p l a c e m e n t m o n i t o r i n g plays a n i m p o r t a n t role in the o b s e r v a t i o n a l o r " d e s i g n - a s - y o u - g o " a p p r o a c h for u n d e r g r o u n d o p e n i n g s . In p a r t i c u l a r , t u n n e l c o n v e r g e n c e c a n be c o n s i d e r e d as a p r i m a r y field m e a s u r e m e n t , because it is n o t o n l y a readily r e c o r d a b l e i n d i c a t o r o f the overall g r o u n d r e s p o n s e , b u t its m a g n i t u d e c o n s t i t u t e s a very useful p a r a m e t e r for the e v a l u a t i o n o f t u n n e l stability. A l t e r n a t i v e m e a s u r e m e n t s , such as s t r a i n a n d load o b s e r v a t i o n s , are c e r t a i n l y i n f o r m a t i v e b u t o f t e n give results that are difficult to i n t e r p r e t d u e to their sensitivity to i n s t a l l a t i o n p r o c e d u r e s a n d localized rock mass failure. C o n s e q u e n t l y , the effectiveness o f g r o u t e d bolts c a n be assessed best in terms o f c o n v e r g e n c e r e d u c t i o n s . U n t e n s i o n e d , g r o u t e d bolts o r friction bolts d e v e l o p load as the rock m a s s d e f o r m s . Relatively small displacem e n t s ( 4 - 5 m m ) are n o r m a l l y sufficient to m o b i l i z e axial bolt t e n s i o n by s h e a r stress t r a n s m i s s i o n f r o m the rock to the b o l t surface. These bolts o r d o w e l s fall i n t o the c a t e g o r y o f passive s u p p o r t if they are n o t p r e l o a d e d . T h e y have been successfully a p p l i e d in y i e l d i n g g r o u n d , a n d f o u n d to be often m o r e e c o n o m i c a l a n d . m o r e Subscripts effective t h a n active bolts. F o r e x a m p l e , at the W a s h i n g p = plastic; ton. D.C. M e t r o , fully g r o u t e d bolts h a v e s h o w n to t = total; r = radial; reduce 2 - d a y d i s p l a c e m e n t s f r o m 7.6-15.2 to 2.5--5 m m 0 = tangential; [!]. G r o u t e d bolts are also widely used in m i n i n g for the s t a b i l i z a t i o n o f drifts a n d shafts. S i m p l i c i t y o f installatGeomechanics Research Centre, Laurentian University, Sudbury, tion, versatility a n d relatively low cost o f r e b a r s are Ontario P3E 2C6. Canada. **Divisionof Geotechnieal Engineering, Asian Institute of Technology. f u r t h e r benefits o f g r o u t e d bolts in c o m p a r i s o n tO their G.P.O. Box 2754. Bangkok 10501, Thailand. alternative counterparts. 269

270

I N D R A R A T N A and KAISER:

G R O U T E D ROCK BOLTS

The design of fully grouted bolts by either empirical methods or force-equilibrium methods does not properly assess the influence of bolting on rock mass displacements. The solution presented here provides an alternative method based on a convergence control approach. The general elasto-plastic solutions for the design of underground tunnel openings [2] have been extended in the following analysis to accommodate for the influence of bolt/ground interaction, size of opening and the bolt pattern on yielding and tunnel wall displacements.

Eo1~

~0 []

t set ~r.

~ t - ~ -~, S ~ I ~ o e m m , 0 r ÷ O~

Or

Part A: Analytical Solution and Verification by Laboratory Simulation

Fig. 2. Failure criterion and flow rule.

The analytical model considers a deep circular opening excavated in a hydrostatic stress field (Ko = 1), in a homogeneous, isotropic material with time-independent properties. The material behaviour is described by an elastic, brittle-plastic model with an instantaneous strength drop at peak stress (Fig. 1) and following a linear Mohr-Coulomb failure criterion. The post-peak behaviour is described by a simplified non-associated plastic flow rule with a constant rate of dilation (Fig. 2). The parameter :t is the dilation coefficient that characterizes the yielding material. For a reinforced material, the range of 1 < x < m is most appropriate because :t = I and ~ = m represent conditions of zero volumetric change and associated flow rule, respectively. The proposed theory predicts the ultimate convergence (at several tunnel diameters behind the face) where 3-D effects can be neglected. Hence, the deformation pattern near the tunnel is described by plane strain condition. For simplicity, an axisymmetric bolt pattern consisting of identical bolts with equal spacing around the circumference and along the longitudinal axis of the tunnel is considered. The equilibrium of an unreinforced element (Fig. 3a) following a linear Mohr-Coulomb failure criterion [2] can be represented by the following differential equation: do',. (1 -m)cr, ~'. dr "~ = -r " r

where: m*=m(i+/~)

and

a*=tr,(l+/~).

The bolt density parameter (/3) is a dimensionless parameter that reflects the relative density of bolts with respect to the tunnel perimeter. It takes into consideration the shear stresses which oppose the rock mass displacements near the tunnel wall. The following algebraic expression defines the bolt density parameter; rr d2a = sLs---7"

The friction factor (),) relates the mobilized shear stress acting on the grouted bolt to the stress acting normal to the bolt. It is analogous to the coefficient of

~a~

03 + do3

(I)

For a free body diagram of a reinforced element (Fig. 3b) with shear stresses along the borehole wall, a modified equilibrium equation is given by lndraratna and Kaiser [3]: dar

(I-m*)o'r

tr*

-

-

------

o© °

(2)

st. '~

03 +

do3

=T'

r

CT

(3)

unsupported reinforced

A

/

-]

o¢r

Fig. I. Elastic. brittle-plastic model,

Fig. 3. Stress acting on a rock mass element for analysis of bolt/ground interaction.

I N D R A R A T N A and KAISER:

friction or the bond angle used for the analysis of reinforced earth and split-set bolts. The dimensionless ratio fl/;. is the inverse function of the bolt spacing for a given bolt and tunnel geometry. It varies generally between 0.1 and 0,4 [4]. The role of bolt length and shear stress distribution

The effectiveness of a grouted bolt depends on its length relative to the extent of the zone of overstressed rock or yield zone. The shear and axial stress distributions of a grouted bolt are also related to the bolt length because equilibrium must be achieved between the bolt and the surrounding ground. The relation between the shear and axial stress distributions as a function of the bolt length is illustrated in Fig. 4 based on both theory and field observation [5-7]. The influence of the relatively thin grout annulus on rock mass deformation has been ignored. The positive shear stresses on the pick-up length of the bolt restrain the rock from moving into the opening, whereas the negative shear stresses acting along the anchor length ensure equilibrium of the bolt. The change in the direction of shear stresses creates a neutral point on the bolt at a radial distance (p), where the relative displacement at the bolt/grout interface is zero. Increasing the bolt length (L) and the bolt density parameter (fl) has the effect of creating a zone of improved, reinforced rock in the region surrounding the tunnel opening. Within the zone defined by the pick-up length of the bolts, the apparent friction angle and the uniaxial compressive strength are increased, and the stress is transferred to areas of greater confinement as discussed elsewhere by lndraratna and Kaiser [8]. Since the extent of yielding is directly related to the material properties, any improvement of the strength and frictional parameters must reduce the extent of overstressed rock. Therefore the plastic zone surrounding a reinforced tunnel has a smaller radius than that of an unsupported tunnel. This zone is called "equivalent

G R O U T E D ROCK BOLTS

271

plastic zone" (EPZ) because it consists of a material with improved strength properties, representing the yielded, reinforced rock mass. The radius (R*) of the EPZ is primarily a function of/L L, a, p and a o. Its determination is categorized according to the extent of the EPZ relative to the location of the neutral point and the boundary of the reinforced zone (Fig. 5): (I) minimal yielding condition: (II) major yielding condition: (III) excessive yielding condition:

The complete mathematical derivations are given by Indraratna and Kaiser [3] and a summary of the final equations is given in the Appendix. Influence of grouted bolts on tunnel wall stability

The radial strains and displacements at the tunnel wall are the most fundamental quantities required to evaluate the stability of a tunnel opening. In the field, these quantities can be monitored reliably. The radial strain (e*) and displacement (u*) of the reinforced tunnel wall can be predicted from the following equations, after the extent of the EPZ (R*) has been determined for the appropriate Category I-III [3]: (* =

~x(I -- V)O'cr M(~)[(R,/a)m+, _ 1] 2G ~(I - v)a c (i - s ) ( R * / a ) t +~ - vac----S~ 2G 2G'

u~*

2G (1 + - -

{i v)ac

2G

+

(I

g(~)[(R*/a)

-

M(~) =

I + ~ - s i n ¢ ( ~ - I)"

f,

lencJt h ~ I ,

Axial Sucss

ShearStress Tunn¢|

•i

P

-

where:

pick-up -~ ) l

"'''~'''"

"+~

s ) ( R * / a ) " +'),

point

,4

(4a)

(I - v)~r~, =

~ L ~J ~ anchot~ lengthen,

,,"" L

R* < p < (a + L), p < R* < (a + L), R* > (a + L).

L P = ln(I + L) - halt

I

Fig. 4. Stress distribution model for grouted bolts.

I]} (4b)

_7.,

INDRARATNA and KAISER: GROUTED ROCK BOLTS L

/

P

Category (I):

R" < p < (a + L) (minimal yielding)

/

% %%

/ _

Category (ll):

f

p < R" < (a + L) (major yielding)

%

Category (Ill): R"

/

/

.J

(a + L) (excessive yielding)

•

Fig. 5. Categories of extent of yielding.

1

The derivation of the above expressions is based on the assumption that both radial displacements and radial stresses are continuous across the elasto-plastic boundaries, regardless of the field stress magnitude. In addition, the post-peak parameters ~ and s are assumed to

,E

IF

D

C

I

E3

/ /

/

/

/ / /

o

/ / / / / / / / / Vl I A: no b o l l s .

•

= 0

8: B = 0.07:3 C: ,8 = 0 . 1 4 5 D: p

-

0.220

E: 15' = 0.291

F: no b o i l s ,

elastic,

be constant irrespective of the bolt density: whereas, the elastic parameters G and v are considered to be characteristic of the original intact material prior to yielding. Figure 6 illustrates the variation of the predicted wall displacements for an externally loaded model tunnel with different bolt patterns and a range of field stresses up to 14 MPa as discussed later in the test procedure. The representative properties of this reinforced opening ( a / L = 0.8) are: q~ = 32 ~, E / a c= 420, v = 0.25, s = 0.9 and :~ = 2. As the bolt density increases, the displacement of the reinforced tunnel wall (u*) decreases and varies between the upper and lower bounds of the unsupported tunnel displacements (ua: dashed line A) and the response of an opening in linear elastic rock (uo: dashed line F). For reinforced openings, a sudden increase in convergence (shift to the right as shown in Fig. 6) occurs for '8 = 0.073-0.22 for ~ro > I 1 MPa. This transition occurs when all bolts become completely embedded in the plastic zone, when Category III: R* > (a + L) is reached. For field stresses in excess of a o = 6 MPa, the normalized convergence ratio (u*/ua) is plotted against the bolt density parameter (8) in Fig. 7. It clearly illustrates the reduction in convergence that can be achieved by bolting. As '8 tends to zero (unsupported tunnel) u* approaches u~. If the intensity of bolting is excessively large, the convergence of the reinforced tunnel approaches the elastic value, thereby curtailing the plastic displacements severely (discussed later in detail). At the lower range of the field stresses (~,, ~< I I MPa), the normalized convergence ratio decreases steadily with increasing /L The curves become gradually flatter at higher bolt densities. This trend is to be expected because excessive yielding or Category III is not encountered at relatively low stresses, hence only a gradual variation of u*."u, occurs as Category II approaches Category I with increasing '8. However, at higher loads (ao~> 12MPa), the initial relatively flat response for / / < 0.15 is due to excessive yielding associated with Category III. A sudden decrease in the normalized convergence ratio is experienced when /3' exceeds 0.15 for ~ro = 12 and 13 MPa. This change is again associated with the transition from Category III to Category II, when some portion of the bolts is anchored in the outer elastic zone. As expected, for even greater loads (no > 13 MPa), a higher bolt density ( / / > 0.22) is required for the above transition to occur. For very high bolt densities ('8 = 0.3), convergence reductions of about 60% can be obtained. The magnitude of convergence reductions is almost independent of the stress level. It is also observed from Fig. 7 that, even for a wide range of field stresses (no = 6-14 MPa), the band width (variation) of predicted normalized convergence ratios is relatively small. These characteristics are of great advantage for design based on convergence control as discussed further in Part B (p. 276).

o 0

l

2

S

4

5

6

7

8

9

tO

TunnelWall Displacements(ram)

Fig. 6. Convergence predicted for model tunnels with various bolt patterns.

LABORATORY SIMULATION AND VERIFICATION The use of physical models helps to expand g¢otechnical research, because scaled models facilitate the

INDRARATNA and KAISER: GROUTED ROCK BOLTS

friction angle:

100

' ~ 9o .o o n-

~bm= ~p;

critical strain:

(°i)

8O

uniaxial strength ratio:

u

g ,o > e0

273

\ ,...\

m

The strain similitude for yielding and failure of bolts can be represented by the following expressions:

6O

50

: = = = - 6 MPo , N .-6 8 MPo :::;; 1 0 MPa

40

~ ..... .....

30

i

0 Z 0.00

0.04

12 MPo 1 3 MPa 1 4 MPa

I 0,08

i

i

0.12

0.16

i 0.20

0.24

0.2S

Geometric shnilitude requirements for a reinforced opening Similitude laws based on dimensionless analysis can be formulated by algebraic operations [1 !] to obtain a general form of equations common to both the model and the prototype. The scale factor (r/) represents the ratio of any specific dimension of the prototype to that of the model. The most relevant dimensions are: the tunnel radius (a), the bolt spacing (S), the bolt length (L) and the diameter (d) of the bolts. Therefore, the geometric scale factor must be the same for the following dimensionless ratios: dp

amSmLmdm' where, the subscripts p and m designate the prototype and the model, respectively.

Material similitude parameters The following dimensionless geotechnical parameters must be accurately established for the artificial rock in order to simulate the real behaviour: Poisson's ratio:

failure:

(cry) = (a~) Ill

investigation of the influence of specific variables (e.g. in situ field stresses, intact rock and joint properties, bolt pattern, etc.) on the response of the rock mass to excavation of a tunnel. Furthermore, physical modelling with synthetic materials can be conducted with test equipment of lower capacity and less sophisticated cutting tools can be employed due to the lower strength and hardness of model materials. In addition, the use of artificial rock reduces the cost, hence a large number of tests can be conducted. This approach was adopted to verify the analytical solution introduced earlier. Hence, an acceptable model had to be constructed with a material that represents the properties of a real rock mass. In the case of a reinforced opening, the geometry and the material properties of the bolts must also be properly scaled. In order to reproduce the behaviour of a real opening accurately, geometrical and material similitude criteria must be satisfied by the model [9, 10].

Lp

(-~b)m=(-~b)p;

vm= %;

p"

0.32

density parameter.

Sp

yielding:

i

Bolt Density Parameter Fig. 7. Variation of the normalized conversence ratio with the bolt

a~

p

The comparison of relative deformation characteristics between the bolts and the ground yields the following similitude equation:

Similitude criteria for the modelling of linear discontinuities can be satisfied by the following dimensionless parameters: friction angle of joints: joint spacing:

q~j.m= ~j.p,

Si'° Sj.m

orientation angle:

=

I~,

0m = 0p.

MODEL TUNNEL TESTING IN ARTIFICIAL ROCK

A model material that obeys all similitude laws simultaneously for a given rock is practically impossible to manufacture. However, the simulation of a general rock class or a range of different rocks is feasible. In order to verify the analytical predictions, a synthetic soft rock (GYPSTONE) was developed with a mixture of gypsum cement (10%), fine uniform Ottawa sand (75.8%), water (14.15%) and a retarder, NazHPO4 (0.05%). The principles of similitude require that for a typical soft rock: 0 . 2 5 < v < 0 . 3 5 , 25 ° < q b < 4 5 °, 2 0 0 < E / a : < 5 0 0 and 0.05 < ~r,/crc < 0.1. Uniaxial, triaxial compression and split cylinder tests indicated that "gypstone" (4 = 32°, v = 0.25, E = 1450 MPa, ~rc = 3.5 MPa, crt = 0.26 MPa) satisfies the similitude criteria for modelling relatively weak sedimentary rocks. Figures 8-10 provide evidence to support the suitability of gypstone for modelling the behaviour of a wide range of sedimentary rocks. The granular structure of gypstone physically resembles a typical sandstone. Further details of this synthetic rock and the manufacturing technique is documented elsewhere [12]. Figure 11 shows a typical test specimen reinforced with resin-bonded brass rods, which were found to be most appropriate for modelling prototype steel bolts after careful consideration of the pertinent similitude criteria. The circumferential arrangement of eight identical trapezoidal segments defines a complete circular opening. Hence the installation of two rows of bolts per sample represents a pattern of 16 bolts around the

274

INDRARATNA and KAISER:

• I R." ./B •

g~1.

-

• .9

i / • / /.."E

/ A/

c, ["

i

. /•

/ .t

•'I

o

/

:/ .'."

4"/

"I ,.:,

,.:" /

~..i

/I

t~

LEGEND

i:'/

....

R; 011 Creek SS

----"

B: C: D: E:

--

t -

~

r/

....

;k ~

'

o

Lunlng Dolomite tJalfcaap LS -Muddy ShaLe Berea LS 'G: ' G y p g t o n e '

Rock Data: I

I

• 00

I

Heuer and Hendron

I

0.50

i

i

I .OO

LEGEND

-- -- O= WolFcm~o LS - - - B= ~ Sl~le

1,."/ //: ~

,

:1/

~>

0

I

,r

-- --" ..--

!

-

C: tuning I~l~.l.te

D: O i l Creek .o..'Gy~t,~"

SS

Rock Data: Heuerand Elendron[9] I

I

0.50

I

i

I

I

1.50

I

I

I

2.00

2.50

0 s/O'~

1

2.00

I

l.O0

[9J

i

l .50

,

-

0,00

!/I

.-"

../.. /

°

31 ~/ .i]

c~

/

,) ...."

/.::;

~ "

::If ,-'/

:/

/

//"...D

/

< / . .. <: .5. /

~ ~,

I

"1":/ Ll"l"i l" / .'tl

/

/

/

/./.f"i /

...

/

/1

/

I'"

o/i.."

~ ~,~.

' '//.!. /. / ../

;,

/

C .""

B / /

/

/ " Iu

.:,

/

z/

o G,

i1...)

I

"/

g

G R O U T E D ROCK BOLTS

Fig. 10. Failure of gypstone and sedimentary rocks in triaxial compression.

2.SO

G,,/G~

Fig. 8. Comparison of gypstone with sedimentary rocks in triaxial compression.

complete circumference (equal spacing) of the opening. This particular sample shape has the advantage of enhancing the scale (r/) of the physical model, since under hydrostatic field stress, the deformation anywhere around a circular opening can be ideally simulated by the behaviour of one representative segment as a result of perfect axisymmetry. The longitudinal bolt spacing (SL) i

i

i

i

i

i

•

n."

/

!

i

)

/t

i

./

B///C

O/

~/

//

/

j

was varied (25, 50, 75 and 100 mm) while maintaining a constant transverse spacing (ST = 50 mm) which provided different bolt densities (/~ = 0.073, 0.145, 0.22 and 0.291). In order to assess the influence of bolt length on the behaviour of the model tunnel, 50- and 100-mm long brass rods (2.5 mm in dia) were installed in two different samples for each bolt density. A typical sample reinforced with 100-mm long rods at a density of/~ = 0.145

..

•

.

..'F

: / /, /

r

T r

?

r-130mm

.-

/ I /I.. /' / //.."

."

t

."

: /,,///:..

% b

J,

i

E

I~ ~/ .' .. /,'/° ,.,'.. // /."/ /,'/.,;:.

..

O"o /. ,t, I. t

(A) PLAN: GEOMETRY OF A TYPICAL TEST SAMPLE

/ I .,'""

,'

100 mm

I,'1//.'" . I;I .'/.'" ..

,,'/,;

:

I¢

St.

;Y

--

:171//." • I/Ill/-

:y

0; .~ble

-

....

-

Rock Data: Hock a n d

g .00

O.SO

0.60

0.90

E: Dolomite

ST

°' .o,°.,.,,.

Io

F: L l e e a t o n e

Brown

I .20

I

[14]

1.50

Gs/G=

Fig. 9. Failure envelopes of gypstone and various rocks.

.80

•

Bolt positions

•

Cony=fixate m ~ n ' i n I

points

- 0.145

{a) ELEVATION A-A: REINFORCED

TUNNEL SECTION

Fig. 11. Typical reinforced gypstone test sample.

INDRARATNA and KAISER:

(Fig. 1i) stimulates a sector of a real tunnel of 5.2 m in diameter, reinforced with 2-m long bolts at a spacing of I x ! m (i.e. r/= 20: l).

The reinforced gypstone samples were subjected to plane strain loading by the process simulation test (PST) apparatus developed by Kaiser and Morgenstern [13]. The application and transfer of load to the test specimen is illustrated in Fig. 12, where the simulated hydrostatic field stress (~o) was applied (Y-direction) at the outer edge of the sample. The magnitude of ao was increased by increments of I MPa, and the longitudinal stress (a:) was adjusted accordingly to maintain plane strain conditions (i.e. zero displacements in the Z-direction). The frictional resistance at the loading boundaries was minimized by a pair of teflon sheets inserted between the sample and the steel frame.

Simulation of reinforced openings in jointed rock When a rock mass contains four or more sets ofjoint planes, its behaviour can be considered to be homogeneous and isotropic with reduced equivalent strength and deformation parameters, as explained by Hock and Brown [14] and Bray [i 5]. Isotropic behaviour cannot be assumed if one of the joint planes becomes more pronounced (weaker or stronger) than the others. In order to model a heavily jointed rock mass, four sets of joint planes (inclined at 45 ° to each other) were introduced into several samples during casting. A typical jointed specimen is shown in Fig. 13. The joint density is reduced

.k,.,,on

Y

i ~

LoadLng

111

Wedge

GROUTED ROCK BOLTS

275

beyond a distance greater than 275 mm from the tunnel wall for convenience during casting and also because this outer region remains elastic for the range of boundary stresses applied by the PST apparatus. The joints were simulated by pairs of smooth plaster plates (~ = 22°). Further details of modelling jointed rock and the determination of its equivalent isotropic properties are discussed in detail by Indraratna and Kaiser [4].

Verification of the analytical model A complete comparison of strains and displacements (measured vs predicted) at various distances from the tunnel wall for a wide range of stress levels (ao = 0-14 MPa) has been presented by Indraratna [12] and summarized by Indraratna and Kaiser [16]. The observed data convincingly support the theoretical predictions. Only a comparison of the tunnel convergence data is presented here. A summary of normalized convergence data for various bolt patterns (L/a = 0.8) is illustrated in Fig. 14 for intact samples. The range of measured convergence data (dashed and vertically shaded) is plotted for several selected stress levels (t7o = 5-14 MPa) and is compared with the corresponding range of the predicted results (solid lines and horizontally shaded). The band of observed data closely overlaps with that of the predictions, indicating excellent agreement with the theory. Figure 15 compares the band of normalized convergence measurements with that of the predictions (vertically shaded) for the same stress levels and bolt patterns for the jointed samples. The predicted results fall within the slightly wider band of observed data, and support the analytical equations. In both Figs 14 and 15, the theoretical elastic response is also plotted for ao = 5 and 14 MPa. For unjointed rock (Fig. 14), the elastic response is reached when the bolt density reaches 0.2 at ao = 5 MPa and more than 0.4 for ao = 14 MPa. On the other hand, in jointed rock (Fig. 15), a higher bolt density of 0.3 for tro= 5 MPa was needed to approach elastic behaviour. Bolt densities far

l

111 ptAN

o

I L I I I I

+

I /

x

\

I /

I

~l\ /l\t/

" " " -

I Tunnel

~7

IT

I TT

T

I

Reaction Read

ELEVATION

; l~lT[/

\717

C o n c r e t a Block

I

J o I N'r

= AA

Fig. 12. Transfer of load to the test specimens.

Fig. 13. Test sample with four sets of joint planes.

276

I N D R A R A T N A and KAISER:

LEGEND N

-

THEORY: EL~TI~

"~" ~

TEST: ~

~

~qNGE

= B to 14 MPa

~,

o=

=o

....

x J ~ ~ L / a

= O.8

.~'~s. " ~

C L3

x

g

0..2;4. . . . . . . . . .

0

1

o.00

I ,

! .ME,

I

O.lO

8t~

I

I

0,20

I

I

0.~0

0.40

Bolt Density Parameter. /~ 14. Variation

Fig.

of the convergence with L = 100 mm.

bolt density

for

in excess of 0.4 would be required at ao = 14 MPa to read convergence values equal to those of a tunnel in elastic rock. Obviously, a total convergence of less than the computed elastic response is unrealistic. However, the range of bolt density parameters (,8) employed in reality seldom exceeds the upper bounds of ,8 as stated above, hence, the applicability of the analytical model is not limited in general practice. The use of these normalized convergence curves in design is explained in the following section. Part B: Application of Analytical Model for Design

Normalized convergence ratio as a design aid

The total wall displacement (u*) is a function of the properties of the rock, the field stress level and the

!

i

~

i

l

l

G R O U T E D ROCK BOLTS

reinforcement configuration. However, the normalized convergence ratio (u*/u~) is relatively insensitive to moderate changes of the fundamental similitude parameters (ok, v, E/a:, etc.) for a given reinforcement configuration (,8, L). The relatively thin band width of u*/u~ in Figs 14 and 15 also reflects the fact that moderate changes of the field stress level do not significantly influence the computed magnitude of the normalized convergence ratio. Consequently, the normalized convergence ratio is applicable for design, even if the material properties of the rock cannot be clearly defined (poorly defined) or if the in situ stress field cannot be accurately determined. The relation illustrated in Fig. 14 can be used in design as explained by the following example. For instance, consider a tunnel of 5.2 m in diameter, excavated in a fractured weak rock (~# = 33 °) at a depth of 400m(a,,= 10MPa). If the opening is reinforced by 2-m long grouted bolts (L/a =0.8), the tunnel convergence can be reduced by 30-40% for a bolt density parameter (,8) of 0.15-0.2. This could be achieved by installing 30 mm dia, rough (shaped) rebars (2 >/0.5) with a spacing of 0.9 x 0.9 m. Influence of bolt length on tunnel convergence

Figure 16 illustrates the normalized convergence relationships for predicted (vertically shaded) and measured (between dashed lines) data bands for short bolts (L/a = 0.4). The predicted range of data for longer bolts (L/a = 0.8, horizontally shaded) has been repeated on the same figure for ease of comparison. The close correlation between the predicted and measured convergence data for short bolts emphasizes further the capacity of the analytical model. The reduction in total convergence attained is less pronounced for short bolts, because the excessive yielding condition (Category III) becomes increasingly predominant as the bolt length decreases, particularly at relatively high field stresses (ao>>, 12MPa). For example, at a high bolt density i

l

....

.

THEORY: E L R S T I C THEORY=

-- --

i

RFI'GE

EU~+I~J~

t

1

i

I

1

LEGEND

LEGEND

--'--"

THEORY: ELASTIC

-- --

THEORY; E L f l S ÷ P L R S RANGE TEST= r1F_flSURED RRNGE

'

\

~

RF~GE

TEST= rlEP,S L J l E ~ R B N ~

~,

-

5

to

L/a

14

l'IPa

-

0.8

.%

i

n.-

E

elastic range Stress

elastic range

*

0

x z =

Level 5 MPa e l~°m /IPa 1"2 rlPa 14 I I P I

Stress •

x = Y •

U

(rPi4

Level 5 nPm 8 IIPa I0 t l P a t 2 MPa 14 rlPa

;.:iZ'n~a . . . . . . . . . . . . I

0.00

1

i

O.t0 BoLt

I

l

0.20

Denslty

I

I

I

0.30

.40

Parameter. #

Fig, 15. Variation of the convergence with bolt density for jointed samples (L = 100 ram).

0.00

J

0.10

I

I

0.20

I

0.B0

I

0.40

Bolt Density Parameter. # Fig. 16. Variation ofthe convergence with bolt density for L = 50 and 100mm.

I N D R A R A T N A and KAISER:

fl = 0.3 and field stress (#o = 14 MPa), a convergence reduction of approx. 60% can be achieved by long bolts. However, if the short bolts are installed at the same density and field stress, only about 35% convergence reduction can be expected. The bolts are completely embedded within the plastic zone (R* > a + L) for this case and Category III prevails. At lower bolt densities (fl < 0.15) the effectiveness of the short bolts is even less, as indicated by the flat slope of the initial portion of the uppermost solid line. It is also shown in Fig, 16 that excessively high bolt densities (fl > 0.5) would be required for short bolts in order to approach tunnel wall displacements close to the elastic response. These densities would have to be almost double the magnitudes of fl required for the same convergence reduction by the longer bolts. Magnitudes of fl > 0.5 are both impractical and uneconomical. It follows that at high densities, increasing the bolt length may be more effective than further increasing the bolt density. In cases where fl exceeds 0.3, effective convergence reductions can only be attained by increasing the bolt lengths. Based on the results of this study, it may be concluded that L/a ratios less than 0.8 may not be recommended for efficient convergence control, in order to prevent excessive bolt densities. The influence of bolt length on tunnel convergence decreases significantly, if the plastic zone propagates much beyond the reinforced zone (i.e. R* >>a + L). For example at fl = 0.145 and a,, = 14MPa, the convergence reductions associated with the long (L/a = 0.8) and the short (L/a = 0.4) bolts are 20.4 and 17.6% respectively. In such situations, increasing the bolt density rather than the length would be more effective. ANALYSIS OF ENASAN TUNNEI.,.--A CASE HISTORY The application of the elasto-plastic analytical solution to the Enasan tunnel project in Japan [17] will be demonstrated here. In this project, two parallel highway tunnels were constructed through Mount Enasan in the vicinity of the active Andrea fault of the central AIps in the Japanese island of Honshu. These tunnels are 8.6 km long and the centre lines are 60 m apart. The tunnel sections (North Enasan Highway) considered in this study have an average radius of 5.1 m and are located at a mean overburden depth of 450 m. The Enasan tunnel was constructed in three sequential excavation stages (heading, bench and invert) by the New Austrian Tunnelling Method. It was driven through heavily folded and fractured granite characterized by the following average material properties:
G R O U T E D ROCK BOLTS

277

sented by a dilation coefficient (:t) of 1.5, which was assumed to be typical for highly overstressed rock. Figure 17a illustrates a typical transverse section at Station B of the northern tunnel [17]. The initial support system consisted primarily of 9-m long grouted bolts (24mm dia) installed at a spacing of I x 1.05m. A friction factor (2) in the order of 0.35--0.4 can be regarded as typical of smooth grouted rebars installed in heavily fractured rock. Shotcrete and steel sets were also placed and longitudinal slots were provided to permit large radial convergence. Supplemental reinforcement with 9- and 13.5-m long bolts was employed when unacceptable convergences continued to occur, resulting --Rock ---Rock

Bolts Installed Initially Bolts Installed Later

i / 1 . 3 . 5 ~ . ~ , m Rock Bolts !

a ~j

\

\

!

,

~

,

,

~l~ ',|! I I ' r ' l . , '

,

I

"

", , , '',,,\,.,\'. ,V, I'/,,,,'/, '1;/,'1, , ' /,I,,,' \ ' , ' \ ' ,'\~ ',/, "./'.

".

~

%~% \ ~ l " ~"~'~

~s

.

..

i

It I l l l l l l

~

%,x ,x ~% ~ ~ ii t ' *,s I I II,,,, I d el ""

•~"

l

,,

nit

I s II

Ii/

• It

I ,111

*

•

.

•

"i{

•

• II

."

.

•

• ~

.

,,--5.1rr, 9.0 m Rock Bolts

b)

l"

I/I/ t

. . . ~ ,:,, ,,-,', s,,,,,,.,,,,,:.,

,

,"

ii I

6.0 m Rock Bolts

I00 90-

1/ .......... ? .....

BO-

'-

70-

~J

l~ ,o~

302010° 0o

~

"--"•I

O.4-

",

[ legend | *

I ",

I

~

",

Ix T Ak PArT(II~

"

~ O.$Q. ~

0,2-

3 ~

0.1-

0 0

. 3

0 II

S

12

IS

~ 16

21

24

17

$0

ROLT L(NGTH, L Fig. 17. (a) Pattern of grouted bolts at Enasan Tunnel (Station B): and (b) calculated relation between bolt length, bolt density and effectiveness.

278

I N D R A R A T N A and KAISER:

in a mean circumferential bolt spacing as small as 0.26 m. In fact, this corresponds to an increase in bolt density from fl =0.13 to fl =0.52. Deformation of the tunnel roof section was continuously monitored.

G R O U T E D ROCK BOLTS

given stress level, the bolt effectiveness (i) can be defined as:

i = - - u- ~ u7 oo = I - ;l*,,,~ % l l a - - lie

Conrergence of the tunnel roof An unsupported tunnel section excavated in this heavily fractured rock mass would theoretically create a plastic zone of 12.4 m in radius, with a wall convergence in excess of 2 m (i.e total collapse). If yielding could be completely prevented, a convergence of 0.35 m would be expected which is in close agreement with the elastic convergence of 0.37 m estimated by Barlow [18]. Both elasto-plastic and elastic solutions indicate the necessity for intensive support to control these unacceptable large displacements. For the initial bolt pattern (fl =0.13, L = 9m), a maximum estimated tunnel roof convergence of 1.07 m ( R * = 10.9m) is predicted by the analytical solution. The ultimate measured closure at the roof after supplemental bolting (increase in fl from 0.13 to 0.52) at a much later stage was 0.92 m. In comparison with the radial displacement predicted for the initial bolt pattern, the subsequent reduction attained due to supplemental bolting does not seem to be very considerable. However, a much greater convergence reduction could have been achieved if these supplemental bolts were installed immediately. Barlow [18] has in fact highlighted the same finding. This was to be expected, since bolts installed in an already developed large plastic zone have much less effect than if they had been installed before or during the propagation of the yielded zone. Station A of the North Enasan highway is separated by 35 m from Station B. The r o o f o f t h e tunnel at Station A was reinforced primarily with alternating 6- and 9-m long bolts with a bolt density parameter of 0.27. The invert was reinforced by the same bolt pattern as in Station B (Fig. 17a). A maximum closure of 0,71 m was measured at the roof of Station A. The predicted roof displacements for a tunnel with 9- or 6-m long bolts only 0.48m (F* = 9 . 3 3 m ) and 0.65m (R* = 9.89 m), respectively, for the given bolt density (fl = 0.27). These magnitudes slightly underestimate the measured convergence. This discrepancy may be attributed to the variation in material properties between Stations A and B, which can be expected in a heavily-folded and fractured rock mass. Considering these factors, the tunnel convergences predicted by the analytical solution are indeed realistic. Furthermore, it shows that convergence could have been reduced effectively by increasing the bolt density near the tunnel face.

Bolt effectit'eness Optimum effectiveness of a bolt system is achieved when the minimum tunnel convergence within economic and practical limitations is reached. In reality, the total convergence of a yielding, reinforced tunnel wall (u*) must be less than that of the unsupported opening (u~) but greater than the convergence of a tunnel in linear elastic rock (u~). Considering these conditions, for a

(5)

1 - - l l e ':ll a

where: IIe < l l *a < I t a .

Figure 17b illustrates the calculated variation of the bolt effectiveness as a function of the bolt length at station B, Maintaining a constant total quantity of steel, the corresponding variation of the bolt density, with the bolt length is also shown in the lower diagram. A total steel quantity of approx 90 m/m tunnel length was initially used at the roof (solid circle). However, the final steel quantity was greater than 315 m/m tunnel length after supplemental bolting (open circle), which corresponds to an increase of fl from 0.13 to 0.52 (at L = 9 m). An effectiveness of nearly 100% could have been achieved, if 9-m long bolts had been installed immediately with the same final density (dashed line). Alternatively, the installation of 13.5-m long bolts at the beginning with a lower bolt density parameter of fl = 0.32 (solid line, lower graph), the bolt effectiveness would have been as much as 95% instead of 58% (solid line. upper graph). In reality, as a result of delayed installation, a much lower effectiveness in the order of 65-70% was actually achieved. This highlights the importance of installing grouted bolts as early as possible near the tunnel face. Barlow and Kaiser [19] came to the same conclusion based on convergence rates. it is observed from Fig. 17b, that at a bolt length of less than about 6 m, the bolt effectiveness (initial pattern) drops dramatically, as a consequence of the plastic zone propagating beyond the bolts. Nevertheless, for bolt lengths less than 4.5 m, the effectiveness increases again due to the corresponding dramatic increase in ft. In this region, the bolt density fl dominates over the effect of the bolt length, but the corresponding bolt densities are unrealistically high. COMPARISON WITH EMPIRICAL DESIGN METHODS The NGI rock mass classification proposed by Barton

et al. [20] has introduced a support design guide based on the rock mass quality (Q). However, for many ground categories, particularly in poor, yielding rock it does not generally recommend the installation of untensioned grouted bolts. Therefore, it is not meaningful to compare the proposed analytical approach with the NGI method. However, the geomechanics classification or rock mass rating (RMR) system [21] is applicable to fully grouted bolts in all types of rock. The design tables and recommendations proposed by Bieniawski [21] are intended for tunnel openings in the order of I0 m width, excavated by the drill and blast method at depths of less than 1000 m and reinforced by 20 mm dia grouted bolts. Supplemental support by shotcrete, wire mesh and steel sets are also suggested for poorer ground.

INDRARATNA and KAISER:

The recommended bolt lengths (L) and grid spacings (St x ST) for the different rock classes are tabulated in the first three columns of Table I. The ratio/3/~, for these rock classes can be deduced from this information and is tabulated in the fourth column. The magnitude of 2 may be estimated from the effective bond angle of the bolt/grout interface to determine/3. The corresponding bolt density parameters for an assumed ,;. = 0.5 are given in the last column of Table 1. Several interesting aspects evolve from this table. The bolt densities (/3) recommended for poor to very-poor rock are relatively insensitive to rock quality changes and the advocated range of/3 for some rock classes (RMR less than 40) is very wide. Furthermore, the results from Enasan indicate that the magnitude of the recommended bolt densities seems to be too low for the poorest ground (RMR < 20). The fact that only a change in bolt length is recommended to control the weakest rock does not agree with the findings from this study and practical experience [22]. For instance, the bolt density parameter (fl) more than doubles as the spacing is decreased from 1.5 to I m. Hence, a further reduction of the bolt spacing for the weakest rock class would provide a sufficiently high magnitude for 13 to curtail displacements more effectively than by increasing the bolt lengths. This is supported by Laubscher and Taylor [23] who proposed a bolt spacing less than 0.75 m for poor ground at RMR < 30. This bolt spacing corresponds to a ]/-value of about 0.28 for 2 = 0.5, and seems to be in good agreement with the densities proposed earlier for effective convergence reduction. The influence of friction (bolt/grout interaction) as a very important design parameter is also ignored in the empirical method. On the basis of these observations, we conclude that the RMR system may not provide a sufficiently sensitive guide to properly designed grouted bolts in weak, yielding rock. For classes of poor rock (RMR < 40), a rational design method for grouted bolts should be based on the proposed analytical approach, which provides a sound basis for effective convergence control.

Observations from the Kielder experimental tunnel The behaviour of different support systems has been studied extensively at the Kielder experimental tunnel [7, 24]. The excavation of this tunnel was specifically selected to be in a weak mudstone layer (8 m thick). The rock movements were measured near the 3.3 m dia opening and the influence of fully grouted resin bolts on ground displacements was investigated.

GROUTED ROCK BOLTS

279

The Four Fathom mudstone was extremely fissured with multidirectional fractures and abundant mica partings. It was characterized by a rock quality designation (RQD) of less than 8% indicating its highly fissured nature. Houghton [25] describes this mudstone as a material which when exposed is prone to rapid deterioration. Based on the geomeehanics classification (RMR) system, support recommendations for fully grouted bolts have been compiled by Hoek and Brown [14] for the different rocks surrounding the experimental tunnel. Table 2 summarizes the calculated/3/2 ratios, considering the effects of the relatively small size of the experimental tunnel. Accordingly, the recommended bolt pattern for mudstone indicates a/3/2 ratio of 0.05-0.1 for a bolt length of 3-4 m, which provides a bolt density parameter of 0.025-0.05 for an assumed friction factor of 0.5. In comparison with competent limestone and sandstone, this recommended range of/3 by the RMR system does not seem to be sufficient for optimum convergence control in the much weaker mudstone. However, these RMR recommendations may still be acceptable, if the role of bolting is purely based on stability considerations rather than on convergence control. The bolt pattern recommended by the Geomechanics Classification (RMR) was not installed during the construction of the experimental tunnel. The actually installed bolt pattern consisted of 1.8-m long fully-grouted bolts (25mm dia) at a spacing of 0.9 x 0.9m. This provides a/~/2-ratio of 0.16 or a bolt density parameter (/~) of 0.08 for an assumed 2 of 0.5. Application of the proposed method of analysis reveals that/3 must be at least 0.15 to obtain effective convergence reductions for bolt lengths equal to the tunnel radius. Field measurements have indicated that for an unsupported tunnel section, radial displacements in excess of 20 mm at 0.3 m above the roof have occurred. As expected, the pattern of grouted bolts installed in the experimental tunnel has, on average, achieved convergence reductions less than 20%. A greater effectiveness in convergence control could have been achieved if a greater bolt density had been employed with the same bolt length. For instance, if these 1.8-m long grouted bolts were installed at a spacing of 0.6 x 0.6 m (/3/2 = 0.36), convergence reductions close to 40% would be predicted by the analytical solution. Therefore, it may be concluded that the portion of the Kielder experimental tunnel driven through mudstone was not sufficiently reinforced with fully grouted bolts for optimum convergence control. However, this does not imply that the tunnel was inadequately supported with respect to stability considerations.

Table I. Recommended bolt densities according to Geomechanics Classification (RMR) Rock class RMR Condition 81-100

Very good

61-80 41-60 21-40 < 20

Good Fair Poor Very poor

RMMS 27 4 - - D

L

SL and ST

(m)

(m)

/~ ;.

Generally no support required 2-3 3-4 4-5 5-6

2.5 1.5-2.0 1.0-1.5 1.0-1.5

0.05 0.08-0.14 0.14-O.31 0.14-0.31

/~ (at 2 =0.5) 0.00 0.10 0.04--0.07 0.07~).16 0.07-0.16

Table 2. Recommended bolt densities for Kielder experimental tunnel based on the RMR classification system

L

Sk and ST

Rock type

(m)

(m)

Great limestone Four Fathom limestone Four Fathom mudstone Natrass Gill sandstone

2 2 3-4 3

1.5 1.5 1.5 1.5-2.0

P

p/2

(at ,;.= 0.5)

0.046 0.023 0.046 0.023 0.05--0.10 0.025-0.05 0.03--0.05 0.015-0.025

280

[NDRARATNA and KAISER:

SUMMARY AND CONCLUSIONS The application of the convergence control design method in practice is summarized in the flowchart presented in Fig. 18, and the relevant computational steps are as follows: Step I. For the unsupported opening (fl = 0, L = 0), the plastic zone radius (R) and hence, the associated radial displacement of the tunnel wall (ua) are determined from the general elasto-plastic equations, for the given material properties and in situ field stress. Step 2. The equivalent plastic zone (R*) and the displacement of the reinforced tunnel wall (u~*) are determined for an initial combination of the bolt length (L) and density (/~), using the equations of the appropriate yield category (see Appendix) in conjunction with equation (4b). Step 3. The normalized convergence ratio (u*/u,) is computed and compared with the allowable (design) convergence ratio. Step 4. If this value of u*/u, satisfies the required convergence reduction, then the intial combination of bolt length and density is appropriate. The required bolt spacing (SL and ST) can then be calculated from equation (3). Step 5. The convergence of the reinforced tunnel wall after the installation of this bolt configuration is measured to verify the efficacy of the design. If either the predicted convergence ratio (u*/u~) or the observed displacements do not provide the desired or allowable convergence reduction, then the

Field

~.,Observatlons

Stress

I

I

Install Bolt Con f~guratlon

Fig. 18. Computational steps of the analytical model.

GROUTED ROCK BOLTS

computation must be repeated from Step 2, for a different combination of L and ft. The proposed convergence controlled approach for the design of fully-grouted bolts is applicable to rock that can be described by elasto-plastic material models. The evaluation of the extent of yielding as a function of the bolt density and length, provides a rational approach for the determination of the tunnel wall convergence in a reinforced tunnel. The influence of the bolt density parameter (/~) on the apparent strength of the rock mass profoundly reflects the importance of the bolt spacing and bolt/grout interaction in design. The role of fullygrouted bolts is best assessed on the basis of convergence control. This assists the designer in selecting the optimum reinforcement configuration. The normalized convergence ratio (u*/zq) and the resulting bolt effectiveness (i) are the fundamental design aids introduced in this paper. The normalized convergence ratio is most appropriate as a design parameter because it is relatively insensitive to moderate changes of rock mass properties. The use of bolt effectiveness (i) is particularly advantageous in design, where the bolt density and bolt length can be selected simultaneously, to achieve an economic quantity of steel. Laboratory simulations, the analysis of the Enasan tunnel and the observations from the Kielder tunnel all verify the applicability of the proposed analytical method for the prediction of the convergence of reinforced tunnels in relatively poor ground conditions (RMR <40). In such weak rock the Geomechanics Classification System (RMR) does not provide a sufficiently sensitive guide for bolt design. The results presented here indicate that in weak fractured rock, installation of rough rebars with a dense bolt pattern (//~>0.15) and adequate bolt lengths (L/a>>.O.8) are recommended for effective curtailment of displacements. These supports should be provided near the tunnel face soon after excavation. The study of the Enasan tunnel demonstrates that the installation of an optimum number of grouted bolts immediately behind the tunnel face contributes much more to control wall displacements than the provision of supplemental bolting at a latter stage. It is indeed the initial bolt configuration that is predominant in controlling the extent of yielding around an opening and hence, the final wall convergence. The mathematical treatment of the elasto-plastic analysis is based on several simplifying assumptions, which result in an axisymmetric yield zone around a deep circular opening. The adequacy of the proposed design method may be questionable if the geological conditions and the tunnel geometry become very complex. Furthermore, since the behaviour of the rock mass is represented by an elastic, brittle-plastic material model with a linear Mohr-Coulomb failure criterion, materials with a pronounced non-linear and strain-dependent post peak stress-strain behaviour are not modelled accurately. Under such circumstances, alternative solutions may have to be considered in conjunction with the proposed analytical approach in order to reach reliable solution.

INDRARATNA and KAISER: GROUTED ROCK

Acknowledgements--This research work was conducted while both authors were at the Department of Civil Engineering, University of Alberta, Edmonton, Canada, and was financially supported by a grant from the National Sciences and Engineering Research Council of Canada. The authors also express their sincere gratitude to the technical support staff at the University of Alberta for assistance during the extensive laboratory test program. Accepted for publication 30 October 1989.

REFERENCES

I. Van Sint Jan M. L. Ground and lining behaviour of shallow underground rock chambers for the Washington D.C. subway. Ph.D. Thesis, University of Illinois, Urbana (1982). 2. Kaiser P. K., Guenot A. and Morgenstern N. R. Deformation of small tunnels--Part IV. Behaviour during failure. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 22, 141-152 (1985). 3. Indraratna B. and Kaiser P. K. Analytical model for the design of grouted rock bolts. Int. J. Numer. Analyt. Methods Geomech. ( 1989). 4. lndraratna B. and Kaiser P. K. Stabilization of underground openings in jointed rock by fully grouted bolts. 15th Can. Syrup. in Rock Mech., Toronto, pp. 195-206 (1988). 5. Sun Xueyi. Grouted rock bolts used in underground engineering in soft surrounding rock or in highly stressed regions. Int. Syrup. on Rock Bolting. pp. 93-100. Balkema, Brookfield (1984). 6. Tap Zhen Yu and Chen Jie Xian. Behaviour of rock bolts as tunnel support. Int. Syrup. on Rock Bolting, pp. 87-92. Balkema, Brookfield (1984). 7. Freeman T. J. The behaviour of fully bonded rock bolts in the Kidder experimental tunnel. Tunnels Tunnelling 10, 37-40 (1978). 8. Indraratna B. and Kaiser P. K. Wall convergence in tunnels supported by fully grouted bolts. 28th U.S. Syrup. on Rock Mech., Tuscon, Arizona. pp. 843-852 (1987). 9. Heuer R. E. and Hendron A. J. Geomechanical model study of the behaviour of underground openings in rock subjected to static loads. U.S. Corps of Engineers, Report N-69-1, 2, Contract DACA 39-67-C-0009 ( 1971 ). 10. Clark G. B. Geotechnical centrifuges for model studies and physical property testing of rock and rock structures. Colarado School Mines Q. 76, (1981 ). I I. Obert L. and Duvall W. 1. Rock Mechanics and the Design of Structures in Rock. Wiley, New York (1967). 12. Indraratna B. Application of fully grouted bolts in yielding rock. Ph.D. Thesis. Department of Civil Engineering, University of Alberta (1987). 13. Kaiser P. K. and Morgenstern N. R. Time-dependent deformation of small tunnels--Part I. Experimental facilites. Int. J. Rock Mech. M#I. Sci. & Geomech. Abstr. 18, 129-140 (1981). 14. Hock E. and Brown E. T. Underground Excavations in Rock. Institute of Mining and Metallurgy, London (1980). 15. Bray J. W. A study of jointed and fractured rock, Parts I and 2. Felsmech. Ingenie,~rgeol. 5, 117-136; 197-216 (1967). 16. Indraratna B. and Kaiser P. K. Control of tunnel convergence by grouted bolts. Proc. of Rapid Excavation and Tunnelling Conf., New Orleans. Vol. I. Chap. 22, pp. 329-348 ([987). 17. Ito Y. Design and construction by NATM through Chogiezawa Fault Zone for Enasan Tunnel on Central Motorway (in Japanese). Tunnels & Umlerground 14, 7-14 (1983). 18. Barlow J. P. Interpretation of tunnel convergence measurements. M.Sc. Thesis, Department of Civil Engineering, University of Alberta (I986).

281

BOLTS

19. Barlow J. P. and Kaiser P. K. Interpretation of tunnel convergence measurements. 6th Congr. Int. Soc. Rock Mech., Montreal, Vol. 2, pp. 787-792 (1987). 20. Barton N., Lien R. and Lunde J. Estimation of support requirements for underground excavations. Proc. of 16th Syrup. on Rock Mech., Minneapolis, pp. 99-113 (1975). 21. Bieniawski Z. T. Rock mass classification in rock engineering. Proc. of Syrup. on Exploration for Rock Engineering, Johannesburg, Vol. !, pp. 97-106 (1976). 22. Golscr J. Personal communication (1987). 23. Laubscher D. H. and Taylor H. W. The importance of Geomechanics Classification of jointed rock masses in mining operations. Proc. Syrup. on E.rploration for Rock Engineering, Johannesburg, pp. 119-128 (1976). 24. Ward W. H., Coats D. J. and Tedd P. Performance of tunnel support systems in Four Fathom mudstone Proc. of Tunnelling "76, pp. 329-340. Institution of Mining and Metallurgy, London (1976). 25. Houghton D. A. The role of rock quality indices in the assessment of rock masses. Proc. of Syrap. on Exploration for Rock Engineering, Johannesburg, pp. 129-135 (1976).

APPENDIX Solutions of the Equivalent Plastic Zone Radius Category (1): R* < p < (a + L) (minimal yielding),

R_~ = [ l + l _ ( m * - [ ~ ( 2 ~ , -

"

m* =m(i + ~). Category Ill): p < R ° < (a + L) (major yielding),

R* a

p(I+B,~'"-" a

]

where

l {m'-

1\/2o,,

m'--i

\

I+~

,,,- - i

m' = m(l -,8). Category (Ill):R* > (a + L) (excessiveyielding),

R* --

a

= {i +

(Lfa)]

(

I -'~-B, ~'lm- I) I +A:+A~J

where

B - //"m - I'~/"2~0- I).

'-st-,+ i7 t,,,<

A.,= ~ A~=(I+~)

([(a + L ) / p ] " - ' - I),

(°-'/ ~

(a+L)/ol'~-'{(o/a~"-'-l}.