TUNNEL ANALYSIS

Stability Analysis of Tunnel Keyblock: a Case Study

I.-M. Lee and J.-K. Park

Abstract-The probabilistic block theory which was suggested by Hatzor and Goodman (1992) was
a plied to an telecommunication tunnel by taking into consideration the individual joints obtained from
f%ce mapping during tunnel construction. Using the actual unrolled joint trace that was developed in the
Banpo telecommunication tunnel during tunnelling, a statistical analysis of observed discontinuity data
was performed. The o!eterministic analysis and probabilistic keyblock concept were applied and their
results were compared with theobserved failure modes. The result obtained from the deterministic analysis
resulted in a large difference compared with the observed failure pattern; the analysis method which
considers the joint combination probability gave more similar results to the observed data than the
deterministic analysis. Furthermore, individual keyblock analysis was performed by means of the newly
developedprogram. While thepositions ofthe keyblocks werepredictedproperly by the individual keyblock
analysis, the predicted sizes of the keyblocks were found to be different from the observed data. It was
acknowledged from the individual keyblock analysis that most keyblocks are composed of steeply inclined
joints in the crown or in the side wall of a tunnel. But, in the fault-zone, any method based upon the block
theory could not predict the failure pattern properly. It can be concluded that the continuous careful
investigations of geological conditions were very important in tunnelling through a rock mass that
experiences mainly structure-induced failures. 0 2001Published by Eleevier Science Ltd. All righta reserved.

Key words: block theory, unrolled joint truce, statistical analysis, o!eterministic analysis, probabilistic
keyblock concept, individual keyblock analysis

1. Introduction In this paper, the probabilistic block theory, which was

T
here are now many tunnelling projects underway, suggested by Hatzor and Goodman (1992), was applied to
including subway tunnels, railway tunnels, rapid the example site. Their theory was summarized first, and
railway tunnels, roadway tunnels, electricity tun- their results were compared with ours. Moreover, failure
nels, telecommunication tunnels, etc. The support design shapes observed in the field were compared with the
for tunnels is so far mostly empirical. The typical support results obtained based on the block theory. Lastly, the
design pattern is adopted based on the rock/soil types, or individual keyblock stability analysis coded by the authors
based on RMR (Rock Mass Rating) values. However, both was performed using actual unrolled joint trace amps and
stress- and structure-induced failures should be considered compared with failure shapes observed in the field.
in the design of rock support for tunnel design 111.As for the
assessment of the structure induced failures, the so-called 2. Block Theory in Tunnel Design
block theory was suggested by Goodman and Shi (1985). In general, the analysis of tunnel stability using block
Block theory is a useful tool to determine the removability theory is classified into two parts, the kinematic analysis
and stability of rock blocks that were created by the inter- and the stability analysis. In the kinematic analysis,
section of joints 151. discontinuities of rock masses are analyzed to determine
During the construction of the Banpo telecommunica- whether the orientation of discontinuities could result in
tion tunnel located in Seoul, Korea, unrolled joint trace instability of the tunnel using the spherical projection
maps were drawn by measuring strikes and dips of all the technique [ll]. However, this analysis is restricted be-
joints. Moreover, locations and shapes of all the structure cause it does not consider the loading conditions. Once it
induced failures were observed and recorded. has been determined that a kinematically possible failure
mode is present, a limit equilibrium stability analysis is
performed to compare the resisting forces with the result-
Dr.In-MOLee,DepartmentofCivil Engineering,
Present addresses: ant forces. The stability analysis of a tunnel depends on
Korea University,Seoul, 136-701 Korea; and Jun-Kyung Park, several conditions, such as failure modes, loading condi-
Department of Geotechnical Engineering, SamBo Engrg. Co, Ltd., tions, block morphology, and analytical me@hodology.
Seoul, Korea.

www.elsevier.comAocate/tust
lhnelling and Chdergmund Space !l&chnolqgy,Vol. 16, No. 4, pp. 452-492,2GQ0
0996779&DW S - me front matter 0 2001 Published by Elswier Science Ltd. Pergamon
All
Ii&d
lwewed.
PII: s9996-7798(01xHK)141
3. Description of the Example Tunnel and Site and the tunnel is located at about 30-35 m depth below
Characteristics ground surface [ 121.
In the preliminary site investigation, a standard pen-
The example site (Fig. 1) selected for this study is the
etration test and several in-situ tests were performed.
Banpo telecommunication tunnel located at Seo-Cho Ku,
Except for andesite intrusion to some extent, the tunnel
Seoul. The tunnel is about 922 m long, the diameter is 3.5 m,
traverses a rock mass composed of granitic gneiss and

Geological Distribution
(planview and cross sectional view)

ZoWd LO”. width (mm)

Geological Distribution
(planvibwand cross sectionsi view)
P

Figure 1. Geological distribution of the Banpo telecommunication tunnel.

454 TUNNELLING
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SPACETECHNOLOGY Volume 15, Number 4,200O
schist, From an engineering point of view, the tunnel runs general tendency, as shown in Figure 2(b). The 922-m-long
through various rock types from hard rock to slightly tunnel is divided into several small sections, which show a
weathered rock. The longitudinal and the cross-sectional general tendency of discontinuity orientations. For each
geologic distributions obtained from the observation dur- small section, a constant bearing and rise of tunnel axis is
ing tunnelling are presented in Figure 1, and the typical used for removability analysis, as described in Table 1. The
tunnel section is also presented in Figure 2(a). global rock mass structures and their characteristics for
The orientations of discontinuities do not show any each section are also presented in Table 1.

Figure 2. Characteristics of the Banpo telecommunication tunnel.

Table 1. Global rock mass structures.

_ _

1 <8O%P> GI1 : 3/76 [114, 1.751 , Glz : 80/117 [156, 0.391, G13 : 40/Z:
(from 0 to 105 m) 12171, 0.371
2 <75vq”> GPI : 7/80 [230, 1.961 , G22 : 77/49 [144, 0.291 , G23 : 84167 [218
(from 105 to 203m) 0.231
3 <73”/0”> Grl : 28/246 [136, 1.611 , Gj2 : 70/208 [477,0.21] , Grr : 44171 [1166, 0.281

(from 203 to 301m) 1 Gjg : 68/67 [89,0.21]

4 <68”/0”> G41 : 3/140 123, 1.671 , G42 : 86/61 [57, 0.261, G43 :.72/178 [47
(from’301 to 400111) 0.27 1
5 <67’/0”> Gsl : 13/84 [30, 1.721 , G52 : 88/232 Q99, 0.271, G53 : 85/80 I263
I
(from 400 to 505m) 0.251
6 <65’/0”>
G61: l/310 [70, 1.821 , Gaz : 74/298 [85,0.36] , G6) : 70/47 [223,0.34]
(Corn 505 to 603m)
7 <65”/0”>
G,I : 2/282 [120,2.08], G,r : 301143 [790,0.33] , Gn : 79/25 [199,0.22]
(Corn 603 to 706m) I
8 <47*/O”> ‘GsI : 6/222 [56, 1.961 , Gaz : 85/270 [66, 0.261 , G83 : 82/2 [122
(from 706 to 804m) 0.351
9
Ggl : 6/47 [ 114,O. 171 , Gg2 : 7/205 [ 156,0.20]
(from 804 to 922m)

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455
 A Joint Pyramid (JP) is the infinite volume simulta-
neously on the block side of each joint of a block 161.A system
of three joint sets will produce six removable JPs in a
cylindrical tunnel; however, not all combinations of joints
have the same likelihood of intersecting any given volume,
and vice versa. Mauldon (1995) derived a general expres-
sion that would fit any joint spacing distribution, which also
addresses the inherent dependence of the distribution on
the joint orientations:
P(JC)=(&il,&;h)lG-, 06, xn’, 1 (1)
where ri, indicates a unit vector in the direction of the joint
normal and 1, is the joint set frequency, a rock mass
property that can be obtained from a scanline survey. In
this analysis, Mauldon’s generalized equation was used to
compute the joint combination probability. The P(JC)s of
all 86 combination cases are shown in Figure 3.

Shape Parameter, [K]

This parameter evaluates the likelihood that a remov-
H~urdourJC
able block will actually release from the excavation pyra-
mid, using primarily kinematical considerations. It distin-
Figure 3. Joint combination probability for all Jc’s. guishes between kinematically free and kinematically con-
strained sliding vectors of different removable JPs. The
shape parameter(K) measures the ratio ofthe JP spherical
4. Probabilistic Keyblock Concept to Tunnelling triangle area, assumingtetrahedral blocks, on the stereonet
Recently, several efforts have been made in applied block to the surface area of the projection sphere:
theory approach, using field observation and geostatistical
K ,(A +B +C +R’=(A +B +C -TT)
methods [3,4,6, 7,8, 13, 14, 15, 161. Goodman and Hatzor (2)
(1992) used block moulds to correlate observed block fail- 4iTR2 41c
ures with the theoretically removable blocks determined by where A, B, C are the JP interplanar angles and R is the
block theory. They found that the observed block failures radius of the stereographic projection sphere, as shown in
are correlated with a small subset of the removable block Figure 4. Consequently, it is based on the observation that
outcome space, i.e. block moulds. This observation has the JP shape has an effect on the degree of freedom, and the
triggered an effort to try to predict the failure likelihood of governing factor is the sum of the internal angles between
theoretically removable blocks, and to determine the criti- the JP planes. The [K] values for all removable JPs are now
cal keyblocks in excavation. computed by equation (2). Figure 5 shows a plot of the [Kl
values for all removable JPs among the hazardous JCs.
Joint Combination Probability, P(JC)
In this paper, only tetrahedral blocks with a three JP Instability Parameter, [F]
and a free face, which had been observed as a general trend This parameter evaluates the freedom of the removable
in this site, were considered. A Joint Combination (JC) was JP to slide out of the excavation pyramid, using limit
defined as a subsystem of joint sets that intersect over a equilibrium analysis [6, 71. Goodman and Shi (1985) dis-
small region in space [2]. The number of JCs that need to cussed the limit equilibrium analysis of removable JPs for
be considered is 86 cases, i.e. 6C3 + 5C3 + 4C3 + 6C3 + 3C3 different sliding modes, which include lifting, single-face
+3C3 + 5C3 , calculated by using Table 1. sliding and double-face sliding. The force required to bring

Figure 4. JP shape parameter for all Jc’s. Figure 5. JP shape parameter for all JCs.

456 TUNNELLINGANDUNDERGROUNDSPACE'~CHNOLOGY Volume 15, Number 4,200O

 Hsntdws X liuardous JC

Figure 6. JP instability parameter for all JCs. Figure 7. Block failure likelihood for all JCs.

an unsafe block back to limit equilibrium is the net sliding Cumberland Gap tunnel in Kentucky and Tennessee [7,8]
force (F*). This utilizes the sliding force F that is required to were compared with that of the present study. The greater
keep the JP in place, and is based upon a limit equilibrium the block failure likelihood, the greater the frequencies of
analysis: failures observed. However, the slopes of the regression line
F=2WR’-I
(31 are somewhat different, as shown in Figure 8. This result
could be related to the different lithologies in the three sites.
where R is the magnitude of the active resultant. The block In the Hanging Lake tunnel, 47 block moulds were
instability parameter F is a mapping of the F* to a range observed during tunnelling. The Hanging Lake tunnel was
between 0 and 1 where F = 0 corresponds to a block having excavated through a crystalline rock mass, where the basic
no failure mode, F = l/2 corresponds to a state of limit assumption of block theory of infinite joint planes seems
equilibrium, and F = 1 corresponds to a falling mode, as valid. In the Cumberland Gap tunnel, there were 110 block
shown in Table 2. [7] moulds. The Cumberland Gap tunnel was excavated through
Using equation (31, the instability parameter could be a sedimentary sequence of carbonate rocks where joints end
computed for all removable JPs in tunnel crown and sidewall. towards mechanical layer boundaries. It is possible, there-
The results are shown in Figure 6.

Block Failure Likelihood

This likelihood represents a relative
risk offered by the different joint combi-
nations in the rock mass when a free face
of fixed orientation is excavated through l”‘oo
I ______- - - - - - - - - - - - - - - -_- -_--
it 16, 71. The block failure likelihood is +
the product of the JP density, shape and
instability parameters: ~,oo__----------------____i_____

P (B) = P (JC) W El (4)

__- --------- ----
The relative block failure likelihood
is computed for all JCs and thus a dis-
tribution of P(B) values for all JCs is
obtained as plotted in Figure 7. The
relative block failure likelihood distri- __- -------------

butions indicate that JCs 2, 3, 22, 36,

39,56 and 77 will create critical blocks.
A better insight can be gained by
probing into the correlation between each
P(B) parameter and observed block __ - - - - -

moulds. Figure 8 shows the relationship

between percent block mould of the total
number found in each tunnel and the
corresponding predicted P(B) values. A
linear regression model of the least
squares estimate yields a determination
coefficient of 0.92. This correlation indi-
cates a good agreement between predic-
0.0040 0.0080 0.0120 0.0160
tions and field observations. Block Failure Likelihood, P(B)
The P(B) parameters of the Hanging
Lake tunnel in Colorado and the
Figure 8. Correlation between P(B) and observed block moulds in all tunnels.

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TUNNELLING SPACETECHNOLOGY
457
fore, that the limited extent ofjoint persistence in sedimen- Some characteristics for each small section are as follows:
tary rock masses violates fundamental block theory as- In the first and second sections, the failures were
sumptions. As a result, the shape parameter hypothesis observed to occur in the crown of the tunnel. It was
regarding closed and open JPs loses its validity because proved that the analysis based upon the block failure
joints never extend far enough to produce maximum-sized likelihood considering the variability of the joint sets
keyblocks, and the block apex is never at the maximum was more correct than the deterministic analysis (Fig.
distance calculated by block theory where infinite joint 10(a),(b)).
planes are assumed. In the third section, larger failure volumes were
The Banpo telecommunication tunnel was excavated observed than in the analysis results, mainly because
through granitic gneiss and schists classified as crystalline of poor rock mass conditions, which have small RMR
rock. Figure 8 shows that the percent block moulds for a values (Fig. 10(c)).
given P(B) value are larger in the sedimentary rock than in
In the fourth section, similar results to observed
the crystalline rock. Furthermore, for the given P(B) value,
failure shapes could be obtained only by using deter-
the percent block moulds are larger in the sedimentary rock
ministic analysis in the region where G45, G46 joint
than the crystalline rock.
sets were dominant (Fig. 10(e)). However, in the
region where joint sets with high block failurslikeli-
5. Comparison between Observed Data and hood were dominant, the analysis based upon proba-
Analysis Results bilistic keyblock concept gave more correct results
In view of the geological conditions of the zones where (Fig. 10(d)).
failure occurred, most failed blocks contained gouges be- Small-sized failures were observed in the crown and
tween joints, and the orientation of the tunnelling excava- sidewall in the fifth section, and it was also found that
tion was against the dip of the joint plane. Moreover, the the analysis which was based upon probabilistic
secondary block failures were observed in many parts ofthis keyblock concept gave more reasonable results (Fig.
site. The consideration of the secondary keyblocks might be lo(n).
needed for analyzing progressive failures. In the sixth and seventh sections, no prediction meth-
Most foliations did not develop into discontinuity planes ods mentioned above gave correct results due to the
in the example site composed of biotite granitic gneiss, existence of fault zones. Analyses based upon block
except for the zone influenced by geological movement. The theory were impossible in the sixth section, where
general failure patterns for each small section, which were large-sized fault zones appeared in the crown of the
observed in the tunnel construction stages, were compared tunnel (Fig. 10(h)), and in the seventh section where
with the results of the deterministic and probabilistic key fault zones appeared on the left sidewall of the tunnel
block analysis (see Fig. 10). The probabilistic analysis was (Fig. 10(i)). However, the analysis using probabilistic
performed for the case representing the highest values of keyblock concept gave reasonable results in the re-
the block failure likelihood in each section. For entire gion where the fault zone did not appear (Fig. 10(g)).
sections, except section 5, the highest values ofblock failure In the eighth section, failures were observed in the
likelihood were obtained when the joint combination prob- crown and upper part of the right sidewall, and it was
ability was maximum. found that the design using probabilistic keyblock
concept was also more correct than that based upon
the deterministic analysis.

60.00 6. Keyblocks in Unrolled Joint

Traces
- Banpo Telecommunication Tunnel (+)
It is assumed here that the joints
having traces in the tunnel surface ex-
tended sufficiently far behind the tun-
nel surface to form blocks by their mu-
tual intersections. It has been proved
that if the joints do extend behind the
tunnel surface, the maximum 3D
keyblocks of the tunnel can be delim-
ited by operating only with the joint
traces exposed on the tunnel surface. It
also has been proved that the 3D maxi-
mum-curved keyblock corresponding to
any maximum keyblock loop on the
unrolled joint trace map can be con-
structed completely from the informa-
tion on the unrolled map [l, 171.
An actual unrolled joint trace map,
which was observed in the Banpo tele-
communication tunnel surface, was
used to find keyblocks through the indi-
vidual keyblockanalysis, and the analy-
sis results were compared with the slid-
ing wedge that actually occurred in the
0.0100 0.0200 0.0300 field. The actual joint traces on un-
Block Failure Likelihood, P(B) rolled tunnel surfaces in the regions
where major failure occurred are shown
Figure 9. Comparison of correlation between P(B) and observed frequency of in Figure 11(a).
block moulds among three sites. The stability analysis was performed
and individual block volumes were ob-

458 TUNNELLINGANDUNDERGROUNDSPACETECHNOLOGY Volume 15, Number 4,200O

tained using the information from each joint. For the In the failure region, as shown in Figure 11(a), a total of
individual block analysis, the strength parameters of the 32 joints exist; the orientations of each joint are summa-
discontinuities are assumed to be: friction angle (8) = 30”, rized in Table 3. The result from the individual keyblock
cohesion (C) = 0, and the unit weight of the rock mass is analysis is presented in Figure 11(b). From the individual
assumed to be 27 kN/ms from the various in-situ and keyblock stability analysis, a total of 35 blocks were found,
laboratory tests (e.g. tilt test, direct shear test etc.) 1181. but most of them were infinite.

(a> Section 1 (b) Section 2

(c) Section 3 (d) Section 4(l)

(f> Section 5

Figure 10. Comparison between observed failure patterns and analysis results (continued on following page).

Volume 15, Number 4,200O TUNNELING

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459
 The joint planes of each removable block, its volume, the calculated volume from the individual keyblock stabil-
sliding mode, and the factor of safety are summarized in ity analysis was too small compared with the observed
Table 4. Once a keyblock failed, continuous block failures failure shape.
were observed in the tunnelling due to various reasons, Another reason may be that only well-developed joint
including the secondary keyblock effect and geological con- planes with lengths over several meters were included in the
ditions. However, these effects were not considered in the calculations. The local minor foliations around the joint were
current analysis based upon the block theory. Therefore, not considered. Based upon these considerations, the total

(g) Section 6(l)

m Observed Failure
. .. .. .. . .. . . Pattern
Deterministic Analysis
D-,.,1+
(h) Section 6(2)

(j) Section 8 (i> Section 7

Figure 10 (continued from previous page).

460 TUNNELLING AND UNDERGROUND SPACE TECHNOLOGY Volume 15, Number 4,200O
 (a) unrolled trace map

Distance from the portal of the tunnel (rn)

185 190 195 200 205 ‘210 215 220 225 230 295 240

(b) individual key block

Figure 11. Unrolled trace map of 184-240 m from the portal and result of individual keyblock analysis.

volume of the unstable wedge would be much larger than the acquire these data during a tunnel construction stage,
calculated results. Additionally, it was observed that most they will be very useful in constructing another
keyblocks were composed of steeply inclined joints in the tunnel in the neighborhood.
crown or in the sidewall of a tunnel. cl While the locations of the keyblocks were predicted
correctly by the individual keyblock analysis, the
Conclusion predicted sizes of the keyblocks were mostly found to
be smaller than the observed block mould data.
The final goal of this study is to apply probabilistic block
Furthermore, most keyblocks were composed of
theory suggested Hatzor [6,7] to the Banpo telecommunica-
steeply inclined joints in the crown or in the sidewall
tion tunnel, taking into consideration the individual joints
obtained from face mapping during tunnel construction. of a tunnel. Even though the morphology of a block
was considered to be an infinite block in the indi-
For the sake of this goal, the deterministic analysis, the
vidual keyblock analysis, failure might occur due to
probabilistic analysis based upon the block failure likeli-
the effect of other geological features that could not
hood, and the individual keyblock analysis were performed
be observed from the inner surface of the tunnel.
using the unrolled trace map of the Banpo telecommunica-
tion tunnel located in Seoul. Comparisons were made among d) It was proved that the probabilistic keyblock analy-
various analysis methods and observed failures in the field sis, which considers the joint combination probabil-
for each small section. ity, gave better results to the observed failure shapes
Conclusions drawn from this study can be summarized than the deterministic analysis, as shown in Figure
as follows: 10. From this result, it can be concluded that continu-
a) The results ofthe deterministic analysis showed some- ous and careful investigations of geological condi-
what of a difference compared with the observed tions during tunnel construction are very important
in a rock mass that experiences mainly structure-
failure pattern as shown in Figure 10.
induced failures.
b) The probabilistic keyblock concept was applied to the
example site by using the observed block moulds
data. The greater the block failure likelihood, the References
greater the frequencies offailures observed. Ifwe can 1. Goodman, R.E. 1989.Introductionto Rock Mechanics, 2nd ed.
New York:John Wiley & Sons.

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461
Table 2. The relationship between components of the Table 3. Zndividual joint orientation in the region of 184-
instability parameter [F]. 240 m.

80 5 17 250 30
F=l
2 1 80 5 250 30
1.0 Falling
3 320 70 19 55 70

4 320 70 20 55 70
0 < F*/R < Sliding
5 1 255 1 85 1 21 1 55 1 70 1
1
Limit 6 260 80 22 250 30
1.0
F=1/2 7 250 70 23 250 30

0 Equilibrium 8 250 70 24 250 30

O<F< 9 250 70 25 70 70

<o Safe 10 1 80 5 1 26 1 70 1 70 1
l/2
+ -00 No mode
F+O

14 140 80 30 250 30

15 250 30 31 250 30
2. Goodman, R.E. and G. Shi. 1985. Block Theo~ wzd Its
Applications to Rock Engineering. Englewood Cdfi 1, N.J.: 16 1 250 1 32 1 60 80 1
Prentice-Hall.
3. Chern, J. C. and M.T. Wang. 1993. Computing 3-D key blocks
delimited by joint traces on tunnel surfaces., Znt. J. Rock Mech.
Min. Sci. & Geomech. Abstr. 30 (7), 1599-1604.
4. Cravero, M., A. Mahtab and P. Grasso. 1991. Key block analysis Table 4. Keyblocks in the region of 184-240 m.
with statistical characterization ofjointa in tunnelling-a case
study. Tunnelling 91, 93-100.
5. Goodman, R.E. 1995. Block theory and its application.
Geotechnique 45 (3), 383-423.
6. Hatzor, Y. and R. E. Goodman. 1992. Applicationofblock theory
and the critical key block concept to tunneling: two case histories.
Proceedings ofZSRM Conference on Fractured and Jointed Rock
Mass, Luke Tahoe, CA, 1992.
7. Hatzor, Y. 1993. The block failure likelihood: a contribution to 1 2354 0.80 1.35 3&5
rock engineering in blocky rock masses. Znt. J. Rock Mech. Min.
Sci. & Geomech. Abstr. 30 (7), 1591-1597.
8. Hatzor, Y. and R. E. Goodman. 1997. Three-dimensional back- 2 3456 0 0.62 Falling
analysis of saturated rock slopes in discontinuous rock-a case
study. Geotechnique 47 (4), 817-839.
9. Heliot, D. 1988. Generatinga blockyrockmass.Znt. J. Rock Mech.
Min. Sci. & Geomech. Abstr. 25 (31, 127-138.
3 2637 0.95 1.77 2&6
10. Hoek, E. and E. T. Brown. 1980. Underground Excavations in
Rock. London: The Institution of Mining and Metallurgy.
11. Lee, Myung-Jae. 1996. Reliability-Based Optimization for Rock 4 39 15 0.67 2.01 3
Slopes. Ph. D. Thesis, Korea University, Seoul, Korea.
12. Lee, In-MO, Jun-Kyung Park and Seung-Won Song. 1998.
Statistical analysis and observed discontinuity data and case 5 14 17 19 0.31 0.81 14
study of tunnel stability using block theory. Proceedings of the
KGS Spring National Conference, 329-346. in Korean.
13.Leung, C. F. 1990. Computer-aided design of underground
6 27293032 0.84 1.41 27 & 29
excavations in jointed rock. Rock Mechanics and Rock
Engineering 23,71-89.
14. Leung, C. F. and S. T. Quek. 1995. Probabiliticstability analysis
of excavations in jointed rock. Can. Geotech. J. 32,397-407.
15. Mauldon, M. 1995. Keyblockprobabilities andsizedistributions: 17. Shi, Gen-Hua and R. E. Goodman. 1989. The key blocks of
a first model for impersistent 2-D fractures. Znt. J. Rock Mech. unrolled joint traces in developed maps of tunnel walls. Int. J.
Min. Sci. & Geomech. Abstr. 32 (6), 575-583. for Numerical and Analytical Methods in Geomechanics 13,
16,Quek, S. T. and C. F. Leung. 1995. Reliability-based stability 131-158.
analysis of rock excavations. Znt. J. Rock Mech. Min. Sci. & 18. Park, Jun-Kyung. 1999. A Case Study of Tunnel Keyblock
Geomech. Abstr. 32 (6), 617-620. Stability and Support Optimization Using Block Theory. M. SC.
Thesis, Department of Civil Engineering, Korea University.

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