doi:10.1016/j.tust.2005.05.001Underground Space Technology
incorporating Trenchless
Technology Research
Tunnel roof deflection in blocky rock masses as a function of joint
spacing and friction – A parametric study using
discontinuous deformation analysis (DDA)
Department of Geological and Environmental Sciences, Ben Gurion
University of the Negev, P.O. Box 653, Beer Sheva 84105,
Israel
Received 2 September 2004; received in revised form 15 February
2005; accepted 30 May 2005 Available online 2 August 2005
Abstract
The stability of underground openings excavated in a blocky rock
mass was studied using the discontinuous deformation analysis (DDA)
method. The focus of the research was a kinematical analysis of the
rock deformation as a function of joint spacing and friction. Two
different opening geometries were studied: (1) span B = ht; (2) B =
1.5ht; where the opening height was ht = 10 m for both
configurations. Fifty individual simulations were performed for
different values of joint spacing and friction angle. It was found
that the extent of loosening above the excavation was predominantly
controlled by the spacing of the joints, and only secondarily by
the shear strength. The height of the loosening zone hr was found
to be dependent upon the ratio between joint spac- ing and
excavation span Sj/B: (1) hr < 0.56B for Sj/B 6 2/10; (2) stable
arching within the rock mass for Sj/B P 3/10. The results of this
study provide explicit correlation between geometrical features of
the rock mass, routinely collected during site investigation and
excavation, and the expected extent of the loosening zone at the
roof, which determines the required support. 2005 Elsevier Ltd. All
rights reserved.
Keywords: Roof deflection; Discontinuities; Numeric analysis;
DDA
1. Introduction
Most rock masses are discontinuous over a wide range of scales,
from macroscopic to microscopic. In sedimentary rocks the two major
sources of discontinu- ities are bedding planes and joints, the
intersection of which form the so-called ‘‘blocky’’ rock mass
(Terzaghi, 1946).
Excavation of an underground opening in a blocky rock mass disturbs
the initial equilibrium, and the stres- ses in the rock mass tend
to readjust until new equilib- rium is attained. During
readjustment of internal
0886-7798/$ - see front matter 2005 Elsevier Ltd. All rights
reserved.
doi:10.1016/j.tust.2005.05.001
* Corresponding author. Present address: Faculty of Civil and
Environmental Engineering, Technion, Israel Institute of
Technology, Haifa 3200, Israel. Tel.: +972 4 8292462.
E-mail address:
[email protected] (M. Tsesarsky).
stresses, and consequently rearrangement of load resist- ing
forces, some displacements of rock blocks occurs. Joints and
beddings are sources of weakness in the otherwise competent rock
mass and therefore large dis- placements and rotations are only
possible across these discontinuities.
Failure occurs when the stresses can no longer readjust to form a
stable, load resisting structure. This may occur either when the
material strength is exceeded at some locations, or when movements
of rock blocks preclude the development of a stable geometric
configuration.
Terzaghi (1946) in his rock load classification scheme estimated
that for tunnels excavated in stratified rock the maximum expected
over-break, if no support is in- stalled, is 0.25B to 0.5B, where B
is the tunnel span. For tunnels excavated in moderately jointed
rock the maximum expected over break is 0.25B. For tunnels
30 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space
Technology 21 (2006) 29–45
excavated in blocky rock mass the expected over break is 0.25B to
1.1(B + ht), where ht is the height of tunnel, pending on the
degree of jointing. However, no particu- lar reference to the
mechanical and geometrical proper- ties of the discontinuities was
discussed by Terzaghi.
Hatzor and Benary (1988) have used both the classic Voussoir model
(Evans, 1941; Beer and Meek, 1982) and the discontinuous
deformation analysis (DDA, Shi (1988, 1993)) in back analysis of
historic roof collapse in an underground water storage system
excavated in a densely jointed rock mass. Hatzor and Benari coined
the term ‘‘laminated Voussoir beam’’ for an excavation roof
comprised of horizontally bedded and vertically jointed rock mass.
Their research showed that: (1) the classic Voussoir model is
unconservative for the given rock mass structure; (2) the stability
of a laminated Voussoir beam is dictated by the interplay between
fric- tion angle along joints and joint spacing.
Lee et al. (2003) showed that when two joint sets are encountered
at a tunnel excavation face, the most criti- cal joint combination
is when a set of horizontal joints (bedding planes) intersects
vertically dipping joints. Fur- thermore, they have shown that the
displacement of a key block at the roof tends to increase as the
block size decreases. However, no particular reference to joint
spacing or tunnel dimensions is given.
Park (2001) studied the mechanics of rock masses containing
inclined joints during tunnel construction using a physical trap
door model. Whu et al. (2004) rep- licated these experiments
numerically using DDA, show- ing very good agreement between the
physical and the numerical models. The results of both models
showed that the distribution of arching stresses above the open-
ing is a function of joint inclination. Huang et al. (2002) studied
the development of stress arches above large caverns and evaluated
the effects of different rock bolt types upon the size and shape of
the arch.
Broch et al. (1996) stressed out the importance of vir- gin
horizontal stress on the stability of large span open- ings, up to
65 m, excavated in Norway. However, these high stresses are of
tectonic origins which are predomi- nantly active along convergent
tectonic boundaries. In areas found at some distance from such
boundary, or in different tectonic setting, the magnitude of
tectonic stresses is diminished, and the arching stresses are
devel- oped due to excavation induced displacements. Which are in
most cases structurally controlled.
The main objective of the study presented herein is to investigate
the stability of underground openings exca- vated in horizontally
layered and vertically jointed rock masses. The effects of joint
spacing and shear resistance along joints on the height of the
loosening zone above the excavation are studied using the discrete
numerical model of DDA.
The focus of this study is rock mass kinematics, rather than stress
distribution. Monitoring of displace-
ments at and behind the excavation face is a routine practice in
rock engineering. Displacement measure- ments are relatively simple
comparing to in situ stress measurements, and in most cases is
cheaper. Analytical models for displacements around tunnels
excavated in a continuous rock-mass (e.g., Sulem et al., 1987) and
numerical models for displacements around tunnels excavated in a
rock-mass transected by a single fault (e.g., Steindorfer, 1997)
are currently available. How- ever, reliable models for
displacements around tunnels excavated in a blocky rock-mass are
less common, and those that exist still require validation. In this
research, we present numerical analysis of displacements at an
excavation face as a function of rock mass structure and opening
geometry.
2. Outline of DDA theory
The discontinuous deformation analysis, a member of the discrete
element models family, was developed by Shi (1988, 1993) for
modeling large deformations in blocky rock masses. Shi presented
DDA in an explicit matrix form; the following description is rather
more general, and is based on recent works by Jing (1998), and
Doolin and Sitar (2002).
In DDA the motion of a homogenously deformable discrete element
(block) is computed using series expan- sion of the displacement U
= TD. For two-dimensional formulation the displacement (u, v) at
any point (x, y) in a block can be related to six displacement
variables
½D ¼ ðu0 v0 r0 ex ey cxyÞ T ; ð1Þ
where (u0, v0) are the rigid body translations of a specific point
(x0, y0) within the block, (r0) is the rotation angle of the block
with a rotation center at (x0, y0), and ex, ey and cxy are the
normal and shear strains of the block. Assuming complete
first-order approximation of dis- placement, the expansion term T
takes the following ex- plicit form:
.
ð2Þ By the second law of thermodynamics, a mechanical
system under load must move or deform in the direction that
produces the minimum total energy of the system. For a discrete
element the energy balance may be writ- ten in terms of kinetic
energy R and potential energy V:
E ¼ R V ¼ 1
2 _DM 0
_DPðDÞ; ð3Þ
where M0 is the mass matrix quantifying the mass distri- bution
around the center of rotation. Body forces, loads, and displacement
constraints are expressed in terms of
Table 1 Geometry, material properties and numeric control
parameters used in DDA Voussoir models
Item Value
Numeric parameters
Penalty stiffness 1000 MN/m Time step size 0.00025 s Penetration
control parameter (g2) 0.00025
M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space
Technology 21 (2006) 29–45 31
the potential P(D). Explicit matrix form derivation of P(D) is
found in Shi (1988, 1993).
The minimization of the total energy is performed by first-order
differentiation with respect to the displace- ment vector U:
oE oU
oU ¼ 0. ð4Þ
Eq. (4) yields a weak equilibrium equation describing the motion of
the block:
M0 €U þ C _U þ KU ¼ F ; ð5Þ
where C and K are generalized damping and stiffness terms,
respectively.
The equation of motion is discretized using a New- mark type time
integration scheme (Newmark, 1959) with collocation parameters b =
1/2, c = 1:
Uðt þ DtÞ ¼ UðtÞ þ Dt _UðtÞ þ ð1 2 bÞDt2 €UðtÞ
þ bDt2 €Uðt þ DtÞ; _Uðt þ DtÞ ¼ _UðtÞ þ ð1 cÞDt €UðtÞ þ cDt €Uðt þ
DtÞ.
ð6Þ
This approach is implicit and unconditionally stable. The local
equilibrium equations are then assembled
to yield a global stiffness matrix [K], which for a block system
defined by n blocks is
K11 K12 K1n
or ½KfDg ¼ fF g ð7Þ
where Kij are sub-matrices defined by the interactions of blocks i
and j, Di is a displacement variables sub-matrix, and Fi is a
loading sub-matrix. For two-dimensional for- mulation Kij is a 6 ·
6 sub matrix, and Di, Fi are 6 · 1 sub-matrices. In total the
number of displacement un- knowns is the sum of the degrees of
freedom of all the blocks.
The solution of the system of equations (7) is con- strained by
inequalities associated with block kinemat- ics. All constraints,
including inter-block displacement constraints, are imposed using
penalty functions. At each time step the no-tension and
no-penetration condi- tions between blocks are enforced before
proceeding to the next time step: the so-called open–close
iterations. The reader is referred to Shi (1988, 1993) and Doolin
and Sitar (2002) for further reading on open–close
iterations. The accuracy of DDA and its applicability to
prob-
lems of rock engineering was studied by many research-
ers, for a thorough review of DDA validation the reader is referred
to MacLaughlin and Doolin (2005).
3. Kinematics of single and laminated Voussoir beams
Before proceeding to analysis of full-scale problems we have
performed a series of parametric studies of a single Voussoir beam
and of a layered Voussoir beam. The purpose of these studies was to
explore the kinemat- ics, with special focus on deflection as a
function of joint spacing and shear strength. The geometry and the
mechanical properties of the rock mass chosen for these analyses
are of the ancient water reservoir of Tel Beer Sheva, previously
investigated by Hatzor and Benary (1988) and Tsesarsky and Hatzor
(2003), refer to Table 1.
The effect of joint friction was studied for a constant joint
spacing of Sj = 0.25 m, the average spacing in situ, while the
friction along joints (/av) was varied from /av = 20 to /av = 80.
The effect of joint spacing was studied for /av = 47, the peak
friction angle obtained from direct shear tests of natural joints,
while joint spac- ing was changed from Sj = 0.25 m to Sj = 4 m. For
a single Voussoir beam the displacements were measured at selected
points along the lower fiber of the beam at intervals of 0.5 m. For
the layered Voussoir configura- tion the displacements were
measured at five locations: (1) m1 at (x1, y1) – mid-span of
immediate roof; (2) m2
at (x1, y1 + 2.5 m); (3) m3 at (x1, y1 + 5 m); (4) m4 at (x1 + 4 m,
y1 + 2.5 m); (5) m5 at (x1 4 m, y1 + 2.5 m), refer to Fig. 1.
3.1. The three-hinged beam problem
First, a simple two-block system was analyzed, typi- cally referred
to as the ‘‘three hinged beam’’ (Fig. 2). In order to simplify the
analysis and to preclude vertical (shear) displacements at the
abutments the two blocks were constrained by assigning fixed points
at base verti- ces (Fig. 2). Similar analysis, under somewhat
different
Sj
S
t
m3
m2
m1
a
b
Fig. 1. Geometry of DDA Voussoir models: (a) single; (b) laminated.
Stiff abutments are represented by two non-deformable blocks, each
containing three fixed points.
Fig. 2. Mid-span deflection time histories: three-hinged beam
configuration.
32 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space
Technology 21 (2006) 29–45
M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space
Technology 21 (2006) 29–45 33
boundary conditions was performed by Yeung (1991), showing very
good agreement between the analytical and numerical solutions. The
aim of this analysis was not to reproduce analytical or
semi-analytical solutions, but rather to study the behavior of the
DDA solution over time.
The following input parameters were used for DDA: E = 10 GPa, m =
0.25, and q = 2.7 · 103 kg/m3. The block dimensions were: S/2 = 5
m, t = 0.5 m. The numerical control parameters were: normal
penalty
20 30 40
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.5
0.75
1
1.25
1.5
1.75
S
φ
c
b
a
Fig. 3. DDA prediction for mid-span deflection of a single Voussoir
beam: histories for different values of friction angle, for joint
spacing of Sj = 0.25 friction angle of /av = 47.
stiffness PN = 1 · 109 N/m and time step size
Dt = 0.001 s. The behavior of the system is shown in Fig. 2.
Two
types of analyses were performed: (1) fully dynamic; (2)
pseudo-static, achieved by zeroing the initial velocity at every
time step. For both types of analysis the system attained
equilibrium after mid-span deflection of d = 0.003455 m. The
dynamic analysis shows a typical oscilla- tory behavior decaying
towards the equilibrium state. This phenomenon is known as
algorithmic damping,
50 60 70 80
e (sec)
j = 0.25m
av = 47o
φ Sj
(a) as a function of friction angle (/) and joint spacing (Sj); (b)
time m; (c) time histories for different values of joint spacing,
for available
-0.3
-0.2
-0.1
0
Distance from mid-span (m)
-0.4 -0.3 -0.2 -0.1
0 0.1 0.2 0.3
30o
= 45o
= 75o
= 80o
a
b
c
Fig. 4. Deformation profiles of single Voussoir beam by DDA,
measured at the lowermost fiber of the beam: (a) horizontal
displacement (u); (b) vertical displacement (v); (c) rotation
(x).
Fig. 5. DDA graphic outputs of single Voussoir beam deformation for
different values of joint friction angle: (a) undeformed; (b) /av =
45; (c) /av = 75; (d) /av = 80.
34 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space
Technology 21 (2006) 29–45
M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space
Technology 21 (2006) 29–45 35
and is typical to the Newmark type time integration used in
DDA.
This simple analysis shows that the DDA solution of a three-hinged
beam problem is oscillatory before con- verging to the equilibrium
position.
3.2. Kinematics of a single Voussoir beam
DDA results for the Voussoir beam are presented in Fig. 3(a),
showing mid-span deflection (d), friction angle (/av), and joint
spacing (Sj). Figs. 3(b) and (c) are time histories of mid-span
deflection for different values of friction angle and spacing,
respectively.
With a fixed joint spacing of 0.25 m and friction an- gles smaller
than 78 the beam progressively deflects, and eventually fails. For
friction angles greater than 78 the beam attains stable equilibrium
after small initial deflection. Figs. 4(a)–(c) show the
displacements, u, v, and the rotation x for selected values of
friction angle and for joint spacing of 0.25 m. Fig. 5 presents
graphic output for the four realizations described in Fig. 4.
At low values of /av = 30 and /av = 45 deforma- tion is dominated
by inter-block shear, which is maxi-
i - 1
Fig. 6. Schematic representation of the forces actin
mum at mid-span and minimum at the abutments (Fig. 4(a)). Block
rotation is mostly uniform and anti- symmetric. The deformation
characteristics are changed when the friction angle is greater than
/av = 75, shear displacement is reduced by an order of magnitude,
while the rotation at the beam ends rises significantly.
The rotation data implies that at low values of fric- tion angle
the moment generated by the lateral thrust within the beam, does
not develop effectively. Beam deformation is dominated by vertical
shear, which con- sequently leads to structural failure. Where the
available shear resistance along joints is sufficiently high to
pre- clude excessive vertical displacements, and to induce block
rotation, consequent build-up of effective lateral thrust within
the beam equilibrates the overturning mo- ment and the beam attains
equilibrium position. Sche- matic representation of the forces
acting on a block within the beam is given in Fig. 6.
Increasing block size by setting Sj = 0.5 m lead to de- creased
mid-span deflection. The beam however does not attain equilibrium,
and eventually fails. Examina- tionof the deformation time
histories for beams with dif- ferent joint spacing (Fig. 3(c))
reveals that an oscillatory
i
g on a block within the multi-fractured beam.
36 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space
Technology 21 (2006) 29–45
solution marks equilibrium. In this particular analysis equilibrium
is attained when Sj P 1.25. The style of deformation is similar:
shear dominates unstable geome- tries, with relatively small
rotations, while block rotation is exhibited for stable
geometries.
3.3. Kinematics of a laminated Voussoir beam
DDA results for laminated Voussoir beam (Fig. 1(b)) are presented
in Fig. 7(a), which is a plot of mid-span deflection (d) versus
friction angle (/) and joint spacing (Sj). The deflection data are
given at measurement points m1,m2 andm3. Time histories form1,m2
andm3 for dif- ferent valuesof joint spacingarepresented inFig.
7(b)–(d).
For joint spacing of 0.25 m the deflections are exces- sive for all
the analyzed friction angle values, and failure
20 30 40 5
-0.25 -0.2
-0.15 -0.1
-0.05 0
a
b
c
d
Fig. 7. (a) DDA prediction for mid-span deflection of the laminated
Voussoir angle (/) and joint spacing (Sj); (b–d) are time histories
for different values of available friction angle of /av = 47.
is expected. Furthermore, as expected, measurement points data
indicate that dm1 > dm2 > dm3, suggesting vertical load
transfer. The lowermost layer carries most of the vertical load and
consequently deflects most. Deflection is decreased with increasing
vertical distance from the immediate roof. For /av < 50
measurement point deflection are similar: dm1 0.43 m; dm2 0.26 m;
and dm3 0.18 m. Deformation is achieved by shear as the lateral
thrust is not fully developed.
When shear resistance is increased to /av = 60 mea- surement point
deflections are reduced. For /av > 60 shear resistance is
increased and downward displace- ment is restrained. Consequently,
the dominant defor- mation mechanism changes from shear to block
rotation, and downward displacements are reduced to dm1 < 0.15
m, dm2 < 0.13 m and dm3 < 0.11 m. By
0 60 70 80
2.5 3 3.5 4
φav = 47o
Sj = 0.25m
beam, at measurement points m1, m2, and m3 as a function of
friction joint spacing at measurement points m1, m2, and m3,
respectively, for
M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space
Technology 21 (2006) 29–45 37
increasing frictional resistance the laminated Voussoir beam
behaves like a coherent beam throughout its thick- ness.
Nevertheless, complete stabilization is not attained as indicated
by the measured deflections.
Deflections are very much restrained when joint spac- ing is
increased to Sj = 0.5 m, dm1 = 0.131 m, dm2 < 0.116 m and dm3
< 0.106 m, compared with dm1 = 0.434 m, dm2 < 0.246 m and dm3
< 0.155 m for Sj =
a
c
b
me
Sj
Sb
0,0
Y
Fig. 8. Geometry of DDA model for parametric study. Fixed
boundaries are fixed points.
Table 2 Matrix of DDA parametric study
Model / () Sj/B
B = ht 20 1.5/10 2/10 3/10 4 30 40 50 60
B = 1.5ht 20 2/15 3/15 4/15 5 30 40 50 60
B = opening span; ht = opening height; / = friction angle; Sj =
mean joint bridge.
0.25 m. With further increase in joint spacing to Sj P 0.75 m
complete beam stability is obtained with negligible deflections:
dm1 < 0.05 m, dm2 < 0.03 m and dm3 < 0.01 m. As before the
equilibrium solution is oscil- latory. The beam behaves as a
coherent element and the vertical loads are transmitted effectively
to the abut- ments. As a result, the deflections are homogenized
throughout the bulk of the beam.
asurment point
X
represented by four fixed blocks, each containing a minimum of
three
Dr Lj (m) bj (m)
/10 5/10 0 25 1
/15 6/15
spacing; Dr = degree of randomness; Lj = joint trace length; bj =
joint
38 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space
Technology 21 (2006) 29–45
4. The general case – a parametric study
4.1. Geometry and material properties of the analysis
domain
A representative underground opening in horizon- tally bedded and
vertically jointed rock mass is shown in Fig. 8, where a horseshoe
section with span B = 2a and height ht = b + c is presented. Two
different open- ing geometries are studied: (1) B = ht; (2) B =
1.5ht, where the tunnel height is 10 m for both cases. The ver-
tical dimension of the domain is set such that the under-
Fig. 9. Two types of rock masses: (a) not containing cantilever
beams; (b) containing cantilever beams.
0 500 1000
) +28.5m
+23.5m
+18.5m
+13.5m
+8.5m
+4.5m
crown
Fig. 10. Time histories of vertical displacement (d) above an
underground available friction angle /av = 20.
ground opening is located at depth greater than 1.1(B + ht),
conforming with Terzaghis rock load expectation in blocky rock
masses. The displacements within the rock mass are measured at
seven measure- ment points along the tunnel centerline (Fig.
8).
Two joint sets are generated using the synthetic trace line
generation algorithm of Shi and Goodman (1989). The horizontal
bedding planes are assumed of infinite persistence, with average
spacing of Sb = 0.1ht and de- gree of randomness of Dr = 0.25
(spacing may vary by 25% of the mean value during random joint
trace gener- ation). Vertical joints are generated for different
values of mean spacing. The input spacing, trace length, bridge
length and degree of randomness are given in Table 2. Vertical
joints were generated such that the number of potential cantilever
beams within the rock mass was minimized. Fig. 9 shows a schematic
representation of two types of rock masses: with and without
cantilever beams. The presence of cantilever beams reduces the dis-
placements in the rock mass and enhances stability (Ter- zaghi,
1946), due to enhanced arching. Therefore, larger displacements are
expected when the number of cantile- ver beams is minimized.
Mechanical properties for intact rock material are chosen to
conform with ‘‘average’’ sedimentary rocks: specific gravity c =
24.525 kN/m3; Elastic modulus E = 10 GPa, and Poissons ratio m =
0.25. The shear resistance along discontinuities is assumed to be
purely frictional, cohesion and tensile strength are neglected. The
input discontinuities represent clean planar joints without surface
roughness, wall annealing or infilling. The friction angle for
bedding planes and vertical joints is assumed equal for simplicity;
this is by no means a limitation of the DDA method or its numeric
implementation.
1500 2000 2500
steps
opening of span B = ht, vertical joint spacing of Sj/B = 1.5/10
and
Fig. 11. DDA graphic output for tunnel of span B = ht, vertical
joint spacing of Sj/B = 1.5/10 and available friction angle /av =
20: (a) initial configuration; (b) deformed configuration.
M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space
Technology 21 (2006) 29–45 39
4.2. Results of the parametric study
4.2.1. B = ht = 10 m
Representative time histories of vertical displace- ments are given
in Fig. 10, which shows DDA results for joint spacing of Sj/B =
1.5/10 and available friction angle along joints of /av = 20. The
crown of the exca- vation is in a state of progressive failure,
clearly marked by the progressive downward displacement. The
vertical displacements at points located at y > 0.45ht above the
crown are oscillatory, and are con- fined to values of d < 0.1
m, implying stable arching. Graphic output for this particular
realization is shown in Fig. 11.
Vertical displacement profiles for selected values of joint spacing
are given in Fig. 12(a)–(d). For joint spac- ing Sj/B 6 2/10 the
displacement at the crown is d < 0.3 m, depending upon friction.
The displacements die out with vertical distance from the crown and
at y > 0.85ht the displacements are smaller than 0.1 m. For
joint spacing of Sj/B P 3/10 displacements are reduced to d <
0.1 m, approaching minimum values of d 0.05 m, followed by
homogenization of displacements.
The vertical displacement differences Dd/Dy calcu- lated between
pairs of measurement points within the vertical profile are
presented in Fig. 12(e)–(h). For joint spacing of Sj/B 6 2/10
homogenization of displacements begin at y > 0.45ht above the
crown, and the difference approaches zero with greater distance
from the crown. For joint spacing of Sj/B P 3/10 the displacement
differ- ence is reduced to 0.005, with very little variation from
the crown up.
From the described above, it can be concluded that for a tunnel
span of B = ht = 10 m the height of the loosening zone above the
excavation is about 0.5ht for joint spacing of Sj/B 6 2/10. For
joint spacing of Sj/B P 3/10 the rock mass above the opening
attains stable arching. The rock mass response is governed by the
joint spacing and to a lesser extent by the joint friction. Only in
one case, Sj/B = 2/10 and /av = 60, the friction angle inhibits
excessive deflections, and induces stable arching. Where joint
spacing is large enough stable arching is independent of friction
angle. The findings of this sec- tion are summarized in Table
3.
4.2.2. Random joint trace generation
Modeling the vertical joints as persistent with con- stant spacing
results in a rock mass structure with a minimum number of
cantilever blocks. In this configu- ration the deflections above
the underground opening are expected to attain maximum values.
However, joints are seldom persistent, and statistical variations
of length and spacing are to be expected. In order to study the
effect of joint randomness on rock mass response the simulations
for joint spacing of Sj/B =
1.5/10 are repeated, but with the following changes: trace length
Lj = 5 m, bridge length Bj = 0.5 m and degree of randomness D r =
0.25. All other mechanical and geometrical parameters of the
analysis are kept equal. Comparison of the vertical displacements
and of the displacement difference between the two models are
described in Fig. 13.
0
-0.1
-0.2
-0.3
( m
0
-0.1
-0.2
-0.3
( m
0
0.01
0.02
0.03
0.04
0.05
y
a
b
c
d
e
f
g
h
3
Fig. 12. Vertical displacement d (plots (a–d)) and vertical
displacement difference Dd/Dy (plots (e–h)) profiles above an
underground opening of span B = 10 m, for different values of joint
spacing (Sj) and friction along joints (/).
40 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space
Technology 21 (2006) 29–45
Random joint trace generation reduces vertical dis- placements and
enhances deformation homogenization. The crown displacement is
reduced from d < 0.3 m for non-random joints to d 6 0.06 m for
the same opening geometry but with random joint statistics,
independent
of /av. Similarly the vertical displacement difference immediately
above the crown is reduced from Dd/ Dy < 0.04 to Dd/Dy <
0.005.
The restrained displacements and their homogeniza- tion, are
attributed to the combined action of two factors:
Table 3 Normalized height (hr = h/ht) of the loosening zone above
an underground opening
Friction angle ()
20 30 40 50 60
Sj/B 1.5/10 <0.45 <0.45 <0.45 <0.45 <0.45 2/10
<0.45 <0.45 <0.45 <0.45 Stable 3/10 Stable Stable
Stable Stable Stable
M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space
Technology 21 (2006) 29–45 41
1. enlargement of block length; 2. greater abundance of cantilever
blocks in the strati-
fied roof.
It is concluded that random joint generation im- proves the overall
performance of the rock mass. This effect is less evident with
decreasing mean joint spacing.
4.2.3. B = 1.5ht = 15 m
The vertical displacement profiles for the different values of
joint spacing are given in Fig. 14(a)–(d), and the displacement
difference profiles are given in
4/10 Stable Stable Stable Stable Stable 5/10 Stable Stable Stable
Stable Stable
Geometry: horseshoe tunnel of width B = 10 m and height ht = 10
m.
0 5 10 15 20 25 30
0
-0.1
-0.2
-0.3
no random joints
5 10 15 20 25 30 vertical distance from crown (m)
0
0.02
0.04
0.06
δ
/ v
δ
/ v
a
b
c
d
Fig. 13. Rock mass response above an underground opening of span B
= 1 random joint statistics; (b) vertical displacement difference –
non-random (d) vertical displacement difference – random joint
statistics.
Fig. 14(e)–(h). Clearly, enlarging the opening span by 50% while
keeping the height unchanged degrades the stability of the rock
mass. For joint spacing of Sj/B = 2/15 and /av = 20 the crown
displacement is d = 1.8 m. At y = 0.45ht the displacement is d =
0.6 m, and approaching d = 0.2 m at y > 2.5ht, which is the
magnitude of crown displacement for opening span of B = ht. The
graphic output for this particular case is gi- ven in Fig. 15. It
is clearly seen that the rock mass imme- diately above the crown is
sagging, and inter-bed separation is clearly evident.
Enlarging the joint spacing to Sj/B = 3/15 reduces the vertical
displacement at the crown to d = 0.54 m for /av = 20, d = 0.38 m
for /av = 30, and d < 0.25 m for /av P 40. The displacements are
dying out with in- creased distance from the crown, approaching a
value of d = 0.1 m. For joint spacing of Sj/B P 4/15 the dis-
placements are homogenized, decreasing to d 0.1 m, pending on joint
spacing value. Vertical displacement differences (Dd/Dy) reveal
similar trends: decreasing with increasing joint spacing, and
homogenization of dis- placements for Sj/B P 4/15. The findings of
this section are summarized in Table 4.
le (deg.) 20
0
-0.1
-0.2
-0.3
random joints
5 10 15 20 25 3 vertical distance from crown (m)
0
0
0.02
0.04
0.06 random joints
0 m, and joint spacing of Sj = 1.5 m: (a) vertical displacements –
non- joint statistics; (c) vertical displacements – random joint
statistics;
0
0
0
0.02
0.04
0.06
0.08
0.1
δ /
y
a
b
c
d
e
f
g
h
Fig. 14. Vertical displacement d (plots (a–d)) and displacement
difference Dd/Dy (plots (e—h)) profile above an underground opening
of span B = 15 m, for different values of joint spacing (Sj) and
friction along joints (/).
42 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space
Technology 21 (2006) 29–45
5. Discussion
From the described above, it can be concluded that for the modeled
geometries the prime factor controlling
the stability of underground openings excavated in hor- izontally
layered and vertically jointed rock masses is the spacing of
vertical joints. The effect of friction along joints is secondary,
and is evident only when vertical
Fig. 15. DDA graphic output for tunnel of span B = 1.5ht, vertical
joint spacing of Sj/B = 2/15 and /av = 20: (a) initial
configuration; (b) deformed configuration.
Table 4 Normalized height (hr = h/ht) of the loosening zone above
an underground opening
Friction angle ()
20 30 40 50 60
Sj/B 2/15 <0.85 <0.85 <0.45 <0.45 <0.45 3/15
<0.85 <0.85 <0.45 <0.45 <0.45 4/15 Stable Stable
Stable Stable Stable 5/15 Stable Stable Stable Stable Stable 6/15
Stable Stable Stable Stable Stable
Geometry: horseshoe tunnel of width B = 15 m and height ht = 10
m.
M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space
Technology 21 (2006) 29–45 43
joint spacing is lower than a certain threshold. For the
underground openings modeled here this threshold is Sj/B 6
1/5.
When joint spacing is sufficiently large, the gravita- tional
moment acting within each block is equilibrated by the lateral
moments generated by rotation and reac- tions with neighboring
blocks, thus leading to a stable,
load-resisting structure. However, when joint spacing is bellow the
threshold value the stability is determined by the interaction
between joint spacing and friction.
For an underground opening of span B = ht, joint spacing of Sj/B 6
1/5 and /av smaller than 60 he shear resistance along joints is not
sufficient to preclude verti- cal displacements near the excavation
crown. Stable arching is only met at h > 0.45ht. However, when
joint friction is greater than 60 shear resistance is sufficient to
induce effective arching at the crown is developed.
For an underground opening of span B = 1.5ht and joint spacing of
Sj/B 6 1/5 stable arching is attained at h > 0.85ht for /av 6 30
and at h > 0.45ht for /av > 30. WhenSj/B > 1/5 stable
arching begins at the crown. In terms of span B the height of the
loosening zone for both geometries considered is hr < 0.56B when
Sj/B 6 1/5.
Given the modeled rock mass structure these esti- mates are clearly
conservative. The synthetically gener- ated rock mass is designed
such that resistance to downward displacement is provided only by
shear resis- tance along joints, no cantilever beams are formed
within the modeled rock mass. Cantilever beams however are expected
in a natural rock mass, where joint geometry is characterized by a
certain statistical distribution. The presence of cantilever beams
provides further resistance to downward displacement. Simple linear
perturbation of joint spacing and bridge leads to reduction of
vertical displacements and induces stable arching from the crown
up. In a similar configuration with homogenous spacing distribution
arching is only achieved at h > 0.45ht. Given the uncertainties
associated with rock mass geometry, and its extrapolation, the
extent of cantilever action in the rock mass cannot be determined
accurately. There- fore, the displacement values reported here
should be considered as upper bound.
According to Terzaghi (1946) for tunnels excavated in a blocky rock
mass (consists of ‘‘chemically intact or almost intact rock
fragments, which are entirely sep- arated from each other and
imperfectly interlocked’’) the expected over break ranges from
0.25B to 1.1(B + ht), pending on the ‘‘degree of jointing’’.
How-
44 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space
Technology 21 (2006) 29–45
ever, a quantitative description of the ‘‘degree of join- ting’’ is
not given.
Rose (1982) revised Terzaghis classification and de- scribed the
degree of jointing in terms of RQD (Deere et al., 1967). According
to Rose for a moderately blocky rock mass (RQD = 75–85) the
expected over break ranges from 0.25B to 0.2(B + ht), whereas for a
very blocky rock mass (RQD = 30–75) the expected over break is
(0.2–0.6)(B + ht). This reduction was achieved by ignoring the
level of water table, which according to Brekke (1968) has little
effect on rock load. The draw- backs of this revision are: (1) the
friction along joints is neglected; (2) correlation with RQD.
RQD provides a quantitative estimate of rock mass quality from
drill cores and is defined as the percentage of intact rock pieces
longer than 10 cm in the total length of the core. RQD is a
directionally dependent parameter and its value may change
considerably depending on borehole orientation. In a horizontally
layered and verti- cally jointed rock mass the RQD will be
determined by the spacing between beds rather than the spacing be-
tween joints. Furthermore, RQD is not sensitive to spac- ing
greater than 10 cm. For example, a drill core of say 3 m comprised
of intact rock pieces each 10 cm long will yield the same RQD
estimate as a similar drill core com- prised of three pieces each 1
m long. Correlation with RQD is therefore problematic, especially
for rock masses comprised of horizontal layers with vertical
joints.
Comparison of our research results with Terzaghis prediction shows
that the latter is conservative. Ter- zaghis rock load
classification scheme lacks a consistent treatment of
discontinuities. While this research pro- vides a systematic
treatment of both joint spacing and friction. Both joint spacing
and friction are readily obtainable, either in the field or in
laboratory. There- fore, roof deflection prediction based on these
parame- ters is straightforward and explicit.
6. Conclusions
The stability of underground openings excavated in horizontally
layered and vertically jointed rock masses is studied using the DDA
method with special emphasis on joint spacing and friction.
The reported displacements herein are assumed con- servative due to
the synthetic nature of the modeled rock mass, which does not allow
block interlocking due to irregular joint traces.
Introduction of random joint trace generation reduces
displacements, due to formation of cantilever beams and the
development of longer blocks.
It is found that the height of the loosening zone above an
underground excavation is controlled primarily by the ratio between
joint spacing and excavation span (Sj/B).
For the two geometries studied (B = ht and B = 1.5ht) the following
results are obtained: 1. When Sj/B 6 1/5 the height of the
loosening zone is
smaller than 0.5ht. When B = 1.5ht and / 6 30 the height of the
loosening zone extends to hr = 0.85 ht.
2. In general, the height of the loosening zone is found to be
smaller than 0.56B for both geometries.
3. When Sj/B P 1/3 the rock mass at the roof attains stable
arching, and the height of the loosening zone is negligible.
Acknowledgment
This research was funded by the US – Israel Bina- tional Science
Foundation through Grant 98-399.
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Tunnel roof deflection in blocky rock masses as a function of joint
spacing and friction -- A parametric study using discontinuous
deformation analysis (DDA)
Introduction
The three-hinged beam problem
The general case ndash a parametric study
Geometry and material properties of the analysis domain
Results of the parametric study
B=ht=10m