tunnel roof deflection in blocky rock masses as a function
TRANSCRIPT
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
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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.
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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.
References
Beer, G., Meek, J.L., 1982. Design curves for roofs and hanging walls in bedded rock based on voussoir beam and plate solutions. Trans. Inst. Min. Metal. 91, A18–A22.
Broch, E., Myrvang, A.M., Stjern, G., 1996. Support of large rock caverns in Norway. Tunn. Undergr. Space Technol. 11 (1), 11– 19.
Brekke, T.L., 1968. Blocky and seamy rock mass in tunneling. Bull. Assoc. Eng. Geologist. 5 (1).
Deere, D.U., Hendron, A.J., Patton, F.D., Cording, E.J., 1967. Design of surface and near-surface construction. In: Fairhurst, C. (Ed.), Failure and Breakage of Rock. Society of Mining Engineers of AIME, New York.
Doolin, D., Sitar, N., 2002. Displacement accuracy of discontinuous deformation analysis applied to sliding block. Journal of Engi- neering Mechanics, ASCE 128 (11), 1158–1168.
Evans, W.H., 1941. The strength of undermined strata. Trans. Inst. Min. Metal. 50, 475–500.
Hatzor, Y.H., Benary, R., 1988. The stability of a laminated voussoir beams: back analysis of a historic roof collapse using DDA. Int. J. Rock. Mech. Min. Sci. 35 (2), 165–181.
Huang, Z.P., Broch, E., Lu, M., 2002. Cavern roof stability – mechanism of arching and stabilization by rock bolting. Tunn. Undergr. Space Technol. 17 (3), 249–261.
Jing, L., 1998. Formulations of discontinuous deformation analysis for block systems. Eng. Geol. 49, 371–381.
Lee, D.-H., Choo, S.-K., Hong, C.-S., Park, Y.-J., Kim, G.-J., 2003. A parametric study of the discontinuity in rock mass and its influence on tunnel behavior. In: Proceedings of the 10th ISRM Congress. South African Institute of Mining and Metallurgy, pp. 745–748.
MacLaughlin, M.M., Doolin, D.M., 2005. Review of validation of the discontinuous deformation analysis (DDA) method. Int. J. Numer. Anal. Meth. Geomech. 29.
Newmark, N.M., 1959. A method of computation for structural dynamics. J. Eng. Mech. Div. ASCE. 85 (EM3).
Park, S.-H., 2001. Mechanical behavior of ground with inclined layers during tunnel excavation. Ph.D. Thesis, Department of Civil Engineering, Kyoto University, Japan.
Rose, D., 1982. Revising Terzaghis tunnel rock load coefficients. In: Goodman, R.E., Hueze, F. (Eds.), Proceedings of the 23rd Symposium on Rock Mechanics. Society of Mining Engineers, AIMMP, Berkeley, CA, pp. 951–960.
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Shi, G.-h., 1988. Discontinuous deformation analysis – a new numerical method for the statics and dynamics of block system. Ph.D. Thesis, Department of Civil Engineering, University of California, Berkley.
Shi, G.-h., 1993. Block system modeling by discontinuous deformation analysisTopics in Engineering, vol. 11. Computational Mechanics Publications, Southampton, 209 p.
Shi, G.-h., Goodman, R.E., 1989. The key block of unrolled joint traces in developed maps of tunnel walls. Int. J. Numer. Anal. Methods Geomech. 13, 131–158.
Steindorfer, A., 1997. Short-term prediction of rock mass behavior in tunneling by advanced analysis of displacement monitoring data. Ph.D. Thesis, Department of Civil Engineering, Graz University of Technology.
Sulem, J., Panet, M., Guenot, A., 1987. Closure analysis in deep tunnels. Int. J. Rock. Mech. Min. Sci. 24, 145–154.
Terzaghi, K., 1946. Rock defects and loads on tunnel support. In: Proctor, R.V., White, T. (Eds.), Rock Tunneling with Steel Supports. Commercial School and Stamping Co., Youngstown, pp. 15–99.
Tsesarsky, M., Hatzor, Y.H., 2003. Deformation and kinematics of vertically jointed rock layers in underground openings. In: Ming, L. (Ed), Proceedings of the Sixth International Con- ference on Analysis of Discontinuous Deformation, pp. 93– 102.
Whu, J.-H., Ohnishi, Y., Nishiyama, S., 2004. Simulation of the mechanical behavior of inclined rock mass during tunnel construc- tion using discontinuous deformation analysis (DDA). Int. J. Rock. Mech. Min. Sci. 41 (5), 731–743.
Yeung, M.R., 1991. Application of Shis discontinuous deformation analysis to the study of rock behavior. Ph.D. Thesis, Department of Civil Engineering, University of California, Berkley.
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
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