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International Journal of Rock Mechanics & Mining Sciences 46 (2009) 262– 271
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International Journal ofRock Mechanics & Mining Sciences
1365-16
doi:10.1
� Corr
E-m
journal homepage: www.elsevier.com/locate/ijrmms
Modeling mechanical layering effects on stability of underground openingsin jointed sedimentary rocks
Dagan Bakun-Mazor a, Yossef H. Hatzor a,�, William S. Dershowitz b
a Department of Geological and Environmental Sciences, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israelb Golder Associates, 18300 NE Union Hill Road, Redmond, WA 98052, USA
a r t i c l e i n f o
Article history:
Received 9 January 2008
Received in revised form
16 March 2008
Accepted 2 April 2008Available online 22 May 2008
Keywords:
DFN
DDA
Keyblock
Tunneling
Archeology
Kinematics
Arching
Geo-statistics
09/$ - see front matter & 2008 Elsevier Ltd. A
016/j.ijrmms.2008.04.001
esponding author. Tel.: +972 8 6472621; fax:
ail address: [email protected] (Y.H. Hatzor).
a b s t r a c t
This paper examines the significance of mechanical layering for ‘‘blocky’’ rock mass deformation around
underground openings excavated through sedimentary rocks. The analysis is based on an integration of
geologically based discrete fracture models (‘‘geoDFN’’), which incorporate ‘‘mechanical layering’’, with
the numerical discrete element method—the discontinuous deformation analysis (DDA). We begin with
addressing limitations of classical solutions for mine roof stability in layered and jointed rock masses
via the analysis of the free standing, unsupported, 2000-year-old underground quarry known as
Zedekiah’s cave below the old city of Jerusalem, Israel. We show that both the ‘‘clamped beam’’ model
and the ‘‘Voussoir beam analogue’’ fail to predict the observed roof stability. Only application of discrete
element modeling, which allows for interactions between multiple blocks in the rock mass, can capture
correctly the arching mechanism which takes place in the roof and which properly explains the long-
term stability of this underground opening.
We continue with examining the effect of joint trace geometries on ‘‘blocky’’ rock mass deformation
using the hybrid geoDFN-DDA approach. We show that with increasing joint length and decreasing
bridge length vertical deformations in the rock mass are enhanced. We explain this by the greater
number of distinct blocks in the rock mass due to the greater joint intersection probability in such
geometries. We find that rock bridge length is particularly important when considering the stability of
the immediate roof. With increasing rock bridge length the number of blocks in immediate roof
decreases and consequently individual block width is increased. Increased block width in immediate
roof layers enhances stable arching development, thus improving their load carrying capacity and
overall stability of the underground structure.
& 2008 Elsevier Ltd. All rights reserved.
1. Introduction
Sedimentary rock masses exhibit a geological structure knownas ‘‘mechanical layering’’ [1–3] where vertical to sub-verticaljoints are bounded by bedding plane boundaries (Fig. 1a), and aratio between bed thickness and joint spacing is typically defined[1,4,5]. This paper demonstrates the use of discrete fracturemodels, which incorporate the ‘‘mechanical layering’’ concept toimprove stability analysis for underground opening. This poten-tially represents a significant advance over earlier rock engineer-ing approaches which relied on simplified, statistically based,fracture patterns. These simplified models have typically beenparameterized in terms of for example joint persistence [6] jointtrace length [7] and bridge [8,9] (Fig. 1b).
ll rights reserved.
+972 8 6472997.
This paper presents an approach, which combines the‘‘mechanical layering’’ fracture spatial model [10] for sedimentaryrock (referred to below as a geologic discrete fracture network orgeoDFN) with the discrete element discontinuous deformationanalysis (DDA) method [11]. The DDA approach is applicable forrock masses in which the significant fractures affecting stabilitymust be modeled explicitly using mean joint attitude, length,spacing, and bridge. This includes rock masses with morefractures than can be analyzed using the clamped beam model[12] or the Voussoir bean analogue [13–16] for roof stability inmines, and rock masses where the number of fractures isinsufficient for application of particle flow codes [17] or plasticcontinuum approximations [18].
We begin by examining the stability of the free standingimmediate roof at the 2000-year-old cave of Zedekiah, locatedbelow the old city of Jerusalem and excavated through horizon-tally bedded and vertically jointed Upper Cretaceous limestone.We first study the expected deflection and tensile stresses at mid-section of the 30 m span, 50 cm thick immediate roof layer
ARTICLE IN PRESS
MLT
MLT
MLT
Joint Spacing
SL
b
Degenerate tips
D.R=0.5
B.W.
B.W.
B.W.
Fig. 1. (a) Schematic diagram illustrating ‘‘mechanical layering’’ in sedimentary
rocks (MLT—mechanical layer thickness). (b) Definition of terms used in synthetic
generation of joint trace maps, mesh generated in DDA line generation code DL,
and (c) output of DDA block cutting code (DC).
Local roof slab collapsePortal
“Freemasons Hall”
A
A’
NLocal roof slab collapsePortal
“Freemasons Hall”
A
A’
N
0 5m 10m 15m
A’ AA’ A
N
Fig. 2. The 2000-year-old cave of Zedekiah underneath the old city of Jerusalem.
(a) Layout of Zedekiah cave superimposed on the old city of Jerusalem, (b) plan of
Zedekiah cave, ‘‘Freemasons’ Hall’’ delimited by dashed square, and (c) a cross-
section through Freemasons’ Hall (for location see Fig. 2b).
D. Bakun-Mazor et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 262–271 263
assuming a continuous beam clamped on both ends and showthat the developed tensile stresses at the lowermost fiber exceedthe available tensile strength of the rock by an order of magnitude,implying failure. Since the cave roof has not, in fact, failed, thisapproach is shown to be unrealistically over-conservative. Wecontinue with the Voussoir beam analogue [13] and show that byapplication of this approach a ‘‘snap-through’’ mechanism is anti-cipated. Along with demonstrating the limitations of the Voussoirbeam analogue for the case study at hand we show in Appendix Athat the suggested iterative procedure [16,19,20] is in factredundant as the magnitude of the maximum compressive stressthrough the beam can be determined analytically for differentthrust line geometries. We then apply DDA and show that it nicelycaptures the developed arching mechanism in the layered andjointed roof at the site. We conclude that application of discreteelement approaches (DEM) is essential for correct stability ana-lysis in such rock masses, provided that the rock mass structure ismodeled correctly.
To demonstrate the sensitivity of numerical modeling resultsto geological structure we explore the influence of joint and bridgelength on rock mass stability by incorporating geoDFN modelsinto the block cutting algorithm of DDA and studying the resultingrock mass deformations. We conclude that adding such enhancedcapabilities to the existing block cutting code of DDA is importantfor accurate prediction of both roof deflections and surfacesettlements due to underground mining in fractured rock masses.
2. Zedekiah Cave, Jerusalem, Israel
Zedekiah Cave has been used as an underground quarry belowthe city of Jerusalem from ca. 700 to 800 BCE, and continuouslyuntil the end of the late Byzantine period, in order to extract high-quality building stones for monumental construction in Jerusalem
and vicinity. The quarry is excavated underneath the old city ofJerusalem (Fig. 2a) in a sub-horizontally bedded and moderatelyjointed, low strength, upper Cretaceous limestone, belonging tothe Bina formation [21] of central Israel. The underground quarryis 230 m long, with maximum width and height of 100 m and15 m, respectively (Fig. 2b).
The most striking feature of the quarry is certainly the 30 mspan, unsupported central chamber, sometimes referred to as‘‘Freemasons’ hall’’ (Fig. 2c). Site investigations revealed that largeroof slabs in several side chambers have collapsed over the years(see Fig. 2b), but that the free standing roof of the central chamberhas remained intact to the present day.
2.1. Continuum mechanics approach
Preliminary analysis was carried out using a continuummechanics approach. Obert and Duvall [12] review an elasticsolution for a continuous clamped beam which provides deflec-tions, shear forces and bending moments across the beam. Thissolution may be applicable for the immediate roof of Freemasons’hall provided that the limestone bed comprising the immediateroof material is completely continuous with no intersecting joints.An accurate cross-section through Freemasons’ hall is shown inFig. 2c and its location in the underground monument is shown in
ARTICLE IN PRESS
D. Bakun-Mazor et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 262–271264
Fig. 2b (section A-A0). The concept of Obert and Duvall [12] for aclamped beam model as applied to underground openingsexcavated in rock masses containing planes of weakness parallelto the roof is illustrated in Fig. 3. Assumed geomechanicalparameters for the case of Freemasons’ hall are listed in Table 1,based on field mapping and laboratory testing. Results of theclamped beam analysis are listed in Table 2.
Application of the analytical clamped beam model for Free-masons’ hall predicts that the maximum axial stresses that willdevelop in the beam will be s ¼710.5 MPa. While the upper fiberof the beam is safe against failure by crushing, the lowermost fiberof the beam is unsafe against failure in tension; consequently, theclamped beam model predicts that a tensile fracture will initiateat the centerline and propagate upwards, disjointing the beaminto two blocks, each of 15 m length. The stability of such a three-hinged beam can be estimated by application of the Voussoirbeam analogue as discussed in Section 2.2 below.
2.2. The Voussoir beam analogy
The Voussoir beam model can be invoked to assess the stabilityof the immediate roof at Freemasons’ hall assuming a tensioncrack has formed at the centerline due to tensile strength failureat the lowermost fiber of the immediate roof layer, as discussed inSection 2.1 above. The concept of the Voussoir beam analogue andsign convention are shown graphically in Fig. 4 and the assumed
Fig. 3. Deflection of a single layer on elastic pillars [12].
Table 1Physical and mechanical properties of immediate roof in Freemasons’ Hall—Ze-
dekiah cave, Jerusalem
Free span (L) 30 m
Beam thickness (t) 0.85 m
Unit weight (g ) 19.8 kN/m3
Elastic modulus (E0) 8�103 MPa
Uniaxial compressive strength (sc) 16.4 MPa (bedding parallel)
Tensile strength (st) 2.8 MPa
Table 2Results of clamped beam analysis for immediate roof in Freemasons’ Hall
Maximum deflection at centerline (Z) 8.67 cm
Maximum shear stress at abutments (tmax) 0.445 MPa
Maximum axial stress (s) 10.5 MPa
S
tT
T
Mid-Span Crackfc
n
fc
ZT
Z
Fig. 4. Definition of terms used in the analysis of the Voussoir beam analogue:
S—beam span, t—beam thickness, h—height of compressive zone in beam,
Z—lever arm, and T—thrust.
geomechanical parameters are listed in Table 1, based on fieldmapping and laboratory tests. The determination of the maximumcompressive stress in the beam (fc) is typically obtained byiteration [14,19]. Application of the iterative solution for the roofof Freemasons’ hall reveals that the value of the beam thicknessratio (n) does not converge, since after a single iteration n
becomes greater than 1.0. Using the modified approach suggestedby Diederichs and Kaiser [16] by introducing incremental steps inn, reveals that when the system is supposed to attain equilibriumthe thickness of the compressive arch (Z) is negative. The meaningof this result is that under the given loads, geometry, and materialproperties the beam will undergo buckling deformation leading toa ‘‘snap-through’’ mechanism [19]. Indeed, in some roof sectionsat the site snap-through mechanism may be responsible for theobserved failures (see Fig. 2b). This is certainly not the case in theroof of Freemasons’ hall, which still stands unsupported. There-fore, this approach also proves over-conservative and inapplicablein this case. An accurate understanding of the mechanics of theroof in Freemasons’ hall seems to require an approach whichallows for interaction between blocks and which incorporatesfriction laws for the discontinuities, namely a discrete elementapproach (DEM).
Recall that the Voussoir beam analogue is a statically indeter-minate problem because the stress distribution at the boundaryand the geometry of the thrust line are unknown [19]. Assuminglinear stress distribution at the boundary and an elliptical thrustline geometry, iterative procedures have been proposed todetermine the axial thrust (T) and the deflection of the beam atmid-section [16,19]. It has been argued, but not proven, thatiterations are not necessary if a linear stress distribution is assumedat the boundary, for the value of n must be 0.75 in such a case [15].We prove this in Appendix A and extend the solution for a generalstress distribution function at the boundary.
2.3. The discontinuous deformation analysis (DDA) method
The elastic solution for the roof predicts tensile fracture ifmodeled as a clamped, continuous beam. Application of theVoussoir beam analogy for the roof predicts a snap-throughmechanism. Both scenarios did not materialize in the roof ofFreemasons’ hall during its �2500 year history. Field inspectionsindicate that the current immediate roof is original as testified bypreserved chisel marks on the free surface of the roof. Thesefindings suggest that interactions between distinct blocks in therock mass above the immediate roof must stabilize, rather thanweaken, the roof. To further explore this possibility the rock massaround Freemasons’ hall is analyzed using a discrete elementapproach, the DDA method.
The rock mass structure at Zedekiah cave consists of one set ofsub-horizontal beds and three sets of inclined joints (Table 3).Three-dimensional (3D) analysis of the blocks formed due to jointintersections in the roof [2] is discussed elsewhere [22] in terms ofclassic block theory [23], and is beyond the scope of this paper.Since our discussion in this paper is restricted to a two-dimensional (2D) analysis, the fracture pattern has been simpli-fied to two discontinuity sets: sub-vertical joints and horizontal
Table 3Rock mass structure at Zedekiah cave
Discontinuity set Genetic type Mean orientation Mean spacing (m)
1 Bedding 08/091 0.85
2 Shears 71/061 0.79
3 Shears 67/231 1.48
4 Joints 75/155 1.39
ARTICLE IN PRESS
Table 4Input parameters for DDA analysis of Freemasons’ Hall in Zedekiah cave, Jerusalem
Mechanical properties
Elastic modulus 10 GPa
Poisson’s ratio 0.184
Density 2500 kg/m3
Numerical control parameters
Dynamic control parameter 0.98
Number of time steps 20000
Time interval 0.00025 (s)
Assumed max. disp. ratio 0.0004
Penalty stiffness 5�109 N/m
Friction angle 411
Ver
tical
Dis
plac
emen
t (m
)or
izon
tal C
ompr
essi
veSt
ress
( k
Pa)
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
00
0 0.5-0.025
m point 1m point 2m point 3m point 4
-700-600-500-400-300-200-100
0
D. Bakun-Mazor et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 262–271 265
bedding planes. The three sub-vertical joint sets are thereforereplaced by a single representative vertical joint set with meanspacing of 1.5 m, trace length 8 m, and bridge length 0.2 m. Tracelines for the joint sets are generated synthetically by the linegeneration code of DDA (DL), in this case with a degree ofrandomness of 0.7 (for definition of the degree of randomness andthe trace length generation algorithm employed in DL see [8]). Thetrace lines for the bedding planes are inserted manually to avoidundesired bedding plane locations, maintaining a mean bedthickness of 2 m in the mesh. The reason both joint set andbedding plane spacing is larger in the mesh than in the field is ourdesire to minimize the total number of blocks which will becomputed without compromising on geometrical block character-istics. The resulting DDA mesh is therefore, to some extent, anidealized picture of the rock mass in the field with a smallernumber of individual blocks than in reality; 368 individual blocksform the DDA mesh used for forward numerical analysis (Fig. 5a).
The response of the DDA mesh to gravitational loading ismodeled for duration of 5 s, an equivalent of 20,000 DDA timesteps (time step size and other numerical control parameters arelisted in Table 4). A friction angle value of 411 is assumed for alldiscontinuities based on tilt tests of saw cut planes and measuredjoint surface profiles in the field. The deformed DDA meshconfiguration is shown in Fig. 5b, where principal stresstrajectories at the end of the computation are marked as well.Inspection of Fig. 5b reveals the deformation mechanism thattakes place in the discontinuous roof: following initial verticalshear along the abutments effective arching is obtained whicharrests all further vertical deflections. The measured verticaldeflections in the immediate roof (measurement point 1) withrespect to the rock mass above it (measurement points 2–4) areplotted in Fig. 6a. Note that arching is obtained following verylittle vertical displacement in the rock mass above the cavern
Fig. 5. DDA simulation of ‘‘Freemasons’ Hall’’: (a) block cutting configuration with
four measurement point location; (b) deformed configurations after 5 s of
gravitational loading, with principal stress trajectories.
H
Time (sec)
-8000 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Fig. 6. Results of numerical forward modeling for the ‘‘Freemasons’ Hall’’, obtained
from four measurement points: (a) vertical displacement vs. time and (b)
horizontal compressive stress vs. time.
(measurement points 2–4) whereas in the immediate roof asignificant amount of vertical shear (34 cm) is required beforestable arching is obtained, 3 s after gravity turn on. Thestabilization of the immediate roof by an arching mechanism isobtained by consequent development of a significant axial thrustthrough the disjoint roof beam, in this case of a magnitude of750 kPa (Fig. 6b).
We have seen in this section that attaining stable arching indiscontinuous rock masses around underground openings is acomplex process, which involves dynamic interactions betweenblocks and requires simultaneous modeling of individual blockstrains, displacements, and rotations. This can only be achievedthrough robust numerical analyses suitable for handling distinctelements. Results of recent numerical studies [24,25] suggest thatrock mass structure, and specifically joint set spacing value, haveparamount effect on rock mass deformation. We proceed with astudy of the influence of structural parameters on rock massdeformation, focusing this time on joint length and bridgedistributions, by employing a hybrid geoDFN-DDA approach.
3. A hybrid geoDFN-DDA approach for modeling rock massdeformation
We have seen, through the discussion of the case study above,that correct numerical simulation of the structural pattern is an
ARTICLE IN PRESS
Fig. 8. (a) FracMans simulation of mechanically layered rock mass (for structural
parameters see text); (b) DDA mesh using line coordinate input from FracMans.
D. Bakun-Mazor et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 262–271266
important prerequisite for accurate stability assessment of under-ground openings. As our discussion is limited here to twodimensions, line, rather than plane, generation schemes areexplored, via the existing FracMans [10] and DDA [11] software.The statistical trace line generation code in the DDA environment(DL) is based on the work of Shi and Goodman [8,26] where jointtraces are characterized and simulated using mean joint length(L), bridge (b), and spacing (s) (Fig. 1b). Line generation in DL isbased on a simple Poisson process described by normal distribu-tions for each of the three simulated parameters with userspecified degree of randomness (DR), a parameter which describesthe degree of deviation from the mean which is allowed duringthe simulations (see [8] for details). After line generation iscomplete all line data is provided to the DDA block cutting code(DC). The block cutting process results in a DDA mesh consistingof distinct blocks with known area, center of mass, and edgecoordinates (see Fig. 1c). This block system is used by the forwardmodeling DDA code (DF), to obtain rock mass deformation.
The key to successful application of this approach is thegeneration of a realistic fracture pattern. In sedimentary rocks,such as those at the studied site, it has been found that thefracture pattern is well described by Gross’s concept of ‘‘mechan-ical layering’’ [1]. This cannot be achieved with the standard DLcode, which is based on a simple Poisson spatial process. Inmechanically layered rock masses joint trace lengths are con-strained by bedding plane boundaries. The fracture pattern infractured rock masses with ‘‘mechanical layering’’ is described bythe fracture spacing index (FSI), defined as the ratio between themechanical layer thickness and median joint spacing [5].
A hybrid geoDFN-DDA approach is presented here to addressexactly such cases. The hybrid approach begins by generating a3D, mechanically layered fracture pattern using FracMans, whichallows the simulation of realistic fracture patterns includingspatial correlations and geological processes such as mechanicallayering. For the purposes of the 2D DDA analysis, a 2D trace planewas cut through the 3D DFN model to provide a 2D trace modelwhich can be simulated with the DL code. The DDA block cuttingalgorithm (DC) was then applied to generate a mesh of finiteblocks. Once the DDA mesh is constructed forward modeling ofdeformation can be performed with the DF code. A flow chartshowing the essentials of this procedure is shown in Fig. 7.
SamEdit SoftwareDefine borehole and trace-plane
FracWorks SoftwareDefine box region
and single fractures
SAB.file XML Surface file
FracWorks SoftwareDefine statistical distributions
for structural parameters
Downscaling
X1 Y1 Z1 X2 Y2 Z2
DL format
DDA DL codeDrawing lines + tunnel
x1 y1 x2 y2
DDA DC codeCutting blocks
DDA DF codeForward analysis
dcdt file
blck filedf file
FracMan®
DDA
Legend:
SamEdit SoftwareDefine borehole and trace-plane
FracWorks SoftwareDefine box region
and single fractures
SAB.file XML Surface file
FracWorks SoftwareDefine statistical distributions
for structural parameters
Downscaling
X1 Y1 Z1 X2 Y2 Z2
DL format
DDA DL codeDrawing lines + tunnel
x1 y1 x2 y2
DDA DC codeCutting blocks
DDA DF codeForward analysis
dcdt file
blck filedf file
FracMan®
DDA
Legend:
FracMan®
DDA
Legend:
Fig. 7. Flow chart diagram showing implementation of the hybrid geoDFN-DDA
pre-processor.
Consider for example a mechanically layered rock massconsisting of one set of horizontal layers (beds) and one set ofvertical joints as presented schematically in Fig. 1 but withspecified statistical distributions. The layer thickness in ourexample will be described by log normal distribution with thefollowing parameters: log[mean(m)] ¼ 0, log[deviation(m)]¼ �0.2 and minimum layer thickness of 0.7 m. A minimum layer
thickness is imposed to eliminate generation of unrealisticallyslim blocks due to the application of a constant FSI, which in ourexample will be set at FSI ¼ 1.3 for al layers, a common value forsedimentary rocks [5,27]. A 3D visualization of the mechanicallylayered rock mass obtained in FracMans environment is shown inFig. 8a. The computed 2D block mesh for a selected cross-sectionobtained with the DDA DC code is shown in Fig. 8b. This hybridFracMan-DDA procedure therefore, brings together the power oftwo different geo-engineering tools, one for diverse statisticalsimulations of geological fracture patterns and the other forrobust mechanical deformation analysis.
4. Structural analysis of simulated rock masses
To compare between the hybrid geoDFN-DDA and the standardDDA joint trace simulation approaches we will discuss structuralcharacteristics in meshes obtained with the hybrid procedure,where mechanical layering is imposed on the simulation (meshdenoted FMML from now on for Frac-Man-Mechanical-Layering),and in meshes obtained with earlier simple Poisson fracturemodels (DL and DC codes in DDA), where mean joint length,bridge, and spacing can be varied within some bounds defined bythe degree of randomness. In particular, we will study howvariations in joint length and bridge effect block size distribution
ARTICLE IN PRESS
L=5
0200400600800
10001200140016001800
Block Width [m]
Sam
ples
Bridge=1Bridge=2Bridge=3Bridge=4Bridge=5FMML
L=10
0
500
1000
1500
2000
2500
Sam
ples
Bridge=2
Bridge=4
Bridge=6
Bridge=8
Bridge=10
FMML
L=15
0
500
1000
1500
2000
2500
3000
Sam
ples
Bridge=3
Bridge=6
Bridge=9
Bridge=12
Bridge=15
FMML
0 1 2 3 4 5 6
Block Width [m]
0 1 2 3 4 5 6
D. Bakun-Mazor et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 262–271 267
in meshes obtained in the two different approaches. In bothapproaches the same identical layer thickness distribution,obtained using FMML, is used, with an imposed minimum layerthickness of 0.7 m. Since we impose an FSI value of 1.3 in thegeneration of the FMML mesh, the joint spacing value for theminimum thickness layer in the FMML mesh is set at 0.54 m. Thisjoint spacing value is used as the mean joint spacing value for theentire rock mass, and a degree of randomness of DR ¼ 0.5 isapplied for L, b, and s in all DL simulations. The complete matrixfor DDA simulations in this study is provided in Table 5 and theoutline of the mesh is shown in Fig. 9 with measurement pointlocation for future reference.
Preliminary analysis of the computed block systems enables usto obtain some important structural rock mass characteristicssuch as number of blocks, block width, and block area distribu-tions, utilizing the powerful integration scheme implemented inthe DC block cutting code. This analysis is performed beforeforward modeling is conducted and relates primarily to structuralcharacterization of the rock mass.
Any important geometric characteristic of the simulated rockmass can be studied quantitatively by analyzing the generatedmeshes, and its effect on the mechanical response can be analyzedafter forward modeling is complete.
Consider for example the block width and block area distribu-tions obtained from DL simulations in comparison to FMML(Figs. 10 and 11, respectively). In both FMML and DL simulationsthe obtained block size distributions are similar. The total numberof blocks however, while fixed in the FMML mesh, clearlyincreases with increasing mean joint length and decreases withincreasing mean bridge length. Namely, with increasing mean
Table 5Matrix of DDA simulations
Model L (m) b (m) s (m) DR
1
5
1
0.54 0.5
2 2
3 3
4 4
5 5
6
10
2
0.54 0.5
7 4
8 6
9 8
10 10
11
15
3
0.54 0.5
12 6
13 9
14 12
15 15
L ¼ mean joint length, b ¼ mean rock bridge, s ¼ mean joint spacing, and
DR ¼ degree of randomness [8].
Fixed PointM.P.5
M.P.4
M.P.3
M.P.2
M.P.1
Zoom In Window
Fixed Point
3 30 15 30 3
30
6
63
M.P.5
7.47.4
7.57.5
7.57.5
7.47.4
M.P.4
M.P.3
M.P.2
M.P.1
Zoom In Window
Fig. 9. Outline of the mesh used for forward DDA simulations.
Block Width [m]
0 1 2 3 4 5 6
Fig. 10. Block width distribution obtained from preliminary analysis.
trace length and decreasing mean bridge length the number ofblocks cut by the DC code out of the DL trace maps increases. Thisobservation is intuitive when we consider that the probability forline intersections in a randomly selected unit area in the rockmass should increase with increasing trace length and withdecreasing bridge length. The 2D line intersection probability, aprerequisite for block cutting in the DC code as well as for blockformation in the real rock mass, is discussed elsewhere [2]. Sincelayer thickness distribution is fixed in all meshes, with increasingbridge length individual blocks cut by the DC code are expected tobe wider, since less blocks will be cut in each layer. This effect isshown graphically in Fig. 12 using results of all DL simulations.This result has significant effect on rock mass deformation as willbe discussed in the following section.
Figs. 10 and 11, which describe quantitatively structuralcharacteristics of the rock mass, can be used to obtain someconstraints on the expected rock mass geomechanical res-ponse, and can enhance engineering judgment concerning the‘‘quality’’ of the rock mass, a parameter which otherwise mustbe based on empirical classification methods such as the GSI, Q,and RMR.
5. Mechanical response of simulated rock mass structures
To compare between the deformation of a mechanicallylayered rock mass and a rock mass simulated by mean jointspacing (s), length (L), and bridge (b) values, the forward modelingcode in the DDA environment is employed once for the meshobtained using the hybrid procedure (FMML), and then for models1–15 obtained using DL code (see Table 5). The assumed
ARTICLE IN PRESS
L=5
0100200300400500600700800900
Block Area [m^2]
Sam
ples
Bridge=1Bridge=2Bridge=3Bridge=4Bridge=5FMML
L=10
0
200
400
600
800
1000
1200
Sam
ples
Bridge=2Bridge=4Bridge=6Bridge=8Bridge=10FMML
L=15
0
200
400
600
800
1000
1200
1400
Sam
ples
Bridge=3Bridge=6Bridge=9Bridge=12Bridge=15FMML
0.0 1.0 2.0 3.0 4.0 5.0
Block Area [m^2]
0.0 1.0 2.0 3.0 4.0 5.0
Block Area [m^2]
0.0 1.0 2.0 3.0 4.0 5.0
Fig. 11. Block size distribution obtained from preliminary analysis.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
bridge/Length
Ave
rage
Blo
ck W
idth
[m
]
L=5L=10L=15
0 0.2 0.4 0.6 0.8 1
Fig. 12. Obtained average block width as a function of joint bridge/joint length.
Table 6Input parameters for DDA simulations
Mechanical properties
Elastic modulus 15.3 GPa
Poisson’s ratio 0.21
Density 2300 (kg/m3)
Numerical control parameters
Dynamic control parameter 0.99
Number of time steps 10000
Time interval 0.0005 s
Assumed max. disp. ratio 0.0005
Penalty stiffness 100�106 N/m
Friction angle 301
D. Bakun-Mazor et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 262–271268
geometrical and mechanical parameters for DDA forward model-ing are listed in Table 6.
The response of a mechanically layered rock mass to anunderground opening with a rectangular geometry is shown inFig. 13. The immediate roof, which includes measurement point 1(Fig. 9), collapses and the opening attains a new equilibrium. Notethat the height of the loosened zone is 0.5B, where B is theopening width, exactly as predicted by Terzaghi [28] for such ablocky rock mass. Note also that above the loosened zone (aroundmeasurement point 2) several individual Voussoir beams aredeveloped and attain a new state of equilibrium following somepreliminary vertical deflection. The vertical displacement andaxial stress developed in the five measurement points are plottedin Figs. 13c and d, respectively. The stabilization of the roofsegment containing measurement points 2–5 is indicated by thearrest of the downward vertical deflection (Fig. 13c) and by thedevelopment of stable arching stresses in the beams (Fig. 13d).
The deformation pattern of the roof for rock structuresobtained using standard line generation is shown in Figs. 14 and15 for mean joint trace length of 5 m and 10 m, respectively(graphical simulation outputs for L ¼ 15 m are omitted forbrevity). Note that the deformed meshes presented in Figs. 14and 15 are confined to the zone of interest above the immediateroof, as delineated in Figs. 9 and 13a. The vertical displacementdata obtained for the rock mass above the immediate roof(measurement points 3–5—Fig. 9) are shown in Fig. 16 in termsof the bridge over length (b/L) ratio, where the FMML results areplotted as well for reference.
6. Discussion
6.1. The influence of bridge length on the stability of the immediate
roof
With increasing bridge length the intersection probability oftwo joints belonging to two different sets in a randomly selectedunit area in the rock mass naturally decreases. Therefore, withincreasing bridge length fewer blocks are expected in the rockmass, as discussed above. In the layered rock mass configurationmodeled here this also implies wider blocks in each layer.Previous studies indicate that jointed layer stability increaseswith increasing block width [29,30] up to an optimal widthbeyond which vertical shear along the abutments will dominateover stable arching—as the dead weight of the overlying conti-nuous beams becomes too high [24]. The average block widthwith respect to beam span in the simulations performed in thispaper is well within the range for which Hatzor and Benary [24]found increasing stability with increasing block length (Fig. 17).We see this effect here for the two simulated joint trace lengths inthe graphical outputs of the deformed meshes in the immediateroof zone (Figs. 14 and 15) but this is particularly evident for theL ¼ 10 m set of plots (Fig. 15). It can be appreciated by visualinspection of the graphical outputs that with increasing bridgelength individual layers behave more rigidly, as they are consistedof a smaller number of blocks, and consequently of widerindividual blocks. This effect is particularly important for thestability of the immediate roof area (measurement points 1 and 2)where failure of entire roof slabs is possible.
6.2. The influence of joint length on rock mass deformations
In Fig. 16 the vertical deformations in the rock mass away fromthe immediate roof zone and all the way up to the surface(measurement points 3–5) are plotted as a function of mean jointtrace length as well as b/L ratio. Inspection of Fig. 16 reveals thatthe influence of the b/L ratio on rock mass deformation above the
ARTICLE IN PRESS
-2500
-2000
-1500
-1000
-500
00 1 2 3 4 5
0 1 2 3 4 5
time (sec)
v (m
m) m.p.1
m.p.2m.p.3m.p.4m.p.5
-0.14
-0.12
-0.1
-0.08
-0.06
-0.04
-0.02
0
Time (sec)
σx (
MPa
/m) m.p.1
m.p.2m.p.3m.p.4m.p.5
Fig. 13. Mechanical layering model (FMML) response: (a) whole deformed model; (b) zoom-in on the loosened zone (see location in Figs. 9 and 12a); (c) accumulated
vertical displacement for 5 s; and (d) horizontal compressive stress vs. time.
L = 5, B = 1
33
22
11
L = 5, B = 3
33
22
11
L = 5, B = 5
33
22
11
Fig. 14. Deformation pattern of the roof for models 1, 3, and 5 (see Fig. 9 for
perspective location).
L = 10, B = 10
33
22
11
L = 10, B = 2
33
22
11
L = 10, B = 6
33
22
11
Fig. 15. . Deformation pattern of the roof for models 6, 8, and 10 (see Fig. 9 for
perspective location).
D. Bakun-Mazor et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 262–271 269
immediate roof zone is not significant as can be appreciated fromthe flat curves in this plot. The parameter which seems to be themost significant for rock mass deformation above the immediateroof zone seems to be the simulated mean joint trace length. Withincreasing length of through-going joints [3], more vertical shear
deformation is possible in the rock mass in comparison tomechanically layered rock masses where the vertical extent ofcross joints is bounded by bedding plane boundaries. Our studyclearly indicates that mechanically layered rock masses exhibit lessvertical deformation, and consequently less surface settlements,
ARTICLE IN PRESS
-400
-300
-200
-100
0
Dis
plac
emen
t [m
m]
-500
-400
-300
-200
-100
0
Dis
plac
emen
t [m
m]
Bridge/Length-1000
-800
-600
-400
-200
0
Dis
plac
emen
t [m
m]
L=5 L=10 FMMLL=15
m.p.5
m.p.4
m.p.3
0 0.2 0.4 0.6 0.8 1
Fig. 16. Final vertical displacement (after 5 s) above immediate roof as a function
of joint bridge/joint length. FMML results shown for reference.
Fig. 17. Required friction angle for stability vs. ratio between block width and
beam (opening) span (after [24]). Dashed ellipse shows the relevant block widths/
opening ratio for the simulations performed in this study.
D. Bakun-Mazor et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 262–271270
than a rock mass with persistent, through-going joints, even whenthe total number of blocks in the rock mass is equal.
γ tS/2
S/4
ZTh=nt
xT
x
y
)(xfy =xTT
tZ
T
Fig. A1. Definition of terms used in the analysis of the Voussoir beam analogue.
Only half span is shown due to symmetry: S—beam span, t—beam thickness,
h—height of compressive zone in beam, Z—lever arm, T—mthrust, XT—point of
application of the resultant thrust, g—unit weight, y ¼ f(x)—compressive stress
distribution function at the boundaries.
7. Summary and conclusions
Arching in discontinuous rock masses around undergroundopenings is a complex process that requires robust numericalanalyses suitable for handling distinct elements, namely DEM.
The combination of the geologically realistic fracture models ofmechanical layering provided by FracMans, and the advancedblock cutting algorithm and forward modeling capabilities of DDAcan provide a powerful design tool in geomechanics analyses.
Classic mechanically layered rock masses, as simulated in theFMML model, produce results consistent with Terzaghi’s rock loadprediction of 0.5B.
Rock masses that exhibit through-going joints must bemodeled differently and parameters such as joint length andbridge must be considered. In such rock masses with increasingjoint length and decreasing rock bridges, the joint intersection(and block formation) probability increases, and consequently thetotal number of blocks in such rock masses increases. This resultsin greater vertical deformations as expressed in greater surfacesettlements above the excavated opening.
We find that a rock mass rich in through-going joints exhibitsgreater vertical deformation and surface settlements above deepunderground openings when compared to mechanically layeredrock masses with the same total number of rock blocks. This isbecause through-going joints provide continuous surfaces forvertical shear displacements whereas in mechanically layeredrock masses the extent of vertical joints is limited by beddingplane boundaries and so are the vertical displacements. It is alsoreasonable to assume that mechanically layered configurationsattain better interlocking between blocks, a process which furtherrestricts vertical displacements in such rock masses.
With increasing bridge length the number of blocks per layerdecreases. This is particularly significant in the stabilization of theimmediate roof, because immediate roof beam stability increaseswith increasing block width, up to an optimal block width beyondwhich the load due to beam weight dominates over arching (seeFig. 17).
Acknowledgments
Research of the case study was funded by Israel Ministry ofNational Infrastructure, the quarries rehabilitation fund. RoniEimermacher is thanked for field work and preliminary numericalanalysis. Gony Yagoda Biran assisted in developing the solutionsin Appendix A. Data for the hybrid geoDFN-DDA models weretaken from field inspections performed for Israel CementEnterprises Ltd. Yaakov Mimran, Uri Mor, and Ilia Wainshteinare thanked for discussions. Mark Diederichs is thanked forclarifying aspects relating to publication [16]. Comments receivedby two anonymous reviewers improved the quality of thismanuscript.
Appendix A. Note about the Voussoir beam analogue
Let the stress distribution function at the boundary of theVoussoir beam (Figs. 4 and A1) have the following general form:
y ¼ aðx� hÞb (A.1)
for which a linear stress distribution (b ¼ 1.0) is a private case.From the boundary conditions (Fig. A1) x ¼ 0, y ¼ fc and thereforea ¼ fc(�h)�b. The resultant axial thrust T given by the shaded areain Fig. A1 is
T ¼ A ¼
Z h
0yðxÞdx ¼
Z h
0
f c
ð�hÞbðx� hÞbdx ¼
f ch
bþ 1(A.2)
and the point of application XT is at:
XT ¼ Xc:m: ¼1
A
Z h
0yðxÞxeldx ¼
1
T
Z h
0
f c
ð�hÞbðx� hÞbxdx ¼
h
bþ 2
(A.3)
ARTICLE IN PRESS
D. Bakun-Mazor et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 262–271 271
For the typically assumed linear stress distribution at theboundary (b ¼ 1.0) XT ¼ h/3.
The lever arm Z between the two resultant forces is
Z ¼ t � 2nt
bþ 2¼ t 1�
2n
bþ 2
� �(A.4)
The driving moment, Md, and the resisting moment, Mr, are
Md ¼gtS2
8(A.5)
Mr ¼ TZ ¼f cnt2
bþ 11�
2n
bþ 2
� �(A.6)
The maximum compressive force fc at equilibrium in terms of n
and b can be found by equating (A.5) and (A.6):
f cðn; bÞ ¼bþ 1
8
gS2
ntð1� ð2n=bþ 2ÞÞ(A.7)
Following the procedure suggested by Brady and Brown [19],but analytically and for a general stress distribution, the value of fc
and n at equilibrium are determined here by finding the minimumof (A.7) for a constant value of b:
qf c
qn¼ �ðbþ 1Þ
8
gS2ðt � ð4nt=bþ 2ÞÞ
ðnt � ð2n2t=bþ 2ÞÞ(A.8)
And the obtained value of n for equilibrium is
n ¼bþ 2
4(A.9)
Inserting (A.9) into the second derivative of fc with respect to n
will be used to verify a minimum for fc:
q2f c
qn2¼ðbþ 1Þ
ðbþ 2Þ332gS2
t(A.10)
Note from Eq. (A.9) that for the private case of a linear stressdistribution at the boundary (b ¼ 1.0) indeed the value of thestress ratio n is 0.75, as discussed by Sofianos [15,31] andadmitted by Diederichs and Kaiser [32] in their reply. In reality,the stress distribution along the boundary is not necessarily linerand experimental studies are necessary to determine its exactgeometry [33].
Although mathematically the minimum value is satisfied forevery �1ob, physically b can not be a negative number since themaximum compressive stress (fc) must be at x ¼ 0 (Fig. A1).Moreover, b can not be greater than 2 because n can not be greaterthan 1.0, namely the compressive zone can not be higher than h.Therefore the physically meaningful range for b is: 0obo2.
An interesting outcome of this analysis is that the lever arm (Z)and the resultant force position (XT) are independent of b, asobtained by inserting (A.9) into (A.4):
Zeq0 ¼ t 1�2neq0
bþ 2
� �¼ t 1�
2ðbþ 2Þ
4ðbþ 2Þ
� �¼
1
2t (A.11)
XTðeq0 Þ ¼ 1�bþ 1
bþ 2
� �neq0 t ¼
1
4t (A.12)
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