ARTICLE IN PRESS. International Journal of Rock Mechanics & Mining Sciences

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1 International Journal of Rock Mechanics & Mining Sciences 46 (9) 6 7 Contents lists available at ScienceDirect International Journal of Rock Mechanics & Mining Sciences journal homepage: Modeling mechanical layering effects on stability of underground openings in 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 845, Israel b Golder Associates, 8 NE Union Hill Road, Redmond, WA 985, USA article info Article history: Received 9 January 8 Received in revised form 6 March 8 Accepted April 8 Available online May 8 Keywords: DFN DDA Keyblock Tunneling Archeology Kinematics Arching Geo-statistics abstract 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, -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 longterm 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. & 8 Elsevier Ltd. All rights reserved.. Introduction Sedimentary rock masses exhibit a geological structure known as mechanical layering [ ] where vertical to sub-vertical joints are bounded by bedding plane boundaries (Fig. a), and a ratio between bed thickness and joint spacing is typically defined [,4,5]. This paper demonstrates the use of discrete fracture models, which incorporate the mechanical layering concept to improve stability analysis for underground opening. This potentially represents a significant advance over earlier rock engineering approaches which relied on simplified, statistically based, fracture patterns. These simplified models have typically been parameterized in terms of for example joint persistence [6] joint trace length [7] and bridge [8,9] (Fig. b). Corresponding author. Tel.: ; fax: address: hatzor@bgu.ac.il (Y.H. Hatzor). This paper presents an approach, which combines the mechanical layering fracture spatial model [] for sedimentary rock (referred to below as a geologic discrete fracture network or geodfn) with the discrete element discontinuous deformation analysis (DDA) method []. The DDA approach is applicable for rock masses in which the significant fractures affecting stability must be modeled explicitly using mean joint attitude, length, spacing, and bridge. This includes rock masses with more fractures than can be analyzed using the clamped beam model [] or the Voussoir bean analogue [ 6] for roof stability in mines, and rock masses where the number of fractures is insufficient for application of particle flow codes [7] or plastic continuum approximations [8]. We begin by examining the stability of the free standing immediate roof at the -year-old cave of Zedekiah, located below the old city of Jerusalem and excavated through horizontally bedded and vertically jointed Upper Cretaceous limestone. We first study the expected deflection and tensile stresses at midsection of the m span, 5 cm thick immediate roof layer 65-69/$ - see front matter & 8 Elsevier Ltd. All rights reserved. doi:.6/j.ijrmms.8.4.

2 D. Bakun-Mazor et al. / International Journal of Rock Mechanics & Mining Sciences 46 (9) MLT MLT Joint Spacing MLT Degenerate tips L b S D.R=.5 Portal Local roof slab collapse N A B.W. B.W. Freemasons Hall B.W. A Fig.. (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). A A assuming a continuous beam clamped on both ends and show that the developed tensile stresses at the lowermost fiber exceed the available tensile strength of the rock by an order of magnitude, implying failure. Since the cave roof has not, in fact, failed, this approach is shown to be unrealistically over-conservative. We continue with the Voussoir beam analogue [] and show that by application of this approach a snap-through mechanism is anticipated. Along with demonstrating the limitations of the Voussoir beam analogue for the case study at hand we show in Appendix A that the suggested iterative procedure [6,9,] is in fact redundant as the magnitude of the maximum compressive stress through the beam can be determined analytically for different thrust line geometries. We then apply DDA and show that it nicely captures the developed arching mechanism in the layered and jointed roof at the site. We conclude that application of discrete element approaches (DEM) is essential for correct stability analysis in such rock masses, provided that the rock mass structure is modeled correctly. To demonstrate the sensitivity of numerical modeling results to geological structure we explore the influence of joint and bridge length on rock mass stability by incorporating geodfn models into the block cutting algorithm of DDA and studying the resulting rock mass deformations. We conclude that adding such enhanced capabilities to the existing block cutting code of DDA is important for accurate prediction of both roof deflections and surface settlements due to underground mining in fractured rock masses.. Zedekiah Cave, Jerusalem, Israel Zedekiah Cave has been used as an underground quarry below the city of Jerusalem from ca. 7 to 8 BCE, and continuously until the end of the late Byzantine period, in order to extract highquality building stones for monumental construction in Jerusalem and vicinity. The quarry is excavated underneath the old city of Jerusalem (Fig. a) in a sub-horizontally bedded and moderately jointed, low strength, upper Cretaceous limestone, belonging to the Bina formation [] of central Israel. The underground quarry is m long, with maximum width and height of m and 5 m, respectively (Fig. b). The most striking feature of the quarry is certainly the m span, unsupported central chamber, sometimes referred to as Freemasons hall (Fig. c). Site investigations revealed that large roof slabs in several side chambers have collapsed over the years (see Fig. b), but that the free standing roof of the central chamber has remained intact to the present day... Continuum mechanics approach 5m m 5m Fig.. The -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 crosssection through Freemasons Hall (for location see Fig. b). Preliminary analysis was carried out using a continuum mechanics approach. Obert and Duvall [] review an elastic solution for a continuous clamped beam which provides deflections, shear forces and bending moments across the beam. This solution may be applicable for the immediate roof of Freemasons hall provided that the limestone bed comprising the immediate roof material is completely continuous with no intersecting joints. An accurate cross-section through Freemasons hall is shown in Fig. c and its location in the underground monument is shown in

3 64 D. Bakun-Mazor et al. / International Journal of Rock Mechanics & Mining Sciences 46 (9) 6 7 Fig. b (section A-A ). The concept of Obert and Duvall [] for a clamped beam model as applied to underground openings excavated in rock masses containing planes of weakness parallel to the roof is illustrated in Fig.. Assumed geomechanical parameters for the case of Freemasons hall are listed in Table, based on field mapping and laboratory testing. Results of the clamped beam analysis are listed in Table. Application of the analytical clamped beam model for Freemasons hall predicts that the maximum axial stresses that will develop in the beam will be s ¼ 7.5 MPa. While the upper fiber of the beam is safe against failure by crushing, the lowermost fiber of the beam is unsafe against failure in tension; consequently, the clamped beam model predicts that a tensile fracture will initiate at the centerline and propagate upwards, disjointing the beam into two blocks, each of 5 m length. The stability of such a threehinged beam can be estimated by application of the Voussoir beam analogue as discussed in Section. below... The Voussoir beam analogy The Voussoir beam model can be invoked to assess the stability of the immediate roof at Freemasons hall assuming a tension crack has formed at the centerline due to tensile strength failure at the lowermost fiber of the immediate roof layer, as discussed in Section. above. The concept of the Voussoir beam analogue and sign convention are shown graphically in Fig. 4 and the assumed geomechanical parameters are listed in Table, based on field mapping and laboratory tests. The determination of the maximum compressive stress in the beam (f c ) is typically obtained by iteration [4,9]. Application of the iterative solution for the roof of Freemasons hall reveals that the value of the beam thickness ratio (n) does not converge, since after a single iteration n becomes greater than.. Using the modified approach suggested by Diederichs and Kaiser [6] by introducing incremental steps in n, reveals that when the system is supposed to attain equilibrium the thickness of the compressive arch (Z) is negative. The meaning of this result is that under the given loads, geometry, and material properties the beam will undergo buckling deformation leading to a snap-through mechanism [9]. Indeed, in some roof sections at the site snap-through mechanism may be responsible for the observed failures (see Fig. b). This is certainly not the case in the roof of Freemasons hall, which still stands unsupported. Therefore, this approach also proves over-conservative and inapplicable in this case. An accurate understanding of the mechanics of the roof in Freemasons hall seems to require an approach which allows for interaction between blocks and which incorporates friction laws for the discontinuities, namely a discrete element approach (DEM). Recall that the Voussoir beam analogue is a statically indeterminate problem because the stress distribution at the boundary and the geometry of the thrust line are unknown [9]. Assuming linear stress distribution at the boundary and an elliptical thrust line geometry, iterative procedures have been proposed to determine the axial thrust (T) and the deflection of the beam at mid-section [6,9]. It has been argued, but not proven, that iterations are not necessary if a linear stress distribution is assumed at the boundary, for the value of n must be.75 in such a case [5]. We prove this in Appendix A and extend the solution for a general stress distribution function at the boundary... The discontinuous deformation analysis (DDA) method Fig.. Deflection of a single layer on elastic pillars []. Table Physical and mechanical properties of immediate roof in Freemasons Hall Zedekiah cave, Jerusalem Free span (L) m Beam thickness (t).85 m Unit weight (g ) 9.8 kn/m Elastic modulus (E ) 8 MPa Uniaxial compressive strength (s c ) 6.4 MPa (bedding parallel) Tensile strength (s t ).8 MPa Table Results of clamped beam analysis for immediate roof in Freemasons Hall Maximum deflection at centerline (Z) Maximum shear stress at abutments (t max ) Maximum axial stress (s) n T f c Z S f c Mid-Span Crack T t 8.67 cm.445 MPa.5 MPa 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. The elastic solution for the roof predicts tensile fracture if modeled as a clamped, continuous beam. Application of the Voussoir beam analogy for the roof predicts a snap-through mechanism. Both scenarios did not materialize in the roof of Freemasons hall during its 5 year history. Field inspections indicate that the current immediate roof is original as testified by preserved chisel marks on the free surface of the roof. These findings suggest that interactions between distinct blocks in the rock mass above the immediate roof must stabilize, rather than weaken, the roof. To further explore this possibility the rock mass around Freemasons hall is analyzed using a discrete element approach, the DDA method. The rock mass structure at Zedekiah cave consists of one set of sub-horizontal beds and three sets of inclined joints (Table ). Three-dimensional (D) analysis of the blocks formed due to joint intersections in the roof [] is discussed elsewhere [] in terms of classic block theory [], and is beyond the scope of this paper. Since our discussion in this paper is restricted to a twodimensional (D) analysis, the fracture pattern has been simplified to two discontinuity sets: sub-vertical joints and horizontal Table Rock mass structure at Zedekiah cave Discontinuity set Genetic type Mean orientation Mean spacing (m) Bedding 8/9.85 Shears 7/6.79 Shears 67/.48 4 Joints 75/55.9

4 D. Bakun-Mazor et al. / International Journal of Rock Mechanics & Mining Sciences 46 (9) bedding planes. The three sub-vertical joint sets are therefore replaced by a single representative vertical joint set with mean spacing of.5 m, trace length 8 m, and bridge length. m. Trace lines for the joint sets are generated synthetically by the line generation code of DDA (DL), in this case with a degree of randomness of.7 (for definition of the degree of randomness and the trace length generation algorithm employed in DL see [8]). The trace lines for the bedding planes are inserted manually to avoid undesired bedding plane locations, maintaining a mean bed thickness of m in the mesh. The reason both joint set and bedding plane spacing is larger in the mesh than in the field is our desire to minimize the total number of blocks which will be computed without compromising on geometrical block characteristics. The resulting DDA mesh is therefore, to some extent, an idealized picture of the rock mass in the field with a smaller number of individual blocks than in reality; 68 individual blocks form the DDA mesh used for forward numerical analysis (Fig. 5a). The response of the DDA mesh to gravitational loading is modeled for duration of 5 s, an equivalent of, DDA time steps (time step size and other numerical control parameters are listed in Table 4). A friction angle value of 4 is assumed for all discontinuities based on tilt tests of saw cut planes and measured joint surface profiles in the field. The deformed DDA mesh configuration is shown in Fig. 5b, where principal stress trajectories at the end of the computation are marked as well. Inspection of Fig. 5b reveals the deformation mechanism that takes place in the discontinuous roof: following initial vertical shear along the abutments effective arching is obtained which arrests all further vertical deflections. The measured vertical deflections in the immediate roof (measurement point ) with respect to the rock mass above it (measurement points 4) are plotted in Fig. 6a. Note that arching is obtained following very little vertical displacement in the rock mass above the cavern Table 4 Input parameters for DDA analysis of Freemasons Hall in Zedekiah cave, Jerusalem Mechanical properties Elastic modulus GPa Poisson s ratio.84 Density 5 kg/m Numerical control parameters Dynamic control parameter.98 Number of time steps Time interval.5 (s) Assumed max. disp. ratio.4 Penalty stiffness 5 9 N/m Friction angle 4 Vertical Displacement (m) Horizontal Compressive Stress ( kpa) m point 4 m point m point m point Time (sec) 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 4) whereas in the immediate roof a significant amount of vertical shear (4 cm) is required before stable arching is obtained, s after gravity turn on. The stabilization of the immediate roof by an arching mechanism is obtained by consequent development of a significant axial thrust through the disjoint roof beam, in this case of a magnitude of 75 kpa (Fig. 6b). We have seen in this section that attaining stable arching in discontinuous rock masses around underground openings is a complex process, which involves dynamic interactions between blocks and requires simultaneous modeling of individual block strains, displacements, and rotations. This can only be achieved through robust numerical analyses suitable for handling distinct elements. Results of recent numerical studies [4,5] suggest that rock mass structure, and specifically joint set spacing value, have paramount effect on rock mass deformation. We proceed with a study of the influence of structural parameters on rock mass deformation, focusing this time on joint length and bridge distributions, by employing a hybrid geodfn-dda approach.. A hybrid geodfn-dda approach for modeling rock mass deformation 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. We have seen, through the discussion of the case study above, that correct numerical simulation of the structural pattern is an

5 66 D. Bakun-Mazor et al. / International Journal of Rock Mechanics & Mining Sciences 46 (9) 6 7 important prerequisite for accurate stability assessment of underground openings. As our discussion is limited here to two dimensions, line, rather than plane, generation schemes are explored, via the existing FracMan s [] and DDA [] software. The statistical trace line generation code in the DDA environment (DL) is based on the work of Shi and Goodman [8,6] where joint traces are characterized and simulated using mean joint length (L), bridge (b), and spacing (s) (Fig. b). Line generation in DL is based on a simple Poisson process described by normal distributions for each of the three simulated parameters with user specified degree of randomness (DR), a parameter which describes the degree of deviation from the mean which is allowed during the simulations (see [8] for details). After line generation is complete all line data is provided to the DDA block cutting code (DC). The block cutting process results in a DDA mesh consisting of distinct blocks with known area, center of mass, and edge coordinates (see Fig. c). This block system is used by the forward modeling DDA code (DF), to obtain rock mass deformation. The key to successful application of this approach is the generation of a realistic fracture pattern. In sedimentary rocks, such as those at the studied site, it has been found that the fracture pattern is well described by Gross s concept of mechanical layering []. This cannot be achieved with the standard DL code, which is based on a simple Poisson spatial process. In mechanically layered rock masses joint trace lengths are constrained by bedding plane boundaries. The fracture pattern in fractured rock masses with mechanical layering is described by the fracture spacing index (FSI), defined as the ratio between the mechanical layer thickness and median joint spacing [5]. A hybrid geodfn-dda approach is presented here to address exactly such cases. The hybrid approach begins by generating a D, mechanically layered fracture pattern using FracMan s, which allows the simulation of realistic fracture patterns including spatial correlations and geological processes such as mechanical layering. For the purposes of the D DDA analysis, a D trace plane was cut through the D DFN model to provide a D trace model which can be simulated with the DL code. The DDA block cutting algorithm (DC) was then applied to generate a mesh of finite blocks. Once the DDA mesh is constructed forward modeling of deformation can be performed with the DF code. A flow chart showing the essentials of this procedure is shown in Fig. 7. SamEdit Software FracWorks Software Define borehole Define box region and trace-plane and single fractures SAB.file XML Surface file FracWorks Software Define statistical distributions for structural parameters df file blck file DDA DF code Forward analysis dcdt file DDA DC code Cutting blocks DL format X Y Z X Y Z Downscaling x y x y DDA DL code Drawing lines + tunnel Legend: FracMan DDA Fig. 7. Flow chart diagram showing implementation of the hybrid geodfn-dda pre-processor. Fig. 8. (a) FracMan s simulation of mechanically layered rock mass (for structural parameters see text); (b) DDA mesh using line coordinate input from FracMan s. Consider for example a mechanically layered rock mass consisting of one set of horizontal layers (beds) and one set of vertical joints as presented schematically in Fig. but with specified statistical distributions. The layer thickness in our example will be described by log normal distribution with the following parameters: log[mean(m)] ¼, log[deviation(m)] ¼. and minimum layer thickness of.7 m. A minimum layer thickness is imposed to eliminate generation of unrealistically slim blocks due to the application of a constant FSI, which in our example will be set at FSI ¼. for al layers, a common value for sedimentary rocks [5,7]. A D visualization of the mechanically layered rock mass obtained in FracMan s environment is shown in Fig. 8a. The computed D block mesh for a selected cross-section obtained with the DDA DC code is shown in Fig. 8b. This hybrid FracMan-DDA procedure therefore, brings together the power of two different geo-engineering tools, one for diverse statistical simulations of geological fracture patterns and the other for robust mechanical deformation analysis. 4. Structural analysis of simulated rock masses To compare between the hybrid geodfn-dda and the standard DDA joint trace simulation approaches we will discuss structural characteristics in meshes obtained with the hybrid procedure, where mechanical layering is imposed on the simulation (mesh denoted from now on for Frac-Man-Mechanical-Layering), and in meshes obtained with earlier simple Poisson fracture models (DL and DC codes in DDA), where mean joint length, bridge, and spacing can be varied within some bounds defined by the degree of randomness. In particular, we will study how variations in joint length and bridge effect block size distribution

6 D. Bakun-Mazor et al. / International Journal of Rock Mechanics & Mining Sciences 46 (9) in meshes obtained in the two different approaches. In both approaches the same identical layer thickness distribution, obtained using, is used, with an imposed minimum layer thickness of.7 m. Since we impose an FSI value of. in the generation of the mesh, the joint spacing value for the minimum thickness layer in the mesh is set at.54 m. This joint spacing value is used as the mean joint spacing value for the entire rock mass, and a degree of randomness of DR ¼.5 is applied for L, b, and s in all DL simulations. The complete matrix for DDA simulations in this study is provided in Table 5 and the outline of the mesh is shown in Fig. 9 with measurement point location for future reference. Preliminary analysis of the computed block systems enables us to obtain some important structural rock mass characteristics such as number of blocks, block width, and block area distributions, utilizing the powerful integration scheme implemented in the DC block cutting code. This analysis is performed before forward modeling is conducted and relates primarily to structural characterization of the rock mass. Any important geometric characteristic of the simulated rock mass can be studied quantitatively by analyzing the generated meshes, and its effect on the mechanical response can be analyzed after forward modeling is complete. Consider for example the block width and block area distributions obtained from DL simulations in comparison to (Figs. and, respectively). In both and DL simulations the obtained block size distributions are similar. The total number of blocks however, while fixed in the mesh, clearly increases with increasing mean joint length and decreases with increasing mean bridge length. Namely, with increasing mean Table 5 Matrix of DDA simulations Model L (m) b (m) s (m) DR L ¼ mean joint length, b ¼ mean rock bridge, s ¼ mean joint spacing, and DR ¼ degree of randomness [8]. 6 6 Fixed Point Zoom In Window M.P.5 M.P.4 M.P. M.P. M.P. 5 Fig. 9. Outline of the mesh used for forward DDA simulations trace length and decreasing mean bridge length the number of blocks cut by the DC code out of the DL trace maps increases. This observation is intuitive when we consider that the probability for line intersections in a randomly selected unit area in the rock mass should increase with increasing trace length and with decreasing bridge length. The D line intersection probability, a prerequisite for block cutting in the DC code as well as for block formation in the real rock mass, is discussed elsewhere []. Since layer thickness distribution is fixed in all meshes, with increasing bridge length individual blocks cut by the DC code are expected to be wider, since less blocks will be cut in each layer. This effect is shown graphically in Fig. using results of all DL simulations. This result has significant effect on rock mass deformation as will be discussed in the following section. Figs. and, which describe quantitatively structural characteristics of the rock mass, can be used to obtain some constraints on the expected rock mass geomechanical response, and can enhance engineering judgment concerning the quality of the rock mass, a parameter which otherwise must be 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 mechanically layered rock mass and a rock mass simulated by mean joint spacing (s), length (L), and bridge (b) values, the forward modeling code in the DDA environment is employed once for the mesh obtained using the hybrid procedure (), and then for models 5 obtained using DL code (see Table 5). The assumed L=5 Block Width [m] L= Bridge= Bridge= Bridge= Bridge=4 Bridge=5 Bridge= Bridge=4 Bridge=6 Bridge=8 Bridge= Block Width [m] L=5 Bridge= Bridge=6 Bridge=9 Bridge= Bridge= Block Width [m] Fig.. Block width distribution obtained from preliminary analysis.

7 68 D. Bakun-Mazor et al. / International Journal of Rock Mechanics & Mining Sciences 46 (9) geometrical and mechanical parameters for DDA forward modeling are listed in Table 6. The response of a mechanically layered rock mass to an underground opening with a rectangular geometry is shown in Fig.. The immediate roof, which includes measurement point (Fig. 9), collapses and the opening attains a new equilibrium. Note that the height of the loosened zone is.5b, where B is the opening width, exactly as predicted by Terzaghi [8] for such a blocky rock mass. Note also that above the loosened zone (around measurement point ) several individual Voussoir beams are developed and attain a new state of equilibrium following some preliminary vertical deflection. The vertical displacement and axial stress developed in the five measurement points are plotted in Figs. c and d, respectively. The stabilization of the roof segment containing measurement points 5 is indicated by the arrest of the downward vertical deflection (Fig. c) and by the development of stable arching stresses in the beams (Fig. d). L=5 Block Area [m^] L= Bridge= Bridge= Bridge= Bridge=4 Bridge=5 Bridge= Bridge=4 Bridge=6 Bridge=8 Bridge= Block Area [m^] L=5 Bridge= Bridge=6 Bridge=9 Bridge= Bridge= Block Area [m^] Fig.. Block size distribution obtained from preliminary analysis. Average Block Width [m] bridge/length L=5 L= L=5 Fig.. Obtained average block width as a function of joint bridge/joint length. Table 6 Input parameters for DDA simulations Mechanical properties Elastic modulus 5. GPa Poisson s ratio. Density (kg/m ) Numerical control parameters Dynamic control parameter.99 Number of time steps Time interval.5 s Assumed max. disp. ratio.5 Penalty stiffness 6 N/m Friction angle The deformation pattern of the roof for rock structures obtained using standard line generation is shown in Figs. 4 and 5 for mean joint trace length of 5 m and m, respectively (graphical simulation outputs for L ¼ 5 m are omitted for brevity). Note that the deformed meshes presented in Figs. 4 and 5 are confined to the zone of interest above the immediate roof, as delineated in Figs. 9 and a. The vertical displacement data obtained for the rock mass above the immediate roof (measurement points 5 Fig. 9) are shown in Fig. 6 in terms of the bridge over length (b/l) ratio, where the results are plotted as well for reference. 6. Discussion 6.. The influence of bridge length on the stability of the immediate roof With increasing bridge length the intersection probability of two joints belonging to two different sets in a randomly selected unit area in the rock mass naturally decreases. Therefore, with increasing bridge length fewer blocks are expected in the rock mass, as discussed above. In the layered rock mass configuration modeled here this also implies wider blocks in each layer. Previous studies indicate that jointed layer stability increases with increasing block width [9,] up to an optimal width beyond which vertical shear along the abutments will dominate over stable arching as the dead weight of the overlying continuous beams becomes too high [4]. The average block width with respect to beam span in the simulations performed in this paper is well within the range for which Hatzor and Benary [4] found increasing stability with increasing block length (Fig. 7). We see this effect here for the two simulated joint trace lengths in the graphical outputs of the deformed meshes in the immediate roof zone (Figs. 4 and 5) but this is particularly evident for the L ¼ m set of plots (Fig. 5). It can be appreciated by visual inspection of the graphical outputs that with increasing bridge length individual layers behave more rigidly, as they are consisted of a smaller number of blocks, and consequently of wider individual blocks. This effect is particularly important for the stability of the immediate roof area (measurement points and ) where failure of entire roof slabs is possible. 6.. The influence of joint length on rock mass deformations In Fig. 6 the vertical deformations in the rock mass away from the immediate roof zone and all the way up to the surface (measurement points 5) are plotted as a function of mean joint trace length as well as b/l ratio. Inspection of Fig. 6 reveals that the influence of the b/l ratio on rock mass deformation above the

8 D. Bakun-Mazor et al. / International Journal of Rock Mechanics & Mining Sciences 46 (9) v (mm) m.p. m.p. m.p. m.p.4 m.p.5-5 time (sec) σx (MPa/m) Time (sec) m.p. m.p. m.p. m.p.4 m.p.5 Fig.. Mechanical layering model () response: (a) whole deformed model; (b) zoom-in on the loosened zone (see location in Figs. 9 and a); (c) accumulated vertical displacement for 5 s; and (d) horizontal compressive stress vs. time. L = 5, B = L =, B = L = 5, B = L =, B = 6 L = 5, B = 5 Fig. 4. Deformation pattern of the roof for models,, and 5 (see Fig. 9 for perspective location). immediate roof zone is not significant as can be appreciated from the flat curves in this plot. The parameter which seems to be the most significant for rock mass deformation above the immediate roof zone seems to be the simulated mean joint trace length. With increasing length of through-going joints [], more vertical shear L =, B = Fig. 5.. Deformation pattern of the roof for models 6, 8, and (see Fig. 9 for perspective location). deformation is possible in the rock mass in comparison to mechanically layered rock masses where the vertical extent of cross joints is bounded by bedding plane boundaries. Our study clearly indicates that mechanically layered rock masses exhibit less vertical deformation, and consequently less surface settlements,

9 7 D. Bakun-Mazor et al. / International Journal of Rock Mechanics & Mining Sciences 46 (9) 6 7 Displacement [mm] Displacement [mm] Displacement [mm] m.p m.p m.p. Bridge/Length L=5 L= L=5 Fig. 6. Final vertical displacement (after 5 s) above immediate roof as a function of joint bridge/joint length. results shown for reference. We find that a rock mass rich in through-going joints exhibits greater vertical deformation and surface settlements above deep underground openings when compared to mechanically layered rock masses with the same total number of rock blocks. This is because through-going joints provide continuous surfaces for vertical shear displacements whereas in mechanically layered rock masses the extent of vertical joints is limited by bedding plane boundaries and so are the vertical displacements. It is also reasonable to assume that mechanically layered configurations attain better interlocking between blocks, a process which further restricts vertical displacements in such rock masses. With increasing bridge length the number of blocks per layer decreases. This is particularly significant in the stabilization of the immediate roof, because immediate roof beam stability increases with increasing block width, up to an optimal block width beyond which the load due to beam weight dominates over arching (see Fig. 7). Acknowledgments Research of the case study was funded by Israel Ministry of National Infrastructure, the quarries rehabilitation fund. Roni Eimermacher is thanked for field work and preliminary numerical analysis. Gony Yagoda Biran assisted in developing the solutions in Appendix A. Data for the hybrid geodfn-dda models were taken from field inspections performed for Israel Cement Enterprises Ltd. Yaakov Mimran, Uri Mor, and Ilia Wainshtein are thanked for discussions. Mark Diederichs is thanked for clarifying aspects relating to publication [6]. Comments received by two anonymous reviewers improved the quality of this manuscript. Appendix A. Note about the Voussoir beam analogue Fig. 7. Required friction angle for stability vs. ratio between block width and beam (opening) span (after [4]). Dashed ellipse shows the relevant block widths/ opening ratio for the simulations performed in this study. than a rock mass with persistent, through-going joints, even when the total number of blocks in the rock mass is equal. 7. Summary and conclusions Arching in discontinuous rock masses around underground openings is a complex process that requires robust numerical analyses suitable for handling distinct elements, namely DEM. The combination of the geologically realistic fracture models of mechanical layering provided by FracMan s, and the advanced block cutting algorithm and forward modeling capabilities of DDA can provide a powerful design tool in geomechanics analyses. Classic mechanically layered rock masses, as simulated in the model, produce results consistent with Terzaghi s rock load prediction of.5b. Rock masses that exhibit through-going joints must be modeled differently and parameters such as joint length and bridge must be considered. In such rock masses with increasing joint length and decreasing rock bridges, the joint intersection (and block formation) probability increases, and consequently the total number of blocks in such rock masses increases. This results in greater vertical deformations as expressed in greater surface settlements above the excavated opening. Let the stress distribution function at the boundary of the Voussoir beam (Figs. 4 and A) have the following general form: y ¼ aðx hþ b (A.) for which a linear stress distribution (b ¼.) is a private case. From the boundary conditions (Fig. A) x ¼, y ¼ f c and therefore a ¼ f c ( h) b. The resultant axial thrust T given by the shaded area in Fig. A is Z h Z h f T ¼ A ¼ yðxþdx ¼ c ð hþ ðx b hþb dx ¼ f ch (A.) b þ and the point of application X T is at: X T ¼ X c:m: ¼ Z h yðxþx A el dx ¼ Z h f c T ð hþ ðx b hþb xdx ¼ h b þ (A.) h=nt y y = f (x) x T T x S/4 Z γts/ Fig. A. 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, X T point of application of the resultant thrust, g unit weight, y ¼ f(x) compressive stress distribution function at the boundaries. T x T t

10 D. Bakun-Mazor et al. / International Journal of Rock Mechanics & Mining Sciences 46 (9) For the typically assumed linear stress distribution at the boundary (b ¼.) X T ¼ h/. The lever arm Z between the two resultant forces is Z ¼ t nt n ¼ t (A.4) b þ b þ The driving moment, M d, and the resisting moment, M r, are M d ¼ gts 8 (A.5) M r ¼ TZ ¼ f cnt n (A.6) b þ b þ The maximum compressive force f c at equilibrium in terms of n and b can be found by equating (A.5) and (A.6): f c ðn; bþ ¼ b þ gs (A.7) 8 ntð ðn=bþþþ Following the procedure suggested by Brady and Brown [9], but analytically and for a general stress distribution, the value of f c and n at equilibrium are determined here by finding the minimum of (A.7) for a constant value of b: qf c qn þ Þ gs ðt ð4nt=b þ ÞÞ ¼ ðb 8 ðnt ðn t=b þ ÞÞ (A.8) And the obtained value of n for equilibrium is n ¼ b þ (A.9) 4 Inserting (A.9) into the second derivative of f c with respect to n will be used to verify a minimum for f c : q f c ðb þ Þ gs ¼ (A.) qn ðb þ Þ t Note from Eq. (A.9) that for the private case of a linear stress distribution at the boundary (b ¼.) indeed the value of the stress ratio n is.75, as discussed by Sofianos [5,] and admitted by Diederichs and Kaiser [] in their reply. In reality, the stress distribution along the boundary is not necessarily liner and experimental studies are necessary to determine its exact geometry []. Although mathematically the minimum value is satisfied for every ob, physically b can not be a negative number since the maximum compressive stress (f c ) must be at x ¼ (Fig. A). Moreover, b can not be greater than because n can not be greater than., namely the compressive zone can not be higher than h. Therefore the physically meaningful range for b is: obo. An interesting outcome of this analysis is that the lever arm (Z) and the resultant force position (X T ) are independent of b, as obtained by inserting (A.9) into (A.4): Z eq ¼ t n eq ðb þ Þ ¼ t ¼ b þ 4ðb þ Þ t (A.) 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