Tunnelling and Underground Space Technology 21 (2006) 29 45

Transcription

1 Tunnelling and Underground Space Technology 21 (26) Tunnelling and Underground 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) Michael Tsesarsky *, Yossef H. Hatzor Department of Geological and Environmental Sciences, Ben Gurion University of the Negev, P.O. Box 653, Beer Sheva 8415, Israel Received 2 September 24; received in revised form 15 February 25; accepted 3 May 25 Available online 2 August 25 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 = h t ; (2) B = 1.5h t ; where the opening height was h t =1m 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 h r was found to be dependent upon the ratio between joint spacing and excavation span S j /B: (1) h r <.56B for S j /B 6 2/1; (2) stable arching within the rock mass for S j /B P 3/1. 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. Ó 25 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 discontinuities 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 stresses in the rock mass tend to readjust until new equilibrium is attained. During readjustment of internal * Corresponding author. Present address: Faculty of Civil and Environmental Engineering, Technion, Israel Institute of Technology, Haifa 32, Israel. Tel.: address: michaelt@technion.ac.il (M. Tsesarsky). stresses, and consequently rearrangement of load resisting forces, some displacements of rock blocks occurs. Joints and beddings are sources of weakness in the otherwise competent rock mass and therefore large displacements 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 installed, is.25b to.5b, where B is the tunnel span. For tunnels excavated in moderately jointed rock the maximum expected over break is.25b. For tunnels /$ - see front matter Ó 25 Elsevier Ltd. All rights reserved. doi:1.116/j.tust

2 3 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space Technology 21 (26) excavated in blocky rock mass the expected over break is.25b to 1.1(B + h t ), where h t is the height of tunnel, pending on the degree of jointing. However, no particular reference to the mechanical and geometrical properties 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 friction angle along joints and joint spacing. Lee et al. (23) showed that when two joint sets are encountered at a tunnel excavation face, the most critical joint combination is when a set of horizontal joints (bedding planes) intersects vertically dipping joints. Furthermore, 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 (21) studied the mechanics of rock masses containing inclined joints during tunnel construction using a physical trap door model. Whu et al. (24) replicated these experiments numerically using DDA, showing very good agreement between the physical and the numerical models. The results of both models showed that the distribution of arching stresses above the opening is a function of joint inclination. Huang et al. (22) 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 virgin horizontal stress on the stability of large span openings, up to 65 m, excavated in Norway. However, these high stresses are of tectonic origins which are predominantly 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 developed 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 excavated 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 displacements at and behind the excavation face is a routine practice in rock engineering. Displacement measurements 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. However, 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 (22). In DDA the motion of a homogenously deformable discrete element (block) is computed using series expansion 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 ½DŠ ¼ ðu v r e x e y c xy Þ T ; ð1þ where (u, v ) are the rigid body translations of a specific point (x, y ) within the block, (r ) is the rotation angle of the block with a rotation center at (x, y ), and e x, e y and c xy are the normal and shear strains of the block. Assuming complete first-order approximation of displacement, the expansion term T takes the following explicit form: ½T Š¼ 1 ðy y Þ ðx x Þ ðy y Þ=2 1 ðx x Þ ðy y Þ ðx x Þ=2 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 written in terms of kinetic energy R and potential energy V: E ¼ R V ¼ 1 2 _ DM _D PðDÞ;. ð3þ where M is the mass matrix quantifying the mass distribution around the center of rotation. Body forces, loads, and displacement constraints are expressed in terms of

3 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space Technology 21 (26) 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 displacement vector U: oe ou ¼ o 1 DM _ 2 _D PðDÞ ¼. ð4þ ou Eq. (4) yields a weak equilibrium equation describing the motion of the block: M 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 Newmark type time integration scheme (Newmark, 1959) with collocation parameters b = 1/2, c = 1: Uðt þ DtÞ ¼UðtÞþDt _UðtÞþð 1 2 bþdt2 UðtÞ þ bdt 2 Uðt þ DtÞ; _Uðt þ DtÞ ¼ _UðtÞþð1 cþdt UðtÞþcDt 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 2 3 K 11 K 12 K 1n K >< sym 5>: K nn D 1 D 2.. D n 9 >= >< ¼ or ½KŠfDg ¼fF g ð7þ >; 8 >: F 1 F 2.. F n 9 >= >; where K ij are sub-matrices defined by the interactions of blocks i and j, D i is a displacement variables sub-matrix, and F i is a loading sub-matrix. For two-dimensional formulation K ij is a 6 6 sub matrix, and D i, F i are 6 1 sub-matrices. In total the number of displacement unknowns is the sum of the degrees of freedom of all the blocks. The solution of the system of equations (7) is constrained by inequalities associated with block kinematics. All constraints, including inter-block displacement constraints, are imposed using penalty functions. At each time step the no-tension and no-penetration conditions 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 (22) for further reading on open close iterations. The accuracy of DDA and its applicability to problems of rock engineering was studied by many researchers, for a thorough review of DDA validation the reader is referred to MacLaughlin and Doolin (25). 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 kinematics, 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 (23), refer to Table 1. The effect of joint friction was studied for a constant joint spacing of S j =.25 m, the average spacing in situ, while the friction along joints (/ av ) was varied from / av =2 to / av =8. The effect of joint spacing was studied for / av =47, the peak friction angle obtained from direct shear tests of natural joints, while joint spacing was changed from S j =.25 m to S j = 4 m. For a single Voussoir beam the displacements were measured at selected points along the lower fiber of the beam at intervals of.5 m. For the layered Voussoir configuration the displacements were measured at five locations: (1) m 1 at (x 1, y 1 ) mid-span of immediate roof; (2) m 2 at (x 1, y m); (3) m 3 at (x 1, y m); (4) m 4 at (x 1 +4m,y m); (5) m 5 at (x 1 4m,y m), refer to Fig The three-hinged beam problem First, a simple two-block system was analyzed, typically 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 vertices (Fig. 2). Similar analysis, under somewhat different Table 1 Geometry, material properties and numeric control parameters used in DDA Voussoir models Item Value Beam span 8 m Block properties Density 19 kg/m 3 Elastic modulus 784 MPa PoissonÕs ratio.17 Numeric parameters Penalty stiffness 1 MN/m Time step size.25 s Penetration control parameter (g 2 ).25

4 32 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space Technology 21 (26) S a t Sj m3 b m4 m2 m5 m1 Fixed point 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.

5 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space Technology 21 (26) 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 = 1 GPa, m =.25, and q = kg/m 3. The block dimensions were: S/2 = 5 m, t =.5 m. The numerical control parameters were: normal penalty stiffness P N =1 1 9 N/m and time step size Dt =.1 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 =.3455 m. The dynamic analysis shows a typical oscillatory behavior decaying towards the equilibrium state. This phenomenon is known as algorithmic damping, a φ δ (m) b S j (m) φ Sj -.1 φ( o ) 2 δ (m) Sj =.25m c -.5 Sj (m) δ (m) φ av = 47 o Time (sec) Fig. 3. DDA prediction for mid-span deflection of a single Voussoir beam: (a) as a function of friction angle (/) and joint spacing (S j ); (b) time histories for different values of friction angle, for joint spacing of S j =.25 m; (c) time histories for different values of joint spacing, for available friction angle of / av =47.

6 34 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space Technology 21 (26) a u (m) φ = 3 o φ = 75o φ= 45 o φ = 8 o b v (m) c ω (rad.) Distance from mid-span (m) 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 =8.

7 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space Technology 21 (26) 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 converging to the equilibrium position 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 (S j ). 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.25 m and friction angles 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.25 m. Fig. 5 presents graphic output for the four realizations described in Fig. 4. At low values of / av =3 and / av =45 deformation is dominated by inter-block shear, which is maximum at mid-span and minimum at the abutments (Fig. 4(a)). Block rotation is mostly uniform and antisymmetric. 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 friction angle the moment generated by the lateral thrust within the beam, does not develop effectively. Beam deformation is dominated by vertical shear, which consequently leads to structural failure. Where the available shear resistance along joints is sufficiently high to preclude excessive vertical displacements, and to induce block rotation, consequent build-up of effective lateral thrust within the beam equilibrates the overturning moment and the beam attains equilibrium position. Schematic representation of the forces acting on a block within the beam is given in Fig. 6. Increasing block size by setting S j =.5 m lead to decreased mid-span deflection. The beam however does not attain equilibrium, and eventually fails. Examinationof the deformation time histories for beams with different joint spacing (Fig. 3(c)) reveals that an oscillatory Non-effective thrust line Effective thrust line i - 1 i i+1 i S i+1 R i-1 R i+1 S i-1 W Fig. 6. Schematic representation of the forces acting on a block within the multi-fractured beam.

8 36 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space Technology 21 (26) solution marks equilibrium. In this particular analysis equilibrium is attained when S j P The style of deformation is similar: shear dominates unstable geometries, with relatively small rotations, while block rotation is exhibited for stable geometries 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 (S j ). The deflection data are given at measurement points m1, m2 and m3. Time histories for m1, m2 and m3 for different values of joint spacing are presented in Fig. 7(b) (d). For joint spacing of.25 m the deflections are excessive for all the analyzed friction angle values, and failure is expected. Furthermore, as expected, measurement points data indicate that d m1 > d m2 > d m3, 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 <5 measurement point deflection are similar: d m1.43 m; d m2.26 m; and d m3.18 m. Deformation is achieved by shear as the lateral thrust is not fully developed. When shear resistance is increased to / av =6 measurement point deflections are reduced. For / av >6 shear resistance is increased and downward displacement is restrained. Consequently, the dominant deformation mechanism changes from shear to block rotation, and downward displacements are reduced to d m1 <.15 m, d m2 <.13 m and d m3 <.11 m. By a.1 Sj (m) φ av = 47 o δ (m) friction m1 m2 m3 S j =.25m spacing m1 m2 m φ ( ) b -.1 δ 1 (m) c δ 2 (m) d δ3 (m) Time (sec) Fig. 7. (a) DDA prediction for mid-span deflection of the laminated Voussoir beam, at measurement points m1, m2, and m3 as a function of friction angle (/) and joint spacing (S j ); (b d) are time histories for different values of joint spacing at measurement points m1, m2, and m3, respectively, for available friction angle of / av =47.

9 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space Technology 21 (26) increasing frictional resistance the laminated Voussoir beam behaves like a coherent beam throughout its thickness. Nevertheless, complete stabilization is not attained as indicated by the measured deflections. Deflections are very much restrained when joint spacing is increased to S j =.5 m, d m1 =.131 m, d m2 <.116 m and d m3 <.16 m, compared with d m1 =.434 m, d m2 <.246 m and d m3 <.155 m for S j =.25 m. With further increase in joint spacing to S j P.75 m complete beam stability is obtained with negligible deflections: d m1 <.5 m, d m2 <.3 m and d m3 <.1 m. As before the equilibrium solution is oscillatory. The beam behaves as a coherent element and the vertical loads are transmitted effectively to the abutments. As a result, the deflections are homogenized throughout the bulk of the beam. measurment point Y, X S j c S b b a fixed point Fig. 8. Geometry of DDA model for parametric study. Fixed boundaries are represented by four fixed blocks, each containing a minimum of three fixed points. Table 2 Matrix of DDA parametric study Model / ( ) S j /B D r L j (m) b j (m) B = h t 2 1.5/1 2/1 3/1 4/1 5/ B = 1.5h t 2 2/15 3/15 4/15 5/15 6/ B = opening span; h t = opening height; / = friction angle; S j = mean joint spacing; D r = degree of randomness; L j = joint trace length; b j = joint bridge.

10 38 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space Technology 21 (26) The general case a parametric study 4.1. Geometry and material properties of the analysis domain Fig. 9. Two types of rock masses: (a) not containing cantilever beams; (b) containing cantilever beams. A representative underground opening in horizontally bedded and vertically jointed rock mass is shown in Fig. 8, where a horseshoe section with span B =2a and height h t = b + c is presented. Two different opening geometries are studied: (1) B = h t ; (2) B = 1.5h t, where the tunnel height is 1 m for both cases. The vertical dimension of the domain is set such that the underground opening is located at depth greater than 1.1(B + h t ), conforming with TerzaghiÕs rock load expectation in blocky rock masses. The displacements within the rock mass are measured at seven measurement 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 S b =.1h t and degree of randomness of D r =.25 (spacing may vary by 25% of the mean value during random joint trace generation). 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 displacements in the rock mass and enhances stability (Terzaghi, 1946), due to enhanced arching. Therefore, larger displacements are expected when the number of cantilever beams is minimized. Mechanical properties for intact rock material are chosen to conform with average sedimentary rocks: specific gravity c = kn/m 3 ; Elastic modulus E = 1 GPa, and PoissonÕs ratio m =.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. -.1 δ (m) m +23.5m +18.5m +13.5m +8.5m +4.5m crown time steps Fig. 1. Time histories of vertical displacement (d) above an underground opening of span B = h t, vertical joint spacing of S j /B = 1.5/1 and available friction angle / av =2.

11 4.2. Results of the parametric study M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space Technology 21 (26) B = h t =1m Representative time histories of vertical displacements are given in Fig. 1, which shows DDA results for joint spacing of S j /B = 1.5/1 and available friction angle along joints of / av =2. The crown of the excavation is in a state of progressive failure, clearly marked by the progressive downward displacement. The vertical displacements at points located at y >.45h t above the crown are oscillatory, and are confined to values of d <.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 spacing S j /B 6 2/1 the displacement at the crown is d <.3 m, depending upon friction. The displacements die out with vertical distance from the crown and at y >.85h t the displacements are smaller than.1 m. For joint spacing of S j /B P 3/1 displacements are reduced to d <.1 m, approaching minimum values of d.5 m, followed by homogenization of displacements. The vertical displacement differences Dd/Dy calculated between pairs of measurement points within the vertical profile are presented in Fig. 12(e) (h). For joint spacing of S j /B 6 2/1 homogenization of displacements begin at y >.45h t above the crown, and the difference approaches zero with greater distance from the crown. For joint spacing of S j /B P 3/1 the displacement difference is reduced to.5, with very little variation from the crown up. From the described above, it can be concluded that for a tunnel span of B = h t = 1 m the height of the loosening zone above the excavation is about.5h t for joint spacing of S j /B 6 2/1. For joint spacing of S j /B P 3/1 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, S j /B = 2/1 and / av =6, 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 section are summarized in Table Random joint trace generation Modeling the vertical joints as persistent with constant spacing results in a rock mass structure with a minimum number of cantilever blocks. In this configuration 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 S j /B = Fig. 11. DDA graphic output for tunnel of span B = h t, vertical joint spacing of S j /B = 1.5/1 and available friction angle / av =2 : (a) initial configuration; (b) deformed configuration. 1.5/1 are repeated, but with the following changes: trace length L j = 5 m, bridge length B j =.5 m and degree of randomness D r =.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.

12 4 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space Technology 21 (26) S j /B = 1.5/1 a.5.4 e (m) y S j /B = 2/1 b.5.4 f (m) y S j /B = 3/1 c.5.4 g (m) y S j /B = 4/1 d.5.4 h (m) y y / h t y / h t 3 Y (deg.) h t x B 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 = 1 m, for different values of joint spacing (S j ) and friction along joints (/). Random joint trace generation reduces vertical displacements and enhances deformation homogenization. The crown displacement is reduced from d <.3 m for non-random joints to d 6.6 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 <.4 to Dd/Dy <.5. The restrained displacements and their homogenization, are attributed to the combined action of two factors:

13 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space Technology 21 (26) Table 3 Normalized height (h r = h/h t ) of the loosening zone above an underground opening Friction angle ( ) S j /B 1.5/1 <.45 <.45 <.45 <.45 <.45 2/1 <.45 <.45 <.45 <.45 Stable 3/1 Stable Stable Stable Stable Stable 4/1 Stable Stable Stable Stable Stable 5/1 Stable Stable Stable Stable Stable Geometry: horseshoe tunnel of width B = 1 m and height h t =1m. 1. enlargement of block length; 2. greater abundance of cantilever blocks in the stratified roof. It is concluded that random joint generation improves the overall performance of the rock mass. This effect is less evident with decreasing mean joint spacing B = 1.5h t =15m 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 Fig. 14(e) (h). Clearly, enlarging the opening span by 5% while keeping the height unchanged degrades the stability of the rock mass. For joint spacing of S j /B = 2/15 and / av =2 the crown displacement is d = 1.8 m. At y =.45h t the displacement is d =.6 m, and approaching d =.2 m at y > 2.5h t, which is the magnitude of crown displacement for opening span of B = h t. The graphic output for this particular case is given in Fig. 15. It is clearly seen that the rock mass immediately above the crown is sagging, and inter-bed separation is clearly evident. Enlarging the joint spacing to S j /B = 3/15 reduces the vertical displacement at the crown to d =.54 m for / av =2, d =.38 m for / av =3, andd <.25 m for / av P 4. The displacements are dying out with increased distance from the crown, approaching a value of d =.1 m. For joint spacing of S j /B P 4/15 the displacements are homogenized, decreasing to d.1 m, pending on joint spacing value. Vertical displacement differences (Dd/Dy) reveal similar trends: decreasing with increasing joint spacing, and homogenization of displacements for S j /B P 4/15. The findings of this section are summarized in Table 4. friction angle (deg.) a -.3 no random joints c -.3 random joints δ (m) δ (m) b.6 no random joints d.6 random joints δ/ v.4.2 δ/ v vertical distance from crown (m) vertical distance from crown (m) Fig. 13. Rock mass response above an underground opening of span B = 1 m, and joint spacing of S j = 1.5 m: (a) vertical displacements nonrandom joint statistics; (b) vertical displacement difference non-random joint statistics; (c) vertical displacements random joint statistics; (d) vertical displacement difference random joint statistics.

14 42 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space Technology 21 (26) Sj/B = 2 /15 a.3 e δ (m ) δ / y.2.1 δ (m) Sj/B = 3/15 b δ / y f δ (m ) Sj/B = 4/15 c σ / y g δ (m ) Sj/B = 5/15 d δ / y h Y φ av (deg. ) 2 x h t B 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 (S j ) and friction along joints (/). 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 horizontally 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

15 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space Technology 21 (26) Table 4 Normalized height (h r = h/h t ) of the loosening zone above an underground opening Friction angle ( ) S j /B 2/15 <.85 <.85 <.45 <.45 <.45 3/15 <.85 <.85 <.45 <.45 <.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 h t =1m. Fig. 15. DDA graphic output for tunnel of span B = 1.5h t, vertical joint spacing of S j /B = 2/15 and / av =2 : (a) initial configuration; (b) deformed configuration. joint spacing is lower than a certain threshold. For the underground openings modeled here this threshold is S j /B 6 1/5. When joint spacing is sufficiently large, the gravitational moment acting within each block is equilibrated by the lateral moments generated by rotation and reactions 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 = h t, joint spacing of S j /B 6 1/5 and / av smaller than 6 he shear resistance along joints is not sufficient to preclude vertical displacements near the excavation crown. Stable arching is only met at h >.45h t. However, when joint friction is greater than 6 shear resistance is sufficient to induce effective arching at the crown is developed. For an underground opening of span B = 1.5h t and joint spacing of S j /B 6 1/5 stable arching is attained at h >.85h t for / av 6 3 and at h >.45h t for / av > 3. WhenS j /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 h r <.56B when S j /B 6 1/5. Given the modeled rock mass structure these estimates are clearly conservative. The synthetically generated rock mass is designed such that resistance to downward displacement is provided only by shear resistance 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 >.45h t. Given the uncertainties associated with rock mass geometry, and its extrapolation, the extent of cantilever action in the rock mass cannot be determined accurately. Therefore, 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 separated from each other and imperfectly interlocked ) the expected over break ranges from.25b to 1.1(B + h t ), pending on the degree of jointing. How-

16 44 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space Technology 21 (26) ever, a quantitative description of the degree of jointing is not given. Rose (1982) revised TerzaghiÕs classification and described 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.25b to.2(b + h t ), whereas for a very blocky rock mass (RQD = 3 75) the expected over break is (.2.6)(B + h t ). This reduction was achieved by ignoring the level of water table, which according to Brekke (1968) has little effect on rock load. The drawbacks 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 1 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 vertically jointed rock mass the RQD will be determined by the spacing between beds rather than the spacing between joints. Furthermore, RQD is not sensitive to spacing greater than 1 cm. For example, a drill core of say 3 m comprised of intact rock pieces each 1 cm long will yield the same RQD estimate as a similar drill core comprised 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 TerzaghiÕs prediction shows that the latter is conservative. TerzaghiÕs rock load classification scheme lacks a consistent treatment of discontinuities. While this research provides a systematic treatment of both joint spacing and friction. Both joint spacing and friction are readily obtainable, either in the field or in laboratory. Therefore, roof deflection prediction based on these parameters 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 conservative 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 (S j /B). For the two geometries studied (B = h t and B = 1.5h t ) the following results are obtained: 1. When S j /B 6 1/5 the height of the loosening zone is smaller than.5h t. When B = 1.5h t and / 6 3 the height of the loosening zone extends to h r =.85 h t. 2. In general, the height of the loosening zone is found to be smaller than.56b for both geometries. 3. When S j /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 Binational Science Foundation through Grant References Beer, G., Meek, J.L., 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., Support of large rock caverns in Norway. Tunn. Undergr. Space Technol. 11 (1), Brekke, T.L., 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., 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., 22. Displacement accuracy of discontinuous deformation analysis applied to sliding block. Journal of Engineering Mechanics, ASCE 128 (11), Evans, W.H., The strength of undermined strata. Trans. Inst. Min. Metal. 5, Hatzor, Y.H., Benary, R., The stability of a laminated voussoir beams: back analysis of a historic roof collapse using DDA. Int. J. Rock. Mech. Min. Sci. 35 (2), Huang, Z.P., Broch, E., Lu, M., 22. Cavern roof stability mechanism of arching and stabilization by rock bolting. Tunn. Undergr. Space Technol. 17 (3), Jing, L., Formulations of discontinuous deformation analysis for block systems. Eng. Geol. 49, Lee, D.-H., Choo, S.-K., Hong, C.-S., Park, Y.-J., Kim, G.-J., 23. A parametric study of the discontinuity in rock mass and its influence on tunnel behavior. In: Proceedings of the 1th ISRM Congress. South African Institute of Mining and Metallurgy, pp MacLaughlin, M.M., Doolin, D.M., 25. Review of validation of the discontinuous deformation analysis (DDA) method. Int. J. Numer. Anal. Meth. Geomech. 29. Newmark, N.M., A method of computation for structural dynamics. J. Eng. Mech. Div. ASCE. 85 (EM3). Park, S.-H., 21. Mechanical behavior of ground with inclined layers during tunnel excavation. Ph.D. Thesis, Department of Civil Engineering, Kyoto University, Japan. Rose, D., Revising TerzaghiÕs 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

17 M. Tsesarsky, Y.H. Hatzor / Tunnelling and Underground Space Technology 21 (26) Shi, G.-h., 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., Block system modeling by discontinuous deformation analysistopics in Engineering, vol. 11. Computational Mechanics Publications, Southampton, 29 p. Shi, G.-h., Goodman, R.E., The key block of unrolled joint traces in developed maps of tunnel walls. Int. J. Numer. Anal. Methods Geomech. 13, Steindorfer, A., 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., Closure analysis in deep tunnels. Int. J. Rock. Mech. Min. Sci. 24, Terzaghi, K., 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 Tsesarsky, M., Hatzor, Y.H., 23. Deformation and kinematics of vertically jointed rock layers in underground openings. In: Ming, L. (Ed), Proceedings of the Sixth International Conference on Analysis of Discontinuous Deformation, pp Whu, J.-H., Ohnishi, Y., Nishiyama, S., 24. Simulation of the mechanical behavior of inclined rock mass during tunnel construction using discontinuous deformation analysis (DDA). Int. J. Rock. Mech. Min. Sci. 41 (5), Yeung, M.R., Application of ShiÕs discontinuous deformation analysis to the study of rock behavior. Ph.D. Thesis, Department of Civil Engineering, University of California, Berkley.