PFC3D modeling of a jointed rock block behavior near anunderground excavation and comparison with physical model test results P.H.S.W. Kulatilake Rock Mass Modeling and computational Rock Mechanical Laboratory, University of Arizona, Tucson, AZ 85721, USA; E-mail address:
[email protected]
X.X. Yang Rock Mass Modeling and computational Rock Mechanical Laboratory, University of Arizona, Tucson, AZ 85721, USA and State Key Laboratory for Geo-mechanics& Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China
ABSTRACT: Mechanical behavior of a jointed rock mass with non-persistent joints located adjacent to a free surface on the wall of an excavation was simulated under compression without and with support stress on the free surface using approximately 0.5 m cubical synthetic jointed rock blocks having 9 non-persistent joints of length 0.5 m, width 0.1 m and a certain orientation arranged in an en echelon and a symmetrical pattern using PFC3D software package. The joint orientation was changed from one block to another to study the effect of joint orientation on strength, deformability and failure modes of the jointed blocks. First the micro-mechanical parameters of the PFC3D model were calibrated using the macro mechanical properties of the synthetic intact standard cylindrical specimens and macro mechanical properties of a limited number of physical experiments performed on synthetic jointed rock blocks of approximately 0.5 m cubes. Under no support stress, the synthetic jointed rock blocks exhibited the same three failure modes:(a) intact rock failure, (b) step-path failure and (c) planar failure under both the physical experiments and numerical simulations for different orientations. The jointed blocks which failed under intact rock failure mode and planar or step-path failure mode produced high and low jointed block strengths, respectively. Three phases of convergence of free surface were discovered. The joint orientation and support stress played important roles on convergence magnitude.The average increment of jointed block strength turned out to be about 10, 7.9 and 6.6 times the support stresswhen support stresses of 0.06 MPa, 0.20 MPa and0.40 MPa were applied, respectively. The modeling results offer some guidelines in support design for underground excavations.
1 INTRODUCTION Underground excavations are extensively made for varying purposes such as mining, transportation, and nuclear waste and fluid storage. Stabilization of these excavations is of great scientific interest for engineers due to its significant effects on the safety of workers as well as the operation of underground facilities. Stability of underground excavations is dominated by the mechanical behavior of adjacent rock massessubjected to excavation-induced stress redistribution. Stability conditions around excavations in discontinuous rock masses heavily depend on the discontinuity geometry pattern. The reinforcement with lining, or grouted rock bolts or cables or steel setsis effective to keep underground excavations stable in poor rock mass conditions, especially in rock masses with discontinuities. Kulatilake et al. (2013)conducted an investigation to obtain the optimum stability conditions of a tunnel in a deep coal mine in China by analyzing different tunnel shapes and support patterns. They demonstrated that an appropriate support pattern improves the strength of the rock mass and significantly reduce the plastic zone around the tunnel and convergence of the tunnel. Wang et al. (2012)through stability investigations around a mine tunnel using three-dimensional discontinuum and continuum stress analyses showed that
the applied rock support decreased the wall deformations by about 43-51%, roof deformations by about 29-39% and floor deformations by 9-10%. Due to the presence of discontinuities such as joints, bedding planes, shear zones and faults, rock masses show anisotropic mechanical behavior.A number of investigators have performed research on anisotropic mechanical behavior of discontinuous rock massesunder two categories of discontinuities: (a)persistent discontinuities and (b) non-persistent discontinuities. Yang et al. (1998)have investigated the effect of persistent joint sets on the unconfined compressive strength and deformation of synthetic rock mass models and found that the failure modes can be divided into three types, namely splitting failure through the rock blocks, sliding along joint planes and mixed splitting-sliding mode. They also have demonstrated that the failure strength and deformability exhibit pronounced anisotropy mainly due to the effect of joint orientation. Kulatilake et al. (2001)haveobtained similar results on failure modes by conducting physical and particle flow modeling onsynthetic persistent-jointed rock blocks under uniaxial loading. Both the laboratory experiments and the numerical simulations have shown that the uniaxial block strength decreases in a nonlinear manner with increasing values of the fracture tensor component (Kulatilake et al., 1993)which quantifies the combined directional effect of the joint geometry parameters including joint density, orientation and size distributions and the number of joint sets. Non-persistent joints commonly exist in underground rock masses.Due to the significant effect of non-persistent joints in rock engineering, some investigators have focused their efforts on mechanical properties of rock mass models with non-persistent joints. Prudencio and Van Sint Jan (2007)performed biaxial tests on physical models of rock with non-persistent joints and concluded that the failure modes and strengths depend on the geometry of the joint system, the orientations of the principal stresses, and the ratio between the intermediate principal stress and intact material compressive strength. Their tests showed three basic failure modes, namely failure through a planar surface, step-path failure and failure by rotation of new blocks. Bahaaddini et al. (2013)have numerically investigated the effect of joint geometrical parameters on the mechanical properties of a non-persistent jointed rock block under uniaxial compression and have found that the failure mode is principally determined by the step angle and the joint orientation with respect to the applied principal stress direction.Wu and Kulatilake (2012)developed relations between strength and deformability parameters of jointed blocks and fracture tensor components by performing three dimensional stress analyses on different size jointed blocks having non persistent joints. Using the developed relations they finally developed an orthotropic constitutive model. In their study, they also showed the effect of scale on strength and deformability of jointed blocks. In this paper, mechanical behavior of a typical jointed rock block having non-persistent joints located adjacent to an underground excavation is investigated through numerical simulation by applying appropriate three-dimensional boundary conditions using the PFC3D software package(Itasca, 2008). First, the micro-mechanical parameters of the selected synthetic intact rock material are calibrated using the macro-mechanical properties of the same synthetic intact rock. Then the micro-mechanical parameters of the joints in the synthetic jointed rock blocks are calibrated using the macro-mechanical results of the physical experiments performed on the same synthetic jointed rock blocks of approximately 0.5 m cubes.Next the capabilities of the calibrated numerical model on synthetic jointed rock blocks are validated by comparing the failure modes of the numerical simulations with the corresponding experimental physical jointed block test results. Then, the effect of joint orientation on strength, deformability and failure modes of jointed rock blocks are demonstrated by performing numerical simulations on the validated synthetic jointed rock model under different rock support stress conditions applied on the free surface of the excavation. 2 PHYSICAL MODELING OF JOINTED ROCK BLOCKS To have a better understanding of the mechanical behavior of jointed rock masses around anexcavation in an underground mine, an experimental study on strength and deformation behavior of large scale synthetic jointed rock blocks was performed by Zhang(2013)and Jing et al. (2014). The large scale model experimental system has been designed based on the following
consideration. An anchored unit body of jointed rock mass of size 2.5 m × 2.5 m × 2.4 mhas been separated from the wall of atunnel in a coal mine as shown in Figure 1. For this selected rock mass, the major principal stress, σ1, acts in the vertical direction. The intermediate principal stress, σ2, acts in the lateral direction perpendicular to the shown rock mass in Figure 1. The minor principal stress, σ3, acts in the lateral direction perpendicular to the free face. The minor principal stress direction has been used to apply support stress to the rock mass. According to the above description, the conceptual rock block model shown in Figure 2a has been developed. Note that in this conceptual model the rock block face opposite to the free face is considered as a fixed face. The size of the synthetic jointed rock block used in the large scale physical experiment has been 0.5 m × 0.5 m × 0.48 m (see Figure 2b) to simulate the real rock mass of size 2.5 m × 2.5 m × 2.4 m. With respect to the similarity ratios used, the reader is referred to Zhang (2013). Aspecially designed loading frame has been used to apply a stress system on the large scale synthetic jointed rock block (Figure 2b)which is similar to the stress system shown in Figure 2a for the conceptual rock block model.
Cable Bolt σ1
Free surface
2500
Anchored unit body 2500
Figure 1.Selected anchored unit body of jointed rock mass on the wall of an underground excavation(Zhang, 2013; Jing et al., 2014).
σσ11
(a)
(b)
Topplate plate Top
σ2
σ2
yy
Force sensor Force sensor σ1
Lateral plate
Lateral plate
Freesurface surface Free
xx
σ2σ2
σσ22
z
z
σσ22 σσ1 1
Figure 2. (a) Conceptual rock block model; and (b) loading system used for physical experiments (Zhang, 2013; Jing et al., 2014).
For the large scale experiments,syntheticrock blocks were fabricated by mixingC42.5 cement, plaster, 200M silica flour and water in themass ratio of 7:3:5:2.5. The unit weight of the synthetic rock-like sample was 16.35kN/m³.The uniaxial compressive strength, UCS, and the elastic modulus,E,determined by conducting unconfined compressive strength tests on cylindrical spe-
cimensof diameter 50 mm and height 100mm in the laboratoryfor the synthetic rock material were3.77MPa and 0.44GPa,respectively. After determining the basic properties of the experimental synthetic material, synthetic jointed rock blocks have been fabricated using amold. The synthetic jointed rock blockswere fabricated with nine joints arranged in an en echelon and a symmetrical pattern as shown in Figure 3. In order to study the effect of joint orientation on the mechanical behavior of the synthetic jointed rock blocks, the joint geometry was designed by varying the dip angle, α, using three values: α=30º, 45º and 60º and keeping the other joint parameters as constants (see Figure3). The joints have been fabricated by inserting steel plates of thickness 2mm and length 100mm into the synthetic rock block when pouring the synthetic model material mixture. The mixing, molding and curing of the model have been carefully controlled for the production of repeatable specimens.
500
500
60°
500
1 00
500
0 10 45°
500
0 10
1 00 0 10 30°
100
500
Figure 3.Joint geometry configurations used in the fabricated synthetic jointed rock blocks (Unit: mm).
The aforementioned specimens then have been carefully cured in water under 20±5ºCtemperature for seven days before pulling out the steel plates to create some joints. After another eight days, the moldshave been removed and then the jointshave been filled with talcum powder and plugged with plaster to simulate smooth joints. Before testing, the specimenshave beencured for 60 days in water. The prepared specimens have been then tested usingtheTYS-500 testing machine and the axial load, axial displacement and horizontal displacementshave been collected using a data acquisition system. The data obtained from the aforementioned large scale physical experiments(Zhang, 2013; Jing et al., 2014)have been used to calibrate and validate a numerical model developed on the same synthetic jointed rock blocks using PFC3D, a commercially available software package developed by Itasca (2008),based on the discrete element method. 3 CALIBRATION AND VALIDATION OF THE PFC3D NUMERICAL MODEL 3.1 Calibration of intact material micro-mechanical parameters The intact synthetic model material selected in this study is represented by compacted particles cemented with the parallel bond model(Potyondy and Cundall, 2004; Pierce et al., 2007; Itasca, 2008).Therefore, the micro-mechanical parameters consist of two categories: one for particles and the other for parallel bonds. The PFC3D code allows one to simulate the macromechanicalbehavior of the selected synthetic model material by selecting appropriate values forthe micro-mechanicalparameters such as the particle size distribution and packing, particle and parallel bond stiffnesses, particle friction coefficient and bond strengths(Potyondy and Cundall, 2004;Pierce et al., 2007; Itasca, 2008).To select appropriate micro-mechanical parameter values for the synthetic intact materialone needs to basically go through a trial and error procedure iteratively varying the micro-mechanical parameter values to match the required macromechanical behaviors of the selected synthetic material. This procedure is known as the calibration of the intact synthetic material micro-mechanical parameters. Herein, the required macromechanical behaviors are chosen as the strength response through theUCS and the deformability response through theYoung's modulus and Poisson's ratio of a cylindrical specimen having 50mm diameterand length/diameter ratio =2:1 under unconfined compression.
In this study, a specialcalibration sequence was followed to rationalize the micro-mechanical parameter calibration and to minimize the number of iterations.First, the particle and parallel bond moduli and the ratios of normal to shear stiffness were set equal between the particles and parallel bonds to reduce the number of independent parameters(Potyondy and Cundall, 2004).Then, the Young's modulus was calibrated by setting the material strengths to a large value and varying E(particleYoung’s modulus) and E (parallel bond Young’s modulus) to match the Young’s modulus between the numerical and laboratory specimens. Next, by changing k n /k s (particle normal stiffness/particle shear stiffness) and k n / k s (parallel bond normal stiffness/parallel bond shear stiffness),the Poisson's ratio of the numerically simulated intact synthetic cylindrical specimenwas matched to that of the laboratory specimen. After calibrating the aforementioned deformation micro-mechanical parameters, the peak strength between the numerical and laboratory specimens were matched by gradually reducing the normal and shear bond strengthsofthe parallel bonds. During thisprocedure, it is important to fix the ratio of normal to shear bond strength ( n / s )because it affects the failure mode of the specimen. Therefore, a series of numerical simulationswere conducted to match the failure mode of the cylindrical specimen between the numerical and laboratory by varying n / s while keeping other parameters unchanged.The obtained failure modeswith varying n / s for cylindrical specimens used in the numerical simulations are displayed in Figure4. c / c 0.5
c / c 1. 0
c / c 1.5
c / c 2.0
c / c 2.5
Laboratory specimen
Figure 4. Comparison of the failure modes obtained between the laboratory specimen and the cylindrical specimens used in the numerical simulations with different ratios of normal to shear bond strength of parallel bonds. (The red and blue imply tensile cracks and shear cracks, respectively.)
As can be seen from Figure4, as the specimen is compressed under uniaxial stress, numerous cracks(tensile and shear cracks) are produced through breakage of parallel bonds. At first, the cracks mainly concentrate on one end of the specimen, and then extend to the middle area of the specimen. If the ratio of normal to shear bond strength is under one or over one, the crack distributiondoes not seem to form a complete macro fracture plane in the specimen.On the other hand, the crack distributionseems to form a single macro-fracture plane in the caseof c / c 1.0 . The failure mode of the numerically simulated synthetic cylindrical specimen with c / c 1.0 agrees well with that of the laboratory specimen(Zhang, 2013). Thus, setting the ratio of normal to shear bond strength equal to one seems to increase the confidence of the numerical modelin simulating the appropriate failure behavior of the synthetic material. The friction coefficient μmainly affects the post-peak behavior of the synthetic material. To select an appropriate μ value, a series of numerical testswere conducted on standard cylindrical specimens under uniaxial condition using different μ values (μ=0.40, μ=0.60, μ=0.80) to obtain stress-strain curves. The numerically obtained stress-strain curves were compared with the stress-strain curve obtained from the laboratory specimen (Figure5). In this comparison, in addition to comparing the post peak behavior, it is necessary to match the macro UCS and Young's modulus values obtained for different numerical simulations of the synthetic materialwith that obtained for the laboratory specimen.Therefore, the E( E ) and σ c ( c ) values need to be modified with μvalue at the same time. Figure 5 shows that the post-peak trend of the laboratory stress-strain curve is approximately parallel to that of all the stress-strain curves of cylindrical numerical specimens.Therefore, the post peak behavior of numerical specimens did not provide any guidance to select the particle friction value. On the other hand, the coefficient of friction obtained from the laboratory triaxial tests on synthetic intact rock was around 0.64. Therefore, the friction coefficient value of 0.60 was selected to represent the particle friction.
4.00
Stress/MPa
3.20 2.40 1.60
Laboratory specimen Experiment[26,27] μ=0.40 Numerical modeling μ=0.60 specimen μ=0.80
0.80 0.00 0
4
8
12
16
20
Strain/10-3
Figure 5.Comparison between the stress-strain curves obtained from the numerical simulations and the stress-strain curve from the laboratory specimen.
Table 1 lists the determined micro-mechanical parameter values to simulate the intact synthetic rock with the parallel bond model and Table 2 shows the calibration results. Table 1.Micro-mechanical parameter values usedfor the synthetic rock material. Parameter of particle Value Parameter of parallel bond ρ (kg/m3) 1635
kn / ks
Value 1.0
1.70 kn / ks Ec (GPa) 0.37 0.37 E c (GPa) 0.60 3.00±0.75 σ c (mean± std.dev., MPa) R R /R 1.66 3.00±0.75 c ( mean± std.dev., MPa) 3 3 Rmax 1.079 10 , Rmin 0.650 10 (for cylindrical specimen) R (m) Rmax 7.280 10 3 , Rmin 4.550 10 3 (for jointed rock block) Note that ρ is the density of the synthetic rock material; is the radius multiplier used to set the parallel bond radii; R is the radius of the particle; R and R are the maximum radius and minimum radius of the particle, respectively. rat
1.70
max
min
max
min
Table 2.Results of calibration. Strength, UCS Deformation modulus,E Poison's ratio, ν MPa GPa Numerical simulation result 3.78 0.433 0.185 Laboratory experimental result 3.77 0.440 0.16 - 0.25 Deviation 0.3% 1.6% N/A Note that the Poisson's ratioin the laboratory experimental result is not obtained by the experiment, but calculated according to a similarity criterion(Zhang, 2013; Jing et al., 2014). Mechanical parameter
3.2 Calibration of joint micro-mechanical parameters 3.2.1 Setup of jointed rock model The micro-mechanical parameters of joints are calibrated in this section through an inverse modeling using the results of laboratory physical experiments conducted on jointed rock blocks. First, a numerical modelof size:500mm×500mm×480mm, which is identical to the jointed rock block fabricated in the laboratory (Figure2)was generated by using PFC3D software package (Itasca, 2008). Then, a series of trial and error numerical simulationswereperformed varying mi-
cro-mechanical parameters of joints in a systematicmanner and comparing the strength values and failure modesobtained between the numerical simulations and the laboratory test results. Figure 6 shows the synthetic jointed rock block having joint dip angle α=45ºused in the numerical simulation as an example. σ1
Top Wall Free Surface Rear Wall σ2
Lateral Wall
Bottom Wall Lateral Wall
Figure 6.Numerical model of synthetic jointed rock block.
In the synthetic jointed numerical model, nine joints are simulated with the smooth joint model and arranged as an en echelon as shown in Figure 6. Note that every joint is placed fully across the model inσ2direction and the joint is thus a rectangle discontinuity with length 480mm and width 100mm. In PFC3D, a rectangular smooth joint plane cannot be created directly because the smooth joint contact model can only be defined as a circular disk-shaped area having aspecified radius. Herein, each joint plane is obtained by geometricalcombination of multiple smooth joint disksas displayed in Figure7.
50mm 100mm
57.7mm
Length of the joint plane Figure 7.Creation of a smooth joint plane in the numerical model.
In the numerical simulation, both the rear wall and bottom wall were fixed with no displacement; whereas the velocities of two lateral walls were controlled by a servo-control mechanism to maintain a certain confining stress σ2. During the numerical compression test, the synthetic rock blocks are loaded by the top wall using a constant velocity. In the laboratory physical experimentσ2has been set to 1.15MPa throughout the compression test (Zhang, 2013; Jing et al., 2014). Besides the aforesaid five walls, a front wall was also created to form a vessel for generation of the synthetic rockmaterial. After the whole model was set to be in static state under the hydrostatic pressure of 1.15MPa by moving the four movable walls, the front wall was deleted to expose a free surface to simulate the unsupported inner face of the underground excavation. 3.2.2 Smooth joint model The joints createdinthe intact synthetic material were simulated with the smooth-joint model. The advantages of having a smooth joint model are explained in Itasca (2008). A typical smooth joint contact model is shown in Figure8. Once a joint plane is defined, a smooth joint is assigned
at contacts between balls whose centers lie on the opposite sides of the defined joint plane. At these contacts, first, the existed bonds are removed and smooth joints are defined in a direction parallel to the joint plane regardless of the contact orientations. These contacts behave according to the rules defined by the smooth joint model with specified parameters given by the user.The particles having such contacts may overlap or pass through each other rather than be forced to roll around one another. y x
Physical analog Closed joint
θp
Surface2 Joint plane Surface1
Joint plane ball1
Smooth joint contact
nˆ c
ˆt j
nˆ c
nˆ j
ball2
Surface2
Cross-section
Surface1
Figure 8. Smooth joint contact model (Itasca, 2008).
These newly created contacts act mechanically as a set of elastic springs uniformly distributed over a circular cross-section centered at the contact point. The area of the smooth joint section is given by 2 (1) AR where R is the radius of the circular cross-section and the magnitude of it is given by Equation 2 given below: (2) R min( R A , R B ) A B where is a radii multiplier (equals to 1.0 by default) and R , R are the particle radii. In this study, the smooth joint was treated as a non-bonded unit to match the smooth joint used in the laboratory physical experiment.Therefore, the smooth joint properties include the normal stiffness k nj , shear stiffness k sj , coefficient of friction j and angle of dilation j . The effective joint geometry of a single smooth joint consists of two initially coincident planar surfaces. The two contacting particles are permanently associated with the two surfaces, one per side. During each time step, the relative translational displacement increment between the two surfaces is decomposed into components along the joint and perpendicular to the joint and these components are multiplied by the smooth joint shearstiffness and normal stiffness to produce increments of joint force. The force-displacement law operates in the coordinate system of the joint plane and provides Coulomb sliding behavior with no dilation in the present study. 3.2.3 Determination of joint deformation and strength parameters Once the smooth-joint model is created on these contacts lying on the opposite side of the joint plane, the following operations occur: a. The contact model and parallel bond are deleted and replaced by the smooth-joint contact with no parallel bond. b. The mechanical properties are inherited from the properties of the contact and the two contacting entities as follows: (3) k nj kn / A k n k sj ks / A k s j
j 0 c. The force, displacement and gap are set to zero.
(4) (5) (6)
The aforementioned inherited values could be over-written by the users according to the experimental setup or field conditions. The surface of joints in the physical experiments is relatively smooth and fresh(not weathered) (see Yang et al., 2015). Thus, the normal stiffness and friction coefficient of smooth joints should be set to values less than that of the intact material. Herein, at first, the normal stiffness and shear stiffness are set to 1.36×1010N/m³and 0.45×1010N/m³, respectively (Table 3). However, the appropriate values of the joint stiffnesses as well as friction coefficient should be determined through a calibration process using the strength of jointed rock blocks determined in the laboratory. Figure9 indicates the dependence of the jointed block strength values on the joint friction coefficient and deformation parameters. Table 3.Micro-mechanical parameter values used for the smooth joint model Mechanical Normal stiffness, k nj Shear stiffness, k sj parameter N/m³ N/m³ Inherited materi10.30×1010 6.06×1010 al value (mean) Assigned joint 1 1 1 k nj / k nj / k nj 1.36×1010/1.36×109/1.36×108 value 3 30 300
j
0.60
0
0.30/0.35/0.40/0.45
0
As can be seen from Figure9(a), the variation of the friction coefficient value has a dramatic influence on the strength of jointed rock block with dip angle of 30º. The strength increases sharply with the friction coefficient applied on smooth joint model at dip angle of 30º. Forjointed rock model with dip angle of 45º, the strength alsoincreases with the friction coefficient, butwith less significance compared to that of dip angle of 30º. For dip angle of 60º, there is very little difference observed on strength values as the friction coefficient is varied from 0.30 to 0.45. (a)
4.5 Laboratory experiment μ =0.30
Laboratory experiment
σ1 (MPa)
4.0
μj=0.30 j
μj=0.35
μj=0.35
3.5
μj=0.40 μj=0.45 μj=0.45 μj=0.40
3.0 2.5 2.0
25 (b)
30
35
40 45 50 55 Dip angle, α (°)
60
65
4.0 3.5 σ1 (MPa)
3.0 2.5 2.0
Laboratory experiment Laboratory experiment
1.5
knj=1.36e10 k nj 1.36 1010
1.0
knj=1.36e9 k nj 1.36 10 9
0.5
knj=1.36e8 k 1.36 10 8 nj
0.0 25
30
35
40 45 50 55 Dip angle, α (°)
60
65
(c)
5.0 Laboratory experiment Laboratory experiment
4.5
ksj=1/3knj k sj / k nj 1 / 3
σp (MPa)
4.0
ksj=1/30knj k sj / k nj 1 / 30
3.5
ksj=1/300knj k sj / k nj 1 / 300
3.0 2.5 2.0 1.5 25
30
35
40 45 50 55 Dip angle, α (°)
60
65
Figure 9.Strength values of jointed blocks from numerical simulations:(a)Variation of strength of numerically simulated jointed blocks with friction coefficient of smooth joint;(b) Variation of strength of numerically simulated jointed blocks with normal stiffness of smooth joint for k sj / k nj 1 / 3 ;(c)Variation of strength of numerically simulated jointed blocks with k sj / k nj of smooth joint for k nj 1.36e10 (stiffness unit: N/m3).
Figure9(b)displays the variation of strength of jointed blocks with the normal stiffness values. The normal stiffness of smooth joint was varied from 1.36×1010N/m³to 1.36×108N/m³.Figure9(b) shows that the normal stiffness mainly affects the strength value of jointed rock block with dip angle of 30º. For jointed rock blocks with dip angle of 45º and 60º, the effect of joint normal stiffness on strength is much less significant. Figure9(c)also demonstrates that the ratio of joint shear stiffness to normal stiffness( k sj / k nj )mainly induces a dramatic change on strength of jointed block values only when dip angle equals 30º; the strength values of jointed rock blocks increase with k sj / k nj for dip angle of 30º.For jointed blocks having α=45 º and 60º, the strength values show only a little increase with respect to the increase of k sj / k nj . Above all, the joint mechanical parameters have more influence on the strength of rock blocks with low joint dip angle. It might be because at low joint dip angle there is more normal stress applied on the joint plane to make the joint surfaces interact with each other very well. Therefore, according to the above calibration process, the smooth joint deformation and strength parameterscan be determined as k nj 1.36 1010 N/m? , k sj 0.45 1010 N/m? , j 0.40 and 0 . With the determined smooth joint deformation and strength parameters, the strength values of the numerically simulated synthetic jointed rock models have a good agreement with the results of the laboratory physical experiments. Table 4shows the comparison of the strength values obtained between the laboratory physical experiments and numerical simulations. Table 4.Comparison of strength between the laboratory experiment and numerical simulationresults Joint angle Laboratory experiment value Numerical simulation value Deviation º MPa MPa 30 3.69 3.70 0.3% 45 2.52 2.69 6.7% 60 2.28 2.47 7.3%
3.3 Validation of the calibrated PFC3D numerical model through comparison of the failure modes Figure10displays the comparison of failure modes between the laboratory physical experimental results and numerical simulations for rock blocks with dip angles of 30º, 45º and 60º.For α=30º, the jointed synthetic rockblockin the physical experiment mainly failed through a step-path failure involving joints 1, 3 and 6, which arelocated close to the free surface. The corresponding numerical model also shows a step-path failure mode through the coalescence of the three outside joints with the failed parallel bonds in the intact material segments.
α=30º
α=45º
α=60º 4
1
2
4
1
3
7
5
5
7
4
2
1
2
7 5
3 3
6
9
8
6
9
8
9
8 6
Figure. 10 Comparison of failure modes between the laboratory physical experimental results and numericalsimulations(The yellow and green imply the particles and joints, respectively;the red and blue imply tensile cracks and shear cracks, respectively.).
For the physical experimental rock block with α=45º, the main failure path involves joints 2, 5 and 6. Macro fractures can be seenin the synthetic rock bridge between the inner tip of joint 2 and the top wall as well as in the synthetic rock bridge between the outside tip of joint 2 and outside tip of joint 5. With another fracture forming in the synthetic rock bridge between joints 5 and 6 canlead to forming a complete failure path to separate the whole block into two parts. The numerical model with the dip angle of 45º shows a main failure path similar to that of the physical experimental specimen. Also note that the numerical model has produced fractures similar to the physical experimental specimen at the bottom left andright corners of the block. For α=60º, three failure planes can be seen on the physical experimental specimen: (1) along the direction of joint 1, (2) along the direction of joints 2 and 3, and (3) along the direction of joints 4, 5 and 6. The jointed rock block is separated into multiple beams along the dipping direction of joints under compressive loading. In the numerical model with the dip angle of 60º,the parallel bond breakages can be seen in the synthetic rock bridges between the adjacent joints located on the same plane. This failure phenomenon can easily produce inclined synthetic rock beams which slide along these failure planes.Thereforethe aforementioned comparison shows that the calibrated numerical modelhas the capability toreproduce thefailure modes that can be seen on the blocks used for the physical experiments.This implies that the calibrated numerical model has the capability of producing accurate values for jointed block strength and proper failure modes for the jointed blocks. 4 EFFECT OF JOINT ORIENTATION ON MECHANICAL BEHAVIOR OF JOINTED ROCK BLOCKS The aforementioned calibrated model is used in this section to investigate the effect of the joint dip angle on the failure mode, jointed block strength and deformation. The dip angle of joints are varied from 0º (horizontal)to 90º(vertical)with an increment of 15º. In this study, no support stress is applied on the free surface (in σ 3 direction). 4.1 Effect of joint orientation on failure behavior of jointed rock blocks Figure11shows the failure modes ofthe synthetic jointed rock blocks for different dip angles. Besides the blocks with joints (Blocks B-H), an intact synthetic rock block (Block A)was also generated and tested under the loading conditions illustrated in Section 3.2.1. All the cracks of the whole model at certain state along the σ2 direction are displayed in the visible surface. For
each case, four images display the bond breakage level at different stages of the stress-strain curve. The first imagewas captured when the micro cracks initiate from the block and the second image shows the bond breakage level at the peak stress level during compressive loading (i.e. 1 p ). The third and fourth images display the bond breakage levelwhen the stress drops to 0.85σpand 0.55σpin the post peak phase, respectively.As can be seen from Figure11, the joint orientation plays a major role on failure modes of rock blocks adjacent to underground excavations.Threedistinct failure modes are observed as follows: ModeⅠ:Intact rock failure-Block A(no joints), Block B(α=0º), Block C(α=15º) and Block H (α=90º). The rock materialin Block A is intact with no joints. Whenthe model is loaded at the top wall, bond breakages begin to appear in the top-left corner near the free surface. These bond breakages then combine into a macro failure plane that propagates downwards and inwards. The macro failure plane inclines at an angle of approximately 32º with respect to the free surface or the loading direction.As the loading stress reaches 0.85σp, the macro failure plane propagates to the bottom wall. Afterwards, bond breakages occur at the top-right corner of the model. At the same time, the width of the initial failure plane gradually increases and the propagation continues again from the bottom wall to the top-right corner. Crack initiation Block A No Joints
Block B α =0º
Block C α=15º
Block D α=30º
Block E α=45º
Peak-stress(σp) 32 °
Post-peak1(0.85σp)
Post-peak2(0.55σp)
Block F α=60º
Block G α=75º
Block H α=90º
Figure 11. Failure modes of synthetic jointed rock blocks under no support stress through numerical simulation (Note that the free surface is on the left hand side of each block;the yellow and green imply the particles and joints, respectively;the red and blue imply tensile cracks and shear cracks, respectively.).
When an underground excavation is made, the created free surface(Figure1) changes the insitu stress condition. This leads to rock mass deformation and dilation. The stress concentration first occursnear the free surface and when the stress exceeds the strength of the material, the rock mass fails. After that, the failure develops from the free surface (outside) to inside until new mechanical balance is formed. If the rock material is intact, the propagation of failure may be relatively stable. Block Ashows the failure process of the intact rock mass adjacent to an underground excavation in detail. Herein, this kind of failuremode is named as the intact rock failure mode. The intact rock failure modeis also observed in Blocks B, C and is shown inFigure11. InBlocks B and C with dip angle of 0ºand 15º, respectively, the bond breakages initiateat the top-left corner of the models. Contrary to Block A,although these bond breakages group into one macro failure plane at peak stress, the failure plane propagates through the joint tips. As the macro failure plane extends to the bottom of the model at σ1=0.85σp, the next tendency of bond failure is then towards the direction of the top-right corner by crossing joint tips one by one(see the third image in Block C). After that, at the top-right corner of the model, the bond breakagestake placeto accelerate the rock block to lose its stability. In the numerical model with vertical joints (Block H, α=90 º), the bond breakages start to develop in the rock segment between the free surface and the adjacent joint. Then, two inclined macro failure planes propagate to the top wall and the bottom wall, respectively. After that, both of the two failure planes propagate symmetrically, crossing the tips of joints, to the joint near the rear wall.Although the failure propagation involves the joints in Blocks B, C and H, the failure process is quite similar to Block A and the failure planes always cross the joint plane obliquely. Mode Ⅱ:Step-path failure-Block D (α=30 º). For Block D with dip angle of 30º, the failure mode is quite different from the above models with relatively lower dip angles or no joints. The bond breakages initiate from the tips of the middle set of joints. Then numerous bond breakages are produced from the tips of the joints which are located in the upper part of the model and propagate in the direction perpendicular to the joint sets. When the loading stress decreases to 85% of the peak stress, the bond breakages in the intact material together with the three joints adjacent to the free surface group into a macro failure plane which separates the top-left part of the model from the rest of the block. Once the top-left part of the model loses its capacity of resistance, the bond breakages begin to appear in the inner part of the rock block. Even though the failure process is complex, the sliding on a
joint segment and stepping between adjacent parallel joints in the top-left part of the model is dominant. Therefore, the step-path failure mode is applicable for Block D. Mode Ⅲ: planar failure-Block F (α=60 º)and Block G (α=75 º). When the dip angle increases to 60º (Block F), the bond breakages are produced initially from the tips of all the joints at the same time. Thesebond breakages propagate along the direction of each joint. These non-persistent joints which lie on each planeare then easily connected with each other by the bond breakages of the intact material. This leads to formingmultiple macro failure planes along the different planes of the joint set. These macro failure planes divide the jointed block into pieces ofsynthetic rock beams and some of the beams break into smallersynthetic rock blocks(see fourth image of Block F in Figure11). In Block G with dip angle of 75º, the bond breakages first occurat the outside tip of the joint closest to the free surface. Then, in the intact material bridges between the joints belonging to the second and third rows of joints some bond breakagesgrow and propagate parallel to the dipping direction of the joints. After that, some bond breakages continue to develop from outside to inside and at σ1=0.85σpthe first macro failure plane appears across the whole model from top to bottom. At σ1=0.55σp, the second macro failure plane emerges along the fourth row of joints. In the synthetic rock block with dip angle of 45º(Block E), it can be seen that both the steppath failure and planar failure occur in the model at the same time. Thus, it is a more complex failure mode. During the test, bond breakages nearly initiate from the tips of all the nine joints. These bond breakagescorresponding to wing cracks propagate perpendicular to the dip direction of the joint set until the loading stress reaches the peak stress. Afterwards, thesebond breakages coalescence with joints to form multiple macro failure planes in the model. 4.2 Effect of joint orientation on strength behavior of jointed rock blocks
Normalized block strength
The strength assessment of rock masses is of importance to provide guidelines for support design and stabilizing the underground excavations. This section presents a modeling method to estimate the strength of jointed rock blocks adjacent to underground excavations. Figure12displays the effect of joint orientation on synthetic rock block compressive strength σp. To assess the strength reduction in synthetic jointed rock blocks compared to that of the intact rock, the jointed block compressive strength σ p is normalized with respect to the corresponding value for Block A, i.e. the rock block with no joints. 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0
15
30 45 60 Dip angle, α ( °)
75
90
Figure 12. Variation of normalized jointed block strength with the dip anglefrom numerical simulations.
As can be seen from Figure12, the strength of jointed rock blocks has a very significant dependence on the joint orientation. The Block B with the joint dip angle of 0ºhas the highest compressive strength value of 97.3% of the intact block strength. That is because both Block A and Block B show the same failure mode, ModeⅠ:intact rock failure, under compression. Even though the bond breakages corresponding to wing cracks at joint tips affect the propagation direction of failure, these closed joints cannot dramatically reduce the resistance capacity of the jointed block. Besides Block B, the block models with dip angles of 15º and 90º also have relatively high strength values of 94.8% and 91.5% of the intact block strength, respectively.These jointed blocks also display a failure process similar to that of the intact material block.
It should be noted thatforα=90º, even though the strength value is high,it does not reach the value corresponding to that of the block having the horizontal joints(α=0º), just like for persistent joints. For thejointed modelhavingα=0º, the increase in normal stress on a joint is greater than the increase in shear stress; thus, the bond failures are unlikely to develop along the joints. The failure behavior and mode of bond failure propagation for the jointed block having horizontal joints are almost identical to the isotropic rocks subjected to compressive loads. This principle is suitable for persistent joint models as well as for non-persistent joint modelswith closed joints. For numerical models with vertical non-persistent joints(Figure13), the rock segment between the free surface and the joint closest to the free surface can easily detach from the rest of the jointed block and the stress concentration can occur in this zone. Numerous cracks initiate with breakages of parallel bonds in this rock segment until it loses the resistance; thus, a defect appears near the free surface in the jointed block. It is this defect that makes it possible for the strength value of thejointed block with vertical joints to be lower than that of jointed block with the horizontal joints. In practice, it is useful to take measures to eliminate the detachment of rock segments between the free surface and the vertical joints for stability control of underground excavations. Free surface
Rock segment
Figure 13.Synthetic rock block with vertical non-persistent joints.
Whenthe dip angle changes from 15ºto 30º, the compressive strength reducesdramatically from 0.95 to 0.72 times that of the intact material block. That is due to the fact that the failure mode changes from ModeⅠto ModeⅡ. As the block with joints havingα=30º is loaded, a steppath failure occurs throughthe coalescence of intact material bond breakages with the existing joints. This separates the top-left part of the jointed model from the rest of the block and makes it slide outwards to the free surface. The three joints adjacent to the free surface contribute to the localized failure and decrease the strength value. As the dip angle increases to 45º (Block E), the increase of shear stress on a joint is significant such that the bond failures corresponding to wing cracks initiate at tips of all nine joints(see the second image for Block E in Figure11)compared to that at tips of joints near the free surface like for Block D(α=30º). These fully distributed bond failurescan induce a strength decrease on the whole rock block rather than only in a localized area. This is the reason why Block E displays a lower strength value than that of Block D. The minimum σp occurs around α=60º, which is 0.48 times that of the intact material block strength. As can be seen from Figure11, Block F fails inplanar failuremode. The resistance in this case is mainly provided by the intact material bridges along the joint plane. According to Jennings' strength theory, the combined strength of the joint and intact material bridges can be computed from the simple linear weighing of the strength contributed by each fraction of material(Jennings, 1970).In this case, the joint segments which have no bond strength dramatically reduce the resistance of the whole model. Once the intact material bridges fail, the jointed block breaks into pieces of intact material beams and slide along the failure plane into the excavation space.This may induce severe disasters for both the workers and equipment. Therefore, particular measures should be taken for surrounding rock masses with joints inclined around α=60º. At α=75º, even though the jointed block fails in a planer failure mode, the failure mainly occurs in a localized area near the free surface when the compressive stress reaches the peak.Thus, the strength value for the jointed block with dip angle of 75º is much higher than that of the jointed block with dip angle of 60º. The aforementioned discussions clearly indicate thatthe strength behavior of jointed rock blocks adjacent to underground excavations show great dependence on the joint orientation.
Some joint dip angles induce failures in localized areas at the peak stress; while some other joint dip angles induce failures in the whole model at the peak stress. The models with the former joint dip angles show higher strength than the models with the latter joint dip angles. 4.3 Effect of joint orientation on deformation behavior of jointed rock blocks In engineering practice, the convergenceof surrounding rock mass is one of the most important indicators to estimate the stability of underground excavations. The measurement of convergence for a surrounding rock massis commonly recognized as an effective method to know the working conditions of the excavation. In this study, the deformation of the free surface in the numerical modelsisused as the convergence of surrounding rock mass.Thus, the effect of nonpersistent jointson convergence behavior of surrounding rock mass can be evaluated. To obtain the deformation magnitude of the free surface, a new wall which clings on to the free surface was generated before conducting the compression tests (see Figure 14). As the free surface deforms, this wall movesparallelto the free surface from beginning to end.It does not get separated from the free surface. The horizontal displacement of the vertical wall can represent the deformation of the free surface. As can be seen from Figure14, the particles on the free surface cannot deform uniformly such that the displacement of the measuring wall equals to themaximum deformation value of the free surface. Measuring wall
Free surface
Figure 14.The measuring wall and free surface.
The deformation behavior of the free surface is related to the displacement of the top loading wall.Therefore, to clearly present the effect of joint orientation on the deformability of the free surface, the linear strain of the free surface (ε 3) is normalized by the linear strain of the loading wall(ε1). ε3/ε1is plotted against ε1 in Figure15.The variation of ε3/ ε1under compression can be explained through three phrases as given below: PhaseⅠ: Whenthe front wall is deleted and replaced by a measuring wall, the in situ stress is released and the intact material will deform elastically. In this phase, the deformation of the free surface results from a combined effect due to lateral stress release at the free surface and due to the Poisson’s ratio effect from low σ1(low ε 1) and σ2 stresses. Thus, the ε3/ε 1 is high for all rock models with different dip angles, ranging from 4.5 to 6.5 at the beginning. Then this value drops dramatically because thiselastic deformation finishes quickly. 8.0 7.0
ε3/ε1
6.0 5.0 4.0
Phase Ⅰ
3.0 2.0 1.0
Phase Ⅲ
Phase Ⅱ
0.0
0
2
4
6
8
10
α=0º α=15º α=30º α=45º α=60º α=75º α=90º No Joint
12
ε1 (10-3)
Figure 15.Variation of ε3/ε1 with ε1for different jointed blocks having different joint dip angles.
PhaseⅡ: As ε1 increases, the bond breakages increase in a steady manner and thejointed block model deforms in a stable manner.The ε3/ε1 value turns out to be a constant around 1.0 in this phase.According to Figure15, the ε3/ ε1in this phase varies with the joint orientation to some extent. For the intact material block, the ε3/ε1is around 0.5, quite lower than 1.0. Due to the fact that the deformation in the σ2 direction(ε2) is low, which is ignored in this study, the volumeof the model diminishes (ε3+ε1
