a numerical study on 2-d compressive wave propagation in rock masses with a set of joints along the...

8/13/2019 A Numerical Study on 2-D Compressive Wave Propagation in Rock Masses With a Set of Joints Along the Radial Di

1/16

A numerical study on 2-D compressive wave propagationin rock masses with a set of joints along the radial direction

normal to the joints

W.D. Lei a, *, A.M. Hefny b, S. Yan c, J. Teng a

a Department of Urban and Civil Engineering, Harbin Institute of Technology Shenzhen Graduate School,HIT Campus Shenzhen University Town, Xili, Shenzhen, Guangdong, 518055, China

b School of Civil and Environmental Engineering, Nanyang Technological University, 639798 Singapore, Singaporec School of Civil Engineering, Shenyang Jianzhu University, Shenyang, 110168, China

Received 24 April 2006; received in revised form 13 January 2007; accepted 18 January 2007Available online 7 May 2007

Abstract

An explosion in a borehole or an accidental explosion in a tunnel will generate a two-dimensional (2-D) compressive wave that travelsthrough the surrounding rock mass. For the problem of 2-D compressive wave propagation in a rock mass with parallel joints in theradial direction normal to the joints, parametric studies on the transmission ratio and the maximum rebound ratio are performed inuniversal distinct element code. The variation of the transmission ratio with the ratio of joint spacing to wavelength is generalized intoa characteristic curve, which can be used as a prediction model for estimating the transmission ratio. The relationship between the max-imum rebound ratio and the inuence factors is obtained in charts. The charts can be used as a prediction model for estimating the max-imum rebound ratio. The proposed prediction models for estimating the transmission ratio and the maximum rebound ratio are appliedto a eld explosion test, Mandai test in Singapore. The minimum possible values of peak particle velocity (PPV) at the monitoring pointsare estimated by using the proposed prediction model for estimating the transmission ratio along the radial direction normal to the joints.On the other hand, the maximum possible values of PPV are estimated by using the proposed prediction model for estimating the max-imum rebound ratio along the same radial direction. The comparison shows a good agreement between the eld-recorded PPVs and theestimated range of PPVs given by the minimum possible PPVs and the maximum possible PPVs at the monitoring points in Mandai test.The good agreement between the estimated and eld-recorded values validates the proposed prediction models for estimating peak par-ticle velocity in a rock mass with a set of joints due to application of a compressive wave at the boundary of a tunnel or a borehole.

2007 Elsevier Ltd. All rights reserved.

Keywords: 2-D compressive wave; Transmission ratio; Maximum rebound ratio; Stiffness of joint; Field explosion test

1. Introduction

1-D wave (such as seismic wave) passing across single joint or parallel joints has been wildly studied for P-waveand S-wave, respectively. The investigation of effects of sin-gle joint in 1-D wave problem is a good basis for the prob-lem of wave propagation in jointed rock masses. The effects

of single joint on wave propagation have been wildly stud-ied with full consideration of different joint deformationalbehavior [16]. In literature [3], for the case of a pulse wavenormally hitting a joint, the transmitted waves were calcu-lated for a range of joint stiffness. In literature [5], Gu et al.conducted a study on wave reection, transmission andconversion of harmonic wave incidence upon a joint atarbitrary angles. For the case of parallel joints in 1-D wavepropagation in jointed rock masses, the effects of joint stiff-ness, joint spacing and the number of joints in the joint sethave been studied [711]. Because it is difficult to explicitly

0266-352X/$ - see front matter 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compgeo.2007.01.002

* Corresponding author. Tel.: +86 755 26033306; fax: +86 75526033349.

E-mail address: [email protected] (W.D. Lei).

www.elsevier.com/locate/compgeo

Available online at www.sciencedirect.com

Computers and Geotechnics 34 (2007) 508523
mailto:[email protected]:[email protected]


2/16

consider the multiple reections, a simplied method wasproposed by ignoring the multiple reections, where thetransmission coefficient for normally incident wave acrossparallel joints jT nj is the product of the transmission coef-cients of the individual joints. It has been found that thissimplied formula is applicable when the rst arriving

wave is not contaminated by the multiple reections (suchas for the case where the joint spacing is large enough com-paring with the wavelength) [79]. Also for the case of nor-mally incident 1-D wave passing through parallel joints,Cai & Zhao and Zhao et al. conducted studies on P-waveattenuation across parallel joints with linear deformationalbehavior by considering the interfracture reections. Bysolving a set of recurrence equations which were developedby using the method of characteristics, the magnitude of the transmission coefficient for parallel joints jT nj has beenobtained. It has been found that jT nj varies with the nor-malized normal stiffness of the joints, the ratio of the jointspacing over wavelength and the number of joints. It hasalso been found that the solutions from the studies byCai & Zhao and Zhao et al. are applicable for both casesof large joint spacing and small joint spacing.

For the above-mentioned studies on 1-D wave propaga-tion in jointed rock masses, there are two points to benoted. First, in those studies, P- and S-waves were studied,respectively. Second, the method of characteristics wasused as the key technique.

However, for a practical dynamic problem in a rockmass involving wave propagation from an undergroundexplosion source [1215], the problem could be treated asa 2-D problem, if the cavity or borehole for the dynamic

loads is long enough. (It should be noted that the problemis a 3-D problem. The 2-D treatment in this paper for thisproblem is a simplication. However, if the tunnel or bore-hole where the dynamic load originates is considerablylong, this simplication would be acceptable in the senseof engineering). In 2-D wave propagation, the combinedwave is of signicance in the sense of engineering. The sta-bility of the underground structure depends heavily onpeak particle velocity (PPV) or peak particle acceleration(PPA), in terms of the combined wave. Therefore, corre-sponding to the rst point needing to be noted in 1-D prob-lem, in this paper, the combined wave will be studied,rather than P- or S-wave, respectively. Corresponding tothe second point needing to be noted in 1-D problem, themethod of characteristics is very difficult to be applied in2-D wave problem in this paper, due to the complexity of the axis transition in the mathematical processing causedby the difference between the P-wave velocity and theS-wave velocity. Furthermore, the experimental study is dif-cult to conduct due to the potential difficulties, complex-ities and dangers. So, a numerical analysis method has beenthus proposed to address the problem of 2-D compressivewave propagation through parallel joints in rock masses[15]. As a part of that research, this paper aims to study2-D compressive wave propagation through a set of paral-

lel joints in rock masses.

When a 2-D wave propagates through a joint in a rockmass, part of the wave passes through the joint, and part of the wave reects back. In the process of 2-D compressivewave propagation through a set of parallel joints in theradial direction normal to the joints, the minimum possiblewave amplitude at the points behind the joints is controlled

by wave attenuation measured by the transmission ratio.On the other hand, the maximum possible wave amplitudeat the points between two adjacent joints is controlled bysuperposition of the multiple transmitted and the reectedwaves, which is measured by the maximum rebound ratio.In this paper, a prediction model for estimating the trans-mission ratio and a prediction model for estimating themaximum rebound ratio are developed in the radial direc-tion normal to the joints, in the process of 2-D wave prop-agation through a set of parallel joints in rock masses. Forthe purpose of verication of the proposed prediction mod-els, a reported explosion test in a jointed rock mass is intro-duced which was conducted in a quarry site in Mandai,Singapore [14]. For any individual monitoring point inthe verication example, both the maximum possiblePPV and the minimum possible PPV are estimated by usingthe proposed models. The eld record at any individualmonitoring point is compared with the predicted PPVrange, which is dened by the maximum possible PPV(i.e. the upper limit) and the minimum possible PPV (i.e.the lower limit).

2. Fundamental backgrounds

2.1. UDEC model

The geometry of the UDEC model is shown in Fig. 1,where a circular cavity with radius of 5 m is existing inan innite rock mass with parallel joints. The width is200 m and the height is 150 m. The center of cavity fordynamic input is positioned at ( 40,0). The rst joint inthe joint set is perpendicular to x-axis crossing through

1 0 m 15 m

Finite difference zone150 m

200 m

Y

X

Joint set with n joints

Symmetrical boundary

Viscous boundaries

Point A

Fig. 1. Sketch UDEC model.

W.D. Lei et al. / Computers and Geotechnics 34 (2007) 508523 509


3/16

( 25,0), with a normal distance of 15 m from the center of the cavity. Three of the four boundaries are viscous bound-aries (left, top and right) for dynamic problems to absorbthe outgoing waves, while the lower boundary is a symmet-rical boundary. For the boundary condition in dynamicproblems in rock engineering, it has been noted that there

are several articially truncated boundaries in Finite Ele-ment Method codes, such as the consistent boundaries(boundary element and nite element boundary [16]) andapproximately quiet boundaries (viscous boundary[17,18], paraxial axial approximation boundary [19], Hig-don boundary [20], transmitting boundary [21] and viscou-spring boundary [22]). It has been found that the consistentboundary has several advantages in rock dynamics. How-ever, as the most commonly used numerical code in analyz-ing engineering problems in jointed rock masses [23],UDEC only gives viscous boundary for rock dynamics,which is efficient for the computational domain of the cir-cular shape and approximately efficient for rectangularshape. For the boundary conditions in a numerical model,it is also noted that the dynamic innite element [24,25] isan efficient and effective representation for the far eld inan elastic wave propagation problem in an innite medium.The coupled method of nite element and dynamic inniteelement has been used to solve many wave propagationproblems in engineering eld [2629].

The properties of Bukit Timah granite in Singaporeshown in Table 1 are considered for the study. The bulkmodulus ( K ), shear modulus ( G ), and density ( q ) of theBukit Timah granite in Singapore were obtained fromextensive laboratory tests [30]. The wave propagation

velocities in the rock for P-wave ( cp) and S-wave (cs) arecalculated as: c p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 4G =3=qp and c s ffiffiffiffiffiffiffiffiffiG =qp [18]. 2.2. Denitions of the transmission ratio T n and radial angel a

In this paper, the wave amplitude at a grid-point fromthe case of jointed rock mass ( A joint ) is frequently com-pared with that from the case of intact rock ( Aintact ). Thetransmission ratio is dened as:

T n A joint A intact

1

The radial angle a at a point (such as point A in theUDEC model shown in Fig. 1) located behind the existing

joints is dened as the angle between the joint and the ra-dial line passing through the point in clockwise direction.

2.3. Review of the preliminary study on characteristics of T n

In literature [31], the following research strategies, which

are related to the research in this paper, have been pro-posed and justied:

(1) It is reasonable for the study on T n at the points alongthe radial angle of 90 by using one single pointbehind the joint and keeping the same shear stiffnessratio of jks/G and the normal stiffness ratio of jkn/K.

(2) Parametric studies on T n in 2-D wave propagationcan be conducted by using any typical rock materialand any reasonable radius of tunnel or borehole,from which the wave originates.

3. A study on T n

This section aims to study T n after parallel joints in theprocess of 2-D compressive wave propagation in the radialdirection of 90 , based on the preliminary study on charac-teristics of T n. A general prediction model for estimating T nis developed.

3.1. Modeling conditions and results

Parametric studies are conducted on the UDEC modelshown in Fig. 1. A sinusoidal velocity pulse wave (P-wave)

is input in the cavity with unity amplitude and a frequencyof 200 Hz. The mesh size of 0.79 m is chosen to keepnumerical error less than 2% [15]. The following combina-tions of normal stiffness of joint (jkn) and shear stiffness of joint (jks) are considered: (0.03K, 0.03G), (0.07K, 0.07G),(0.1K, 0.1G), (0.2K, 0.2G) and (0.4K, 0.4G) (Noted bythe authors: In this paper, jks and jkn are digitally set asvalue of K or G times a certain factor, where K , G arethe bulk and shear modulus of the rock material. The unitsof jkn and jks are stress per length, different from the unitsof K and G ). The values of joint spacing are generally from0.002 wavelength to 0.7 wavelength. The numbers of the joints are: 2, 4 and 8.

Fig. 2 shows the variations of T 2, T 4 and T 8 versus theratio of joint spacing over wavelength (spacing/wave-length) along the radial direction of 90 , for different givenvalues of jkn/( x z). The term jkn/( x z) is the normalizednormal stiffness of joint. x is the angular frequency of the wave, z equivalent to q c is wave impedance, q for den-sity of the rock and c for wave velocity. The parameter zdepends upon the type of waves. For compressive waveincidence, z equivalent to q cp is the seismic impedancefor P-wave, where cp is the P-wave velocity in the rock.Likewise, for shear wave incidence, z equivalent to q cs isthe seismic impedance for S-wave, in which cs is the S-wave

velocity in the rock.

Table 1Mechanical properties and wave propagation velocity of Bukit Timahgranite

Properties Value

Density (q ) (kg/m3) 2650Bulk Modules (K) (GPa) 39.5Shear Modules (G) (GPa) 26.0Velocity of compression wave propagation ( cp) (m/s) 5292

Velocity of shear wave propagation ( cs) (m/s) 3133

510 W.D. Lei et al. / Computers and Geotechnics 34 (2007) 508523


4/16

3.2. Development of prediction model for estimating T n

From Fig. 2, for every given quantity of joints and everygiven value of jkn/( x z), the curve of T n versus spacing/wavelength can be simplied as the characteristic curve inFig. 3. There are two critical points ( S peak , T peak ) and(S at , T at ), where S stands for spacing/wavelength and T stands for the transmission ratio. These two critical pointsare dependent on quantity of joints and jkn/( x z).

From, Fig. 2, it is found that T n changes with spacing/wavelength in three different modes, similar to the generalmodes in 1-D wave propagation, although for P-wave or S-

wave, respectively [10,11]:

(1) If spacing/wavelength > S at , T n is independent of the joint spacing.

(2) If S at > spacing/wavelength P S peak , T n increasesfrom T at up to a maximum value T peak as the jointspacing decreases.

(3) If spacing/wavelength < S peak , T n decreases from themaximum value as joint spacing decreases.

As a general prediction model for estimating T n in theprocess of 2-D compressive wave propagation through par-allel joints in the radial direction of 90 , it is necessary todetermine ( S peak , T peak ) and (S at , T at ) for any number of joints and any value of jkn/( x z).

3.2.1. Relationship between (S peak ,T peak ) and (S at ,T at )and jkn/( x z) for the given numbers of joints

The relationship between the two critical points ( S peak ,T peak ) and (S at , T at ) and jkn/( x z) for all the given num-bers of joints can be directly obtained from Fig. 2. There-fore, S peak , T peak , S at and T at as a function of jkn/( x z)are shown in Fig. 4 for different given numbers of joints:2, 4 and 8. The legends of the four charts in Fig. 4 arethe same, as shown in Fig. 4a.

3.2.2. Relationship between (S peak ,T peak ) and (S at ,T at )and jkn/( x z) for arbitrary numbers of joints

From Fig. 4, it can be seen that the two critical points(S peak , T peak ) and (S at , T at ) can be determined for any jkn/( x z) for the three given numbers of joints 2, 4 and 8.However, as a prediction model for estimating T n, it is nec-essary to determine the relationship between the two criti-cal points ( S peak , T peak ) and (S at , T at ) and jkn/( x z) forarbitrary numbers of joints. This relationship is implicitlyincorporated in Fig. 2. In order to apparently determinethis relationship, the parameters S peak , T peak , S at and T atmust be generalized as a function of the number of jointsfor different given values of jkn/( x z).

Fig. 5 shows S peak , T peak , S at and T at versus the number

of joints for different given values of jkn/( x z). The legends of

00.1

0.20.3

0.4

0.50.60.7

0.80.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7spacing/wavelength

T r a n s m

i s s i o n r a

t i o

T 2

jkn/( z)=0.067

jkn/( z)=0.157

jkn/( z)=0.224

jkn/( z)=0.448

jkn/( z)=0.897

0

0.1

0.2

0.30.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8spacing/wavelength

T r a n s m

i s s i o n r a

t i o T 4

jkn/( z)=0.157 jkn/( z)=0.067

jkn/( z)=0.224

jkn/( z)=0.448

jkn/( z)=0.897

0

0.1

0.2

0.3

0.40.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8spacing/wavelength

T r a n s m

i s s i o n r a

t i o T 8

jkn/( z)=0.067

jkn/( z)=0.224 jkn/( z)=0.157

jkn/( z)=0.448

jkn/( z)=0.897

Fig. 2. T n versus spacing/wavelength for different values of given jkn/( x z):(a) n = 2; (b) n = 4; (c) n = 8.

T

S

(Speak , Tpeak )

(Sflat, T flat)

Notes: S for spacing/wavelength

T for transmission ratio

Fig. 3. Characteristic curve for T n versus spacing/wavelength.



5/16

the fourcharts in Fig. 5 are the same, as shown in Fig.5a. Thereadings of S peak , T peak , S at , T at , for the three given num-bers of joints and the ve given values of jkn/( x z) are directly

taken from each curve shown in Fig. 2. It is noted that the

readings of S peak , T peak , S at and T at for the case of 6 jointsin the gure are obtained from additional numerical results,for the purpose of validating the approximately linear rela-

tionships when the number of joints is larger than 4.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

S p e a

k

2 joints4 joints8 joints

jkn/( z)

00.10.20.3

0.40.50.60.70.80.9

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

T p e a k

jkn/( z)

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

S f l a t

jkn/ ( z)

00.10.20.30.40.50.60.70.80.9

0 0.1 0.20 .3 0.40 .5 0.60 .7 0.80 .9 1

T f l a t

jkn/ ( z)

Fig. 4. (S peak , T peak ) and (S at , T at ) versus jkn/(x z) for the given numbers of joints of 2, 4 and 8: (a) for S peak ; (b) for T peak ; (c) for S at ; (d) for T at .

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 1 2 3 4 5 6 7 8Number of joints

S p e a k

jkn/( z)=0.067

jkn/( z)=0.157

jkn/( z)=0.224

jkn/( z)=0.448

jkn/( z)=0.897

00.10.20.30.4

0.50.60.70.80.9

0 1 2 3 4 5 6 7 8

Number of joints

T p e a k

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0 1 2 3 4 5 6 7 8

Number of joints

S f l a t

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1 2 3 4 5 6 7 8Number of joints

T f l a t

Fig. 5. S peak , T peak , S at and T at as a function of the number of joints for different jkn/( x z): (a) for S peak ; (b) for T peak ; (c) for S at ; (d) for T at .



6/16

3.2.3. Prediction model for estimating T nBased on the analysis shown in Fig. 5, S peak , T peak , S at

and T at versus jkn/( x z) for arbitrary number of joints canbe obtained by linear interpolation or extrapolation asshown in Fig. 6, where the legends of the four charts arethe same as shown in Fig. 6a. To clearly show the crowded

curves, the cases of the odd numbers of joints are notincluded in the gure. The coarse continuous lines aredirectly obtained from Fig. 2, the ne continuous linesfrom linear interpolation and the dash lines from linearextrapolation.

Therefore, ( S peak , T peak ) and (S at , T at ) in Fig. 3 can beobtained from Fig. 6 for any number of joints and any jkn/(x z) in the radial direction of 90 . These four charts can beused as the model for estimating T n after parallel joints inthe process of 2-D compressive wave propagation in theradial direction of 90 .

3.3. Discussion

In the UDEC, modeling in this paper, the cohesion of the joint is assumed to be of a high value in the parametricstudies, where the prediction model for estimating T n (pro-posed in this section) and the subsequent prediction modelfor estimating the maximum rebound ratio (to be proposedin Section 3.2.3) are developed. In this scenario, a linearelastic behavior without opening and frictional failure isassigned to the joints with high cohesion value, and thestress-dependency of the joint stiffness is not considereddue to this assumption for simplicity. Therefore, the pro-posed prediction models are applicable for the wave eld

where the wave amplitude is relatively small (such as the

ground motion in the far-eld for a normal explosion orthe whole wave eld for a small-scale blasting).

The joints in this paper are assumed to be planar, dryand large in extent. The rock material between any twoadjacent joints is ideally elastic rock, therefore, no dampingis taken into account in the UDEC modeling.

4. A study on the maximum rebound ratio

In the process of 2-D compressive wave propagationthrough parallel joints, superposition of multiple reectedand transmitted waves occurs between two adjacent joints.This section aims to study the wave amplitude ratio aftersuperposition of multiple reections and transmissions inthe process of 2-D compressive wave propagation throughparallel joints along the radial direction of 90 . A predic-tion model for estimating the maximum rebound ratio isobtained based on UDEC results.

4.1. Denitions of the rebound ratio and the maximumrebound ratio

In the process of 2-D compressive wave propagationthrough parallel joints along the radial direction of 90 ,the superposition of the multiple reected and transmittedwaves between two adjacent joints is dened as reboundwave. The rebound ratio is dened as:

Reb n A reb;n

A intact2

where Areb, n is the amplitude of the rebound wave at the

point between the ( n 1)th and nth joints. Aintact is the

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

S p e a k

2joints 4 joints

6 joints 8 joints

10 joints 12 joints

20 joints

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

T p e a k

0

0.1

0.2

0.3

0.4

0.5

0.6

S f l a t

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

T f l a t

Fig. 6. S peak , T peak , S at and T at versus jkn/(x z) for arbitrary number of joints: (a) for S peak ; (b) for T peak ; (c) for S at ; (d) for T at .



7/16

wave amplitude at the corresponding point in a model of intact rock. The maximum rebound ratio (Max. Reb n) isthe maximum value of Reb n at all grid-points betweenthe (n 1)th and nth joints.

In order to compare the denitions of the transmissionratio dened in Eq. (1) and the rebound ratio dened in

Eq. (2), two cases of 2-D wave propagation in a jointedrock mass are introduced, as shown in Fig. 7. In case A,a 2-D wave propagates in an innite rock mass with a jointset. In case B, the same wave propagates only through(n 1) joints, while the other conditions are the same asin case A. The left plot in Fig. 7 shows the typical compar-ison between Reb n at the points between the ( n 1)th andnth joints obtained from case A and T n 1 at the samepoints obtained from case B , where the joint spacing isquite big. (It should be noted the transmission ratio is inde-pendent of the radial distance from the cavity center [31],therefore, T n 1 is a at line. It should also be noted thatthere is a critical distance from the nth joint on the varia-tion curve of Reb n against the radial distance from the cav-ity center. However, so far, the critical distance could notyet be dened exactly). When the point is before the nth joint within the range of a critical distance, T n 1 (lineb d ) is always smaller than Reb n (curve b c), because inthis range, the wave reection has signicant effects. Bycontrast, when the point before the nth joint is beyondthe range of the critical distance, Reb n is approximatelyequal to T n 1 (line a b). In this range, the wave reection

has negligible effects. Therefore, Max. Reb n is always big-ger than T n 1. It has been found that for the case of small joint spacing, Reb n is always bigger than T n 1 at all thegrid-points between the two joints.

This means that Max. Reb n is the upper limit of the ratioof the wave amplitude from the model with joints to the

wave amplitude from the model of intact rock, whileT n 1 is the lower limit of the mentioned ratio.

4.2. Problem statement and modeling conditions

The investigation of Max. Reb n is based on the UDECmodel shown in Fig. 1, but the geometry is enlarged to adimension of 150 m 250 m, where the length of 200 m isincreased to 250 m, while the height is unchanged. A cavitywith a radius of 5 m exists in an innite rock mass with 14 joints, which are parallel to the y-axis with a normal dis-tance of 15 m from the center of the cavity to the rst jointin the joint set. In the geometry, three boundaries (left, topand right) are viscous boundaries, while the lower bound-ary is a symmetrical boundary.

A sinusoidal velocity pulse wave (P-wave) with unityamplitude and a frequency of 200 Hz is applied at theboundary of the cavity. The mesh size of 0.79 m is chosento keep the numerical error less than 2% for the compres-sive wave of 200 Hz [15]. Properties of Bukit Timah granitein Singapore shown in Table 1 are considered for the study.The stiffness of the joints (jkn and jks) considered in this

14 16 18 20 22 24 26 280.0

0.1

0.2

0.30.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

d

c

ba

I n f l u e n c e z o n e

f o r r e

f l e c

t i o n

N o n - i n

f l u e n c e z o

n e

f o r r e

f l e c

t i o n

c r i t i c a l d i s t a n c e

Tn-1

Max. Reb n

P o s

i t i o n o

f t h e n

t h j

o i n t

P o s

i t i o n o

f t h e

( n - 1

) t h

j o

i n t

T n - 1

o r

R e

b n

Radial distance from the cavity center (m)

R eb n Tn-1

Symmetrical line

The (n-1) th joint The n th joint

Case A

Symmetrical line

The (n-1) th joint

Case B

Fig. 7. Comparison between T n 1 and Reb n.



8/16

part are the following combinations (0.03K, 0.03G),(0.07K, 0.07G), (0.1K, 0.1G), (0.2K, 0.2G) and (0.4K,0.4G). The values of spacing/wavelength are 0.04, 0.07,0.1, 0.13, 0.17, 0.20, 0.25, 0.30, 0.40 and 0.50.

The velocity wave at all the grid-points within the rstand the last joint in the joint set, along the radial direction

of 90 , are recorded and processed for the study on Max.Reb n.

4.3. Development of prediction model for estimating Max.Reb n and discussion

Fig. 8 shows Max. Reb n before each joint in the radialdirection of 90 for the given jkn/( x z) and spacing/wave-length. The legends of the ve charts are the same, as

shown in Fig. 8a. It can be seen that the effects of all thefactors inuencing Max. Reb n, jkn/( x z) and joint spacing,are comprehensively generalized into Fig. 8 for every jointfrom which the wave rebounds. The charts can be used asthe prediction model for estimating Max. Reb n for any joint in the joint set in the radial direction of 90 .

It can be seen that for the rst joint ( n = 1), Max. Reb 1is determined by jkn/( x z). For any two adjacent joints inthe joint set, Max. Reb n is determined by jkn/( x z), spac-ing/wavelength and the joint from which the wave isrebounded.

From Fig. 8, there are the following observations:

(1) In the rst joint, Max. Reb 1 is controlled by jkn/( x z).Max. Reb 1 decreases as jkn/( x z) increases.

0

0.2

0.4

0.6

0.8

1

1.2

1.41.6

1.8

2

1 2 3 4 5 6 7 8 9 10 11 12 13 14Joint number

M a x . R

e b n

(spacing/wavelength)=0.04

(spacing/wavelength)=0.07(spacing/wavelength)=0.1(spacing/wavelength)=0.13


(spacing/wavelength)=0.20(spacing/wavelength)=0.25

(spacing/wavelength)=0.30(spacing/wavelength)=0.40


00.20.40.60.8

11.21.41.61.8

2

1 2 3 4 5 6 7 8 9 10 11 12 13 14Joint number

M a x . R

e b n

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Joint number

M a x . R

e b n

0

0.2

0.4

0.6

0.8

1

1.2

1.41.6

1.8

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Joint number

M a x . R

e b n

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 2 3 4 5 6 7 8 9 10 11 12 13 14Joint number

M a x . R

e b n

Fig. 8. Max. Reb n before different joints for different values of spacing/wavelength and for different jkn/( x z): (a) for jkn/(x z) = 0.067; (b) for jkn/

(x z) = 0.157; (c) for jkn/( x z) = 0.224; (d) for jkn/( x z) = 0.448; (e) for jkn/( x z) = 0.987.



9/16

(2) For the other joints (other than the rst one), there arethree observations. First, Max. Reb n along the radialdirection of 90 is inuenced by jkn/( x z), spacing of the joints and the joint from which the wave rebounds.Second, when the spacing of joints and the joint fromwhich thewave rebounds are xed, Max. Reb n increases

as jkn/( x z) increases. Third, for each given jkn/( x z)for any specic joint from which the wave rebounds,Max. Reb n decreases as joint spacing increases.

It has been noted that the shape of the transmitted waveor rebound wave form in the UDEC results is often dis-

torted by some unknown factors. This might be causedby the wave reection and conversion. Therefore, in thepaper, although these wave phenomena are not explicitlyconsidered, their effects are comprehensively incorporatedin the amplitude of the transmitted wave and the reboundwave.

5. Verication of the proposed prediction models

In this section, a reported explosion test in jointed rockmass is introduced, which was conducted in a quarry site inMandai, Singapore [14]. There were ve monitoring bore-holes (BHs 1, 2, 3, 4 and 5) located along a line in the rockmass. PPVs at the monitoring points were recorded in allthe ve monitoring boreholes. The PPVs at the locationsof the monitoring points in the eld are estimated by themethods proposed in this paper. The estimated PPVs arecompared with the eld records and the solutions fromtwo empirical formula. The comparison serves as the veri-cation of the proposed prediction models for estimatingT n and Max. Reb n.

5.1. The Mandai explosion test

The site of Mandai test was in the rock mass containingBukit Timah granite in Singapore, as shown in Fig. 9 [14].A vertical charge hole of about 11 m in depth was drilled.As shown in Fig. 10, the charge chamber was 4.2 m inheight and 0.8 m in diameter. The explosive charge wasplaced at the center of the charge chamber. Before theexplosion, the charge chamber was covered by a steel plate

and concrete blocks. The explosive charge was 41.5 kg

BH 1, 2.5 m BH 2, 5.0 m BH 3, 10 m BH 4, 25 m BH 5, 50 m

2.1 m

2.1 m

6 m

Radial distance from the center line of charge chamber

Ground surface

Legend

Uniaxial stress (radial)

Triaxial acceleration (vertical, radial and transverse)

Biaxial accele ration (vertical and radial)

Uniaxial acceleration (radial)

Explosive

Fig. 10. Instruments in boreholes for monitoring the eld explosion (after Chen et al. 2000 [14]).

230

Rock Mass

BH 1 (2.5 m)BH 2 (5 m)BH 3 (15 m)

BH 4 (25 m)

BH 5 (50 m)

050

Charge hole

N

Fig. 9. Conguration of test (after Chen et al. 2000 [14]).



10/16

TNT in charge weight, equivalent to 20 kg/m 3 in loadingdensity (the weight is divided by the chamber volume). Fivevertical instrumentation boreholes of 0.2 m in diameterwere drilled along the bearing of 230 . Three of the veboreholes were 14 m deep and the other two were 8.5 mdeep. The locations of the boreholes (BHs 1, 2, 3, 4 and

5) were 2.5 m, 5 m, 10 m, 25 m and 50 m from the centerof the charge hole. Three types of instruments, includingtriaxial accelerometers, biaxial accelerometers and uniaxialaccelerometers, were installed in the boreholes as shown inFig. 10. Table 2 shows the eld recorded PPVs, for the vemonitoring points.

The Mandai test is a factually 3-D problem, it is roughlytreated as a simplied 2-D problem in this paper, as it wastreated in literature [14]. Similar simplied treatments canbe seen in literatures [12,13].

5.2. Explosion load

In order to estimate the PPV at any point in the rockmass due to the explosion by using the proposed method,the properties of the compressive wave, generated fromthe explosion, at the borehole boundary should be known.The nite difference program Autodyn is used to model theexplosion test and output the particle velocity wave at theboundary of the charge hole.

Chen et al. [14] modeled the same explosion test inAutodyn, using rock deformation parameters obtainedfrom preliminary test results. After Chens analysis, exten-sive laboratory test has been done on the properties of therock material from the explosion test site. It has been found

that the Youngs modulus and Poissons ratio are 64 GPaand 0.23, respectively [30], instead of 72 GPa and 0.16 usedin literature [14]. These new parameters are used in Auto-dyn modeling in this paper.

5.2.1. Estimating properties of the rock mass involved inAutodyn modeling 5.2.1.1. Properties of the in situ rock mass. The rock mass atthe test site (Bukit Timah granite) contains two dominantsub-vertical joint sets, with an average joint spacing of 2.5 m. The uniaxial compression strength of the intact rockmaterial is 180 MPa, with a density of 2650 kg/m 3. Therock quality designation (RQD) at the site is about 90%.The rock joint surface is mostly fresh, some stained, closed,rough and undulating. The ratio of the average horizontal

principle stress to the vertical principle stress is 34. Therock mass is mostly dry.

Table 3 shows the properties of the intact rock, deter-mined from extensive laboratory testing program. FromTable 3 , the shear modulus of the intact rock ( G r) can becalculated as:

G r E r

21 l

6421 0:23

26 GPa 3

where E r is the Youngs modulus of the intact rock, and lis Poissons ratio.

5.2.1.2. Properties of the equivalent continuous rock. Auto-dyn is a continuum-based program that requires the jointedrock mass to be converted into its equivalent continuousrock. Firstly, the RMR value of the in situ rock mass is cal-culated based on rock mass rating system (RMR) devel-oped by Bieniawski [32]. And then, Youngs modulus(E e) of the in situ equivalent continuous rock is estimatedby the relationship between E e and RMR value proposedby Seram and Pereira [33]. Finally, the shear modulus

(G e) of the in situ equivalent continuous rock is calculatedaccordingly.According to the conditions of the in situ rock mass, the

ratings of (1) strength of the intact rock, (2) RQD, (3) spac-ing of joints and (4) condition of joints, are determined as12, 17, 25 and 20, respectively. The RMR value is the sumof the four ratings: RMR = 12 + 17 + 25 + 20 = 74.

According to the relationship between E e and RMRvalue proposed by Seram and Pereira [33], E e for thein situ equivalent continuous rock can be calculated as:

E e 10RMR-10

40 40 GPa 4

Assuming the equivalent continuous rock has the samePoissons ratio l of 0.23 as the intact rock, G e can be cal-culated as:

G e E e

21 l

4021 0:23

16:3 GPa 5

The equivalent continuous rock is assumed to obey Mo-horCoulomb strength criteria. The properties of theequivalent continuous rock involved in Autodyn modelingare shown in Table 4 .

In the process of converting the jointed rock mass intoits equivalent continuous rock mass, the jointed rock massat the test site, containing two perpendicular sets of joints,

would be more properly represented by a transverse elas-

Table 2Mandai test records

Radial distance from thecenter of the charge hole (m)

Field records(PPVs) (m/s)

2.5 0.615.0 0.468

10 0.19225 a

50 0.021

a Datum not available.

Table 3Properties of the intact rock material

Property Value

Density ( q ) (kg/m3) 2650Youngs Modulus ( E r) (GPa) 64Poissons ratio 0.23Cohesion (MPa) 24Friction angle ( ) 54Tensile strength (MPa) 16



11/16

ticity with elastic constants for two principle directions.However, the code Autodyn can only consider isotropicrock mass. In order to use Autodyn to determine the out-put of the explosion, a compromise is made to treat thetransverse rock mass as an isotropic continuous rockmass. An independent numerical study performed inUDEC shows that an acceptably higher result (usuallyabout 10% higher, depending on the relative position of

the recorded point to the adjacent joints) is generateddue to the compromise.

5.2.1.3. Properties of the explosive and the concrete blocksinvolved in Autodyn modeling. The properties of the explo-sive (TNT) are obtained from the library in Autodyn asshown in Table 5 , where TNT is assumed to obey theJonesWilkinsLee state equation. The concrete blockscovering the charge chamber is treated as a linear continu-ous medium and assumed to obey the porous equation of state and MohorCoulomb strength criteria. The proper-ties are obtained from the material library in Autodyn asshown in Table 6 . The air in the charge chamber is treatedas ideal gas.

5.2.2. Explosion load modeled in AutodynAs shown in Fig. 11, the Autodyn model is a 2-D sym-

metrical model with only half of the domain involved inthe model. The ground surface at the top of the modelis treated as free boundary. The right and the bottomboundaries are viscous boundaries. The left boundary isa symmetrical boundary. The measuring point is locatedin the surrounding rock close to the boundary of the

charge chamber at the charge level to record the proper-ties of the particle velocity wave. The Autodyn resultsshow that the amplitude of the compressive velocity pulseis 3.07 m/s, with a frequency of 700 Hz.

5.3. Comparison results of the verication example

The proposed prediction models for estimating T n in2-D compressive wave propagation along the radial direc-tion of 90 is applied to the Mandai test to estimate theminimum possible PPVs at the monitoring points. The pro-posed prediction model for estimating Max. Reb n isapplied as well to estimate the maximum possible PPVs.The estimated PPV range dened by the minimum andmaximum possible PPVs is compared with the eld recordsand two empirical formula.

x

Z

Viscous boundary

0.75 m

Containment

41.5 Kg TNT

Measurement point

0.4 m

4.2m

8.1 m

50m

60 m

Symmetrical boundary

Rock Mass

Ground surface

Fig. 11. Sketch of the computational model in Autodyn.

Table 4Properties of equivalent continuous rock mass

Property Value

Density (q ) (kg/m3) 2650Youngs modulus ( E e) (GPa) 40Poissons Ratio 0.23Cohesion (MPa) 24Friction angle ( ) 54Tensile strength (MPa) 16

Table 5Properties of TNT explosive

Property Value

Reference density (kg/m 3) 1630Parameter A (GPa) 373.77Parameter B (GPa) 3.7471Parameter R 1 4.15Parameter R 2 0.9Parameter W 0.35C J detonation velocity (m/s) 6930C J pressure (GPa) 21Burn on compression fraction 0Pre-burn bulk modulus (GPa) 0

Adiabatic constant (GPa) 0

Table 6Properties of concrete material

Property Value

Reference density (kg/m 3) 2454Sound velocity (m/s) 2693Density #1 (kg/m 3) 2368Density #2 (kg/m 3) 2376Density #3 (kg/m 3) 2412Density #4 (kg/m 3) 2447Pressure # 1 (MPa) 0Pressure # 2 (MPa) 14.8Pressure # 3 (MPa) 59Pressure # 4 (MPa) 111.1Shear modulus (GPa) 4.5Yield stress #1 (MPa) 23.6Yield stress #2 (MPa) 73.8Yield stress #3 (MPa) 270Yield stress #4 (MPa) 270



12/16

5.3.1. Relative position between the monitoring points and the joint set

In Mandai test, the joints in the granite were mapped inthe quarries and inferred from boreholes using impressionpacker data. There are two predominant sub-vertical jointsets, with strike directions of 140 and 230 , respectively,

perpendicularly intersecting each other. The strike of themost dominant joint set is 140 [34], perpendicular to theline along which the monitoring boreholes are located.

In the test, when the 2-D compressive wave from thecharge hole travels in the in situ rock mass, it is reasonableto assume that the peak particle velocities at the pointsalong the monitoring line with the bearing of 230 is dom-inated by the joint set with strike of 140 , which is perpen-dicular to this line. The other joint set (with strike of 230 ),parallel to the monitoring line, is assumed to have negligi-ble effects on the PPVs at the monitoring points (It shouldbe noted that an independent study using UDEC con-rmed this assumption). Therefore, the relative positionof the monitoring points and the joint set with strike of 140 can be shown in Fig. 12.

5.3.2. The factors involved in estimating T nIn order to estimate T n at the monitoring points in the

eld, the values of the factors involved in estimating T nmust be determined. The factors include spacing/wave-length, the properties of the input compressive wave, theradial angle, the number of the joints to be crossed bythe wave before reaching the monitoring point and the stiff-ness of the joints.

From the site investigation, the spacing of the joints is

determined to be 2.5 m. The properties of the input com-pressive wave have been determined by Autodyn modeling.The amplitude of the wave is determined as 3.07 m/s, and

the frequency is 700 Hz. The radial angle for all the moni-toring points is 90 , since they lie along the radial directionperpendicular to the joint set ( Fig. 12). The number of the joints to be crossed for every monitoring point can bedetermined based on the spacing of the joints and the rela-tive position of the monitoring points to the joint set. The

stiffness of the joints can be determined by the formula pro-posed by Hart [35].For a rock mass containing uniformly spaced joints,

Hart proposed the following relations to estimate the nor-mal and shear stiffness of the joints (in the formula, s is thespacing of the joints):

jkn E e E r

s E r E e 6

jks G eG r

sG r G e 7

According the properties of the intact rock, shown in Table3 for E r and in Eqs. (3)(5) for G r, E e and G e, respectively, jkn and jks are calculated as: jkn E e E r s E r E e

40 642:564 40 42:7

GPa =m and jks G eG r sG r G e 26 16:32:526 16:3 17:5 GPa =m.

For the input compressive velocity wave with a frequencyof 700 Hz, the normalized normal and shear stiffness of the joint are: jkn/( x z) = 0.695 and jks/( x z) = 0.475.

For estimation purpose, a sketch of the Mandai testshowing the relative position between the eld monitoringpoints and the joints in the rock mass is provided inFig. 13. Based on the relative position shown in Fig. 13and the determined radial angle of 90 at the monitoringpoints, the numbers of joints that the compressive wavecrosses before reaching the monitoring points are tabulatedin Table 7 .

The parameters involved in estimating T n for the moni-toring points are summarized as:

(1) The properties of the input velocity wave: frequencyis 700 Hz, amplitude is 3.07 m/s.

(2) jkn is 42.7 GPa/m, jks is 17.5 GPa/m, hence the cor-responding values of normalized stiffness of jointsare: jkn/( x z) = 0.695, jks/( x z) = 0.475.

(3) The spacing of the joints is 2.5 m, hence spacing/wavelength is 0.33.Rock Mass

N

230

050

Charge holeBH 1 (2.5m)

BH 2 (5.0 m)BH 3 (10.0m)

BH 4 (25.0m)

BH 5 (50.0 m)

90

140

320

Fig. 12. Relative position of the monitoring points to the joint set.

1st, 2 nd, 3 rd and 4 th joint

20 th joint

9th joint

Charge holeMonitoring point

Fig. 13. Sketch of the Mandai test for the purpose of estimating T n.



13/16

5.3.3. Comparison between the eld records and theestimated PPVs5.3.3.1. The minimum possible PPVs from the proposed prediction models for estimating T n . The rst monitoringpoint in the eld is located after the rst joint close tothe boundary of the charge camber. For the case of single joint, in literature [15], the value of T 1 can be calculated as0.78 for jkn/( x z) of 0.695. The estimated PPV at any mon-

itoring point is equal to the estimated T n times the ampli-tude of the wave at the same point from the case of intact rock. The amplitude of the wave at any point forthe case of intact rock can be determined using Fig. 15 inAppendix I . (It is noted the amplitude from Fig. 15 is forthe input wave with unity amplitude. Therefore, in thispaper, the amplitude value from Fig. 15 is multiplied by3.07, i.e. the amplitude of the input wave in Mandai test).At the location of the rst monitoring point, the amplitudeof the wave for the case of intact rock is obtained fromFig. 15 as 0.768 m/s. This leads to a value of estimatedPPV of 0.599 m/s at the rst monitoring point.

To calculate T n for the case of parallel joints for othermonitoring points (rather than the rst monitoring point),the two critical points ( S peak , T peak ) and (S at , T at ) asdened in Fig. 3 for a specic jkn/( x z) have to be deter-mined. The values of S peak , T peak , S at and T at can bedetermined using Fig. 6. After determination of S peak ,T peak , S at and T at , the values of T n is calculated asfollows:

(1) If S at 6 the target value of spacing/wavelength, theestimated T n is obtained as:T n T flat 8

(2) If S at > the target value of spacing/wavelength(>S peak ), the estimated T n is obtained as:

T n T peak The target value of spacing wave length S peak

S flat S peak T peak T flat

Using the target values for the Mandai test (jkn/( x z) of 0.695, the value of spacing/wavelength of 0.33), in Fig. 6,T n and PPVs at the ve monitoring points can be obtainedas shown in Table 8 .

5.3.3.2. The maximum possible PPVs from the proposed prediction model for estimating Max. Reb n . The estimatedT n is the ratio after the wave travels through n joints,assuming there are no more joints afterwards. However,in practical cases, the point of interest is most probablylocated at any point between two adjacent joints in the joint set. The PPV at a point between the nth and(n + 1)th joints is the PPV of the wave after superpositionof the transmitted wave and the reected wave. Dependingon spacing/wavelength and the relative location of thepoint between the two adjacent joints, the PPV at the pointwould have a minimum possible value and a maximumpossible value. The minimum possible PPV is estimatedfrom T n, while the maximum possible PPV is estimatedfrom Max. Reb n+1 between the nth and the ( n + 1)th joints.In this part, the maximum possible PPVs at the monitoringpoints will be estimated from the proposed predictionmodel for estimating Max. Reb n.

From Fig. 13, it can be seen that the 1st, 2nd, 3rd, 4thand 5th monitoring point in Mandai test is located before

the 2nd, 3rd, 5th, 10th and 21st joint, respectively, fromthe boundary of the borehole. In Mandai test, the joint

Table 7Numbers of joints to be penetrated to reach at the monitoring points forthe Mandai test

BH No. No. of joints to be crossed

BH 1 1BH 2 2BH 3 4BH 4 9BH 5 20

Table 8The minimum possible PPVs (including the parameters involved in the calculation) and eld records

Monitoring point (Row 1) BH 1 BH 2 BH 3 BH 4 BH 5

No. of joints to be crossed n (Row 2) 1 2 5 9 20S at (Row 3) 0.215 0.32 0.39 0.51T at (Row 4) 0.63 0.43 0.25 0.04S peak (Row 5) 0.061 0.053 0.043 0.021T peak (Row 6) 0.76 0.7 0.665 0.59Amplitude from the model without joint (m/s) (From Fig. 15) (Row 7) 0.768 0.503 0.322 0.166 0.091Estimated T n (Row 8) 0.78 0.63 (Eq. (8)) 0.43 (Eq. (8)) 0 .322 (Eq. (9)) 0.214 (Eq. (9))Minimum possible PPVs (m/s) (Row 7 by Row 8) 0.599 0.317 0.139 0.053 0.019Field records (PPV) (m/s) 0.61 0.468 0.192 a 0.021

a

Datum not available.

Table 9Max. Reb 10 for spacing/wavelength = 0.3, 0.4 and (jkn/ x z) = 0.897, 0.448

Max. Reb 10 for different values of spacing/wavelength and jkn/( x z)

Value

Max. Reb 10 for spacing/wavelength of 0.3 and jkn/( x z) = 0.897 0.518Max. Reb 10 for spacing/wavelength of 0.4 and jkn/( x z) = 0.897 0.45Max. Reb 10 for spacing/wavelength of 0.3 and jkn/( x z) = 0.448 0.4Max. Reb 10 for spacing/wavelength of 0.4 and jkn/( x z) = 0.448 0.315

http://-/?-http://-/?-


14/16

spacing is 0.33 wavelength and jkn/( x z) is equivalent to0.695. In this part, the prediction of several maximumrebound ratios, Max. Reb 2, Max. Reb 3, Max. Reb 5,Max. Reb 10 and Max. Reb 21 , are estimated based on thecurves as shown in Fig. 8. Among these maximum rebound

ratios, the procedure of predicting Max. Reb 10 is intro-duced in detail.

First, according to the target value of jkn/( x z) of 0.695,Fig. 8d and e are chosen for linear interpolation, becauseFig. 8d and e correspond to jkn/( x z) of 0.448 and 0.897with a range covering the target value of jkn/( x z) of 0.695.

Table 9 shows the data drawn from Fig. 8d and e,involved in estimating Max. Reb 10 for spacing/wavelengthof 0.33 and jkn/( x z) of 0.695 in the radial direction of 90 .

In Table 9 , xing the number of joints and jkn/( x z),treading spacing/wavelength as a variable, for the targetvalue of spacing/wavelength equivalent to 0.33, by linearinterpolation, Max. Reb 10 can be calculated for jkn/( x z)of 0.897 and jkn/( x z) of 0.448, respectively as: 0.498and 0.375. Hence Max. Reb 10 for spacing/wavelength

Table 10Several maximum rebound ratios

Max. Reb n Value

Max. Reb 2 1.08Max. Reb 3 0.8Max. Reb 5 0.63Max. Reb 10 0.44Max. Reb 21 0.35a

a Reb 21 is obtained from extension of the curves.

0.001

0.01

0.1

1

10

100

0.1 1 10 100Scaled range

P e a k p a r t

i c l e v e

l o c i

t y ( m / s )

Field records

The minimum possiblePPVsThe maximum possiblePPVsAmbraseys & Hendron(1968)Wu (1975)

Fig. 14. Comparison among the minimum possible PPVs and maximum possible PPVs from the model along the radial direction with a of 90 , eld

records and PPVs from two empirical formula (unit of scaled range is m/kg 1/3 ).

Table 11The minimum possible PPVs, maximum possible PPVs and the eld records

Monitoring point (Row 1) BH 1 BH 2 BH 3 BH 4 BH 5

Amount of the joints to be crossed n (Row 2) 1 2 4 9 20Max. Reb n+1 (From Table 10) (Row 3) 1.08 0.8 0.63 0.44 0.35Amplitude from the model of intact rock (m/s) (From Fig. 15) (Row 4) 0.768 0.503 0.322 0.166 0.091Maximum possible PPV (m/s) (Row 3 by Row 4) 0.829 0.402 0.203 0.073 0.032Minimum possible PPV (m/s) (From Table 8 ) 0.599 0.317 0.139 0.053 0.019Filed records (PPV) (m/s) 0.61 0.468 0.192 a 0.021

a Datum not available.



15/16

equivalent to 0.33 and jkn/ x z of 0.695 can be calculated bylinear interpolation as 0.443.

Repeating the same procedure for estimating Max.Reb 10 , other maximum rebound ratios, Max. Reb 2, Max.Reb 3, Max. Reb 5 and Max. Reb 21 can be obtained. Table10 shows Max. Reb 2, Max. Reb 3, Max. Reb 5, Max.

Reb 10 and Max. Reb 21 before the 2nd, 3rd, 5th, 10th and21st joints in a joint set spaced at 0.33 wavelength, with jkn/( x z) of 0.695, in the radial direction of 90 .

The maximum possible PPVs at the monitoring point isthe product of the estimated Max. Reb n by the amplitudeof the compressive pulse wave at the same monitoringpoint from the model of intact rock. Table 11 shows theminimum possible PPVs (form the proposed predictionmodel for estimating T n), the maximum possible PPVs(from the proposed prediction model for estimating Max.Reb n) and the eld records.

5.3.3.3. Comparison between the eld records and theestimated PPVs. Fig. 14 visually shows the estimatedminimum possible and maximum possible PPVs, togetherwith the eld records from Mandai test. The scaled rangeis termed as r/w1/3 , as dened in literature [36,37], where ris the radial distance in m from the explosion source, w isthe charge weight in kg. The solutions from two empiricalformula in literature [36,37] are also included in Fig. 14.In literature [36], Wu suggested a formula for estimatingPPV as a function of the scaled range as: PPV = 1.8(r/w1/3 ) 2.5. While in literature [37], Ambrasseys andHendron suggested the following formula: PPV =

11.45(r/w1/3

) 2.8, if r/w1/3

< 4.14 and PPV = 2.08(r/w1/3 ) 1.6, if r/w1/3 P 4.14.In the eld, every monitoring point is between two adja-

cent joints, the eld records are PPVs of the superpositionof the multiple reected and transmitted waves, not only of the transmitted wave. Because of the uncertainty of the rel-ative position of the monitoring point to the two adjacent

joints in the joint set, the eld record at a monitoring pointshould be within the range given by the minimum possiblePPV as the lower limit and the maximum possible PPV asthe upper limit.

From Fig. 14, it can be seen that the estimated range of PPV for the monitoring point agrees well with the eld-

recorded PPV. Three of the four available eld-recordedPPVs lie exactly between the estimated range, while thefourth available eld-recorded PPV lies close to the maxi-mum estimated value.

6. Conclusions

In this paper, the prediction models for estimating T nand Max. Reb n are proposed, as shown in Figs. 6 and 8,respectively. For these two prediction models, the follow-ing conclusions can be made:

(1) In the radial direction normal to the joints, if spacing/wavelength is greater than a critical value S at , thevalue of T n becomes a constant and independentof spacing/wavelength. As spacing/wavelengthdecreases from the critical value S at , T n increasesand reaches the peak value T peak at another criticalvalue of spacing/wavelength S peak . As spacing/wave-length further decreases from S peak , T n decreases fromits peak value T peak . The variation of T n with spacing/wavelength was generalized into a characteristiccurve, which is determined by two critical points(S peak , T peak ) and (S at , T at ). The two critical points

are controlled by quantity of joints and jkn/( x z).(2) The variations of S peak , T peak , S at , and T at with jkn/x z for arbitrary number of joints are generalized intofour charts. These four charts can be used as the pre-diction model for estimating T n in the radial directionof 90 .

(3) In the process of 2-D compressive wave propagationin a joint set along the radial direction of 90 , themaximum possible PPV is controlled by superposi-tion of the multiple reected and transmitted wavesbetween two adjacent joints, which can be measuredby Max. Reb n. The relationship between Max. Reb nand the inuence factors are generalized into vecharts. Theses charts can be used as the predictionmodel for estimating Max. Reb n.

The proposed models in this paper are applied to a prac-tical explosion test conducted in Mandai in Singapore forthe purpose of verication. The verication example shows:

(1) The proposed models for estimating T n and Max.Reb n for the radial direction of 90 are reliable.

(2) Taking all the factors inuencing T n and Max. Reb nin the process of 2-D compressive wave propagationin a jointed rock mass into account, the solutions

from the prediction models proposed in this paper

0

0.2

0.4

0.6

0.8

1

1.2

0 50 100 150 200 250

Scaled distance

A m p l

i t u d e o f v e

l o c i

t y ( m

/ s )

Fig. 15. Amplitude of velocity wave versus scaled distance R R 0= R 1:50for wave frequency of 700 Hz.



16/16

agree with the eld test records better than empiricalformula. This is the signicance of the currentresearch.

Acknowledgements

The rst author thank for the suggestions and guidancein the research given by Prof. Zhao Jian in Ecole Polytech-nique Federale de Lausanne (EPFL), Rock MechanicsLaboratory, Lausanne, Switzerland.

Appendix I. 2-D compressive wave propagation in an intactrock

In literature [15], several charts have been prepared forchecking the amplitude at any point in the surroundingintact rock. In the prepared charts, the wave amplitude atany point in the surrounding intact rock can be deter-mined, for an input compressive velocity wave with unityamplitude for any frequency. Fig. 15 shows the amplitudeof the velocity wave at any point in the surrounding intactrock versus the scaled distance R R 0= R 1:50 for the fre-quency of 700 Hz, where R is the radial distance from thecavity center and R0 is the radius of the cavity from whichthe wave originates.

References

[1] Schoenberg M. Elastic wave behavior across linear slip interfaces. JAcoust Soc Am 1980;68(5):151621.

[2] Miller RK. The effects of boundary friction on the propagation of elastic waves. Bull Seismol Soc Am 1978;68(4):98798.

[3] Myer LR, Pyrak-Nolte LJ, Cook NGW. Effects of single fracture onseismic wave propagation. In: Proceedings of ISRM symposium onrock fractures. Loen; 1990. p. 46773.

[4] Pyrak-Nolte LJ. The seismic response of fractures and the interrela-tions among fracture properties. Int J Rock Mech Mining SciGeomech Abs 1996;33(8):787802.

[5] Gu B, Sua0 rez-Rivera R, Nihei KT, Myer LR. Incidence of planewaves upon a fracture. J Geophys Res 1996;101(B11):2533746.

[6] Zhao J, Cai JG. Transmission of elastic P-waves across singlefractures with a nonlinear normal deformational behavior. RockMech Rock Eng 2001;34(1):322.

[7] Pyrak-Nolte LJ, Myer LR, Cook NGW. Anisotropy in seismicvelocities and amplitudes from multiple parallel fractures. J GeophysRes 1990;95(B7):1134558.

[8] Hopkins DL, Myer LR, Cook NGW. Seismic wave attenuation

across parallel fractures as a function of fracture stiffness and spacing.EOS Trans AGU 1988;69(44):142738.[9] Myer LR, Hopkins D, Peterson JE, Cook NGW. Seismic wave

propagation across multiple fractures. Fract Joint Rock Masses1995:10510.

[10] Cai JG, Zhao J. Effects of multiple parallel fractures on apparentwave attenuation in rock masses. Int J Rock Mech Mining Sci2000;37(4):66182.

[11] Zhao J, Zhao XB, Cai JG. A further study of P-wave attenuationacross parallel fractures with linear deformational behaviour. Int JRock Mech Mining Sci 2006;43(6):77688.

[12] Ma M, Brady BH. Analysis of the dynamic performance of anunderground excavation in jointed rock under repeated seismicloading. Geotech Geol Eng 1999;17(1):120.

[13] Senseny PE, Simons DA. Comparison of calculation approaches forstructural deformation in jointed rock. Int J Numer Anal MethGeomech 1994;18:32744.

[14] Chen SG, Cai JG, Zhao J, Zhou YX. Discrete element modeling of anunderground explosion in jointed rock mass. Geotech Geol Eng2000;18(2):5978.

[15] Lei WD. Numerical studies on 2-D compressional wave propagationin jointed rock masses. Ph.D thesis, Nanyang Technological Univer-sity, Singapore; 2005.

[16] Medina F, Tailao R. Finite element technicals for problems of unbounded domains. Int J Numer Meth Eng 1983;19(4):120926.

[17] Lysmer J, Kuemyer RL. Finite dynamic model for innite media. JEng Mech Div, ASCE 1969;95(EM4):87795.

[18] Itasca consulting group. UDEC manual. Version 3.0. USA: Minne-apolis, MN; 1996.

[19] Clayton R, Engquist B. Absorbing boundary conditions for waveequation migration. Geophysics 1980;45(5):895904.

[20] Hogdon RL. Absorbing boundary conditions for elastic waves.Geophysics 1991;56(2):23141.

[21] Liao ZP, Wong HL, Yang BP, Yuan YF. A transmitting boun-dary for transient wave analysis. Sci China, Serial A 1984;27(10):106376.

[22] Liu JB, Lu YD. A direct method for analysis of dynamic soil-structure interaction. China Civil Eng J 1998;31(3):5564. in Chinese.

[23] Jing LR. A review of techniques, advances and outstanding issues innumerical modeling for rock mechanics and rock engineering. Int JRock Mech Mining Sci 2003;40(3):283353.

[24] Zhao CB, Valliappan S, Wang YC. A numerical model for wavescattering problems in innite media due to P- and SV-waveincidences. Int J Numer Meth Eng 1992;33(8):166182.

[25] Zhao CB, Valliappan S, A dynamic innite element for 3D in-nite domain wave problems. Int J Numer Meth Eng 1993;36(15):256780.

[26] Valliappan S, Zhao CB. Dynamic response of concrete gravity damsincluding damwater-foundation interaction. Int J Numer Anal MethGeomech 1992;16(2):7999.

[27] Zhao CB, Valliappan S. Incident P and SV wave scattering effectsunder different canyon topographic and geological conditions. Int JNumer Anal Meth Geomech 1993;17(2):7394.

[28] Zhao CB, Valliappan S. Dynamic analysis of a reinforced retainingwall using nite and innite element coupled method. Comput Struct1993;47(2):23944.

[29] Zhao CB, Valliappan S. Seismic wave scattering effects under differentcanyon topographic and geological conditions. Soil DynamicsEarthquake Eng 1993;12(3):12943.

[30] Lee CB. Fracturing characteristics of Bukit Timah granite. M.Eng.thesis. Nanyang Technological University, Singapore; 2002.

[31] Lei WD, Hefny AM, Zhao J. Pilot studies on two dimensional wavepropagation in rock masses. Trans Nonferrous Met Soci China2005;15(4):94955.

[32] Bieniawski ZT. Rock mass classication in rock engineering.Proceedings of symposium in exploration for rock engineering, vol.

1. Cape Town: Balkema; 1976. p. 97106.[33] Seram JL, Pereira JP. Consideration of the geomechanical classi-cation of Bieniawski. In: Proceedings of International Symposium OnEngineering Geology and Underground Construction. Lisbon 1(II);1983. p. 3344.

[34] Zhao J, Hefny AM, Zhou YX. Hydrofracturing in situ stressmeasurement in Singapore Granite. Int J Rock Mech Mining Sci2005;42(4):57783.

[35] Hart RD. An introduction to distinct element modeling for rockengineering. In: Comprehensive rock mechanics 2; 1993. p. 24561.

[36] Wu TH. Soil Dynamics. Allyn and Bacon; 1975.[37] Ambraseys NR, Hendron AJ. Dynamic behavior of rock masses. In:

Proceedings of Rock Mechanics in Engineering Practice. London:Wiley; 1968. p. 20327.


a numerical study on 2-d compressive wave propagation in rock masses with a set of joints along the...

Documents