Estimation of damping ratio of rock mass for numerical simulation of blast induced vibration propagation

Jae-Kwang Ahn i), Duhee Park ii) and Jin-Kwon Yoo i)

i) Ph.D Student, Department of Civil and Environmental Engineering, Hanyang University, Korea.. ii) Associate professor, Department of Civil and Environmental Engineering, Hanyang University, 506 Jaesung Civil Engr Bldg,

Wansimni-ro, Sungdong-gu, Seoul, 133-791, Korea.

ABSTRACT Numerical evaluation of the effect of blast induced vibration on adjacent structures requires determination of the damping ratio. Selection of the dampring ratio for the rock mass is very difficult because it is heavily influenced by the joint layout. We propose a simple yet robust method to estimate the damping ratio of rock mass that fits the target attenuation curve. The damping ratio can be selected as functions of the shear wave velocity of the rock mass and the attenuation parameter of the target attenuation relationship. Keywords: blast load, attenuation relationship, numerical modeling, peak particle velocity

1 INTRODUCTION

The prediction of blast induced vibration is an important issue in the blasting design (Lu et al., 2011). The effect of blasting on nearby structures and facilities is evaluated with the ground motion parameter PPV (peak particle velocity). Most often, an empirical attenuation relationship is used to predict the blast induced PPV (Dowding, 1996; Singh and Singh, 2005; Wiss, 1981).

Numerical simulation of blast induced vibration propagation is performed if PPV is not sufficient to evaluate the stability of a nearby structure and a dynamic analysis is needed. To perform a numerical analysis of blasting, blast load and dynamic properties of the medium are needed as input parameters. The dynamic properties of soil or rock include Young's modulus, Poisson's ratio and damping ratio. Whereas the dynamic properties of soil is well defined, the properties of jointed rock mass is not as well documented. The property with the highest uncertainty is the damping ratio. Because the damping ratio is heavily influenced by the joint layout and properties, determining the damping ratio is quite difficult (Hao et al., 2001). It is even more difficult to determine the damping ratio for a 2D plane analysis, since the damping ratio has to be additionally adjusted for the different attenuation characteristics of a 2D analysis compared to a true 3D propagation of a spherical induced blast load.

We performed a series of 2D nonlinear dynamic analyses to investigate the characteristics of blast induced vibration attenuation. From the analyses,

equivalent damping ratios for various types of rock masses were proposed that correspond to a range of empirical attenuation relationships.

2 ATTENUATION OF BLAST INDUCED VIBRATION

The amplitude of vibration induced by blasting decreases with increasing distance from the source. The attenuation is produced by two phenomena, which are geometrical spreading and material damping. The geometrical spreading can described by the following equation:

12 1

2

mr

A Ar

(1)

where A1 is the amplitude of vibration at distance r1 from the source, A2 the amplitude of vibration at distance r2 from the source and m is a geometric damping coefficient. For blasting, m=1.0. Table 1. lists the values of m. For blasting, m=1.0.

Table 1. Geometric damping coefficient for source and wave type (Kim and Lee, 2000) Source location

Source Type

Wave Type

m Physical

Surface Point Body 2.0 Single footing Load Surface 0.5 Single footing Load Under ground

Point Body 1.0 Spherical load Line Body 0.5 Cylindrical load

The 15th Asian Regional Conference on Soil Mechanics and Geotechnical Engineering

Japanese Geotechnical Society Special Publication

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Material damping is caused by nonlinear hysteretic behavior, friction and cohesion between soil particles or rock joints (Hao et al., 2001; Kim and Lee, 2000). The attenuation due to material damiping is affected by soil or rock type and frequency of loading:

A2= A

1e-a ( r

2-r

1) (2)

where α is the material damping coefficient. Kim and Lee (2000) reported that α =0.026 for one set of measured blast induced vibrations. However, the ragne of value of α has not been investigated.

A simple empirical equation combining the geometrical spreading and material damping has been proposed by Hino (1956):

0

m

RP P

R (3)

where R0 is the blasthole radius, P is the pressure at distance R from the center of a blasthole, Pm is the maximum pressure at the blasthole wall, and β is the decay coefficient The form of the equation is similar to Eq. (1) for geometrical spreading, but it includes the effect of the material damping. Hino (1956) proposed that β =-1.5 (for rocks in general). Liu and Tidman (1995) proposed representative values of β for different types of rock, as summarized in Table 2.

Table 2. Decay coefficient β for various rock types (Liu and Tidman, 1995)

Rock Type Explosive Type

β Rock velocity (Vp, km/sec)

Limestone Emulsion -1.65 3.5 ANFO -1.54 Granite Emulsion -1.48 4.82 ANFO -1.39 Granodiorite Emulsion -1.24 5.6 ANFO -1.32 Pyrite Emulsion -1.38 6.0 ANFO -1.29

Wiss (1981) proposed the following empirical

blasting vibration attenuation equation, as shown below:

nPPV K SD (4)

where K and n is curve fitting coefficients, SD(=R/W1/2) is the scaled distance. R is the distance from the blast source, W is maximum charge weight per delay. K is proportional to the charge weight and also dependent on the blast and rock types. n is an attenuation parameter which also depends on the blast and rock types. This equation is most widely used in practice to predict the vibration under blasting. Representative

values for K and n have been recommended and widely used in design.

3 NUMERICAL ANALYSIS

Numerical analyses were performed with FLAC2D v. 7.0 (Itasca Consulting Group, 2011), a commercial finite difference analysis (FD) program. Modeling of blast at underground is shown in Fig1 and material properties are summarized in Table 3.

BlastLoad

Mesh

60 m

130 m

40 m

20 m

R=1

0.25 m

Fig. 1 Modeling

Table 3. Material properties Type Shear Velocity

(m/sec) Unit Weight

(kN/m3) Poisson

ratio Hard Rock 2500 25 0.25

2000 1500

Soft Rock 1200 23 0.27 1000 800

Weathered Rock

650 21 0.3 500 450

We assumed that blasting creates a fractured zone

and the elastic waves are propagated beyond this crushed section. The fractured zone was assumed to be a circle with a radius of 1 m (Lu et al., 2011). The blast load was applied in a direction perpendicular to the elastic boundary.

Damping was modeld using the Rayleigh damping formulation, which is defined as follows:

[ ] [ ] [ ]C M K (5)

where [C] is the damping matrix, [M] is the mass matrix, [K] is the stiffness matrix, ξ is the damping ratio,

and are the Rayleigh coefficients that determine the frequency dependence of the damping formulation. The formulation matches the target damping ratio only at the frequencies of fm and fn (Fig. 2). The damping of soils or rocks is known to be independent of the loading frequency. However, the Rayleigh damping formulation is frequency-dependent and introduces numerical damping. The frequencies of fm and fn in blast simulation should be selected to minimize the frequency dependence of damping. In the analyses, the

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fn was used for the predominant frequency of the input blast wave and fm was set to1/2 fn.

0

0.05

0.1

0.15

0.2

0.25

0.3

1 10 100

ξ(%

)

frequency (Hz)

Target damping ratio

Mass propotionaldamping

Stiffness propotionaldamping

fm fn

Fig. 2 Frequency dependency of the full rayleigh damping formulation

The viscous damper proposed by Kuhlemeyer and Lysmer (1973) was applied at the model boundary. Reflected wave at model boundaries is absorbed by specifying viscous boundary. The size of the elements was selected to accurately model the wave propagation, as suggested by (Lysmer and Kuhlemeyer, 1969):

10

pVl

f (6)

where l is element size, f is frequency of load, Vp is P wave velocity. Typical frequency range of underground blast induced vibration is between 50 and 100 Hz (Dowding, 1996; Yang et al., 2003). The blast load used in the analysis is a sine pulse with a duration of 0.01 s, as shown in Fig. 3.

-10

-5

0

5

10

0 0.005 0.01 0.015

Vel

ocity

(cm

/sec)

Time (sec)

Fig. 3 Blast load time series

4 RESULTS

4.1 Particle motion under blast induced vibration The calculated particle velocity time series at

selected points at the surface are shown in Fig. 4. Fig. 4a shows the vertical and longitudinal velocity time series and Fig. 4b shows the vertical vs. horizontal velocity hysteretic curves. It is shown that the principal direction of the response is parallel to the direction of propagation. It also shows that the major component of vibration is compressional wave. The results are similar to the trend observed in field recordings, as reported by by Kim and Lee (2000).

-1

0

1

Vel

ocity

(c

m/se

c) Point A

-0.8

0

0.8

Vel

ocity

(c

m/s

ec) Point B

-0.6

0

0.6

0 0.05 0.1

Vel

ocity

(c

m/se

c)

Time (sec)

Point C

-1

0

1Point A

-0.8

0

0.8Point B

-0.6

0

0.6

0 0.05 0.1Time (sec)

Point C

Longitudinal (X-direction)Vertical (Y-direction)(a)

10 m

Point C

Blast source

40 m

40 m

Point BPoint A

80 m

Z

X

Y

TransverseVertical

Longitudinal

source

-0.5

-0.25

0

0.25

0.5

-0.5 -0.25 0 0.25 0.5

Y-Vel(cm/sec)

X-Vel(cm/sec)

-0.6

-0.3

0

0.3

0.6

-0.6 -0.3 0 0.3 0.6

Y-Vel(cm/sec)

X-Vel(cm/sec)

-0.9

-0.45

0

0.45

0.9

-0.9 -0.45 0 0.45 0.9

Y-Vel(cm/sec)

X-Vel(cm/sec)

(b)

Fig. 4. Particle velocity of underground blast

4.2 Influence of damping ratio and rock stiffness Fig. 5 plots the PPV against distance for weathered

rock. It is shown that the PPV quickly decays with increasing distance from the source. The rate of decrease in PPV for damping ratio = 5 % is significantly higher than that for damping ratio = 1 %. It is shown that the damping ratio is the dominant factor in the attenuation of the PPV. Also shown in the figure is the effect of the shear wave velocity. The shear velocity has negligible influence on the attenuation for damping = 1 % case, but its influence is higher for the case of damping = 5 %. However, its influence is greatly less than the damping ratio.

0.05

0.5

20 200Distance from source (m)

(b) ξ=5%

0.2

2

20 200

PPV

(cm

/sec)

Distance from source (m)

(a) ξ=1%

Vs=450Vs=500Vs=650

Fig. 5 Effect of damping ratio and shear wave velocity

4.5 Damping ratio design chart We performed a series of analyses to develop a

correlation between the damping ratio and the attenuation coefficient. Damping ratios used range from 1 % to 10 %. Corresponding attenuation coefficient, n,

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was selected from log PPV versus log scaled distance plot. The best fit empirical attenuation relationship was found between distance of 40 m to 100 m. We performed analyses for a range of shear wave velocities to account for its influence. Fig. 6 plots the damping ratio vs. the attenuation coefficient. The chart is very robust and easy to use. If the shear wave velocity of the rock mass and attenuation coefficient is known, the corresponding equivalent damping ratio can easily be selected.

0

2

4

6

8

10

12

1 1.5 2 2.5 3 3.5 4

Vs ≤ 300300 ≤ Vs ≤ 650650 ≤ Vs ≤ 900900 ≤ Vs ≤ 1,2001,200 ≤ Vs ≤ 1,8001,800 ≤ Vs

0

2

4

6

8

10

12

1 1.5 2 2.5 3 3.5 4

Dam

ping

(%)

n Fig. 6 Damping – attenuation coefficient

5 CONCLUSIONS

We performed a series of 2D nonlinear dynamic analyses to determine equivalent damping ratio of rock mass for a 2D plane strain numerical simulation of blast induced vibration propagation. The results showed that the vibrations induced blast is dominantly a compressional wave. It is also shown that the damping ratio is both dependent on the shear wave velocity of the rock mass. The results of the analyses were integrated to an attenuation relationship coefficient-shear wave velocity-equivalent damping ratio correlation chart. The proposed chart is quite robust and yet simple to use, and can be easily applid in practice.

REFERENCES 1) Dowding, C. (1996): Construction vibrations, Prentice Hall. 2) Hao, H., Wu, Y., Ma, G. and Zhou, Y. (2001): Characteristics

of surface ground motions induced by blasts in jointed rock mass, Soil Dynamics and Earthquake Engineering. 21(2), 85-98.

3) Hino, K. (1956): Fragmentation of rock through blasting and shock wave; theory of blasting, Quarterly of the Colorado School of Mines.

4) Itasca Consulting Group, Inc. (2011): Flac (fast lagrangian analyses of continua), user's manual, Ontario, Canada.

5) Kim, D.S. and Lee, J. S. (2000): Propagation and attenuation

characteristics of various ground vibrations, Soil Dynamics and Earthquake Engineering. 19(2), 115-126.

6) Kuhlemeyer, R. L. and Lysmer, J. (1973): Finite element method accuracy for wave propagation problems, Journal of Soil Mechanics & Foundations Div. 99(Tech Rpt.).

7) Liu, Q. and Tidman, P. (1995): Estimation of the dynamic pressure around a fully loaded blast hole, Retrieved from Canmet Mrl Experimental Mine.

8) Lu, W., Yang, J., Chen, M. and Zhou, C. (2011): An equivalent method for blasting vibration simulation, Simulation Modelling Practice and Theory. 19(9), 2050-2062.

9) Lysmer, J. and Kuhlemeyer, R. (1969): Finite element model for infinite media, Journal of Engineering Mechanics Division. ASCE. 95, 859-877.

10) Singh, T. and Singh, V. (2005): An intelligent approach to prediction and control ground vibration in mines, Geotechnical & Geological Engineering. 23(3), 249-262.

11) Wiss, J. F. (1981): Construction vibrations: State-of-the-art, Journal of the Geotechnical Engineering Division. 107(2), 167-181.

12) Yang, H. S., Lim, S. S. and Kim, W. B. (2003): Tunnel blasting design with equation obtained from borehole and crater blasting, Journal of Korean Geotechnical Society. 19(5), 327-333.

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