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Analytical study of dynamic friction mechanism in blocky rock systems
ARTICLE IN PRESS International Journal of Rock Mechanics & Mining Sciences 46 (2009) 946–951
Contents lists available at ScienceDirect
International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
Technical Notes
Analytical study of dynamic friction mechanism in blocky rock systems G.W. Ma a,, X.M. An a, M.Y. Wang b a b
School of Civil and Environmental Engineering, Nanyang Technological University, Singapore 639798 Engineering Institute, PLA University of Science and Technology, China 210007
a r t i c l e in f o Article history: Received 12 May 2008 Received in revised form 3 April 2009 Accepted 7 April 2009 Available online 7 May 2009
1. Introduction Underground excavations such as tunnels, storage caverns, and underground power plants have been constructed throughout the world. Rock masses in underground constructions always contain discontinuities such as faults, joints, bedding planes, and fractures. These discontinuities intersect with each other to create rock blocks [1]. One of the most important problems in underground excavation is the accidental falling of rock blocks at the working faces [2]. To predict this, the removability and stability of the rock blocks around the underground tunnels and caverns must be evaluated based on the characteristics of the discontinuities in the rock masses. The most commonly used method is the block theory developed by Goodman and Shi [3]. A complete block theory analysis consists of removable block identification based on geometric and topological methods, mode analysis to determine whether the removable block has a mode of failure or not, and stability evaluation to identify the key blocks by incorporating comparatively simple mechanics analysis. After that, the supporting force and directions are designed to make sure that the key blocks are stable. If the key blocks are stable, the entire rock mass will be stable [4]. Block theory has been applied widely in rock engineering since early 1990s [5]. However, its stability evaluation is based on static analysis. Recent studies demonstrate that sometimes an external dynamic disturbance may dominate the system behavior. The traditional stability analysis based on static analysis may underestimate the rock instability if not taking these dynamic factors into account. Earlier researchers [6–8] observed that when a pulse loading is applied to a blocky rock system, friction between the blocks in the orthogonal direction to the pulse loading may be significantly reduced or even disappear, which is known as an ultra low friction
Corresponding author.
E-mail address: [email protected] (G.W. Ma). 1365-1609/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2009.04.001
phenomenon. Such phenomenon may have significant influence on the stability of underground excavation or large slopes, which is illustrated in Fig. 1. With a horizontal dynamic disturbance, the ceiling of an underground tunnel or cavern may easily fall down under the gravity (Fig. 1a), while with a vertical external pulse, the two sides of the underground tunnel or cavern may fail under a small lateral pushing force (Fig. 1b). The external disturbance can be caused by earthquake, nearby blasting, construction disturbance and so on. With the external disturbance, the rock blocks at the excavation faces will more likely meet the conditions of slumping. After the falling of the first block, the boundary conditions of the block system are changed, the system forces are redistributed, progressive collapse may happen. The ultra low friction phenomenon requires developing rock mass dynamic stability theory during underground mining and constructions. In the present paper, the ultra low friction phenomenon is analytically investigated and a preliminary interpretation of this phenomenon is provided. The blocky rock system is simplified into a multiple-degree-of-freedom mass–spring–dashpot system and the interactions of the rock blocks are considered. The stabilities of a single-block system and a multiple-block system are, respectively, discussed. The parametric studies on the ultra low friction phenomenon regarding the pulse width, pulse amplitude, pulse shape and damping ratio are carried out. Newmark’s sliding block theory [9,10] is extended to calculate the lateral displacement of the blocky rock system, which is caused by the lateral force and the axial pulse load.
2. Analytical model for a blocky rock system In the present study, a blocky rock system with n identical blocks sequentially connected with each other as shown in Fig. 2a is considered. The blocky rock system is subjected to a dynamic pulse loading F V ðtÞ along its axis in the vertical direction. The deformable blocky rock system is simplified into a multiple-
ARTICLE IN PRESS G.W. Ma et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 946–951
947
Pulse loading
Excavation
Blocks at
Excavation
ceiling
Pulse loading
Blocks at side walls Fig. 1. Dynamic instability of underground excavation: (a) with a horizontal pulse, the blocks at ceiling may easily fall down under gravity; and (b) with a vertical pulse, the blocks at two side walls of the excavation may easily fail under a small pushing force.
FV (t)
Fi,i−1 (t) FHi (t)
Block 1
mi
Block 2
Fi,i+1 (t)
FH (t)
degree-of-freedom mass–spring–dashpot system seen in Fig. 2b. F H ðtÞ is a horizontal pushing force which represents a lateral force applied to the blocky rock system. In the vertical direction, it is treated as a mass–spring–dashpot system with n degrees-offreedom (DOFs) (Fig. 2b), where mi ði ¼ 1 nÞ is the mass, ci ði ¼ 1 nÞ is the viscous damping coefficient, ki ði ¼ 1 nÞ is the stiffness, ui ði ¼ 1 nÞ is the vertical displacement, and the subscripts correspond to the block number. Taking the static equilibrium position as the initial position, the equilibrium equation of the nDOF mass–spring–dashpot system can be expressed as
Block 3
€ þ cuðtÞ _ þ kuðtÞ ¼ pðtÞ muðtÞ Block i Block n
FV (t) m1 c1
u1 k1
m2 k2
c2 FH (t)
u2
m3 c3
where, m is the mass matrix, c is the damping matrix, k is the stiffness matrix, u is the displacement vector, expressed as u ¼ fu1 ; u2 ; u3 ; . . . ; un gT , pðtÞ is the vertical loading vector, _ is the velocity vector expressed as pðtÞ ¼ fF V ðtÞ; 0; 0; . . . ; 0gT , u defined as the derivative of the displacement vector with respect € is the acceleration vector. to time, u The solution of Eq. (1) gives the vertical displacement response of the system, which will be discussed in Sections 3 and 4 for a single-block system and a multiple-block system, respectively. In the horizontal direction, a rock block is approximated as a single-degree-of-freedom (SDOF) system subjected to friction forces from its adjacent blocks and a lateral pushing force, as plotted in Fig. 2c, where F i;i1 is the friction force on block i from block i1, F i;iþ1 is the friction force on block i from block i+1, F Hi ðtÞ is the lateral pushing force applied on block i. The friction force F i;i1 and F i;iþ1 can be obtained based on the vertical displacement response of the block system. The horizontal displacement of block i can be determined by using Newmark’s block sliding theory [9,10]. Detailed calculation of the horizontal displacement will be presented in Section 5.
u3
3. Dynamic friction mechanism of a single-block system
k3 mi
ci
ui ki
To study the dynamic friction mechanism, a single block resting on the ground is first investigated by simplifying it into a single-degree-of-freedom (SDOF) system. The equilibrium equation is expressed as € þ cuðtÞ _ þ kuðtÞ ¼ F V ðtÞ muðtÞ
mn cn
(1)
un kn
Fig. 2. Multiple-block system: (a) n-block system; (b) analytical model in vertical direction; and (c) analytical model in the horizontal direction.
(2)
where m is the mass, c is the viscous damping coefficient, k is the _ stiffness, uðtÞ is the vertical displacement of the block, uðtÞ is the € velocity, uðtÞ is the acceleration, F V ðtÞ is the applied pulse loading in the vertical direction. Eq. (2) can be normalized as _ þ o2n uðtÞ ¼ € þ 2Bon uðtÞ uðtÞ
F v ðtÞ m
(3)
ARTICLE IN PRESS G.W. Ma et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 946–951
where B ¼ c=cr is the damping pffiffiffiffiffiffiffiffiffiffi ratio, cr ¼ 2mon is the critical damping coefficient, on ¼ k=m is the natural frequency of the SDOF system. Consider a half-sine pulse loading expressed as ( P 0 sin ot; 0ptpT (4) F v ðtÞ ¼ 0; t4T where T is the duration of pulse loading, P0 is the amplitude of the loading, o is the loading frequency given by o ¼ p=T. Eq. (3) is solved for the displacement uðtÞ subjected to the initial conditions _ ¼ u_ 0 , where u0 and u_ 0 are the initial of uð0Þ ¼ u0 and uð0Þ displacement and velocity, respectively. In the current study, the initial condition is assumed as u0 ¼ 0 and u_ 0 ¼ 0. The differential Eq. (3) can be solved by using the convolution integral method (or so called Duhamel’s integral method) [11,12] or the Laplace transform and the inverse Laplace transform method [13–15]. The response of the SDOF system consists of two phases: (1) Forced vibration phase. During this phase, the system is subjected to a half-sine pulse loading. The response of the system is given by uðtÞ ¼ eBon t ðA cos oD t þ B sin oD tÞ þ ðC sin ot þ D cos otÞ; tpT
(5a)
where oD isp the damped natural frequency of vibration, expressed ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi as oD ¼ on 1 B2 , the first two items in the right hand side of Eq. (5a) are the homogeneous solution with the coefficients A and B to be determined by satisfying initial conditions, i.e. uð0Þ ¼ u0 _ and uð0Þ ¼ u_ 0 , the last two items in the right hand side are the particular solution, with coefficients C and D expressed as C¼
D¼
P0 1 ðo=on Þ2 mk ½1 ðo=on Þ2 2 þ ½2Bðo=on Þ2
(5b)
P0 2Bo=on mk ½1 ðo=on Þ2 2 þ ½2Bðo=on Þ2
(5c)
For a case study, the dimension of the block is assumed to be 0.4 m 0.2 m with a unit thickness. The density, Young’s modulus, and friction angle are 2650 kg/m3, 80 GPa, and 301, respectively. With the gravity load, the static friction is F hs ¼ mg tan f ¼ 1:224 kN
3
(2) Free vibration phase. After the pulse loading ends at time T, the system undergoes damped free vibration with the solution given by uðtÞ ¼ eBon ðtTÞ uðTÞ cos oD ðt TÞ _ uðTÞ þ Bon uðTÞ þ sin oD ðt TÞ t4T (6)
Dynamic frictional force Static frictional force
2.5 2 1.5 1.224 1 0.5 0.29 0 0
oD
1
2
3
4
5
6
Time (ms)
(7)
where F hd is the dynamic friction in the horizontal direction, f is the static friction angle, N is the normal force at the interface between the block and ground, varies with time t, expressed as NðtÞ ¼ kuðtÞ, where k is the equivalent stiffness of the block. Here, F hd is referred to as the dynamic friction because its value varies with time, not because the block is already moving. In the dynamic case, a pulse loading is applied in the vertical direction, and its effect on the friction in the horizontal direction is reflected by the variation of the normal force N. In Eq. (7), we choose the Mohr–Coulomb model among several available friction models to describe the relationship between the normal force and friction because this model has been widely used in rock mechanics and the parameters involved in this model can be easily determined by experiments. In fact, no matter what kind of friction model we use, similar results will be obtained.
Fig. 3. Comparison between static friction force and dynamic friction force for a single-block system.
Smallest force to move the block (kN)
_ where uðTÞ and uðTÞ are the displacement and velocity of the block at time of t ¼ T, respectively, obtained from Eqs. (5a) and (5b). Once the vertical displacement response is obtained, the friction between the block and ground can be obtained as F hd ¼ NðtÞ tan f
(8)
where m is the mass of the block, and g is the gravitational acceleration. The dynamic friction force from Eq. (7) and the static friction force from Eq. (8) are plotted and compared in Fig. 3 for the case with pulse amplitude of P 0 ¼ 1:5 kN and the damping ratio of B ¼ 0:1. For a static case, the friction force is a constant, i.e. 1.224 kN. The block will move in the horizontal direction if the pushing force exceeds 1.224 kN. For a dynamic case with a pulse loading applied in a vertical direction, the dynamic friction force vibrates around the static friction force. The block will move in the horizontal direction as soon as the pushing force exceeds the minimum dynamic friction force, which is 0.29 kN, decreased by 76% from the static value. This case shows that with a vertical pulse loading, we can move the block in the horizontal direction more easily with a much smaller pushing force. The reason is the vibration of the block in the vertical direction induced by the applied pulse loading. Then, the pulse width effect and the pulse amplitude effect are investigated by varying the loading frequency ratio b from 0.05 to 6.95 and the pulse amplitude from 0 to 1.9 kN. Here, the loading frequency ratio b is defined as the loading frequency divided by the natural frequency of the block system, expressed by b ¼ o=on, where o is the loading frequency, calculated by o ¼ p=T, T is the duration of the pulse loading. The relationships between the smallest force required to move the block (smallest force in short
Frictional force (kN)
948
1.2 1 0.8
Pulse amplitude: 0.7kN
0.6
1.0kN 1.5kN
0.4 Critical loading frequency ratio
0.2 0
0.73 1
2
3
4
5
6
7
Loading frequency ratio Fig. 4. Relationship between smallest force to move the block and the loading frequency ratio for a single-block system.
ARTICLE IN PRESS G.W. Ma et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 946–951
hereafter) and the loading frequency ratio b for the three pulse amplitudes of 0.7, 1.0 and 1.5 kN are compared in Fig. 4. All the three curves firstly decrease with the increase of the loading frequency, reach their minimum values, and then gradually increase. It is worth noting that all the three curves reach their minimum values at a same loading frequency ratio of b ¼ 0:73. The smallest force required to move the block is 1.244 kN for the static case; however, for the dynamic cases with the loading frequency b ¼ 0:73, the smallest force to move the block decreases to 0.788 kN for the pulse amplitude of 0.7 kN, 0.601 kN for the amplitude of 1.0 kN, and 0.29 kN for the amplitude of 1.5 kN, respectively. Further study shows that for the dynamic cases with b ¼ 0:73, the smallest force required to move the block decreases linearly with the increase of the pulse amplitude. In order to examine the influence of the pulse shape effect, three kinds of pulse loadings, namely, triangular, half-sine and rectangular pulses with various loading frequency ratios b from 0.05 to 6.95 and various pulse amplitudes from 0 to 1.9 kN are applied to the block system. The triangular loading can be expressed as 8 2P 0 T > > t; tp > > 2 < T 2P 0 T F V ðtÞ ¼ (9) t; otpT 2P > > > 0 T 2 > : 0; t4T and the rectangular loading is given as ( P 0 ; 0ptpT F V ðtÞ ¼ 0; t4T
uðtÞ ¼
3 X
fn qn ðtÞ
949
(12)
n¼1
Once the vertical displacement response of each block is obtained, the dynamic friction between any two blocks is subsequently derived. As a case study, the dimension of each block is again assumed to be 0.4 m 0.2 m with a unit thickness. The density, Young’s modulus, and friction angle are again 2650 kg/m3, 80 GPa, and 301, respectively. Without applying the pulse load, the static friction forces for the top block, the middle block and the bottom block are F s1 ¼ m g tan f ¼ 1:224 kN F s2 ¼ 3 m g tan f ¼ 3:672 kN F s3 ¼ 5 m g tan f ¼ 6:120 kN
(13)
respectively, where m is the mass of each block. Fig. 5 shows the dynamic friction forces of the three blocks with pulse amplitude of P0 ¼ 3:0 kN, damping ratio of B ¼ 0:1.
(10)
Similar results have been obtained for the three kinds of loadings with different pulse shapes. There exist critical frequency ratios for all the three cases. However, the critical loading frequency ratio varies with the pulse shape, which is 0.67 for the triangular loading, 0.73 for the half-sine loading, and 1.00 for the rectangular loading, respectively. As a parametric study, the damping effect is also investigated. It is found that the damping ratio has very little effect on the critical frequency ratio, which is probably because the viscous damping does not have enough time to make a strong influence on the critical force for the short-duration transient problem considered in the current study.
4. Dynamic friction mechanism of a multiple-block rock system The dynamic friction mechanism of a multiple-block rock system is subsequently investigated in this section. Without loss of generality, a rock block system with three blocks is studied here. As discussed in Section 2, the vertical response of a threeblock rock system can be simplified into a three-degree-offreedom (3DOF) mass–spring–dashpot system described by the equilibrium equation given by Eq. (1) with the rank of each matrix as three. Solving the zero-damping eigenvalue equation gives the natural frequencies on ðn ¼ 1; 2; 3Þ and the corresponding mode shapes fn ðn ¼ 1; 2; 3Þ. Then, Eq. (1) can be decoupled in modal coordinates as M n q€ n ðtÞ þ C n q_ n ðtÞ þ K n qn ðtÞ ¼ Pn ðtÞ
fTn mfn
(11)
fTn cfn
where, M n ¼ is the generalized mass, C n ¼ is the T generalized damping, K n ¼ fn kfn is the generalized stiffness, and T P n ¼ fn pðtÞ is the generalized loading. Following the similar procedure with the SDOF system, qn ðtÞ ðn ¼ 1; 2; 3Þ can be obtained. Then, the displacements for each block can be calculated as
Fig. 5. Dynamic friction and contribution of each mode for a three-block system: (a) top block; (b) middle block; and (c) bottom block.
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Similar to the phenomenon observed for the single-block system in Section 3, the dynamic friction force for each block vibrates around their corresponding static friction force. The smallest force required to move each block decreases from their static friction force given in Eq. (13) to the minimum value of the dynamic friction force shown in Fig. 5. It becomes easier to move the blocks in the horizontal direction because of the vibration of the blocks in the vertical directions induced by the applied pulse loading. Fig. 5 also shows the first mode has the most significant contribution to the system response while the higher order modes (i.e. the second mode and the third mode) has only some influences on the first peak of the vibration and very little influence on the following vibrations, and the smallest force required to move the block is governed by the first vibration mode of the MDOF system. Parametric studies with respect to the pulse width, pulse amplitude and pulse shape have also been conducted. In order to investigate the pulse width effect, a series of pulse loadings with identical amplitude but different pulse frequencies are applied to the multiple-block system. Fig. 6 shows the relationship between the smallest force required to move each block and the loading frequency ratio b, which is defined as b ¼ o=o1 , where o is the loading frequency, o1 is the first natural frequency of the MDOF system. It is found that each curve reaches their minimum value at a critical loading frequency ratio, which is also 0.73 for a half-sine pulse loading, same as that observed for the single-block system. In order to investigate the influence of the pulse amplitude, a series of pulse loadings with an identical loading frequency but different amplitudes are applied to the system. The smallest force required to move each of the three blocks decreases linearly with the pulse amplitude, which also has similar trend with the result of the single-block system. The pulse shape effect is studied by using the three kinds of pulse loadings, i.e. the triangular loading, the half-sine loading, and the rectangular loading. Again it is found that there exist critical loading frequency ratios for all the three loadings which are the same as the corresponding results of the single-block system, i.e., 0.67 for the triangular loading, 0.73 for the half-sine loading, 1.00 for the rectangular loading. The only difference is the definition of the loading frequency ratio. For a single-block system, the ratio is defined as the loading frequency over the natural frequency of the system, while it is defined as the loading frequency over the first natural frequency for a multiple-block system.
5. Horizontal displacement
Smallest force to move each block (kN)
Newmark’s block sliding theory [9,10] was originally proposed to predict the displacement of a block subjected to seismic 7 Bottom block
6
Middle block Top block
5 4 3 2 1
Critical loading frequency ratio
0 0
0.73 1
2
3 4 Loading frequency ratio
5
6
7
Fig. 6. Relationship between smallest force to move each block and loading frequency ratio for a three-block system.
motion. We extend it here to determine the horizontal displacement of blocks in the lateral direction. An example for the displacement prediction by Newmark’s block sliding theory is shown in Fig. 7, where a half-sine loading with an amplitude of P 0 ¼ 3:4 kN, a loading frequency ratio of b ¼ 0:74, and a horizontal constant pushing force of F H ¼ 3:3 kN are used. The constant pushing acceleration (i.e. pushing force divided by the mass of the block) and the varied frictional acceleration (i.e. dynamic friction force divided by the mass of the block) are plotted in Fig. 7a. When the pushing acceleration is larger than the frictional acceleration (region indicated by gray color below the straight dash line in Fig. 7a), the velocity of the block increases from zero (Fig. 7b) and the block starts to move with the displacement increasing (Fig. 7c). The velocity of the block can be calculated by integrating the acceleration over time in the region marked by gray color while the permanent displacement is obtained by integrating the velocity over time. When it comes to the intersection point between the two acceleration curves in Fig. 7a, the velocity reaches its first peak. After that, the pushing acceleration becomes smaller than the frictional acceleration (region marked by gray color above the straight dash line in Fig. 7a), the velocity of the block decreases but the displacement keeps increasing until the velocity decreases to zero. When the pushing acceleration becomes larger than the frictional acceleration again, the velocity increases from zero again, the block continues moving in the horizontal direction. When the pushing acceleration is smaller than the frictional acceleration, the velocity decreases while the displacement increases until the velocity reduced to zero. The above process repeats until the pushing acceleration is no longer larger than the frictional acceleration. The velocity of the block becomes zero while the horizontal displacement reaches its maximum value. Parametric studies with respect to both the horizontal pushing force and the pulse loading are carried out. The maximum horizontal displacement increases with both the pushing force and the pulse loading amplitude.
6. Conclusions In the present study, the dynamic friction mechanism of blocky rock system has been analytically investigated by adopting a multiple-degree-of-freedom mass–spring–damper system. In a static case, the friction force applied on a rock block is a constant value, and the rock block can move when the pushing force exceeds this value. However, in a dynamic case with a pulse loading applied, the dynamic friction force will vibrates around the static friction force, and the rock block can move as soon as the pushing force is lager than the minimum dynamic friction force, which is definitely smaller than that in the static case. In short, with a pulse loading applied in the axial direction, we can move the block in the lateral direction with a much smaller pushing force than the static case. The result is consistent with the experimentally observed ultra low friction phenomenon. It is also found that it is easiest to move the block in the lateral direction when the loading frequency ratio b reaches a critical value, which is 0.67 for a triangular loading, 0.73 for a half-sine loading, and 1.0 for a rectangular loading. Same results are obtained for both a single-block system and a multiple-block system. With the same loading frequency, it will be relatively easier to move out a block with a higher loading amplitude. The damping ratio has very minor influences on this phenomenon. Lastly, Newmark’s block sliding theory is extended to determine the displacement of each block in the horizontal direction. Analytical results presented in this paper demonstrate that the pulse loading or the external disturbance in an orthogonal direction has a significant influence
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951
Horizontal acceleration (m/s2)
50 Acceleration by frictional force
40
Acceleration by horizontal pushing force
30 20 10
Horizontal velocity (mm/s)
0 0
1
2
3
4
0
1
2
3
4
5 Time (ms)
6
7
8
9
10
10
5
0
Horizontal displacement (um)
-5 5 Time (ms)
6
7
8
9
10
8 6 4 2 0 0
1
2
3
4
5 Time (ms)
6
7
8
9
10
Fig. 7. Horizontal displacement prediction based on Newmark’s sliding block theory: (a) acceleration time history; (b) velocity time history; and (c) displacement time history.
on stability of a blocky rock system. So, it is essential to take this factor into account when carrying out stability analysis of slopes and underground excavations. The traditional stability method based on static analysis (e.g., block theory) may underestimate the instability of a blocky system. References [1] Murakami O, Yokoo A, Ohnishi Y, Nishiyama S. Study on application of key block theory to rock engineering projects. In: Proceedings of the eighth international conference on analysis of discontinuous deformation, Beijing, 2007. p. 281–4. [2] Ahn SH, Lee CI. Removability analysis of rock blocks by block theory and a probabilistic approach. Int J Rock Mech Min Sci 2004;41(3):429. [3] Goodman RE, Shi GH. Block theory and its application to rock engineering. Englewood Cliffs, NJ: Prentice-Hall; 1985. [4] Yeung MR, Jiang QH, Sun N. Validation of block theory and three-dimensional discontinuous deformation analysis as wedge stability analysis methods. Int J Rock Mech Min Sci 2003;40:265–75. [5] Zhang QH, Wu AQ. Study on geometrical identification of stochastic block in block theory. In: Proceedings of the eighth international conference on analysis of discontinuous deformation, Beijing, 2007. p. 255–63.
[6] Wang HL, Ge T, Wang DR, Lu YS, Wang MY, Qian QH. Comparison of theoretical and experimental analysis of dynamic characteristics of block rock mass. Chin J Rock Mech Eng 2007;26(5):952–8 (in Chinese). [7] Wang HL, Zhou ZP, Wang MY. Study on characteristics of block surface. Protective Eng 2006;28(2):30–5 (in Chinese). [8] Wang MY, Qi CZ, Qian QH. Study on deformation and motion characteristics of blocks in deep rock mass. Chin J Rock Mech Eng 2005;24(16):2825–30 (in Chinese). [9] Newmark NM. Effect of earthquakes on dams and embankments. Geotechnique 1965;15(2):787–830. [10] Cai Z, Bathurst RJ. Deterministic sliding block methods for estimating seismic displacements of earth structures. Soil Dynam Earthq Eng 1996;15: 255–68. [11] Clough RW, Penzien J. Dynamics of structures. 2nd ed. New York: McGrawHill; 1993. [12] Chopra AK. Dynamics of structures: theory and applications to earthquake engineering. 2nd ed. Englewood Cliffs, NJ: Prentice-Hall; 2001. [13] Schiff JL. The Laplace transform. New York: Springer; 1999. [14] Shu DW, Shi BJ, Meng H, et al. The pulse width effect of single half-sine acceleration pulse on the peak response of an actuator arm of hard disk drive. Mater Sci Eng A 2006;423:199–203. [15] Shu DW, Shi BJ, Meng H, et al. Shock analysis of a head actuator assembly subjected to half-sine acceleration pulses. Int J Impact Eng 2007;34:253–63.
Contents lists available at ScienceDirect
International Journal of Rock Mechanics & Mining Sciences journal homepage: www.elsevier.com/locate/ijrmms
Technical Notes
Analytical study of dynamic friction mechanism in blocky rock systems G.W. Ma a,, X.M. An a, M.Y. Wang b a b
School of Civil and Environmental Engineering, Nanyang Technological University, Singapore 639798 Engineering Institute, PLA University of Science and Technology, China 210007
a r t i c l e in f o Article history: Received 12 May 2008 Received in revised form 3 April 2009 Accepted 7 April 2009 Available online 7 May 2009
1. Introduction Underground excavations such as tunnels, storage caverns, and underground power plants have been constructed throughout the world. Rock masses in underground constructions always contain discontinuities such as faults, joints, bedding planes, and fractures. These discontinuities intersect with each other to create rock blocks [1]. One of the most important problems in underground excavation is the accidental falling of rock blocks at the working faces [2]. To predict this, the removability and stability of the rock blocks around the underground tunnels and caverns must be evaluated based on the characteristics of the discontinuities in the rock masses. The most commonly used method is the block theory developed by Goodman and Shi [3]. A complete block theory analysis consists of removable block identification based on geometric and topological methods, mode analysis to determine whether the removable block has a mode of failure or not, and stability evaluation to identify the key blocks by incorporating comparatively simple mechanics analysis. After that, the supporting force and directions are designed to make sure that the key blocks are stable. If the key blocks are stable, the entire rock mass will be stable [4]. Block theory has been applied widely in rock engineering since early 1990s [5]. However, its stability evaluation is based on static analysis. Recent studies demonstrate that sometimes an external dynamic disturbance may dominate the system behavior. The traditional stability analysis based on static analysis may underestimate the rock instability if not taking these dynamic factors into account. Earlier researchers [6–8] observed that when a pulse loading is applied to a blocky rock system, friction between the blocks in the orthogonal direction to the pulse loading may be significantly reduced or even disappear, which is known as an ultra low friction
Corresponding author.
E-mail address: [email protected] (G.W. Ma). 1365-1609/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijrmms.2009.04.001
phenomenon. Such phenomenon may have significant influence on the stability of underground excavation or large slopes, which is illustrated in Fig. 1. With a horizontal dynamic disturbance, the ceiling of an underground tunnel or cavern may easily fall down under the gravity (Fig. 1a), while with a vertical external pulse, the two sides of the underground tunnel or cavern may fail under a small lateral pushing force (Fig. 1b). The external disturbance can be caused by earthquake, nearby blasting, construction disturbance and so on. With the external disturbance, the rock blocks at the excavation faces will more likely meet the conditions of slumping. After the falling of the first block, the boundary conditions of the block system are changed, the system forces are redistributed, progressive collapse may happen. The ultra low friction phenomenon requires developing rock mass dynamic stability theory during underground mining and constructions. In the present paper, the ultra low friction phenomenon is analytically investigated and a preliminary interpretation of this phenomenon is provided. The blocky rock system is simplified into a multiple-degree-of-freedom mass–spring–dashpot system and the interactions of the rock blocks are considered. The stabilities of a single-block system and a multiple-block system are, respectively, discussed. The parametric studies on the ultra low friction phenomenon regarding the pulse width, pulse amplitude, pulse shape and damping ratio are carried out. Newmark’s sliding block theory [9,10] is extended to calculate the lateral displacement of the blocky rock system, which is caused by the lateral force and the axial pulse load.
2. Analytical model for a blocky rock system In the present study, a blocky rock system with n identical blocks sequentially connected with each other as shown in Fig. 2a is considered. The blocky rock system is subjected to a dynamic pulse loading F V ðtÞ along its axis in the vertical direction. The deformable blocky rock system is simplified into a multiple-
ARTICLE IN PRESS G.W. Ma et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 946–951
947
Pulse loading
Excavation
Blocks at
Excavation
ceiling
Pulse loading
Blocks at side walls Fig. 1. Dynamic instability of underground excavation: (a) with a horizontal pulse, the blocks at ceiling may easily fall down under gravity; and (b) with a vertical pulse, the blocks at two side walls of the excavation may easily fail under a small pushing force.
FV (t)
Fi,i−1 (t) FHi (t)
Block 1
mi
Block 2
Fi,i+1 (t)
FH (t)
degree-of-freedom mass–spring–dashpot system seen in Fig. 2b. F H ðtÞ is a horizontal pushing force which represents a lateral force applied to the blocky rock system. In the vertical direction, it is treated as a mass–spring–dashpot system with n degrees-offreedom (DOFs) (Fig. 2b), where mi ði ¼ 1 nÞ is the mass, ci ði ¼ 1 nÞ is the viscous damping coefficient, ki ði ¼ 1 nÞ is the stiffness, ui ði ¼ 1 nÞ is the vertical displacement, and the subscripts correspond to the block number. Taking the static equilibrium position as the initial position, the equilibrium equation of the nDOF mass–spring–dashpot system can be expressed as
Block 3
€ þ cuðtÞ _ þ kuðtÞ ¼ pðtÞ muðtÞ Block i Block n
FV (t) m1 c1
u1 k1
m2 k2
c2 FH (t)
u2
m3 c3
where, m is the mass matrix, c is the damping matrix, k is the stiffness matrix, u is the displacement vector, expressed as u ¼ fu1 ; u2 ; u3 ; . . . ; un gT , pðtÞ is the vertical loading vector, _ is the velocity vector expressed as pðtÞ ¼ fF V ðtÞ; 0; 0; . . . ; 0gT , u defined as the derivative of the displacement vector with respect € is the acceleration vector. to time, u The solution of Eq. (1) gives the vertical displacement response of the system, which will be discussed in Sections 3 and 4 for a single-block system and a multiple-block system, respectively. In the horizontal direction, a rock block is approximated as a single-degree-of-freedom (SDOF) system subjected to friction forces from its adjacent blocks and a lateral pushing force, as plotted in Fig. 2c, where F i;i1 is the friction force on block i from block i1, F i;iþ1 is the friction force on block i from block i+1, F Hi ðtÞ is the lateral pushing force applied on block i. The friction force F i;i1 and F i;iþ1 can be obtained based on the vertical displacement response of the block system. The horizontal displacement of block i can be determined by using Newmark’s block sliding theory [9,10]. Detailed calculation of the horizontal displacement will be presented in Section 5.
u3
3. Dynamic friction mechanism of a single-block system
k3 mi
ci
ui ki
To study the dynamic friction mechanism, a single block resting on the ground is first investigated by simplifying it into a single-degree-of-freedom (SDOF) system. The equilibrium equation is expressed as € þ cuðtÞ _ þ kuðtÞ ¼ F V ðtÞ muðtÞ
mn cn
(1)
un kn
Fig. 2. Multiple-block system: (a) n-block system; (b) analytical model in vertical direction; and (c) analytical model in the horizontal direction.
(2)
where m is the mass, c is the viscous damping coefficient, k is the _ stiffness, uðtÞ is the vertical displacement of the block, uðtÞ is the € velocity, uðtÞ is the acceleration, F V ðtÞ is the applied pulse loading in the vertical direction. Eq. (2) can be normalized as _ þ o2n uðtÞ ¼ € þ 2Bon uðtÞ uðtÞ
F v ðtÞ m
(3)
ARTICLE IN PRESS G.W. Ma et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 946–951
where B ¼ c=cr is the damping pffiffiffiffiffiffiffiffiffiffi ratio, cr ¼ 2mon is the critical damping coefficient, on ¼ k=m is the natural frequency of the SDOF system. Consider a half-sine pulse loading expressed as ( P 0 sin ot; 0ptpT (4) F v ðtÞ ¼ 0; t4T where T is the duration of pulse loading, P0 is the amplitude of the loading, o is the loading frequency given by o ¼ p=T. Eq. (3) is solved for the displacement uðtÞ subjected to the initial conditions _ ¼ u_ 0 , where u0 and u_ 0 are the initial of uð0Þ ¼ u0 and uð0Þ displacement and velocity, respectively. In the current study, the initial condition is assumed as u0 ¼ 0 and u_ 0 ¼ 0. The differential Eq. (3) can be solved by using the convolution integral method (or so called Duhamel’s integral method) [11,12] or the Laplace transform and the inverse Laplace transform method [13–15]. The response of the SDOF system consists of two phases: (1) Forced vibration phase. During this phase, the system is subjected to a half-sine pulse loading. The response of the system is given by uðtÞ ¼ eBon t ðA cos oD t þ B sin oD tÞ þ ðC sin ot þ D cos otÞ; tpT
(5a)
where oD isp the damped natural frequency of vibration, expressed ffiffiffiffiffiffiffiffiffiffiffiffiffi ffi as oD ¼ on 1 B2 , the first two items in the right hand side of Eq. (5a) are the homogeneous solution with the coefficients A and B to be determined by satisfying initial conditions, i.e. uð0Þ ¼ u0 _ and uð0Þ ¼ u_ 0 , the last two items in the right hand side are the particular solution, with coefficients C and D expressed as C¼
D¼
P0 1 ðo=on Þ2 mk ½1 ðo=on Þ2 2 þ ½2Bðo=on Þ2
(5b)
P0 2Bo=on mk ½1 ðo=on Þ2 2 þ ½2Bðo=on Þ2
(5c)
For a case study, the dimension of the block is assumed to be 0.4 m 0.2 m with a unit thickness. The density, Young’s modulus, and friction angle are 2650 kg/m3, 80 GPa, and 301, respectively. With the gravity load, the static friction is F hs ¼ mg tan f ¼ 1:224 kN
3
(2) Free vibration phase. After the pulse loading ends at time T, the system undergoes damped free vibration with the solution given by uðtÞ ¼ eBon ðtTÞ uðTÞ cos oD ðt TÞ _ uðTÞ þ Bon uðTÞ þ sin oD ðt TÞ t4T (6)
Dynamic frictional force Static frictional force
2.5 2 1.5 1.224 1 0.5 0.29 0 0
oD
1
2
3
4
5
6
Time (ms)
(7)
where F hd is the dynamic friction in the horizontal direction, f is the static friction angle, N is the normal force at the interface between the block and ground, varies with time t, expressed as NðtÞ ¼ kuðtÞ, where k is the equivalent stiffness of the block. Here, F hd is referred to as the dynamic friction because its value varies with time, not because the block is already moving. In the dynamic case, a pulse loading is applied in the vertical direction, and its effect on the friction in the horizontal direction is reflected by the variation of the normal force N. In Eq. (7), we choose the Mohr–Coulomb model among several available friction models to describe the relationship between the normal force and friction because this model has been widely used in rock mechanics and the parameters involved in this model can be easily determined by experiments. In fact, no matter what kind of friction model we use, similar results will be obtained.
Fig. 3. Comparison between static friction force and dynamic friction force for a single-block system.
Smallest force to move the block (kN)
_ where uðTÞ and uðTÞ are the displacement and velocity of the block at time of t ¼ T, respectively, obtained from Eqs. (5a) and (5b). Once the vertical displacement response is obtained, the friction between the block and ground can be obtained as F hd ¼ NðtÞ tan f
(8)
where m is the mass of the block, and g is the gravitational acceleration. The dynamic friction force from Eq. (7) and the static friction force from Eq. (8) are plotted and compared in Fig. 3 for the case with pulse amplitude of P 0 ¼ 1:5 kN and the damping ratio of B ¼ 0:1. For a static case, the friction force is a constant, i.e. 1.224 kN. The block will move in the horizontal direction if the pushing force exceeds 1.224 kN. For a dynamic case with a pulse loading applied in a vertical direction, the dynamic friction force vibrates around the static friction force. The block will move in the horizontal direction as soon as the pushing force exceeds the minimum dynamic friction force, which is 0.29 kN, decreased by 76% from the static value. This case shows that with a vertical pulse loading, we can move the block in the horizontal direction more easily with a much smaller pushing force. The reason is the vibration of the block in the vertical direction induced by the applied pulse loading. Then, the pulse width effect and the pulse amplitude effect are investigated by varying the loading frequency ratio b from 0.05 to 6.95 and the pulse amplitude from 0 to 1.9 kN. Here, the loading frequency ratio b is defined as the loading frequency divided by the natural frequency of the block system, expressed by b ¼ o=on, where o is the loading frequency, calculated by o ¼ p=T, T is the duration of the pulse loading. The relationships between the smallest force required to move the block (smallest force in short
Frictional force (kN)
948
1.2 1 0.8
Pulse amplitude: 0.7kN
0.6
1.0kN 1.5kN
0.4 Critical loading frequency ratio
0.2 0
0.73 1
2
3
4
5
6
7
Loading frequency ratio Fig. 4. Relationship between smallest force to move the block and the loading frequency ratio for a single-block system.
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hereafter) and the loading frequency ratio b for the three pulse amplitudes of 0.7, 1.0 and 1.5 kN are compared in Fig. 4. All the three curves firstly decrease with the increase of the loading frequency, reach their minimum values, and then gradually increase. It is worth noting that all the three curves reach their minimum values at a same loading frequency ratio of b ¼ 0:73. The smallest force required to move the block is 1.244 kN for the static case; however, for the dynamic cases with the loading frequency b ¼ 0:73, the smallest force to move the block decreases to 0.788 kN for the pulse amplitude of 0.7 kN, 0.601 kN for the amplitude of 1.0 kN, and 0.29 kN for the amplitude of 1.5 kN, respectively. Further study shows that for the dynamic cases with b ¼ 0:73, the smallest force required to move the block decreases linearly with the increase of the pulse amplitude. In order to examine the influence of the pulse shape effect, three kinds of pulse loadings, namely, triangular, half-sine and rectangular pulses with various loading frequency ratios b from 0.05 to 6.95 and various pulse amplitudes from 0 to 1.9 kN are applied to the block system. The triangular loading can be expressed as 8 2P 0 T > > t; tp > > 2 < T 2P 0 T F V ðtÞ ¼ (9) t; otpT 2P > > > 0 T 2 > : 0; t4T and the rectangular loading is given as ( P 0 ; 0ptpT F V ðtÞ ¼ 0; t4T
uðtÞ ¼
3 X
fn qn ðtÞ
949
(12)
n¼1
Once the vertical displacement response of each block is obtained, the dynamic friction between any two blocks is subsequently derived. As a case study, the dimension of each block is again assumed to be 0.4 m 0.2 m with a unit thickness. The density, Young’s modulus, and friction angle are again 2650 kg/m3, 80 GPa, and 301, respectively. Without applying the pulse load, the static friction forces for the top block, the middle block and the bottom block are F s1 ¼ m g tan f ¼ 1:224 kN F s2 ¼ 3 m g tan f ¼ 3:672 kN F s3 ¼ 5 m g tan f ¼ 6:120 kN
(13)
respectively, where m is the mass of each block. Fig. 5 shows the dynamic friction forces of the three blocks with pulse amplitude of P0 ¼ 3:0 kN, damping ratio of B ¼ 0:1.
(10)
Similar results have been obtained for the three kinds of loadings with different pulse shapes. There exist critical frequency ratios for all the three cases. However, the critical loading frequency ratio varies with the pulse shape, which is 0.67 for the triangular loading, 0.73 for the half-sine loading, and 1.00 for the rectangular loading, respectively. As a parametric study, the damping effect is also investigated. It is found that the damping ratio has very little effect on the critical frequency ratio, which is probably because the viscous damping does not have enough time to make a strong influence on the critical force for the short-duration transient problem considered in the current study.
4. Dynamic friction mechanism of a multiple-block rock system The dynamic friction mechanism of a multiple-block rock system is subsequently investigated in this section. Without loss of generality, a rock block system with three blocks is studied here. As discussed in Section 2, the vertical response of a threeblock rock system can be simplified into a three-degree-offreedom (3DOF) mass–spring–dashpot system described by the equilibrium equation given by Eq. (1) with the rank of each matrix as three. Solving the zero-damping eigenvalue equation gives the natural frequencies on ðn ¼ 1; 2; 3Þ and the corresponding mode shapes fn ðn ¼ 1; 2; 3Þ. Then, Eq. (1) can be decoupled in modal coordinates as M n q€ n ðtÞ þ C n q_ n ðtÞ þ K n qn ðtÞ ¼ Pn ðtÞ
fTn mfn
(11)
fTn cfn
where, M n ¼ is the generalized mass, C n ¼ is the T generalized damping, K n ¼ fn kfn is the generalized stiffness, and T P n ¼ fn pðtÞ is the generalized loading. Following the similar procedure with the SDOF system, qn ðtÞ ðn ¼ 1; 2; 3Þ can be obtained. Then, the displacements for each block can be calculated as
Fig. 5. Dynamic friction and contribution of each mode for a three-block system: (a) top block; (b) middle block; and (c) bottom block.
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G.W. Ma et al. / International Journal of Rock Mechanics & Mining Sciences 46 (2009) 946–951
Similar to the phenomenon observed for the single-block system in Section 3, the dynamic friction force for each block vibrates around their corresponding static friction force. The smallest force required to move each block decreases from their static friction force given in Eq. (13) to the minimum value of the dynamic friction force shown in Fig. 5. It becomes easier to move the blocks in the horizontal direction because of the vibration of the blocks in the vertical directions induced by the applied pulse loading. Fig. 5 also shows the first mode has the most significant contribution to the system response while the higher order modes (i.e. the second mode and the third mode) has only some influences on the first peak of the vibration and very little influence on the following vibrations, and the smallest force required to move the block is governed by the first vibration mode of the MDOF system. Parametric studies with respect to the pulse width, pulse amplitude and pulse shape have also been conducted. In order to investigate the pulse width effect, a series of pulse loadings with identical amplitude but different pulse frequencies are applied to the multiple-block system. Fig. 6 shows the relationship between the smallest force required to move each block and the loading frequency ratio b, which is defined as b ¼ o=o1 , where o is the loading frequency, o1 is the first natural frequency of the MDOF system. It is found that each curve reaches their minimum value at a critical loading frequency ratio, which is also 0.73 for a half-sine pulse loading, same as that observed for the single-block system. In order to investigate the influence of the pulse amplitude, a series of pulse loadings with an identical loading frequency but different amplitudes are applied to the system. The smallest force required to move each of the three blocks decreases linearly with the pulse amplitude, which also has similar trend with the result of the single-block system. The pulse shape effect is studied by using the three kinds of pulse loadings, i.e. the triangular loading, the half-sine loading, and the rectangular loading. Again it is found that there exist critical loading frequency ratios for all the three loadings which are the same as the corresponding results of the single-block system, i.e., 0.67 for the triangular loading, 0.73 for the half-sine loading, 1.00 for the rectangular loading. The only difference is the definition of the loading frequency ratio. For a single-block system, the ratio is defined as the loading frequency over the natural frequency of the system, while it is defined as the loading frequency over the first natural frequency for a multiple-block system.
5. Horizontal displacement
Smallest force to move each block (kN)
Newmark’s block sliding theory [9,10] was originally proposed to predict the displacement of a block subjected to seismic 7 Bottom block
6
Middle block Top block
5 4 3 2 1
Critical loading frequency ratio
0 0
0.73 1
2
3 4 Loading frequency ratio
5
6
7
Fig. 6. Relationship between smallest force to move each block and loading frequency ratio for a three-block system.
motion. We extend it here to determine the horizontal displacement of blocks in the lateral direction. An example for the displacement prediction by Newmark’s block sliding theory is shown in Fig. 7, where a half-sine loading with an amplitude of P 0 ¼ 3:4 kN, a loading frequency ratio of b ¼ 0:74, and a horizontal constant pushing force of F H ¼ 3:3 kN are used. The constant pushing acceleration (i.e. pushing force divided by the mass of the block) and the varied frictional acceleration (i.e. dynamic friction force divided by the mass of the block) are plotted in Fig. 7a. When the pushing acceleration is larger than the frictional acceleration (region indicated by gray color below the straight dash line in Fig. 7a), the velocity of the block increases from zero (Fig. 7b) and the block starts to move with the displacement increasing (Fig. 7c). The velocity of the block can be calculated by integrating the acceleration over time in the region marked by gray color while the permanent displacement is obtained by integrating the velocity over time. When it comes to the intersection point between the two acceleration curves in Fig. 7a, the velocity reaches its first peak. After that, the pushing acceleration becomes smaller than the frictional acceleration (region marked by gray color above the straight dash line in Fig. 7a), the velocity of the block decreases but the displacement keeps increasing until the velocity decreases to zero. When the pushing acceleration becomes larger than the frictional acceleration again, the velocity increases from zero again, the block continues moving in the horizontal direction. When the pushing acceleration is smaller than the frictional acceleration, the velocity decreases while the displacement increases until the velocity reduced to zero. The above process repeats until the pushing acceleration is no longer larger than the frictional acceleration. The velocity of the block becomes zero while the horizontal displacement reaches its maximum value. Parametric studies with respect to both the horizontal pushing force and the pulse loading are carried out. The maximum horizontal displacement increases with both the pushing force and the pulse loading amplitude.
6. Conclusions In the present study, the dynamic friction mechanism of blocky rock system has been analytically investigated by adopting a multiple-degree-of-freedom mass–spring–damper system. In a static case, the friction force applied on a rock block is a constant value, and the rock block can move when the pushing force exceeds this value. However, in a dynamic case with a pulse loading applied, the dynamic friction force will vibrates around the static friction force, and the rock block can move as soon as the pushing force is lager than the minimum dynamic friction force, which is definitely smaller than that in the static case. In short, with a pulse loading applied in the axial direction, we can move the block in the lateral direction with a much smaller pushing force than the static case. The result is consistent with the experimentally observed ultra low friction phenomenon. It is also found that it is easiest to move the block in the lateral direction when the loading frequency ratio b reaches a critical value, which is 0.67 for a triangular loading, 0.73 for a half-sine loading, and 1.0 for a rectangular loading. Same results are obtained for both a single-block system and a multiple-block system. With the same loading frequency, it will be relatively easier to move out a block with a higher loading amplitude. The damping ratio has very minor influences on this phenomenon. Lastly, Newmark’s block sliding theory is extended to determine the displacement of each block in the horizontal direction. Analytical results presented in this paper demonstrate that the pulse loading or the external disturbance in an orthogonal direction has a significant influence
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951
Horizontal acceleration (m/s2)
50 Acceleration by frictional force
40
Acceleration by horizontal pushing force
30 20 10
Horizontal velocity (mm/s)
0 0
1
2
3
4
0
1
2
3
4
5 Time (ms)
6
7
8
9
10
10
5
0
Horizontal displacement (um)
-5 5 Time (ms)
6
7
8
9
10
8 6 4 2 0 0
1
2
3
4
5 Time (ms)
6
7
8
9
10
Fig. 7. Horizontal displacement prediction based on Newmark’s sliding block theory: (a) acceleration time history; (b) velocity time history; and (c) displacement time history.
on stability of a blocky rock system. So, it is essential to take this factor into account when carrying out stability analysis of slopes and underground excavations. The traditional stability method based on static analysis (e.g., block theory) may underestimate the instability of a blocky system. References [1] Murakami O, Yokoo A, Ohnishi Y, Nishiyama S. Study on application of key block theory to rock engineering projects. In: Proceedings of the eighth international conference on analysis of discontinuous deformation, Beijing, 2007. p. 281–4. [2] Ahn SH, Lee CI. Removability analysis of rock blocks by block theory and a probabilistic approach. Int J Rock Mech Min Sci 2004;41(3):429. [3] Goodman RE, Shi GH. Block theory and its application to rock engineering. Englewood Cliffs, NJ: Prentice-Hall; 1985. [4] Yeung MR, Jiang QH, Sun N. Validation of block theory and three-dimensional discontinuous deformation analysis as wedge stability analysis methods. Int J Rock Mech Min Sci 2003;40:265–75. [5] Zhang QH, Wu AQ. Study on geometrical identification of stochastic block in block theory. In: Proceedings of the eighth international conference on analysis of discontinuous deformation, Beijing, 2007. p. 255–63.
[6] Wang HL, Ge T, Wang DR, Lu YS, Wang MY, Qian QH. Comparison of theoretical and experimental analysis of dynamic characteristics of block rock mass. Chin J Rock Mech Eng 2007;26(5):952–8 (in Chinese). [7] Wang HL, Zhou ZP, Wang MY. Study on characteristics of block surface. Protective Eng 2006;28(2):30–5 (in Chinese). [8] Wang MY, Qi CZ, Qian QH. Study on deformation and motion characteristics of blocks in deep rock mass. Chin J Rock Mech Eng 2005;24(16):2825–30 (in Chinese). [9] Newmark NM. Effect of earthquakes on dams and embankments. Geotechnique 1965;15(2):787–830. [10] Cai Z, Bathurst RJ. Deterministic sliding block methods for estimating seismic displacements of earth structures. Soil Dynam Earthq Eng 1996;15: 255–68. [11] Clough RW, Penzien J. Dynamics of structures. 2nd ed. New York: McGrawHill; 1993. [12] Chopra AK. Dynamics of structures: theory and applications to earthquake engineering. 2nd ed. Englewood Cliffs, NJ: Prentice-Hall; 2001. [13] Schiff JL. The Laplace transform. New York: Springer; 1999. [14] Shu DW, Shi BJ, Meng H, et al. The pulse width effect of single half-sine acceleration pulse on the peak response of an actuator arm of hard disk drive. Mater Sci Eng A 2006;423:199–203. [15] Shu DW, Shi BJ, Meng H, et al. Shock analysis of a head actuator assembly subjected to half-sine acceleration pulses. Int J Impact Eng 2007;34:253–63.