Can tilt tests provide correct insight regarding frictional behavior of sliding rock block under seismic excitation?

Engineering Geology 122 (2011) 84–92

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Engineering Geology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e n g g e o

Can tilt tests provide correct insight regarding frictional behavior of sliding rock block under seismic excitation? Yo-Ming Hsieh a, Kuo-Chen Lee b, Fu-Shu Jeng b,⁎, Tsan-Hwei Huang b a b

Department of Construction Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan Department of Civil Engineering, National Taiwan University, Taipei, Taiwan

a r t i c l e

i n f o

Article history: Received 4 February 2010 Received in revised form 15 November 2010 Accepted 30 November 2010 Available online 15 December 2010 Keywords: Friction Seismic Earthquake Landslides Tilt test Shaking table test

a b s t r a c t Earthquake induced dip-slope sliding or rock-block sliding is usually analyzed using friction angles or friction coefficients measured at the sliding interface. A tilt test is a convenient test for measuring the required friction angle. However, a tilt test is a test under static conditions, and the applicability of measured friction parameters to analyze slopes under dynamic excitation requires further discussion. This study conducts a static tilt test and a dynamic shaking table test to simulate block sliding with base excitation, compares differences in measured sliding thresholds, and discusses the cause of these differences. Tests on three different materials (aluminum, sandstone, and synthetic sandstone) show that friction coefficients measured by tilt tests are always larger than the ones derived by shaking table tests. Moreover, high frequency tests yield larger friction coefficients, suggesting the sliding threshold is non-constant under excitation. In addition, tests with varying normal stresses on the sliding block show that with increasing contact stresses, sliding thresholds decrease, implying that sliding threshold varies with normal stress. Instantaneous friction coefficients, μi(t), during sliding are also studied in this work. It has been found that frictional behavior of synthetic sandstone deviates from the idealized Coulomb friction model. The instantaneous friction coefficient varies with relative displacement and relative velocity during sliding. Finally, reasons for differences between a static tilt test and a dynamic shaking table test are discussed. This study preliminarily identifies the limitations of the tilt test when applied to dynamic problems, and concludes that realistic sliding thresholds can only be obtained using dynamic tests such as shaking table tests. © 2010 Elsevier B.V. All rights reserved.

1. Introduction The friction angle at the sliding interface is a key parameter for analyzing earthquake-induced slope failures. One of the most damaging types of earthquake-induced landslide is slope-failure in dip slopes, which can cause massive damages and casualties, as seen in Tsaoling and Jiufengershan during Chi-Chi earthquake in 1999 (Kamai et al., 2000; Huang et al., 2002; Wang et al., 2003; Chang et al., 2005a). Both infinite-slope and finite-slope models are often used for analyzing safety factors in these problems, and influences of earthquakes are often considered through a pseudo-static method (Terzaghi, 1950; Chen et al., 2003; Jeng et al., 2004a) or a blocksliding method (Newmark, 1965; Wilson and Keefer, 1983; Jibson and Keefer, 1993; Kramer and Matthew, 1997; Wartman et al., 2000; Huang et al., 2001; Ling, 2001; Bray and Travasarou, 2007; Kokusho and Ishizawa, 2007; Chiang et al., 2009; Dong et al., 2009). All these methods use friction angle or friction coefficient at the sliding interface to deduce the sliding threshold.

⁎ Corresponding author. Tel./fax: +886 2 2364 5734. E-mail address: [email protected] (F.-S. Jeng). 0013-7952/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.enggeo.2010.11.008

Friction angle or friction coefficient obtained from a tilt test is questionable when applied to dynamic problems. For slope stability analyses considering only the gravity effect, i.e. no base excitation or static loading condition, a tilt test is a convenient and reasonable test method for measuring friction coefficient or friction angle (Barton and Choubey, 1977). However, its applicability in Newmark analysis for determining sliding threshold requires further investigation. This study intends to identify quantitatively the difference between measured versus sliding thresholds for a block on a slope subjected to base excitation through small-scale laboratory experiments. It is often found that friction coefficients obtained using backanalyses on large-scale earthquake-induced rockslides on dip slopes tend to be smaller than the ones obtained using laboratory tests such as direct shear test, cyclic shear test or tilt test (Hencher, 1980; Barbero et al., 1996; Shou and Wang, 2003; Jafari et al., 2004; Chang et al., 2005b). Besides the difference in the scale (large versus small scale), the deviation in loading conditions (dynamic versus static loading) may also play a role in the difference between backcaculated analysis and laboratory measurements of friction angles. This will be investigated in this work. Furthermore, the applicability of tilt test measured friction coefficients in determining the sliding threshold of earthquake-

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induced landslides is discussed. The discussion is facilitated by comparing the sliding thresholds obtained from those 1) measured with the proposed small-scale shaking table tests versus 2) tilt tests. Both tests have similar scales of normal stresses, eliminating the scale effect, and helping to focus on the effect of static versus dynamic loading conditions. 2. Methodology In this work, two types of experiments were conducted: 1) tilt test and 2) small-scale shaking table test to facilitate the discussion of the applicability of the frictional parameters obtained using a tilt test in analyzing earthquake-induced rockslides and rock block slides. In the following sections: 1) these two types of experiments are briefly described, 2) frictional parameters obtained from these types are derived from measurements, and 3) overall laboratory test programs are presented. Furthermore, this paper defines the sliding threshold as the friction angle at sliding. 2.1. Tilt test Barton and Choubey (1977) proposed a tilt test to measure the maximum static friction angle ϕt at the sliding interface under low normal stress condition. The friction coefficient μt can then be obtained by μt = tanϕt. This study follows the same procedure to obtain these parameters and to compare with the ones obtained from the next test. It is noted the friction coefficient μt derived from the maximum static friction angle ϕt is the sliding threshold obtained using tilt tests. 2.2. Shaking table test The setup of the shaking table test, shown in Fig. 1, follows related works (Yegian and Lahlaf, 1992; Wartman et al., 2003; Park et al., 2006; Mendez et al., 2009) that induce sliding between the block and the base block using the shaking table. The shaking table has controlled excitation through the servo-control system with fixed frequency of sinusoidal vibration ranged from 1.0 Hz to 9.0 Hz; the bearing allows a free adjustment to the slope angle; and the lubricated sliding guide ensures the sliding block moves one-dimensionally on the base block. Auxiliary uniaxial accelerometers (PCB-3711D1FA3G) at A1, A2, and A3 positions record temporal accelerations of the sliding block, the base block, and the rigid plate driven by the servocontrol system. High precision laser displacement sensors (Keyence

Fig. 2. Some critical quantities used in this work: (a) the critical acceleration, Ac, defined by Newmark (1965) and Ẍ ðt Þ is the temporal acceleration; (b) to define the initiation time of sliding, tc, and Xr(t) is the temporal relative displacement of the sliding block to the base block; (c) to define instantaneous critical friction coefficient, μ c = μ i(tc), as the measured instantaneous friction coefficient at the initiation of sliding, i.e. t = tc.

LK-G155) with accuracy of 0.01 mm are installed in L1 and L2 positions to measure temporal displacement of the base and the sliding block on the base block, respectively. Rather than using the relative acceleration (A2−A1) between the sliding block and the base block to define the initiation time of sliding, tc, this study proposes the use of direct measurement of relative displacement by L2 to obtain tc. Through laboratory tests conducted by authors, the proposed method identifies the tc more accurately

Fig. 1. The setup of shaking table test and instrumentation used in this study.

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Fig. 3 illustrates the force equilibrium for the sliding block on the ! slope, in which: N is the normal force exerted on the slope by the sliding block, m is the mass of the sliding block, g is the gravitational ⋅⋅ → acceleration, θ is the slope angle, Xg ðt Þ is the absolute temporal ⋅⋅ → acceleration at base and Xb ðt Þ is the absolute temporal acceleration of the sliding block. ! The derivation of Ac starts from Eq. (1) to compute N , and then substitute into Eq. (2), which states the driving force equals the resistance force, which is satisfied at initiation of sliding. By rearranging Eq. (2), the Ac can then be obtained as in Eq. (3).  
! ! ! ̈ N = m g cosθ + Xg ðt Þ sinθ

ð1Þ


!  : !
! ! ̈ m g sinθ−m Xg ðt Þ cosθ = μc N sign Xr ðt Þ

ð2Þ

Fig. 3. Illustration of the force equilibrium for the sliding block on a slope.


!   ̈  Ac =  Xg ðtc Þ = than the former method when the relative displacement between the sliding block and base block is small. Furthermore, it should be emphasized that the temporal accelerations at the excitation base and the sliding block are also derived from these temporal displacement measurements (by taking time derivatives twice). The friction angle and friction coefficient derived (to be detailed in Section 2.3) at tc are then the sliding thresholds defined using this test.

2.3. Instantaneous friction coefficient μ i(t) Fig. 2 illustrates some critical quantities used in this work. Instantaneous friction coefficient, μ i(t), defines temporal friction coefficient derived from shaking table tests and differentiates itself from μt measured from tilt tests, as proposed by following Chaudhuri and Hutchinson (2005) and Mendez et al. (2009). This work additionally defines instantaneous critical friction coefficient, μ c = μ i(tc), as the measured instantaneous friction coefficient at the initiation of sliding, i.e. t = tc. The critical acceleration, Ac, defined by Newmark (1965), is the acceleration measured at t = tc. These quantities are derived from measurements of the shaking table test using the formulations described in the subsequent paragraphs.

! ! μ g cosθ− g sinθ c  ! ! − g cosθ + μc g sinθ

ð3Þ

The derivation of μ c = μ i(tc) can be similarly obtained by the force equilibrium in the x direction, as in Eq. (4), and then rearrange Eq. (4) to obtain μ i(t): :  !
!
! ! ̈ ̈ m g sinθ−m Xg ðt Þ cosθ−μi ðt Þ N sign xðt Þ = m Xr ðt Þ

ð4Þ


! 
! ! ̈ ̈ m − Xg ðt Þ cosθ− Xb ðt Þ + g sinθ
! ; in which Xb̈ ðt Þ μi ðt Þ ¼ ! N
!
! = X ̈ ðt Þ + Ẍ ðt Þ

ð5Þ

g

r

2.4. Laboratory test program Fig. 4 shows the complete laboratory test program for this study, and how each test result is used to discuss various aspects of friction behavior during excitation. It is seen in Fig. 4 that there are three primary aspects studied in this work: 1) differences of μ c and μ t, 2) factors affecting μ c, and 3) factors affecting μ i(t).

Fig. 4. Schematic illustration of the laboratory test program of this research.

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be used to discuss if friction measured in a static test can be applied to dynamic loading conditions.

Table 1 Friction coefficients of different specimens. (a) Measured by tilt test Type

μt

Aluminum MS4 sandstone Synthetic sandstone

0.35 0.45 0.47

0.33 0.44 0.47

0.36 0.45 0.47

0.35 0.45 0.46

0.32 0.43 0.48

Mean

STD

0.34 0.44 0.47

0.044 0.013 0.015

Mean

STD

0.26 0.19 0.21

0.006 0.010 0.012

(b) Measured by shaking table Type

μc

Aluminum MS4 sandstone Synthetic sandstone

0.26 0.18 0.20

0.25 0.18 0.22

0.27 0.18 0.20

0.26 0.21 0.22

0.27 0.19 0.20

2.4.1. Differences of μ c and μ t The differences of μ c and μ t are identified by conducting both a tilt test and a shaking table test on selected materials: aluminum, MS4 sandstone (Jeng et al., 2004b), and synthetic sandstone (Jafari et al., 2003). In addition, the dimension of the base block is 180 mm (L) × 160 mm (W) × 30 mm (H), and that of the sliding block is 100 mm × 100 mm × 30 mm, respectively. The surfaces of each block were polished by grinder, and the physical and mechanical properties of specimen are summarized in Table 3. The friction coefficients obtained can be then compared and discussed to understand the effect of dynamic excitation on sliding threshold in selected materials. Identifying differences between μ c and μ t clarifies whether the sliding threshold obtained in a tilt test is the same as the threshold obtained under base excitation conditions. These results can further

2.4.2. Factors affecting μ c In this study, excitation frequencies and normal stresses are two studied factors that influence μ c based on shaking table tests. Previous studies show frictional behaviors of sliding blocks are affected by: material (Dieterich, 1972; Byerlee, 1978), roughness (Jafari et al., 2003), normal stress (Kato and Hirasawa, 1996; Masao and Miromine, 1996), duration of contact (Dieterich, 1979), and temperature (Atkinson, 1980; Blanpied et al., 1991). Results measured from varying excitation frequencies can suggest if characteristics of excitation would affect friction behaviors. Previous studies show the dominant frequency range of earthquake depends on characteristics of the source and propagation medium. Measurements of earthquake motions on rock sites indicate that dominant frequencies are normally in the range of 0.1 Hz to 10.0 Hz (Ashford and Sitar, 2002; Bhasin and Kaynia, 2004). Therefore, the designed tests were performed with three distinct excitation frequencies 4 Hz, 6 Hz, and 8 Hz with sine wave form under slope angles θ of 0°, 5°, and 10° to see if effects of excitation frequencies vary with slope angles. All of the tests were conducted using synthetic sandstones for their excellent repeatability and close resemblance to real sandstones. Normal stresses at the sliding interface also vary by adding weight on the sliding block. Three different normal stresses are applied: 1.00σn, 1.93σn, and 3.52σn, where σn = (σn)A1 is the normal stress exerted on the slope by the sliding block and added weight (Table 2). The tests were conducted at slope angle θ of 0° under excitation frequencies of 4 Hz, 6 Hz, and 8 Hz. The test results will suggest how significantly normal stresses can affect measured μ c. 2.4.3. Factors affecting μ i(t) In addition to understanding how sliding thresholds are affected by various factors, factors affecting μ i(t) are also studied. Understanding μ i(t) is important to estimate reasonably final accumulative permanent sliding displacements of blocks. The tests were designed to observe how dynamic friction coefficient changes with developing relative displacements and relative velocities. The measured friction coefficients are also compared with the Coulomb friction model. 3. Differences on μt and μc Table 1 shows both measured μt and μc on aluminum, sandstone, and synthetic sandstone. Reasonable repeatability of five different measurements on each selected material can be observed. These specimens are then tested using the shaking table to measure μ c with

Fig. 5. μt and μc of aluminum, sandstone, and synthetic sandstone, and these results show that the μc is less than μt in different material specimens (θ = 0°).

Fig. 6. The measured μc/μt was affected by excitation frequency of 4 Hz, 6 Hz, and 8 Hz (Synthetic sandstone).

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Y.-M. Hsieh et al. / Engineering Geology 122 (2011) 84–92 Table 3 Physical and mechanical properties of specimen. Properties

Unit

Aluminum

MS4 sandstone

Synthetic sandstone

Apparent specific gravity Uniaxial compressive strength Young's modulus Poisson's ratio Roughness (JRC range) a

– MPa

2.71 –

2.65 23.20

2.40 17.80

GPa – –

74.00 0.31 0

2.61 0.27 0

1.38 0.23 0

Remark: a The surfaces of these three kinds of speciment are polished by grinder.

5. Observed μ i(t) after initiation of sliding Fig. 7. The measured μc/μt was affected by normal stresses (σn)A1, (σn)A2, and (σn)A3 (θ = 0°).

θ = 0° and excitation frequency of 4 Hz. From these tests, it is seen in Fig. 5 that the measured μ c is consistently smaller than μ t, i.e., μ c/μ t is around 76% for aluminum, 43% for sandstone, and 45% for synthetic sandstone. Moreover, sandstone and synthetic sandstone show larger deviation between μ c and μ t with reduction close to 50%. These results suggest that the sliding threshold can be reduced by the presence of base excitation.

4. Factors affecting μc How excitation frequency and normal stress affect the sliding threshold is studied through a series of tests with synthetic sandstone. The effect of excitation frequency is studied at three different slope angles: 0∘, 5∘, and 10∘ (which are smaller than the measured friction angle, ϕt = tan − 1(μ t)). At each slope angle, three different frequencies 4 Hz, 6 Hz, and 8 Hz are used in the shaking table test to measure μc. These results are summarized in Fig. 6 showing how excitation frequencies affect the measured μ c/μ t. It is seen in Fig. 6 that μ c/μ t is strongly related to the excitation frequency regardless the slope angle, and μc/μ t increases with excitation frequency. Higher excitation results in higher sliding threshold. From these experiments, an increase of 2 Hz in excitation frequency causes an increase in μ c/μ t by 5%–7%. Three different normal stresses are used to study the effect of normal stresses on μ c by adding mass blocks on top of the sliding block. These normal stresses (σn)A1, (σn)A2, and (σn)A3 are respectively 1.0, 1.93, and 3.52 times the normal stress, σn, exerted on the slope by the sliding block and adding weight, and their corresponding μ t are summarized in Table 2. With three different normal stresses, shaking table tests are then conducted with θ = 0° and three different excitation frequencies 4 Hz, 6 Hz, and 8 Hz. The test results are summarized in Fig. 7 and Table 2, and show that increasing normal stresses decreases μ c/μ t regardless of the excitation frequency. With each increase of normal stress by 100%, μ c/μ t is decreased by roughly 5%.

Table 2 Static friction coefficients of synthetic sandstone under different normal stresses. Different normal stresses (σn)A1 (σn)A2 (σn)A3

block block + 580 g (weight) block + 1580 g (weight)

Mass (g)

Normal stress (kPa)

μt

628.1 1211.2 2212.9

0.616 1.187 2.169

0.468 0.475 0.464

After the block starts to slide on the slope, its frictional behavior μ i(t) controls the final relative displacement of the sliding block relative to the base block. The measured μ i(t) on synthetic sandstones is compared with the Coulomb friction model to understand the validity of such an idealized model. The tests were conducted with excitation frequency of 4 Hz and horizontal slope θ = 0°. The relationship between μ i(t) and Vr(t), the temporal relative velocity of the sliding block to the base block, is depicted in Fig. 8. Fig. 8(a) and (b) shows the measured Vr(t) and μ i(t). The interpretation using the ideal Coulomb friction model is shown in Fig. 8(c). The model assumes only one constant friction coefficients that are independent from Vr(t), and therefore Fig. 8(c) shows that the μ i(t) stays constant with changing Vr(t). However, based on measured data shown in Fig. 8(a) and (b), μ i(t) versus Vr(t) can be derived as Vr(t) shown in Fig. 8(d). It is seen after the sliding block reaches the sliding threshold of μ c = 0.25 (calculated) at point A, μ i(Vr) first rapidly increases, and then stays roughly constant to point B. It then continues to point C with decreasing Vr(t), and μ i(Vr) decreases nonlinearly. This deviates from the Coulomb friction model significantly. Points D through F show behavior similar to that in points A through C. From these observed behaviors, it is suggested the friction coefficient is not constant with respective to the relative velocity along the slope. Similarly, the relationship between μ i(t) and s(t), the temporal relative displacement of the sliding block to the slope, is depicted in Fig. 9. Fig. 9(a) and (b) shows the measured s(t) and calculated μ i(t). Their interpretation using the ideal Coulomb friction model is again depicted in Fig. 9(c), which shows a constant friction coefficient during sliding. The calculated μ i(s) is shown in Fig. 9(d). Before point A, s(t) = 0, i.e. there is no sliding, and the μ i(t) increases from zero to the sliding threshold μ c. From point A to point C, close to linear increase in μ i(s) is shown; at reverse motion, highly linear μ i(s) can be observed. These observations suggest the Coulomb friction model cannot reasonably describe the measured behaviors. 6. Effect of excitation frequency on μi(t) Fig. 10(a) and (b) shows measured μi with respect to Vr and to s, respectively with slope angle θ = 10°, four excitation cycles, and two different excitation frequencies (4 Hz and 6 Hz). Consistent behaviors can be observed between different cycles under the same testing conditions. It is also seen that higher excitation frequency results in higher μ i(Vr), in Fig. 10(a). It can also be seen from Fig. 10(b) that the accumulative permanent displacement under low excitation frequency is larger than the one obtained with higher excitation frequency, consistent with the observation from Fig. 10(a). However, the reasons for the accumulative permanent displacement (or maximum relative velocity) under low excitation frequency is larger include two parts: 1) μ c, which is greater under high frequency

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Fig. 8. (a) Vr (t); (b) μi(t); (c) μi(Vr) of the Coulomb friction model (θ = 0°); (d) μi(Vr) of synthetic sandstone, which is measured by shaking table test (θ = 0°).

excitation in each cycle; 2) μi, which is also greater under high frequency excitation in each cycle. For these reseasons, we infered that higher frequency has higher μi, thus higher resistance to movements.

7. Discussion Through the presented experimental results, it is consistently shown the measured μc using shaking table tests is significantly

Fig. 9. (a) s(t); (b) μi(t); (c) μi(s) of the Coulomb friction model (θ = 0°); (d) μi(s) of synthetic sandstone, which is measured by shaking table test (θ = 0°).

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Fig. 10. The measured μ i(t) with respect to Vr and to s, respectively with slope angle θ = 10°, the first four excitation cycles of shaking table test, and two excitation frequencies (4 Hz and 6 Hz).

Fig. 11. Assume the normal stress distribution on the sliding interface is linear: (a) A slope under static condition; (b) A slope under seismic excitation condition.

smaller than μt measured in tilt tests. This suggests, under the same static normal stress, the additional inertia forces induced by excitation at the base could be the key factor causing μc ≪ μt. Therefore, this study sets out to identify how normal stress distribution on the sliding interface affects the μ c by a simple model described subsequently. Assume the normal stress distribution is linear, in Fig. 11, along the sliding interface, and the resultant normal force is the only factor that yields eccentricity, e. By using force equilibrium, the e can be derived for the following two conditions:

At initiation of sliding, Ẍr ðt Þ is 0. Therefore, ec, eccentricity at initiation of sliding, can be obtained using Eq. (11).

7.1. The sliding block on the slope at rest

ec =

Using Fig. 11(a), the normal force N acting on the slope by the sliding block is a component of gravity, as shown in Eq. (6). As the block does not rotate, the moment around the central point (chosen for convenience) of sliding interface should be zero, as in Eq. (7). Besides, h is the half height of the sliding block. The eccentricity e can then be obtained in Eq. (8) by substituting Eq. (6) into Eq. (7). mg cosθ = N

ð6Þ

M = mg sinθ × h = N × e

ð7Þ

estatic = e = tanθ × h

ð8Þ

7.2. The sliding block on the slope at excitation Using Fig. 11(b), the same derivation as in the previous case can be used with normal force in Eq. (10), moment equilibrium in Eq. (10), and eccentricity e in Eq. (11). mg cosθ + m Ẍg ðt Þ sinθ = N

ð9Þ

h  i mg sinθ−m Ẍg ðt Þ cosθ −m Ẍr ðt Þ × h = N × e

e=

½g sinθ− Ẍg ðt Þ cosθ− Ẍr ðt Þ × h g cosθ + Ẍg ðt Þ sinθ

½g sinθ− Ẍg ðt Þ cosθ × h g cosθ + Ẍg ðt Þ sinθ

ð10Þ

ð11Þ

ð12Þ

From Eq. (11) or Eq. (12), it is seen that e varies with the acceleration at base of excitation, Ẍg ðt Þ. Based on Eqs. (8) and (12), and Fig. 12(a) it can be estimated how variations of slope angles (0°, 5°, and 10°) and excitation frequencies affect the eccentricity of block at rest (estatic) and at excitation (ec, shaking frequency are 4 Hz, 6 Hz and 8 Hz). It is seen that ec is at least twice as much as estatic. Therefore, under the sliding block, the normal stress distribution at excitation is less uniformed than under the one at rest. The higher eccentricity causes larger eccentric moment of block, i.e. this means the block under base excitation less stable than the one at rest, because ec is greater than estatic. Furthermore, using the shaking table tests that were conducted, eccentricity versus μc can be plotted as in Fig. 12(b). It shows ec is larger than estatic, while μc is smaller than μt. As previously discussed, larger ec increases the non-uniformity of stress distribution, and this non-uniformity may explain why μc is smaller than μt. It should be emphasized, based on current results, that the effects of excitation frequency on ec and μc are yet to be identified, and so far no definite trends can be concluded from experimental data. Also,

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Additionally, based on experiments on aluminum, sandstone, and synthetic sandstone, it has been shown that the material strongly dictates the observed frictional behavior on both sliding threshold and instantaneous friction coefficient during sliding. Therefore, in order to establish rock block or rock slope sliding models during excitation, it is necessary to use a frictional model based on measured behaviors on real rock materials or synthetic rock materials. We plan to continue experiments on real rock specimens (e.g. sandstone, shale, siltstone, etc.) to build up representative mechanical models for rocks during sliding under excitation, which can be applied to analyses for real cases.

Fig. 12. (a) Variations of slope angles (0°, 5°, and 10°) and excitation frequencies affect the eccentricity at rest and at excitation; (b) Using conducted shaking table tests, eccentricity versus (μc/μt) * 100%.

other factors such as surface roughness, contact duration, temperature, material characteristics, etc. influencing friction behavior of sliding blocks at excitation require further discussion. It can only be concluded from this study that during excitation, the inertia forces change the uniformity of stress distribution under the sliding block, which results in a reduction of μc as compared to μt.

8. Conclusion Through this study, the applicability of a tilt test measured sliding threshold, μt, in dynamic analyses is clarified. The sliding threshold measured by a tilt test in static conditions is much higher than the sliding threshold in dynamic conditions measured by a shaking table test. Using μt in sliding block analyses under excitation underestimates the accumulative relative displacement due to the overestimated frictional resistance. Therefore, the sliding threshold should be measured using dynamic tests such as shaking table tests used in this study. This study suggests the reason for such overestimated sliding threshold measured by tilt tests is primarily due to the normal stress distribution under the sliding block. During excitation, the inertial forces of the sliding block cause non-uniformity of normal stresses, and indirectly cause reductions to friction coefficients and, thus, larger displacements. The authors will continue studying how normal stresses of different scales apply on sliding blocks, and researching excitation characteristics (such as excitation duration and direction of the base excitation such as vertical motion etc.) of sliding threshold and frictional behaviors during sliding. Although only a dry condition sliding surface is considered in this study, the influence of pore water pressure acting on the sliding surface is also a key issue of a natural slope. The study is going to be extended to consider a block sliding on a lubricated surface and a wet surface.

9. Nomenclature ϕt The static friction angle measured by tilt test; μt The friction coefficient is obtained by μt = tan ϕt; tc The initiation time of block sliding; μi(t) The instantaneous friction coefficient; μc The instantaneous critical friction coefficient and μc = μi(tc); Ac The critical acceleration, defined by Newmark (1965); g The gravitational acceleration; m The mass of the sliding block; ! N The normal force exerted on the slope by the sliding block; θ :: The slope angle;
! X:: g ðt Þ The absolute temporal acceleration at base;
! The absolute temporal acceleration of the sliding block; X:: b ðt Þ
! The relative temporal acceleration of the sliding block to the Xr ðt Þ :: :: ::
!
!
! base; block, Xr ðt Þ = Xb ðt Þ− Xg ðt Þ; σn The normal stress exerted to the slope by the sliding block and adding weight; Vr(t) The temporal relative velocity of the sliding block to the base block; μi(Vr) The instantaneous friction coefficient function of the temporal relative velocity of the sliding block to the base block; s(t) The temporal relative displacement of the sliding block to the slope; μi(s) The instantaneous friction coefficient function of the temporal relative displacement of the sliding block to the base block; e The eccentricity; ec The eccentricity at initiation of sliding; estatic The eccentricity of block at rest

Acknowledgements The authors wish to express their appreciation to the National Science Council of Taiwan for the financial support (project: NSC 972221-E-002-187 and NSC 97-2116-M-002-015), which makes this research possible. References Ashford, S.A., Sitar, N., 2002. Simplified method for evaluating seismic stability of steep slopes. Journal of Geotechnical and Geoenvironmental Engineering 128 (2), 119–128. Atkinson, B.K., 1980. Stress corrosion and the rate-dependent tensile failure of a FineGranite quartz rock. Tectonophysics 65, 522–557. Barbero, M., Barla, G., Zaninetti, A., 1996. Dynamic shear strength of rock joints subjected to impulse loading. International Journal of Rock Mechanics and Mining Sciences 33 (2), 505–575. Barton, N., Choubey, V., 1977. The shear strength of rock joints in theory and practice. Rock Mechanics 10, 1–54. Bhasin, R., Kaynia, A.M., 2004. Static and dynamic simulation of a 700-m high rock slope in western Norway. Engineering Geology 71, 213–226. Blanpied, M.L., Lockner, D.A., Byerlee, J., 1991. Fault stability inferred from granite sliding experiments at hydrothermal conditions. Geophysical Research Letters 18, 609–612.

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