study on dynamic response of the “dualistic” structure...

J. Mt. Sci. (2013) 10(6): 996–1007 e-mail: [email protected] http://jms.imde.ac.cn DOI: 10.1007/s11629-012-2490-7

996

Abstract: Currently, scant attention has been paid to the theoretical analysis on dynamic response mechanism of the “Dualistic” structure rock slope. The analysis presented here provides insight into the dynamic response of the “Dualistic” structure rock slope. By investigating the principle of energy distribution, it is shown that the effect of a joint plays a significant role in slope stability analysis. A dynamic reflection and transmission model (RTM) for the “Dualistic” structure rock slope and explicit dynamic equations are established to analyze the dynamic response of a slope, based on the theory of elastic mechanics and the principle of seismic wave propagation. The theoretical simulation solutions show that the dynamic response of the “Dualistic” structure rock slope (soft-hard) model is greater than that of the “Dualistic” structure rock slope (hard-soft) model, especially in the slope crest. The magnifying effect of rigid foundation on the dynamic response is more obvious than that of soft foundation. With the amplitude increasing, the cracks could be found in the right slope (soft-hard) crest. The crest failure is firstly observed in the right slope (soft-hard) during the experimental process. The reliability of theoretical model is also investigated by experiment analysis. The conclusions derived in this paper could also be used in future evaluations of Multi-layer rock slopes. Keywords: Reflection and transmission waves; Dualistic structure; Seismic wave; Dynamic response

Introduction

Normally earthquakes are a major trigger for

instability of natural and man-made slopes in seismically active regions. For complicated geological slopes, one of the parameters affecting the stability analysis is the structural surface (joint). After Wenchuan Earthquake on May 12th, 2008 (Huang and Li 2008), a large number of landslides are found in these slopes constructed by two different geological formations, and these slopes can be called “Dualistic” structure slopes. The “Dualistic” structure rock slope (soft-hard) means the upper part of the slope is soft rock element (for example: mudstone) and the lower part of the slope is composed by hard rock element (for example: limestone). Similarly, the upper part of the slope is hard rock element and the lower part of the slope is soft rock element is determined as the “Dualistic” structure rock slope (hard-soft).

Various procedures are available for stability analysis, including the pseudo-static approach, method-of-slices approach and the use of the finite element method (FEM). The pseudo-static approach is widely accepted as a method to evaluate slope stability (Newmark 1965; Seed 1979; Hong et al. 2005). In particular, the PS method has been adopted in limit equilibrium analysis (Baker et al. 2006) and limit analysis (Loukidis et al. 2003) to provide chart solutions for soil slopes. The development of the method-of-slices for identifying the critical failure surface corresponds to the minimum factor of safety (Malkawi et al. 2001; Zolfaghari et al. 2005). But there is no guarantee that an identified failure surface will correspond to the global minimum safety factor. The FEM has advantages over some classical methods as the

Study on Dynamic Response of the “Dualistic” Structure Rock Slope with Seismic Wave Theory

CHEN Zhen-lin*, XU Qiang, HU Xiao

State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu 610059, China

* Corresponding author, e-mail: [email protected]; Tel.: 0086-28-84078955

© Science Press and Institute of Mountain Hazards and Environment, CAS and Springer-Verlag Berlin Heidelberg 2013

Received: 12 September 2012 Accepted: 27 April 2013

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dynamic response and variable discipline of the stability factor. However, seismic wave input and the boundary condition assumption are required in the FEM procedure (Liu et al. 2003; Tang et al. 2004).

Recently, more research has been focused on the dynamic response analysis of complicated geological slopes including the “Dualistic” structure slope and the multi-layered slope (Qi et al. 2004; Jiang et al. 2007; Huang et al. 2007). In a plane failure, the translational block slides along a single plane of weakness or a complicated geological interface (David and Eefer 2002). The influence of shape and reinforced measures on seismic response is investigated in Large-Scale shaking table model tests (Wu et al. 2008; Xu et al. 2008). Their experiments show that the self-vibrating frequency of a model will decrease after repetitive vibration and the seismic acceleration of soil shows a larger amplification. For geotechnical analysis, the vertical ground motions may have the greatest influence on the seismic induced displacements.

In some earthquake-induced landslides, especially near-field cases, the vertical acceleration is found to be a relevant factor affecting landslide initiation (Xu et al. 2009). They investigated the genetic mechanism of the geohazards induced by strong earthquake, the influence of the factors such as the direction of the seismic force, texture and shape characteristics of slopes. Yuan (2003) has done some research on amplification effect of seismic wave caused by site soil, according to the principle of seismic wave propagation of the layered elastic medium. Jing et al. (2002) and Fan et al. (2006) have studied the dynamic characteristic of weak interlayer in layered space with joints. The thickness of the soft interlayer is much smaller than that of its surrounding layers and it is shorter than the seismic wavelength. So its dynamic response model would be a scattering model rather than a direct transmission model. Based on some theories of plane elastic wave propagation ， the transmission model of weak intercalations for P wave fields is established and used to analyze propagation properties varying with incidence angles，depth，impedance ratio of the weak intercalations. Finally，the mechanism of waveform distortion of transmission wave is also discussed.

Various analytical procedures have been used

to study the dynamic response of earth structures, the theory of seismic wave propagation and attenuation has become increasingly popular in solving seismic slope stability problems (Fan et al. 2007; Gulyayev 2006). Based on modal perturbation method, Pan and Lou, 2008) developed an approach for solving the random seismic response of horizontal layered soil. In this approach, the dynamic responses of layered soil are calculated approximately in the modal subspace spanned by several lower models of equivalent homogeneous shear beam with the same depth, average shear modulus, and mass density. The interaction between blast waves with arbitrary impinging angles and a rock joint are analyzed in detail in Li and Ma 2010. Li et al. (2009, 2011) employs the principle of conservation of momentum and the displacement discontinuous conditions along a rock joint to investigate the interaction of an obliquely incident P- or S-wave with a rock joint. The analytical solutions for the coefficients of transmission and reflection of stress waves across the joints could be derived from the fractal damage joint model (Zhao et al. 2008; Li et al. 2011) based on fractal damage theory. The analytical solution of the stress wave interaction with a rock joint is obtained by simplifying the wave propagation equation.

All the above research about slope stability analysis including numerical simulation and experimental investigation are focused on the dynamic response of slope caused by angle of slope inclination, direction of excitation, soft interlayer, but less analytical algorithms for dynamic response of the “Dualistic” structure slope ever before. In this paper, a rigorous but simplified methodology is presented to analyze the influence of a joint in the “Dualistic” structure slope. The explicit dynamic response formulae and distribution of energy intensity for the “Dualistic” structure slope are also investigated based on seismic wave propagation.

1 Experimental Analysis on the “Dualistic” Structure Slope

The main purpose of a Larger-Scale shaking table test of the “Dualistic” structure slope model reported here is to investigate the reliability of the

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theoretical analysis method proposed in this paper. Hence, we briefly introduce the design of this experiment. The law of similitude is based on the study (Wang 2005) with the same unit weight and gravity. The scaling parameters between the prototype and the model for this experiment will follow the theory (Lin and Wang 2006). The scaling factor λ is the linear ratio between the prototype and the model. Then, the factor of stress and moduls will be λ; the factor of time and shear wave velocity will be 2/1λ ; and the factor of frequency will be 2/1−λ .

The scaling factor λ used in this experiment is 100. Based on the similarity requirement, the controlling factor used in the model test is frequency. The slope height of 1.7 m is used in the model in order to simulate a 170 m high prototype slope. The slope surface was compacted by a modeling tool to keep the slope angle at the designated value. The final slope specimen is shown in Figure 1 with a height of 1.7 m, a width of 1.5 m, a length of 1.65 m, and a slope angle of 60°. The left slope (hard-soft) is composed by limestone (upper part) and mudstone (foundation); at the same time, the lithology combination of the right slope (soft-hard) is opposite.

Similar material is a mixture of barite powder, quartz sand, gypsum, glycerin, and water. The proportion of mixture of mudstone is: barite powder: quartz sand: gypsum: glycerol: water=33:55.5:2.5:2:6.5; the pro-portion of mixture of limestone is: 32:53:5:1:9. Thin layer (1.2 mm thick) clay is capped on joint, and then covered with silver sand. The density of mudstone and limestone is 2,000 kg/m3 and 2,200 kg/m3, respectively. The model should be maintained in quiescence for 30 days before starting the shake table test to guarantee the design strength of model material. The material properties were obtained from laboratory tests. The modulus, Poisson rate, frictional angle and cohesive strength of the mudstone slope model is 15.2 MPa, 0.3, 35° and 0.3 MPa, respectively. Similarly, the material properties of limestone slope model, the modulus, Poisson rate, frictional angle and cohesive strength, is 56.3 MPa, 0.2, 50° and 0.5 MPa. The natural

seismic wave (“Wenchuan” wave) (Huang and Li 2008; Xu et al. 2008) and sinusoidal acceleration excitation with a frequency of (5Hz-15Hz) and a peak acceleration of (0.1 g-0.8 g) were applied to the foundation.

The experimental results show the amplification effect of the right slope (soft-hard) model is more obvious than that of the left slope (hard-soft) model and the PGA (peak of ground acceleration) will enlarge with the amplitude increase when the vertical seismic wave acts on the slope foundation. Figure 2 and Figure 3 show the variation law of acceleration response when the amplitude is 0.2 g. It is found that the variations of horizontal acceleration response of two kinds of slopes are similar under different seismic excitations in Figure 2. From Figure 3, when the frequency of sine wave is 10 Hz, the vertical acceleration responses of the upper part of two slope models are little changed, and the magnification is less than 1.1. The elevation amplification effect on both slopes is more pronounced when frequency is 15 Hz. Comparing with sine wave excitation (10 Hz), the variations of vertical dynamic response of slopes under sine wave excitation (15 Hz) is closer to that of nature wave excitation. From the experimental observation, it is found that the amplification effect

Figure 1 Experimental model and distribution of measuring instrument.

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of vertical dynamic response on the slopes will increase with frequency increasing, but the amplification effect of horizontal dynamic response will decrease when the frequency increases.

The destruction mode of two kinds of slopes with different lithology combination is also different: cracks occur on the crest and slope surface of the right slope (soft-hard) firstly and the loose sliding body is composed by shattered rock mass as shown in Figure 4 and Figure 5; while the collapse landslide occurs in the left slope (hard-soft) and shear destruction is induced by horizontal seismic force in upper slope as shown in Figure 5. In order to further investigate the variations of dynamic response of the “Dualistic” structure slope comprehensively, a more precise theoretical analysis model should be established.

2 Dynamic RTM for the “Dualistic” Structure Slope

For theoretical analysis, the following two assumptions should be considered: (1) Every part of different geological formations in a slope is isotropic; (2) the seismic wave is plane wave. If a joint is between two geological formations, the assumption of the continuity of the joint will be correct. Based on these assumptions, the principle of seismic wave propagation could be employed to analyze the dynamic response of layered slopes. Figure 6 shows a seismic wave propagation model on a joint. The positive direction of displacement of SV wave, d, follows Right-hand rule. It is important to note that total reflection occurs if the angle of incidence is larger than or equal to the critical angle.

2.1 Seismic motion synthesis algorithm

In order to analyze the dynamic response of slope precisely, the influence of reflection and transmission waves could not be ignored. Normally, the seismic motion synthesis function of an arbitrary point can be given as:

∑=

−⋅=N

iii ttfTtF

1

)()(rv

(1)

where )( ittf −r is a seismic wave function of the ith

seismic wave ray; ti is the delayed time; N is the count of all arrived rays. Ti is the transfer coefficient of the ith seismic wave ray and could be determined by Zoeppritz equation which will be given in next section. The accuracy of equation (1)

0.5

1.5

2.5

3.5

4.5

5.5

6.5

0.2 0.5 0.8 0.95 1.1 1.4 1.7

Sine wave (10Hz,left slope)

Sine wave (15Hz, left slope)

Nature wave (left slope)

Sine wave (10Hz, right slope)


Nature wave (right slope)

(m)

Coe

ffic

ient

s of A

mpl

ifica

tion

for

Hor

izon

tal P

GA

Figure 2 Variations of amplification coefficients for horizontal PGA (amplitude at 0.2 g) of the slope model

0.5

1.5

2.5

3.5

4.5

5.5

0.2 0.5 0.8 0.95 1.1 1.4 1.7

Sine wave (10Hz,left slope)

Sine wave (15Hz, left slope)

Nature wave (left slope)



Nature wave (right slope)

(m)

Coe

ffic

ient

s of

Am

plif

icat

ion

for

Hor

izon

tal P

GA

Figure 3 Variations of amplification coefficients for vertical PGA (amplitude at 0.2 g) of the slope model

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is controlled by coefficient N and delay time ti.

2.2 Reflection and transmission coefficients on a joint

Generally, the reflection and transmission coefficients are determined by the Zeoppritz equation. Hence, we should establish the suitable Zeoppritz equation for the dynamic RTM proposed in this study.

From Figure 6 , the Shell-Law can be written as:

s

s

p

p

s

s

p

p

s

s

p

p

ccccccc ′=

′===== θθββαα sinsinsinsinsinsin1

(2)where cp and cs (or c’p and c’s) are velocity of

longitudinal wave and shear wave in medium I (mudstone) (or medium II: limestone), respectively.

Let φ and ψ be longitudinal and shear wave function, respectively. From the theory of elastic wave (Li 1993), we have:

where λ and μ are Lamé coefficients. According to the assumption previously, both

sides of the joint in slope are continuous. Then,

.;;; zxzxzzwwuu ττσσ ′=′=′=′= (5)

where u(or u’) and w(or w’) are displacements of the point of joint in medium I (or medium II); σz (or σ’z) and τzx (or τ’zx) are normal stress and shear stress of the point of joint in medium I (or medium II), respectively.

If the incident wave is longitudinal wave, submitting equations (3) and (4) into equation (5), after some simplification we thus have:

}{][}{ pp TZA ⋅= (6)

where

{ }Tspppp ββββ 2cos2sincossin}{ −−=A (7a)

{ }Tsspppsppp TTRR=}{T (7b)

zxu

∂∂−

∂∂= ψϕ

, xzw

∂∂+

∂∂= ψϕ

(3)

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂−∂∂+

∂∂=

zxztc pz

ψϕμϕλσ2

2

2

2

2

22

(4a)

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂−

∂∂∂+

∂∂=

2

22

2

2

221

xzxtcszx

ψϕψμτ

(4b)

Medium II

Medium I

Joint

P wave

SV wave

α: Angle of incidence

θ: Angle of refraction

β: Angle of reflection

αs αp

βs βp

θs

θp

d: Positive direction of displacement of SV wave

d d

z

x

Figure 6 Scheme of incident, reflected and transmitted waves on a joint

Figure 4 Initiation cracks in the right slope model

Figure 5 Destruction character of the experimental slope model

Left slope Right slope

Cracks

Left slope Right slope

Shear movement Destruction region

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⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⋅′⋅′−

⋅′⋅′

−−

⋅′⋅′

−′⋅

′⋅′−

−

=

sp

ss

p

ps

p

ss

ss

spp

sp

sps

s

pp

spsp

spsp

cc

cc

cc

ccc

cccc

cc

θρρθ

ρρ

βθ

θρρ

θρρ

ββ

θθββθθββ

2sin2cos2sin2cos

2cos)(

2sin)()(

2cos2sin

sincossincoscossincossin

][ 22

2

Z

(7c)

where ρ (ρ’) means the density of medium I (medium II).

Similarly, if the incident wave is shear wave (SV), the equation (6) is rewritten as:

}{][}{ ss TZA ⋅= (8)

where T

sp

ss

s

psss c

ccc

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−−−= ββββ 2sin2cossincos}{A(9a)

{ }Tssspsssps TTRR=}{T (9b)

2.3 Reflection coefficients on free surface

For an incident P-wave, the reflection coefficients on free surface (Wang 2005) can be given as following:

s

ppspp c

cqqq

qqRqqqqqqR ⋅

−+−−=

−+−−=

22221

221

22221

22221

)1(4)1(4;

)1(4)1(4

(10)

where sp ctgqctgq ββ == 21 ; Similarly, for an incidence S-wave, the

reflection coefficients on free surface can be given as:

.)1(4)1(4;

)1(4)1(4

22221

22221

22221

222

qqqqqqR

cc

qqqqqR ss

p

ssp −+

−−=⋅−+

−=

(11)

2.4 Computing formulae for delay time

As shown in Figure 7a, when an incident P-wave beam, in most general case (

pCαα ≤≤0 ), two separate waves are generated, i.e., reflected P- and S-waves, where pCα is the critical angles of the

incident P-wave. From Figure 7a, the delay time tp and ts for two rays (PP and PS), respectively, can be given as:

jj

Aj

p

jj c

yy

cy

t⋅+

−+=

)cos( βα , ( j=p, s) (12)

where

( )( ))(1

))((

j

AjNANj tgk

ytgxxkyy

βαβα

+⋅−⋅+−−+

=

)()(

DN

DN

xxyyk

−−=

, ( j=p, s), (13)

Similarly, when an incident S-wave beam with

an angle α (SCαα ≤≤0 ) impinges the free surface,

there are also two waves emitted from the free interfaces, the reflected P- and S-waves, where

SCα is the critical angles of the incident S-wave, as shown in Figure 7b. The formulae of delay time tp and ts for two rays (SP and SS), respectively, could be obtained from the following equation:

jj

Bj

s

jj c

yy

cy

t⋅+

−+=

)cos( βα , ( j=p, s) (14)

where yj could be obtained from equation (13) with (xB, yB ) instead of (xA, yA). It should be noted that a similar algorithm could also be employed to compute the delay time if the analytical point is

x

z

P wave SV wave

(xA, yA) A

β α

Medium I: ρ,cp,cs

Medium II: ρ’,c’p,c’s

D E

O F

M N (xN, yN)

(xD, yD)

(a) Incident P-wave

x

z

P wave SV wave

(xB, yB) B

β α

Medium I: ρ,cp,cs

Medium II: ρ’,c’p,c’s

D E

O F

M N (xN, yN)

(xD, yD)

(b) Incident S-wave

Figure 7 Access of (a) incident P-wave and (b) incident S-wave

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Table 1 Path, delay time, transfer coefficients of P-wave of the left slope

No. Incident wave Position Path Delay

time Transfer coefficients

1 P1 Upper part PP t1

MNPPTT =1

Lower part P T1=1

2 P1 Upper part PPP

t2 GCPP

MNPP RTT ⋅=2

Lower part PP MNPPRT =2

3 P1 Lower part PPPP t3 NMPP

GCPP

MNPP TRTT ⋅⋅=3

4 P2 Upper part PPS

t3 CNPS

MNPP RTT ⋅=4

Lower part PPP MNPP

NDPP RRT ⋅=4

5 P3 Upper part PPP

t5 CNPP

MNPP RTT ⋅=4

Lower part PS NDPSRT =5

6 P4 Upper part PPSS

t6

GCSS

MNPS

NDPP RTRT ⋅⋅=6

Lower part PPS MNPS

NDPP RRT ⋅=6

7 P5 Upper part PPS

t7

MNPS

NDPP TRT ⋅=7

Lower part PP NDPPRT =7

Notes: Transfer coefficients are comprised by reflection and transmission (R

and T) coefficients; MN

PPT is the transmission coefficient when the incident

wave comes from lower slope; and NM

PPT is the transmission coefficient when the incident wave comes from upper slope.

located in upper slope.

3 Application of Dynamic RTM

3.1 Dynamic response of the “Dualistic” structure slope

The splitting and stacking field of seismic wave rays can be determined directly while the angle of reflection and transmission can be computed by equation (2). For example, considering an incident P-wave first, the principle of seismic wave propagation of most points in the left slope will follow the description of Figure 8(a-b). The path, delay time, transfer coefficients of incident P-wave in the left slope (hard-soft) model are given in Table 1, respectively. The dynamic response function of the points in a dangerous region can be obtained directly. Then, we evaluate the value of the peak of ground accelerate of the points located in the dangerous region by time-history analysis. Hence, the potential dangerous region and potential sliding surface can be theoretically determined.

In order to expatiate on the theoretical model, the dynamic response functions of two emblematical points are given here. According to Figure 8a and Table 1, there are six seismic wave rays including all reflection and transmission wave rays through the point H1. The explicit seismic motion synthesis function of the point H1 in upper part of the left slope could be obtained as:

)2sin()()cos()()(

144

1332

ααβ

ttfTttfTtF sx

−⋅−+−⋅−=

(15a) )cos()()cos()( 266255 ss ttfTttfT θθ −⋅+−⋅−

)sin()()()()(

1333

2211

αβ +−⋅+−⋅−−⋅=

s

z

ttfTttfTttfTtF

)sin()()sin()()2cos()(

266

255144

s

s

ttfTttfTttfT

θθα

−⋅+−⋅+−⋅−

(15b) Similarly, from Figure 8b and Table 1, the

explicit seismic motion synthesis function of the

point H2 in lower part of the left slope could also be written as:

)cos()()sin()()( 1255244 αββ +−⋅−−⋅−= spx ttfTttfTtF )2cos()()cos()( 177266 αβ ttfTttfT s −⋅+−⋅− (16a)

)cos()()()()()(

244

332211

p

z

ttfTttfTttfTttfTtF

β−⋅−−⋅−−⋅−−⋅=

)2sin()()sin()()sin()(

177

2661255

αβαβ

ttfTttfTttfT ss

−⋅+−⋅++−⋅+

(16b)

The delay time, reflection and transmission coefficients can be evaluated according to the theory presented in Section 2. Hence, the dynamic response (displacement, velocity and acceleration) of the point H1 (or H2) could be derived from equations (15a-b) (or (16a-b)). Furthermore, the synthesis functions of other points in the left slope could also be obtained from equations (15a-b) (or (16a-b)) by adjusting the computing parameters.

For the sake of simplicity, the enantiomorphism of the right slope (soft-hard) is considered. Then, the computing formula of the points in the left slope (hard-soft) can be used to

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compute the dynamic response of the points in the right slope directly. The principle of seismic wave propagation in the enantiomorphism of the right slope is given in Figure 9(a-b). The path, arrival time, transfer coefficient of incident P-wave in the enantiomorphism of the right slope are given in Table 2.

3.2 Validation of theoretical modeling

Considering a sine wave excitation (0.3 g and 15 Hz) along z direction, the theoretical simulation solutions of the “Dualistic” structure rock slope could be obtained from the dynamic response function described in Section 3.1. In order to compare with the experimental results, the dynamic responses of some characteristic points (where the acceleration sensors are placed in) are evaluated by the theoretical method. Figures 10, 11 and 12 show the comparison between the

theoretical simulation solutions and the experimental results. Some conclusions are summarized as following:

(1) From Figures 10 and 11 we find that the variation of dynamic response is discontinuous around the structural surface (joint). With the division and the superposition of seismic waves, the dynamic response of the region near the structural surface (joint) is very complex. Notable influence on the dynamic response in the upper part of the slope is induced by the existence of the structural surface (joint) in the rock slope.

(2) According to Figures 10 and 11, the theoretical results and the experimental results all show that the coefficients of amplification for PGA increase gradually along slope elevation. The dynamic response of the right slope (soft-hard) model is much greater than that of the left slope (hard-soft) model. It is noted that the soft foundation of the left slope model is useful for

x

z

C

E

F

G

O

D

H1

P1 P2 P3 P5

α1

θ2s

β3s

α1

α2p

M N

(I)

(II)

P4

x

z

H2 β1s

D E

O F P1 P2 P3 P4 P5

M N

α1

β2s

(I)

(II β2p

(a) Upper part (b) Lower part

Figure 8 Access of seismic wave of (a) upper part and (b) lower part of the left slope

x

z

C

E

F

G

O

D

P1 P2 P3 P4

θ2p

α4s

M N (I)

(II)

P5 P6

x

z D

E

O F P1 P2 P3 P4 P5

M N

α1

(I)

(II)

(a) Upper part (b) Lower part

Figure 9 Access of seismic wave of (a) upper part and (b) lower part in the enantiomorphism of the right slope

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shock absorption, at the same time the amplification effect on the dynamic response is very obvious in the right slope model with rigid foundation.

(3) Figures 10 and 11 show that the theoretical results coincide with the experimental results on the field away from the joint in both slope models. But we also found that the theoretical solutions of the amplification coefficients for vertical PGA near joint (h = 0.95 m) are quite different with the experimental results. In particular, the influence of the joint on the left slope is more obvious than that on the right slope. As the boundary of the experimental model may be very rough, the splitting and stacking field of seismic wave rays in the experimental model would be more complex than that in theoretical model so that the difference of both methods is inevitable. But the varying tendency of the solutions derived by theoretical model is consistent with the experimental results.

(4) Figure 11 denotes the variations of the amplification coefficients for vertical PGA. The maximum of the amplification coefficients for vertical PGA in the left slope is less than 1.6 and the minimum in the right slope is larger than 1.7. It indicates that the energy intensity of transmission wave in the left slope is much weaker than that of the right slope. In Figure 11, the theoretical result reaches a maximum at height of 1.4 m. In other words, the middle-upper part of the right slope model would be the dangerous region. The failure phenomena are observed in the right slope during the experimental process.

(5) From Figures 11 and 12, it is noted whether in the experimental study or the theoretical

simulation， the dynamic response of the right slope model is greater than that of the left slope model，especially in the slope crest. For example, in the slope crest, the amplification coefficient for vertical PGA of the point (AL15) and the point (AR15) is about 1.4 and 2.0, respectively. With the amplitude increasing, the cracks could be found in the right slope crest firstly (Figure 4).

Table 2 Path, delay time, transfer coefficients of P-wave of the right slope

No. Incident wave Position Path Delay

time Transfer coefficients

1 P1 Upper PP t1

MNPPTT =1

Lower P T1=1

2 P1 Upper PPP

t2 GCPP

MNPP RTT ⋅=2

Lower PP MNPPRT =2

3 P1 Lower PPPP t3 NMPP

GCPP

MNPP TRTT ⋅⋅=3

4 P2 Upper PPS

t4 CNPS

MNPP RTT ⋅=4

Lower PPP MNPP

NDPP RRT ⋅=4

5 P3 Upper PPP

t5

CNPP

MNPP RTT ⋅=5

Lower PPS MNPS

NDPP RRT ⋅=5

6 P4 Upper PPSSS

t6 GCSS

CNSS

MNPS

NDPP RRTRT ⋅⋅⋅=6

Lower PS NDPSRT =6

7 P5 Upper PPP

t7

MNPP

NDPP TRT ⋅=7

Lower PP NDPPRT =7

8 P6 Upper PPS t8 MNPS

NDPP TRT ⋅=8

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

0.2 0.5 0.95

Experimental Results (left slope)

Theoretical Results (left slope)

Experimental Results (right slope)

Theoritical Results (right slope)

(m)

Am

plif

icat

ion

Coe

ffic

ient

s for

V

ertic

al P

GA

Elevation

Figure 10 Variations of amplification coefficients for vertical PGA of different measuring points (From AL4 (AR4) to AL9 (AR9), see Figure 1) along the height of the right slope

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3.3 Distribution of energy intensity

Based on the previous study, we find: (1) the theoretical RTM can be employed to analyze the dynamic response of the “Dualistic” structure rock slope, though the splitting and stacking field of seismic wave will be more complicated in natural layered slopes due to the rough slant surface and joint; (2) the stability of the right slope (soft-hard) model is very poor in comparison with the left slop (hard-soft) model.

In order to further investigate the stability mechanism of the “Dualistic” structure slope, the distribution principle of seismic energy intensity should be studied carefully when the seismic wave impinges the joint. Without loss of generality, the initial energy intensity of incident seismic wave could be defined as one unit. The energy intensity could be obtained by explicit seismic motion synthesis function proposed in Section 3.1. Figures 13 and 14 show the distribution of energy intensity and surface topography effect (slope inclination). Comparison of the distribution of seismic wave energy in Figure 13 with Figure 14 demonstrates that the energy intensity of transmission waves of right slope is larger than that of the left slope and the energy intensity of reflection waves of the right slope is smaller than that of the left slope.

From Figure 13, the value of energy intensity of transmission P-wave in the left slope (hard-soft) model stays about 0.4567 when the angle of slope inclination is of 0° to 65° and it becomes 0.5683 at 66° Then, the value of energy intensity of transmission P-wave increases gradually with the inclination angle increasing. For reflection S-wave, the significant increment of energy intensity occurs at 59°, changing from 0.0636 to 0.5302.

Figure 14 denotes that the energy intensity of transmission P-wave in the right slope (soft-hard)

model is about 0.9164 for different angles of slope inclination. The energy intensity of reflection S-wave is near zero till the angle of a slope inclination is 60°, it becomes 0.9539.

According to Figures 13 and 14, the energy density intensifies sharply when the angle of slope inclination is greater than 60° for this kind of “Dualistic” structure rock slope. For current analytical model, the angle of slope inclination is 60°, the total energy intensity of transmission waves is about 0.9164 and the total energy intensity of reflection is1.0429 in the right slope, while the total energy intensity of transmission waves is about 0.4825 and the total energy intensity of reflection is 0.5723 in the left slope.

0.2

1.2

2.2

3.2

4.2

5.2

0.95 1.4 1.7




Theoretical Results (right slope)

Am

plif

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ion

Coe

ffic

ient

s for

V

ertic

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(m)Elevation

Figure 11 Variations of amplification coefficients for vertical PGA of different measuring points (From AL10 (AR10) to AL15 (AR15), see Figure 1) along the height of the Model

1.20

1.40

1.60

1.80

2.00

2.20

2.40

2.60

2.80

3.00

3.20

-1.33 -1.19 -1.04


Theoretical Results (right slope)



(m)

Am

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r V

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Horizontal distance

Figure 12 Variations of amplification coefficients for vertical PGA of different measuring points (From AL14 (AR14) to AL16 (AR16), see Figure 1) along the crest of the right slope

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The total energy intensity of transmission waves (or reflection waves) in the right slope is almost twice of that in the left slope. The results indicate that the left slope model is more stable than the right slope model; and this conclusion is consistent with the previous analysis and experimental observation.

4 Conclusions

This study proposed a theoretical dynamic RTM to analyze the dynamic response of a layered slope based on the theory of elastic mechanics and the principle of seismic wave propagation. The explicit seismic motion synthesis function for the “Dualistic” structure rock slope is established. The dynamic response of the “Dualistic” structure rock slope can be calculated directly when the suitable Zeoppritz equation for computing reflection and transmission coefficients is investigated. The dynamic characters of the “Dualistic” structure slope obtained from the theoretical simulation are consistent with the experimental results. The distribution of energy intensity is also employed to analyze the stability of the “Dualistic” structure rock slopes (hard-soft and soft-hard). Some preliminary conclusions derived in this study would be helpful to reveal effect of the lithology combination on slope stability under earthquake excitation, and provide valuable reference for seismic design of slope engineering.

Acknowledgement

This research is financially supported by Project of the National Natural Science Foundation of China (Grant No. 41002126); Project of State Key Laboratory of Geohazard Prevention and Geoenvironment Protection (Grant No.

SKLGP2009Z010). We thank PH.D student LIU Hanxiang for her valuable suggestions and providing experiment data and anonymous reviewer for his critiques and comments.

10 20 30 40 50 60 70 80 90

1.2

1

0.8

0.6

0.4

0.2

0

Angle of slope inclination (Degree) En

ergy

Inte

nsity

ETP

0.5285

ERP

ERS

ETS

ETP: Total energy intensity of transmitted P-wave ERP: Total energy intensity of reflected P-wave ETS: Total energy intensity of transmitted S-wave ERS: Total energy intensity of reflected S-wave

Figure 13 Relationship between the energy distribution of seismic waves in the left slope (hard-soft) and angles of slope inclination

10 20 30 40 50 60 70 80 90Angle of slope inclination (Degree)

Ener

gy In

tens

ity

0.9539 ETP

ERP

ETS

1.2 1 0.8 0.6 0.4 0.2

0

ERS

Figure 14 Relationship between the energy distribution of seismic waves in the right slope (soft-hard) and angles of slope inclination

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