Slope topography effects on ground motion in the presence of deep soil layers

Transcription

1 Slope topography effects on ground motion in the presence of deep soil layers Tripe, R. a, Kontoe, S. b* & Wong T. K. C. c a Atkins Limited, London, formerly Imperial College London b Department of Civil & Environmental Engineering, Imperial College London. c Geotechnical Engineering Office, Civil Engineering and Development Department, The Government of the Hong Kong Special Administrative Region, formerly Imperial College London Corresponding author: Stavroula Kontoe Civil & Environmental Engineering Imperial College London South Kensington campus London SW7 2AZ, UK Tel: Fax: : Abstract An extensive investigation has been made into the interaction between topographic amplification and soil layer amplification of seismic ground motion. This interaction is suggested in the literature as a possible cause for the differences between topographic amplification magnitudes observed in field studies and those obtained from numerical analysis. To investigate this issue a numerical finite element (FE) parametric study was performed for a slope in a homogeneous linear elastic soil layer over rigid bedrock subjected to vertically propagating in-plane shear waves (Sv waves). Analyses were carried out using two types of artificial time history as input excitation, one mimicking the build-up and decay of shaking in the time histories of real earthquake events, and the other to investigate the steady-state response. The study identified topographic effects as seen in previous numerical studies such as modification of the free-field horizontal motion, generation of parasitic vertical motion, zones of alternating amplification and de-amplification on the ground surface, and dependence of topographic amplification on the frequency of the input motion. For the considered cases, topographic amplification and soil layer amplification effects were found to interact, suggesting that in order to accurately predict topographic effects, the two effects should not be always handled separately. 1

2 Introduction Topographic amplification of earthquake ground motion has been well documented in various destructive earthquake events in the literature. Unusually severe earthquake induced damage has been attributed to topographic amplification of earthquake motion, such as in the 1985 Canal Beagle Chile earthquake (Celebi, 1987 & 1991), Whittier Narrows 1987 earthquake (Kawase & Aki, 1990), Aegion Greece 1995 earthquake (Bouckovalas et al 1999) and Athens Greece 1999 earthquake (Gazetas et al 2002). Instrumented field studies of topographic effects during earthquakes have directly observed modification of earthquake ground motion. Typically these studies measure earthquake motion on the surface of the topography (e.g. slope or hill) relative to a base station. The magnitude of topographic effects reported from instrumented studies varies, some studies reporting amplification in the range of 2-3 times (Pedersen et al 1994a), whilst other studies observe amplifications of up to 30 times (Geli et al 1988). Topographic amplification has also been the focus of numerous numerical studies employing the Finite Element Method (FEM) (e.g. Gazetas et al 2002, Assimaki & Gazetas 2004), the Finite Difference Method (FDM) (e.g. Bouckovalas & Papadimitriou, 2005, Stamatopoulos et al 2007), the Spectral Element Method (SEM) (e.g. Paolucci et al 1999, Paolucci 2002, Assimaki & Gazetas 2004), the direct Boundary Element Method (BEM) (e.g. Hisada et al 1993, Semblat et al 2002), the indirect BEM (e.g. Sanchez-Sesma & Campillo 1993, Pedersen et al 1994a, 1994b) and hybrid combinations of BEM/FEM (e.g. Kamalian et al 2006, 2007, 2008) and FEM/SEM (e.g. Havenith et al 2003). Comprehensive reviews of the numerous numerical studies on topographic amplification can be found in Beskos (1997) and Semblat & Pecker (2009). The numerical studies investigating the mechanism and behaviour of topographic amplification typically do not report those very high values of amplification that are observed in some instrumented studies (Geli et al, 1988, Paolucci et al 1999, Paolucci 2002, Semblat et al 2002). Topographic amplification is reported to vary as a function of frequency (Ashford et al 1997; Bouckovalas & Papadimitriou, 2005; Geli et al 1988), type (Ashford et al, 1997; Ashford & Sitar, 1997; Kamalian et al, , 2008), orientation (Pedersen et al 1994b), and angle of incidence (Ashford & Sitar, 1997) of incident wave motion, and shape of the topography, both in two dimensional (2D) (Nguyen & Gatmiri 2007; Boore, 1972; Kamalian, 2007 & 2008; Ashford et al 1997; Bouckovalas & Papadimitriou, 2005) and three dimensional (3D) cases (Bouchon et al 1996). Typically, topographic amplification of free-field motion of two to three times is reported in these numerical studies. 2

3 The considerable difference between the amplification observed in instrumented field studies and numerical studies has been attributed to a number of factors, such as the location of the base station (the base station itself being affected by topographic effects) (Geli et al, 1988; Pedersen et al, 1994a; Chavez-Garcia et al 1996), 3D effects of topography (Bouchon & Barker, 1996, Bouchon et al 1996), effects of adjacent topography (Geli et al, 1988), or in some cases the additional effects of amplification due to soil layer effects (Graizer, 2009). This study sets out to investigate the last factor, the relationship and interaction between topographic amplification and soil layer amplification, for the particular case of slopes. This case has been the subject of relatively few systematic investigations reported in the literature. The majority of these numerical studies of slopes identified in the literature considered the case of a slope in a homogenous half-space, whereas the studies by Ashford et al (1997) and Sitar and Clough (1983) were the only systematic studies identified in the literature of slopes within a soil layer. Analysis Methodology To investigate the interaction between topographic amplification and soil layer amplification, a time-domain two-dimensional (2D) numerical finite element (FE) parametric study was performed for a slope in a soil layer over rigid bedrock subjected to vertically propagating in-plane shear waves (Sv waves). The soil was modelled as a homogenous elastic material and the geometry of the FE model consisted of a soil slope of height H, inclination angle i, and bedrock depth Z, as schematically shown in Figure 1. A fixed slope height of H=50 m and a fixed mesh width of L=1000m (L 1 =L 2 =500m in Figure 1) was adopted in the analyses. The following factors influencing the seismic motion at the ground surface in the model were investigated, by carrying out a parametric analysis: variation of the ratio of slope height (H) to input motion wavelength (λ) (referred to as the normalised frequency H/λ), by varying the input motion frequency. The ratio of topography dimension to wavelength is identified in the literature as a key parameter affecting the magnitude of topographic effects. variation of the slope angle (i) identified in the literature as affecting topographic amplification. variation of the soil layer amplification and soil layer s fundamental frequency, by consideration of different bedrock depths (Z). To focus the investigation on the effects of topographic amplification and soil layer amplification and their interaction, other parameters identified in the literature as affecting topographic effects were not assessed in the current study, such as wavefield incident angle and orientation, wave type, and 3D topography geometry. Similarly a relatively simple homogeneous elastic soil model was adopted, with a shear wave velocity, Vs=500m/s and a target damping ratio of 5% (see Table 1) 3

4 using the Rayleigh damping formulation. It is well established that Rayleigh damping is frequency dependant, while in reality damping in soils is almost independent of frequency. To alleviate this shortcoming, it is common practice to try to get the right target damping for the important frequencies of the problem within a frequency range to. The selection of, is highly dependent on the problem analysed and on the frequency content of the input motion. In this study was taken as the first natural frequency of the 1D system corresponding to a soil column behind the slope crest and was the predominant frequency of the input motion. Input motion generated by Sv waves was considered in the current study assuming plane strain conditions. Previous studies comparing the effects of different wave types (Sh and P waves) identified Sv waves as causing the greatest topographic effect (Ashford et al, 1997; Ashford & Sitar, 1997; Kamalian et al, 2008). To allow the investigation of single frequency input motion, two types of artificial acceleration (a(t)) time histories of input excitation were adopted. To mimic the build-up and decay of shaking in time histories of real earthquake events, a modified Gabor wavelet (Gabor 1946, Mavroeidis & Papageorgiou 2003), which is cited as Chang s time history in Bouckovalas and Papadimitriou (2005), was considered, given by Equation (1). The constants in the wavelet time history were varied to give a maximum amplitude of unity, and the same number of cycles was used for the different frequencies considered. A typical example of the acceleration-time history of the wavelet motion is shown in Figure 2. ( ) ( ) (1) where α, β, and γ are constants controlling the shape and amplitude of the acceleration-time history, T p is the predominant period of the pulse and t is the time. To assess the steady state conditions, a sinusoidal time history was used as input excitation in the analyses, with sufficient duration to approach a steady state response. The analysis was carried out using the Imperial College finite element program ICFEP (Potts and Zdravkovic, 1999). The time integration was performed with the generalised-α method (Chung & Hulbert, 1993; Kontoe et al., 2008a) which is an unconditionally stable implicit method, with second order accuracy and controllable numerical damping. The time step of the analysis was taken as a fraction (Δt= T p /40) of the predominant period T p of the input pulse. To accurately represent the wave transmission through the finite element mesh, it is necessary to ensure that the element size is small relative to the transmitted wavelengths. Thus, the element side length ( ) was chosen based on recommendations by Kuhlemeyer and Lysmer (1973) as: 4

5 (2) where input pulses. is the wavelength corresponding to the highest frequency of the considered To effectively model the lateral free-field conditions, the domain reduction method (DRM) (Bielak et al, 2003) in conjunction with the standard viscous boundaries (SVB) of Lysmer and Kuhlemeyer (1969) were used as boundary conditions in the model. A similar approach, employing an earlier version of DRM, was also followed by Gazetas et al (2002) to study topography effects. The DRM is a two-step substructuring procedure that was originally developed for seismological applications to reduce the domain that has to be modelled numerically by a change of governing variables. However, the DRM, in conjunction with a conventional absorbing boundary (i.e. the SVB), can also be efficiently used in the numerical modelling of geotechnical earthquake engineering problems as an advanced absorbing boundary condition. In this method, the analysis is divided into two steps. In the first step a simplified model is analysed, considering the earthquake source and entire domain excluding the feature to be modelled. The resulting motion is determined at a surface, Γ, within the first phase model. The motion determined at the surface Γ is then applied to a second model that includes the feature to be modelled. Further description of the implementation of this method and its use as a boundary condition is presented in Kontoe et al (2008b, 2009). In this study, the step I model consisted of a soil column of thickness Z and width of 20m, with the acceleration-time history applied at the base in the horizontal direction, while restricting the vertical movement along the lateral boundaries and base which is consistent with the rigid bedrock assumption. This column analysis, which will be referred to as the 1D model (since it models 1D wave propagation), gave the freefield ground motion in which any modification of the input motion is purely due to soil layer effects and damping of the soil layer. During the step I analyses, the incremental displacements were calculated at various depths of the 1D model. These were then used in the step II analyses to calculate the equivalent forces that were applied to the corresponding nodes of the step II model, located between the boundaries Γ e and Γ (see Figure 1). This procedure allows the seismic excitation, in the form of equivalent forces calculated in the first step, to be directly introduced into the step II computational domain allowing flexibility in the choice of absorbing boundary conditions. In this study, the SVB was applied along the lateral boundaries of the step II model and both horizontal and vertical displacements were restricted along the bottom boundary (in accordance with the rigid bedrock assumption). The different geometry and time history cases analysed in the study are summarised in Table 2. It is important to note that the fictitious boundary Г separates the computational domain into two areas Ω and +. The formulation of the DRM is such 5

6 that the perturbation in the external area + is only outgoing and corresponds to any deviation of the step II 2D model from the step I 1D model. Hence in this study the wave-field in area and + will be solely induced by the presence of the slope. Numerical examples by Yoshimura et al (2003) and Kontoe et al (2009) showed that the ground motion in the external area and + is generally small compared to the corresponding motion of the free-field model. Therefore when the absorbing boundaries are used in conjunction with DRM they are required to absorb less energy than they would have to absorb in a conventional analysis. Typical Results From the results of the FE analysis, the acceleration-time histories at the ground surface and particularly at the slope crest were obtained, and the effects of variation of the parameters presented in Table 2 on ground motion were assessed. It should be noted that due to the asymmetry of the geometry of the problem under consideration, the free-field motion corresponding to the crest stratigraphy was used for the step I column analysis. This does not impact the accuracy of the predicted response next to the crest, but it can affect the predicted response between the toe and the left hand side boundary. Therefore it was decided to focus the investigation on the response adjacent to the crest, where previous studies have shown topographic amplification to be more severe. Typical results are shown in Figures 3 and 4 for a single input time history, corresponding to the wavelet input motion for H/ for two bedrock depths (Z=125m and 500m) which have a similar free field response at the selected H/λ value. The figures plot the maximum absolute horizontal and vertical acceleration values (a hmax and a vmax ) for a series of points on the ground surface behind the slope crest (distance x c in Figure 1) (for i=90 ), determined from the acceleration-time histories at each point. Also shown in Figure 3 are the maximum absolute acceleration values at free field (a hff ), which is purely horizontal, for the 1D model. Although the response in both figures attenuates with increasing distance from the crest, exact free-field conditions were not reached even at large distances (e.g. x c =450m) from the crest. The difficulty of attaining free-field conditions was also highlighted by Bouckovalas & Papadimitriou (2005) who found that topography effects decrease asymptotically with distance from the slope. This was attributed to the reflection of the incoming SV waves on the inclined free surface of the slope which can result (a) in P, SV waves impinging obliquely at the free ground surface behind the crest and (b) in the generation Rayleigh waves propagation away from the slope (i.e. in the positive x c direction in Figure 1). In this study rotational near surface motion, moving out from the crest of the slope, was identified in the results, indicating the generation of outward travelling surface waves. It should be noted that 6

7 for the analyses carried out with sinusoidal input motion, the steady state maximum absolute responses were considered, ignoring the initial transient response. With the presence of a soil/rock interface in the model, the accelerations at the ground surface behind the slope crest are subjected to both topographic and soil layer amplifications. Therefore, the soil layer amplification effects have to be removed first. For this purpose, the maximum absolute slope model surface accelerations are normalised by the maximum absolute free-field motions (a hff ), which is purely horizontal, obtained from the 1D model. These normalised acceleration values are notated as A hmax for horizontal motion, and A vmax for vertical motion. As the free-field ground motion is purely horizontal, the vertical accelerations are also normalised by the horizontal motion. The normalised horizontal and vertical acceleration values at the slope crest were determined for the different bedrock depths, slope inclination and input motion cases listed in Table 2. These are presented in Figures 5 to 10 for the three different bedrock depths considered, where wv represents an analysis considering the wavelet time history described by Equation (1), and ss a sinusoidal time history. Also plotted are published numerical results by Bouckovalas and Papadimitriou (2005) and Ashford et al (1997), for the case of a slope in a homogeneous halfspace. Effects of Topography on Surface Ground Motion The results of this study show several typical topographic effects on seismic motion which are broadly in agreement with those reported in previous studies of slopes and other types of topography: - Modification of the horizontal free-field motion, as seen in the difference between free-field accelerations and ground surface accelerations in Figure 3, and the A hmax values differing from 1, in Figures 5, 7 and 9. - Generation of parasitic vertical motion, as seen in the vertical accelerations in Figure 4, and the non-zero A vmax values in Figures 6, 8 and Spatial variation of the ground motion, with zones of alternating amplification and de-amplification of the free-field motion across the model surface behind the slope crest, as seen in Figures 3 and 4. The magnitude of the deamplification and amplification was typically seen to reduce away from the crest of the slope, and the number of zones to vary with normalised input frequency (H/λ). - Dependency of the normalised amplification on normalised frequency, H/λ, as seen in Figures 5 to 10. For low H/λ values, below approximately H/λ of 0.05, topographic effects are insignificant. Hence it is seen that when the input 7

8 motion wavelength is large relative to the slope height, there is no observable topographic effect. Effects of Input Motion Types The limited duration wavelet motion and the long duration steady-state motion showed similar trends in behaviour. However, the analyses with the wavelet motion typically developed smaller magnitudes. This is considered to be due to the short duration input not allowing the full steady-state response to develop. The amplification values from the analyses using the wavelet motion are considered to better represent amplification values that may occur during real earthquakes. Effects of Slope Inclination Increasing the slope inclination was found to result in greater topographic effects. In Figures 5 to 8 the normalised crest amplification (A hmax and A vmax ) generally increase with steeper slopes, with the greatest amplification occurring for the vertical slope (i=90). This agrees with the findings from Ashford et al (1997) and Bouckovalas and Papadimitriou (2005), also presented in Figures 5 to 8, for the case of a slope in a homogeneous half space. Interaction between Topographic Effects and Soil Layer Effects Previous parametric studies conducted by Bouckovalas & Papadimitriou (2005) and Nguyen and Gatmiri (2007) considered slopes in a homogeneous half-space and thus they did not consider the combined effects of topographic amplification and soil layer amplifications. In the current study, the presence of a soil/rock interface results in soil layer effects, and the interaction between topographic effects and soil layer effects was investigated. This study attempted to separate the effects of soil layer amplification and topographic amplification by normalising the ground motion at the slope crest, which is influenced by both topographic and soil layer effects, by the free-field motion, which is influenced by soil layer effects only. However comparison of the variation of normalised acceleration values (A hmax and A vmax ) at slope crest with normalised frequency of input motion H/λ (Figures 5 to 10) for the three examined bedrock depths shows considerable differences. Similarly, comparing the normalised acceleration values at slope crest of this study to the results of previous studies for slopes in a homogeneous half-space by Bouckovalas and Papadimitriou (2005) and Ashford et al (1997), different patterns of amplification are observed. These differences in amplification values show that the topographic amplification is still affected by the presence of soil/rock interface even when this is removed by 8

9 normalising the acceleration values at slope crest against the free-field motion. This indicates that, for the considered cases, there is an interaction between the two effects (in other words, the two effects are coupled). Hence when a rigorous assessment of topographic effects is required such interaction should not be ignored and the two effects should not be treated separately. This is in contrast to the study by Ashford et al (1997) on the interaction between soil layer and topographic effects, in which they concluded that the two effects are uncoupled and could be considered separately. It was found that the majority of the overall modification of the input motion at the slope crest was generally due to soil layer effects. This is seen in Figures 11 to 13, where the amplification of the input motion in the 1D model and the 2D slope model are compared. Amplification in the 1D model (A hin ), taken as the maximum absolute free-field motion in the 1D model normalised by the maximum acceleration of the input motion, is due to soil layer amplification effects. Amplification at the slope crest in the 2D model, taken as the maximum absolute acceleration at the slope crest normalised by the maximum acceleration of the input motion, is due to the combined effects of soil layer and topographic effects. Comparing the two amplification values, the amplification at the slope crest in the 2D model is seen to be typically similar or marginally greater than the amplification in the 1D model, showing that the additional amplification due to topographic effects is relatively small compared to the amplification due to soil layer effects. A similar conclusion was reached by Sitar & Clough (1983) and by Ashford et al (1997) in their studies of slopes within soil layers. They identified the natural period of the soil deposit as being the dominant factor in affecting the magnitude of accelerations in the slope. Figure 14 plots the maximum absolute vertical acceleration as a function of H/, for the wavelet input motion and a slope angle of i=90 for the three bedrock depths. The corresponding free-field maximum horizontal accelerations are also plotted in Figure 14 to indicate the soil layer effects. Although the maximum free-field response across the normalised frequencies for the three bedrock depths is of comparable magnitude, the parasitic vertical motion is significantly higher for the shallow layer (i.e. Z=125m) for normalised frequencies. Furthermore the greatest deviation in amplification values between the current study and the previous results of studies of slopes in homogeneous half-space occurs for the range of frequencies near the natural frequencies of the soil layer. Figure 15 to Figure 17 show the topographic amplification values at slope crest obtained from this study, the topographic amplification values for slopes in a homogeneous half-space (from Ashford et al, 1997; Bouckovalas and Papadimitriou, 2005) and the theoretical elastic soil layer amplification values. These were determined by the amplification function of a damped elastic soil layer of depth Z over rock as given by Equation 3 9

10 (from Kramer, 1996), while the natural frequencies of soil layer are determined by Equation 4 (from Kramer, 1996). ( ) where k is the wave number. (3) ( ), n=0, 1, 2,, (4) The dimensionless frequency (H/ ) can be expressed in terms of the predominant period of the input motion (T p ), the slope height and the shear wave velocity of the soil ( ) as follows: which for resonance conditions with the natural site periods, substituting (4) into (5) becomes: (5) ( ) (6) Consequently, for example, the values of corresponding to the fundamental (n=0) and second (n=1) site periods of the 250 m thick soil layer are 0.05 and 0.15 respectively (see also Figure 16). From the abovementioned figures it is seen that the maximum topographic amplification at slope crest occurs not necessarily when the normalised frequency is equal to 0.2 (as suggested by previous studies), but when the normalised frequency of input motion is between the values corresponding to the natural frequencies of the soil layer, while at natural frequencies of the soil layer, topographic de-amplification relative to free-field motion occurs (A hmax < 1). Hence the peak soil layer amplification and peak topographic amplification do not occur at the same normalised frequency of the input motion. For higher natural frequencies, which have smaller soil layer amplification, the topographic amplification from this study begins to approach the results for topographic amplification of a slope in a homogenous half-space. It is noteworthy that the previous numerical studies conducted by Ashford et al (1997) and Bouckovalas & Papadimitriou (2005) of slopes in a homogeneous half-space did not report any de-amplification of ground motion at slope crest compared to free-field response for almost all the cases they considered. It is also interesting to note that such topographic de-amplification against free-field response occurs when the normalised frequency of the input motion is equal to the values corresponding to the natural frequencies of the soil layer, which means that soil layer amplification occurs. 10

11 Figures 15 to 17 also show that for the cases of slopes within a soil layer, when the range of natural frequency of the soil layer is outside the range of frequencies, reported in Ashford et al (1997) and Bouckovalas & Papadimitriou (2005), for which topographic amplification occurs, there will be little interaction between the two effects, and topographic amplification will approach the case of a slope in a homogeneous half-space. Furthermore Figures 15 to 17 indicate that topographic amplifications increase with decreasing bedrock depths, and much larger amplification factors are obtained in this study compared with the previous studies conducted by Ashford et al (1997) and Bouckovalas & Papadimitriou (2005) who studied slopes in a homogeneous halfspace. For example, Figure 15 shows that an amplification of more than 150% was obtained for 90 slope in this study for a bedrock depth of 125m, but the amplification value reported in Ashford et al (1997) and Bouckovalas & Papadimitriou (2005) was only about 50%. With increasing bedrock depth the deviation between the topographic amplification values of this study and the topographic amplification of a slope in a half-space was seen to decrease. In the case of the Z = 500 m bedrock depth analyses (Figure 17), little difference is seen. This might be due to either: - reduction in topographic effects with increasing bedrock depth gradually approaching the half-space case; or - the natural frequencies of the Z = 500 m soil layer being outside of the range of frequencies over which considerable topographic effect occur, as discussed above. This study has demonstrated that with the presence of a soil/rock interface, much larger topographic amplification than those previously reported in literature for simple cases of slopes in a homogeneous half-space is possible and the interaction between soil layer effects and topographic effects is complex. This means that a rigorous prediction of topographic amplification in the presence of soil layers would require numerical analysis of the system considering the interaction between soil layer and topographic effects. Conclusions The interaction between topographic amplification effects and soil layer amplification effects has been investigated by carrying out finite element modelling of a slope in a homogeneous soil layer overlying bedrock. A parametric analysis was performed varying depth to bedrock, input motion frequency, slope inclination, considering both a long duration and a short duration input motion. Typical effects of topographic amplification were observed, as seen in previous studies of slopes, and for other types of topography: 11

12 - Modification of the horizontal free-field motion - Generation of parasitic vertical motion - Zones of amplification and de-amplification on the ground surface - Frequency dependency, with the amplification being a function of normalised frequency of the input motion H/λ - Increase in topographic effects with slope inclination For the considered cases, it was found that soil layer effects have a greater influence to the ground motion than topographic effects. Comparing the overall amplification of the incoming ground motion at the slope crest (which includes topographic and soil layer effects), and the amplification of the incoming motion in the free field (soil layer effects only), it was seen that the soil layer effects dominate the overall response. More importantly, for the considered cases, topographic effects and soil layer effects were found to interact. The pattern of topographic amplification for the three bedrock depth cases considered differs, and also differs to the case of a slope in a halfspace, as seen in the results of Bouckovalas & Papadimitriou (2005) and Ashford et al (1997). It was found that the greatest interaction happens over the range of the soil layer s natural frequencies. This suggests that the contribution of topographic amplification and soil layer amplification are not easily separated. It should be though recognised that this study examined only cases of rigid bedrock and this assumption could have aggravated the interaction between topographic and soil layer effects. Furthermore it is important to note that the peak soil layer amplification and peak topographic amplification do not occur for the same normalised frequency of the input motion. It was observed that this interaction reduces with increasing bedrock depth. The pattern of topographic amplification for the Z=500 m case was seen to approach the results of Bouckovalas & Papadimitriou (2005) and Ashford et al (1997) for slopes in homogeneous half-space. It is considered this might be due to a reduction in topographic amplification with depth, or in part due to the soil layer s natural frequencies being outside of the range of frequencies where considerable topographic effects occur. This complicated interaction between soil layer effects and topographic amplification adds another variable that should be considered in making detailed assessments of topographic effects, along with other parameters which influence topographic effects, such as wave field nature, topography shape, etc. Acknowledgements This paper is published with the permission of the Head of the Geotechnical Engineering Office and the Director of Civil Engineering and Development, the 12

13 Government of the Hong Kong Special Administrative Region, who provided financial support to the third author during his studies at Imperial College. The authors are grateful for the anonymous reviews of this paper which helped to improve the clarity and completeness of the manuscripts considerably. 13

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17 Paolucci R., Faccioli E. & Maggio F. (1999). 3D Response analysis of an instrumented hill at Matsuzaki, Japan by a spectral method. Journal of Seismology, Vol.3, pp Pedersen H., Le Brun B., Hatzfield D., Campillo M., and Bard P.-Y. (1994a). Ground motion amplitude across ridges. Bulletin of the Seismological Society of America, Vol. 84, No. 6, pp Pedersen H.A., Sanchez-Sesma F.J. & Campillo M. (1994b). Three-dimensional scattering by two-dimensional topographies. Bulletin of the Seismological Society of America, Vol. 84, No. 4, pp Potts D.M. and Zdravkovic L. (1999). engineering. Thomas Telford, London. Finite element analysis in geotechnical Sanchez-Sesma, F.J. & Campillo, M. (1993). Topographic effects for incident P, SV and Rayleigh waves. In: F. Lund (Editor), New Horizons in Strong Motion: Seismic Studies and Engineering Practice. Tectonophysics, 218: Semblat J.F. & Pecker A. (2009). Waves and Vibrations in Soils: Earthquakes, Traffic, Shocks, Construction works. IUSS Press, Pavia, Italy. Semblat J.F., Duval A.M. & Dangla P. (2002), Seismic site effects in a deep alluvial basin: numerical analysis by the boundary element method, Computers and Geotechnics, Vol. 29, No. 7, pp Sitar N. and Clough G.W. (1983). Seismic response of steep slopes in cemented soils. Journal of Geotechnical Engineering, Vol. 109, No. 2, pp Stamatopoulos C.A., Bassanou M., Brennan A.J., Madabhushi G. (2007). Mitigation of the seismic motion near the edge of cliff-type topographies, Soil Dynamics and Earthquake Engineering, Vol. 27, No. 12, December 2007, Pages Yoshimura C., Bielak J., Hisada Y. & Fernández A. (2003). Domain Reduction Method for Three-Dimensional Earthquake Modeling in Localized Regions, Part II: Verification and Applications. Bulletin of the Seismological Society of America, Vol. 93, No 2, pp

18 Figures Figure 1: Geometry of the Two Dimensional Finite Element Model. 18

19 Figure 2: Wavelet time history for normalised frequency, H/λ =

20 Figure 3: Maximum absolute horizontal acceleration (a hmax ) behind the slope crest (i=90 ), for Z = 125 m and 500m, using the wavelet input motion of normalised frequency H/λ = 0.2 and corresponding 1D response. 20

21 Figure 4: Maximum absolute vertical acceleration (a vmax ) behind the slope crest (i=90 ), for Z = 125 m and 500m, using the wavelet input motion of normalised frequency H/λ =

22 Figure 5: Normalised horizontal acceleration (A hmax ) at the slope crest for Z = 125m ( wv stands for wavelet and ss for steady state). The results of Bouckovalas and Papadimitriou (2005) (referred to as B&P) and Ashford et al (1997) are also shown. 22

23 Figure 6: Normalised vertical acceleration (A vmax ) at the slope crest for Z = 125 m ( wv stands for wavelet and ss for steady state). The results of Bouckovalas and Papadimitriou (2005) (referred to as B&P) and Ashford et al (1997) are also shown. 23

24 Figure 7: Normalised horizontal acceleration (A hmax ) at the slope crest for Z = 250m ( wv stands for wavelet and ss for steady state). The results of Bouckovalas and Papadimitriou (2005) (referred to as B&P) and Ashford et al (1997) are also shown. 24

25 Figure 8: Normalised vertical acceleration (A vmax ) at the slope crest for Z = 250m ( wv stands for wavelet and ss for steady state). The results of Bouckovalas and Papadimitriou (2005) (referred to as B&P) and Ashford et al (1997) are also shown. 25

26 Figure 9: Normalised horizontal acceleration (A hmax ) at the slope crest for Z = 500m ( wv stands for wavelet). The results of Bouckovalas and Papadimitriou (2005) (referred to as B&P) and Ashford et al (1997) are also shown. 26

27 Figure 10: Normalised vertical acceleration (A vmax ) at the slope crest for Z = 500m ( wv stands for wavelet). The results of Bouckovalas and Papadimitriou (2005) (referred to as B&P) and Ashford et al (1997) are also shown, adopted from Bouckovalas and Papadimitriou (2005). 27

28 Figure 11: Free-field amplification ( 1D plot) and slope crest amplification ( i = 90 plot) of input motion for Z = 125 m ( wv stands for wavelet and ss for steady state). 28

29 Figure 12: Free-field amplification ( 1D plot) and slope crest amplification ( i = 90 plot) of input motion for Z = 250 m ( wv stands for wavelet and ss for steady state). 29

30 Figure 13: Free-field amplification ( 1D plot) and slope crest amplification ( i = 90 plot) of input motion for Z = 500 m ( wv stands for wavelet). 30

31 Figure 14: Maximum absolute vertical acceleration (a vmax ) at the crest (i=90 ) for Z=125m, 250m and 500m for the wavelet input motion and corresponding free-field response in term of maximum absolute horizontal acceleration (a hmax ). 31

32 Figure 15: Topographic amplification at slope crest and amplification function ( 1D analytical plot, values on right hand side scale), for Z = 125 m. The results of Bouckovalas and Papadimitriou (2005) (referred to as B&P) and Ashford et al (1997) for slopes in a homogeneous half-space are also shown ( wv stands for wavelet and ss for steady state). 32

33 Figure 16: Topographic amplification at slope crest and amplification function ( 1D analytical plot, values on right hand side scale), for Z = 250 m. The results of Bouckovalas and Papadimitriou (2005) (referred to as B&P) and Ashford et al (1997) for slopes in a homogeneous half-space are also shown ( wv stands for wavelet and ss for steady state). 33

34 Figure 17: Topographic amplification at slope crest and amplification function ( 1D analytical plot, values on right hand side scale), for Z = 500 m. The results of Bouckovalas and Papadimitriou (2005) (referred to as B&P) and Ashford et al (1997) for slopes in a homogeneous half-space are also shown ( wv stands for wavelet and ss for steady state). 34

35 Tables Table 1: Soil Parameters. Modulus of elasticity, E 1,333 MPa Mass density, ρ 2.0 Mg/m 3 Poisson s ratio, 1/3 Horizontal coefficient of earth pressure, K 0 Damping ratio, ξ 5% (achieved by varying Rayleigh constants) 1.0 Table 2: Summary of geometry and time history cases analysed in this study. Soil layer thickness, Z (m) Slope angle, i (degrees) Type of time history Normalised frequency (H/λ) of time histories considered Wavelet 0.01, 0.05, 0.1, 0.2, 0.3, 0.5, 1 Sinusoidal 0.01, 0.05, 0.1, 0.2, 0.3, 0.5, 1 45 Wavelet 0.05, 0.1, 0.2, 0.3, 0.5, 1 30 Wavelet 0.05, 0.1, 0.2, 0.3, 0.5, 1 10 Wavelet 0.05, 0.1, 0.2, 0.3, 0.5, Wavelet 0.01, 0.025, 0.05, 0.1, 0.15, 0.2, 0.3, 0.5, 1 Sinusoidal 0.01, 0.05, 0.1, 0.15, 0.2, 0.3, 0.5, 1 45 Wavelet 0.01, 0.025, 0.05, 0.075, 0.1, 0.15, 0.2, 0.3, 1 30 Wavelet 0.01, 0.025, 0.05, 0.075, 0.1, 0.15, 0.2, 0.3, 1 20 Wavelet 0.01, 0.025, 0.05, 0.075, 0.1, 0.15, 0.2, 0.3, Wavelet 0.01, 0.025, 0.05, 0.075, 0.1, 0.15, 0.2, 0.3, 1 35