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: +44 20-7594-5996 Fax: : +44 20-7594-5934

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.

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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, 2006 2007, 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.

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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 (L1=L2=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 wave-

field 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)

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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, Tp 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= Tp

/40) of the predominant period Tp 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:

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(2)

where is the wavelength corresponding to the highest frequency of the considered

input pulses.

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 sub-

structuring 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 free-

field 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 Γe 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

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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 (ahmax and avmax) for a series of points on the ground surface behind the slope

crest (distance xc 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 (ahff), 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.

xc=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 xc 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

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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 (ahff),

which is purely horizontal, obtained from the 1D model. These normalised

acceleration values are notated as Ahmax for horizontal motion, and Avmax 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 half-

space.

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 Ahmax 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 Avmax values in Figures 6, 8 and 10.

- 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 de-

amplification 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

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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 (Ahmax and Avmax) 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 (Ahmax and Avmax) 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

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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 (Ahin), 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

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(from Kramer, 1996), while the natural frequencies of soil layer are determined by

Equation 4 (from Kramer, 1996).

| ( )|

√ (3)

where k is the wave number.

( ) , n=0, 1, 2,…, ∞ (4)

The dimensionless frequency (H/ ) can be expressed in terms of the predominant

period of the input motion (Tp), the slope height and the shear wave velocity of the

soil ( ) as follows:

(5)

which for resonance conditions with the natural site periods, substituting (4) into (5)

becomes:

( )

(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 (Ahmax < 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.

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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 half-

space. 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:

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- 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 half-

space, 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

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.

14

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Figures

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

19

Figure 2: Wavelet time history for normalised frequency, H/λ = 0.2.

20

Figure 3: Maximum absolute horizontal acceleration (ahmax) 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.

21

Figure 4: Maximum absolute vertical acceleration (avmax) behind the slope crest (i=90°), for Z = 125 m and 500m, using the wavelet input motion of normalised frequency H/λ = 0.2.

22

Figure 5: Normalised horizontal acceleration (Ahmax) 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.

23

Figure 6: Normalised vertical acceleration (Avmax) 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.

24

Figure 7: Normalised horizontal acceleration (Ahmax) 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

Figure 8: Normalised vertical acceleration (Avmax) 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.

26

Figure 9: Normalised horizontal acceleration (Ahmax) 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.

27

Figure 10: Normalised vertical acceleration (Avmax) 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).

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).

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).

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).

31

Figure 14: Maximum absolute vertical acceleration (avmax) 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 (ahmax).

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).

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).

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).

35

Tables

Table 1: Soil Parameters.

Modulus of elasticity, E 1,333 MPa

Mass density, ρ 2.0 Mg/m3

Poisson’s ratio, 1/3

Horizontal coefficient of earth pressure, K0

1.0

Damping ratio, ξ 5% (achieved by varying Rayleigh constants)

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

125 90 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, 1

250 90 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, 1

500 90 Wavelet 0.01, 0.025, 0.05, 0.075, 0.1, 0.15, 0.2, 0.3, 1