Identification of Topographic Seismic Site Periods in Sloping Terrains

Abstract

The fundamental period of a terrain is a key parameter for characterizing the maximum soil amplification. Since the 1960s, research has been conducted for sloping terrains with a focus on evaluating topographic effects. However, few studies have focused on identifying whether the site topography induces an amplification peak that is associated with a characteristic period of sloping terrain. This study conducts a parametric analysis to identify a potential amplification pattern attributable to terrain geometry, using two-dimensional finite element models subjected to the action of a dynamic signal. The periods in which amplification peaks are generated are evaluated and compared with the amplification response recorded in the free field on horizontal terrain. The results reveal that the dynamic response of sloping terrain is a combination of the response from the surrounding terrain to the sloping zone and vice versa, and a distinctive amplification peak linked to the topography is identified. A new expression is proposed to define a topographic seismic site period in terms of shear wave velocity and the total soil thickness from the bedrock to the crest of sloping terrain. This study advances the processes of characterizing the seismic response of sloping terrains by demonstrating that the topographic seismic site period is consistent regardless of the slope angle. This provides engineers with a new dimension of analysis for the practical definition of criteria to determine topographic effects in design spectra.

1. Introduction

For the seismic design of buildings, defining the maximum demand for acceleration and displacement and identifying the periods during which these maximums occur play an important role in the development of the design response spectra for a given condition. The fundamental period of a terrain, T₀, is one of the parameters used to define the period of vibration in which the largest peak soil amplification can be expected. Therefore, for terrains with a horizontal surface, different seismic design codes use T₀ [1,2], either directly or indirectly, as one of the parameters to define the design response spectra according to the seismic site classification. Among other factors, it has been established that steep or complex terrain may be subject to potential seismic response amplifications, which would represent an additional demand on the structure compared to that which occurs in a low-slope condition. Recent studies have indicated that seismic amplification is significantly influenced by the input signal frequency and that current standards, such as Eurocode, might underestimate this effect. It suggests that a greater emphasis on topographic characteristics, along with soil properties, should be considered to accurately assess potential earthquake damage [3].

Since the 1960s, seminal works by Idriss and Seed [4] and Boore [5] established the importance of understanding topographic effects on seismic site response, particularly in sloping terrains, which has led to both numerical and experimental studies on the topographic effects on seismic site response [6,7,8,9,10,11]. Among other factors, it has been established that steep or complex terrain may be subject to potential seismic response amplifications, which would represent an additional demand on the structure compared to that which occurs in a low-slope condition [8,11,12,13,14]. However, few studies have focused on identifying whether the topography of a site induces an amplification peak associated with a characteristic period on the sloping terrain, which is not observed in the free field on horizontal terrain. In this regard, Idriss and Seed [4] suggest that the natural period of the soil column behind the crest of a slope can condition the amplification of motion, even more than the terrain geometry can. On the other hand, several studies agree with the formulation proposed by Boore [5] and Davis and West [15], who indicate that the periods of amplification depend on the wavelength-to-ridge dimension ratio. Thus, if the wavelength of the seismic waves is significantly greater than the geometric dimensions of the terrain, there would be no topographic effects associated with it.

To identify a characteristic period, T₀, in an earth dam, Dakoulas and Gazetas [16] developed an analytical solution based on a two-dimensional “shear-beam analysis”. For the case of a dam with a triangular cross-section with a height H and shear wave velocity Vs, they found that $T_0~2.61{\displaystyle \frac{H}{Vs}}$. Similarly, based on the study by Géli et al. [14], Paolucci [17] used the Rayleigh method to propose an analysis for the case of a symmetric triangular terrain with a height of H and an inclination angle of β, where he proposed that $T_0~{\displaystyle \frac{\left(Htantan \beta \right)}{\left(0.7Vs\right)}}$. For the case of a sloping terrain, which is the focus of the present study, Ashford et al. [18] used the term “topographic frequency, f_t” to describe the frequency of ground motion at which the maximum topographic amplification occurs, which is equal to $f_t={\displaystyle \frac{Vs}{5H}}$. They observed this occurs when the H/λ ratio is 0.2, where H represents the slope height, and λ denotes the wavelength of the motion.

On the other hand, Diaz-Segura [19], based on 2D and 3D numerical analysis and experimental measurements using the horizontal to vertical spectral ratio (HVSR) method, identified periods in which the highest spectral amplitudes of the response occur repeatedly along the slope of the terrain, and their magnitude is mainly conditioned by the stiffness and height of the surrounding terrain, rather than by the slope angle, β, of the terrain.

While some of the variables that affect the seismic response of sloping terrains have been identified through previous research (e.g., [10,19,20,21,22]), a characteristic period associated with a peak amplification that can be attributed mainly to the slope of the terrain has not been identified yet. The identification of such a characteristic period represents a contribution to the seismic characterization of terrains under potential topographic effects, as well as to the identification of the amplification sources in the response spectra for seismic design purposes.

The purpose of this study is to identify and characterize a topographic seismic site period T_t in sloping terrains that could aid in evaluating the variation in seismic response due to topographic effects. Therefore, to identify an amplification pattern attributable to the geometry of the terrain, a parametric analysis was carried out using two-dimensional finite element models subjected to the action of a dynamic signal. The periods in which amplification peaks were generated were evaluated and compared with the amplification response recorded in the free field on horizontal terrain with equal stiffness.

Based on the results, it was observed that the frequency content of the dynamic response of the sloping terrain is a combination of the response from the surrounding terrain to the sloping zone and vice versa, and an amplification peak attributable to the topography, i.e., to the discontinuity generated by the slope of the terrain was identified. Thus, a simple expression with a high determination coefficient, R², is proposed to define a topographic site period, whose equation form takes the same shape as the equation used for determining the soil fundamental period in terms of Vs and the total thickness of the soil measured from the bedrock to the crest of the sloping terrain, H_T. This research aims to provide a basis for improving seismic response analysis criteria in areas with complex topography by evaluating the relationship between the characteristic site period and the resulting amplification patterns.

2. Material and Methods

2.1. Terrain Characteristics Used

In this study, stratigraphic conditions and lateral heterogeneities were not considered to compare the spectral response using a uniform shear wave velocity, Vs, for the entire model domain. This simplification was intentional to isolate and focus on the topographic effects on seismic response, allowing for a clearer analysis of how terrain features influence spectral amplification. The uniformity of Vs provides a controlled environment to assess the pure impact of topography, free from the additional complexities introduced by stratigraphic variability and lateral heterogeneities.

By standardizing the shear wave velocity, it becomes easier to compare the findings with previous studies that adopted similar modeling assumptions. This approach aims to make the results more relevant to understanding topographic effects while also providing a basis for future studies to potentially incorporate more complex variables for further validation.

The parametric analysis focused on the inclination angle and stiffness of the terrain as principal variables, also allowing for a comparison of the potential variation of the period recorded on the sloping terrains, with respect to the period calculated in a one-dimensional soil column, where, for homogeneous and slope-free terrains, it is known that the period is equal to T₀ = 4H/Vs. Thus, the terrain was modeled with linear elastic behavior characterized mainly by the soil stiffness variation of the shear wave velocity, Vs.

A sensitivity analysis of the initial stress conditions on the principal analysis variables, amplification ratio and period, was performed. From this, no significant influence of the variables associated with the initial stress state was observed, for which geostatic conditions with k₀ = 0.4, γ = 18 kN/m³, and ν = 0.3 were defined for all models.

The range of values for β and H were defined based on criteria from various seismic design codes [23,24], which converge for angles below 20° and at heights below 30 m. Therefore, for the present study, the finite element parametric analysis was performed using β angles of 20°, 30°, 40°, and 50°; H_a values of 30 m, 50 m, and 100 m; and H values of 30 m, 60 m, 90 m, and 120 m. In this range, the potential influence of topography on the seismic response may not necessarily be significantly different from what is expected on a horizontal terrain.

The terrains were characterized with Vs values of 150 m/s, 350 m/s, and 500 m/s to cover a broad range of soil stiffness. These values were chosen to represent typical soil conditions encountered in engineering practice, ranging from softer, more deformable soils to stiffer, less deformable soils. By selecting this range of Vs values, the study aims to reflect the variability in soil conditions that can significantly influence seismic response. This approach aligns with the general guidelines found in standard seismic design codes, which often account for such variations in soil stiffness. Considering the combination of parameters, a total of 144 cases were modeled.

2.2. Overview of the Numerical Model and Terrain Characteristics Used

Since the present study is aimed at identifying the periods associated with a variation in the amplification ratio on a sloping terrain, dynamic response modeling was carried out using finite elements in the Plaxis 2D CONNECT Edition V22 [25], based on the geometric configuration shown in Figure 1. To achieve this, amplification ratios were determined by comparing Fourier acceleration spectra at different study points with the spectra recorded in the free field in horizontal zones and at the level of the bedrock, shown in Figure 1.

A geometric configuration as shown in Figure 1 and Figure 2 was used, whose main variables were the inclination angle, β, the shear wave velocity, Vs, and the heights of the base and crest of the slope up to the bedrock, H_a and H_b, respectively.

For the finite element model, a mesh composed of 6-node triangular elements with a maximum size of λ/15 was used, where λ is the wavelength associated with a maximum frequency of 20 Hz [25]. The free-field boundary was employed for the lateral boundary conditions, which allows the model surface to behave as a free surface, without any restrictions or influence from structures or other elements [25].

A time-domain, two-dimensional, finite element model was used, with its base resting on bedrock subjected to vertically propagating in-plane shear waves. This represented a conservative analysis from the point of view of topographic amplification compared to analyses performed using SH waves (out of plane) [26,27,28,29].

Since the study focused on analyzing the identification of fundamental vibration periods, a controlled excitation was applied to the model to decouple the influence of other frequency sources and identify the fundamental vibration modes of the terrain [21,22,24]. Following this approach, a sensitivity analysis was conducted on models activated with a single pulse, recording the free vibration response of the terrain until it ceased. The analysis included variations in period (0.1 to 0.5 s), amplitude (0.2 g to 1 g), and signal type, considering both rectangular and sinusoidal pulses. Based on this, no significant influence of the signal characteristics was observed in the magnitude and identification of the vibration periods of interest in this study.

Based on the above, the models were triggered with a single acceleration rectangular pulse characterized by an amplitude of 1 g and a period of 0.1 s, applied at the bottom boundary in the horizontal direction, and the vertical displacements were restricted, which was consistent with the rigid bedrock assumption.

The ratio of topography dimension to wavelength was identified in the literature as a key parameter affecting the magnitude of topographic effects [5,14,18,30,31]. However, it was recorded that the H/λ ratio does not show significant changes in the identified periods in terrains characterized as homogeneous [19]. Nevertheless, a wide range of H/λ values from 0.1 to 0.6 were used.

The wave incidence angle may affect the amplification effects. However, in the present study, aiming to identify the periods in which amplification peaks occur, site response analysis is conducted by evaluating the accelerations, and it is reasonable to consider vertically propagating waves, as acceleration magnitudes are generally not significantly affected by the incident wave angle [18,27].

3. Results and Discussion

3.1. Identification of Periods or Frequencies of Interest in a Sloped Terrain

For horizontally homogeneous terrain, the periods at which amplification peaks associated with different vibration modes are generated are estimated by using the well-known one-dimensional approximation:

$$T_n={\displaystyle \frac{4H_T}{\left(n+1\right)V_s}}$$

(1)

where H_T is the total thickness of the soil measured from the bedrock, and n is the vibration mode. Therefore, if the domain shown in Figure 1 is simplified and decomposed into three blocks, specifically as two lateral rectangular blocks and one central trapezoidal block, as shown in Figure 2, for points in free-field conditions in the lateral zones with horizontal surfaces, such as points A and B, it is possible to identify the vibration periods using Equation (1), provided these points are sufficiently far from the sloping area of the terrain and also that a constant Vs condition is maintained throughout the domain. However, since the dynamic response cannot be decoupled, the central block, which represents the sloping area, is influenced by the vibration of the surrounding blocks, and vice versa, both in terms of the amplitude and frequency content. While this coupled response and interaction among all blocks is present, there is an influence zone beyond which any analyzed point will exhibit spectral behavior analogous to the response in free-field conditions generated on a horizontal terrain. This influence zone is delimited by the distances X_a and X_b in Figure 2.

Focusing on the identification of the periods in which amplification ratio peaks occur, the periods were recorded in the different conducted models, with their representative response shown in Figure 3 for a reference case. Notably, this response was observed in the different modeled cases. In Figure 3, the presence of two main amplification peaks can be observed for different points located on the slope of the terrain, specifically the central trapezoidal block, whose associated period coincides with the primary mode of vibration of the lateral blocks, calculated using Equation (1) with H_T = H_a and H_T = H_b. Additionally, other peaks are recorded, some associated with values close to the period of the second mode of the vibration of the lateral blocks, although they are not the only peaks present, such as the case of the period identified as T_? in Figure 3. It is worth emphasizing that the periods identified in the studied models were practically independent of the inclination angle β or the location of the point on the slope.

Schematically, the observed response in the modeled cases is presented in Figure 4, which illustrates that the spectral behavior of a point that is located on the slope of the terrain is predominantly a composition of the dynamic response of the three blocks together, where the self-vibration of the lateral blocks, in particular, induces the highest amplification peaks. However, a consistent pattern was also observed; the presence of a peak that is not associated with the vibration of the lateral soil blocks, as shown in Figure 3, and highlighted in red in Figure 4. This peak may be related to the discontinuity caused by the inclination of the terrain.

While this study does not specifically focus on the magnitude of the amplification ratio peaks, a trend of alternating maximum values is observed based on the position of the analysis point, as shown in Figure 4. Specifically, for points near the base, the maximum occurs at the period associated with the first mode of the vibration of the lower block. Conversely, for points approaching the crest, the maximum amplification corresponds to the period of the tallest soil column.

Considering that the response at points X, Y, and Z in Figure 1 is a composition with significant influence from the lateral blocks, for the purpose of analysis, a simplified approach is used to subtract the free-field response at points A and B from the response of the points on the slope. Thus, if ARi defines the amplification ratio at point i on the slope of the terrain and ARa and ARb are the amplification ratios at points A and B in the horizontal free field (Figure 2), it is possible to calculate a dimensionless ratio between them, i.e., ${\displaystyle \frac{ARi}{\left(ARa\times ARb\right)}}$. This calculation allows for an indirect evaluation of the behavior of the response resulting from the vibration of the central block. Based on this analysis, as depicted in Figure 5, even after removing the potential effect of the lateral blocks, the amplification peak highlighted in red in Figure 4 persists and maintains the same period T_t for any point located on the slope of the terrain. Therefore, it is inferred that this response is primarily generated by the discontinuity caused by the slope of the terrain.

Considering the above, the response of points X, Y, and Z is analyzed for all the cases derived from the set of variables, totaling 144 combinations. In all cases, the period Tt associated with the previously mentioned amplification peak is identified. Therefore, in Figure 6 the identified periods are plotted as a function of the $H_T/Vs$ ratio, aiming to find a correlation while maintaining the general form of Equation (1). A fitted line with an R²-value of 99% is obtained, which allows for the determination of the T_t period, referred to as the topographic site period for practical purposes, using the following expression:

$$T_t={\displaystyle \frac{3H_T}{V_s}}$$

(2)

The results obtained highlight the impact of topographic features on seismic response, especially in urban environments where variable terrain can pose particular challenges for structural design. The identification of a characteristic period resulting from terrain slope is an analytical element that needs to be evaluated when performing a seismic classification of terrains near or on steep slopes, particularly in cases involving Chilean regulations, where the incorporation of the measured ground vibration period as a seismic classification parameter alongside propagation velocity is currently under discussion.

The use of 2D finite element models in this study was primarily aimed at isolating and understanding the fundamental effects of topography on seismic response. These models offer a straightforward approach to identifying primary amplification mechanisms due to terrain features. While this simplicity is a strength in terms of clarity and computational efficiency, it is clear that there is an inherent limitation in evaluating complex topographic conditions, for which it is recommended that future research extend to three-dimensional cases under different soil conditions. However, recognizing this limitation, it is important to note that 2D models, under more controlled conditions, contribute significantly to exploring topographic effects.

3.2. Identification of Limits X_a and X_b beyond Which T_t Is No Longer Present

Considering the presented results, it is clear that the central block is influenced by the vibration of the adjacent blocks; in turn, they will be influenced by the self-vibration of the central block. The limits of this influence towards each lateral block would be confined within distances X_a and X_b (Figure 2), beyond which the amplification peak associated with the T_t period no longer exists.

Given that, towards the base of the slope, the amplitude of the amplification ratio linked to T_t is reduced compared to that observed at the crest (Figure 3), it was noted that, even for distances X_a less than H, the T_t amplification peak was not perceptible.

On the other hand, towards the crest, this information can be of most interest for analyzing the response of structures located at the crest of a sloping area, since the identification of the distance X_b can be a relevant parameter for evaluating potential topographic effects. For this case, and as observed in a representative analysis case shown in Figure 7, it is found that, from an equivalent distance of approximately 3H, measured from the crest of the sloping area (X_b), the amplification peak associated with T_t is significantly reduced. It should be noted that, for the cases corresponding to H values of 90 m and 120 m, the presence of the amplification peak associated with T_t decreases for distances shorter than 3H, which is expected considering the significant magnitude of the distance X_b in these cases.

4. Conclusions

In the present study, the dynamic response of an inclined terrain was evaluated with the objective of identifying peaks in the amplification ratio that may be associated with the discontinuity generated by the slope of the terrain to obtain a vibration period linked to its site topography.

A parametric analysis focused on the influence of the terrain slope and identified a characteristic period at which an amplification peak occurs, which is not observed under the same soil stiffness conditions in horizontal terrains. Given that topographic amplification effects are frequency dependent, this finding opens the door to defining period ranges, in conjunction with the periods associated with the vibration of the surrounding soil, where structures located on sloping terrains may experience a variation in demand compared to that recorded on horizontal terrains. Therefore, it is recommended that future research evaluates the implications that the period T_t could have on the response spectra for the design of structures located on sloping terrains, extending to three-dimensional cases under different soil conditions.

For all the analyzed cases that corresponded to homogeneous terrains, a pattern was observed in the response of the points located on the slope of the terrain, which included the presence of an amplification peak. This peak was generated due to the interaction between the vibration of the sloped area and the surrounding terrain and was associated with a characteristic seismic site period, T_t, which is independent of the slope angle of the terrain.

To determine the period T_t, referred to as the topographic period T_t, an expression was fitted with a coefficient of determination (R²) of 99%. This expression was defined as $T_t={\displaystyle \frac{3H}{Vs}}$, where H_T represented the height of the slope, and Vs represented the shear wave velocity of the soil.

Finally, it was observed that, beyond a distance equivalent to 3H, measured from the crest of the inclined zone, X_b (Figure 2), the amplification peak associated with T_t significantly decreases.