Quantification of the effects of the spatial variation of ground motions on the seismic response of highway bridges

Abstract

The stochastic responses of highway bridges to spatial variation of ground motions (SVGM) are analysed in this paper. A model of spatially varying ground motions is used to investigate the relative importance of the incoherency effect, the wave passage effect and the site effects on the stochastic dynamic response of an asymmetrical R.C box girder highway bridge with variable inertia. In this study, the incoherency effect is investigated using two widely used models while the wave-passage effect is incorporated using various wave velocities. Then, the random vibration theory is applied to study the effect of the non-uniform seismic excitations on the bridge structure. The bridge response is evaluated in terms of the mean values of the maximum displacements and the bending moments. Analyses of both stationary and transient response are performed. The results show that the stochastic dynamic responses related to site effects are mostly much greater than those calculated using uniform, delayed and incoherent seismic excitation assumptions. As a result, analytical models used for the stochastic dynamic analysis of long span highway bridges should take into account all the SVGM components, particularly the site-response effects.

Keywords

Stochastic highway bridge 3-D FEM random vibration incoherency wave-passage effect site effect

1 Introduction

Previous studies showed that significant variations in the earthquake ground motions would occur at the support points of long span structures such as bridges, dams and pipelines. These variations may be caused by a variety of Spatial Variation of Ground Motion (SVGM) sources. Physically, the SVGM is the result of the combination of three different components: the incoherency effect, the wave-passage effect and the site effects. The incoherency effect may result from seismic wave reflections and refractions through the soil during the wave propagation while the wave-passage effect may be caused by the difference in the arrival times of seismic waves at different points. However, the site effects are due to the spatially varying site conditions resulting from the differences in local soil conditions at different points [1, 2]. Several researchers have early investigated the responses of long span structures to multi-support excitations. For example, but not limited to, Haroun and Abdel Hafiz [3] and Zerva and Zervas [4] investigated the effects of the earthquake motion amplitude and phase difference on the seismic response of long earth dams. They concluded that the dam’s response to wave passage can be greatly amplified. Various bridge configurations have also been studied by various researchers [5 –7]. It was shown that the use of differential support ground motions significantly increased the seismic response of structures, especially for more rigid bridges and for bridges resting on soils with different local soil conditions. Ouanani et al. [8, 9] investigated the nonlinear dynamic response of a cable-stayed bridge considering spatially varying site conditions and the incoherency effect. They concluded that the SVGM has a greater impact on the seismic response of cable-stayed bridge structures than the synchronous motion. Adanur et al. [10] studied the SVGM on the dynamic behavior of a suspension bridge using various random vibration methods. They found that the structural responses, for each method, highly depend on the intensity and frequency contents of ground motions. The SVGM can be important over relatively short support distances due to variations in seismic source radiation and velocity. Recently, Ghannoum. et al. [11], showed the importance of the SVGM effects on a frame structure, which was modeled is a frame of four columns linked with beams where the distance between the columns was considered of 10 m.

The SVGM is more important on stiff structures and it does not significantly affect the response of flexible structures [12]. The pseudo-static component of the structural response is responsible for the increased influence of the SVGM on stiff structures, as opposed to flexible structures in which the total response is dominated by the dynamic component [13].

The goal of the present study is to determine the importance of the SVGM components on the stochastic dynamic responses of an asymmetrical R.C box girder highway bridge with variable inertia under spatially varying earthquake ground motions. The random vibration theory is used to investigate the impact of the different components of the SVGM on the seismic responses of the highway bridge’s superstructure and substructure. For this aim, the analytical soil amplification model of Safak [14] associated to two spatial coherency function models (Luco and Wong [15]; Harichandran and Vanmarcke [16]) are integrated in a random vibration formulation. For uniform ground motion assumption, the incoherency effect, the wave-passage effect and the site effects on the bridge response in terms of the mean values of the maximum vertical displacements and bending moments in the bridge’s superstructure and substructure are evaluated. Furthermore, the numerical results of the stationary and transient analysis are compared and some engineering significance conclusions are provided.

2 Spatial variation of ground motions (SVGM)

First of all, the seismic excitation at the bedrock is considered as a white noise with the filter of Clough and Penzien [17]: $S (ω) = \frac{(ω_{g}^{4} + (2 ξ_{g} ω_{g} ω)^{2}) ω^{4}}{((ω_{g}^{2} - ω^{2})^{2} + (2 ξ_{g} ω_{g} ω)^{2}) ((ω_{b}^{2} - ω^{2})^{2} + (2 ξ_{b} ω_{b} ω)^{2})} S_{0}$ (1) where ω _g and ξ _g are, respectively, the circular frequency and damping ratio of the first filter while ω _b and ξ _b are those of the second filter. In the present study, the first filter parameters are taken as ω_g = 4π rad/s and ξ_g = 0.6, which correspond to firm soil conditions and the second filter parameters are ω_b = 1.636 and ω_b = 0.619 are. S₀ is a scale factor.

The seismic excitation at the surface is obtained considering the site effect as follow [18]: $S^{surface} (ω) = {| H_{S} (ω) |}^{2} S (ω)$ (2) where H_S(ω) is the soil amplification function [14]: $H_{S} (ω) = \frac{2 (1 + r - \frac{i}{4 Q}) exp [- i ω τ_{S} (1 - i / 2 Q)]}{1 + (r - \frac{i}{4 Q}) exp [- 2 i ω τ_{S} (1 - i / 2 Q)]}$ (3)

In Equation (3), Q represents the quality factor and τ_s the time propagation, which is function of the soil layer depth h and the shear wave velocity V_S as follow

$τ_{S} = \frac{h}{V_{S}}$ (4)

The reflection coefficient r is given by Aki and Richard [19] as: $r = \frac{ρ_{R} V_{R} - ρ_{S} V_{S}}{ρ_{R} V_{R} + ρ_{S} V_{S}}$ (5)

Where ρ_R, ρ_s and V_R, V_S are the densities and shear wave velocities for the bedrock and the soil, respectively.

After that, the incoherency effect is captured by the coherency function, γ_ij, which is the ratio between the acceleration cross-spectral density S_ij(ω) of the i and j stations and the square root of the product of the spectral density functions of accelerations at the site i and the site j, S_i(ω) and S_j(ω), respectively, as $γ_{ij} = \frac{S_{ij} (ω)}{\sqrt{S_{i} (ω) S_{j} (ω)}}$ (6)

Several empirical and theoretical models have been developed by leading researchers such as Der Kiureghian [2], Harichandran and Vanmarcke [16], Loh and Yeh [20] and Luco and Wong [15].

The global coherency function model incorporating the different components of the SVGM (incoherency, wave-passage and site effects) takes one of the following two forms: $γ (d, ω) = γ^{i} (d, ω) \times γ^{w} (d, ω) \times γ^{s} (d, ω)$ (7) $γ (d, ω) = γ^{i} (d, ω) \times \exp [i (θ^{s} (d, ω) + θ^{w} (d, ω))]$ (8) where γⁱ (d, ω) , γ^w (d, ω) andγ^s (d, ω) expresses, respectively, the incoherency effect, the wave passage effect and the local site effects.

The wave passage effect due to the difference in the arrival time of the waves at the supports is defined as: $θ^{w} (d, ω) = exp (- i \frac{ω \times d}{v_{app}})$ (9)

In Equation (9), d denotes the distance between two stations i and j and v_app represents the apparent wave velocity.

The site-effect is modelled by the soil transfer functions, H_i(ω) and H_j(ω) at the two points i and j with the phase shift given by: $θ_{AB}^{s} (ω) = {tan}^{- 1} \frac{Im | H_{i} (ω) H_{j}^{*} (ω) |}{Re | H_{i} (ω) H_{j}^{*} (ω) |}$ (10) where Im and Re are, respectively, the imaginary and the real parts of a complex number.

For the incoherency effect, two models are usually used. They are those proposed by Luco and Wong [15] and Harichandran and Vanmarcke [16] as shown in Table 1. In this table, ω is the circular frequency and d is the distance between two support points.

Table 1

Coherency models for incoherency effect

Model name	Model equation
Luco and Wong (1986)	(1) $γ^{i} (d, ω) = exp [- {(\frac{α \times ω \times d}{V_{s}})}^{2}]$
Harichandran and Vanmarcke (1986)	(2) $\begin{matrix} γ^{i} (d, ω) = A exp [\frac{- 2 α}{α \times θ (ω)} (1 - A + α \times A)] + \\ (1 - A) \times exp [\frac{- 2 d}{θ (ω)} (1 - A + α \times A)] \end{matrix}$
$θ (ω) = k {[1 + {(\frac{ω}{2 π f_{0}})}^{b}]}^{- 1}$

⁽¹⁾α is factor representing the loss of coherency, Vs the shear wave velocity of the soil. The value α/V_S = 2.5 × 10^-4 proposed by Luco and Wong is used for this study. ⁽²⁾A, α, k, f₀ and b are model parameters. In this study the values obtained by Harichandran and Vanmarcke (A = 0.636, α=0.0186, k = 31200, f₀ = 1.51 Hz and b = 2.95) are used.

Figure 1 displays the incoherency effect for two support separations, d = 100 m and d = 512 m. The Luco and Wong model is fully correlated at low frequencies, which means that it will underestimate the effect of the spatial variation of the ground motions for the highway bridges. However, there is only a partial correlation in the Harichandran and Vanmarcke model meaning that this model may overestimate the impact of the SVGM on the response of highway bridges because it shows a greater loss of coherence at low frequencies for large separation distance. In general, the incoherency effect decreases as both frequency and separation distance increase.

Fig. 1

Schematic illustration of the incoherency effect for: (a) Luco and Wong model, (b) Harichandran and Vanmarcke.

3 Stochastic response

3.1 Stationary response

For the multiple support excitations, the total mean-square responses can be calculated using the formula z = z_d + z_s, which is commonly used to decompose any response into a dynamic component and a pseudo-static component [17, 21] as: $σ_{z}^{2} = σ_{z_{d}}^{2} + σ_{z_{s}}^{2} + 2 cov (z_{s}, z_{d})$ (11)

Where σ_zs and σ_zd are the variances of the pseudo-static response and the dynamic response, respectively, and the term Cov(z_s, z_d) is the covariance between the pseudo-static component (σ _zs ) and the dynamic component (σ _zd ). These quantities can be written, respectively, as $\begin{matrix} σ_{z_{d}}^{2} = \sum_{j = 1}^{n} \sum_{k = 1}^{n} ψ_{j} ψ_{k} \int_{- \infty}^{\infty} \\ (\sum_{l = 1}^{r} \sum_{m = 1}^{r} Γ_{lj} Γ_{mk} H_{j}^{*} (ω) H_{k} (ω) S_{lm} (ω)) d ω \end{matrix}$ (12) $σ_{z_{s}}^{2} = \sum_{l = 1}^{r} \sum_{m = 1}^{r} B_{l} B_{m} \int_{0}^{\infty} \frac{1}{ω^{4}} S_{lm} (ω) d ω$ (13)

and $cov (z_{s}, z_{d}) = \sum_{j = 1}^{n} \sum_{l = 1}^{r} ψ_{j} B_{l} \int_{0}^{\infty} \frac{1}{ω^{2}} \sum_{m = 1}^{r} Γ_{mj} H_{j} (ω) S_{lm} (ω) d ω$ (14)

In the above equations, r is the number of the support degrees of freedom where the ground motion is appliedand n the number of the modes used in the analysis. B_l is the response z due to a unit displacement of the support DOF l. S_lm is the cross spectral density function of the accelerations between the supports l and m and Γ is the modal participation factor. Ψ _j is the response z from the j^th mode and H(ω) the modal frequency response function.

3.2 Transient response

The transient responses effect can be studied using the Heaviside modulating function [21, 22].

$\begin{matrix} H_{k} (ω, t) = H_{k} (ω) [1 - e^{- ξ_{k} ω_{k} t} e^{- i ω t} \times \\ (cos ω_{kd} t + \frac{(ξ_{k} ω_{k} + i ω}{ω_{kd}} sin ω_{kd} t)] \end{matrix}$ (15)

In Equation (15), $ω_{kd} (ω_{kd} = ω_{k} \sqrt{1 - ξ_{k}^{2}})$ is the damped circular frequency. The transient responses is calculated by substituting the normal frequency response function with the function H_k(ω,t) in the expressions of the stationary response.

3.3 Mean response

According to the random vibration theory, the mean of the maximum response value may be written as $E [max X (t)] = p σ$ (16)

Where p is the peak factor and σ is the standard deviation of the total response [23, 24]. $p = {\begin{matrix} 1.253 + 0.001 ν_{e} T 0 < ν_{e} T \leq 2.1 \\ {(2 ln ν_{e} T)}^{1 / 2} + \frac{0.5772}{{(2 ln ν_{e} T)}^{1 / 2}} 2.1 \leq ν_{e} T \leq 1000 \end{matrix}$ (17) $\frac{v_{e}}{v} = {\begin{matrix} 2 δ 0 < δ \leq 0.1 \\ (1.63 δ^{0.45} - 0.38) 0.1 \leq δ \leq 0.69 \\ 1 0.69 \leq δ \leq 1 \end{matrix}$ (18) where v_e is the equivalent mean zero-crossing rate and T the duration of the ground motion process (T is taken equal to 20 s in the present study). v is the frequency of occurrence and δ the form factor expressed, respectively, as $v = \frac{1}{π} {(\frac{λ_{2}}{λ_{0}})}^{1 / 2}$ (19) $δ = {(1 - \frac{λ_{1}^{2}}{λ_{0} λ_{2}})}^{1 / 2}$ (20)

The spectral moments λ_m, appearing in the preceding equations are expressed in terms of the power spectral density of the ground motion as follows $λ_{m} = 2 \int_{0}^{\infty} ω^{m} S (ω) d ω$ (21)

4 Validation of the calculation program

Before to carry a case study and in order to ensure the accuracy of the calculation program, elaborated in the Matlab environment, translating the different steps presented in the previous section, an example already treated by Harichandran and Wang [25] is examined here. The example consists in the evaluation of the response components of a two-span continuous beam. The incoherency and wave passage effects are assumed to cause seismic excitations in the three beam supports (Fig. 2).

Fig. 2

Two span beam under SVGM ( $L = 30 m, V = 1000 \frac{m}{s}, ω_{1} = 18.84 rad / sec$ ).

The results are obtained in terms of the normalized displacements variance over the left span of the beam. The normalization is performed by dividing by the maximum total response. It is clear from this figure that the results obtained by the Matlab program elaborated in this study produce identical results as those of Harichandran and Wang [25] (Fig. 3).

Fig. 3

Normalized displacements variance.

5 Case study: Azazga highway bridge

It is aimed in this case study to assess the dynamic response of the Azazga highway bridge under SVGM. Firstly, the Azazga bridge will be described, then its response will be presented.

5.1 Description of the Azazga bridge

The Azazga highway bridge (Fig. 4(a)) is selected as a case study. It is located in Northern Algeria in a seismic zone IIa [26] characterized by a Peak Ground Acceleration (PGA) equal to 0.25 g. The studied bridge is 512 m with four main spans in prestressed concrete of 100 m and two side spans of 56 m as indicated in Fig. 4(a). The bridge deck consists of a unicellular box girder with 9.50 m width and variable height of 5.95 m at the piers and 2.70 m at abutments (Fig. 2(b)). The bridge piers possess variable heights and have identical hollow rectangular cross sections (Fig. 4(c)).

Fig. 4

Description of the Azazga highway bridge: (a) bridge elevation, (b) segment cross sections, (c) Pier cross sections.

The considered bridge is modelled as a lumped mass system divided into a number of small discrete 3D frame elements. The 3D Finite Element Model (FEM) of the bridge is divided into 49 small discrete 3-D frame elements, 7 rigid elements and two spring elements modelling the Normal Rubber Bearings (N.R.B.) at two seat abutments. The 3D FEM of the studied bridge is represented by 292 degrees of freedom. In addition, the first three vibration frequencies (i.e. the 1^st mode in the vertical direction, the 2^nd mode in the lateral direction and the 3^rd mode in the longitudinal direction) have been obtained equal to 1.74 Hz, 1.83 Hz and 1.918 Hz, respectively. The discrete model of the studied bridge is approximated by the 3-D FEM model as presented in Fig. 5.

Fig. 5

Discrete model of the highway bridge.

5.2 Azazga bridge responses

The bridge is studied stochastically considering the separate and the combined effects of the SVGM components. Both stationary and transient bridge responses are evaluated and compared.

The bridge response in terms of the mean values of the maximum vertical displacements and the bending moments in the superstructure and the substructure are evaluated taking into account of the incoherency effect, the wave-passage effect and the site-response effects. The case of uniform ground motion where the two abutments and intermediate piers bridge are assumed to be founded on soil type S₁ (firm site condition) as defined in RPOA [26] is also considered for comparison. For the site-response effect, the two abutments, the first and the last piers, are assumed to be founded on soil type S₁ (firm site conditions) while the intermediate piers (2,3,4 and 5) are assumed to be founded on soil type S₂ (medium site conditions). The shear wave velocities for medium (S₂) and firm (S₁) soil types are considered equal to 400 m/s and 800 m/s, respectively. The soil layer depth (h) and the quality factor (Q) are taken equal to 150 m and 30, respectively.

5.2.1 Stationary responses

Figure 6 shows the effect of the wave propagation on the mean of the maximum value of the vertical displacement for different apparent wave velocities (300 m/s, 600 m/s, 1200 m/s). It is seen from Fig. 4, that the deck displacement is generally overestimated under the uniform ground motion assumption. The wave passage effect decreases with increasing velocities.

Fig. 6

Mean of maximum values of vertical displacement along the deck of the Azazga bridge under uniform excitation and wave passage effects.

Figure 7 displays the mean of the maximum vertical displacements calculated for both incoherency models and uniform ground motion. It can be observed from this figure that the uniform ground motion assumption and the two-incoherency models (Harichandran and Vanmarke and Luco and Wong) produce nearly the same responses. This result is expected because the frequencies of the studied bridge are situated in the low frequency range and the separation distance d = 100 m where the two models of coherency present the full correlation.

Fig. 7

Mean of maximum values of vertical displacements along the deck of the Azazga bridge under uniform excitation and incoherency effect.

In order to compare the response under the combined effects to that produced by the uniform ground motion, the normalized three components of the vertical deck displacement (dynamics, pseudo-static and total) are presented in Fig. 8. The normalization is performed by dividing the displacement values by the maximum total displacement. It is observed from Fig. 8(a) that the pseudo-static displacement component for the uniform ground motion model remains constant because of the rigid body motion assumption. This component has a significant contribution (i.e. 50% of the maximum total response). At the bridge deck where the maximum total displacement takes place, it can be observed that the contribution of the dynamic component is 48% of the maximum total response. When considering the combined effects (Fig. 8(b)), it is found that the contribution of the dynamic component increases at the central spans of the bridge (58%) while the pseudo-static component still providing a significant contribution (42%).

Fig. 8

Normalized mean of maximum values of vertical displacements along the deck of the Azazga bridge: a) Uniform excitation, b) Combined effect (SVGM).

Now, it is intended to investigate the relative importance of the incoherency effect, the wave passage effect and the site effects on each component of the bridge response. For this end, the mean of the maximum displacements along the bridge deck for the three components (pseudo-static, dynamic and total) are calculated and compared in Fig. 9 with the case of uniform ground motion.

It can be observed from the Fig. 9(a) that the dynamic displacements along the bridge are generally overestimated under the assumption of site-response effects, whereas, the displacements when considering the incoherency effect and the wave-passage effects are smaller than the displacement given under the assumption of identical seismic excitation. Similar conclusions are also observed for the total displacement Fig. 9(c).

Fig. 9

Mean of maximum values of displacements along the Azazga bridge deck under uniform excitation and SVGM components.

For the different ground motion models, the mean of the maximum vertical pseudo-static deck displacements is shown in Fig. 9(b). The displacements given by the incoherency and wave-passage effects are very close to those obtained for the uniform ground motion model, as the displacements obtained from this model are constant along the bridge due to the rigid body motion. The results, however, are much larger when the site-response effect is taken into account.

The mean of the maximum bending moments along the bridge deck are calculated and presented in Fig. 10.

Fig. 10

Mean of maximum values of bending moment along the Azazga bridge deck under uniform excitation and SVGM components.

From Fig. 10, it is clearly seen that the bending moment at the central spans are maximum regardless of the used ground motion (uniform ground motion or SVGM). It is found that, compared to the uniform ground motion, the site effects contribute by an increment of 214%. Nevertheless, the uniform ground motion and the SVGM incorporating the incoherency effects develop close bending moments to one another, whereas the wave passage produces generally the lower responses.

Figure 11 depicts the variations of the mean of the maximum values of the displacement along the bridge piers under the uniform excitation as well as under the SVGM. It is clear from this figure, that the displacements under investigation are more important in the tallest pier whatever the used ground motion model (uniform excitation or SVGM components). It is also observed that the mean of the maximum bridge piers displacements due to site effects and those due to combined effects are more important than those produced by the uniform ground motion or other SVGM components. The mean of the maximum total bending moments along the pier heights are calculated and compared in Fig. 12 for the uniform ground motion and for the three main components of the SVGM defined above. It can be seen from this figure that the concerned bending moments are much larger in the tall piers than those in the short piers regardless of the used ground motion model. One may note from Fig. 12 that the largest mean maximum values at the tall piers are those brought by site effects and combined effects.

Fig. 11

Mean of maximum values of bridge piers displacements under uniform excitation and SVGM excitation: (a) shortest pier, (b) tallest pier.

Fig. 12

Mean of maximum values of bending moments on bridge piers under uniform excitation and SVGM excitation: (a) shortest pier (b) tallest pier.

5.2.2 Transient responses

This section is dedicated to obtain the transient responses under the combined effects of the SVGM for several durations of the strong shaking (10 s, 15 s and 20 s). The SVGM is characterized by the Luco (1986) coherency model with the parameters α/V_S = 2.5 × 10^-4 and V_app = 600 m/s. The two abutments, the first and the last piers (2 and 6) are assumed to be founded on soil type S₁ (firm site conditions) while the intermediate piers (3, 4 and 5 piers) are assumed to be founded on soil type S₂ (medium site conditions).

The mean of the maximum values of the transient response along the bridge deck in terms of the vertical displacements and the bending moments are plotted in Fig. 13(a) and Fig. 13(b), respectively. In the same figures, the transient responses are compared to stationary ones.

Fig. 13

Mean of maximum values of transient response along the bridge deck: (a) vertical displacement, (b) bending moment.

From Fig. 13, it should be noted that the transient displacements and moments along the bridge deck are relatively close to the stationary ones at side-spans. However, the central span transient displacement and the bending moment values are smaller than the stationary ones especially for the 10^th second. It’s important to note that after 10, 15 and 20 seconds of the transient response, 72, 83 and 91% of the stationary response are reached at the deck point where the maximum displacement occurs.

The mean of the maximum transient displacements at the short and the tall pier heights of the Azzazga bridge are illustrated in Figs. 14.

Fig. 14

Mean of maximum values of transient displacements along the bridge piers: a) shortest pier, b) tallest pier.

It is evident from Figs. 12 that, at the top of the bridge piers, the stationary displacements are the greatest which is in accordance with the trends discussed in Fig. 13.

6 Conclusions

In this paper, the stochastic dynamic responses of a highway bridge under spatially varying earthquake ground motions are analysed. In order to illustrate the different sources of SVGM on the bridge responses, a 3-D FEM model of an asymmetric bridge: the Azazga highway bridge (Northern Algeria) is performed. The bridge response in terms of mean values of the maximum vertical displacements and the bending moments in the superstructure and the substructure of the studied bridge are evaluated under the uniform ground motion as well as under the spatially varying ground motions (SVGM) incorporating the incoherency effect, the wave-passage effect, the site-effects and their combination

In the light of the obtained results, the following conclusions may be drawn:

The stochastic dynamic responses obtained under the assumption of SVGM incorporating site-effects are in general much larger than those obtained under the assumption of uniform ground motion. This study mainly implies that the site-effects should be considered in the stochastic seismic analysis of long span highway bridges

The obtained results show that the incoherency and the wave passage has no significant effect on the maximum values of the dynamic vertical displacement along the deck and the bridge piers because the sstructure’s displacement to non-uniform excitation is a response to averaged excitations of all support. For the case of the wave passage, this averaged excitation considering the time delay in the arrival of the ground motion at neighboring supports is automatically less than the uniform excitation. For the incoherency effect, the frequencies of the studied bridge are situated in the low frequency range and the coherency model produces nearly the same responses as the uniform excitation model. In this case, the uniform seismic excitation model leads to conservative results. It is interesting to note that the results obtained by uniform ground motion can be conservative for the displacements and may be unconservative for the moments due to the contribution of the pseudo-static component in the case of spatially varying ground motions.

It appears that the stationary assumption of the highway bridge is reasonable for the ground motion durations greater than 15s based on a comparison between the transient response, obtained for various durations of the strong shaking, and the stationary responses.

Because of the complex nature of the problem, it is difficult to make general conclusions based on this study of a single highway bridge. However, as different highway bridge models show typically similar structural dynamics, this study mainly implies that long span bridges are sensitive to SVGM and in the stochastic analysis of such type of structures, the incoherency effect, the wave-passage effect and, especially, the site effects should be considered.

Statements & Declarations

Funding

The authors have no known competing financial interests to declare.

Conflict of interest

There is no conflict of interest.

Author contributions

All authors contributed to this study. Data collection and analysis were performed by [Nassira Belkheiri] and [Boualem Tiliouine]. The first draft of the manuscript was written by [Nassira Belkheiri] under the supervision of [Boualem Tiliouine] and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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