# Influence of Multiple-Support Excitation on Seismic Response of Reinforced Concrete Arch Bridges
## 3 Methodology  
### Model of Spatial Variability of Ground Motion  
The ground motion spatial variability is often probabilistically described using the space-time random field of ground motion, with data obtained from the series of dense instrument arrays. Thus, this model can be expressed through cross-power spectral density of the motions \(S(\xi_{jk},\omega)=S(\omega)\gamma_{jk}(\xi_{jk},\omega)\) where \(S(\omega)\) is the power spectral density function of the motions, \(\xi_{jk}\) is the distance of supports \(j\) and \(k\), and \(\gamma_{jk}(\xi_{jk},\omega)\) is the complex coherency. The complex coherency \(\gamma_{jk}(\xi_{jk},\omega)\)for each frequency range contains the amplitude term representing coherency loss \(|\gamma_{jk}(\xi_{jk,\omega})|\) and the phase term \(\theta_{jk}(\xi_{jk,\omega})\) representing the wave-passage effect. The coherency loss is usually described by means of the lagged coherency, which is the measure of linear statistical dependence between seismic motions. For the perfectly linearly dependent processes the value is one, while it is zero for the completely independent processes. It is expected that at short distances and at low frequencies the motions will be similar, while for large distances and at large frequencies the motions will become uncorrelated [8]. Most lagged coherency functions have been obtained empirically and semi-empirically using recorded data at strong motion arrays. Some of the empirical functions, developed mostly from the SMART-1 array, are those of [PERSON] and [PERSON] [43], [PERSON] et al. [44], [PERSON] et al. [45], and [PERSON] [46]. The functional form of semi-empirical models is analytically based, but parameters are evaluated from recorded data. Some of them are those proposed by [PERSON] et al. [47], [PERSON] [48] and [PERSON] [49]. The selection of coherency model significantly affects the seismic response of structures subjected to multi-support excitation [7]. Hence, the recommendations for choosing the appropriate model from a large number available are necessary. The comprehensive discussion on this topic is presented in [8], where simulations of the motions from the two models very common in engineering applications, those of [PERSON] and [PERSON] [46] and of [PERSON] and [PERSON] [49], are compared. An important consideration in the selection of the coherency model is the site classification. The lagged coherency at rock sites has a flatter exponential decay than the lagged coherency at the softer soil sites. According to [7], coherency models at rock sites are limited. [PERSON] [50] proposed an empirical coherency model for rock sites based on recordings at the Pinyon Flat array. [PERSON] and [PERSON] coherency model with a range of slower exponential decays, compatible with the trend of recorded data, can also be used as representative for the coherency at rock sites. Therefore, in this study the loss of coherency was modelled using this function, given by the expression:  
\[\left|\gamma_{jk}(\xi_{jk},\omega)\right|=e^{-\alpha^{2}\omega^{2}\xi_{jk}^{2}} \tag{1}\]  
in which parameter \(\alpha\) controls the exponential drop of the coherency function with an increase of distance and frequency. Parameter \(\alpha\) was taken with value of \(2\times 10^{-4}\), the lowest value of those proposed by [PERSON] and [PERSON], to be appropriate for the rock site. The wave-passage effect depends on the apparent velocity of the waves. According to [PERSON] [51], studies of motions recorded in several arrays showed that the apparent velocity of S-waves is typically in the range 2.0-3.5 km/s. Hence, the value of 2 km/s was chosen in this study. It was assumed that seismic wave propagated from the left abutment as the reference station to the other bridge supports in the direction of the main bridge axis.