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The characteristics of the site response on the surface during an earthquake are described by using several methods, taking into consideration the effect of the local site (Schnabel et al., 1972; Kramer1996). The results obtained are then used to perform the structural dynamic analyses. These methods are usually available and commonly used in the geotechnical earthquake engineering practice. Each method is specific and has its own formula and program. However, the concept is the same (Schnabel et al., 1972; Kramer, 1996; Kottke & Rathje, 2008).
Regarding the linear method, the most applied approach is the traditional method (TM) in contrast to the random vibration theory (RVT), which is less commonly used (Schnabel et al., 1972; Kramer, 1996; Kottke & Rathje, 2008). The theory in the traditional method is based on the one-dimensional wave propagation by considering an homogeneous and isotropic soil profile, while taking properties such as shear modulus (G), damping factor (ξ), and mass density (ρ) constant, for each layer (Schnabel et al., 1972). The simulation procedure of a response at a free surface is generated from the acceleration time history. The foremost program used is SHAKE91, developed by Idriss and Sun in 1992. This program is based on several studies carried out by Kanai (1951), Cooley and Tukey (1965), Seed (1968), Seed and Idriss (1970), and Schnabel, Lysmer and Seed (1972) (Schnabel et al., 1972; Kramer, 1996; Kottke & Rathje, 2008; Afra & Pecker, 2002). The concept of the aforementioned program resides on the calculation of the response spectrum specified at any point for any sublayer from the input motion. The first step consists to compute the new motion at the top of the sublayer (free surface), using the transfer function. In the second step, the response spectrum is determined from the surface motion (see Figure 1). The non-linearity of the shear modulus and damping is considered by using the equivalent linear soil properties, which give an approximated strain that is compatible with the nonlinear response of the soil (Rathje & Ozbey, 2006). The above-mentioned procedure is the most commonly applied in practice; however, it is very cumbersome.
Conversely, and according to Albert et al. (2013) the random vibration theory (RVT), using a linear approach is considered to be the simplest method to evaluate the site response. However, with this alternative method, the time-domain input motion is not required; rather, a single input motion is specified as the Fourier Amplitude Spectrum (FAS), which can be directly derived from the seismological theory or from the acceleration response spectrum motion as described in Figure 1 (Albertet al., 2013; Beresnev & Atkinson, 1998; Boore, 2003). Firstly, the concept of the random vibration theory is based on the calculation of the propagation of the Fourier Amplitude Spectrum at the ground surface by using a transfer function. Secondly, the response spectrum at the free surface is calculated from Fourier Amplitude Spectrum by using the random vibration theory (Beresnev & Atkinson, 1998; Boore, 2003; Wang & Rathje, 2016). The nonlinear analysis is performed by using a discrete model such as finite elements or lumped mass models (Boore, 2003; Wang & Rathje, 2016; Menzer et al., 2016). The random vibration theory method is potentially a powerful tool for site response analysis that can provide fast and accurate estimates of the surface ground motion at the site (Menzer et al., 2016).