During of a soil during an earthquake.

During the earthquakes, the effects of soil conditions on ground shaking are assessed through simulation of wave propagation called seismic site response analysis that becomes an important method to predict the surface response from bedrock ground motions. The important property of the local soil conditions that influence ground shaking is the shear wave velocity ( ) and the modulus reduction and damping (MRD) curves. Characteristics of soil conditions have been carried out in many works.

Seed and Idriss (1969) proposed constant values of both the shear modulus and the damping factor of a soil during an earthquake. Seed et al. (1986) reviewed numerical models of nonlinear shear modulus reduction and material damping increase curves for sand and gravelly soils. Effects of nonlinear dynamic properties are available by other authors (Hardin and Drnevich, 1972 – HD72; Anderson and Woods, 1975 – AW75; Darendeli, 2001 – Da01). HD72 proposed an analytical model of nonlinear soil behavior with shear strain, namely hyperbolic model. Da01 also used a modified hyperbolic model to model the relationship between material damping ratio and strain.

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The other model is a Ramberg-Osgood model which used by AW75 to describe the variation of shear modulus with shearing strain. Ishibashi and Zhang (1993) collected available experimental data on dynamic shear modulus and damping ratios of soil properties including strain-dependent shear modulus and damping ratio, and proposed formulas for properties.Based on the previous researchers, Menq (2003) developed and presented a multi-mode device to evaluate the dynamic properties of sandy and gravelly soils.The soil profiles with the corresponding properties such as thickness, , density, shear modulus also have an important impact on the soil behavior. Many researchers also present probabilistic approaches in practical earthquake engineering applications (Koutsourelakis et al., 2002; Popescu et al., 2006; Rathje et al., 2010 – Ra10).

Koutsourelakis (2002) used non-Gaussian random fields and ground motion as a non-stationary random process for soil properties model. The variation of  profile and nonlinear soil properties was considered to evaluate the effects of soil characterization (Ra10). Additional work by Bazzurro and Cornell (2004) for considering the influence of stochastic soil layers on ground motion intensity at the surface of soil. Andrade, J.

E., & Borja, R. I. (2006) investigated the soil response by comparing the equivalent linear (Idriss and Sun, 1992 – IS92) and time domain nonlinear (Borja et al, 2000) model.

Kwok (2008), who evaluated the soil behavior of specific site in Turkey Flat using the nonlinear and equivalent-linear ground-response computer code DEEPSOIL (Hashash et al., 2012 – Ha12) and compared with the prediction to measurements. Nour et al. (2003) presented a finite element model for the probabilistic seismic response, which the  is randomized by using the non-Gaussian distribution.

Usually, in engineering practice, there is no data available about the stochastic variable, example the layer thickness, . Therefore, it is necessary to develop a simulation technique of uncertainty processes. An essential part of the probabilistic methods is the selection of probability distribution functions to represent the uncertainty of the random variables considered. In this study, the good solution namely PSHAKE based on the original SHAKE91 framework of site response analysis is proposed.

The site response analyses are conducted using randomized soil deposit (soil profile and nonlinear soil properties). The property randomization includes the variation of the shear wave velocities and thickness of soil deposit from the surface to bedrock using the different probabilistic distribution. The random  is assumed to be station stochastic process but with the different probabilistic distribution including normal and log-normal distribution. In Toro (1995) – To95 process, the  is described as a non-Gaussian distribution.

The layer thickness can be modelled using either a uniform, normal or log-normal distribution. The results of fifty randomized profiles are used to confirm the influence of layering and  on the site response analyses. The results of maximum peak ground motion at each layer, the amplification function and the response spectrum of ground motion are presented for evaluating the effects of probabilistic distribution.

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