r - The Hong Kong Polytechnic University

phenomenon, however, cannot be considered in Der Kiureghian’s model. It does not explicitly model the
influences of the sites effects on ground motion spatial variations.
The ground motion power spectral density functions and spatial variation models can be used directly as inputs
at multiple supports of structures in spectral analysis of structural responses. This approach, however, is usually
applied to relatively simple structural models and for linear response of the structures owing to its complexity.
For complex structural systems and for nonlinear seismic response analysis, only the deterministic solution can
be evaluated with sufficient accuracy. In this case, the generation of artificial seismic ground motions is required.
Many methods are available to generate artificial spatially correlated time histories at different structural
supports. Hao et al. (1989) presented a method of generating spatially varying time histories at different
locations on ground surface based on the assumption that all the spatially varying ground motions have the same
intensity, i.e., the same power spectral density or response spectrum. The variation of the spatial ground motions
is modelled by an empirical coherency loss function and a phase delay depending on a constant apparent wave
propagation velocity. If the considered site is flat with uniform soil properties, the uniform ground motion
intensity assumption for spatial ground motions in the site is reasonable. However, for a canyon site or a site
with varying soil properties, because local site conditions affect the wave propagation hence the ground motion
intensity and frequency contents as discussed above, the uniform ground motion power spectral density
assumption is no longer valid. Deodatis (1996) developed a method to simulate spatial ground motions with
different power spectral densities at different locations. The method is based on a spectral representation
algorithm (Shinozuka 1972; Shinozuka and Jan 1972) to generate sample functions of a non-stationary,
multivariate stochastic process with evolutionary power spectrum. Similar to the Der Kiureghian (1996) model,
the considered varying spectral densities are filtered white noise functions with different central frequency and
damping ratio. This method thus can only approximately represent local site effects on ground motions.
Moreover, trying to establish an analytical expression for a realistic ground motion evolutionary power spectrum
related to the local site conditions is quite difficult since very limited information is available on the spectral
characteristics of propagating seismic waves (Shinozuka and Deodatis 1988).
The first part of this paper extends the work by Der Kiureghian (1996) by using the 1D wave propagation theory
(Wolf 1985) to more realistically model the influence of local site conditions on seismic waves. The spectral
representation method (Shinozuka 1972; Shinozuka and Jan 1972) and 1D wave propagation theory are
combined to derive the power spectral density functions of the spatially varying ground motions on surface of a
canyon site with multiple soil layers. The ground motion spatial variations are modelled in two steps: firstly, the
spatially varying base rock ground motions are assumed to consist of out-of-plane SH wave or in-plane
combined P and SV waves and propagate into the layered soil site with an assumed incident angle. The spatial
base rock motions are assumed to have the same intensity and frequency contents and are modelled by the
filtered Tajimi-Kanai power spectral density function (Tajimi 1960). The spatial variation effect is modelled by a
theoretical coherency loss function (Sobczky 1991). The surface motions of a canyon site with multiple soil
layers are derived based on the deterministic 1D wave propagation theory. The auto power spectral density
functions of ground motions at various points on ground surface and the cross power spectral density functions
between ground motions at any two points are derived by neglecting the wave scattering on the uneven canyon
surface. The spectral representation method is then used to generate spatially varying ground motion time
histories. Compared to the work by Deodatis (1996), in this study the power spectral density functions at
different locations of a canyon site are derived based on the 1D wave propagation theory, which directly relates
the local soil conditions and base rock motion characteristics with the surface ground motions, thus local site
effect can be realistically considered. The current approach also allows for a consideration of different incoming
wave types and incident angles to the soil site, which have great influence on the surface motions. The proposed
approach can be used to simulate ground motion time histories at an uneven site with known non-uniform site
conditions.
For bridge structures with conventional expansion joints, a complete avoidance of pounding between bridge
decks during strong earthquakes is often impossible since the separation gap of an expansion joint is usually a
few centimetres to ensure a smooth traffic flow. Therefore, pounding damages of adjacent bridge structures have
always been observed in almost all the previous major earthquakes. Pounding is an extremely complex
phenomenon involving damage due to plastic deformation, local cracking or crushing, fracturing due to impact,
and friction when two adjacent bridge decks are in contact with each other. To simplify the analysis, many
researchers modelled a bridge girder as a lumped mass (e.g. Malhotra 1998; Jankowski et al. 1998;
Ruangrassamee and Kawashima 2001; DesRoches and Muthukumar 2002; Chouw and Hao 2005, 2008) or
beam-column element frame (e.g. Jankowski et al. 2000; Chouw et al. 2006). Based on theses simplified
lumped mass model or beam-column element model, only 1D point to point pounding, usually in the axial
direction of the structures, can be considered. In a real bridge structure under seismic loading, pounding could
-157-