r - The Hong Kong Polytechnic University

take place along the entire surfaces of the adjacent structures. Moreover, it was observed from previous
earthquakes that most poundings actually occurred at corners of adjacent bridge decks, this is because torsional
responses of the adjacent decks induced by spatially varying transverse ground motions at multiple bridge
supports resulted in eccentric poundings. To more realistically model the pounding phenomenon between
adjacent bridge structures, a detailed 3D finite element analysis is necessary. However, previous studies of
seismic pounding responses of bridge structures based on the detailed 3D FEM are limited. To the best
knowledge of the authors, the only two studies in which bridge girders were modelled as 3D finite elements
were reported by Zanardo et al. (2002) and Zhu et al. (2002). These two studies, however, either neglected the
surface to surface and eccentric poundings (Zanardo et al. 2002) or the algorithm for searching for the pounding
locations is quite complicated (Zhu et al. 2002).
Pounding between adjacent bridge decks occurs because of large relative displacement responses. Ground
motion spatial variation, besides differences in vibration properties of adjacent bridge structures, is a source of
relative displacement responses. Owing to the difficulty in modelling ground motion spatial variation, many
studies assumed uniform excitations (Malhotra 1998; Ruangrassamee and Kawashima 2001; DesRoches and
Muthukumar 2002; Zhu et al. 2002) or assumed variation was caused by wave passage effect only (Jankowski et
al. 1998, 2000). Only a few studies considered the combined wave passage effect and coherency loss effect in
analyzing relative displacement responses of adjacent bridge structures (Chouw and Hao 2005, 2008; Chouw et
al. 2006; Zanardo et al. 2002). It should be noted that all these studies mentioned above assumed that the
analyzed structures locate on a flat-lying site, the influence of local site effect, which further intensifies ground
motion spatial variation at multiple structural supports, are neglected. Studies revealed that local site effect not
only causes further phase difference (Der Kiureghian 1996; Bi et al. 2010), but also affects the coherency loss
between spatial ground motions (Bi and Hao 2010). These differences will significantly affect the structural
responses. Consequently, neglecting local soil effect on the spatial ground motion variations at multiple supports
of a bridge structure crossing a canyon site may lead to inaccurate estimation of bridge responses.
In the last part of this study, seismic pounding responses between the abutment and the adjacent bridge deck and
between two adjacent bridge decks of a two-span simply-supported bridge located on a canyon site are
investigated. The three-dimensional spatially varying ground motions at different supports of the bridge
structure are stochastically simulated as inputs based on the methodology proposed in the first section of the
present paper. A detailed 3D finite element model of the bridge is constructed in ANSYS, and then LS-DYNA is
employed to calculate the structural responses. The influences of pounding effect and local soil conditions on
ground motion spatial variation and on structural responses are investigated in detail.
SPATIALLY VARYING GROUND MOTION SIMULATION
Ground Motion Generation
Consider a canyon site with horizontally extended multiple soil layers resting on an elastic half-space as shown
in Figure 1, in which h m , G m , ρ m , ξ m and υ m are the depth, shear modulus, mass density, damping ratio
and Poisson’s ratio of layer m. The spatially varying base rock motions are assumed to consist of out-of-plane
SH wave or in-plane combined P and SV waves and propagating into the layered soil site with an assumed
incident angle. The incident motions at different locations on the base rock are assumed to have the same power
spectral density, and are modelled by a filtered Tajimi-Kanai power spectral density function. The spatial
variation of ground motions at base rock is modelled by an empirical coherency function for spatial ground
motions on a flat site. The cross power spectral density functions of surface motions at n locations of the layered
site can be written as:
⎡ S11(
ω)
S12
( iω)
⋅⋅⋅ S1n
( iω)
⎤
⎢
⎥
⎢
S21(
iω)
S22
( ω)
⋅⋅⋅ S2n
( iω)
S ( iω)
=
⎥
(1)
⎢ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⎥
⎢
⎥
⎣Sn1(
iω)
Sn2
( iω)
⋅⋅⋅ Snn
( ω)
⎦
where
S
S
jj
jk
( ω)
= H ( iω)
S
*
k
( ω)
= H ( iω)
H
j
j
2
g
( ω)
j = 1,2, L,
n
( iω) S ( ω)
γ ( d , iω)
j = 1,2, L,
n
g
are the auto and cross power spectral density function respectively. In which S (ω)
is the ground motion power
'
spectral density on the base rock; γ j k d j k , iω)
is the coherency function between location j and k ' on the
' '( ' '
j'
k'
j'
k'
g
(2)
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