r - The Hong Kong Polytechnic University

a mean recurrence interval with extreme value theories based on short-term monitoring data of structural
response.
VEHICLE-INDUCED FATIGUE RELIABILITY ASSESSMENT OF EXISTING BRIDGES
Prediction of vehicle induced bridge vibration
To predict vehicle-induced vibrations for fatigue assessment, the vehicle is modeled as a combination of several
rigid bodies connected by several axle mass blocks, springs and damping devices (Cai and Chen 2004). The tires
and suspension systems are idealized as linear elastic spring elements and dashpots. The equations of motion for
the vehicle-bridge coupled system are as follows:
⎡Mb ⎤⎧⎪d&&
⎫
b⎪ ⎡Cb + Cbb Cbv⎤⎧⎪d&
⎫ K
b⎪
⎡ b
+ Kbb Kbv⎤⎧db⎫ ⎧ Fb
⎫
⎢ G
M
⎥⎨ ⎬+ ⎢
v d C
v
vb
C
⎥⎨ ⎬+ ⎢ ⎨ ⎬=
⎨ ⎬
v d K
v
vb
K
⎥
⎣ ⎦⎪ &&
⎪ ⎣ ⎦⎪ &
(1)
⎩ ⎭ ⎩ ⎪⎭
⎣ v ⎦⎩dv⎭ ⎩Fc + Fv
⎭
where, [M b ] is the mass matrix, [C b ] is the damping matrix and [K b ] is the stiffness matrix of the bridge, {F b } is
wheel-bridge contact forces on bridge; [M v ], is the mass matrix, [C v ], is the damping matrix and [K v ] is the
stiffness matrix of the vehicle; {F G v } is the self-weight of vehicle; and {F c }is the vector of wheel-road contact
forces acting on the vehicle. The terms C bb , C bv , C vb , K bb , K bv , K vb , F b and F v in Eq. (1) are due to the interactions
between the bridge and vehicles.
To demonstrate the methodology, a 12m long and 13 m wide slab-on-girder bridge is analyzed, which is designed
in accordance with AASHTO LRFD bridge design specifications (AASHTO 2007). In the present study, after
conducting a sensitivity studying by changing the meshing, 27543 solid elements and 43422 nodes are used to
build the finite element model of the bridge. The damping ratio is assumed to be 0.02. The present study focuses
on the fatigue analysis at the longitudinal welds located at the conjunction of the web and the bottom flange at
the mid-span. Since the design live load for the prototype of the bridge is HS20-44 truck, this three-axle truck is
chosen as the prototype of the vehicle in the present study. In addition, only one vehicle in one lane is
considered to travel along the bridge for fatigue analysis due to its short span length.
Road surface roughness is an important parameter that causes dynamic effect and fatigue problem. It is
generally defined as an expression of irregularities of the road surface and it is the primary factor affecting the
dynamic response of both vehicles and bridges (Deng and Cai 2010; Shi et al. 2008). Road roughness condition
is classically quantified using Present Serviceability Rating (PSR), Road Roughness Coefficient (RRC) or
International Roughness Index (IRI). Various correlations have been developed between the indices (Paterson
1986; Shiyab 2007). Based on the studies carried out by Dodds and Robson (1973) and Honda et al. (1982), the
road surface roughness was assumed as a zero-mean stationary Gaussian random process and it could be
generated based on the RRC through an inverse Fourier transformation as (Wang and Huang 1992):
N
rx ( ) = ∑ 2 φ( nk) Δ ncos(2 πnx
k
+ θk)
(2)
k = 1
where θ k is the random phase angle uniformly distributed from 0 to 2π; φ()
is the power spectral density (PSD)
function (m 3 /cycle/m) for the road surface elevation; n k is the wave number (cycle/m). Wang and Huang (1992)
also suggested a concise power-spectrum-density function that was used in the present study:
n −2
φ( n) = φ( n0
)( )
(3)
n0
where φ( n)
is the PSD function (m 3 /cycle) for the road surface elevation; n is the spatial frequency (cycle/m);
n 0 is the discontinuity frequency of 1/2π (cycle/m); and φ( n0
) is the RRC (m 3 /cycle) and its value is chosen
depending on the road condition.
In order to consider the road surface damages, a progressive deterioration model for road roughness is necessary.
More specifically, it is essential to have such a model for RRC in order to generate the random road profile.
Therefore, the RRC at any time after construction is predicted using the progressive deterioration model for IRI
and the relationship between the IRI and RRC(Paterson 1986; Shiyab 2007):
− 9 η
( { 5 ( ) }
6
0) 6.1972 10 exp 1.04 t
− −
φ n
t
= × × ⎡
⎣ e ⋅ IRI0
+ 263(1 + SNC) CESAL ⎤/ 0.42808 + 2×
10
t ⎦ (4)
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