Civil Engineering Reference
In-Depth Information
c
is the viscous damping operator with respect to the spatial variables
y
(
x
,
t
) is the vertical deflection of the bridge along longitudinal
x
-direction at time
t
f
(
x
,
t
) is the force exerted by the vehicle on the bridge
To obtain a unique solution, the boundary conditions and the initial dis-
placement
y
(
x
,0) and velocity
y
(
x
,0) must be defined. The eigenvalues and
eigenvectors (modal shapes), all in the vertical direction, of Equation 17.5
can be easily handled by close-form solution or through mathematical
modeling. Based on Green and Cebon (1994), Equation 17.5 can be solved
with the convolution integral of
−∞
∫
y x t
( , )
h x x t
( ,
,
) (
f x
, )
d
=
−
τ
τ τ
(17.6)
f
f
∞
where
h x x t
f
( ,
,
)
− τ =
impulse response function at position
x
for an impulse
applied at position
x
f
, which is related to the mode shapes. Therefore, the
bridge response is determined by the mode shapes and the forcing function.
The main factors affecting vehicle-induced bridge dynamics are bridge
surface roughness, speed, frequency matching, and vehicle suspension type
(Cantieni and Heywood 1997). Solving the problem can be described in the
following steps (MacDougall et al. 2006):
Step 1: Simulate the vehicle within a routine to solve for the vehicle's nat-
ural frequency and the wheel static load. This routine is used to predict
the tire forces of articulated vehicles where, for example, Figure 17.5
shows an 11-DOF vehicle model used by Cole and Cebon (1992).
Step 2: Apply the vehicular loads on the bridge model to calculate the
force
f
(
x
,
t
) exerted on the bridge at certain location
x
and time
t
due
to the moving vehicle.
Step 3: Use the calculated force
f
(
x
,
t
) from step (2) and the bridge's
impulse response function
h x x t
f
( ,
,
)
− τ to determine the bridge's
deflection
y
(
x
,
t
).
Step 4: Based on the calculated bridge's deflection
y
(
x
,
t
), the equiva-
lent external loading applied on the bridge is equal to the sum of the
bridge's self-weight and the bridge's inertia force:
F
mg ma mg m x t mg
ÿ
(
f x t
,
cy ky
applied
=
+
=
+
,
)
=
+
(
)
−
−
(17.7)
In the case of highway bridges, moving vehicles on bridge are arranged
randomly in terms of speeds, loads, direction, and location; however, for
railway bridges, train vehicles generally provide uniformly distributed load
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