Civil Engineering Reference
In-Depth Information
The bridge model used here is a simply supported 15 m Finite Element beam that consists of twenty discretized beam
elements with four degrees of freedom. The beam therefore has a total of
n
¼
42 degrees of freedom. It has a constant
10
10
Nm
2
, mass per unit length,
28 125 kg m
1
and second moment of area,
modulus of elasticity
E
¼
3.5
μ ¼
0.5273 m
4
. The first natural frequency of the beam is 5.65 Hz. The response of a discretized beam model to a series of
moving time-varying forces is given by the system of equations:
J
¼
M
b
€
y
b
þ
C
b
_
y
b
þ
K
b
y
b
¼
N
b
f
int
(10.9)
y
b
where
M
b
,
C
b
and
K
b
are the (
n
n
) global mass, damping and stiffness matrices of the beam model respectively and
y
b
,
and
y
b
are the (
n
1) global vectors of nodal bridge displacements and rotations, their velocities and accelerations
respectively. The product
N
b
f
int
is the (
n
1) global vector of forces applied to the bridge nodes. The vector
f
int
contains
the interaction forces between the vehicle and the bridge and is described using the following vector:
f
int
¼ P þ F
t
(10.10)
where
P
is the static axle load vector and
F
t
contains the dynamic wheel contact forces of each axle. The matrix
N
b
is a
(
n
n
f
) location matrix that distributes the
n
f
applied interaction forces on beam elements to equivalent forces acting on
nodes. This location matrix can be used to calculated bridge displacement under each wheel,
y
br
:
N
b
y
b
y
br
¼
(10.11)
, is varied in simulations to assess the system's potential as an indicator of changes in
damping. Although complex damping mechanisms may be present in the structure, viscous damping is typically used for
bridge structures and is deemed to be sufficient to reproduce the bridge response accurately. Therefore, Rayleigh damping is
adopted here to model viscous damping:
The damping ratio of the bridge,
ξ
C
b
¼ α
M
b
þ β
K
b
(10.12)
where
α
and
β
are constants. The damping ratio is assumed to be the same for the first two modes [
20
] and
α
and
β
are
obtained from
α ¼
2
ξω
1
ω
2
= ω
1
þ ω
2
ð
Þ
and
β ¼
2
ξ= ω
1
þ ω
2
ð
Þ
where
ω
1
and
ω
2
are the first two natural frequencies of the
bridge [
21
].
The dynamic interaction between the vehicle and the bridge is implemented in Matlab. The vehicle and the bridge are
coupled at the tire contact points via the interaction force vector,
f
int
. Combining (
10.1
) and (
10.9
), the coupled equation of
motion is formed as
M
g
u
þ
C
g
u
þ
K
g
u
¼
F
(10.13)
where
M
g
and
C
g
are the combined system mass and damping matrices respectively,
K
g
is the coupled time-varying system
stiffness matrix and
F
is the system force vector. The vector,
u
y
b
T
is the displacement vector of the system. The
equations for the coupled system are solved using the Wilson-Theta integration scheme [
22
,
23
]. The optimal value of the
parameter
¼
y
v
;
1.420815 is used for unconditional stability in the integration schemes [
24
]. The scanning frequency used for
all simulations is 1,000 Hz.
θ ¼
10.3 Concept of Subtracting Axle Accelerations to Detect Bridge Damping
The vehicle is simulated traveling over a 100 m approach length followed by a 15 m simply supported bridge at 20 m s
1
.
The Class 'A' road profile is generated according to the ISO standard [
25
]. This is extrapolated into a number of different
paths, where each path is correlated with those adjacent to it. Figure
10.2
illustrates two paths through this road profile. The
vehicle is simulated crossing each path separately to assess the sensitivity of the algorithm to transverse position. This is
repeated six times, once for each level of damping (from 0% to 5%), representing different degrees of damage.
The trailer vertical accelerations are transformed from the time domain into the frequency domain using the Fast Fourier
Transform. Plots of Power Spectral Density (PSD) against frequency can be seen in Fig.
10.3
. In the PSD for an individual
