Civil Engineering Reference
In-Depth Information
the ratio of rise to span as well as the performance of steel and concrete
could not be ignored.
9.3.3 Measuring of hanger-cable force
For arch bridge (or any other cable-related bridges), hanger-cable force can
be measured to provide an indication on the damage degree of bridge. The
measurement methods of hanger force were studied by many researchers.
Among them, the frequency method was the most commonly used one in
practical projects. But in some measurement formula, the bending stiffness
of cable must be recognized. While the true bending stiffness was difficult
to measure because the hanger cables are made up of several steel wires and
the stiffness of wires will be changed constantly. The measurement formula
of hanger-cable force, which was based on the function of deflection curve
shape, is introduced here (Li et al. 2014).
According to the conservation principle of energy, the total energy (the
kinetic energy and the strain energy) of a free-damped vibrating elastic
body should be unchanged at any time. Transverse vibration curve of a
uniform cross-sectional cable is assumed to be a deflection curve under
uniform load. The deflection curves satisfy the boundary conditions, and
the maximum kinetic energy
E
k
max
of the cable is
l
1
2
∫
[
]
2
2
E
max
=
ω
m Y x
( )
dx
(9.4)
k
0
The maximum strain energy
V
max
of the cable is
l
l
1
2
1
2
∫
[
]
2
∫
[
]
2
V
max
=
EI Y x
′′
( )
dx
+
T
Y x
′
( )
dx
0
(9.5)
0
0
l
l
1
2
1
2
∫
∫
′
2
=
q x Y x dx
( ) ( )
+
T Y x dx
[
( )]
0
0
0
According to the conservation principle of energy
E
=
V
(9.6)
k
max
max
Natural vibration frequency ω
can be obtained as follows:
l
l
∫
∫
[
]
2
q x Y x dx T
( ) ( )
+
Y x
′
( )
dx
0
ω
2
=
0
0
2
l
(9.7)
∫
[
]
m Y x
( )
dx
0
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