Civil Engineering Reference
In-Depth Information
H
0
+
H
q
H
0
+
H
q
y
(
x
)
f
q
(
x
)
η(
x
)
x
η(
x
)
M
(
x
)
Span length
Figure 12.12
A single-span suspension bridge.
early nineteenth century, deflection theory in the late nineteenth century,
and finite deformation (large displacement) theory nowadays.
Considering one span of the simple support girder as shown in the upper
part of Figure 12.12, its moment distribution is
M x M x
( )
=
( )
(12.19)
0
If the girder is multiple supported by hangers from the cable as shown in
the lower part of Figure 12.12, when loads that cause moment distribu-
tions as shown in Equation 12.19 are applied on the girder, the cable will
be tensioned and the moment distribution on the girder will be reduced as
M x M x H x y x
q
( )
=
( )
−
( ) ( )
(12.20)
0
where
H x
q
( )
is the horizontal component of cable tension due to loads
distribution of
q
(
x
). Equation 12.20 represents the elastic theory. Its dif-
ferential equation form is
4
2
EI
d
dx
η
H x
d y
dx
( )
( )
q x
=
−
(12.21)
q
4
2
One example of using Equation 12.20 is to calculate the cable tension in
the middle of the span. Assuming the loads distribution
q
(
x
) is constant as
q
0
and is completely distributed to the cable. Thus, the moment distribution
in the girder in Equation 12.20 will be zero. The cable tension in the hori-
zontal component at the middle of the span can be derived by considering
the moment in the middle span of a simple support beam.
2
l
=
q l
f
0
H
(12.22)
q
2
8
Assume the structure is balanced before
q
(
x
) applies and the horizontal com-
ponent of its initial cable tension is
H x
0
( )
. After
q
(
x
) applies, the increase of
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