Civil Engineering Reference
In-Depth Information
and approach track) is (see
Figure E4.6)
d
2
x
i
(x)
d
x
2
−
E
i
A
i
+
k
i
x
i
(x)
=
q
i
(x)
,
(4.26)
where
E
i
A
i
is the axial stiffness of the member (rail or span).
The
r
esulting system of equations m
ay
be solved for the longitudinal displace-
ments,
x
i
(x)
, and forces,
N
i
(x)
E
i
A
i
(
d
x
i
(x)/
d
x)
, in the bars. The solution may be
obtained using transformation methods (Fryba, 1996) and the appropriate boundary
conditions (e.g.,
Table E4.2
in Example 4.7).
The longitudinal traction and braking forces transferred to the bearings may be
determined from equilibrium following computation of the rail and span axial forces,
N
i
(x)
. However, as seen in Example 4.7, even the simplest bridge models will involve
considerable calculation.
=
Example 4.7
Develop the equations of longitudinal forces and boundary conditions for
the open deck steel railway bridge shown in Figure E4.6.
L
1
L
2
L
3
L
4
LF
k
1
k
2
k
2
k
3
L
5
L
6
LF
in
E
i
A
i
S
in
L
i
u
i
(
x
)
d
2
u
i
(
x
)
dx
2
+
k
i
u
i
(
x
) =
q
i
(
x
), i = 1, 4
-
E
i
A
i
d
2
u
i
(
x
)
dx
2
+
k
i
[
u
i
(
x
)-
u
i+3
(
x
)] =
q
i
(
x
), i = 2, 3
-
E
i
A
i
d
2
u
i
(
x
)
dx
2
+
k
2
[
u
i
(
x
)-
u
i-3
(
x
)] = 0, i = 5, 6
-
E
i
A
i
N
i
n
=1
d(
x
-
S
in
)
LF
in
q
i
(
x
) = S
FIGURE E4.6
