Civil Engineering Reference
In-Depth Information
TABLE E4.2
Boundary Conditions
Force and Displacement Conditions
Rails
(i
=
1, 2, 3, 4
)
N
1
(
0
)
=
N
4
(L
4
)
=
0
N
1
(L
1
)
−
LF
L1
=
N
2
(0)
N
2
(L
2
)
−
LF
L2
=
N
3
(
0
)
N
3
(L
3
)
−
LF
L3
=
N
4
(
0
)
u
1
(L
1
)
=
u
2
(
0
)
u
2
(L
2
)
=
u
3
(
0
)
u
3
(L
3
)
=
u
4
(
0
)
Span (
i
=
5, 6)
N
5
(L
5
)
=
N
6
(L
6
)
=
0
u
5
(
0
)
=
u
6
(
0
)
=
0
Particular
Expansion joints at end of bridge
L
1
=
L
4
=
0
CWR across bridge
L
1
=
L
4
No longitudinal rail restraint (free rails)
k
2
=
0
Rails fixed (direct fixation to deck)
k
2
The equations of longitudinal forces and boundary conditions are shown
in
Figure E4.6
and Table E4.2, respectively.
Extensive testing and analytical work has been performed (see references Foutch
et al., 1996, 1997; LoPresti et al., 1998; LoPresti and Otter, 1998; Otter et al., 1996,
1997, 1999, 2000; Tobias Otter and LoPresti, 1998; Tobias et al., 1999; Uppal et al.,
2001)toovercomethetheoreticalmodelcomplexitiesandnumericalmodelingefforts.
This work has established relationships for braking and traction dependent on the
length of the portion of the bridge under consideration. Testing in the United States
has provided longitudinal forces for Cooper's E80 design live load that are shown in
Figure 4.12
a
nd Equations 4.27 and 4.28. It appears that, for loaded lengths less than
about 350 ft, longitudinal force due to traction governs. However, locomotive traction
occurs over a relatively small length and braking forces on a loaded length consisting
of the entire bridge may exceed the tractive effort (see Examples 4.9 and 4.10).
∗
The
force due to traction governs for short- and medium-length bridges.
LF
B
=
45
+
1.2
L
,
(4.27)
25
√
L
,
LF
T
=
(4.28)
where LF
B
is the longitudinal force due to train braking (kips), LF
T
is the longitudinal
force due to locomotive traction (kips), and
L
is the length of the portion of the bridge
under consideration (ft).
However, while an estimate of the magnitude of the applied longitudinal trac-
tion and braking forces appropriate for design is readily available, the distribution of
∗
As illustrated by
Figure 4.13
showing the ratio of the longitudinal force transmitted to the bearings,
H
B
,
to the applied longitudinal force for bridges with continuous welded rail and steel bearings (based on
European tests reported by Fryba, 1996).
