Civil Engineering Reference
In-Depth Information
H
LF
Braking
k
Time
Traction
FIGURE 4.11
Time history of braking and traction forces (at fixed bearing) from railroad
equipment.
For the condition of no slippage (complete adhesion),
θ
(t)
=
(x(t)/r)
. Substitution
of
(
d
2
(t)/
d
t
2
)
(
d
2
x(t)/r
d
t
2
)
into Equation 4.23 yields
θ
=
d
2
x(t)
d
t
2
I
p
r
(M
T
(t)
=
−
rH
LF
(t))
.
(4.24)
Substitution of Equation 4.24 into Equation 4.21a provides
m
F
I
p
r
T
F
(t))
1
=
+
≤ μ
H
LF
(t)
(M
T
(t)
R
V
(t)
,
(4.25)
1
−
m
F
I
p
where
is the coefficient of adhesion between locomotive wheels and rail without
slippage (can be as high as 0.35 for modern locomotives with software-controlled
wheel slip).
Equation 4.25 allows the numerical solution for longitudinal force,
H
LF
(t)
, which
remains, however, too arduous for ordinary design. The longitudinal forces described
by Equation 4.25 (including the effects of axle bearing, wheel rim friction, air resis-
tance, rolling friction, and other effects) have been observed and recorded by field
testing in both Europe and the United States. The longitudinal forces exhibit almost
static behavior since maximum traction and braking forces occur at low speeds when
starting and at the end of braking, respectively (Figure 4.11). Therefore, a static
analysis can be performed with
H
LF
= μ
μ
R
V
=
= μ
W
.
For a static longitudinal analysis, the bridge may be modeled as a series of lon-
gitudinal elastic bars (with independent longitudinal and flexural deformations) on
horizontal elastic foundations simply modeled
∗
as equivalent horizontal springs with
stiffness,
k
i
. The static longitudinal equilibrium equations for a system of
i
bars
(spans and rails) on elastic foundations (elastic horizontal stiffness of bridge deck
†
LF
∗
Other models that incorporate different longitudinal restraint at the rail-to-deck and deck-to-
superstructure may be used to provide greater accuracy.
†
Particularly appropriate for modern elastic rail fastening systems.
