Influence of Ballast Track on Vertical Response of Multi-span SS Bridges
5
Table 1. Bridge-track reference parameters
Rail and bridge parameters
Track parameters per rail seat
A r (m 2 )
76.86E−4
K p (N/m)
1E8
E r (Pa)
2.1E11
C p (Ns/m)
7.5E4
I zr (m 4 )
3055E−8
M sl (kg)
300
ρ r (m 3 )
7850
K b (N/m)
1.933E8
D sl (m)
0.60
C b (Ns/m)
5.88E4
L bi (m)
11.93
M b (kg)
317.91
(I z ·E) bi (Nm 2 )
7.09E9
K f (N/m)
7.3987E7
m bi (kg/m)
8727
C f (Ns/m)
3.115E4
ζ bi (%)
1.565
K w (N/m)
7.84E7
K n,bi (N/m) st/dyn
11.165E8/22.33E8 C w (Ns/m)
8E4
the range [40, 100] m/s. A Chebyshev order 3 filter is applied to the acceleration response
filtering contributions below 1 Hz and above 60 Hz. Maximum response envelopes are
obtained for values of the track parameters:
[0.5, 1, 2, 4] · K p [0, 0.5, 1, 2] · C p [0.5, 1, 2, 4] · K b [0, 0.5, 1, 2] · C b . (1)
where the values of K p , C p , K b and C b are those in Table 1. Figure 5 shows the
evolution of the maximum acceleration, which always takes place in the center of the
second span, in absolute value in terms of the velocity for isolated variations of each
track parameter.
In all the plots shown in Fig. 5 three resonant peaks can be detected, corresponding
to second, third and fourth resonances of the fundamental mode, which in the reference
case presents a frequency f 1 = 10.07 Hz.
V
r,j=2
1
=
df 1
2
3.6 = 326.27
km
h
V
r,j=3
1
= 217.5
km
h
V
r,j=4
1
= 163.1
km
h
(2)
The individual variations considered in the track do not affect significantly the reference fundamental frequency. From the observation of Fig. 5(c)–(d) it may be concluded
that the influence of the variations in the rail pads and ballast damping constants is negligible on the maximum acceleration response of the bridge for the reference values of
the remaining parameters. The parameter that seems to affect the most the acceleration
envelope at the most critical section is the rail pad stiffness K p , leading to a decrease
in the maximum acceleration at resonance as K p reduces (for more flexible rail pads).
This effect is more visible for higher resonance orders. The same tendencies, although
in a less pronounced manner, are observed in terms of the variations considered in the
ballast stiffness K b .
5
Table 1. Bridge-track reference parameters
Rail and bridge parameters
Track parameters per rail seat
A r (m 2 )
76.86E−4
K p (N/m)
1E8
E r (Pa)
2.1E11
C p (Ns/m)
7.5E4
I zr (m 4 )
3055E−8
M sl (kg)
300
ρ r (m 3 )
7850
K b (N/m)
1.933E8
D sl (m)
0.60
C b (Ns/m)
5.88E4
L bi (m)
11.93
M b (kg)
317.91
(I z ·E) bi (Nm 2 )
7.09E9
K f (N/m)
7.3987E7
m bi (kg/m)
8727
C f (Ns/m)
3.115E4
ζ bi (%)
1.565
K w (N/m)
7.84E7
K n,bi (N/m) st/dyn
11.165E8/22.33E8 C w (Ns/m)
8E4
the range [40, 100] m/s. A Chebyshev order 3 filter is applied to the acceleration response
filtering contributions below 1 Hz and above 60 Hz. Maximum response envelopes are
obtained for values of the track parameters:
[0.5, 1, 2, 4] · K p [0, 0.5, 1, 2] · C p [0.5, 1, 2, 4] · K b [0, 0.5, 1, 2] · C b . (1)
where the values of K p , C p , K b and C b are those in Table 1. Figure 5 shows the
evolution of the maximum acceleration, which always takes place in the center of the
second span, in absolute value in terms of the velocity for isolated variations of each
track parameter.
In all the plots shown in Fig. 5 three resonant peaks can be detected, corresponding
to second, third and fourth resonances of the fundamental mode, which in the reference
case presents a frequency f 1 = 10.07 Hz.
V
r,j=2
1
=
df 1
2
3.6 = 326.27
km
h
V
r,j=3
1
= 217.5
km
h
V
r,j=4
1
= 163.1
km
h
(2)
The individual variations considered in the track do not affect significantly the reference fundamental frequency. From the observation of Fig. 5(c)–(d) it may be concluded
that the influence of the variations in the rail pads and ballast damping constants is negligible on the maximum acceleration response of the bridge for the reference values of
the remaining parameters. The parameter that seems to affect the most the acceleration
envelope at the most critical section is the rail pad stiffness K p , leading to a decrease
in the maximum acceleration at resonance as K p reduces (for more flexible rail pads).
This effect is more visible for higher resonance orders. The same tendencies, although
in a less pronounced manner, are observed in terms of the variations considered in the
ballast stiffness K b .