Civil Engineering Reference
In-Depth Information
For the moving harmonically varying concentrated force shown in Figure 4.6,
p(x , t) = δ ( ξ )P sin
ω F t ,
(4.14)
where
ω F is the frequency of the harmonic force, P .
The steady state solution for maximum dynamic deflection of relatively long-
span steel railway bridges with light damping, considering the relatively slow speed
of heavy freight traffic, is (Fryba, 1972)
96 EI ω 1
π
cos π
t
e (c/ 2 m)t
PL 3
cos
ω 1 t
V
L
V
L
y(x , t)
=
V/L) 2
(c/ 2 m) 2
(
π
+
2 m sin π
t sin π
c
V
L
x
L .
(4.15)
This solution for mid-span deflection would be applicable for a live load with
harmonically varying frequency, such as a steam locomotive.
Dynamic analyses can be performed with the moving locomotives and trains ide-
alized as multi-degree of freedom vehicles with wheels modeled as unsprung masses,
bodies modeled as sprung masses, and with body and wheels connected by linear
springs with parallel viscous dampers. Track irregularities can be estimated by equa-
tions and a variable stiffness elastic layer can be used to account for open deck or
ballastedtieconditions.Theanalyticalsolutionisonerousandgenerallyaccomplished
by the differential equations using numerical methods, such as the Runge-Kutta
method (Carnahan et al., 1969). Using spectral analysis techniques, a closed form
solution of Equation 4.4, including damping, dynamic vehicle load effect, and sur-
face roughness, has been accomplished for the variation of dynamic deflection due to
live load (Lin, 2006). This is valuable information concerning the statistical behavior
of railway bridge vibrations, but does not provide a definitive mathematical solution
for dynamic load allowance.
As indicated earlier, the natural frequency,
ω n , of the bridge span is a useful
dynamic property that depends on the stiffness and mass of the span. The undamped
natural frequency of various beam spans may be calculated using free vibration analy-
sis [ c
=
0 and p(x , t)
=
0] and some approximations for vibration modes, i , are shown
in Table 4.2.
However, for short- and medium-span steel railway bridges, free vibration cal-
culations that yield the natural frequency of the span must be made considering the
inertial effects of the locomotive and trailing car weights. Some approximations for
the unloaded fundamental ( n
ω 1 for railway bridges, developed from
statistical analysis of measurements on European bridges, are shown in Table 4.3
with L given in ft. These equations are also plotted in Figure 4.8 with a typical esti-
mate for highway bridges. § It can be observed that the fundamental natural frequency
=
1) frequency,
Occurs where forcing frequency equals fundamental frequency, ω F = ω 1 (resonance).
A harmonically varying equation is often used to facilitate solution of the differential equations.
Based on 95% reliability.
§ An approximation for the unloaded fundamental frequency, ω 1 , for highway bridges is 2060/L rad/s
(L in ft) (Heywood, Roberts and Boully, 2001).
 
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