Civil Engineering Reference
In-Depth Information
25
20
15
10
5
0
0
50
100
Length of span (
ft
)
150
200
FIGURE 4.7
Empirical values of viscous damping frequency to fundamental frequency for
steel railway bridges. (After Fryba, L., 1996,
Dynamics of Railway Bridges
, Thomas Telford,
London, UK.)
ω
c
is much less than 1
(which is generally the case for steel railway bridges as illustrated in Figure 4.7),
Equation 4.12a with the concentrated moving force,
P
, at mid-span is
Furthermore, for structures with light damping, where
sin
ω
ω
1
e
−ω
c
t
sin
ω
1
t
sin
π
2
FL
3
x
L
. (4.12b)
y(x
,
t)
=
4
EI
1
ω
1
)
2
ω
t
−
π
−
(
ω
/
The solution of Equation 4.11 is also greatly simplified for simply supported spans
withlightdampingbyneglectingthedamping(
c
=
0)andassumingagener-
alized single degree of freedom system with a sinusoidal shape function of sin
(
2
m
ω
c
=
x/L)
(CloughandPenzien,1975;Chopra,2004).Theforcedvibrationsolutionformid-span
(
x
π
=
L/
2) deflection may then be expressed as
sin
π
ω
1
t
.
2
P
Vt
L
−
V
ω
1
L
sin
π
y(L/
2,
t)
=
mL
ω
V/L)
2
(4.13)
2
1
−
(
π
Equation 4.13 indicates effectively static behavior for very short or stiff spans with
a high natural frequency. However, in the development of Equation 4.13, the inertia
effects of the stationary mass, dynamic effects of the load (vehicle suspension system
dynamics),
∗
damping,andtheeffectsofsurfaceirregularitiesareneglected.Therefore,
Equation4.13willnotprovidesatisfactoryresultsfortypicalrailwayspans,
†
asshown
in Example 4.3.
Example 4.3
Neglecting inertia effects, load dynamics, damping, and higher vibration
modes and assuming a sinusoidal shape function, determine the maximum
∗
Vehicle dynamic loads may be particularly important in medium span superstructures.
†
It may be appropriate to use equation (4.13) as a preliminary design tool for long or complex bridges.
