Civil Engineering Reference
In-Depth Information
TABLE 4.2
Undamped Natural Frequencies of Various Beams
Beam
ω
ni
(rad/s)
i
2
EI
m
2
π
L
2
EI
m
(
4
i
2
2
+
4
i
+
1
)
π
4
L
2
EI
m
(
4
i
2
2
−
4
i
+
1
)
π
4
L
2
EI
m
(
16
i
2
2
+
8
i
+
1
)
π
16
L
2
ω
1
=
=
for the span of Example 4.3,
10.6 Hz, is slightly greater than the
estimate of 9.4 Hz for railway ballasted girder spans in
Figure 4.8.
Due to the inertial effects of the relatively large railway live load on steel
spans, the loaded simply supported beam natural frequencies are required in the
dynamic analysis of steel railway superstructures. Approximate equations for the
loaded simply supported beam fundamental frequency,
66.58 rad/s
ω
L1
, have been proposed
(Fryba, 1972) as
1
ω
L1
= ω
1
for moving uniform continuous loads
1
+
(
2
P/mgL)
+
(m
w
/
2
mg)
(4.16)
and
1
ω
L1
= ω
1
for moving constant concentrated loads.
(4.17)
1
+
(
2
P/mgL)
A similar equation for the loaded simply supported beam fundamental fre-
quency,
ω
L1
, was proposed for a moving harmonically varying concentrated force
TABLE 4.3
Unloaded Fundamental Frequencies of Steel Railway Bridges
(Empirical Equations from Fryba, 1996)
Superstructure
Unloaded Fundamental Frequency,
f
1
(Hz)
1135
(L)
−
1.1
Steel truss spans
135
(L)
−
0.7
Ballasted girder spans
Open deck girder spans
680/
L
Note: L
=
length in feet.
