Civil Engineering Reference
In-Depth Information
Table 1.3
Correspondence between experimental (EMA) and analytical (FEA) mode shapes of the Refined FE Model. Natural frequency errors.
Δ ¼
100
(
p
FEA
-
p
EMA
)/
p
EMA
; w.c.
¼
without correspondence.
EMA frequency
p
r
[Hz]
FEA frequency
p
r
[Hz]
MAC
[%]
EMA mode order
r
FEA mode order
r
Δ
1
2.022
0.001
1
1.985
1.84
99.4
2
3.053
0.003
2
2.857
6.41
87.3
3
3.180
0.002
3
3.165
0.48
89.5
4
3.605
0.002
4
3.369
6.56
100.0
5
4.831
0.011
5
4.611
4.56
94.3
6
6.887
0.046
7
6.809
1.13
97.5
7
6.934
0.015
6
6.605
4.75
83.6
8
7.995
0.005
9
8.829
10.43
89.0
9
9.107
0.020
10
8.903
2.24
73.7
10
12.910
0.025
w.c.
-
-
-
11
14.228
0.081
15
16.488
15.97
56.5
12
14.433
0.100
w.c.
-
-
-
deformed shape of the FEA transverse rigid-body mode shows a not negligible transverse deformability of the deck in the
horizontal plane. Moreover, experimental FRF measurements show appreciable flexibility of the top section of the pier in
the global dynamic behavior of the bridge, both for transverse and longitudinal vibrations. Therefore, it is reasonable to
assume that this additional flexibility may affect the procedure used in previous section to estimate the shearing stiffness
K
x
,K
y
of the isolators.
Taking into account the above considerations, the longitudinal stiffness of the isolators K
y
was estimated by considering
the influence of the compliance introduced by the pier and assuming rigid-body motion along the longitudinal direction of
the bridge. The pier was described by a Timoshenko beam and, by imposing the EMA frequency value 3.61 Hz, it was found
K
y
¼
172.4 MN/m. Concerning the transverse direction, the in-plane vibrations of the bridge were modeled by a flexible
beam under three elastic supports, two at the abutments and one at the pier. By imposing the coincidence of the EMA
frequency value 3.05 Hz it was found K
x
¼
151.2 MN/m.
The difference between the identified stiffness values for motions along the longitudinal and transverse direction is
significantly reduced with respect to the previous estimate, and it is around 12%. This result corroborates the assumption of
considering a unique value for both K
x
and K
y
, corresponding to the average value of about 161.8 MN/m. This optimal value
is about 2.2 the nominal value, in agreement with the results found in [
9
]. Table
1.3
shows the correspondences between
EMA and FEA vibration modes of the Refined FE Model of the bridge, and corresponding natural frequency values. It can be
seen that there is a rearrangement of the modes with respect to the Preliminary FE Model, and that EMA Modes 6,7,8,9,11
now correspond to FEA Modes 7,6,9,10,15, respectively. Modeling errors are generally of order of few points per cent for
the lower vibration modes. Underestimates are of order 4-6% for rigid-body motions, and errors on EMA Modes 8 and 11
are about 10 and 16%, respectively. We refer to the paper [
13
] for a further improvement of this model.
1.6 Conclusions
The results of a series of harmonic forced vibration tests on an isolated bridge have been presented and interpreted in this
paper. An identification procedure based on modal analysis and finite element modeling has been adopted for the
characterization of the bridge and, particularly, for the identification of the stiffness of the bearing devices supporting the
deck superstructure. The research allowed to identify a baseline model of the bridge that will be useful for long-term
maintenance plans and for possible future investigation of diagnostic character.
References
1. Aktan AE, Farhey DN, Helmicki AJ, Brown DL, Hunt VJ, Lee K-L, Levi A (1997) Structural identification for condition assessment:
experimental arts. J Struct Eng ASCE 123(12):1674-1684
2. Benedettini F, Zulli D, Alaggio R (2009) Frequency-veering and mode hybridization in arch bridges. In: Proceedings of IMAC XXVII,
Orlando
