Civil Engineering Reference
In-Depth Information
Table 1.2
Correspondence between experimental (EMA) and analytical (FEA) mode shapes of the Preliminary 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
2.022
0.001
1
1
2.062
1.99
99.4
2
3.053
0.003
2
2.530
17.13
92.8
3
3.180
0.002
4
3.301
3.80
89.6
4
3.605
0.002
3
3.246
9.96
100.0
5
4.831
0.011
5
3.862
20.06
95.8
6
6.887
0.046
7
7.129
3.51
98.0
7
6.934
0.015
8
7.693
10.94
85.2
8
7.995
0.005
10
10.259
28.31
89.7
9
9.107
0.020
9
9.367
2.85
83.5
10
12.910
0.025
w.c.
-
-
-
11
14.228
0.081
16
19.585
37.75
53.3
12
14.433
0.100
w.c.
-
-
-
bending mode of the bridge deck, can be explained by observing that this mode is weakly excited during testing, even along
the transverse X-direction, because the position of the shaker was near a nodal region of this mode. To confirm the
correspondence based on visual comparison, the Modal Assurance Criterion (MAC) matrix between experimental and
analytical mode shapes has been evaluated [
15
]. MAC values collected in Table
1.2
have been evaluated with reference to
the horizontal or vertical modal components for rigid-body and bending/torsional vibration modes, respectively. The
correlation generally is extremely good, with MAC values higher than 0.85. Correlation becomes worse for experimental
Mode 11 and EMA modes 10,12 have no analytical counterpart.
Comparing with the measured modal frequency values, it can be seen that the analytical FEM-based frequencies of the
rigid-body vibration modes are appreciably lower than the corresponding experimental values for EMA Modes 2 and 5, with
differences of about 17% and 20%, respectively. Experimental natural frequencies associated to bending/torsional vibration
modes of the deck are overestimated, and errors generally are of order of few points per cent, with the exception of large
discrepancies on higher order modes, namely about 28% and 38% on EMA Modes 8 and 11, respectively.
1.5 A Refined FE Model of the Bridge
An analysis of the differences in the numerical and experimental natural frequencies of the Preliminary FE Model implied
that there was a problem with uncorrected modeling of the bridge. Moreover, the fact that experimental frequency values of
rigid-body modes are underestimated and that natural frequencies of bending and torsional deck modes are overestimated,
suggests that simultaneous underestimates and overestimates of inertia/stiffness properties may be present in the Preliminary
FE model of the bridge. In addition, the large difference between transverse K
x
and longitudinal K
y
stiffness of the isolators
(that, in principle, should be approximately equal) is hardly justified.
An extensive numerical/analytical investigation has been carried out in order to give an interpretation of the experimental
results and, ultimately, to find an Refined FE model of the bridge. Referring to the paper [
13
] for a detailed description of the
steps and results of the identification procedure, here the main improvements made on the Preliminary FE model and a
discussion on their quantitative effects are summarized.
One of the improvements concerns the influence of the footways, which were neglected in previous stage. A numerical
investigation performed on the basis of the Preliminary FE Model, in fact, showed that the footway mass, although marginal
in comparison with the total mass of the bridge deck (about the 7%), cause no negligible reduction in natural frequencies,
especially on torsional modes. It can be shown that this follows from the fact that the additional mass is located on regions of
high sensitivity for those modes, or, equivalently, on regions with large modulus of modal displacement. The total mass of
the bridge deck, footways included, is now M
0
¼
1,735 t. Conversely, the stiffening contribution offered by the footways
can be considered negligible, due to the presence of large longitudinal voids in the downstream side and because of the small
size dimensions of the footway located on upstream side, see Fig.
1.1
.
Another important point is connected with the influence of the in-plane transverse deformability of the deck and of the
deformability of the pier on the estimate of the isolator stiffness. These two phenomena were neglected in previous stage and
the assumption of rigid-body motion of the deck for fixed supports of the isolators was assumed. Actually, the analysis of the
