Advanced Materials Research
ISSN: 1662-8985, Vols. 446-449, pp 3264-3272
doi:10.4028/www.scientific.net/AMR.446-449.3264
© 2012 Trans Tech Publications, Switzerland
Online: 2012-01-24
Correlation Study on Modal Frequency and Temperature Effects
of a Cable-Stayed Bridge Model
Limin Sun1, a, Yi Zhou2, b and Xuelian Li3, c
1, 2, 3
State Key Laboratory for Disaster Reduction in Civil Engineering,
Tongji University, Shanghai 200092, China
a
b
c
[email protected], [email protected], [email protected]
Keywords: Structural Health Monitoring, Experimental Study, Frequency, Temperature,
Cable-Stayed Bridge
Abstract. In recent years, structural health monitoring has been paid more and more attention in
bridge engineering community. Previous researches showed that ambient temperature was one of
principal factors affecting structural modal parameters in long-term. In this paper, an experimental
study on correlation between dynamic properties of a cable-stayed bridge and its structural
temperature was performed under temperature controlled laboratory environment. Using hammer
impacting method, a dynamic testing was conducted based on a steel cable-stayed bridge model
which had a span layout of 0.9+1.9+0.9m. During the experiment, the first six vertical bending modes
under the environmental temperature of 0, 20 and 40˚C were identified with the consideration of three
kinds of boundary conditions at the deck’s ends as to two degrees of freedom, i.e. the longitudinal
translation (UX) and the rotation about the transverse beam (RotZ). The above boundary conditions
are UX & RotZ not constrained, UX constrained only and UX & RotZ constrained, attempting to
simulate the different conditions of the bridge expansion joints. The efforts were paid to explain the
physical mechanism of the results based on the updated FE model. This experimental study indicates
a tendency that the frequency of the cable-stayed bridge model decreases with the increase of
temperature. And furthermore, the relative difference of frequencies between 0 and 40 ˚C is affected
by boundary conditions; in other words, when the deck is free to expand, the variation of model’s
frequencies is smaller than that when the deck is restrained to expand, which is similar to the
condition of the bridge’s expansion joints cannot work as normal. This experimental study can give
some reference to the research of SHM and damage identification for cable-stayed bridges.
Introduction
In last two decades, the need to apply structural health monitoring technology in bridge engineering
emerged, with the vibration-based method gained more and more attention. The key step of bridge
health monitoring involves the extraction of damage-sensitive features from the measurement of a
structure, the determination of current state of structural health through these features, and the
decision of the condition-based maintenance and management strategy. In theory, damage will change
the structural vibration parameters, e.g., frequency, mode shape and damping. The modal frequency
becomes the most-widely used damage indicator for its ease of measuring.
In recent years, many researchers have concentrated on the influence of environmental conditions
on structural modal frequencies based on monitoring data. Almost all the related literatures found the
environmental conditions such as temperature, humidity, wind properties, traffic flow etc. sometimes
can cause larger variation of dynamic properties than the structural damage. Among these
environmental factors, the temperature seems to be the most important.
Take cable-stayed bridge (CSB) for example. Sun and Min [1] reported that the first four modal
frequencies of the main navigation channel bridge of Donghai Bridge varied during one year by 2.6%,
8.31%, 3.1%, 2.34% respectively. Liu et al. [2] conducted a series of dynamic tests in different
seasons on Sutong Bridge, which has a main span of 1088m, and found the first three vertical
vibration frequencies increase by 0.5% as the ambient temperature decrease. GilJong et al. [3]
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Advanced Materials Research Vols. 446-449
3265
reported the monthly averaged frequencies of Seohae Grand Bridge, with main span length 470m, has
a negative correlation with temperature, and the slope of the linear regression equation is -0.00021.
Other researchers who studied the relationship between modal parameters and environmental
temperature based on monitoring data of CSB include Li et al. [4], Fan et al. [5] and so on. They
presented the fact that the vibration frequencies of cable-stayed bridge decrease with the increase of
temperature.
The actual CSBs suffer comprehensive effects from so many factors with great uncertainty, so it is
hardly possible to establish the exact causal relation between temperature and modal properties. This
paper performed an experimental study on a steel cable-stayed bridge model with the aim to
investigate the relationship between dynamic properties and temperature, boundary conditions. The
experimental results are presented and can serve as a reference for related researches in Bridge Health
Monitoring.
Test Set-Up
As mentioned before, the purpose of this experiment is to learn more about the cable-stayed bridge’s
frequency versus temperature under two different boundary conditions that may be encountered in the
practical engineering, i.e., whether the longitudinal translation degree of freedom is restrained or not,
which corresponds to the expansion joints’ working statuses.
In order to form a uniform temperature field, a temperature-controlled cabinet is constructed which
is equipped the heating and refrigerating devices with the temperature range in this study from 0 to
40˚C, as shown in Fig. 1.
Fig. 1. Insulation Cabinet
Fig. 2. CSB Model
A double pylon and double cable plane steel CSB model is set up having a span arrangement of
0.9+1.9+0.9m. The deck is 0.2m wide and consists of 2 longitudinal beams and 38 cross beams, the
former being 0.09m between centers with the cross-section 0.01*0.008m2 for each; the latter being
0.1m between centers with the cross-section 0.02*0.02 m2. The pylon is 0.88m high and the sections
of the pylon’s leg above and below the deck measure 0.01*0.01m2 and 0.03*0.03m2, respectively.
Cables are made of constantan wire with the diameter of 0.5mm. Because the experimental results
will not be further applied to a bridge design, actually this model is not a strictly scaled one of any
prototype bridge, which simplified the model’s design and manufacturing. Fig. 2 shows the model.
In order to roughly estimate the analysis frequency range, a finite element (FE) model, shown in
Fig. 3, is built in the software package ANSYS and the modal parameters are obtained by performing
a prestressed modal analysis of a large-deflection solution, as listed in Column 2 of Table 2.
The Beam4 element is used to model the deck and pylon, Link10 element for cables and Combin
14 for boundary springs. In the FE analysis, the reference temperature is set to 0 ˚C; the thermal
coefficient of linear expansion for steel is 1.1*10-5/˚C; and the elastic modulus is assumed to decrease
0.036% per degree increase, referring to [6].
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The FE calculation shows that below 40 Hz there are 6 vertical bending, 1 lateral bending and 1
tortional modes for the deck and 2 modes for the pylons. What’s more, above 40 Hz the modes are
spaced closely. To ensure the experiment’s accuracy and simplicity, we only concentrated on the 6
vertical bending modes of interest.
The simulation of boundary conditions is vital for successful testing. At the joint of pylon-deck, the
support fixtures restrain the vertical and lateral translation degrees of freedom; at the two ends of the
deck, three cases are taken into consideration, whose differences lie on the longitudinal translation
DOF UX and rotation DOF RotZ, as listed in Column 2 and 3 of Table 1 and shown in Fig. 4. Case III
is an extreme situation which constrains both the UX and RotZ, and has little chance to happen in
practical structure. However, this case enables us to grasp the limits of the change in frequency and
can provide data for FE model updating.
Fig. 3. FE model
Table 1. Boundary Conditions
DOF Restrained (Y/N)
Case
Stiffness Before Updating
Stiffness After Updating
UX
RotZ
UX
RotZ
UX
RotZ
N /m
N ⋅m
N /m
N ⋅m
Case I
N
N
0
0
0
0
Case II
Y
N
1.00E+10
0
4.29E+04
0
Case III
Y
Y
1.00E+10
1.00E+10
4.29E+04
7.18E+04
*X, Y, Z axes are parallel to the axial lines of the girder, cross-beam and pylon, respectively.
(a) No Restraints
(b) UX Restrained
(c) UX & RotZ Restrainted
Fig. 4. Simulation of Boundary Conditions
Advanced Materials Research Vols. 446-449
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Experimental Approach
Theoretically, the frequency response function (FRF) is independent on the excitation’s
characteristics. In the light of the fact that the hammer impacting method is cheap, easy and quick to
implement and the equipment is within the reach, the steel cable-stayed bridge model was excited
using an impact hammer with the help of DASP-V10 measurement system developed by China Orient
Institute of Noise and Vibration.
There are 28 measuring points uniformly distributed on the deck, and 8 on the tips of the four
pylons. The vibration responses are recorded by 4 accelerometers and the pulse input is recorded by a
load cell mounted on the hammer. So the Single-Input-Multi-Output modal test method is adopted.
The exciting points, denoted by -21 in Fig. 5, is near the 3/8 of the central span to avoid a nodal point
of the 6 concerning modes.
Fig. 5. Layout of Measuring Points
The test uses the Varying-Time-Base mode, a special sampling method where the sampling
frequencies of force pulse signal and response signal are different. In this test, the force’s sampling
rate is 1024 Hz, and the VTB scale is 8, which results in a sampling rate of 128 Hz for acceleration
responses. The cut-off frequency of the low pass anti-aliasing filter is 40 Hz. To reduce the noise that
is uncorrelated with hammer signals for an enhancement of FRF’s quality, the sampling repeat trigger
mode is carried out, with each recording containing three trigger waveforms and then an average is
taken. In every trigger, 4096 response data points are collected, so the signal’s duration is 32s, which
is enough for the response to die away to minimize the leakage.
The whole experiment is performed following the order of Case III, II, I. During each case,
temperature ascends from 0 to 40 ˚C, boundary conditions being changed at 0 ˚C. Make sure every
hammer-impacting is similar to each other in size, in shape and in direction, though these are difficult
to achieve. For each recording, careful assessment of measured data and the resulting FRF must be
undertaken, so as not to acquire poor data with the consequence of loss of quality in the modal
parameters’ estimation.
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Trends in Civil Engineering
Parameter Identification
Fig. 6, Fig. 7 and Fig. 8 are the typical force signal, acceleration signal and the averaged magnitude of
all the measured FRFs.
Fig. 6. Force Signal
Fig. 7. Acceleration Signal
Fig. 8. Average of all the Measured FRFs
After the order of the model is selected manually, the DASP-V10 software automatically obtains
the modal parameters through fitting technique. Because the modes are well separated, the classical
frequency-domain modal analysis method is adopted. The program assumes the residues of each
mode are complex quantity, and compensates for the effects of neighboring modes. The estimates of
natural frequency, damping ratio are weighted averages which take all measured FRFs into account
and the weighting factors are given by the elements in the mode shape.
The Modal Assurance Criterion (MAC) Matrix is used to validate the modal identification. If the
off-diagonal terms in MAC matrix should be small, the modal identification is well done. Fig. 9
presents the 6 vertical bending modes of Case III , with identified parameters in Table 2.
(a) Mode 1 (V1)
(d) Mode 4 (V4)
(b) Mode 2 (V2)
(c) Mode 3 (V3)
(e) Mode 5 (V5)
(f) Mode 6 (V6)
Fig. 9. The First 6 Bending Modes, Case III, 0 ˚C
Advanced Materials Research Vols. 446-449
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Model Updating
The modal analysis’s results describe the properties of the steel CSB model under experimental
condition, but cannot tell the reasons behind the phenomenon. In terms of the explanation of the
results, a good analytical model is needed.
Table 2. Analytical and Measured Results (0 ˚C)
Case III: 0 ˚C
Case II: 0 ˚C
Case I: 0 ˚C
f aini
f aupd
fe
ξe
f aupd
fe
ξe
f aupd
fe
ξe
Hz
Hz
Hz
%
Hz
Hz
%
Hz
Hz
%
V1
4.174
3.971
3.987
0.317
3.766
3.748
0.383
3.766
3.765
0.303
V2
9.784
9.223
9.309
0.652
8.264
8.265
0.543
8.346
8.344
0.427
V3
15.497
14.659
14.64
0.399
10.739
10.731
0.497
10.738
10.701
0.309
V4
17.516
16.469
16.404
0.706
12.64
12.66
0.337
12.652
12.588
0.635
V5
22.627
21.098
21.271
0.342
19.979
20.116
0.319
19.978
19.983
0.437
34.81
32.693
32.816
0.926
30.973
30.949
0.419
30.972
31.073
0.601
Modes
V6
*f
ini
a
, f
upd
a
are the calculated frequencies before and after updating.
* f e , ξ e are the measured frequencies and damping ratios.
Before model updating, the consistence of measured and calculated frequencies is checked, listed
in Column 2 and 4 of Table 2 respectively. If we define the relative difference of natural frequency as
δ=
fa − fe
× 100% , where the subscript a denotes analytical results, while e represents experimental
fe
results, and impose that percentage variation of frequencies cannot exceed 1%, then it is necessary to
modify the FE model’s parameters because δ is about 6% before model updating.
For the reason that the FE model will be used to explain the testing results, the selection of
parameters to be updated should take physical meanings into account. Firstly, we updated the model
according to the measurement results of Case III at 0 ˚C, resulting in a set of optimal parameters. And
then this set of parameters is applied to Case I, Case II, except the boundary spring constants
corresponding to the DOFs without restraints. Considering the plastic deformation of the longitudinal
beams due to straightening process, the loss of section due to the bolt holes and the imperfection of
the support fixtures, we selected the modulus of elasticity of the deck (decreasing 11%), the moment
of inertia of the elements with a bolt hole (decreasing 33%), and the spring stiffness at two ends of the
deck (referring to Table 1) as design variables in the model updating process. The maximal relative
difference of frequencies for Case I to III at 0 ˚C are -0.92%, -0.68% and 0.51%, indicating the
discrepancy has been reduced within the tolerable limit δ = 1% .
It is noted that the experimental data contains the noise, and on the other hand, FE model don’t
consider the damping, so the complete agreement between measured and analytical results is not
necessary.
Results and Discussion
The measured results of each case at 0 ˚C are listed in Table 2, while the measured results at the other
two temperatures will be represented by the relative difference with respect to the value at 0℃. The
discussion section will be based on the experimental measurements. In addition, because the damping
ratios display large variation and cannot be compared with the analytical one, we omitted them in the
following discussion. At the same time, the measured mode shapes seem not to exhibit somewhat
regularity, and is overlooked, too. Only the natural frequency remained.
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Fig. 10. Results of Case I
Fig. 10 shows the relationship between modal frequencies and temperature of Case I, with the
vertical scale the relative ratios of the measured frequencies to those at 0 ˚C. It is clear that the first 6
vertical bending modal frequencies tend to descend when the temperature ascends under the normal
boundary condition, i.e. being free to move longitudinally. Case I is repeated three times and all the
three results, although the variation exists, consistently show the trend of “high temperature, low
frequency” and furthermore the relative difference is approximately 0.5%. The trend of Case I agrees
with the FE results, where the analytical frequencies decrease by 0.7% when the temperature increase
from 0 ˚C to 40 ˚C. In the FE analysis, only two factors, i.e. the drop of the elastic modulus and the
geometric change due to thermal expansion, are taken into account. The geometric change has little
influence on the modal frequency [6], so it is reasonable to attribute the frequency shift at different
temperatures primarily to the temperature-dependence of elastic modulus. As to the quantitative
aspect, the disagreement between measurement and calculation may be caused by the FE modeling
errors, data resolution and a number of practical random effects during experiment, such as the
inevitable electronic noise, the model’s nonlinear behavior, the variation of the temperature in the
cabinet, to name a few.
Fig. 11. Comparison of Frequency Changes for Case I and II
Fig. 11 makes a comparison of the varying amplitudes of frequencies between Case I and II when
suffering a 40℃ temperature change. One can see that with the longitudinal restraint, i.e. Case II, the
absolute frequency variation rates of mode 1~6 are between 0.48% ~ 0.88%, larger than those in Case
I without the longitudinal restraints, except the mode 4. The abnormality for mode 4 may be related to
the uncertainty in this experiment. Fig. 11 gave an idea that as the temperature increase, the varying
amplitude of modal frequency for a cable-stayed bridge is influenced to some extent by the
Advanced Materials Research Vols. 446-449
3271
longitudinal restraint, or the boundary condition. If the longitudinal restraints at the deck’s ends are
stronger, the axial compression force will be larger because of the constraint to the elongation of the
deck. The axial force reduces the total stiffness of the structure in general sense, so it provides an
explanation of the increase of frequency difference from Case I to Case II. Case II reflects the failure
situation of expansion joints in actual cable-stayed bridge. It is necessary to point out that the
discrepancy of measured and analytical frequencies at 40 ˚C for Case II is nearly 2%, larger than the
previously mentioned tolerable limit 1%. This may be caused by the disturbance during the alteration
of the supports at the deck’s ends, resulting in a drop of longitudinal stiffness.
At 0 ˚C, the 1~6 vertical vibration’s frequencies of Case I and Case II are different, and the
variation ranges from -0.28%~0.95%. On the other hand, when the temperature’s change is 40 ˚C, the
corresponding frequency changes of Case I and Case II fall into a range of -0.94%~-0.19% and
-0.98%~-0.48%, respectively. One can see that when the temperature is constant the effects of
boundary conditions on the modal frequencies are in the same order of magnitude compared with the
effects of the temperature change when the supporting conditions are identical. This implication may
be only applicable to this experimental configuration. However, in the FE analysis, the effects of the
translational DOF UX on the modal frequencies are not as spectacular as that of temperature change.
The disagreement of the measured and analytical results reflects the inconstancy on the boundary
conditions, suggesting an improvement on the experiment’s set-up be needed. In addition, although
the rotational DOF RotZ at the ends of deck influence the vertical bending mode much, it is
disregarded in this paper because in practical engineering the failure of expansion joints hardly
supplies the bridge with a considerable rotational constraint.
In this study, model updating with the consideration of physical meaning was performed to
minimize the difference between experimental and analytical models, which of Case I and Case II are
summarized in Column 2~5 of Table 3. Column 6~7 in Table 3 list the relative frequency difference
induced by 40 ˚C temperature change for Case I and II. These data manifested that the numerical
errors of frequency and temperature-induced frequency change are in the same order of magnitude,
which helps us to understand better the difficulty of damage identification based on FE model in
practical application which includes much more unknown and uncertain factors than this experiment.
Table 3. Calculating Errors and Measured Frequency Variation (Case I, II)
Calculating Errors after Model Updating
0 ˚C
Modes
Case I
Measured Frequency Changes
40 ˚C
Case II
Case I
40 ˚C
Case II
Case I
Case II
V1
0.02%
0.49%
-0.37%
-1.97%
-0.32%
-0.51%
V2
0.02%
-0.01%
-0.56%
-2.00%
-0.19%
-0.48%
V3
0.35%
0.07%
0.01%
-1.33%
-0.43%
-0.52%
V4
0.51%
-0.16%
0.69%
-0.69%
-0.94%
-0.88%
V5
-0.03%
-0.68%
-0.43%
-1.42%
-0.35%
-0.64%
V6
-0.33%
0.08%
-0.76%
-0.48%
-0.32%
-0.66%
Conclusions
This paper performed a dynamic testing on a steel cable-stayed bridge model, presented the results of
the first 6 vertical bending modal frequencies with different boundary conditions and ambient
temperature, and attempted to make explanations of the results through the updated FE model. The
concluding remarks can be drawn as follows:
1) As the temperature increases, the modal frequencies have a trend to decrease. The FE analysis
indicates the reason is the negative correlation between the modulus of elasticity of the material and
temperature.
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2) The varying amplitude of the CSB’s modal frequencies at certain temperature changes are
influenced by longitudinal restraints at the deck’s ends, in the other words, the boundary conditions.
When the temperature changes, the deck with expansion constrained has a larger amount of frequency
changes than that without longitudinal restraints, because the axial force introduced by longitudinal
restraints can reduce the total stiffness of the structure.
3) In the set-up of this experiment, the longitudinal restraint and the temperature change have
similar impacts on the vertical vibration frequencies of the CSB model. It may not be a general
conclusion applicable to other situations.
4) In this study, even if the FE model has been updated, the calculation errors of frequencies are in
the same order of magnitude with the frequency variation due to the change of temperature and
boundary conditions. One can realize that the damage identification based on FE model will
encounter great difficulties in practical applications.
There is no doubt that the CSB model presented here suffers a lot of uncertainty. But compared
with the real cable-stayed bridge exposed in the natural environment, the model in the laboratory has
a relatively simpler environment and has a greater chance to obtain the cause-effect relation between
temperature and dynamic properties. The results of this experiment can provide a reference to related
researches on long span bridge health monitoring, and further research is undergoing now.
Acknowledgements
The authors would like to acknowledge financial support from Autonomous Foundation from State
Key Laboratory for Disaster Reduction in Civil Engineering of Tongji University (Grant No.
SLDRCE08-A-05).
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