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Experimental and analytical studies on dynamic characteristics of a large span cable-stayed bridge
Engineering Structures 27 (2005) 535–548 www.elsevier.com/locate/engstruct
Experimental and analytical studies on dynamic characteristics of a large span cable-stayed bridge Wei-Xin Rena,b,∗, Xue-Lin Penga, You-Qin Lina a Department of Civil Engineering, Fuzhou University, Fuzhou, Fujian Province 350002, People’s Republic of China b Department of Civil Engineering, Central South University, Changsha, Hunan Province 410075, People’s Republic of China
Received 7 March 2004; received in revised form 2 November 2004; accepted 23 November 2004 Available online 1 February 2005
Abstract An analytical and experimental modal analysis has been carried out on the Qingzhou cable-stayed bridge in Fuzhou, China. Its main span of 605 m is currently the longest span among the completed composite-deck cable-stayed bridges in the world. An analytical modal analysis is performed on the developed three-dimensional finite element model starting from the deformed configuration to provide the analytical frequencies and mode shapes. The field ambient vibration tests on the bridge deck and all stay cables were conducted just prior to the opening of the bridge. The output-only modal parameter identification is then carried out by using the peak picking of the average normalized power spectral densities in the frequency-domain and stochastic subspace identification in the time-domain. A good correlation is achieved between the finite element and ambient vibration test results. It is demonstrated that the analytical and experimental modal analysis provide a comprehensive study on the dynamic properties of the bridge. The ambient vibration tests are sufficient to identify the most significant modes below 1.0 Hz of this kind of large span cable-stayed bridge. The validated finite element model with respect to ambient vibration test results can serve as the baseline for a more precise dynamic response prediction and long-term health monitoring of the bridge. © 2005 Elsevier Ltd. All rights reserved. Keywords: Cable-stayed bridge; Modal analysis; Parameter identification; Ambient vibration test; Spectra; Frequency; Finite element method
1. Introduction It is the most important issue to ensure an adequate level of safety of both new and existing large span bridges against dynamic loadings such as traffic, wind and earthquakes. This can be achieved if the dynamic properties of the bridge such as natural frequencies, mode shapes and damping ratios are accurately determined. Modal analysis is a technique that estimates the dynamic characteristics (modal parameters) of a structure. These modal parameters serve as a basis for the finite element model updating, structural control, damage detection, condition assessment, and long-term health monitoring of the structure. ∗ Corresponding author at: Department of Civil Engineering, Fuzhou University, Fuzhou, Fujian Province 350002, People’s Republic of China. Tel.: +86 591 7892454; fax: +86 591 3737442. E-mail address: [email protected] (W.-X. Ren). URL: http://bridge.fzu.edu.cn (W.-X. Ren).
0141-0296/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2004.11.013
Modal analysis was originally developed in more advanced mechanical and aerospace engineering disciplines [1–3]. Although its technique is not new, the application of digital computers caused a dramatic increase of the practical possibilities of modal analysis. There is a clear merit in trying to transfer this technology into civil engineering applications where we are dealing with problems which have a completely different scale, logistics and rationale, compared with mechanical and aerospace engineering counterparts. In the context of bridge engineering discipline, for instance, the encountered structures are often complex, large in size and low in frequency. In general, the modal analysis procedure for more complicated structures includes analytical and experimental modal analysis. Analytical modal analysis is the process of determining the dynamic properties of a structure based on the free vibration solution of the equations of motion. The finite element (FE) method is currently a common way
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to perform an analytical modal analysis of cable-stayed bridges [4–10]. Starting from the knowledge of the structural geometry, boundary conditions and material properties, a structure is expressed in a matrix form. It contains sufficient information to determine the structural modal parameters that completely describe the dynamic properties of the structure. However, the success of the FE application strongly depends on the experimental verification of the results since many simplified assumptions are made in the modeling of complicated structures and there are some uncertainties in the material and geometric properties when establishing the FE models of real structures. The reliability of analytical modal analysis results is often questionable if the FE model is not experimentally validated. On-site dynamic testing of a structure provides an accurate and reliable description of its dynamic characteristics. Experimental modal analysis is then a process of identifying the modal parameters based on the dynamic measurements. Starting from the measurements of dynamic input forces and output responses of interests and implementing a modal extraction technique, the dynamic characteristics of the structure can be estimated. For the experimental modal analysis of bridges, there are three main types of dynamic tests: (1) forced vibration tests; (2) free vibration tests; and (3) ambient vibration tests. In the first method, the bridge is excited by artificial means and correlated input–output measurements are performed. Impulse hammers, drop weights and electrodynamic shakers are the main excitation equipments. The successes of forced vibration tests are limited for relatively small structures. In the case of large and flexible bridges like cable-stayed or suspension bridges, it often requires heavy equipment and involves important resources to provide a controlled excitation at sufficiently high levels [11], which becomes difficult and costly. Forced vibration tests are directly related with the standard techniques of modal parameter estimation that were previously developed in mechanical and aerospace engineering. The modal parameter identification in this case is carried out based on both input and output measurement data. By introducing an initial perturbation on the bridge, a condition of free vibration can be induced. Free vibration tests can be done by a sudden release of a heavy load or mass appropriately connected to the bridge deck [12]. Both forced and free vibration tests, however, need the artificial means to excite the bridges and the traffic has to be shut down. This could be a serious problem for intensively used bridges. In contrast, ambient vibrations, induced by traffic, winds, and pedestrians, are the natural or environmental excitations of the bridge. Ambient vibration tests have an advantage of being inexpensive since no equipment is needed to excite the bridge. It corresponds to the real operating condition of the bridge. The service state need not be interrupted to use this technique. Therefore, ambient vibration tests have been successfully applied to several large scale cable-supported bridges, such as the Golden Gate Bridge [13], the Bosporous
Suspension Bridge [14], the Deer Isle Bridge [15], the Quincy Bayview Bridge [16], the Tsing Ma Suspension Bridge [17], the Kap Shui Mun Cable-Stayed Bridge [18], the Maysville Cable-Stayed Bridge [19], and the Roebling Suspension Bridge [20]. In cases of both free and ambient vibration tests, only response data are measured while actual loading conditions are not measured. A modal parameter identification procedure will therefore need to base itself on output-only data. Many challenges exist in the experimental modal analysis of real bridges using ambient vibration measurements. There have been several modal parameter identification techniques available that were developed by different investigators for different uses. The mathematical background for many of these methods is often very similar, differing only from implementation aspects (data reduction, type of equation solvers, sequence of matrix operations, etc.). The benchmark study has been carried out for evaluating the dynamic characteristics from ambient vibration data [21]. The objective of this paper is to demonstrate how the modal analysis on a large cable-stayed bridge is carried out. The field ambient vibration tests on the bridge deck and all stay cables of the Qingzhou cable-stayed bridge were conducted just prior to the opening of the bridge to traffic in order to obtain the baseline dynamic characteristics. The dynamic properties were extracted from the peak picking of the average normalized power spectral densities (ANPSDs) in the frequency-domain and more advanced stochastic subspace identification in the time-domain. The analytical modal analysis was carried out on the developed three-dimensional finite element model. To improve the FE predictions, the FE model was calibrated with respect to the ambient vibration results. The validated finite element model that reflects the built-up structural conditions can serve as the baseline model for the bridge assessment, damage identification, health monitoring and succeeding dynamic response predictions of the bridge. 2. Bridge description The target Qingzhou cable-stayed bridge in this paper is one of the bridges on Luo-Chang Highway over the Ming River in Fuzhou, Fujian Province, China. The bridge is a 5span composite cable-stayed bridge with an overall length of 1186.34 m (41.13 m + 250 m + 605 m + 250 m + 40.21 m). Its main span of 605 m is currently the longest span among the completed composite-deck cable-stayed bridges and lists the fifth longest span among all types of completed cablestayed bridges all over the world. Fig. 1 shows a schematic representation of the bridge with plan, elevation and typical cross sections. The bridge was completed in the year 2000, but it was officially open to the traffic in the year 2002 due to the construction delay of the approach spans. A photograph of the completed Qingzhou cable-stayed bridge is shown in Fig. 2. The main structural features of the bridge are as follows:
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Fig. 1. Schematic representation of Qingzhou cable-stayed bridge. (a) Plan and elevation, (b) typical cross section of composite deck.
Fig. 2. The Qingzhou cable-stayed bridge.
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Table 1 Properties of stay cables Cable no.
Number of strands
Sectional area (cm2 )
Mass per unit length (kg/m)
E (MPa)
σb (MPa)
C1–C4; S1–S4 C5–C6; S5–S6 C7–C9; S7–S8 C10–C12; S9–S10 C13–C14; S11–S13 C15–C18; S14–S17 C19–C20; S18–S19 C21; S20–S21
27-7Φ5 34-7Φ5 37-7Φ5 43-7Φ5 48-7Φ5 55-7Φ5 73-7Φ5 85-7Φ5
37.795 47.593 51.793 60.191 67.190 76.989 102.185 118.983
32.792 41.293 44.937 52.224 58.296 66.798 88.659 103.233
1.95 × 105
1860.0
• The composite deck consists of two I-type main steel girders and a 25 cm thick concrete slab. Each steel girder is 2.45 m high and its maximum plate thickness reaches 80 mm. The precast concrete slab is connected to the steel girders and floor beams by shear studs. The spacing of the steel floor beams is 4.5 m. The bridge has 6 lanes that carry two roadways 29 m wide. • The two diamond-shaped towers are of reinforced concrete. The height of the towers is 175.5 m with a clear navigation of 43 m. The towers are erected on group piles with a diameter of 2 m for Pier 2 and 3 m for Pier 3. The longest pile reaches a length of 71.6 m. • The cable is a fan type in both planes. There are in total 21 × 8 = 168 stay cables. The cables are composed of a number of strands. The strand number varies from 27 to 85 per cable in eight groups. One strand includes 7 high strength wires with a diameter of 5 mm (7Φ5 high strength wires). The density of the cables depends on the number of wires in the individual cables. The material properties of the stay cables are listed in Table 1.
3. Analytical modal analysis To perform the analytical modal analysis of the Qingzhou cable-stayed bridge, a three-dimensional finite element model has been developed using ANSYS® [22]. The model represents the bridge in its current as-built configuration with the geometry and structural properties estimated from the design drawings. Four types of finite elements are used to model the different structural members: • The steel girders, stringers, floor beams and concrete towers are modeled by three-dimensional elastic beam elements (BEAM4). There are in total 1586 beam elements used. • All stay cables are modeled by 3-D tension-only truss elements (LINK10) since they are primarily designed to sustain tension forces. The effect of cable sag is taken into account by using the initial stress matrix due to cable tensions and stress-stiffening capability instead of the cable equivalent modulus of elasticity. Pre-tensions of the cables are included by the initial element strains input.
• The concrete slab is divided into 508 shell elements (SHELL63). • The piers and platforms are modeled by solid elements (SOLID45) where 149 elements are used. The modeling of bridge boundary conditions is an important issue in the modal analysis. Two types of bridge bearings are used in the Qingzhou cable-stayed bridge. Fixed bearings are constructed to connect Pier 2 to the deck, while expansion bearings are used to the rest of piers. In the current model, bridge bearings are modeled by a set of rigid link elements connecting the superstructure and piers. To simulate the actual behavior, the fixed and expansion bearings are simulated by coupling the corresponding translational and rotational degrees of freedom at both end nodes of the link elements. In addition, longitudinal springs are used to account for the restrained action in the longitudinal direction from the adjacent span and abutment at Pier 0 and Pier 5. The soil–pier interaction is not included in the analysis. The constructed three-dimensional finite element model of the Qingzhou cable-stayed bridge is shown in Fig. 3. More details about the model can be found in [23]. Fig. 4 shows some typical vertical, transverse and torsional mode shapes obtained from the analytical modal analysis. Vertical modes are those modes dominated by the vertical bending of the bridge deck. Transverse modes are those modes dominated by the lateral sway of the bridge deck in the horizontal plane, while torsional modes are dominated by the torsional behavior of the bridge deck around the longitudinal axis of the bridge. It is demonstrated that the most significant modes of the Qingzhou cablestayed bridge are below 1.0 Hz. Some dominated modes are close to each other. The first lateral mode of the bridge, for instance, appears right after the second vertical mode. Some dominated modes are coupled with the bending of the tower or the longitudinal drift of the deck. To see the mass contribution of each mode, Table 2 lists the cumulative modal participating factors of the bridge deck for the vertical bending, transverse bending and torsion vibration within 20 modes. Results have demonstrated that more modes are needed to achieve a higher mass participation in the vertical bending of the deck. One of the important features of a large span cable-stayed bridge is that the dead load (self-weight) often contributes
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Fig. 3. Three-dimensional finite element model of the bridge.
Fig. 4. Typical mode shapes obtained from finite element analysis.
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Fig. 4. (continued).
most of bridge loads. It has been realized that the dead load has an influence on the stiffness of cable-supported bridges. This influence can be included by carrying out the modal analysis from the static initial equilibrium position due to dead loading and cable pre-tensions. Therefore, the modal analysis of cable-stayed bridges is a “pre-stress modal analysis” that should include two key steps: • The static analysis is first conducted under dead load and cable pre-tensions. The objective of this process is to achieve the deformed equilibrium configuration of the bridge in which all structural members are “pre-stressed”. • The modal (free vibration) analysis is then followed starting from the deformed equilibrium configuration due to dead load and cable pre-tensions. Many studies [4,6,10] have shown that the linear static analysis is adequate to calculate the static initial equilibrium configuration. Based on the linear static equilibrium
position, the analytical modal analysis has been carried out to study the dynamic characteristics of the Qingzhou cablestayed bridge. To see the effect of dead load on the stiffness of such a cable-stayed bridge, the natural frequencies obtained from the deformed position are compared in Table 4 with those obtained from the non-deformed position. It is observed that all natural frequencies are increased due to the pre-stressed modal analysis. Although the pre-stressed modal analysis might have a minimal effect on increasing the bridge’s natural frequencies, a starting position is essential to determine the responses in succeeding dynamic analyses such as earthquakes, winds and vehicles loadings. 4. Ambient vibration tests on bridge deck Modal testing of a bridge on site provides an accurate and reliable prediction of its global modal parameters.
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Table 2 Cumulative modal participating factors within 20 modes (%) Mode no.
Frequency (Hz)
Vertical bending (%)
Transverse bending (%)
Torsion (%)
Nature of modes of vibration
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0.222 0.266 0.267 0.415 0.454 0.478 0.526 0.535 0.538 0.551 0.571 0.592 0.607 0.622 0.650 0.710 0.712 0.738 0.748 0.752
9.09 9.12 9.12 22.57 22.61 58.04 58.05 58.05 58.05 58.05 59.13 59.64 73.54 73.54 73.56 74.67 74.67 76.02 76.02 91.41
0.0 0.0 27.54 27.54 27.54 27.54 27.54 35.10 73.75 99.24 99.24 99.24 99.24 99.24 99.24 99.24 99.32 99.32 99.51 99.51
0.08 0.08 8.92 9.05 9.05 9.37 9.37 19.58 70.60 99.57 99.58 99.58 99.71 99.71 99.71 99.72 99.73 99.75 99.76 99.90
vertical bending of deck vertical bending of deck transverse bending of deck vertical bending of deck vertical bending of deck vertical bending of deck vertical bending of deck bending of tower bending of tower torsion of deck vertical bending of deck local vertical bending of side span local vertical bending of side span torsion of deck vertical bending of deck vertical bending of deck torsion of deck local vertical bending of side span transverse bending of deck local vertical bending of side span
On November 18–21, 2002, just prior to the official opening of the bridge, ambient vibration tests on the deck of the Qingzhou cable-stayed bridge were carried out. The equipment used for the tests included accelerometers, signal cables, and a 32-channel data acquisition system with signal amplifier and conditioner. Accelerometers convert the ambient vibration responses into electrical signals. Cables are used to transmit these signals from sensors to the signal conditioner. The signal conditioner unit is used to improve the quality of the signals by removing undesired frequency contents (filtering) and amplifying the signals. The amplified and filtered analog signals are converted to digital data using an analog to digital (A/D) converter. The signals converted to digital form are stored on the hard disk of the data acquisition computer; these are the data for modal parameter identification. 4.1. Measurement setup To identify the good mode shapes of the bridge, a dense measurement location on the deck in the vertical, transverse and longitudinal directions was conceived. Measurement stations were chosen near each anchor position on the deck. In addition, several measurement stations were chosen on Span 1 and Span 5. As a result, there were in total 180 measurement stations on the bridge deck. The measurement station arrangement on the bridge deck is shown in Fig. 5, where U and D refer to upstream and downstream directions, respectively. Force-balance accelerometers with measurement frequency ranging from 0.2 to 80 Hz were used to record the ambient vibration responses. The accelerometers were directly placed on the pavement due to the limited access to the
actual floor beams. Fig. 6 shows the accelerometers mounted on the bridge deck in the vertical and transverse directions. 15 accelerometers were used in the tests; 12 accelerometers were moveable, and 3 accelerometers were fixed as reference accelerometers. A reference location was selected according to the information obtained from the mode shapes of the preliminary finite element model. Three base stations were located at the center of a side span (D15), at the deck near Pier 2 (U24) and at the center of the main span (U46). There were 15 test setups to cover all planned measurement locations of the bridge deck. Each setup yielded a total of 15 sets of data, 12 sets from the moveable stations and 3 sets from the base stations. Measurement stations per setup are summarized in Table 3. 4.2. Field measurements and data pre-processing The test was started from the Fuzhou side and progressed to the Changle side. Once the data were collected in one setup, the moveable accelerometers were shifted to the next stations while the base stations remained stationary. This sequence was repeated 15 times to obtain ambient vibration measurements on all stations. Measurements in the vertical, transverse and longitudinal directions were carried out separately in the same stations. The sampling frequency on site was 80 Hz. The ambient vibration measurements were simultaneously recorded for 20 min at all channels. As a result, there were totally 96,000 data points per channel. The typical tri-axial acceleration time history records at the station D46 (main span center) are as shown in Fig. 7. An inspection of the time history measurements shows a significant variation of the deck ambient vibration responses. It is observed that the average
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Fig. 5. Measurement station arrangement for ambient vibration test.
Fig. 6. Accelerometers mounted on the deck. (a) Vertical accelerometer, (b) transverse accelerometer.
Table 3 Measurement stations per setup Setup
Measurement stations
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
U1, U2, U3, U4, U5, U6; D1, D2, D3, D4, D5, D6; D15, U24, U46 U7, U8, U9, U10, U11, U12; D7, D8, D9, D10, D11, D12; D15, U24, U46 U13, U14, U15, U16, U17, U18; D13, D14, D15, D16, D17, D18; D15, U24, U46 U19, U20, U21, U22, U23, U24; D19, D20, D21, D22, D23, D24; D15, U24, U46 U25, U26, U27, U28, U29, U30; D25, D26, D27, D28, D29, D30; D15, U24, U46 U31, U32, U33, U34, U35, U36; D31, D32, D33, D34, D35, D36; D15, U24, U46 U37, U38, U39, U40, U41, U42; D37, D38, D39, D40, D41, D42; D15, U24, U46 U43, U44, U45, U46, U47, U48; D43, D44, D45, D46, D47, D48; D15, U24, U46 U49, U50, U51, U52, U53, U54; D49, D50, D51, D52, D53, D54; D15, U24, U46 U55, U56, U57, U58, U59, U60; D55, D56, D57, D58, D59, D60; D15, U24, U46 U61, U62, U63, U64, U65, U66; D61, D62, D63, D64, D65, D66; D15, U24, U46 U67, U68, U69, U70, U71, U72; D67, D68, D69, D70, D71, D72; D15, U24, U46 U73, U74, U75, U76, U77, U78; D73, D74, D75, D76, D77, D78; D15, U24, U46 U79, U80, U81, U82, U83, U84; D79, D80, D81, D82, D83, D84; D15, U24, U46 U85, U86, U87, U88, U89, U90; D85, D86, D87, D88, D89, D90; D15, U24, U46
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Table 4 Calculated and identified modal parameters Nature of modes of vibration
Finite element analysis (Hz) Starting from Starting from deformed un-deformed position position
Stochastic subspace identification Frequencies Damping (Hz) ratio (%)
Peak-picking identification (Hz)
MAC values
1st vertical bending 2nd vertical bending 1st transverse bending 3rd vertical bending 4th vertical bending 5th vertical bending 1st torsion 7th vertical bending 2nd torsion 3rd torsion 2nd transverse bending 1st longitudinal
0.222 0.266 0.267 0.415 0.454 0.478 0.551 0.571 0.622 0.712 0.748 1.925
0.226 0.272 0.263 0.446 0.480 0.505 0.556 0.653 0.610 0.726 0.628 1.925
0.227 0.271 0.262 0.444 0.482 0.506 0.555 0.609 0.653 0.701 0.612 1.922
0.989 0.755 0.991 0.966 0.783 0.942 0.982 0.882 0.778 0.895 0.919 –
0.217 0.252 0.264 0.405 0.442 0.468 0.548 0.566 0.606 0.701 0.746 1.920
signal levels for all channels in the vertical direction are about 4 times higher than those in the transverse direction and about 10 times than those measured in to the longitudinal direction. That is identical to the ambient vibration measurements on a metallic arch railway bridge reported by Calcada et al. [24]. Before the modal parameter identification is performed, the measured data are first detrended, which enables the removal of the DC-components that can badly influence the identification results. This was accomplished by subtracting the mean values calculated over the full duration of each measurement. A sampling frequency of 80 Hz on site results in a frequency range from 0 to 40 Hz. For such a large span cable-stayed bridge, however, the frequency range of interest lies between 0 and 2.0 Hz, covering the most important frequencies in the vertical, transverse and longitudinal directions. So a resampling of the raw measurement data is necessary. A resampling and filtering from 80 to 4 Hz is the same as decimating (=low-pass filtering and resampling at a lower rate) 20 times which results in 96,000/40 = 4800 data points per channel and an excellent frequency range from 0 to 2.0 Hz. 4.3. Modal parameter identification The classical modal parameter identification techniques that involve frequency- or time-domain analysis of dynamic responses to forced vibration cannot be directly applied to the output-only vibration data because the input excitations are not measured in the test [21]. Two complementary modal parameter identification techniques are implemented in this work. They are the rather simple peak picking (PP) method in the frequency-domain and the more advanced stochastic subspace identification (SSI) method in the time-domain. The data processing and modal parameter identification were carried out by MACEC [25]. The peak picking method is initially based on the fact that the frequency response function goes through an extreme
0.7 0.7 1.0 0.9 2.8 1.5 0.4 0.4 4.6 1.1 0.5 1.9
value around the natural frequency. In the context of ambient vibration measurements, the frequency response function is replaced by the auto spectra of the output-only data [26]. To include the measurement channels of all setups, the average normalized power spectral densities (ANPSDs) are used. A practical implementation of the ANPSDs was realized by Felber [27]. In such a way, the identified natural frequencies are simply obtained from the observation of the peaks on the graphs of ANPSDs. The ANPSDs of vertical and transverse (half-sum) acceleration measurements of all channels of the bridge deck are shown in Fig. 8, where the peaks can be clearly seen and then the frequencies can be picked up. The stochastic subspace identification is a time-domain method that directly works with time data, without the need to convert them to correlations or spectra. The stochastic subspace identification algorithm identifies the system matrix of the state space model based on the measurements by using robust numerical techniques. Details of the theoretical background of the SSI method can be found in the literature [28,29]. Once the mathematical description of the structure is found, it is straightforward to determine the modal parameters by an eigenvalue decomposition. Table 4 summarizes the most significant frequencies of the Qingzhou cable-stayed bridge identified from the test. It can be seen that the identified frequencies agree well between the peak picking and stochastic subspace identification methods. Most of the important frequencies of the bridge deck are below 1.0 Hz, with the lowest frequency 0.223 Hz being a symmetric deck bending in the vertical direction. It is observed that the simple peak picking method cannot identify all important mode shapes for such a large bridge. Vibration mode shapes of the bridge deck have been extracted by the stochastic subspace identification method. Fig. 9 shows the most dominated identified mode shapes in the vertical bending, transverse bending and torsion. The obtained results have demonstrated that the ambient vibration response measurements are sufficient enough to identify the
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Fig. 7. Tri-axial acceleration time histories measured at Station U46.
most significant modes of large span cable-stayed bridges, despite the rather low level of ambient vibration signal captured. The low range (0–1.0 Hz) of natural frequencies of interest, and the relatively dense modes of vibration in that range, are typical characteristics of this kind of bridge. For such a specific cable-stayed bridge, the results of the identified vertical bending mode shapes imply that the stochastic subspace identification method can identify a very good quality of the symmetric vertical bending mode shapes as high as the 9th mode (the 5th symmetric vertical bending mode). However, the quality of the anti-symmetric vertical bending mode shapes is not as good as expected. The damping properties can also be extracted by the stochastic subspace identification method since it is a complex mode based procedure. The identified modal damping ratios are also listed in Table 4. However, the damping property of real large cable-stayed bridges is not fully understood yet due to the complicated damping mechanism of the bridges. It is known that bridge damping in general varies with the amplitude of vibration. Therefore, the applicability of an identified damping ratio through ambient vibration tests is still an issue that needs to be evaluated further by using other identification techniques or other dynamic tests with large vibration amplitudes. 4.4. Correlation between analytical and experimental modal parameters The correlation between modal parameters identified from the test and those calculated numerically can be evaluated by comparing the values of the natural frequencies
Fig. 8. Averaged normalized power spectral densities (ANPSDs). (a) Vertical measurements, (b) transverse measurements.
and corresponding mode shapes. Table 4 shows the comparison of the numerically calculated frequencies from the
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Fig. 9. Typical mode shapes obtained from field tests by stochastic subspace identification.
three-dimensional finite element analysis and experimentally identified frequencies from the field ambient vibration tests. This analysis clearly shows a good correlation between calculated and identified natural frequencies of the bridge, except the 2nd transverse bending frequency. To evaluate the correlation of calculated and identified mode shapes, the Modal Assurance Criterion (MAC) is used. The MAC between a measured mode φm j and an analytical mode φak is defined as MAC j k =
|φmT j φak |2 . T φ )(φ T φ ) (φak ak mj mj
(1)
The MAC values are shown in Table 4. It can been seen that higher values of MAC are found for the most significant modes, which demonstrates a good correlation between the identified and calculated mode shapes, although the anti-
symmetric bending modes of the bridge deck are associated to the lower MAC values. The fact is again verified that the quality of the identified anti-symmetric bending mode shapes in the vertical direction is not so good.
5. Ambient vibration tests on stay cables Stay cables are the most critical structural components in cable-stayed bridges. The tension forces in the stay cables control the internal force distribution in the deck and towers as well as the bridge alignment. Therefore, the cable tension force is an important index in the condition assessment and long-term monitoring of cable-stayed bridges. Experimental vibration measurement is one of the most widely used techniques for tension evaluation of cable-stayed bridges.
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Fig. 9. (continued).
The simplest relationship to calculate the cable forces based on the vibration frequency is the so-called “taut string” chord equation: T =
4m L 2 fn2 n2
(2)
where T = cable tension; m = cable mass per unit length; L = length between cable fixed ends; and fn = nth natural frequency. Right after the tests on the bridge deck, ambient vibration measurements were carried out for all 168 stay cables of the Qingzhou cable-stayed bridge. One accelerometer per cable was used to measure the acceleration in the out-ofplane direction of the stay cables. The accelerometer was mounted securely to the cable as far up as could be reached. The signals were recorded for 5 min at a sampling rate of 40 Hz.
The key to success of vibration-based cable tension evaluation is the accuracy of the identified frequencies of each stay cable. Based on the vibration chord theory (Eq. (2)), the peaks in the cable’s power spectrum are equally spaced when the cable tension is constant. The higher frequencies are integer multiples of the fundamental frequency. Fig. 10 presents the power spectral density (PSD) associated with the ambient acceleration measurement of one of the longest stay cables of the bridge. Response spectra for the Qingzhou bridge cables showed the taut-string characteristics consistently, but it does not always occur in the low frequency range. To measure the spacing between the peaks, neglecting the order information is somewhat easy. So determining the equal spacing between peaks in the power spectrum provides the fundamental frequency of the stay cables. The cable tension force can then be calculated by T = 4m L 2 f 2
(3)
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• The analytical and experimental modal analysis provides a comprehensive investigation of the dynamic properties of bridges. The analytical modal analysis through threedimensional finite element modeling gives a detailed description of the physical and modal characteristics of the bridge, while the experimental modal analysis through the field dynamic tests offers a valuable source of information for validating the drawing-based idealized finite element model.
Fig. 10. Ambient acceleration spectra of Cable S21.
• An ambient vibration based test is a convenient, faster and cheaper way to perform the bridge dynamic test. The ambient vibration response measurements are sufficient enough to identify the most significant modes of large span cable-stayed bridges, despite the rather low level of ambient vibration signal captured. The low range (0–1.0 Hz) of natural frequencies of interest, and the relatively dense modes of vibration in that range, are typical characteristics of this kind of bridge. • The output-only modal parameter identification of large span cable-stayed bridges can be effectively carried out by using two independent numerical techniques: the peak picking of the average normalized power spectral densities (ANPSDs) in the frequency-domain and more advanced stochastic subspace identification in the timedomain. It is essential to verify the identified results by using some other independent identification techniques due to the complexity of the problem. The application of supplementary techniques will increase the reliability of the identified results.
Fig. 11. Fundamental frequencies of all stay cables.
where f is the fundamental frequency of the stay cable. The identified fundamental frequencies determined by the spacing between peaks in the power spectrum density of the ambient vibration measurements are plotted in Fig. 11 for both downstream and upstream cables. The identified fundamental frequencies of the downstream and upstream cables are consistent. The distribution of cable fundamental frequencies is almost symmetric, which is identical to the distribution of the cable pre-tension forces. It is demonstrated that the fundamental cable frequencies of the Qingzhou cablestayed bridge are relatively low, with a range from 0.36 Hz for the longest cable to 1.97 Hz for the shortest cable. Ambient vibration measurement demonstrates a high confidence in identification of the fundamental frequency of the stay cables. 6. Conclusions The following conclusions are drawn from the modal analysis performed on a full sized large span cable-stayed bridge:
• A good correlation is achieved between the numerically calculated modal parameters from three-dimensional finite element analysis and identified ones from field ambient vibration tests. The validated finite element model that reflects the built-up structural conditions can serve as the baseline model for a more precise dynamic response prediction, damage identification and long-term health monitoring of the bridge. • The damping mechanism of this kind of complex bridge is complicated. An analytical evaluation of the bridge damping is extremely difficult. The damping property of real large cable-stayed bridges is not yet fully understood. The applicability of identified damping ratios through ambient vibration tests is still an issue that needs to be evaluated further by using other identification techniques or other dynamic tests with large vibration amplitudes. • The power spectral density analysis of ambient vibration measurements is implemented to identify the fundamental frequencies of all 168 stay cables of the Qingzhou cable-stayed bridge. The fundamental frequencies of the stay cables are relatively low, with a range from 0.36 Hz for the longest cable to 1.97 Hz for the shortest cable. Ambient vibration measurement demonstrated a high identification confidence of the fundamental frequencies of the stay cables in the cable-stayed bridge.
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Acknowledgement Support from the Natural Science Foundation of China (NSFC), under grant number 50378021, is greatly acknowledged. References [1] Ewins DJ. Modal testing: Theory and practice. England: Research Studies Press Ltd; 1984. [2] Juang JN. Applied system identification. Englewood Cliffs (NJ): Prentice-Hall Inc.; 1994. [3] Maia NMM, Silva JMM, editors. Theoretical and experimental modal analysis. England: Research Studies Press Ltd; 1997. [4] Fleming JF, Egeseli EA. Dynamic behavior of a cable-stayed bridge. Earthq Engrg Struct Dyn 1980;8:1–16. [5] Nazmy AS, Abdel-Ghaffar AM. Nonlinear earthquake-response analysis of long-span cable-stayed bridges: Theory. Earthq Engrg Struct Dyn 1990;19(1):45–62. [6] Nazmy AS, Abdel-Ghaffar AM. Nonlinear earthquake-response analysis of long-span cable-stayed bridges: Applications. Earthq Engrg Struct Dyn 1990;19(1):63–76. [7] Wilson JC, Liu T. Ambient vibration measurements on a cable-stayed bridge. Earthq Engrg Struct Dyn 1991;20:723–47. [8] Gardner-Morse MG, Huston DR. Modal identification of cable-stayed pedestrian bridge. J Struct Engrg, ASCE 1993;119(11):3384–404. [9] Karoumi R. Some modeling aspects in the nonlinear finite element analysis of cable supported bridges. Comput & Structures 1999;71: 397–412. [10] Ren WX, Obata M. Elastic–plastic seismic behavior of long span cable-stayed bridges. J Bridge Engrg, ASCE 1999;4(3):194–203. [11] Okauchi I, Miyata T, Tatsumi M, Sasaki N. Field vibration test of a long span cable-stayed bridge using large exciters. J Struct Engrg/Earthq Engrg 1997;14(1):83–93 [in English]. [12] Cunha A, Caetano E, Delgado R. Dynamic tests on large cable-stayed bridge. J Bridge Engrg, ASCE 2001;6(1):54–62. [13] Abdel-Ghaffer AM, Scanlan RH. Ambient vibration studies of Golden Gate Bridge. I: Suspended structure. J Engrg Mech, ASCE 1985; 111(4):463–82.
[14] Brownjohn JMW, Dumanoglu AA, Severn RT, Blakeborough A. Ambient vibration survey of the Bosporus suspension bridge. Earthq Engrg Struct Dyn 1989;18:263–83. [15] Kumarasena T, Scanlan RH, Morris GR. Deer isle bridge: Field and computed vibrations. J Struct Engrg, ASCE 1989;115:2313–28. [16] Wilson JC, Gravelle W. Modeling of a cable-stayed bridge for dynamic analysis. Earthq Engrg Struct Dyn 1991;20:707–22. [17] Xu YL, Ko JM, Zhang WS. Vibration studies of Tsing Ma suspension bridge. J Bridge Engrg, ASCE 1997;2:149–56. [18] Chang CC, Chang TYP, Zhang QW. Ambient vibration of long-span cable-stayed bridge. J Bridge Engrg, ASCE 2001;6(1):46–53. [19] Ren WX, Harik IE. Dynamic testing of a large cable-stayed bridge. In: Proc. 7th int. symposium on struct. engrg for young experts. Tianjin, China, Science Press; 2002. p. 775–80. [20] Ren WX, Harik IE, Blandford GE, Lenett M, Basehearh T. Roebling suspension bridge: II. Ambient testing and live load response. J Bridge Engrg, ASCE 2004;9(2):119–26. [21] Ren WX, Zong ZH. Output only modal parameter identification of civil engineering structures. Struct Engrg Mech 2004;17(3–4): 429–444. [22] ANSYS® . User’s manual, revision 5.6. USA: Swanson Analysis System; 1999. [23] Peng XL. Baseline finite element model of a large span cablestayed bridge. M.Sc. thesis, Department of Civil Engineering, Fuzhou University; 2003 [in Chinese]. [24] Calcada R, Cunha A, Delgado R. Dynamic analysis of metallic arch railway bridge. J Bridge Engrg, ASCE 2000;7(4):214–22. [25] De Roeck G, Peeters B. MACEC2.0—Modal analysis on civil engineering constructions. Department of Civil Engineering, Catholic University of Leuven, Belgium; 1999. [26] Bendat JS, Piersol AG. Engineering applications of correlation and spectral analysis. 2nd ed. New York: John Wiley & Sons; 1993. [27] Felber A. Development of a hybrid bridge evaluation system. Ph.D. thesis, University of British Columbia, Canada; 1993. [28] Peeters B. System identification and damage detection in civil engineering. Ph.D. thesis, Department of Civil Engineering, Catholic University of Leuven, Belgium; 2000. [29] Van Overschee P, De Moor B. Subspace identification for linear systems: Theory, implementation and applications. Dordrecht (Netherlands): Kluwer Academic Publishers; 1996.
Experimental and analytical studies on dynamic characteristics of a large span cable-stayed bridge Wei-Xin Rena,b,∗, Xue-Lin Penga, You-Qin Lina a Department of Civil Engineering, Fuzhou University, Fuzhou, Fujian Province 350002, People’s Republic of China b Department of Civil Engineering, Central South University, Changsha, Hunan Province 410075, People’s Republic of China
Received 7 March 2004; received in revised form 2 November 2004; accepted 23 November 2004 Available online 1 February 2005
Abstract An analytical and experimental modal analysis has been carried out on the Qingzhou cable-stayed bridge in Fuzhou, China. Its main span of 605 m is currently the longest span among the completed composite-deck cable-stayed bridges in the world. An analytical modal analysis is performed on the developed three-dimensional finite element model starting from the deformed configuration to provide the analytical frequencies and mode shapes. The field ambient vibration tests on the bridge deck and all stay cables were conducted just prior to the opening of the bridge. The output-only modal parameter identification is then carried out by using the peak picking of the average normalized power spectral densities in the frequency-domain and stochastic subspace identification in the time-domain. A good correlation is achieved between the finite element and ambient vibration test results. It is demonstrated that the analytical and experimental modal analysis provide a comprehensive study on the dynamic properties of the bridge. The ambient vibration tests are sufficient to identify the most significant modes below 1.0 Hz of this kind of large span cable-stayed bridge. The validated finite element model with respect to ambient vibration test results can serve as the baseline for a more precise dynamic response prediction and long-term health monitoring of the bridge. © 2005 Elsevier Ltd. All rights reserved. Keywords: Cable-stayed bridge; Modal analysis; Parameter identification; Ambient vibration test; Spectra; Frequency; Finite element method
1. Introduction It is the most important issue to ensure an adequate level of safety of both new and existing large span bridges against dynamic loadings such as traffic, wind and earthquakes. This can be achieved if the dynamic properties of the bridge such as natural frequencies, mode shapes and damping ratios are accurately determined. Modal analysis is a technique that estimates the dynamic characteristics (modal parameters) of a structure. These modal parameters serve as a basis for the finite element model updating, structural control, damage detection, condition assessment, and long-term health monitoring of the structure. ∗ Corresponding author at: Department of Civil Engineering, Fuzhou University, Fuzhou, Fujian Province 350002, People’s Republic of China. Tel.: +86 591 7892454; fax: +86 591 3737442. E-mail address: [email protected] (W.-X. Ren). URL: http://bridge.fzu.edu.cn (W.-X. Ren).
0141-0296/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2004.11.013
Modal analysis was originally developed in more advanced mechanical and aerospace engineering disciplines [1–3]. Although its technique is not new, the application of digital computers caused a dramatic increase of the practical possibilities of modal analysis. There is a clear merit in trying to transfer this technology into civil engineering applications where we are dealing with problems which have a completely different scale, logistics and rationale, compared with mechanical and aerospace engineering counterparts. In the context of bridge engineering discipline, for instance, the encountered structures are often complex, large in size and low in frequency. In general, the modal analysis procedure for more complicated structures includes analytical and experimental modal analysis. Analytical modal analysis is the process of determining the dynamic properties of a structure based on the free vibration solution of the equations of motion. The finite element (FE) method is currently a common way
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to perform an analytical modal analysis of cable-stayed bridges [4–10]. Starting from the knowledge of the structural geometry, boundary conditions and material properties, a structure is expressed in a matrix form. It contains sufficient information to determine the structural modal parameters that completely describe the dynamic properties of the structure. However, the success of the FE application strongly depends on the experimental verification of the results since many simplified assumptions are made in the modeling of complicated structures and there are some uncertainties in the material and geometric properties when establishing the FE models of real structures. The reliability of analytical modal analysis results is often questionable if the FE model is not experimentally validated. On-site dynamic testing of a structure provides an accurate and reliable description of its dynamic characteristics. Experimental modal analysis is then a process of identifying the modal parameters based on the dynamic measurements. Starting from the measurements of dynamic input forces and output responses of interests and implementing a modal extraction technique, the dynamic characteristics of the structure can be estimated. For the experimental modal analysis of bridges, there are three main types of dynamic tests: (1) forced vibration tests; (2) free vibration tests; and (3) ambient vibration tests. In the first method, the bridge is excited by artificial means and correlated input–output measurements are performed. Impulse hammers, drop weights and electrodynamic shakers are the main excitation equipments. The successes of forced vibration tests are limited for relatively small structures. In the case of large and flexible bridges like cable-stayed or suspension bridges, it often requires heavy equipment and involves important resources to provide a controlled excitation at sufficiently high levels [11], which becomes difficult and costly. Forced vibration tests are directly related with the standard techniques of modal parameter estimation that were previously developed in mechanical and aerospace engineering. The modal parameter identification in this case is carried out based on both input and output measurement data. By introducing an initial perturbation on the bridge, a condition of free vibration can be induced. Free vibration tests can be done by a sudden release of a heavy load or mass appropriately connected to the bridge deck [12]. Both forced and free vibration tests, however, need the artificial means to excite the bridges and the traffic has to be shut down. This could be a serious problem for intensively used bridges. In contrast, ambient vibrations, induced by traffic, winds, and pedestrians, are the natural or environmental excitations of the bridge. Ambient vibration tests have an advantage of being inexpensive since no equipment is needed to excite the bridge. It corresponds to the real operating condition of the bridge. The service state need not be interrupted to use this technique. Therefore, ambient vibration tests have been successfully applied to several large scale cable-supported bridges, such as the Golden Gate Bridge [13], the Bosporous
Suspension Bridge [14], the Deer Isle Bridge [15], the Quincy Bayview Bridge [16], the Tsing Ma Suspension Bridge [17], the Kap Shui Mun Cable-Stayed Bridge [18], the Maysville Cable-Stayed Bridge [19], and the Roebling Suspension Bridge [20]. In cases of both free and ambient vibration tests, only response data are measured while actual loading conditions are not measured. A modal parameter identification procedure will therefore need to base itself on output-only data. Many challenges exist in the experimental modal analysis of real bridges using ambient vibration measurements. There have been several modal parameter identification techniques available that were developed by different investigators for different uses. The mathematical background for many of these methods is often very similar, differing only from implementation aspects (data reduction, type of equation solvers, sequence of matrix operations, etc.). The benchmark study has been carried out for evaluating the dynamic characteristics from ambient vibration data [21]. The objective of this paper is to demonstrate how the modal analysis on a large cable-stayed bridge is carried out. The field ambient vibration tests on the bridge deck and all stay cables of the Qingzhou cable-stayed bridge were conducted just prior to the opening of the bridge to traffic in order to obtain the baseline dynamic characteristics. The dynamic properties were extracted from the peak picking of the average normalized power spectral densities (ANPSDs) in the frequency-domain and more advanced stochastic subspace identification in the time-domain. The analytical modal analysis was carried out on the developed three-dimensional finite element model. To improve the FE predictions, the FE model was calibrated with respect to the ambient vibration results. The validated finite element model that reflects the built-up structural conditions can serve as the baseline model for the bridge assessment, damage identification, health monitoring and succeeding dynamic response predictions of the bridge. 2. Bridge description The target Qingzhou cable-stayed bridge in this paper is one of the bridges on Luo-Chang Highway over the Ming River in Fuzhou, Fujian Province, China. The bridge is a 5span composite cable-stayed bridge with an overall length of 1186.34 m (41.13 m + 250 m + 605 m + 250 m + 40.21 m). Its main span of 605 m is currently the longest span among the completed composite-deck cable-stayed bridges and lists the fifth longest span among all types of completed cablestayed bridges all over the world. Fig. 1 shows a schematic representation of the bridge with plan, elevation and typical cross sections. The bridge was completed in the year 2000, but it was officially open to the traffic in the year 2002 due to the construction delay of the approach spans. A photograph of the completed Qingzhou cable-stayed bridge is shown in Fig. 2. The main structural features of the bridge are as follows:
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Fig. 1. Schematic representation of Qingzhou cable-stayed bridge. (a) Plan and elevation, (b) typical cross section of composite deck.
Fig. 2. The Qingzhou cable-stayed bridge.
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Table 1 Properties of stay cables Cable no.
Number of strands
Sectional area (cm2 )
Mass per unit length (kg/m)
E (MPa)
σb (MPa)
C1–C4; S1–S4 C5–C6; S5–S6 C7–C9; S7–S8 C10–C12; S9–S10 C13–C14; S11–S13 C15–C18; S14–S17 C19–C20; S18–S19 C21; S20–S21
27-7Φ5 34-7Φ5 37-7Φ5 43-7Φ5 48-7Φ5 55-7Φ5 73-7Φ5 85-7Φ5
37.795 47.593 51.793 60.191 67.190 76.989 102.185 118.983
32.792 41.293 44.937 52.224 58.296 66.798 88.659 103.233
1.95 × 105
1860.0
• The composite deck consists of two I-type main steel girders and a 25 cm thick concrete slab. Each steel girder is 2.45 m high and its maximum plate thickness reaches 80 mm. The precast concrete slab is connected to the steel girders and floor beams by shear studs. The spacing of the steel floor beams is 4.5 m. The bridge has 6 lanes that carry two roadways 29 m wide. • The two diamond-shaped towers are of reinforced concrete. The height of the towers is 175.5 m with a clear navigation of 43 m. The towers are erected on group piles with a diameter of 2 m for Pier 2 and 3 m for Pier 3. The longest pile reaches a length of 71.6 m. • The cable is a fan type in both planes. There are in total 21 × 8 = 168 stay cables. The cables are composed of a number of strands. The strand number varies from 27 to 85 per cable in eight groups. One strand includes 7 high strength wires with a diameter of 5 mm (7Φ5 high strength wires). The density of the cables depends on the number of wires in the individual cables. The material properties of the stay cables are listed in Table 1.
3. Analytical modal analysis To perform the analytical modal analysis of the Qingzhou cable-stayed bridge, a three-dimensional finite element model has been developed using ANSYS® [22]. The model represents the bridge in its current as-built configuration with the geometry and structural properties estimated from the design drawings. Four types of finite elements are used to model the different structural members: • The steel girders, stringers, floor beams and concrete towers are modeled by three-dimensional elastic beam elements (BEAM4). There are in total 1586 beam elements used. • All stay cables are modeled by 3-D tension-only truss elements (LINK10) since they are primarily designed to sustain tension forces. The effect of cable sag is taken into account by using the initial stress matrix due to cable tensions and stress-stiffening capability instead of the cable equivalent modulus of elasticity. Pre-tensions of the cables are included by the initial element strains input.
• The concrete slab is divided into 508 shell elements (SHELL63). • The piers and platforms are modeled by solid elements (SOLID45) where 149 elements are used. The modeling of bridge boundary conditions is an important issue in the modal analysis. Two types of bridge bearings are used in the Qingzhou cable-stayed bridge. Fixed bearings are constructed to connect Pier 2 to the deck, while expansion bearings are used to the rest of piers. In the current model, bridge bearings are modeled by a set of rigid link elements connecting the superstructure and piers. To simulate the actual behavior, the fixed and expansion bearings are simulated by coupling the corresponding translational and rotational degrees of freedom at both end nodes of the link elements. In addition, longitudinal springs are used to account for the restrained action in the longitudinal direction from the adjacent span and abutment at Pier 0 and Pier 5. The soil–pier interaction is not included in the analysis. The constructed three-dimensional finite element model of the Qingzhou cable-stayed bridge is shown in Fig. 3. More details about the model can be found in [23]. Fig. 4 shows some typical vertical, transverse and torsional mode shapes obtained from the analytical modal analysis. Vertical modes are those modes dominated by the vertical bending of the bridge deck. Transverse modes are those modes dominated by the lateral sway of the bridge deck in the horizontal plane, while torsional modes are dominated by the torsional behavior of the bridge deck around the longitudinal axis of the bridge. It is demonstrated that the most significant modes of the Qingzhou cablestayed bridge are below 1.0 Hz. Some dominated modes are close to each other. The first lateral mode of the bridge, for instance, appears right after the second vertical mode. Some dominated modes are coupled with the bending of the tower or the longitudinal drift of the deck. To see the mass contribution of each mode, Table 2 lists the cumulative modal participating factors of the bridge deck for the vertical bending, transverse bending and torsion vibration within 20 modes. Results have demonstrated that more modes are needed to achieve a higher mass participation in the vertical bending of the deck. One of the important features of a large span cable-stayed bridge is that the dead load (self-weight) often contributes
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Fig. 3. Three-dimensional finite element model of the bridge.
Fig. 4. Typical mode shapes obtained from finite element analysis.
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Fig. 4. (continued).
most of bridge loads. It has been realized that the dead load has an influence on the stiffness of cable-supported bridges. This influence can be included by carrying out the modal analysis from the static initial equilibrium position due to dead loading and cable pre-tensions. Therefore, the modal analysis of cable-stayed bridges is a “pre-stress modal analysis” that should include two key steps: • The static analysis is first conducted under dead load and cable pre-tensions. The objective of this process is to achieve the deformed equilibrium configuration of the bridge in which all structural members are “pre-stressed”. • The modal (free vibration) analysis is then followed starting from the deformed equilibrium configuration due to dead load and cable pre-tensions. Many studies [4,6,10] have shown that the linear static analysis is adequate to calculate the static initial equilibrium configuration. Based on the linear static equilibrium
position, the analytical modal analysis has been carried out to study the dynamic characteristics of the Qingzhou cablestayed bridge. To see the effect of dead load on the stiffness of such a cable-stayed bridge, the natural frequencies obtained from the deformed position are compared in Table 4 with those obtained from the non-deformed position. It is observed that all natural frequencies are increased due to the pre-stressed modal analysis. Although the pre-stressed modal analysis might have a minimal effect on increasing the bridge’s natural frequencies, a starting position is essential to determine the responses in succeeding dynamic analyses such as earthquakes, winds and vehicles loadings. 4. Ambient vibration tests on bridge deck Modal testing of a bridge on site provides an accurate and reliable prediction of its global modal parameters.
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Table 2 Cumulative modal participating factors within 20 modes (%) Mode no.
Frequency (Hz)
Vertical bending (%)
Transverse bending (%)
Torsion (%)
Nature of modes of vibration
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0.222 0.266 0.267 0.415 0.454 0.478 0.526 0.535 0.538 0.551 0.571 0.592 0.607 0.622 0.650 0.710 0.712 0.738 0.748 0.752
9.09 9.12 9.12 22.57 22.61 58.04 58.05 58.05 58.05 58.05 59.13 59.64 73.54 73.54 73.56 74.67 74.67 76.02 76.02 91.41
0.0 0.0 27.54 27.54 27.54 27.54 27.54 35.10 73.75 99.24 99.24 99.24 99.24 99.24 99.24 99.24 99.32 99.32 99.51 99.51
0.08 0.08 8.92 9.05 9.05 9.37 9.37 19.58 70.60 99.57 99.58 99.58 99.71 99.71 99.71 99.72 99.73 99.75 99.76 99.90
vertical bending of deck vertical bending of deck transverse bending of deck vertical bending of deck vertical bending of deck vertical bending of deck vertical bending of deck bending of tower bending of tower torsion of deck vertical bending of deck local vertical bending of side span local vertical bending of side span torsion of deck vertical bending of deck vertical bending of deck torsion of deck local vertical bending of side span transverse bending of deck local vertical bending of side span
On November 18–21, 2002, just prior to the official opening of the bridge, ambient vibration tests on the deck of the Qingzhou cable-stayed bridge were carried out. The equipment used for the tests included accelerometers, signal cables, and a 32-channel data acquisition system with signal amplifier and conditioner. Accelerometers convert the ambient vibration responses into electrical signals. Cables are used to transmit these signals from sensors to the signal conditioner. The signal conditioner unit is used to improve the quality of the signals by removing undesired frequency contents (filtering) and amplifying the signals. The amplified and filtered analog signals are converted to digital data using an analog to digital (A/D) converter. The signals converted to digital form are stored on the hard disk of the data acquisition computer; these are the data for modal parameter identification. 4.1. Measurement setup To identify the good mode shapes of the bridge, a dense measurement location on the deck in the vertical, transverse and longitudinal directions was conceived. Measurement stations were chosen near each anchor position on the deck. In addition, several measurement stations were chosen on Span 1 and Span 5. As a result, there were in total 180 measurement stations on the bridge deck. The measurement station arrangement on the bridge deck is shown in Fig. 5, where U and D refer to upstream and downstream directions, respectively. Force-balance accelerometers with measurement frequency ranging from 0.2 to 80 Hz were used to record the ambient vibration responses. The accelerometers were directly placed on the pavement due to the limited access to the
actual floor beams. Fig. 6 shows the accelerometers mounted on the bridge deck in the vertical and transverse directions. 15 accelerometers were used in the tests; 12 accelerometers were moveable, and 3 accelerometers were fixed as reference accelerometers. A reference location was selected according to the information obtained from the mode shapes of the preliminary finite element model. Three base stations were located at the center of a side span (D15), at the deck near Pier 2 (U24) and at the center of the main span (U46). There were 15 test setups to cover all planned measurement locations of the bridge deck. Each setup yielded a total of 15 sets of data, 12 sets from the moveable stations and 3 sets from the base stations. Measurement stations per setup are summarized in Table 3. 4.2. Field measurements and data pre-processing The test was started from the Fuzhou side and progressed to the Changle side. Once the data were collected in one setup, the moveable accelerometers were shifted to the next stations while the base stations remained stationary. This sequence was repeated 15 times to obtain ambient vibration measurements on all stations. Measurements in the vertical, transverse and longitudinal directions were carried out separately in the same stations. The sampling frequency on site was 80 Hz. The ambient vibration measurements were simultaneously recorded for 20 min at all channels. As a result, there were totally 96,000 data points per channel. The typical tri-axial acceleration time history records at the station D46 (main span center) are as shown in Fig. 7. An inspection of the time history measurements shows a significant variation of the deck ambient vibration responses. It is observed that the average
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Fig. 5. Measurement station arrangement for ambient vibration test.
Fig. 6. Accelerometers mounted on the deck. (a) Vertical accelerometer, (b) transverse accelerometer.
Table 3 Measurement stations per setup Setup
Measurement stations
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
U1, U2, U3, U4, U5, U6; D1, D2, D3, D4, D5, D6; D15, U24, U46 U7, U8, U9, U10, U11, U12; D7, D8, D9, D10, D11, D12; D15, U24, U46 U13, U14, U15, U16, U17, U18; D13, D14, D15, D16, D17, D18; D15, U24, U46 U19, U20, U21, U22, U23, U24; D19, D20, D21, D22, D23, D24; D15, U24, U46 U25, U26, U27, U28, U29, U30; D25, D26, D27, D28, D29, D30; D15, U24, U46 U31, U32, U33, U34, U35, U36; D31, D32, D33, D34, D35, D36; D15, U24, U46 U37, U38, U39, U40, U41, U42; D37, D38, D39, D40, D41, D42; D15, U24, U46 U43, U44, U45, U46, U47, U48; D43, D44, D45, D46, D47, D48; D15, U24, U46 U49, U50, U51, U52, U53, U54; D49, D50, D51, D52, D53, D54; D15, U24, U46 U55, U56, U57, U58, U59, U60; D55, D56, D57, D58, D59, D60; D15, U24, U46 U61, U62, U63, U64, U65, U66; D61, D62, D63, D64, D65, D66; D15, U24, U46 U67, U68, U69, U70, U71, U72; D67, D68, D69, D70, D71, D72; D15, U24, U46 U73, U74, U75, U76, U77, U78; D73, D74, D75, D76, D77, D78; D15, U24, U46 U79, U80, U81, U82, U83, U84; D79, D80, D81, D82, D83, D84; D15, U24, U46 U85, U86, U87, U88, U89, U90; D85, D86, D87, D88, D89, D90; D15, U24, U46
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Table 4 Calculated and identified modal parameters Nature of modes of vibration
Finite element analysis (Hz) Starting from Starting from deformed un-deformed position position
Stochastic subspace identification Frequencies Damping (Hz) ratio (%)
Peak-picking identification (Hz)
MAC values
1st vertical bending 2nd vertical bending 1st transverse bending 3rd vertical bending 4th vertical bending 5th vertical bending 1st torsion 7th vertical bending 2nd torsion 3rd torsion 2nd transverse bending 1st longitudinal
0.222 0.266 0.267 0.415 0.454 0.478 0.551 0.571 0.622 0.712 0.748 1.925
0.226 0.272 0.263 0.446 0.480 0.505 0.556 0.653 0.610 0.726 0.628 1.925
0.227 0.271 0.262 0.444 0.482 0.506 0.555 0.609 0.653 0.701 0.612 1.922
0.989 0.755 0.991 0.966 0.783 0.942 0.982 0.882 0.778 0.895 0.919 –
0.217 0.252 0.264 0.405 0.442 0.468 0.548 0.566 0.606 0.701 0.746 1.920
signal levels for all channels in the vertical direction are about 4 times higher than those in the transverse direction and about 10 times than those measured in to the longitudinal direction. That is identical to the ambient vibration measurements on a metallic arch railway bridge reported by Calcada et al. [24]. Before the modal parameter identification is performed, the measured data are first detrended, which enables the removal of the DC-components that can badly influence the identification results. This was accomplished by subtracting the mean values calculated over the full duration of each measurement. A sampling frequency of 80 Hz on site results in a frequency range from 0 to 40 Hz. For such a large span cable-stayed bridge, however, the frequency range of interest lies between 0 and 2.0 Hz, covering the most important frequencies in the vertical, transverse and longitudinal directions. So a resampling of the raw measurement data is necessary. A resampling and filtering from 80 to 4 Hz is the same as decimating (=low-pass filtering and resampling at a lower rate) 20 times which results in 96,000/40 = 4800 data points per channel and an excellent frequency range from 0 to 2.0 Hz. 4.3. Modal parameter identification The classical modal parameter identification techniques that involve frequency- or time-domain analysis of dynamic responses to forced vibration cannot be directly applied to the output-only vibration data because the input excitations are not measured in the test [21]. Two complementary modal parameter identification techniques are implemented in this work. They are the rather simple peak picking (PP) method in the frequency-domain and the more advanced stochastic subspace identification (SSI) method in the time-domain. The data processing and modal parameter identification were carried out by MACEC [25]. The peak picking method is initially based on the fact that the frequency response function goes through an extreme
0.7 0.7 1.0 0.9 2.8 1.5 0.4 0.4 4.6 1.1 0.5 1.9
value around the natural frequency. In the context of ambient vibration measurements, the frequency response function is replaced by the auto spectra of the output-only data [26]. To include the measurement channels of all setups, the average normalized power spectral densities (ANPSDs) are used. A practical implementation of the ANPSDs was realized by Felber [27]. In such a way, the identified natural frequencies are simply obtained from the observation of the peaks on the graphs of ANPSDs. The ANPSDs of vertical and transverse (half-sum) acceleration measurements of all channels of the bridge deck are shown in Fig. 8, where the peaks can be clearly seen and then the frequencies can be picked up. The stochastic subspace identification is a time-domain method that directly works with time data, without the need to convert them to correlations or spectra. The stochastic subspace identification algorithm identifies the system matrix of the state space model based on the measurements by using robust numerical techniques. Details of the theoretical background of the SSI method can be found in the literature [28,29]. Once the mathematical description of the structure is found, it is straightforward to determine the modal parameters by an eigenvalue decomposition. Table 4 summarizes the most significant frequencies of the Qingzhou cable-stayed bridge identified from the test. It can be seen that the identified frequencies agree well between the peak picking and stochastic subspace identification methods. Most of the important frequencies of the bridge deck are below 1.0 Hz, with the lowest frequency 0.223 Hz being a symmetric deck bending in the vertical direction. It is observed that the simple peak picking method cannot identify all important mode shapes for such a large bridge. Vibration mode shapes of the bridge deck have been extracted by the stochastic subspace identification method. Fig. 9 shows the most dominated identified mode shapes in the vertical bending, transverse bending and torsion. The obtained results have demonstrated that the ambient vibration response measurements are sufficient enough to identify the
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Fig. 7. Tri-axial acceleration time histories measured at Station U46.
most significant modes of large span cable-stayed bridges, despite the rather low level of ambient vibration signal captured. The low range (0–1.0 Hz) of natural frequencies of interest, and the relatively dense modes of vibration in that range, are typical characteristics of this kind of bridge. For such a specific cable-stayed bridge, the results of the identified vertical bending mode shapes imply that the stochastic subspace identification method can identify a very good quality of the symmetric vertical bending mode shapes as high as the 9th mode (the 5th symmetric vertical bending mode). However, the quality of the anti-symmetric vertical bending mode shapes is not as good as expected. The damping properties can also be extracted by the stochastic subspace identification method since it is a complex mode based procedure. The identified modal damping ratios are also listed in Table 4. However, the damping property of real large cable-stayed bridges is not fully understood yet due to the complicated damping mechanism of the bridges. It is known that bridge damping in general varies with the amplitude of vibration. Therefore, the applicability of an identified damping ratio through ambient vibration tests is still an issue that needs to be evaluated further by using other identification techniques or other dynamic tests with large vibration amplitudes. 4.4. Correlation between analytical and experimental modal parameters The correlation between modal parameters identified from the test and those calculated numerically can be evaluated by comparing the values of the natural frequencies
Fig. 8. Averaged normalized power spectral densities (ANPSDs). (a) Vertical measurements, (b) transverse measurements.
and corresponding mode shapes. Table 4 shows the comparison of the numerically calculated frequencies from the
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Fig. 9. Typical mode shapes obtained from field tests by stochastic subspace identification.
three-dimensional finite element analysis and experimentally identified frequencies from the field ambient vibration tests. This analysis clearly shows a good correlation between calculated and identified natural frequencies of the bridge, except the 2nd transverse bending frequency. To evaluate the correlation of calculated and identified mode shapes, the Modal Assurance Criterion (MAC) is used. The MAC between a measured mode φm j and an analytical mode φak is defined as MAC j k =
|φmT j φak |2 . T φ )(φ T φ ) (φak ak mj mj
(1)
The MAC values are shown in Table 4. It can been seen that higher values of MAC are found for the most significant modes, which demonstrates a good correlation between the identified and calculated mode shapes, although the anti-
symmetric bending modes of the bridge deck are associated to the lower MAC values. The fact is again verified that the quality of the identified anti-symmetric bending mode shapes in the vertical direction is not so good.
5. Ambient vibration tests on stay cables Stay cables are the most critical structural components in cable-stayed bridges. The tension forces in the stay cables control the internal force distribution in the deck and towers as well as the bridge alignment. Therefore, the cable tension force is an important index in the condition assessment and long-term monitoring of cable-stayed bridges. Experimental vibration measurement is one of the most widely used techniques for tension evaluation of cable-stayed bridges.
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Fig. 9. (continued).
The simplest relationship to calculate the cable forces based on the vibration frequency is the so-called “taut string” chord equation: T =
4m L 2 fn2 n2
(2)
where T = cable tension; m = cable mass per unit length; L = length between cable fixed ends; and fn = nth natural frequency. Right after the tests on the bridge deck, ambient vibration measurements were carried out for all 168 stay cables of the Qingzhou cable-stayed bridge. One accelerometer per cable was used to measure the acceleration in the out-ofplane direction of the stay cables. The accelerometer was mounted securely to the cable as far up as could be reached. The signals were recorded for 5 min at a sampling rate of 40 Hz.
The key to success of vibration-based cable tension evaluation is the accuracy of the identified frequencies of each stay cable. Based on the vibration chord theory (Eq. (2)), the peaks in the cable’s power spectrum are equally spaced when the cable tension is constant. The higher frequencies are integer multiples of the fundamental frequency. Fig. 10 presents the power spectral density (PSD) associated with the ambient acceleration measurement of one of the longest stay cables of the bridge. Response spectra for the Qingzhou bridge cables showed the taut-string characteristics consistently, but it does not always occur in the low frequency range. To measure the spacing between the peaks, neglecting the order information is somewhat easy. So determining the equal spacing between peaks in the power spectrum provides the fundamental frequency of the stay cables. The cable tension force can then be calculated by T = 4m L 2 f 2
(3)
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• The analytical and experimental modal analysis provides a comprehensive investigation of the dynamic properties of bridges. The analytical modal analysis through threedimensional finite element modeling gives a detailed description of the physical and modal characteristics of the bridge, while the experimental modal analysis through the field dynamic tests offers a valuable source of information for validating the drawing-based idealized finite element model.
Fig. 10. Ambient acceleration spectra of Cable S21.
• An ambient vibration based test is a convenient, faster and cheaper way to perform the bridge dynamic test. The ambient vibration response measurements are sufficient enough to identify the most significant modes of large span cable-stayed bridges, despite the rather low level of ambient vibration signal captured. The low range (0–1.0 Hz) of natural frequencies of interest, and the relatively dense modes of vibration in that range, are typical characteristics of this kind of bridge. • The output-only modal parameter identification of large span cable-stayed bridges can be effectively carried out by using two independent numerical techniques: the peak picking of the average normalized power spectral densities (ANPSDs) in the frequency-domain and more advanced stochastic subspace identification in the timedomain. It is essential to verify the identified results by using some other independent identification techniques due to the complexity of the problem. The application of supplementary techniques will increase the reliability of the identified results.
Fig. 11. Fundamental frequencies of all stay cables.
where f is the fundamental frequency of the stay cable. The identified fundamental frequencies determined by the spacing between peaks in the power spectrum density of the ambient vibration measurements are plotted in Fig. 11 for both downstream and upstream cables. The identified fundamental frequencies of the downstream and upstream cables are consistent. The distribution of cable fundamental frequencies is almost symmetric, which is identical to the distribution of the cable pre-tension forces. It is demonstrated that the fundamental cable frequencies of the Qingzhou cablestayed bridge are relatively low, with a range from 0.36 Hz for the longest cable to 1.97 Hz for the shortest cable. Ambient vibration measurement demonstrates a high confidence in identification of the fundamental frequency of the stay cables. 6. Conclusions The following conclusions are drawn from the modal analysis performed on a full sized large span cable-stayed bridge:
• A good correlation is achieved between the numerically calculated modal parameters from three-dimensional finite element analysis and identified ones from field ambient vibration tests. The validated finite element model that reflects the built-up structural conditions can serve as the baseline model for a more precise dynamic response prediction, damage identification and long-term health monitoring of the bridge. • The damping mechanism of this kind of complex bridge is complicated. An analytical evaluation of the bridge damping is extremely difficult. The damping property of real large cable-stayed bridges is not yet fully understood. The applicability of identified damping ratios through ambient vibration tests is still an issue that needs to be evaluated further by using other identification techniques or other dynamic tests with large vibration amplitudes. • The power spectral density analysis of ambient vibration measurements is implemented to identify the fundamental frequencies of all 168 stay cables of the Qingzhou cable-stayed bridge. The fundamental frequencies of the stay cables are relatively low, with a range from 0.36 Hz for the longest cable to 1.97 Hz for the shortest cable. Ambient vibration measurement demonstrated a high identification confidence of the fundamental frequencies of the stay cables in the cable-stayed bridge.
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