This paper presents an investigation of mitigation of longitudinal buffeting responses of the Jiashao Bridge, the longest multispan cable-stayed bridge in the world. A time-domain procedure for analyzing buffeting responses of the bridge is implemented in ANSYS with the aeroelastic effect included. The characteristics of longitudinal buffeting responses of the six-tower cable-stayed bridge are studied in some detail, focusing on the effects of insufficient longitudinal stiffness of central towers and partially longitudinal constraints between the bridge deck and part of bridge towers. The effectiveness of viscous fluid dampers on the mitigation of longitudinal buffeting responses of the bridge is further investigated and a multiobjective optimization design method that uses a nondominating sort genetic algorithm II (NSGA-II) is used to optimize parameters of the viscous fluid dampers. The results of the parametric investigations show that, by appropriate use of viscous fluid dampers, the top displacements of central towers and base forces of bridge towers longitudinally restricted with the bridge deck can be reduced significantly, with hampering the significant gain achieved in the base forces of bridge towers longitudinally unrestricted with the bridge deck. And the optimized parameters for the viscous fluid dampers can be determined from Pareto-optimal fronts using the NSGA-II that can satisfy the desired performance requirements.
For long-span cable-stayed bridges, the multispan cable-stayed bridges with three or more towers have been a recent design trend [
Compared with a conventional three-span cable-stayed bridge with two towers, there are two major problems in the design of multispan cable-stayed bridges. One is the insufficient longitudinal stiffness that arose from the central tower(s). In the conventional cable-stayed bridges with two towers, each of the towers is connected through outermost stay cables to the fixed anchorage or anchor pier, which can provide effective support to the towers [
To date, an amount of research work had been done on investigating the static performance of the multispan cable-stayed bridge under living loadings and temperature action [
The subject of this study is Jiashao Bridge shown in Figure
Jiashao Bridge. (a) View of Jiashao Bridge, (b) elevation of Jiashao Bridge (unit: cm), and (c) cross section of bridge deck (unit: cm).
A three-dimensional finite element model of the Jiashao Bridge has been developed by use of the commercial software package ANSYS [
Finite element model of Jiashao Bridge.
Detailed finite element modeling for rigid links.
Rigid links connecting two steel box girders
Rigid links connecting stay cables
Rigid links simultaneously connecting two steel box girders and stay cables
The detailed finite element modeling for the constraints between the girders and towers (piers) are described as follows.
(i) Modeling of constraints between the girders and towers: As shown in Figure
Detailed finite element modeling for the constraints between the girders and towers (piers).
Constraints between the girders and side towers number 2 and number 5
Auxiliary pier and end pier in each side span
(ii) Modeling of constraints between the girders and piers in each side span: As shown in Figure
The modal analysis of the Jiashao Bridge is essential to determine the buffeting responses in succeeding dynamic analysis subjected to wind loadings. It should be noted that the modal analysis of bridge is followed starting from the static equilibrium state due to dead load and cable pretension, which is obtained by geometrically nonlinear analysis of the bridge. The LANCZOS eigenvalue solver is then adopted for modal analysis. Main vibrations modes of finite element model are listed in Table
Vibration modes of finite element model of Jiashao Bridge.
Mode number | Frequency/Hz | Description |
---|---|---|
1 | 0.2274 | 1st symmetric vertical bending of bridge deck + symmetric longitudinal bending of bridge tower |
2 | 0.2615 | 1st antisymmetric vertical bending of bridge deck + antisymmetric longitudinal bending of bridge tower |
3 | 0.2894 | 1st symmetric lateral bending of bridge tower |
4 | 0.2907 | 1st antisymmetric lateral bending of bridge tower |
5 | 0.2928 | 2nd symmetric lateral bending of bridge tower |
6 | 0.2950 | 2nd antisymmetric lateral bending of bridge tower |
7 | 0.2965 | 3rd symmetric lateral bending of bridge tower |
8 | 0.2970 | 3rd antisymmetric lateral bending of bridge tower |
9 | 0.3085 | 2nd symmetric vertical bending of bridge deck + symmetric longitudinal bending of bridge tower |
10 | 0.3618 | 2nd antisymmetric vertical bending of bridge deck + antisymmetric longitudinal bending of bridge tower |
21 | 0.7087 | 1st symmetric lateral bending of bridge deck + symmetric lateral bending of bridge tower |
33 | 0.8956 | 1st antisymmetric lateral bending of bridge deck + antisymmetric lateral bending of bridge tower |
43 | 1.1361 | 1st symmetric torsion of bridge deck |
44 | 1.1389 | 2nd symmetric torsion of bridge deck |
Mode shapes of the bridge. (a) 1st symmetric vertical bending of bridge deck + symmetric longitudinal bending of bridge tower, (b) 1st antisymmetric vertical bending of bridge deck + antisymmetric longitudinal bending of bridge tower, (c) 1st symmetric lateral bending of bridge tower, (d) 1st antisymmetric lateral bending of bridge tower, (e) 1st symmetric lateral bending of bridge deck + symmetric lateral bending of bridge tower, and (f) 1st symmetric torsion of bridge deck.
To perform the nonlinear buffeting analysis of the bridge in the time domain, eight one-dimensional independent multivariable stochastic wind fields are simulated based on the structural configuration as listed in Table
Simplified simulation of the wind field.
Number | Location | Component of wind velocity | Spacing | Number of simulation points |
---|---|---|---|---|
1 | Bridge deck | Transverse component |
30 m | 95 |
2 | Bridge deck | Vertical component |
30 m | 95 |
3 | Tower number 1 | Transverse component |
10 m | 16 |
4 | Tower number 2 | Transverse component |
10 m | 16 |
5 | Tower number 3 | Transverse component |
10 m | 16 |
6 | Tower number 4 | Transverse component |
10 m | 16 |
7 | Tower number 5 | Transverse component |
10 m | 16 |
8 | Tower number 6 | Transverse component |
10 m | 16 |
The design mean wind speed Kaimal spectrum: Panofsky spectrum:
where
The following empirical exponential coherence function is adopted for the velocity cross-spectrum [
The wind velocity at each simulation point of bridge deck and towers is the sum of the mean wind velocity and turbulent fluctuation around the mean wind velocity. In the present study, the fluctuating wind velocity is taken as realization of a multivariate Gaussian vector process having zero mean and known fixed spectral densities at all simulation points [
Autospectra of longitudinal fluctuating wind velocities at bridge tower number 3. (a) Autospectrum of Point 1 and (b) autospectrum of Point 15.
Correlation functions of longitudinal fluctuating wind velocities at bridge tower number 3. (a) Autocorrelation of Point 1 and (b) cross-correlation between Points 1 and 15 with a distance of 150 m.
The simulated fluctuating wind speed time-histories are used as inputs to obtain the buffeting responses of the Jiashao Bridge in the time domain. In this section, the buffeting analysis method is introduced briefly. Consider a deck section in a turbulent flow and in motion at an instant time
The self-excited forces are expressed as
The buffeting forces are expressed as
Based on (
The buffeting response analysis of the Jiaoshao Bridge was carried out at the design wind velocity of 49.72 m/s at the bridge deck for the return period of 100 years. The longitudinal buffeting response quantities selected are listed in Table
Longitudinal buffeting response quantities.
Response quantities | Positions on the bridge structure |
---|---|
Longitudinal base shear of the bridge tower | On the bottom of bridge towers number 1~number 6 |
Longitudinal base moment of the bridge tower | On the bottom of bridge towers number 1~number 6 |
Longitudinal top displacement of the bridge tower | On the top of bridge towers number 1~number 6 |
Vertical displacement of the bridge deck | In the middle of the spans between two towers |
RMS vertical displacements of the bridge deck (unit: m).
In the middle of the span between two towers | RMS vertical displacement |
---|---|
Towers number 1 and number 2 | 0.1638 |
Towers number 2 and number 3 | 0.1847 |
Towers number 3 and number 4 | 0.2065 |
Towers number 4 and number 5 | 0.1771 |
Towers number 5 and number 6 | 0.1611 |
Time-history of vertical buffeting displacement.
RMS longitudinal buffeting responses of bridge towers. (a) RMS base shear, (b) RMS base moment, and (c) RMS top displacements of bridge towers.
From Figure
Power spectra of longitudinal buffeting responses. (a) Longitudinal buffeting displacement on the top of bridge tower number 3 and (b) vertical buffeting displacement in the middle of bridge deck.
Passive dampers have attracted the particular attention of structural control engineers due to their reliability and ease of implementation. Previous investigations have demonstrated the effectiveness of passive dampers in reducing the seismic responses of long-span cable-stayed bridges [
Installation of viscous fluid dampers in the bridge.
Comparisons of time-history of vertical buffeting displacement with and without dampers.
A parametric study is performed to investigate the effects of variations in damping coefficient
Figure
Criteria reduction factor
Figure
Criteria reduction factor
Figures
Criteria reduction factor
Criteria reduction factor
Table
Statistical results for the criteria reduction factors
Criteria reduction factor | Maximum (%) | Minimum (%) | Average (%) |
---|---|---|---|
|
−0.29 | −245.89 | −31.25 |
|
57.01 | 15.86 | 42.62 |
|
−0.03 | −169.32 | −16.07 |
|
8.93 | −40.18 | 3.90 |
|
57.88 | 7.63 | 32.48 |
|
28.63 | −2.90 | 14.07 |
|
7.71 | −7.14 | 2.56 |
|
18.20 | 1.04 | 7.01 |
|
23.90 | 1.28 | 8.72 |
|
13.43 | −4.16 | 6.32 |
|
21.74 | 1.21 | 11.89 |
|
24.13 | 1.21 | 12.49 |
The optimization design of vibration control with passive dampers is necessary for practical application [
Trade-off between two competing objectives [
In this section, the optimization design method based on the NSGA-II is proposed to determine the optimized parameters
Six objective functions.
Number | Description |
---|---|
|
|
|
|
|
|
|
|
|
|
|
|
Pareto-optimal fronts. (a)
As shown in Figures
In summary, for mitigation of longitudinal buffeting response of the multispan cable-stayed bridge, a design engineer can choose a set of proper parameters
Control effects using three sets of A, B, and C.
Criteria reduction factor | Set A |
Set B |
Set C |
---|---|---|---|
|
−34.25 | −17.49 | −1.76 |
|
56.95 | 55.59 | 49.04 |
|
−7.47 | −0.41 | −4.32 |
|
8.44 | 7.59 | 6.61 |
|
46.41 | 37.33 | 27.92 |
|
24.92 | 19.27 | 12.22 |
|
5.80 | 7.43 | 5.55 |
|
18.14 | 16.01 | 10.20 |
|
23.76 | 19.86 | 12.59 |
|
12.82 | 12.53 | 8.24 |
|
21.67 | 18.61 | 11.97 |
|
23.89 | 19.75 | 12.42 |
The Jiaoshao Bridge in China is a six-tower cable-stayed bridge, which is the longest multispan cable-stayed bridge in the world. In this paper, the longitudinal buffeting responses of the bridge were obtained using a time-domain procedure for analyzing buffeting responses implemented in ANSYS and the performance of viscous fluid dampers for the mitigation of longitudinal buffeting responses is further investigated. From the analytical results of the present study the following conclusions are drawn: Due to the insufficient longitudinal stiffness of the central towers, the longitudinal top displacements of central towers are significantly larger than those of side towers in the multispan cable-stayed bridge. The installations of viscous fluid dampers are beneficial for the reductions in the longitudinal top displacements of central towers. Due to the coupling modes of vertical bending of the bridge deck and longitudinal bending of bridge tower, the vertical displacements of the bridge deck can also be effectively reduced using viscous fluid dampers. In the multispan cable-stayed bridge, part of bridge towers longitudinally restricted with the bridge deck results in the concentration effect of longitudinal base shears and moments. The base shears and moments of bridge towers longitudinally restricted with the bridge deck are significantly larger than those of bridge towers longitudinally unrestricted with the bridge deck. The large base shears and moments of bridge towers longitudinally restricted with the bridge deck are significantly reduced using viscous fluid dampers. However, inappropriate parameters of the viscous fluid dampers will result in significant increases in the base shears and moments of bridge towers longitudinally unrestricted with the bridge deck. The optimization design objective is to yield maximum reductions in the top displacements of central towers and base forces of bridge towers longitudinally restricted with the bridge deck, with hampering the significant gain achieved in the base forces of bridge towers longitudinally unrestricted with the bridge deck. To this end, a multiobjective optimization design method that uses a nondominating sort genetic algorithm II (NSGA-II) is used to optimize parameters of the viscous fluid dampers. Optimization results reveal that a design engineer can choose a set of proper parameters of the viscous fluid dampers from Pareto-optimal fronts that can satisfy the different performance requirements.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors gratefully acknowledge the support of the National Basic Research Program of China (973 Program) (no. 2015CB060000), the National Science and Technology Support Program of China (no. 2014BAG07B01), the National Natural Science Foundation of China (nos. 51438002 and 51578138), and the Fundamental Research Funds for the Central Universities (2242016K41066).