Keywords

1 Introduction

Due to the advantages of not sharing anchorage, relatively shorter main span and lower cost, more and more multi-tower suspension bridges are being built worldwide, especially for the bridges with the main span >1000 m [1]. The Taizhou and Maanshan Yangtze River bridges with a main span of 1080 m are two excellent examples of super long-span three-tower suspension bridges in China. The structural characteristics and aerodynamic performance of a multi-tower suspension bridge are significantly different from that of a two-tower suspension bridge [2], and they are susceptible to flutter instability under wind loading [3]. Under strong wind loads, the wind-induced vibrations and post-critical flutter behaviors of long-span multi-tower bridges remain challenges for wind engineers.

Because the nonlinear flutter behavior of long-span multi-tower bridges is a complex phenomenon involving aerodynamic nonlinearities and structural nonlinearities due to structural large deformations, which can result in damage of partial components, and ultimately the collapse of the whole bridge structure [4]. In the past decades, most research focused on modeling the self-excited forces using rational or indicial functions in the time domain (e.g., Chen and Kareem [5]; Diana et al. [6]). However, the unsteady and nonlinear effects of the aerodynamic forces produced by the wind–structure interaction were been simultaneously considered in those studies. Recently, Wu and Kareem [7] presented a nonlinear convolution scheme based on Volterra-Wiener theory, and Arena et al. [8] used a nonlinear quasi-steady aerodynamic model for time-periodic oscillations of suspension bridges and conducted global bifurcation analysis of post-critical behaviors. Gao et al. proposed a nonlinear self-excited force model in terms of nonlinear flutter derivatives [9], and Xu et al. studied the flutter performance and hysteresis phenomena of a streamlined bridge deck sectional model using a large-amplitude free vibration test [10]. More importantly, Liu proposed a nonlinear aerodynamic force model (NAFM) based on nonlinear differential equations that could produce aerodynamic hysteresis phenomena [11], and Zhou et al. developed the NAFM by considering the vortex-induced force and then analyzed the nonlinear wind-induced behavior of long-span bridges [12, 13]. Through a series of wind-tunnel tests and three-dimensional nonlinear finite-element (FE) analyses, we further compared the comprehensive wind-resistance performance of a suspension bridge with various slot ratios [12], vertical stabilizers [14, 15], grid porosities [16], guide plates [17, 18], and combination of aerodynamic measures [19,20,21]. Previous studies have shown that nonlinear FE models of a 3D bridge incorporated with the NAFM is a commendable approach to simulating the nonlinear behavior of wind-induced vibration of bridges under strong wind.

This study aimed to understand the nonlinear dynamic behaviors in flutter and post-flutter of multi-tower suspension bridges under strong wind excitation. Firstly, the 2D displacement responses and the aerodynamic forces in the NAFM of a closed-box girder were calculated based on the computational fluid dynamics (CFD) simulation. Subsequently, an integrated numerical approach for a 3D three-tower suspension bridge under different wind excitations using a combination of a NAFM and nonstationary flows was developed. Finally, the nonlinear displacement responses and flutter collapse of the 3D bridge under uniform and turbulent flow, respectively, were analyzed. The present study could potentially contribute to further understanding of the flutter mechanism of multi-tower suspension bridges.

2 NAFM of Bridges

2.1 Three-Tower Suspension Bridge

A typical three-tower suspension bridge with a span arrangement of 360 + 2 × 1080 + 360 m and a sag-to-span ratio of the main cable of 1/9 was studied. All three towers are 176 m high with the middle tower being 128 m above the deck, and each of the side towers was 143 m above the deck. The distance between the two cables was 35 m, and the spacing between two adjacent hangers was 16 m. As shown in Fig. 1, the deck cross-section of the bridge was a closed streamlined box steel girder of 38.5 m wide × 3.5 m deep, in which the vertical and torsional frequencies were 0.08 ad 0.26 Hz, respectively.

Fig. 1
A cross-section of a closed streamlined box steel girder with dimensions of 38.5 meters wide and 3.5 meters deep.

Cross-section of the closed-box girder (unit: m)

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2.2 2D Displacement Responses of the Closed-Box Girder

Based on the fluid–structure interaction in the CFD simulation, an unstructured grid system incorporating large eddy simulation modeling with a Smagorinsky subgrid-scale model was used to discretize the governing Navier–Stokes equations. A steady uniform flow velocity was applied at the inlet boundary, and an opening pressure condition was given at the outlet boundary. The overall mesh division around two closed-box girders in the CFD simulations with the computational domain of a 60B × 40B rectangle is described in Fig. 2a. As illustrated in Fig. 2b, the vertical displacement of the bridge deck remained stable when the wind velocity (U) increased from 0 to ≈78 m/s, and then rapidly increased when the U was >80 m/s. Finally, the maximum vertical and torsional displacement responses were as high as ≈2.5 h/H and 80°, respectively. It can be seen that the torsional displacement reached ≈5° under U = 75.5 m/s as the soft flutter phenomenon. Therefore, the critical flutter wind velocity of the three-tower suspension bridge from the CFD simulation was 147.5 m/s. Moreover, the hysteresis loops of FD, FL, and FM predicted by the NAFM were close to those from the CFD simulation shown in Fig. 2c, d. The parameters of static force, self-excited force, and buffeting force in the NAFM of the closed-box girder were identified.

Fig. 2
An overview of C F D simulations of overall mesh division around two closed-box girders. A graph of Y over H and alpha in degrees versus U in meters per second. Two graphs of dimensionless lift force F subscript L versus torsional angles in degrees for U r = 1, 2, 4, 6, 8, and 10 for C F D and N A F M.

Parameters in the NAFM: a overview of the CFD simulation of the closed-box deck; b vertical and torsional displacement responses; c, d hysteresis curves of lifting force and torsional angle of the CFD and NAFM

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3 Nonlinear Flutter and Post-Flutter Behaviors of a Multi-tower Bridge

3.1 Turbulence Flow at Bridge Site

The reference height and the corresponding average wind velocity at the bridge site were Zref = 57.83 m and Uref = 39.3 m/s, respectively, and z0 = 0.01 and α = 0.12 because the bridge site belongs to the B-type terrain. Based on the combination of weighted amplitude wave superposition and Fast Fourier Transform (FFT) technique, the time histories of the vertical and horizontal (along-bridge) wind velocities at the height of z = 54.8 m at the middle point of the two main spans, are shown in Fig. 3.

Fig. 3
Four graphs of U subscript z in meters per second and U subscript y in meters per second versus time in seconds for vertical and horizontal. The graphs trend in an increasing pattern.

Time histories of turbulent flow of a multi-tower suspension bridge: a, b vertical and horizontal wind velocities at the left main span; c, d vertical and horizontal wind velocities at the right main span

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Nonlinear 3D FE models of the three-tower suspension bridge were established with a total of 1228 elements. The nonlinear governing coupled equations in the integrated FE model were numerically solved using the Newton–Raphson method in combination with the Newmark-β method. Accordingly, the turbulent flow at the bridge site was firstly simulated to reflect the influence of turbulent flow on aerodynamic performance. Then the nonlinear behavior of the displacement responses, structural frequencies, oscillation configurations, and failure modes of the 3D bridge under uniform and turbulent flow, respectively, were obtained.

3.2 Flutter and Post-Flutter Behaviors Under Uniform Flow

As presented in Fig. 4, the relative vertical displacement (Y/H) and torsional displacement responses (α) of the right main span rapidly decreased and approached a balance location at wind velocity of U = 70 m/s. However, there was an obvious soft flutter phenomenon of the bridge when the wind velocity increased to U = 75 m/s. In particular, the vertical and torsional displacements gradually become larger after 100 s, and then maintained a sinusoidal oscillation after 250 s with the relative value of Y/H ≅ 0.5 and α = 3.5°. Subsequently, the displacement responses presented an ever-increasing trend with increasing wind velocity. Both the vertical and torsional displacements rapidly increased after 60 s under U = 82.5 m/s and divergence finally occurred with the extreme values of Y/H ≅ 5 and α = 30°. As a result, the soft flutter phenomenon of the bridge under U = 75 m/s fell into a stable limit cycle. In addition, Fig. 4 shows that the spatially dependent lateral amplitudes along the bridge span were generally very small in comparison with the vertical or torsional components of oscillation, and the motion configuration of the two main spans was an antisymmetrical vertical and torsional coupled oscillation. The failure of the whole bridge occurred after two hangers were finally damaged at the middle ½ L of the right main span under U = 82.5 m/s. Therefore, the whole flutter collapse of the three-tower suspension bridge under uniform flow can be defined as the change from the stable limit cycle of soft flutter to the unstable limit cycle of bending-torsional coupled divergence.

Fig. 4
Six graphs depict Y over H and alpha in degrees versus time in seconds for U = 70, 75, and 82.5 meters per second. The Y over H and alpha in degrees of U = 82.5 meters per second remain constant for 60 seconds. The Y over H of U = 70 meters per second remains constant after 100 seconds.

Time histories of displacement responses: a vertical; b torsional

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3.3 Displacement Responses and Failure Modes Under Turbulent Flow

As shown in Fig. 5, the time-dependent torsional displacement response showed a nonlinear growth rate with increasing wind velocity, in which the maximum α was close to 8° under U = 72 m/s and reached 20° under U = 75 m/s. Additionally, the amplitude of torsional oscillation was greatest in three direction vibrations, and the motion configuration of the two main spans was antisymmetrical coupled oscillation. As for the failure mode of the bridge, the failure of the hangers started from those located in the middle ½ L of the left main span of the bridge. Therefore, the whole flutter collapse of the three-tower suspension bridge under turbulent flow directly shifted from the paroxysmal bifurcation to the chaos of bending-torsional coupled divergence.

Fig. 5
Two illustrations of the bridge indicate the clockwise and anticlockwise rotation of the left main span. A graph of vibration amplitude versus main span in meters for U Y, U X, and b times ROT X. An illustration of the bridge indicates hangers fracture from the 1 over 2 L deck and U = 82.5 meters per second.

Flutter collapse of the bridge: a clockwise rotation of the left main span; b clockwise rotation of the right main span; c oscillation configuration; d failure mode

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3.4 Comparison of Displacement Responses

Finally, the calculated relationship between the displacement responses and wind velocity using the integrated approach were compared with the experimental results of full-bridge aeroelastic model wind-tunnel tests [22]. It can be seen that all the calculated Ucr are generally higher than the checked flutter wind velocity. The minimum experimental Ucr was 74.2 m/s at the wind attack angle of +3° under uniform flow, which was lower than the calculated value of Ucr = 82.5 m/s. However, the experimental measurement (i.e., Ucr = 85.8 m/s) was higher than the calculated value (i.e., Ucr = 75 m/s) under turbulent flow. Moreover, Fig. 6 shows the relationship between the maximum displacement responses and wind velocity at the midspan, the quarter point (1/4 L near the side tower) and the three-quarter point (3/4 L near the middle tower) of the bridge. It demonstrates that all three calculated maximum displacement responses under uniform flow showed a stable upward trend with increasing wind velocity, and the trend became more dramatic when U was over certain threshold (i.e., U = 75 m/s, soft flutter). In addition, the maximum value of torsional displacement (i.e. Ucr = 82.5 m/s) was the largest among the three dispacements, followed by the vertical displacement. It should also be mentioned that there was a sudden increase in all three experimental maximum displacement responses when the value of U approached 90 m/s under 0° wind attack. Further, all three maximum displacement responses at the midspan of the bridge were the largest compared with other bridge locations, while the values of the displacement responses at ¼ L near the side tower were the smallest. Although the values of all three maximum displacement responses under turbulent flow also increased with increasing the wind velocity, the growth rates under turbulent flow were much higher than those under uniform flow (Fig. 7).

Fig. 6
3 graphs of alpha in degrees versus time in seconds for U = 72 meters per second under turbulence flow, U = 75 meters per second, and vibration amplitude versus main span in meters for U Y, U X, and b times ROT X. An illustration indicates hangers fracture from the 1 over 2 L deck, U = 75 meters per second.

Flutter collapse of the bridge under turbulent flow: a, b torsional displacement responses at U = 72 m/s and U = 75 m/s; c oscillation configuration; d failure mode

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Fig. 7
Two graphs of maximum values versus U in meters per second for test-lateral, calculation-lateral, test-vertical, calculation-vertical, test-torsional, and calculation-torsional for 1 over 2 point under uniform flow and turbulent flow. Both graphs trend almost in the same patterns.

Comparison of displacement responses: a, b 1/2 point under uniform flow and turbulent flow

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4 Concluding Remarks

In this study, a time-dependent numerical approach was developed to investigate the nonlinear flutter and post-flutter behaviors of a three-tower suspension bridge under different wind excitations. The major findings were:

  1. (1)

    The oscillation configuration of the bridge was very dependent on the coupled antisymmetrical bending-torsional oscillation of the two main bridge spans. The flutter performance under uniform flow (Ucr = 82.5 m/s) was better than that under turbulent flow (Ucr = 75 m/s) due to the increase in the vertical degree participation in the coupled motion.

  2. (2)

    Under uniform flow, the flutter collapse process of the bridge can be described as the change from the stable limit cycle of soft flutter to the unstable limit cycle with disconnection failure of two hangers at the middle of the right main span.

  3. (3)

    Under turbulent flow, the failure mode of the bridge can be described as the sequential fracture failure of multiple hangers at the middle of the left main span of the bridge, and the flutter collapse process of the bridge can be defined as the direct shift from the paroxysmal bifurcation to chaos.

The present study presents some new sights into nonlinear dynamic behaviors in the flutter collapse process of a three-tower suspension bridge. It should be mentioned that the spatial spanwise effects along the bridge of the aerodynamic forces were not taken into account in this study and should be investigated in future research.