Effects of vertical central stabilizers on nonlinear wind-induced stabilization of a closed-box girder suspension bridge with various aspect ratios

<jats:title>Abstract</jats:title>         <jats:p>Aerodynamic shape of a closed-box girder plays an important role in the wind-induced stabilization of long-span suspension bridges.  The purpose of this study is to investigate the effects of the combination of  five aspect ratios and a downward vertical central stabilizer (DVCS) on nonlinear flutter and aerostatic behaviors of a super long-span suspension bridge with closed-box girders. Through conducting a series of wind-tunnel tests and nonlinear finite element analysis, the results show that the nonlinear self-excited forces and the critical wind speed (  U  cr  ) gradually increases as the increase of the aspect ratio (  i.e.  the width to height ratios). Particularly, the application of 20% deck height DVCS could significantly increase the  U  cr  for these aspect ratios, and the enhancement in the flutter performance of the bridge is more obvious for a relatively small aspect ratio. Furthermore, the installation of the DVCS could change the flutter divergence patterns of the bridge from soft flutter to hard flutter, especially for a small aspect ratio. In addition, the aerostatic force coefficients and torsional divergence critical wind speeds of the larger aspect ratio with DVCS are significantly larger than that without DVCS. A relatively small aspect ratio of the bridge has better aerostatic performance than that of a larger aspect ratio.</jats:p>

section_content: The construction of super long-span suspension bridges with a length of main span more than 1500m long becomes increasingly common in recent years, such as the 1915

Canakkale Bridge (2023m main span) in the Turkey, Nansha Bridge (1688m main span)

and Lingdingyang Bridge (1666m main span ) over the Pearl River in China. However, super long-span suspension bridges with closed-box girders are extremely sensitive to wind excitation, due to their flexibility, lightness and low damping ratio [1] [2]. Wind-induced instability (e. g., flutter and aerostatic instability) under strong wind which could directly lead to the safety of the bridges, is one of the most difficult problems encountered when designing super long-span closed-box girder suspension bridges [3] [4] [5] [6]. 

As one of critical factors in aerodynamic shape of a closed-box girder, the aspect ratio of width to height of the deck (B/H) could significantly affect the wind-induced instability of suspension bridges by determining structural parameters [7] [8] [9] [10]. To further improve the wind-induced stabilization of the bridge, a range of passive aerodynamic countermeasures could be implemented to modify the aerodynamic shape, such as vertical central stabilizer [11], central slot [12], guide plate [13]. Although it has been known that the vertical central stabilizer (VCS) is one of practical measures to alleviative the flutter instability of longspan closed-box girder bridges [14], the degree of its enhancement in nonlinear flutter and aerostatic behaviors of closed-box girders suspension bridges with various aspect ratios are still uncertain. In order to guarantee the wind-induced stabilization of super long-span suspension bridges under strong wind, it becomes necessary to systematically study the effects of different aspect ratios and VCS combination on nonlinear flutter and aerostatic behaviors of the bridges with closed-box girders. 

On the one hand, previous studies have provided a better insight into the flutter instability of bridges with respect to only the aspect ratio [15] or only the VCS [16], but the complex nonlinear behaviors of soft flutter [17] [18] and post-flutter [19] [20] for closed-box girder suspension bridges under strong wind are still challenging. During the last decades, a series of nonlinear aerodynamic force models, such as band superposition model [21] [22],

Volterra model [23], polynomial model [24] [25], nonlinear differential equations model [12] [13] and neural-network-based model [26] [27], were developed to predict the soft-flutter and post-flutter behaviors of bridges. Nevertheless, the effects of the aerodynamic shape modification on the nonlinear flutter behaviors of closed-box girder suspension bridges should be further investigated. On the other hand, the nonlinear three-component displacement-dependent wind loads, the geometric and material nonlinearities should be taken into account in evaluating the aerostatic stabilization of suspension bridges [28] [29]. 

More and more studies have shown that the nonlinear aerostatic instability of suspension bridges was investigated both experimentally [15] [16] 30] and theoretically [31] [32] [33] in recent years. However, the influence of the combination of aspect ratios and VCS on the nonlinear aerostatic behavior of closed-box girder suspension bridges has not been fully understood. 

The purpose of this study is to conduct a series of wind-tunnel tests in conjunction with nonlinear numerical analysis for investigating the effects of various aspect ratios in aspect ratios (i. e., 7. 9 and 8. 9) and a DVCS. The present study is helpful to optimize the aerodynamic shape of super long-span suspension bridges under strong wind.
section_name: INTRODUCTION

section_name: NONLINEAR FLUTTER BEHAVIOR WITH VARIOUS ASPECT RATIOS AND DVCS COMBINATION

section_content: As shown in Fig. 1 (a), a super-long-span suspension bridge with the span arrangement of 580 + 1756 + 630m was selected, in which the overall height of two side towers is 247. 5 m and the longitudinal distance between two adjacent suspenders is 18 m. Table 1 shows geometric and mechanical parameters of a closed-box girder with five typical aspect ratios (i. e., B/H=10. 4, 8. 9, 8. 3, 7. 9, and 7. 1). The details of the SEC2 and SEC4 of the bridge with the aspect ratio of 8. 9 and 7. 9, respectively, are shown in Fig. 1(b-c ). In this study, the vertical frequencies for the five aspect ratios are kept constant (i. e., 0. 112Hz), and the torsional frequency increases as the aspect ratio decreases. A series of flutter tests of sectional models were experimentally conducted for studying the aerodynamic shape modification effectiveness of five aspect ratios on the flutter instability in the closed-box girders without and with a DVCS.
section_name: Structural parameters of a super long-span suspension bridge

section_content: Fig. 2 shows the details of experimental setup of fluttering testing involving eight spring-supported rigid-sectional models of closed-box girders with a DVCS of 20% height of the deck subject to a wind attack angle of +3, 0 and -3 degree, respectively. The geometric scale ratio of SEC1, SEC3, and SEC5 is 1:70, while the geometric scale ratio of SEC2 and SEC4 is 1:65. As shown in Table 2, the flutter testing in this study involves 30 testing cases in total considering five different aspect ratios, without and with DVCS under The critical flutter wind speeds of the sectional models with five aspect ratios without and with DVCS under three wind-attack angles are presented in Fig. 3. It shows that the wind attack angle of α=0 ° is most unfavorable for the closed-box girders among three angles. In particular, the critical wind speed Ucr gradually increases as the aspect ratio increases. For example, the flutter performance of the bridge with SEC1 (aspect ratio=10. 4) is the best among the five aspect ratios, followed by SEC2 with a side ratio of 8. 9, and the flutter performance of SEC5 (aspect ratio=7. 1) is the worst. The ratio of torsional frequency to vertical frequency for the closed-box girders is in inverse proportion to critical flutter wind speed. However, none of the sectional models with these five aspect ratios meets the requirement of wind-resistant design (Ucr is less than the required minimum critical wind speed of 83. 1m/s). Thus, the application of DVCS becomes necessary to further modify the aerodynamic shape. The results in Fig. 3(b ) indicate that the application of 20% deck height DVCS could significantly increase the minimum Ucr for all the aspect ratios, particularly for the flutter performance of closed-box girders with smaller aspect ratios.
section_name: Flutter testing of the aspect ratios and DVCS combination

section_content: As shown in Fig. 4, the aerodynamic shape modification effects of these five aspect ratios in combination of a DVCS on the nonlinear flutter behavior of the closed box girder bridges were theoretical investigated based on our previous developed time-dependent nonlinear self-excited force model (NSFM) [12] [13] and the closed-box girders with upward VCS [11]. The NSFM considers the relative instantaneous wind attack angle, relative instantaneous wind velocity u and their derivatives using nonlinear differential equations that simulating the nonlinear self-excited force of the closed-box girders with different aerodynamic shapes. That is,

Where,, and represent the drag force, lift force and lift moment, respectively. and are the horizontal and vertical displacements, respectively. 

and are the horizontal and vertical component of wind velocities, respectively.
section_name: Nonlinear self-excited force analysis of the aspect ratios and DVCS combination

section_content: The NSFM for the closed-box girder with a DVCS in terms of nonlinear differential equations can be described as follows:

{ ((), (), ) = (, ) + (α, ̇̇, ̇)+ (, ̈, ̈) + (,, )

̇= (,, α, ̇̇, ̇, )

The detailed expression of wind-induced stationary force (), dynamic force (), inertia aerodynamic force () and unsteady aerodynamic force are listed in the Equations (8-12) of reference [12], and the expression of self-excited force can be written in Equation ( 4 ), that is

Where, the, and, denote the contribution terms associated with structural translational acceleration and rotate acceleration.,,,, denote the contribution terms associated with input variables of structural movement, respectively. 

In order to identify the parameters of NSFM for the closed-box girders with five different aspect ratios and DVCS, First, the initial values ( and ()) of model input parameters under various reduced wind speeds and oscillation amplitudes will be obtained by CFD simulations using forced oscillation. Then, for a given input function (), the time evolutions of state variables () can be integrated numerically by using the fourth order Runge-Kutta algorithm. Finally, these contribution terms will be identified by implementing the Levenberg-Marquardt algorithm with the aim of minimizing the difference between the outputs of time-dependent aerodynamic forces (i. e. FH, FV and MZ) obtained from model prediction and the CFD simulation [11] [12] [13]. A series of forced oscillations were obtained based on the large eddy simulation (LES) method in conjunction with a Smagorinsky subgrid scale model incorporated CFD model to determine the initial parameters of the NSFM [34] [35]. The rectangle computational region of 75B×50B was used in CFD simulation, and the Reynolds number is set to be 5. 0×10 5 for SEC1 is the largest among the five aspect ratios. The absolute amplitude of FL gradually decreases with a decrease in aspect ratio. Similarly, the amplitude variation of FM is relatively large for a large aspect ratio. In addition, the area of hysteresis loops of lift force FL for SEC1 is the largest among the five aspect ratios, and the area of hysteresis loops decreases with a decrease in aspect ratio. This indicates that the unsteady features increase with the decrease of wind speed. As shown in Fig. 5 (e-f), there is a nonlinear relationship between the multi-frequencies phenomenon of FL and the self-excited force for all five aspect ratios, and their first frequencies are close to 0. 2Hz. was developed for the flutter divergence over the critical wind speed. Specifically, small limit cycle oscillation (LCO) occurred at a wind speed of U = 71m/s for SEC1, U = 70m/s for SEC2, U = 64m/s for SEC3, U = 58m/s for SEC4, and U = 56m/s for SEC5. The torsional displacement of soft flutter for SEC3 was close to ±1. 5° at the beginning of U = 64m/s, and then reached to more than 20° at the extreme wind speed of U = 66m/s. 

For the closed-box girders with a DVCS, the critical wind speeds at which soft flutter became flutter divergence, were 81m/s, 89m/s, 85m/s, 95m/s, and 93m/s, for SEC1 to SEC5 respectively. In addition, it indicates that, after installation of DVCS, the torsional displacement of the hard flutter initially increases, and then suddenly increased to a threshold (more than about 6°) which leads to structural failure. This indicates that there
section_name: Fig. 4 Details of the nonlinear self-excited force model (NSFM) used in this study

section_name: NONLINEAR AEROSTATIC BEHAVIOR WITH VARIOUS ASPECT RATIOS AND DVCS COMBINATION

section_content: Force-measured testing of aerostatic forces on the closed-box girders with different aerodynamic shapes was performed in the TJ-2 boundary layer wind tunnel at Tongji University. Two sectional models (SEC4 and SEC2 with two aspect ratios of 7. 9 and 8. 9, respectively) were 0. 55m wide (B) and two heights (H) of 0. 061m and 0. 069m, respectively, were used during flutter tests. In addition, the 20% height DVCS for SEC2 and SEC4 was also experimentally investigated. In summary, a total of 100 testing cases As shown in Fig. 10, a force-balance system was installed on one side of the tunnel to vertically support the two ends of a sectional model. A wind speed of 10m/s was adopted in the test with the wind-attack angle ranging from -12° to +12° with an increment of 1°.
section_name: Force-measured tests of the aspect ratios and DVCS combination

section_content: The drag-force coefficient (CD), lift-force coefficient (CL) and lifting-moment coefficient (CM) with different aerodynamic shapes were measured in force-measured tests. 

As illustrated in Fig. 11(a), the value of CD is unsymmetrical about α = 0°, and the value of CD at negative wind-attack angles is much larger than that at positive angles. Although most values of the CD for SEC4 without DVCS are close to those for SEC2 without a DVCS, all the values of CD for SEC4 with a DVCS are smaller than those of SEC2 with a DVCS. In addition, the values of CD for both the models without a DVCS are also much smaller than those with a DVCS from α = -9° to α = 9°. Fig. 11(b) shows that the values of CL for SEC4 without a DVCS are slightly larger than those for SEC2 without a DVCS. This is consistent with the observation that the values of CL for SEC4 with a DVCS are larger than those for SEC2 with a DVCS. Both the values of CM for SEC4 without and with a DVCS are also slightly larger than those corresponding values for SEC2, respectively. 

The value of CM generally become larger after installing the DVCS, as shown in Fig. 11(c ). 

Generally, the values of CL and CM gradually increase as α increases from α = -12° to α = 12°, whereas their values decrease as the aspect ratio increases. Most of the aerostatic force coefficients of both SEC4 and SEC2 with a DVCS are larger than those without a DVCS. 

-
section_name: Static wind loading of the aspect ratios and DVCS combination

section_content: Based on the obtained aerostatic force coefficients, the nonlinear 3D aerostatic instability of a bridge with two aspect ratios without and with a DVCS is calculated for the nonlinear Equation ( 5 ) using the optimum iteration method of reference 29. 

[

Where, Ke and Kg are structural elastic stiffness matrix and the geometrical stiffness matrix, respectively. FH(α), FV(α), MT(α) are drag-force, lift-force and lifting-moment, respectively. {}is structural displacement vector. D and B is the height and width of deck. The torsional divergence critical wind speed (Utd) of the bridge was determined considering geometric nonlinearity and aerodynamic force nonlinearity. Fig. 12 shows the three structural displacement responses at the midpoint of the bridge with two aspect ratios (SEC2 and SEC4) under three wind-attack angles. In addition, the displacement responses of the sectional model with a DVCS installed under a wind-attack angles of +3° were also shown in Fig. 12. It can be seen that the lateral displacements gradually increased with the increase of the wind speed, and α=+3° leads to the largest increase rate. The lateral displacements of SEC2 without a DVCS were slightly larger than those of SEC4 without a DVCS, while the displacements of SEC2 with a DVCS were smaller than those for SEC4 with a DVCS. Furthermore, the vertical and torsional displacements of SEC4 and SEC2 under α=+3° gradually increase with the increase of wind speed. In addition, the vertical and torsional displacements of SEC2 without DVCS were slightly larger than that of SEC4 without DVCS. The installation of DVCS could obviously increase the vertical displacements of SEC2, while the torsional displacements for both SEC4 and SEC2 increase after installing DVCS. For example, after installation of DVCS, the Utd of SEC4 and SEC2 increases from 160m/s to 162m/s and 150m/s to 166m/s, respectively. Therefore, a relatively small aspect ratio could enhance the aerostatic performance of the closed-box girder suspension bridge.
section_name: Nonlinear aerostatic behavior of the aspect ratios and DVCS combination

section_content: This study investigated the nonlinear flutter and aerostatic behaviour of a super longspan closed-box girder suspension bridge with five aspect ratios in combination of a DVCS by conducting a series of wind-tunnel tests conjunction with numerical modelling. The following are some major conclusions:

• The critical wind speed (Ucr) gradually increases as the increase of the aspect ratio due to the smaller ratio of natural torsional and vertical frequency, and the aspect ratio also has effect on the limit cycle oscillation development of soft flutter. For example, the flutter performance of the bridge with an aspect ratio of 10. 4 is much better than that with an aspect ratio of 7. 1. 

(c) (

• The application of 20% deck height DVCS could significantly increase the Ucr for different the aspect ratios, and the enhancement in the flutter performance of the bridge is more obvious for a relatively small aspect ratio. The installation of the DVCS could change the flutter divergence patterns of a bridge from soft flutter to hard flutter. 

• The absolute values of CL and CM increase with the increase of wind-attack angle and aspect ratio of a suspension closed box girder bridge. In addition, the installation of DVCS can significantly improve three static wind-load coefficients. 

• The bridge with a larger aspect ratio has a higher torsional divergence critical wind speed. Installation of a DVCS also significantly improves the aeroelastic performance of the bridge, especially for a larger aspect ratio of 8. 9.
section_name: CONCLUSIONS

section_name: ACKNOWLEDGMENTS

section_content: The authors gratefully acknowledge support for this research provided by the National Natural Science Foundation of China (No. 51908374, U2005216 and 52178503 ), the Guangdong Province Natural Science Foundation (No. 2022A1515010665 and 2020A0505100065 ), the State Key Laboratory of Mechanical Behavior and System Safety of Traffic Engineering Structures, Shijiazhuang Tiedao University (No. KF2020-19 ), the Shenzhen Science and Technology Innovation Program (Nos.
section_name: ACKNOWLEDGMENTS

section_name: DATA AVAILABILITY

section_content: The data used to support the findings of this study are available from the corresponding author upon request.
section_name: DATA AVAILABILITY

section_name: CONFLICTS OF INTEREST

section_content: The authors declare no potential conflicts of interest with respect to the research, authorship, and/or publications of this article.
section_name: CONFLICTS OF INTEREST

section_content: Cities (Shenzhen University), Ministry of Education.
section_name: ZDSYS20201020162400001), Key Laboratory for Resilient Infrastructures of Coastal