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

The 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 (Ucr) gradually increase as the increase of the aspect ratio (i.e. the width to depth ratios). Furthermore, the application of 20% deck depth DVCS could significantly increase the nonlinear self-excited forces and Ucr for small aspect ratios of 7.9 and 7.1. Particularly, the installation of the DVCS could change the flutter divergence patterns of the bridge from soft flutter to hard flutter, especially for a relatively 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 with a larger aspect ratio.


Introduction
The construction of super long-span suspension bridges with a length of main span more than 1500 m long becomes increasingly common in recent years, such as the 1915 Canakkale Bridge (2023 m main span) in Turkey, the Nansha Bridge (1688 m main span) and Lingdingyang Bridge (1666 m 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][7].
As one of the critical factors in the aerodynamic shape of a closed-box girder, the aspect ratio of width to depth of the deck (B/H) could significantly affect the wind-induced instability of suspension bridges by determining structural parameters [8][9][10][11]. 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 a vertical central stabilizer [12], central slot [13], and guide plate [14,15]. Although it has been known that the vertical central stabilizer (VCS) is one of the practical measures to alleviative the flutter instability of closed-box girder suspension bridges [16], the degree of its enhancement in nonlinear flutter and aerostatic behaviors of super long-span suspension bridges with various aspect ratios under strong wind is still uncertain. In order to guarantee the windinduced stabilization of super long-span suspension bridges, it becomes necessary to systematically study the effects of different aspect ratios and VCS combinations on nonlinear flutter and aerostatic behaviors of these bridges with closed-box girders.
On the one hand, previous studies have provided a better insight into the flutter instability of bridges concerning only the aspect ratio [17] or only the VCS [18], but the complex nonlinear behaviors of soft flutter [19,20] and post-flutter [21,22] 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 the band superposition model [23,24], Volterra model [25], polynomial model [26,27], nonlinear differential equations model [13,14] and neural-network-based model [28,29], were developed to predict the softflutter and post-flutter behaviors of bridges. Nevertheless, the effects of the aerodynamic shape modification (e.g. the combination of DVCS and aspect ratio) 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, and the geometric and material nonlinearities should be taken into account in evaluating the aerostatic stabilization of suspension bridges [30,31]. More and more studies have shown that suspension bridges' nonlinear aerostatic instability was investigated both experimentally [17,18,32] and theoretically [33][34][35] 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 have not been fully understood.
The purpose of this study is to conduct a series of wind-tunnel tests in conjunction with nonlinear numerical analysis to investigate the effects of various aspect ratios in the combination of downward VCS (DVCS) on the nonlinear flutter and aerostatic performance of a super long-span suspension bridge. By keeping the vertical frequency constant, a series of sectional-model flutter tests were carried out to obtain the critical wind speed (U cr ) of closed-box girders with five typical aspect ratios ranging from 7.0 to 10.4 and a DVCS of 20% deck depth. Subsequently, a threedimensional (3D) nonlinear FE model of the closedbox girder bridge was developed based on the nonlinear self-excited force model, and their timedependent displacements, frequencies, and oscillation modes of soft-flutter were compared. Finally, static wind-load coefficients and nonlinear aeroelastic behaviors were evaluated based on the force-measured testing results of the two important 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.
2 Nonlinear flutter behavior with various aspect ratios and DVCS combination

Structural parameters of a super long-span suspension bridge
As shown in Fig. 1a, a super-long-span suspension bridge with the span arrangement of 580 ? 1756 ? 630 m 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. The 3D nonlinear finite element models of the bridges with various aspect ratios were established by using ANSYS software, and their natural frequencies and mode shapes of the bridge were calculated by using the block Lanczos algorithm. Table 1 shows the 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. 1b, c. Since the change of aspect ratio could lead to the change of the mechanical parameters (e.g., mass and stiffness), both mass and mass inertia were adjusted to keep the important dynamic characteristics of the vertical frequency constant, in order to effectively study the effect of the aspect ratio as the aerodynamic shape on the nonlinear wind-induced stabilization of a closed-box girder suspension bridge.  In this study, the vertical frequencies for the five aspect ratios are kept constant (i.e., 0.112 Hz), 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. Figure 2 shows the details of the experimental setup of fluttering testing involving eight spring-supported rigid-sectional models of closed-box girders with a DVCS of 20% depth of the deck subject to wind attack angles of ? 3, 0, and -3 degree, respectively. The geometric scale ratio of SEC1, SEC3, and SEC5 is     Table 2, the flutter tests in this study involve 30 testing cases in total considering five different aspect ratios, without and with DVCS under three different wind attack angles (i.e., ? 3°, 0°, and -3°). The tests were conducted in the TJ-1 boundary layer wind tunnel at Tongji University. 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 a = 0°is the Frequency (Hz) most unfavorable for the closed-box girders among three angles for most of the cases. In particular, the critical wind speed U cr 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 (U cr is less than the required minimum critical wind speed of 83.1 m/s). Thus, the application of DVCS becomes necessary to further modify the aerodynamic shape. The results in Fig. 3b show that the wind attack angle of a = ? 3°is the most unfavorable for the closed-box girders among three angles after installing the DVCS for most of the cases. It indicates that the application of 20% deck depth DVCS could significantly increase the minimum U cr for all the aspect ratios, particularly for the flutter performance of closed-box girders with smaller aspect ratios of SEC4 and SEC5.

Flutter testing of the aspect ratios and DVCS combination
upward VCS [12], the NSFM of the closed-box girders with various aspect ratios and a DVCS in Fig. 4 was further developed in terms of nonlinear differential equations, which are listed in the Eqs. (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20) of reference [13]. The relative instantaneous wind velocity (u), relative instantaneous wind attack angle (h), and their derivatives are used as the input variables. u x and u y are the horizontal and vertical components of wind velocities, respectively. _ x _ y and _ a are the horizontal velocity, vertical velocity, and torsional angular velocity of the movement, respectively. F H ; F V and M Z represent the drag force, lift force and lift moment, respectively. B and H are the width and depth of the closed-box girder, respectively.
The expression of nonlinear self-excited force can be written in Eq. (1), that is The / m ; K m and / a ; K a denote the contribution matrix variables associated with structural translational acceleration and rotate acceleration, respectively. G a , G m , H m , G a , H a denote the contribution matrix variables 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 of input variables under various wind velocities and oscillation displacements will be obtained by CFD simulations using forced oscillation. Then, these model parameters will be identified by implementing the Levenberg-Marquardt algorithm and fourth-order Runge-Kutta algorithm to minimize the difference between the outputs of timedependent aerodynamic forces (i.e. F H , F V , and M Z ) obtained from model prediction and the CFD simulation [12][13][14]. A series of forced oscillations were obtained based on the large eddy simulation (LES) method in conjunction with a Smagorinsky subgridscale model incorporated CFD model to determine the initial parameters of the NSFM [34,35]. The rectangle computational region of 75B 9 50B was used in CFD simulation, and the Reynolds number is set to be 5.0 9 10 5 based on the depth of closed-box girders. Figure 5a,b shows the simulation results without and with DVCS, respectively. Seven representative reduced wind velocities (U r ) (i.e., U r = U/fB = 2,4,6,8, 10,12,14), six representative maximum vertical displacements (y) of the deck (i.e., y = ± 1, ± 2, ± 3, ± 4, ± 5, ± 6), and six representative maximum torsional displacements (h) of the deck (i.e., h = ± 1°, ± 5°, ± 10°, ± 15°, ± 20°, ± 25°) were used as the input variables in NSFM. Figure 5c,d compares the predicted dimensionless time-dependent outputs, hysteresis loops, and amplitude spectrums of self-excited force (e.g., lift force F L and lifting moment F M ) of closed-box girders with five different aspect ratios. It shows the dimensionless time-dependent F L and F M of a closed-box girder without DVCS under a large reduced wind speed (U r = 10) and heaving amplitude (h = 6). The absolute amplitude of F L for SEC1 is the largest among the five aspect ratios. The absolute amplitude of F L gradually decreases with a decrease in aspect ratio. Similarly, the amplitude variation of F M is relatively large for a large aspect ratio. In addition, the area of hysteresis loops of lift force F L 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. 5e-f, there is a nonlinear relationship between the multi-frequencies phenomenon of F L and the self-excited force for all five aspect ratios, and their first frequencies are close to 0.2 Hz.
The dimensionless time-dependent F L and F M of the closed-box girder with DVCS under the same reduced wind speed (U r = 10) and heaving amplitude (h = 6) are illustrated in Fig. 6. It shows that the amplitudes of F L for SEC4 and SEC5 with DVCS are larger than those with the other three aspect ratios, and the absolute values of F M for SEC4 and SEC5 with a DVCS are larger than those with the other three aspect ratios. In addition, both the amplitudes of F L and F M of the closed-box girder with a DVCS are much larger than those without a DVCS. The areas of hysteresis loops of the lift force F L for SEC5 and SEC1 are the b Fig. 7 Time-dependent torsional displacement responses at the mid-span of the closed-box girder without DVCS. a, b SEC1; c, d SEC2; e, f SEC3; g, h SEC4; i, j SEC5 (c) (e) (f) largest and smallest, respectively. The area of hysteresis loops for a closed-box girder with a DVCS decreases with an increase in the aspect ratio but is larger than those without DVCS. The frequencies of F L for the closed-box girder with DVCS in Fig. 6d show that the DVCS can significantly change the frequency characteristics of the F L of SEC5 and SEC3.Therefore, the nonlinear aerodynamic force decreases with the increase of the aspect ratio after installing DVCS, and the DVCS can significantly influence the nonlinear characteristics of self-excited force.

Nonlinear flutter responses of the aspect ratios and DVCS combination
A three-dimensional closed-box girder bridge model with a total of 853 elements was developed based on the NSFM to investigate the nonlinear flutter responses of the bridge with different aerodynamic shapes. Based on the self-developed C# program, the 3D beam elements with 14 degrees of freedom (including 6 nodal translational degrees of freedom, 6 nodal rotational degrees of freedom, and 2 transverse shear degrees of freedom) are used to simulate the closed-box girder and main towers, while the 3D truss elements with 7 degrees of freedom (6 nodal translation degrees of freedom and 1 axial force degree of freedom) are used to simulate the main cables and hanger. Furthermore, the nonlinear aerodynamic force elements with 6 nodal degrees of freedom and a set of self-excited force subsystem degrees of freedom are used to simulate the NSFM on the nodes of the main girder. The boundary conditions of the main cables are the constraint in vertical, lateral, and longitudinal direction, respectively. The boundary conditions between the main girder and the left tower is the constraint in vertical, lateral, and torsional direction, respectively, while the boundary conditions between the main girder and the right tower is the constraint in vertical, lateral, longitudinal, and torsional direction, respectively. Figure 7 shows the time-dependent torsional displacement response at the mid-span point of the closed-box girder with five aspect ratios without and with DVCS. The soft flutter of five aspect ratios with the torsional displacement of about a = 2 was developed for the flutter divergence over the critical wind speed. Specifically, small limit cycle oscillation (LCO) occurred at a wind speed of U = 71 m/s for SEC1, U = 70 m/s for SEC2, U = 64 m/s for SEC3, U = 58 m/s for SEC4, and U = 56 m/s for SEC5. The torsional displacement of soft flutter for SEC3 was close to ± 1.5°at the beginning of U = 64 m/s, and then reached more than 20°at the extreme wind speed of U = 66 m/s. However, all the torsional displacements of the bridge without DVCS increase rapidly to the diverges at the flutter onset wind speeds of the closed-box girders with DVCS. Figure 8 shows that all the torsional displacements of the bridge with DVCS decrease rapidly at the flutter onset wind speeds of the closed-box girders without DVCS.After installation of DVCS, the critical wind speeds at which soft flutter became flutter divergence, were 81 m/s, 89 m/s, 85 m/s, 95 m/s, and 93 m/s, for SEC1 to SEC5 respectively. In addition, the torsional displacement responses of the hard flutter initially become larger, and then suddenly increase to a threshold (more than about 6°) which leads to structural failure. This indicates that there are different displacement responses in soft flutter for closed-box girders with various aspect ratios, and the installation of DVCS can change the flutter divergence patterns of a bridge from soft flutter to hard flutter. As shown in Fig. 7, there is no obvious LCO in the nonlinear response of the cross section with stabilizer as DVCS can significantly change the vertical DOF participation in the flutter divergence leading to the significant increase in the torsional displacements of the bridge as shown in Figs. 8 and 9. Figure 9a, b compares the frequencies of torsional displacement responses of closed-box girders with five aspect ratios without and with DVCS. The results demonstrate a single frequency phenomenon (i.e., around 0.16 Hz) for all responses. In addition, the frequency increases modestly with the increase of aspect ratio. Figure 9c, d shows the RMS of torsional displacements for five aspect ratios without and with a DVCS. The results show that the torsional amplitudes generally increase smoothly, and post-flutter instability is a nonlinear soft flutter in nature. Specifically, the largest torsional displacements happen from about 4 degree to 9 degree after installing DVCS. The reduced wind speeds of the SEC1 and SEC2 are relatively high ranging from 6.5 to 8 m, while the reduced wind speed of the SEC2, SEC4, and SEC5 with a DVCS are relatively high ranging from 8 to 10. Furthermore, the simulation results of the three displacement responses of the main girders are shown in Fig. 10. It shows that the torsional displacement responses of SEC2 are the largest among the three displacement responses. Most importantly, the first and second anti-symmetrical torsional modes play a dominant role in the structural coupled bendingtorsional oscillations of the main girder with and without a DVCS, respectively. The failure modes of the overall bridge at flutter divergence are shown in Fig. 10c, d. In summary, the installation of DVCS can significantly change the vertical DOF participation in the flutter divergence of the bridge, and thereby increase the critical flutter wind speed of the bridge.
3 Nonlinear aerostatic behavior with various aspect ratios and DVCS combination

Force-measured tests of the aspect ratios and DVCS combination
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 important sectional models (SEC4 and SEC2 with two aspect ratios of 7.9 and 8.9, respectively) were 0.55 m wide (B) and two depths (H) of 0.061 m and 0.069 m, respectively, were used during flutter tests. In addition, the 20% depth DVCS for SEC2 and SEC4 was also experimentally investigated. In summary, a total of 100 testing cases of two aspect ratios without and with a DVCS under 25 windattack angles were experimentally tested. As shown in Fig. 11, 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 10 m/s was adopted in the test with the wind-attack angle ranging from -12°to ? 12°with an increment of 1°. The geometric details of the two models of (c) Fig. 11 The force-balance system used in this study. a Experimental set-up; b SEC4 without DVCS; c SEC2 with DVCS SEC2 and SEC4 without and with the DVCS are shown in Fig. 11b and c, respectively.

Static wind loading of the aspect ratios and DVCS combination
The drag-force coefficient (C D ), lift-force coefficient (C L ), and lifting-moment coefficient (C M ) with different aerodynamic shapes were measured in forcemeasured tests. As illustrated in Fig. 12a, the value of C D is unsymmetrical about a = 0°, and the value of C D at negative wind-attack angles is much larger than that at positive angles. Although most values of the C D for SEC4 without DVCS are close to those for SEC2 without a DVCS, all the values of C D for SEC4 with a DVCS are smaller than those of SEC2 with a DVCS. In addition, the values of C D for both the models without a DVCS are also much smaller than those with a DVCS from a = -9°to a = 9°. Figure 12b shows that the values of C L 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 C L for SEC4 with a DVCS are larger than those for SEC2 with a DVCS. Both the values of C M for SEC4 without and with a DVCS are also slightly larger than those corresponding values for SEC2, respectively.
The value of C M generally becomes larger after installing the DVCS, as shown in Fig. 12c. Generally, the values of C L and C M gradually increase as the increases from a = -12°to a = 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 DVCS.

Nonlinear aerostatic behavior of the aspect ratios and DVCS combination
Based on the measured 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 Eq. (2-3) using the optimum iteration method described in reference [31].
where K e and K g are structural elastic stiffness matrix and the geometrical stiffness matrix, respectively. F H (a), F V (a), M T (a) are drag-force, lift-force and lifting-moment, respectively. u f g is structural displacement vector. D and B is the depth and width of deck.
The torsional divergence critical wind speed (U td ) of the bridge was determined considering geometric nonlinearity and aerodynamic force nonlinearity. Figure 13 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 the wind-attack angles of ? 3°were also shown in Fig. 13. It can be seen that the lateral displacements -12 -9 -6 -3 0 3 6 9 12   Fig. 13 Displacement responses at the midpoint of the bridge with two aspect ratios without/with a DVCS. a, b lateral displacements; c, d vertical displacements; e, f torsional displacements gradually increased with the increase of the wind speed, and a = ? 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 of SEC4 with a DVCS. Furthermore, the vertical and torsional displacements of SEC4 and SEC2 under a = ? 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 the installation of DVCS, the U td of SEC4 and SEC2 increases from 160 m/s to 162 m/s and 150 m/s to 166 m/s, respectively. Therefore, a relatively small aspect ratio could improve the aerostatic performance of the closed-box girder suspension bridge.

Conclusions
This study investigated the effect of the DVCS on nonlinear flutter and aerostatic behaviors of a super long-span closed-box girder suspension bridge with five aspect ratios by conducting a series of wind-tunnel tests in conjunction with nonlinear finite element analysis. The following are some major conclusions: • The critical wind speed (U cr ) 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 affects 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. • The application of 20% deck depth DVCS could significantly increase the U cr 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 C L and C M 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.