Aerodynamic stability evolution tendency of suspension bridges with spans from 1000 to 5000 m

Abstract

Aerodynamic instability owing to aerostatic and flutter-related failures is a significant concern in the wind-resistant design of long-span suspension bridges. Based on the dynamic characteristics of suspension bridges with spans ranging from 888 to 1991 m, we proposed fitted equations for increasing spans and base frequencies. Finite element models of suspension bridges with increasing span from 1000 to 5000 m were constructed. The structural parameters were optimized to follow the fitted tendencies. To analyze the aerodynamic instability, streamlined single-box section (SBS), lattice truss section (LTS), narrow slotted section (NSS), and wide slotted section (WSS) were considered. We performed three-dimensional (3-D) full-mode flutter analysis and nonlinear aerostatic instability analysis. The flutter critical wind speed continuously decreases with span growth, showing an unlimited approaching phenomenon. Regarding aerostatic instability, the instability wind speed decreases with span to approximately 3000 m, and increases when the span is in the range of 3000 to 5000 m. Minimum aerostatic instability wind speed with SBS or LTS girder would be lower than observed maximal gust wind speed, indicating the probability of aerostatic instability. This study proposes that suspension bridge with span approximately 3000 m should be focused on both aerostatic instability and flutter, and more aerodynamic configuration optimistic optimizations for flutter are essential for super long-span suspension bridges with spans longer than 3000 m.

Graphical abstract

Keywords

suspension bridge / super long-span / finite element model / aerodynamic instability / aerodynamic configuration

1 Introduction

Bridges are irreplaceable in modern transportation systems. Various types of suspension bridges have been designed or constructed to cross wide rivers, deep valleys, and straits owing to their great spanning capability. A few examples are the Akashi Kaikyo Bridge [1] in Japan with a main span of 1991 m, the Shiziyang Bridge, a super long-span bridge to be built in China, which is designed to stretch 2180 m, and the Messina Strait Bridge [2] planned in Italy with a span of 3300 m. Recent research indicated that the typhoon with high wind speeds and specific features [3–5] can be a significant factor in the super long-span bridge design, requiring better aerodynamic stability of suspension bridges.

Flutter is one of the biggest threats to long-span bridges and has attracted the attention of scholars in recent decades. In 1940, the Old Tacoma Suspension Bridge suffered from flutter divergence and was destroyed under medium wind conditions. This was the origin of aerodynamic research on bridges. A series of wind tunnel experiments [6] were conducted on the Old Tacoma Suspension Bridge to determine the reason for wind-induced vibration and destruction. Analytical studies on flutter [7] were conducted based on motion-dependent forces on a thin airfoil in the field of aeronautics [8]. Later, an empirical formulation was introduced to describe the actual aeroelastic flutter behavior of suspension bridges [9] by simplifying the airfoil aerodynamic forces. The analysis method for airfoil flutter was not accurate for bridges because decks were often bluff sections. Expressions using aerodynamic derivatives of motion-dependent lift and moment forces [10] were used to analyze the flutter of the bridge deck sections. Subsequently, researchers developed formulations of two degrees of heaving and torsion extending to three degrees, including sway motion and along-wind drag component. Initially, empirical formulae [11,12] were used to calculate the critical flutter speed of bridges. As expressions of aerodynamic forces and computational power were proposed, a series of frequency-domain flutter methods [13,14] and time-domain methods [15,16] were proposed.

In 1967, aerostatic torsional divergence was first observed in a suspension bridge wind tunnel test [17], indicating that aerostatic instability failure is a vital problem in long-span bridges. A linear analysis method using an airfoil has been employed in wind-resistant design of long-span cable-supported bridges. However, the linear method ignores the geometric nonlinearity and aerostatic force nonlinearity of bridges, resulting in an overestimation of aerostatic stability. Using the finite element method, the aerostatic instability speed can be solved by a method that considers wind load nonlinearity and structural geometric nonlinearity [18,19]. Generally, aerostatic instability occurs at wind speeds higher than flutter. However, in recent years, studies on cable supported bridges [20–23] observed the simultaneous occurrence of aerostatic instability and flutter under certain circumstances. The analysis of flutter and aerostatic instability should be performed at such critical points when designing super-span bridges.

The span length of suspension bridges increased with the development of bridge wind engineering theories, promoting the analysis of suspension bridges with different spans and deck forms. The lattice truss is a classical form of stiffened girder in suspension bridges, and is still popular in mountainous areas. Wind tunnel tests were conducted to guarantee the wind-resistant capability of truss deck bridges [24] in the 1950s. With progress in bridge and wind engineering, streamlined single-box deck suspension bridges were designed and constructed. Refined calculations and comparisons [25] showed that the flutter stability of truss suspension bridges was inferior to that of bridges with streamlined single-box sections (SBSs) when the span was less than 2000 m. In recent years, streamlined single-box decks have been used to ensure aerodynamic stability [26]. When the span of the suspension bridge increases continuously, central slotting can effectively improve the flutter stability of the box girder. As the span length increased over the years, an analysis of 2500 and 4000 m span suspension bridges [27] found that super long-span suspension bridges were not restricted by static problems in the structural design stage, but were controlled by wind-induced dynamic instability. Wind tunnel tests of the entire bridge aeroelastic model of an ultimate span suspension bridge [28] indicated the necessity of wide central slotting to avoid flutter in a span of 5000 m.

For the lack of suspension bridges with main span longer than 2000 m constructed and more and more constructing demand, it is necessary to design and then research the aerodynamic stability of super long span suspension bridges. This paper presents the study on flutter critical wind speed and aerostatic instability wind speed of suspension bridge with span from 1000 to 5000 m. As there were not scalable application of new structural form and material, the research was based on double tower one span and parallel main cable and traditional material such as concrete and steel. Considering there are few bridges with span exceeding 2000 m around the world, fitting equation of natural frequencies of suspension bridges were found out and then finite element models (FEMs) were adjusted to follow the tendency with span. Four types of commonly used deck sections, including streamlined single box, latticed truss, narrow slotted box and wide slotted box, were introduced. Aerodynamic stabilities of suspension bridges using these sections were compared and analyzed. Evolution tendency of aerostatic instability speeds and flutter speeds and spans were suggested. This study proposed relationships between spans, deck forms and aerodynamic stability of suspension bridges, which aims to promote the wind-resistant design of super long-span suspension bridges.

2 Natural frequencies with increasing spans

Dynamic characteristics including the stiffness, mass, and damping information of the structure, reflect vibration features under dynamic loads, such as self-excited aerodynamic forces. Thus, it is vital to determine the dynamic characteristics of super-long-span suspension bridges in design and research. The first-order lateral bending, vertical bending, and torsion modes of the main girder are the predominant factors responsible for wind-induced instability. Till date, no suspension bridge has been constructed with a main span exceeding 3000 m. Hence, in this study, we design a series of FEMs of suspension bridges to study the tendency of dynamic characteristics with increasing span and then analyze the aerodynamic stability of these bridges. The modes and frequencies of 27 long-span suspension bridges worldwide were collected (Tab.1) to discuss the evolution tendencies of the predominant modes. In Tab.1, the symmetric and asymmetric modes are not distinct. Some references have stated that certain bridges have symmetric fundamental mode shapes while others have opposite. For some bridges, no references were found, and their information was obtained from conferences or research reports.

The attenuation of the frequency with respect to the span presents a muted drift. There are simplified formulae in the Chinese code for wind-resistant design of highway bridges (JTG/T 3360-01-2018) [43] to estimate the frequencies of bridges, but they are only applicable to suspension bridges completed design. Lee et al. [37] fitted the evolution tendency with span by using logarithmic function formulae. However, the domain of the logarithmic function limits the application span-length range. The fitting equation is:

(1)f=a\times \mathrm{l}\mathrm{n}L+b,

where f is the natural frequency (Hz), L is the span (m), and a and b are coefficients. However, a is less than 0, meaning that frequency in equation would be less than 0 when span increasing to a critical value, so the domain limits the application span-length range. Considering its simplicity and accuracy, in this study, the power function was selected to fit the law of evolution of mode frequencies with spans. The fitting function is:

(2)f=a{L}^b.

The least square method in the fitting toolbox of commercial software MATLAB [44] was used to complete fitting and evaluate fitting effect. The fitting curves and determination coefficients are shown in Fig.1, and the colored strips have a 95% confidence interval. The R² of all fitting equations is greater than 0.6, indicating that the power function can accurately reflect the relationship between the natural frequencies and spans of the suspension bridges.

3 Construction of finite element models

A series suspension bridges with main span from 1000 to 5000 m were designed based on highway suspension bridges. Two parallel main cables support the main girder via suspenders. The distance between the two cables was 34 m, the spacing between adjacent suspenders was 20 m, the sag-to-span ratio was 1/11, and the distance from the anchorage to the bridge tower was 0.3 times of the main span. Both bridge towers were portal frames in the longitudinal direction, with a height of 0.13 times of the main span (Fig.2).

Four sections were selected to compare the aerodynamic stability of suspension bridges with increasing spans. We kept the main girder parameters of different forms constant to reduce the interference from features, except aerodynamic configurations. The width of the bridge deck was 36 m, as commonly used in constructed highway suspension bridges. However, the four sections had different width. Therefore, the sections were scaled to the same width, and the sections after scaling are shown in Fig.3. The mass and geometric parameters of the girder were also assigned initial values. An SBS (Fig.3(a)), referring to the Jiangyin Yangtze River Bridge [33] adopts a streamlined design. The lattice truss section (LTS, Fig.3(b)) [22] has high air permeability. The narrow slotted section (NSS, Fig.3(c)) from the Xihoumen Bridge [39] has a slot width of 1/6 of the girder width, and the wide slotted section (WSS, Fig.3(d)) from the Second Wuhu Yangtze River Highway Bridge [45] has a slot-girder ratio, which is twice of NSS. In terms of construction materials, the main towers were made of concrete and the main girder was made of structural steel. The main cables and suspenders were composed of high-strength steel wire bundles. Thus, the areas of the main cables and suspenders were fixed by stress, which was determined to be the maximal stress level of approximately 650 MPa. A description of the parameters of the components constituting the long-span suspension bridge is provided in Tab.2.

The three-dimensional (3-D) models of the suspension bridges (Fig.4) were constructed using the finite element analysis software ANSYS [46], based on a preliminary design. The main girder and towers were simulated using the spatial beam element BEAM 4. For the main cables, and suspenders, the 3-D truss element LINK 10 were used, and the element MASS 21 was employed to model the mass distribution of the main girder. All degrees of freedom (DOFs) of the bottom of the towers and anchorages were fully constrained. Regarding the main girder, the DOFs of rigid motion in vertical, lateral, and torsional motion were coupled with towers to simulate conjunctions.

As the main span increases, the parameters of the main girder must be revised; otherwise, the natural frequencies would not follow the fitting tendency. Research by other scholars [47] shows that parameters of suspension bridge structure such as sag to span ratio would influence frequencies and flutter critical wind speed obviously. While the parameters of main girder, such as dead load, influences frequencies but little impact to flutter critical wind speed. To exclude the influence of factors unrelated to the span and deck configuration on aerodynamic stability, only the area, the moment inertia and the mass of the main girder were adjusted without changing the sag-to-span ratio, space of suspenders, and distance between the main cables. The natural frequencies after adjustment and the adjustment ratios of adjustment of the parameters are shown in Fig.5. The torsional frequencies of bridges with span longer than 2000 m are lower than the fitting line. The 5000 m span suspension bridge on plan [28] has four main cables and a width over 80 m of the deck. Larger width of the deck and more main cables cause greater structural stiffness than suspension bridges with double main cables and 36 m width main girder in this paper, leading to higher frequency in fitting result.

4 Aerodynamic stability analysis

4.1 Flutter stability

Recent years, lots of aerodynamic analysis models of bridges were proposed. An integrated FEM [48] was presented to investigate the nonlinear flutter characteristics of twin-box girder suspension bridges. Furthermore, a nonlinear numerical scheme including a generalized nonlinear aerodynamic force model and a 3-D bridge FEM [49] was developed to investigate nonlinear behaviors of multiple wind effects on a twin-box girder suspension bridge under various wind conditions. The nonlinear models have the capability of predicting critical wind velocities and modeling nonlinear and unsteady characteristics of different types of aerodynamic forces of twin-box girder bridges. However, this paper focus on the flutter critical speed instead of full nonlinear behaviors, and the aerodynamic self-excited forces acting on the deck per meter include 18 flutter derivatives [50] based on Scanlan’s model have the ability to analyze, the expressions are:

\begin{array}{rl}L=\rho U^{2}B& (K{H}_1^{\ast }\frac{\dot{h}}{U}+K{H}_2^{\ast }\frac{B\dot{\alpha }}{U}+K^2{H}_3^{\ast }\alpha \\ & +K^2{H}_4^{\ast }\frac{h}{B}+K{H}_5^{\ast }\frac{\dot{p}}{U}+K^2{H}_6^{\ast }\frac{p}{B}),\end{array}

\begin{array}{rl}D=\rho U^{2}B& (K{P}_1^{\ast }\frac{\dot{p}}{U}+K{P}_2^{\ast }\frac{B\dot{\alpha }}{U}+K^2{P}_3^{\ast }\alpha \\ & +K^2{P}_4^{\ast }\frac{p}{B}+K{P}_5^{\ast }\frac{\dot{h}}{U}+K^2{P}_6^{\ast }\frac{h}{B}),\end{array}

\begin{array}{rl}M=\rho U^2{B}^2& (K{A}_1^{\ast }\frac{\dot{h}}{U}+K{A}_2^{\ast }\frac{B\dot{\alpha }}{U}+K^2{A}_3^{\ast }\alpha \\ & +K^2{A}_4^{\ast }\frac{h}{B}+K{A}_5^{\ast }\frac{\dot{p}}{U}+K^2{A}_6^{\ast }\frac{p}{B}),\end{array}

(3)K=2\text{π}\frac{fB}{U},

where \rho is the air density, U is the wind speed, B is the width of the girder, K is the reduced frequency, H_i^{\ast }, A_i^{\ast }, P_i^{\ast }\left(i=1,2,3,\dots ,6\right) are flutter derivatives, and f is the vibration frequency. L, D, and M are the lift, drag and moment components of the self-excited force, respectively. The flutter derivatives H_i^{\ast } and A_i^{\ast }\left(i=1,2,3,4\right) were obtained from wind tunnel tests of the section models (Figs. A1–A3) in the Electronic Supplementary Materials, \alpha is angle of attack (AoA). Other parameters were calculated based on the quasi-steady theory [51]:

P_1^{\ast }=-\frac{1}{K}{C}_{\mathrm{D}},P_2^{\ast }=\frac{1}{2K}{C}_{\mathrm{D}}^{\prime },P_3^{\ast }=\frac{1}{2K^2}{C}_{\mathrm{D}}^{\prime },

P_5^{\ast }=\frac{1}{2K}{C}_{\mathrm{D}}^{\prime },H_5^{\ast }=\frac{1}{K}{C}_{\mathrm{L}},A_5^{\ast }=-\frac{1}{K}{C}_{\mathrm{M}},

(4)P_4^{\ast }=P_6^{\ast }=H_6^{\ast }=A_6^{\ast }=0,

where C_{\mathrm{D}} is the drag coefficient, C_{\mathrm{D}}^{\prime } is the derivative of drag coefficient, C_{\mathrm{L}} is the lift coefficient, C_{\mathrm{M}} is the moment coefficient. The frequency domain analysis method was used to analyze the flutter stability of suspension bridges. The flutter motion equation of the bridge structure is:

(5)\left[M_{\mathrm{s}}\right]\left\{\ddot{\delta }\right\}+\left[C_{\mathrm{s}}\right]\left\{\dot{\delta }\right\}+\left[K_{\mathrm{s}}\right]\left\{\delta \right\}=\left\{F_{\mathrm{a}}\right\},

where [M_s] is the structural mass matrix, [C_s] is the structural damping matrix, [K_s] is the structural stiffness matrix, which includes the elastic stiffness matrix [K_e] and geometric stiffness matrix [K_g], \left\{\delta \right\} is the displacement vector, and \left\{F_{\mathrm{a}}\right\} is the aerodynamic self-excited force matrix.

From Eq. (5), the aerodynamic self-excited force can be expressed as

(6)\left\{F_{\mathrm{a}}\right\}=\left[F_{\mathrm{d}}\right]+\left[F_{\mathrm{s}}\right]=\left[A_{\mathrm{d}}\right]\left\{\dot{\delta }\right\}+\left[A_{\mathrm{s}}\right]\left\{\delta \right\},

where \left[F_{\mathrm{d}}\right] and \left[F_{\mathrm{s}}\right] are aerodynamic damping force and aerodynamic stiffness force matrices, and [A_d] and [A_s] are flutter derivative matrices.

The structural motion equation obtained by substituting Eq. (6) into Eq. (5) is:

(7)\left[M_{\mathrm{c}}\right]\left\{\ddot{\delta }\right\}+\left[C_{\mathrm{c}}\right]\left\{\dot{\delta }\right\}+\left[K_{\mathrm{c}}\right]\left\{\delta \right\}=0,

where \left[M_{\mathrm{c}}\right]=\left[M_{\mathrm{s}}\right] is the fluid−structure coupling system mass matrix, \left[K_{\mathrm{c}}\right]=\left[K_{\mathrm{s}}\right]-\left[A_{\mathrm{s}}\right] is the fluid−structure coupling system stiffness matrix, and \left[C_{\mathrm{c}}\right]=\left[C_{\mathrm{s}}\right]-\left[A_{\mathrm{d}}\right] is the fluid−structure coupling system damping matrix.

A 3-D flutter analysis of the full-mode approach [14] was employed to solve the critical wind speed of flutter of suspension bridges with 0° and ±3° AoA in finite element analysis software ANSYS [52,53]. Aerodynamic self-excited forces were loaded on FEMs in the form of element aeroelastic stiffness and damping matrices realized by the MATRIX 27 element, and the damping ratio was assumed to be 0.003, according to the Wind-resistant Design Specification (JTG/T 3360-01-2018). Recent research [54] shows that additional AoA could effect the flutter critical wind speed of super long-span bridges in some cases, so a aerostatic solution was operated before flutter analysis to get the additional AoA on main girder and then the flutter analysis considering additional AoA was promoted.

The critical wind speeds of flutter of suspension bridges continuously decrease with increase in span (Fig.6). Therefore, there is no minimum instability speed. Considering the characteristics of the gradual decline rate, the relationship between the critical wind speed and span of the flutter was fitted by a power function with two parameters:

(8)U_{\mathrm{f}}=a{\left(\frac{L}{1000}\right)}^b,

where U_{\mathrm{f}} is the critical wind speed of flutter (m/s), L is the span (m), and a and b are fitting parameters. The parameters are listed in Tab.3.

The critical wind speed of flutter of the WSS is the highest at 1000 m span, and that of the NSS is slightly lower. The critical wind speeds of flutter of LTS and SBS are significantly lower than those of the slotted sections. Although the drop rate of the critical wind speed of flutter of WSS is higher than that of NSS and is finally lower than that of NSS, it maintains an obvious advantage over LTS and SBS with increasing span. This shows that central slotting significantly improves aerodynamic stability, and it is difficult to adapt the streamlined SBS to super long-span suspension bridges without more advanced aerodynamic optimization methods. However, the wind speed of flutter of the slotted suspension bridge still decreases rapidly to below 70.9 m/s for a span of over 2000 m, which is a 10-min average wind speed in typhoon measurement [55]. This shows that without additional aerodynamic control measures, flutter can be an extraordinarily serious problem for super long-span suspension bridges. The decline of flutter critical wind speed showed an unlimited approaching phenomenon, which means that the flutter critical wind speed would always decrease with spans increasing but it would never reach or below than 0 m/s. It indicates that flutter problem can be more serious with longer span.

4.2 Aerostatic stability

The wind loads are decomposed into drag force, lift force, and lift moment (Fig.7). The three components of the wind forces acting on the deformed deck per meter can be expressed in the wind axes as:

D\left(\alpha \right)=\frac{1}{2}\rho U^2{C}_{\mathrm{D}}\left(\alpha \right)B,

L\left(\alpha \right)=\frac{1}{2}\rho U^2{C}_{\mathrm{L}}\left(\alpha \right)B,

(9)M\left(\alpha \right)=\frac{1}{2}\rho U^2{C}_{\mathrm{M}}\left(\alpha \right)B^2,

where D\left(\alpha \right), L\left(\alpha \right), and M\left(\alpha \right) are drag force, lift fore, and moment force of wind loads, respectively.

A 3-D nonlinear aerostatic stability analysis of a long-span suspension bridge was performed. The equilibrium equation of the structural system under wind load can be expressed as:

(10)\left[K_{\mathrm{e}}\right(y)+K_{\mathrm{g}}^{G+W}(y\left)\right]Y=F\left[D\right(\alpha ),L(\alpha ),M(\alpha \left)\right],

where K_{\mathrm{e}} is the elastic stiffness matrix of the structure, K_{\mathrm{g}} is the geometric stiffness matrix, \alpha is the effective AoA, G and W are the gravity and wind load, respectively, y is the structural displacement, Y is the structural displacement matrix, F is the wind load matrix, and D\left(\alpha \right),L\left(\alpha \right),\mathrm{a}\mathrm{n}\mathrm{d}M\left(\alpha \right) are the drag force, lift force, and lift moment of wind, respectively. In this study, the coefficients of the drag force, lift force, and lift moment are provided in the Supplementary Materials based on the width. These were obtained from wind tunnel tests of the section model at Tongji University, and are shown in the Electronic Supplementary Materials (Fig. A4).

According to Eq. (10), the structural stiffness and wind loads are functions of structural deformations, which can be calculated using an incremental-two-iterative method [19]. Using this method, a program was compiled to analyze a long-span suspension bridge with a main span of 1000–5000 m and solve the aerostatic instability wind speeds. A previous study [56] showed that the wind load on the main cables influence the girder displacements with increasing wind speed but had little impact on the instability critical wind speed. The suspension load did not influence the wind speed of instability. Therefore, to search for the instability critical wind speed, only the wind loads on the girder were considered in this analysis.

The aerostatic instability wind speed of suspension bridges at 0° and ±3° AoA have the same trend with span (Fig.8). The aerostatic instability wind speed first decreases and then increases with increasing span. In view of this, a quadratic function was used to fit the evolution curve with span increase at three AoAs:

(11)U_{\mathrm{s}}=a{\left(\frac{L}{1000}-b\right)}^2+c,

where U_s is the aerostatic instability critical wind speed (m/s), L is the span (m), and a, b, and c are the fitting parameters to be determined. The parameters are listed in Tab.4. In Eq. (11), a indicates the severity of the evolution of the instability wind speed with span; b × 1000 is the span for the minimum aerostatic instability critical wind speed (m), and c is the minimum instability speed (m/s).

Suspension bridges with different sections exhibit distinct aerostatic instability wind speeds. For 1000 m span, both NSS and WSS have significantly higher instability critical wind speeds than other sections, and the aerostatic stability of SBS is slightly better than that of LTS. When the span increases, the aerostatic stability of the NSS has a faster deterioration rate than the other three sections, but is still significantly higher than that of the SBS and LTS. With continuous increase in span, the aerostatic stability of all sections begins to improve. Finally, at a span of 5000 m, the WSS suspension bridge has the highest aerostatic instability wind speed, followed by the NSS, LTS, and SBS. In addition, the AoA also has a significant effect. The aerostatic instability wind speeds of WSS suspension bridges at different angles of attack is about 50m/s different when span is 1000 m, and the influence from angles of attack almost disappears with the increase of span. However, this influence of SBS and NSS suspension bridges always maintains. The instability wind speed at 0° AoA of NSS bridge is much higher than that at −3° AoA, which is the minimum instability wind speed. For the SBS suspension bridge, the highest instability wind speed appears at 3° or −3° AoA as the span increase. The influence from AoA shows that not only do different sections have different worst angles of attack, but also the same section has different worst angles of attack at different spans.

It is recognized that aerostatic wind instability is not the controlling parameter for span increase in suspension bridges, as few studies observed aerostatic divergence in suspension bridges. However, the minimum aerostatic instability wind speeds of the SBS and LTS are only marginally higher than 70.9 m/s at 3500 m span. Field measurements of typhoons show that 10 min average wind speed [55] can exceed 70.9 m/s, with a maximum gust wind speed [57] of 83.3 m/s. As shown in Fig.8, LTS suspension bridges have instability speeds lower than the maximum gust wind speed in typhoon for spans between of 2000 and 4500 m, and at the same speed, SBS bridge span from 2000 to 5000 m. Central slotted decks presents excellent aerostatic stability. Although WSS and NSS suspension bridges still have lowest aerostatic instability critical wind speeds at span of 3000 m, the minimum speeds are higher than the gust wind speed in measurements. In a word, a double tower suspension bridge might occur aerostatic divergence under extreme circumstances with a span approximately 3000 m, especially LTS or SBS main girder.

The aerostatic instability wind speed of super long-span suspension bridges decreases to a minimum value at a span of approximately 3500 m and then increases with span. To clarify the cause of this phenomenon, an parametric analysis was conducted. The case of 0° AoA of the NSS suspension bridge was selected as an example. The torsional moment inertia of the main girder increases from 6.2 to 7.7 m⁴ with span from 1000 to 3000 m to adapt to the torsional mode frequency evolution. Owing to span length extension but no substantial change in the main cable stress, the area of each main cable is amplified from 0.3 to 6.2 m² rapidly (Fig.9(a)). If the torsional moment inertia of the main girder remains constant in the range of 1000–5000 m, the instability speed is lower than that of the initial condition. Although the area of the main cable remains constant, the unstable wind speed of the suspension bridge shows a continuous decreasing tendency (Fig.9(b)). This implies that the stiffness strengthening of the main cable can improve the aerostatic stability of the suspension bridge, even though the torsional natural frequency is still attenuated.

4.3 Comparison and summary

A comparison of flutter and aerostatic instability with the span and sections was performed (Fig.10). The aerostatic instability wind speed of each section is higher than the flutter speed with span from 1000 to 5000 m, indicating that flutter is more likely to occur on super long-span double-tower suspension bridges, which have a normal deck width and parallel double main cables. Flutter stability of the suspension bridge decreases continuously, leading to flutter appearing more likely at longer spans. The worst aerostatic stability of suspension bridge appears at a span length of approximately 3000 m. For SBS and LTS, the critical speeds are lower than a commonly seen gust wind speed in typhoon process, indicating that the aerostatic stability of SBS or LTS suspension bridges at typhoon-affected regions needs to be improved.

The analysis before determines that the evolution process of aerodynamic instability wind speed and span does not change for different section. However, the fitting equations in Subsections 3.1 and 3.2 required 8 groups of 20 fitting parameters to describe the evolution of flutter and aerostatic instability wind speed of four sections with span. It can be summarized to propose a new fitting equation, excluding section information, and the coefficients of the section were introduced to reflect the differences in the girder forms, which the simplified fitting would help to use the results of analysis. To retain the original data characteristics, all the data were included in the fitting. The evolution equations of the flutter and aerostatic instability critical wind speed with span are:

(12)U_{\mathrm{f}}=C_{\mathrm{f}}\times 107.0{\left(\frac{L}{1000}\right)}^{-0.85},

(13)U_{\mathrm{s}}=C_{\mathrm{s}}\left[6.36{\left(\frac{L}{1000}-3.34\right)}^2+95.65\right],

where U_{\mathrm{f}} and U_{\mathrm{s}} are the flutter and aerostatic instability critical wind speed, respectively; C_{\mathrm{f}} and C_{\mathrm{s}} are the coefficients of the section for flutter and aerostatic instability, respectively. The coefficients for different sections are listed in Tab.5.

It shows that the deviations of the wind speeds between fitting curves and FEMs are lower than 10% (Fig.10), and the turning point of the aerostatic instability wind speed fitting curve is 3340 m. According to the coefficients, the slotted sections exhibit better aerodynamic stability. SBS performs the worst in terms of flutter stability and aerostatic stability. The flutter stability of LTS is good, but its aerostatic stability is relatively poor. NSS has excellent flutter stability and slightly weaker aerostatic stability, and WSS has the best aerodynamic stability. The significant diversity of sections with respect to aerodynamic stability indicates that aerodynamic configuration optimization for the main girder section can be very useful in wind-resistant design of long-span suspension bridge.

The evolution tendency of the aerostatic instability wind speed and the span indicates that the suspension bridge with longer span has better aerostatic stability after the span over the turning point approximately 3340 m, but flutter wind speed of the suspension bridge is always decreasing. It is necessary to take some aerodynamic countermeasures such as Upward vertical central stabilizers (UVCSs) [58] for single-box girder and downward vertical central stabilizers (DVCSs) [59] for slotted box girder in order to improve the aerodynamic behavior, especially after considering the special wind field of tropical cyclone [60] and extreme events [61].

5 Conclusions

Based on the structural mode frequency developing tendency with suspension bridge main spans from 1000 to 5000 m, 3-D FEMs of long-span suspension bridges with increasing span were constructed. Full-mode flutter and 3-D nonlinear aerostatic stability analyses were conducted for four types of sections: SBS, LTS, NSS, and WSS. The conclusions are summarized as follows.

1) Based on highway suspension bridges using normal material and structural form, the natural frequencies of first-order lateral bending, vertical bending, and torsion, which are the predominant factors affecting wind-induced instability in suspension bridges, are proposed to be fitted by a power exponential expression with the span from 1000 to 5000 m.

2) For the suspension bridge constructed in this study, flutter critical wind speed decreases continuously with increasing span, showing an unlimited approaching phenomenon. Aerostatic instability critical wind speed first decreases and then increases with span, and the flutter wind speed of the suspension bridge is always lower than the aerostatic instability wind speed at the same span. Flutter would be a more important concern than aerostatic instability in wind resistant design of the super long-span suspension bridge.

3) The aerostatic stability of the suspension bridge presents a strengthen phenomenon. Parametric analysis of suspension bridges shows that the main cable area is a major factor that influences aerostatic stability. The main cable area increases rapidly with span, as the mass of the cable itself and main girder is considerably heavier at super long spans, causing the strengthen of aerostatic stability with increasing span, showing a turning point of span between 3000 and 3500 m. The phenomenon indicates that the aerostatic instability is not a severe problem of the suspension bridge with a span longer than turning point.

4) The aerodynamic stabilities of suspension bridges with four types of sections were fitted by equations combined with coefficients of section to summarize the evolution with spans and reflect the differences between sections. SBS and LTS suspension bridges with spans near the turning point of aerostatic instability, approximately 3340 m, have instability wind speeds of aerostatic lower than a commonly seen gust wind speed in typhoon, and all four types of sections have flutter wind speeds lower than a super typhoon wind speed with span over the same span. It shows that more advanced aerodynamic configuration optimization or newer bridge structural form should be considered to super long span suspension bridges at typhoon affected regions

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