Probabilistic Assessment Approach of the Aerostatic Instability of Long-Span Symmetry Cable-Stayed Bridges

Abstract

The existing safety analysis methods for the assessment of the aerostatic stability of long-span symmetry cable-stayed bridges have difficulties in meeting the requirements of engineering applications. Based on the finite element method and the inverse reliability theory, an approach for the probabilistic safety analysis of the aerostatic instability of long-span symmetry cable-stayed bridges is proposed here. The probabilistic safety factor of aerostatic instability of long-span symmetry cable-stayed bridges was estimated using the proposed method, with Sutong Bridge as an example. The probabilistic safety factors for the aerostatic instability of Sutong Bridge were calculated using the finite element inverse reliability method, based on the FORM approach. The influences of the mean value and the coefficient of variation of random variables, as well as the iterative step length of finite difference, on the probabilistic safety factors of aerostatic instability of Sutong Bridge were analyzed. The results indicated that it is necessary to consider the uncertainties of random variables in probabilistic safety factor assessments of aerostatic instability in cable-stayed bridges using the proposed method, which could be recommended for the assessment of safety factors involved in the aerostatic instability of long-span symmetry cable-stayed bridges. The randomness of the parameters had an important influence on the probabilistic safety factor of the aerostatic stability of Sutong Bridge. Neglecting the randomness of these parameters may result in instability of the structure.

1. Introduction

As long-span civil engineering structures on large rivers, cable-stayed bridges are made of a variety of structural materials and various structural forms, and so show strong competitiveness and applicability [1,2,3,4,5]. However, there are many types of cable-stayed bridges, such as single-tower cable-stayed bridges, single-tower cable-stayed bridges without backstays, double-tower cable-stayed bridges, and long-span symmetrical structure cable-stayed bridges [5,6,7]. The long-span cable-stayed bridge adopts a symmetrical structure, which has many advantages, such as the better material usage of the symmetrical structure, the more reasonable force performance of the symmetrical structure, the lower construction risk of the symmetrical structure, and the cheaper cost of the symmetrical structure [8,9,10].

Cable-stayed bridges have become the preferred structural form of long-span bridges worldwide for their structural capacity and aesthetics. In recent years, China’s long-span cable-stayed bridges have developed rapidly. Among the top ten cable-stayed bridges in the world, seven of them are in China. For example, the Changtai Yangtze River Bridge, currently under construction with a main span of 1176 m, has the largest span in the world. With increases in the cable-stayed bridge span, the stiffness of the structure decreases sharply, so that the structure is more prone to wind-induced vibration. In various wind vibration forms of long-span symmetry cable-stayed bridges, aerostatic wind instability is a great threat to the safety of cable-stayed bridges, which may cause the collapse of the whole bridge. Due to the uncertainty of the wind environment and the cable-stayed bridge structure, the aerostatic wind instability of long-span symmetry cable-stayed bridges becomes a random event.

Existing studies on aerostatic wind stability focus on deterministic analyses, neglecting the randomness of structural parameters and load parameters. Therefore, some scholars have carried out exploratory research on the aerostatic wind stability safety of long-span cable-bearing bridges with consideration to parameter uncertainties. Liu et al. [11] combined the finite element method and the Monte Carlo method to analyze the uncertainty of structural responses under aerostatic wind load, but did not study the reliability of structural stability under aerostatic wind. Based on the simplified limit state function, Ge [12] used the first-order second-moment method to study the reliability of aerostatic wind stability, and the statistical characteristics of random variables related to aerostatic wind instability wind speed were assumed. Luo [13] and Su et al. [14,15] used the Monte Carlo method to simulate the critical wind speed of aerostatic wind instability based on an empirical formula and analyzed its statistical characteristics, and then studied the reliability of bridge aerostatic wind stability on this basis. Since the critical wind speed of aerostatic wind instability adopts an empirical formula, the accuracy of the reliability analysis is not high. Cheng et al. [16] combined the series method and the Monte Carlo method to consider the randomness of parameters and conducted a probabilistic study on the critical wind speed of aerostatic wind instability of suspension bridges. Meanwhile, a polynomial response surface was used to apply the critical wind speed of aerostatic wind instability to a parameter sensitivity analysis. This method is an approximate method, so the accuracy of the probability analysis of aerostatic wind instability critical wind speed is limited. Cheng and Li [17,18,19,20] combined the series method with the Monte Carlo method to study the reliability of the aerostatic wind stability of bridges. Since the series method is an approximate method for solving the critical wind speed of aerostatic wind instability, the calculation accuracy is also limited in the reliability calculation of aerostatic wind instability.

At present, bridge design codes have been developed to ensure the safety of structures by defining the target reliability of structures. In other words, the reliability level of the structure is given in advance. At present, the inverse reliability analysis method has been widely studied and applied in the civil engineering field, including the bending safety factor of cantilever beams [21], the safety factor of the main cables of suspension bridges [22,23], the safety factor of the arch ribs of concrete-filled steel tubular arch bridges [24,25], the overall overturning stability safety factor of the cantilever construction of continuous beam bridges [26,27] and the stability safety factor of rigid frame bridges [28]. However, research into and applications of the safety factor evaluation method, considering randomness in the static wind stability safety evaluation, of long-span symmetrical cable-stayed bridges have not been reported.

In the probability analysis of the aerostatic wind stability of long-span symmetry cable-stayed bridges, the limit state equation of the structures is highly nonlinear, which is usually an implicit function of random variables. High requirements are put forward for probability calculation methods. Among the existing probability analysis methods for the aerostatic wind stability safety of long-span cable-stayed bridges, the FORM method—based on a simplified formula—has a low calculation accuracy, and the random finite element method and Monte Carlo method are complex and inefficient in modeling, which makes it difficult to meet the requirements of engineering applications. Therefore, how to establish a set of accurate and convenient calculations to meet the requirements of engineering applications of the aerostatic wind stability safety probability analysis method is worth studying. Therefore, it is necessary to correct the pre-determined safety factor of long-span cable-stayed bridges to ensure the reliability of the structural aerostatic wind stability. Hence, a probabilistic evaluation method of aerostatic wind stability safety of long-span cable-stayed bridges is critical.

2. Probabilistic Evaluation Model of Aerostatic Wind Stability Safety

2.1. Selection of Random Variables

There are a lot of uncertainties in the probabilistic analysis of aerostatic wind stability of long-span symmetry cable-stayed bridges, and these uncertainties include geometric parameters, material parameters, aerodynamic parameters, and load parameters. In order to simplify the analysis, the influence of the three components of the displacement-dependent wind loads and design wind speed is the main consideration in the probabilistic analysis of aerostatic wind stability of long-span symmetry cable-stayed bridges.

2.2. Establishment of the Limit State Equation

The structural limit state is the structural resistance minus the structural effect. This is a general expression, which is suitable for the static wind instability modes of all symmetrical cable-stayed bridges. There are also other parameters affecting the results, such as mass, stiffness, etc. However, in these parameters, the randomness of the critical wind speed, the aerostatic force, the conversion coefficient, and the gust factor are the most important parameters in the aero-instability of long span cable-stayed bridges, according to reference [29]. The evaluation expression of the safety factor of the aerostatic wind stability of long-span symmetry cable-stayed bridges can be expressed as:

$$g=C_w\cdot V_{\mathit{cr}}-K\cdot G_s\cdot U_{\mathrm{b}}$$

(1)

where, K is the safety factor, and $V_{\mathit{cr}}$ is the critical wind speed of aerostatic wind instability that is included in the uncertain factors of the structural characteristics, which can be determined by the calculation of the three-dimensional nonlinear finite element. $C_w$ is the critical wind speed conversion coefficient, which is included in the uncertain factors of wind field characteristics; $G_s$ is the gust factor, which considers the influence of maximum fluctuations in wind; $U_{\mathrm{b}}$ is the 10 min time-interval average reference wind speed at the bridge deck height of the bridge site, which can be calculated according to existing wind speed records.

For the sake of simplification, the calculation of the critical wind speed of aerostatic wind instability in this paper only considered the influence of the three components of the displacement-dependent wind loads. The critical wind speed of aerostatic wind instability is an implicit function of the three components of the wind loads, which can be expressed as:

$$V_{cr}=V_{cr}\left(C_h,C_V,C_M\right)$$

(2)

2.3. Selection of Random Variables

The resistance, lift and lift moment effects of cable-stayed bridges under wind loads and torsional divergence are a form of symmetrical structural divergence, which is the most common failure mode of symmetrical long-span cable-stayed bridges. The symmetric reliability back analysis theory was used to calculate the static wind stability of a symmetric long-span cable-stayed bridge structure to calculate the probabilistic safety performance of the structure in symmetric torsional instability mode.

The three-dimensional finite element structural analysis method with nonlinear influence (including geometric nonlinearity, material nonlinearity, aerostatic nonlinearity) was adopted, and the vertical, lateral and torsional displacements of the stiffening beam under the combined action of dead load and aerostatic wind load were calculated throughout the whole process, so as to determine the critical wind speed of the structure under aerostatic wind instability.

The aerostatic wind load acting on the unit length of a stiffening beam of the cable-stayed bridge can be decomposed into transverse wind load $P_H$, vertical wind load $P_V$ and torsional moment $M$. Under the action of aerostatic wind, the attitude of the stiffening beam will change, and the relative wind angle of attack between the aerostatic wind load and the section of the stiffening beam will change. The concept of effective wind angle of attack is introduced, and the aerostatic wind load is expressed as a function of wind speed, the three components of the displacement-dependent wind loads, and the effective wind angle of attack:

$$\begin{array}{c}{P}_H=\left(\frac{1}{2}\rho U^2\right)C_H\left(\alpha \right)D\\ P_V=\left(\frac{1}{2}\rho U^2\right)C_V\left(\alpha \right)B\\ M=\left(\frac{1}{2}\rho U^2\right)C_M\left(\alpha \right)B^2\end{array}\}$$

(3)

where $\rho$ is the air density; $D$ is the lateral projection height of the stiffening beam; $B$ is the width of the stiffening beam; $U$ is the average wind speed; $C_H$ is the drag coefficient; $C_V$ is the lift coefficient; $C_M$ is the pitch moment coefficient; and $\alpha$ is the effective wind angle of attack (the sum of the initial angle of the aerostatic wind and the torsional displacement of the section of the stiffening beam).

By analyzing the whole process of wind speed loading, the accurate aerostatic instability wind speed of the structure can be obtained, considering the double nonlinear factors of geometry, materials and aerostatic wind load. The aerostatic wind stability of long-span cable-stayed bridges can still be solved mechanically according to the stability theory of the second class of truss structures, which can be ultimately reduced to solve the following incremental equilibrium equations in the UL series:

$$\left\{\left[{}_{}{}^{\mathit{T}}\mathit{K}{}_{\mathbf{0}}\right]+\left[{}_{}{}^{\mathit{T}}\mathit{K}{}_{\mathsf{\alpha }}\right]\right\}•\left\{\Delta \delta \right\}=\left\{{}_{}{}^{\mathit{T}+\mathit{\Delta }\mathit{T}}\mathit{F}\mathbf{\left(}\mathit{\alpha }\mathbf{\right)}\right\}-\left\{{}_{}{}^{\mathit{T}}\mathit{R}\right\}$$

(4)

where $\left[{}_{}{}^T\mathit{K}{}_{\mathbf{0}}\right]$ is the elastic stiffness matrix of the structure at time $T$; $\left[{}_{}{}^{\mathit{T}}\mathit{K}{}_{\mathit{\alpha }}\right]$ is the geometric stiffness matrix caused by the combined action of dead load and wind load at time $T$; ${}_{}{}^{\mathit{T}\mathbf{+}\mathbf{\Delta }\mathit{T}}\mathit{F}\mathbf{\left(}\mathit{\alpha }\mathbf{\right)}$ is the wind load when the corresponding effective wind angle of attack is $\alpha$; and ${}_{}{}^T\mathit{R}$ is the equivalent nodal force of the structural internal force at time $T$.

The whole process of the analysis of aerostatic wind stability is as follows:

(1)

Determine the initial state of the cable-stayed bridge structure under a dead load, including its geometric shape, the dead load internal force, and the pretension forces in all the cables.

(2)

Set the initial wind speed V₀.

(3)

Calculate the aerostatic wind load $\left\{\mathit{F}\mathbf{\left(}\mathsf{\alpha }\mathbf{\right)}\right\}$ acting on the bridge structure.

(4)

The Newton–Raphson iterative method is used to solve Equation (4), and the current structure displacement is obtained.

(5)

The torsion displacement of the stiffened beam after deformation is determined by the structural displacement, and the corresponding aerostatic wind load is recalculated.

(6)

Check whether the Euclidean norm of the three components of the displacement-dependent wind loads on each node is less than the allowed value, as shown in the following Formula (5):

$${\left\{\frac{{{\displaystyle \sum }}_{j=1}^{N_a}{\left[C_k\left({\alpha }_j\right)-C_k\left({\alpha }_{j-1}\right)\right]}^2}{{{\displaystyle \sum }}_{j=1}^{N_a}{\left[C_k\left({\alpha }_{j-1}\right)\right]}^2}\right\}}^{1/2}\le {\epsilon }_k\left(k=D,L,M\right)$$

(5)

where $N_a$ is the total number of nodes subjected to aerostatic wind load; $C_k\left(k=D,L,M\right)$ is the coefficient of drag force, lift force and pitch moment; and ${\epsilon }_k\left(k=D,L,M\right)$ is the allowable error of drag force, lift force and lift moment coefficients—generally ranging from 0.0005 to 0.005, which needs to be determined based on experience.

(7)

Increase the wind speed according to the predetermined step length if Formula (5) meets the requirements, and then repeat steps (3)–(5); Otherwise, repeat steps (4) and (5) for a new round of iterative calculation.

(8)

When the structure is close to instability, the incremental step of wind speed is appropriately reduced until the structure is unstable.

(9)

Output results.

3. Probabilistic Evaluation Method for Aerostatic Wind Stability Safety

3.1. Inverse Reliability Method Based on the Finite Element Method

In the process of reliability back-analysis and calculation, the calculation results of reliability indicators need to be used. The reliability indicators are actually in a symmetrical standard normal distribution space, and a symmetrically distributed normal function is used to describe the geometric meaning of the reliability indicators. Secondly, the calculation results of the reliability positive analysis are needed in the calculation process of the reliability back analysis. This kind of algorithm is essentially a symmetric reliability algorithm.

Kiureghian [8] and Zhang et al. [9] defined the structural inverse reliability analysis problem as follows:

$$\Vert \mathit{u}\Vert -{\beta }_T=0$$

(6)

$$\mathit{u}+\frac{\Vert \mathit{u}\Vert }{\Vert {\mathbf{\nabla }}_{\mathit{u}}G\left(\mathit{u},\theta \right)\Vert }{\mathbf{\nabla }}_{\mathit{u}}G\left(\mathit{u},\theta \right)=0$$

(7)

$$G\left(\mathit{u},\theta \right)=0$$

(8)

where $\mathit{u}$ is the vector of the standard normal distribution space; ${\beta }_T$ is the reliability index of the structural target; $\mathit{G}\left(\mathit{u},\theta \right)$ is the functional function of the structure; ${\mathbf{\nabla }}_{\mathit{u}}$ is the gradient operator; and $\theta$ is the design parameters to be determined.

This paper proposes a direct algorithm to solve these parameters; the basic idea of the algorithm is: solve the mean of $\overline{\theta }$ (or the mean value of $\theta$) according to ${\beta }_T$, which makes the parameters satisfy Equations (8) and (9).

$$\mathit{min}({\mathit{u}}^{\mathit{T}}\mathit{u})={\beta }_T^2$$

(9)

$$G=G\left(\mathit{u},\theta \right)=0$$

(10)

According to the theory of primary reliability analysis, the parameter vector satisfies the following formula at the design point:

$$\mathit{u}=[\frac{{({\mathbf{\nabla }}_{\mathit{u}}G)}^T\mathit{u}}{{({\mathbf{\nabla }}_{\mathit{u}}G)}^T{\mathbf{\nabla }}_{\mathit{u}}G}]{\mathbf{\nabla }}_{\mathit{u}}G$$

(11)

Thus, the target reliability index ${\beta }_T$ of the structure can be obtained as:

$${\beta }_T=\frac{-{({\mathbf{\nabla }}_{\mathit{u}}G)}^T\mathit{u}}{[{\left({\mathbf{\nabla }}_{\mathit{u}}G{)}^T{\mathbf{\nabla }}_{\mathit{u}}G\right]}^{1/2}}$$

(12)

Perform Taylor expansion at ${\beta }^j$ for the target reliability index ${\beta }_T$, and obtain:

$${\beta }_T={\beta }^j+{\frac{\partial \beta }{\partial K}|}_{K^j}\left(K^{j+1}-K^j\right)$$

(13)

where $K^{j+1}$ and $K^j$ are the (j + 1)-th and j-th iteration values of the parameters, respectively, and ${\beta }^j$ is the reliability index calculated at the j-th time.

According to Equation (13), the iteration formula of the parameter is:

$$K^{j+1}=K^j+\frac{{\beta }_T-{\beta }^j}{{\frac{\partial \beta }{\partial K}|}_{K^j}}$$

(14)

Formula (15) is used as the convergence criterion of the reliability inverse reliability analysis method adopted in this paper:

$$\left|K^{j+1}-K^j\right|\le \epsilon$$

(15)

In the formula, $\epsilon$ is a small number, which can be taken as 0.0001 in specific calculations.

3.2. Probabilistic Analysis Process of Aerostatic Wind Stability Safety

In this paper, the analysis method of aerostatic wind stability of long-span cable-stayed bridges based on FORM is realized by combining finite element analysis software ANSYS and inverse reliability analysis software MATLAB. The detailed analysis steps of the aerostatic wind stability safety of long-span cable-stayed bridges based on FORM are as follows:

Step 1: Set the initial value of the random variable. In general, the mean value of the random variable can be selected as the initial iteration value.

Step 2: The limit state function is established and the initial iteration value is taken as the search point for the normal quantization probability transformation.

Step 3: Input the random variable value into ANSYS, and calculate the critical wind speed of aerostatic wind stability with the deterministic method.

Step 4: Input the critical wind speed of aerostatic wind stability calculated in step 3 into MATLAB to calculate the limit state function value.

Step 5: The difference method is used to analyze the sensitivity of the critical wind speed in ANSYS and to calculate the gradient value of the limit state function.

Step 6: Calculate the probability safety factor.

Step 7: If convergence is achieved, end the calculation; Otherwise, go to step 2 to continue the calculation.

The process above is shown in Figure 1. From the computational process of the probabilistic analysis of aerostatic wind stability safety, we can see that the inverse reliability analysis needs several cycles of forward reliability analysis, and that one cycle of forward reliability analysis needs several iterations of the finite element analysis of structures. However, one iteration of the finite element analysis of structures took about 1 min, so one cycle of forward reliability analysis took about 10 min, and one calculation of inverse reliability took about 1.5 h. The whole process is a multi-nested iterative calculation process [30].

4. Case Analysis

4.1. Project Overview

Sutong Yangtze River Highway Bridge is China’s first thousand-meter-scale cable-stayed bridge, with a span of 1088 m. The flat streamline steel box girder of the main bridge is 41 m wide and 4 m high. The bridge adopts an inverted Y-shaped cable tower, which is 300.4 m high. The length of the longest cable is 577 m. The overall layout of the main bridge is shown in Figure 2, and the cross section of the stiffened beam is shown in Figure 3.

4.2. Calculation of Critical Wind Speed for Aerostatic Wind Stability

A three-dimensional nonlinear finite element model was used to analyze the structural aerostatic wind stability of Sutong Bridge. In the three-dimensional nonlinear aerostatic wind stability numerical analysis, wind angles of attack of −3°, 0° and +3° were considered. The dead load of the structure was taken as the initial state, and the wind speed was increased step by step. The vertical, lateral and torsional displacements of the bridge structure under the combined actions of aerostatic wind load and dead load were calculated. Through calculation and analysis, it could be concluded that the critical wind speed of the aerostatic wind instability of Sutong Bridge is 160 m/s, 177 m/s and 161 m/s, respectively at −3°, 0° and +3°. In order to facilitate subsequent analyses, 160 m/s was taken as the critical wind speed of the aerostatic wind instability of Sutong Bridge.

4.3. Safety Factor Evaluation Equation

The wind stability safety factor evaluation expression of Sutong Bridge aerostatic wind load is:

$$g=C_w\cdot V_{cr}\left(C_h,C_V,C_M\right)-K\cdot G_s\cdot V_b$$

(16)

The statistical characteristics of each random variable in Formula (14) are shown in

Table 1. Statistical characteristics of random variables.

Random Variables	Distribution Type	Mean	Standard Deviation	Coefficient of Variation	Sources
$C_h$	Normal	1	0.1	0.1	[4,5]
$C_V$	Normal	1	0.1	0.1	[4,5]
$C_M$	Normal	1	0.1	0.1	[4,5]
$C_w$	Normal	1	0.1	0.1	[2]
$G_s$	Normal	1.18	0.12	0.1	[2]
$V_b$	Extreme type I	20.26	4.05	0.2	[11,12]

.

4.4. Determination of the Target Reliability Index

The reliability index for the structural objectives specified in the OHBDC (Highway Bridge Design Code of Ontario) code and AASHTO (American Association of State Highway and Transportation Workers) code is 3.5. Therefore, the structural target reliability index was set as 3.5, unless otherwise noted.

4.5. Probability Evaluation of Aerostatic Wind Stability Safety

When the target reliability index of Sutong Bridge is 3.5, the safety factor of the aerostatic wind stability of Sutong Bridge calculated by the safety factor evaluation method based on the inverse reliability analysis proposed in this paper is 2.17. The safety factor of Sutong Bridge’s aerostatic wind stability calculated based on the deterministic model is shown in Equation (17). The inverse analysis method and the reliability of the aerostatic wind stability safety factor, which was calculated based on the deterministic model comparison analysis, were used. The numerical result calculated through the inverse analysis method was smaller than that calculated through the deterministic model. Thus, the parameter uncertainty has a great influence on the aerostatic wind stability safety factor. That is, if the parameter uncertainty is ignored in the calculation, the aerostatic wind stability safety factor of long-span cable-stayed bridges will be overestimated, which may lead to structural risks.

$$K=\frac{E\left[V_{cr}\left]\cdot E\right[C_w\right]}{E\left[G_s\right]\cdot U_b^{100}}=\frac{160\times 1}{1.18\times 38.9}=3.49$$

(17)

4.6. Parameter Sensitivity Analysis

The main factors affecting the aerostatic wind stability safety factor of long-span cable-stayed bridges are: (1) the target reliability index; (2) the mean value of random variables; (3) the coefficient of variation (or standard deviation) of random variables; (4) the step size of differential calculations; (5) the initial iteration value of the safety factor.

4.6.1. The Influence of the Target Reliability Index

The assessment of the aerostatic wind stability safety factor based on inverse analysis is performed using a reverse calculation of the aerostatic wind stability safety factor at a certain level of reliability. There is a corresponding relationship between the target reliability index and the aerostatic wind stability safety factor. Thus, it is necessary to research the influence of the target reliability index on the aerostatic wind stability safety factor and their qualitative relationship. The target reliability index was set at 2.5, 3.0, 3.5 and 4.0, respectively. The specific calculation results of the aerostatic wind stability safety factor of Sutong Bridge are shown in

Table 2. Influence of the target reliability index on the aerostatic wind stability safety factor.

Target Reliability Index	2.5	3.0	3.5	4.0
Safety Factor	2.92	2.48	2.17	1.91

.

According to Table 2, with increases in the target reliability index, the aerostatic wind stability safety factor decreased. It shows that with increases in the target reliability index, the aerostatic wind stability safety factor and the aerostatic wind stability safety coefficient decreased gradually. The aerostatic wind stability of long-span cable-stayed bridges increased gradually, and the safety reserve decreased gradually. In the calculation results of the aerostatic wind stability safety factor under each target reliability index, the calculation results based on the reliability inverse analysis method were all smaller than those based on the deterministic model. This indicates that the parameter uncertainty has a great influence on the safety factor of aerostatic wind stability. Ignoring the parameter uncertainty will lead to overestimations of the safety factor of aerostatic wind stability.

The safety factor evaluation of aerostatic wind stability of long-span cable-stayed bridges based on inverse reliability analysis can not only consider the influence of parameter uncertainty, but can also determine the aerostatic wind stability safety factor at different reliability levels. By correcting the aerostatic wind stability safety factor of long-span cable-stayed bridges to meet the given reliability level, an evaluation system with dual control indexes of the reliability index and the safety factor was established.

4.6.2. Influence of Mean Random Variables

In order to study the influence of the means of random variables on the aerostatic wind stability safety factor of long-span cable-stayed bridges, the mean of one random variable was changed each time. That is, the mean of the changed random variable was 0.9, 1.0 and 1.1-times that of the original mean, while the mean of the remaining random variables remained unchanged. The specific calculation results of the influence of the mean of each random variable on the safety factor of aerostatic wind stability of Sutong Bridge are shown in Figure 4.

It can be seen from Figure 4 that the aerostatic wind stability safety factor decreased slightly with increases in the average coefficient of drag force and lift force, and the change of coefficients had a slight influence on the aerostatic wind stability safety factor of Sutong bridge. The aerostatic wind stability safety factor decreased with the increase of the mean of the pitch moment coefficient, and the change of the mean of the pitch moment coefficient had a great influence on the safety factor. The aerostatic wind stability safety factor increased with increases in the mean value of the critical wind speed conversion factor. The change of the mean value of the critical wind speed conversion factor had a great influence on the aerostatic wind stability safety factor of Sutong Bridge. The aerostatic wind stability safety factor decreased with increases in the gust factor and mean value of the design wind speed. Changes in the gust factor and design wind speed had a great influence on the aerostatic wind stability safety factor of Sutong Bridge.

4.6.3. Influence of the Random Variable Variation Coefficient

In order to study the influence of the random variable variation coefficient on the aerostatic wind stability safety coefficient of long-span cable-stayed bridges, the variation coefficient of one random variable was changed each time, and the variation coefficient of other random variables remained unchanged. The specific calculation results of the influence of the variation coefficients of each random variable on the aerostatic wind stability safety factor of Sutong Bridge are shown in Figure 5.

It can be seen from Figure 5 that the aerostatic wind stability safety factor decreased slightly with increases in the variation coefficient of the drag coefficient and lift coefficient, and the variation coefficient of the drag coefficient and lift coefficient had a slight influence on the aerostatic wind stability safety factor. The aerostatic wind stability safety factor decreased with increases in the variation coefficient of the pitch moment coefficient. The variation of the variation coefficient of the pitch moment coefficient had a great influence on the aerostatic wind stability safety factor. The aerostatic wind stability safety factor decreased with increases in the critical wind speed conversion factor, gust factor and variation coefficient of design wind speed. The variation of the critical wind speed conversion factor and the variation coefficient of the gust factor had a great influence on the aerostatic wind stability safety factor.

4.6.4. Influence of Differential Calculation Step Size and Initial Value

In order to study the influence of the differential calculation step size and the initial value on the aerostatic wind stability safety factor of long-span cable-stayed bridges, the differential calculation step size was, respectively, taken as $h=\sigma /5$, $h=\sigma /10$ and $h=\sigma /20$, and the initial values were, K⁰ = 1.5, K⁰ = 2.5, K⁰ = 3.5, K⁰ = 4.5 and K⁰ = 5.5. The specific calculation results of the influence of the differential calculation step size and initial value on the aerostatic wind stability safety factor of Sutong Bridge are shown in Figure 6. It can be seen from Figure 6 that the step size and initial value of the differential calculation had almost no influence on the calculation convergence process of the safety factor of Sutong Bridge’s aerostatic wind stability, so it did not affect the accuracy of the final calculation results.

5. Discussion

The probabilistic evaluation method of the aerostatic wind stability safety of long-span symmetry cable-stayed bridges was studied here. Taking Sutong Bridge as an engineering example, the probabilistic safety factors of aerostatic wind stability of Sutong bridge were evaluated, and the sensitivity of the parameters was analyzed. To sum up, the main work of this paper and conclusions are as follows:

(1)

Combining the inverse reliability theory with the finite element method, a probabilistic evaluation method for aerostatic wind stability safety of long-span cable-stayed bridges is proposed. The finite element inverse reliability method based on FORM was used to analyze the reliability of the aerostatic wind stability of Sutong Bridge. The results showed that the probability aerostatic wind stability safety factor is 2.17 when the target reliability index is 3.5.

(2)

It is necessary to consider the influence of parameter randomness in the probabilistic evaluation of aerostatic wind stability safety of long-span cable-stayed bridges, and the method proposed in this paper can be used to evaluate the probabilistic evaluation of aerostatic wind stability safety of long-span cable-stayed bridges. The randomness of parameters has an important influence on the probabilistic aerostatic wind stability safety factor of Sutong Bridge.

(3)

The randomness of drag coefficient and lift coefficient has little influence on the probabilistic safety coefficient of aerostatic wind stability, while the randomness of pitch moment coefficient, critical wind speed conversion coefficient, gust factor and design wind speed has an important influence on the probabilistic safety coefficient of aerostatic wind stability. The differential calculation step has no influence on the accuracy of the calculation results of the probability aerostatic wind stability safety factor of Sutong Bridge.

(4)

In the practical engineering of long-span cable-stayed bridges, the pitch moment coefficient, critical wind speed conversion coefficient, gust factor and randomness of design wind speed can be controlled to ensure the stability and safety of aerostatic wind on long-span cable-stayed bridges.