# Flutter Reliability Analysis of Xiangshan Harbor Highway Cable-Stayed Bridges in Service
## 2 The Flutter Analysis Framework for Flexible Bridge  
### Overview  
Bridge flutter is mainly caused by self-excited forces, and its motion equation can be expressed as [33]:  
\[M\ddot{Z}+C\dot{Z}+KZ=F_{\text{se}} \tag{1}\]  
in which \(M\) is the mass matrix; \(C\) is the damping matrix; \(K\) is the stiffness matrix; \(Z\), \(\dot{Z}\) and \(\ddot{Z}\) are the node displacement vector, the velocity vector and the acceleration vector,respectively; and \(F_{\rm se}\) is the self-excited wind force, including the lift forces, drag forces and torsional moments. Since the drag force is not the main factor causing the flutter of the cable-stayed bridge, the influence of the drag force is ignored in this study. Furthermore, the main factor of bridge flutter is the aerodynamic force on the main girder, while the aerodynamic force on the stay cable and the bridge tower has little effect on bridge flutter and can be ignored [1; 2; 5; 6; 7; 10]. Aerodynamic lift force and the torsional moment of the bridge per unit length can be expressed by the aerodynamic derivative proposed by [PERSON] [34] (lateral movement of the section is ignored):  
\[L=(\frac{1}{2}\rho U^{2})(2B)(KH_{1}^{*}\frac{\dot{h}}{U}+KH_{2}^{*}\frac{B \dot{\alpha}}{U}+K^{2}H_{3}^{*}\alpha+K^{2}H_{4}^{*}\frac{h}{B}) \tag{2}\]  
\[M=(\frac{1}{2}\rho U^{2})(2B^{2})(KA_{1}^{*}\frac{\dot{h}}{U}+KA_{2}^{*}\frac{ B\dot{\alpha}}{U}+K^{2}A_{3}^{*}\alpha+K^{2}A_{4}^{*}\frac{h}{B}) \tag{3}\]  
where \(h\) is the vertical displacement of the bridge; \(\alpha\) is the bridge torsion angle; \(\rho\) is the air density; \(B=2b\) is the bridge width; \(U\) is the air inflow velocity; \(H_{i}^{*}(i=1,2,3,4)\), \(A_{i}^{*}(i=1,2,3,4)\) is the flutter derivative; \(K=\frac{B\dot{\alpha}}{U}\) is the dimensionless reduced frequency; and \(w\) is the vibration circle frequency (rad/s). It can be seen from Equations (2) and (3) that the self-excited force acting on the bridge is a function of \(K\) and \(U\). Therefore, the expression of the self-excited force must be expressed in the time-domain before it can be further used for flutter time-history analysis [35]. In this paper, impulse-response function, combined with rational function, is used to express the self-excited force in time-domain. On the basis of the concept of impulse-response function [36; 37], [PERSON] proposed the time-domain expression of self-excited force by combining the time-domain fitting of flutter derivatives with the impulse-response function. Taking lift force in pure vertical motion as an example, the expression \(L_{sth}(t)\) can be obtained as follows:  
\[L_{sth}(t)=\frac{1}{2}\rho U^{2}[A_{Lh1}h(t)+A_{Lh2}\frac{B}{U}\dot{h}(t)+ \sum_{i=3}^{m}A_{Lhi}\int_{-\infty}^{t}e^{\frac{d_{Lh1}U}{B}(t-\tau)}\dot{h}( t)d_{\tau}] \tag{4}\]  
in which \(A_{Lh1}\), \(A_{Ll2}\), \(A_{Lli3}\), \(A_{Lhi}\), \(d_{Lhi}\) (\(d_{Lhi}\geq 0;i=4,\cdots,m\)) are the parameters to be fitted. The value of \(m\) depends on the approximate accuracy of the expression. In the same way, we can find out \(L_{s\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!where \(\Phi\) represents the standard normal distribution function; \(\varphi\) stands for the standard normal distribution density function; \(F_{X_{i}}\) represents the distribution function of non-normal variables; \(f_{X_{i}}\) represents the distribution density function of non-normal variables; \(\mu^{\prime}_{X_{i}}\) represents the mean value of the equivalent normal variable \(X_{i}^{\prime}\); and \(\sigma^{\prime}_{X_{i}}\) is the standard deviation of the equivalent normal variable \(X_{i}^{\prime}\). The mean value and standard deviation of equivalent normal variable \(X_{i}^{\prime}\) can be obtained from the following two formulas:  
\[\left\{ \begin{array}{c}\mu^{\prime}_{X_{i}}=x^{*}_{i}-\Phi^{-1}\big{[}F_{X_ {i}}\big{(}x^{*}_{i}\big{)}\big{]}\sigma^{\prime}_{X_{i}}\\ \sigma^{\prime}_{X_{i}}=\varphi\Big{(}\Phi^{-1}\big{[}F_{X_{i}}\big{(}x^{*}_ {i}\big{)}\big{]}\Big{)}\big{/}f_{X_{i}}\big{(}x^{\prime}_{i}\big{)}\end{array} \right.