## 3 Wind Loads on Structures
### Wind Forces on the Bridge  
The quasi-steady equations accompanied by the aerodynamic admittance functions are utilized to simulate the buffeting forces [30; 32]. The buffeting forces, which include lift force, drag force, and pitching moment, acting on the bridge deck per unit length are given as Equations (2)-(4). \[L_{b}(t)=\frac{1}{2}\rho U^{2}B\left(2C_{L}\chi_{Lu}\frac{u(t)}{U}+(C_{L}^{ \prime}+C_{D})\chi_{Lw}\frac{w(t)}{U}\right) \tag{2}\]  
\[D_{b}(t)=\frac{1}{2}\rho U^{2}B\left(2C_{D}\chi_{Du}\frac{u(t)}{U}+(C_{D}^{ \prime}-C_{L})\chi_{Dw}\frac{w(t)}{U}\right) \tag{3}\]  
\[M_{b}(t)=\frac{1}{2}\rho U^{2}B^{2}\left(2C_{M}\chi_{Mu}\frac{u(t)}{U}+C_{M}^{ \prime}\chi_{Mw}\frac{w(t)}{U}\right) \tag{4}\]  
where \(\rho\) is the air density; \(U\) is the average wind velocity; \(B\) is the width of the bridge deck; \(C_{L}\), \(C_{D}\), and \(C_{M}\) are steady-state coefficients; \(C_{L}^{\prime}=dC_{L}/da\); \(C_{D}^{\prime}=dC_{D}/da\); \(C_{M}^{\prime}=dC_{M}/da\); \(\alpha\) is the attack angle; \(\chi_{Lu}\), \(\chi_{Lw}\), \(\chi_{Du}\), \(\chi_{Du}\), \(\chi_{Mu}\), and \(\chi_{Mu}\) are the aerodynamic admittance functions; \(u(t)\) and \(w(t)\) are fluctuating wind velocities in along-wind and vertical direction, respectively. Self-excited forces are included in the buffeting analysis to describe the coupled and feedback relationships between structural motions and wind field [32]. The self-excited forces acting on the bridge deck per unit length can be expressed as linear functions of nodal displacement and nodal velocity as Equations (5)-(7). \[L_{ae}(t)=\frac{1}{2}\rho U^{2}B\left[KH_{1}^{*}\frac{\dot{h}(t)}{U}+KH_{2}^{* }\frac{B\dot{\alpha}(t)}{U}+K^{2}H_{3}^{*}\alpha(t)+K^{2}H_{4}^{*}\frac{h(t)} {B}+KH_{5}^{*}\frac{\dot{p}(t)}{U}+K^{2}H_{6}^{*}\frac{p(t)}{B}\right] \tag{5}\]  
\[D_{ae}(t)=\frac{1}{2}\rho U^{2}B\left[KP_{1}^{*}\frac{\dot{p}(t)}{U}+KP_{2}^{* }\frac{B\dot{\alpha}(t)}{U}+K^{2}P_{3}^{*}\alpha(t)+K^{2}P_{4}^{*}\frac{p(t)}{ B}+KP_{5}^{*}\frac{\dot{h}(t)}{U}+K^{2}P_{6}^{*}\frac{h(t)}{B}\right] \tag{6}\]  
\[M_{ae}(t)=\frac{1}{2}\rho U^{2}B^{2}\left[KA_{1}^{*}\frac{\dot{h}(t)}{U}+KA_{2} ^{*}\frac{B\dot{\alpha}(t)}{U}+K^{2}A_{3}^{*}\alpha(t)+K^{2}A_{4}^{*}\frac{h(t) }{B}+KA_{5}^{*}\frac{\dot{p}(t)}{U}+K^{2}A_{6}^{*}\frac{p(t)}{B}\right] \tag{7}\]  
where \(K=\omega B/U\) is the reduced circular frequency; \(h(t)\), \(p(t)\), and \(\alpha(t)\)are vertical, lateral, and torsional displacements, respectively; \(\dot{h}(t)\), \(\dot{p}(t)\), and \(\dot{\alpha}(t)\) are vertical, lateral, and torsional velocities, respectively; \(A_{i}^{*}\), \(H_{i}^{*}\), and \(P_{\dot{i}}^{*}\) (\(i=1\), \(2\),..., \(6\)) are flutter derivatives, which are expressed in terms of reduced wind velocity \(U=U/fB\) and determined by geometric configuration of the deck's cross-section; \(f\) is the natural frequency. Equations (5)-(7) is expressed in a frequency manner for complex sinusoidal structural motions [33], but the self-excited forces can be reshaped as aerodynamic stiffness and aerodynamic damping [6; 9; 34; 35]. Hence, the governing equation by Equation (1) can be further expressed as  
\[\ddot{\mathbf{X}}(t)+(\mathbf{C}-\mathbf{C}_{se})\dot{\mathbf{X}}(t)+(\mathbf{ K}-\mathbf{K}_{se})\mathbf{X}(t)=\mathbf{F}_{G}+\mathbf{F}_{st}+\mathbf{F}_{b}(t) \tag{8}\]  
where \(\mathbf{C}_{se}\)and \(\mathbf{K}_{se}\) are the aerodynamic damping matrix and the aerodynamic stiffness matrix, respectively. In such a case, the self-excited forces can be treated as parts of structural components instead of direct loadings. In this study, self-excited forces are realized in ANSYS with the user-defined element MATRIX27 [19; 35]. Then, the ultimate FE model for the buffeting analysis of Taizhou Bridge is shown in Figure 5.  
In the buffeting analysis of Taizhou Bridge, the steady-state coefficients and flutter derivatives were obtained from the section-model wind tunnel tests conducted in the State Key Lab for Disaster Reduction of Civil Engineering at Tongji University [17; 19].