# Buffeting Response Prediction of Long-Span Bridges Based on Different Wind Tunnel Test Techniques
## 2 Theoretical Analysis
### Calculation of Buffeting Response Based on Sectional Model Vibration Test  
Traditional buffeting response calculation and analysis has always been based on the strip assumption; it divides the bridge into several strip units with independent aerodynamic properties in the spanwise direction, so as to obtain the aerodynamic response of the whole bridge through the integration along the spanwise direction [31]. From another perspective, it is assumed that the spanwise correlation of the buffeting force acting on the structure is equal to the correlation of the fluctuating wind. Based on the one-wavenumber aerodynamic admittance identified through methods such as wind tunnel tests to calculate the structural buffeting response, the calculation and analysis of the buffeting force and the buffeting response can be greatly simplified. However, many scholars [32; 33; 34; 35; 36; 37] found that when the spanwise wavelength of the incoming flow is not much larger than the structure width, the three-dimensional effect of turbulence cannot be ignored, and the spanwise correlation of the buffeting force is much greater than that of the wind. This will lead to the failure of the strip assumption, which means that there will be a certain error in the calculation of the buffeting response of the bridge when using the traditional one-wavenumber aerodynamic admittance. This indicates that the influence of the spanwise wavenumber \(k_{2}\) should not be ignored when calculating the buffeting response of the structure. In turbulent analysis, compared with the autocorrelation of different turbulent fluctuation components, the influence of the cross-correlation is small and can be ignored [38; 39; 40]. To simplify the theoretical elaboration, according to the two-wavenumber buffeting analysis considering the three-dimensional effect of turbulence, and under the premise of ignoring the correlation between the horizontal fluctuating wind velocity and the vertical fluctuating wind velocity, the contribution of the longitudinal and vertical fluctuating wind velocities to the lift is equated to an equivalent aerodynamic admittance that considers the contribution of the fluctuating wind velocity in both directions to the lift. The two-wavenumber buffeting lift spectrum of a long-span bridge is:  
\[S_{L_{i}}(k_{1},k_{2})=(\rho Uk_{1})^{2}|\chi(k_{1},k_{2})|^{2}\bigg{[}4C_{L}^{ 2}\big{|}J_{u_{h_{i}}}(k_{1},k_{2})\bigg{|}^{2}S_{u}(k_{1})+\big{(}C_{L}^{ \prime}+C_{D}\big{)}^{2}\big{|}J_{u_{h_{i}}}(k_{1},k_{2})\bigg{|}^{2}S_{w}(k_{ 1})\bigg{]} \tag{4}\]  
where \(S_{L_{i}}(k_{1},k_{2})\) is the unsteady buffeting lift spectrum corresponding to the \(i\)-th mode of the structure; \(\chi(k_{1},k_{2})\) is the equivalent two-wavenumber aerodynamic admittance; \(C_{D}\) is the drag coefficient of the bridge section; \(C_{L}^{\prime}\) represents the slope of the lift coefficient to the wind attack angle; \(J_{u_{h_{i}}}(k_{1},k_{2})\) and \(J_{u_{h_{i}}}(k_{1},k_{2})\) represent the two-wavenumber joint acceptance function corresponding to the longitudinal and vertical fluctuating wind velocities, respectively; \(S_{w}(k_{1})\) is the vertical one-wavenumber fluctuating wind velocity spectrum. According to recent theoretical and experimental studies [26; 27; 28; 29] of the influence of the aspect ratio on the accuracy of the strip assumption and the three-dimensional effect of turbulence, for long-span bridges and other slender linear structures with large aspect ratios, the strip assumption is accurate enough.