# Buffeting Response Prediction of Long-Span Bridges Based on Different Wind Tunnel Test Techniques
## 2 Theoretical Analysis
### Traditional Buffeting Response Calculation  
As mentioned above, most of the current bridge structure buffeting analysis uses [PERSON]'s correction through the introduction of the aerodynamic admittance function into [PERSON]'s quasi-steady aerodynamic force expression. Although much work has been completed by many scholars to continuously improve the theories, these theoretical approaches are based on the pioneering work of [PERSON] and [PERSON].  
Because the studies on buffeting by [PERSON] et al. [26] and [PERSON] et al. [27; 28; 29] used the wavenumber domain, in order to compare the traditional buffeting response calculation method with the following method, the buffeting response in the frequency domain is converted to the wavenumber domain for expression according to the relationship between the wavenumber and the frequency \(k=2\pi n/U\). Based on the random vibration theory and the buffeting analysis principle, the power spectrum of the bridge buffeting displacement response can be expressed as:  
\[S_{h_{i}}(y,k_{1})=\frac{\varphi_{h_{i}}^{2}(y)}{M_{h_{i}}^{2}}(2\rho UlbC_{L}) ^{2}|\chi(k_{1})|^{2}|H_{h_{i}}(k_{1})|^{2}|J_{h_{i}}(k_{1})|^{2}S_{u}(k_{1}) \tag{1}\]  
where \(S_{h_{i}}(y,k_{1})\) is the power spectral density of the vertical buffeting displacement response at the axial coordinate \(y\) of the bridge; \(\varphi_{h_{i}}(y)\) is the \(i\)-th mode shape of the vertical motion; \(M_{h_{i}}\) is the vertical generalized mass; \(\rho\) is the air density; \(U\) is the mean wind velocity; \(b=B/2\) is the half-width of the bridge, where \(B\) is the width; \(C_{L}\) is the lift coefficient of the bridge section; \(k_{1}=2\pi n/U\) is the longitudinal wavenumber, where \(n\) is the frequency; \(\chi(k_{1})\) is the one-wavenumber aerodynamic admittance function; \(H_{h_{i}}(k_{1})\) is the vertical one-wavenumber frequency response function; \(J_{h_{i}}(k_{1})\) is the vertical one-wavenumber joint acceptance function of the structure; \(S_{u}(k_{1})\) is the longitudinal one-wavenumber fluctuating wind velocity spectrum.  
As mentioned above, the traditional method for calculating the buffeting response is to identify the aerodynamic admittance and the aerodynamic derivative in the frequency response function of the bridge section through sectional model tests, and then obtain the power spectrum of the structural buffeting displacement response based on Equation (1).  
Among them, the aerodynamic admittance function is an important transfer function for calculating the buffeting response, which is identified by the sectional model pressure or force measurement test. The RMS (root mean square) of the buffeting displacement can be obtained by integrating the buffeting response power spectrum:  
\[\sigma_{h_{i}}^{2}(y)=\int_{0}^{+\infty}S_{h_{i}}(y,k_{1})\mathrm{d}k_{1} \tag{2}\]  
Finally, according to the SRSS (Square Root of Sum Square) method, the total buffeting response without considering the coupling effect between modes can be obtained:  
\[\mathrm{SRSS}(h_{i})=\sqrt{\sigma_{h_{i},1}^{2}+\sigma_{h_{i},2}^{2}+\cdots+ \sigma_{h_{i},m}^{2}} \tag{3}\]  
where m is the number of structural modes.