# Dynamic Deflection Measurement on Stiff Bridges with High Piers by Preloaded Spring Method
## 5 Wind Load Effects  
Constructing scaffoldings under a stiff bridge with high piers of more than 10 m is very costly and there are no successful cases yet. The reliability of the numerical simulations and the accuracy of the measurements of the PSM have been verified by indoor and field experiments in Section 4, respectively. Therefore, in this section, numerical analysis was adopted to further analyze the performance of the PSM and SHM in measuring the dynamic deflection of a bridge in a windy environment. A 4 \(\times\) 30 m precast continuous box-girder bridge remains as the object, and the details of the bridge and FE model can be found in Section 3.1. The section stiffness of the stretched wire was assumed to be \(3.45\times 10^{5}\) N, while the stiffness of the compression spring was 50 N/m, with an initial preload of 10\(L\) N. All the above parameters satisfied Equation (10). The parameters of the stretched wire in the SHM were consistent with those of the PSM, the diameter of the hammer was 0.1 m, and the weight of the hammer was optimized according to [19]. The wind load of the suspension hammer was calculated according to the windward area of the suspension hammer. As wind is a stochastic process, it was applied to both measurement systems in the form of a simple harmonic force, as in Equation (13), to simplify the calculation. The wind loads were applied as disturbances to the wire and the suspension hammer in two horizontal directions that are perpendicular to each other. The deflection measurement point was located at the mid-span of the second span. Considering various wind speeds (from 2 m/s to 16 m/s) and various pier heights (from 5 m to 30 m), the dynamic deflections \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Test Serial Number**} & \multicolumn{3}{c}{**NRMSE (\%)**} & \multicolumn{3}{c}{**APVE (\%)**} \\ \cline{2-7}  & **PSM \(\backslash\)2** & **SHM \(\backslash\)2** & **VOM \(\backslash\)2** & **PSM \(\backslash\)2** & **SHM \(\backslash\)2** & **VOM\(\backslash\)2** \\ \hline The first test & 5.83 & 7.09 & 22.00 & \(-0.89\) & \(-5.67\) & \(-0.09\) \\ The second test & 3.73 & 4.41 & 22.55 & \(-2.00\) & \(-3.61\) & \(-12.42\) \\ The third test & 9.17 & 11.91 & 85.28 & \(-4.74\) & \(-11.25\) & \(-102.93\) \\ Average & 6.25 & 7.80 & 43.28 & \(-2.54\) & \(-6.84\) & \(-38.48\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of deflection estimation accuracy. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Test Serial Number** & **SFM \(\backslash\)1** & **PSM\(\backslash\)2** & **SHM \(\backslash\)2** & **Err\({}_{1}\) = (\(\backslash\)2 \(-\)1)** & \(\backslash\)1(\(\backslash\)2(\%) \(\backslash\)2 \(\backslash\)1/\(\backslash\)1 (\%) \(\backslash\)of the bridge measured by both the PSM and SHM methods are shown in Figures 19 and 20, and the DLA is shown in Figure 21.  
\[F_{g}=\frac{1}{2}\rho U_{b}^{2}C_{\text{D}}A_{n} \tag{13}\]  
where \(F_{g}\) is the amplitude of the wind load on the suspension hammer or wire, N; \(\rho\) is the air density, 1.25 kg/m\({}^{3}\); \(U_{b}\) is the wind speed at the reference height, m/s; \(C_{\text{D}}\) is the resistance coefficient of the suspension hammer or wire, m; \(A_{n}\) is the projected area in the downwind direction of the suspension hammer or wire, m\({}^{2}\). Figure 19: Dynamic deflections of the bridge subjected to a wind speed of 8/m: (**a**) deflections obtained by the SHM and (**b**) deflections obtained by the PSM. Figure 20: Dynamic deflections when the bridge with a pier height of 5 m: (**a**) deflections obtained by the SHM and (**b**)deflections obtained by the PSM. Figure 21: Statistical results for wind-affected dynamic load allowances (DLAs): **(a)** dynamic load allowances (DLAs) acquired by the SHM. **(b)** Dynamic load allowances (DLAs) acquired by the PSM. Notes: \(L\) is the length of the wire or pier height; SHM represents the suspension hammer method; PSM represents the preloaded spring method.