# Dynamic Deflection Measurement on Stiff Bridges with High Piers by Preloaded Spring Method
## 3 Parameter Analysis and Optimization
### Dynamic Responses and DLAs  
The deflection DLA of bridges is a barometer in bridge safety evaluation. After obtaining dynamic deflections, researchers focus more on DLA. Moreover, DLA is more appropriate as an error analysis index for dynamic deflection measurement since it is more sensitive to influencing factors. In this paper, DLA is defined as follows:  
\[DLA=\frac{y_{d}(x)-y_{s}(x)}{y_{s}(x)} \tag{9}\]  
where \(y_{d}(x)\) and \(y_{s}(x)\) are the maximum dynamic and static deflection of the bridge at the location \(x\), respectively. Based on the single-variable principle, an extensive parametric study of the DLA of the material point has been carried out, including the stretched-wire section stiffness \(E_{w}A_{w}\), compression spring stiffness \(k_{s}\), initial preload force \(f_{P}\), material point mass \(m_{P}\), and stretched-wire length \(L\). The deflection measurement point is located at the mid-span of the second span. The vehicle crossed the bridge with an unevenness of grade C at a constant speed of 72 km/h. The dynamic displacements of the girder and SWSS are obtained by vehicle-bridge interaction (VBI) [56], and then the DLAs are calculated by Equation (9). The dynamic displacements of the SWSS are determined by the vertical dynamic displacement of the material mass point in the system. (1) Stretched-wire Section Stiffness  
Assuming the mass of the material point was 0.01 kg and the stiffness of the compression spring was 200 N/m with an initial preload of 20 N, and considering various section stiffnesses of the stretched wire, the deflections of the girders and the SWSS can be obtained by VBI analysis as shown in Figure 9, and the corresponding DLAs are drawn in Figure 10.  
Figure 8: Profiles of road roughness. Figure 10: Dynamic load allowances (DLAs) of the PSM influenced by stretched-wire section stiffness. Note: \(L\) is the length of the stretched wire or pier height; PSM is the preloaded spring method. Figure 9: Dynamic deflections of the PSM when the stretched-wire section stiffness \(E_{w}A_{w}=1.53\times 10^{5}\) N. **(a)** deflection response; **(b)** deflections of the shaded area in **(a)**. Note: \(L\) is the length of the stretched wire; PSM is the preloaded spring method. In Figure 9, for bridges with pier heights from 5 m to 30 m, the deflection vibration trends of the SWSS and the girder remain consistent with an error range from \(-\)0.76 to \(-\)3.76% at the peak deflection. Meanwhile, in Figure 10, the DLA of the girder is 0.192, while the maximum DLA obtained by the PSM is 0.190; under the same pier height or stretched-wire length conditions, the difference in the DLAs between the girder and PSM gradually decreases with the section stiffness of the stretched wire, which means that the vibration amplitude at the girder and that at the SWSS gradually converge. When the stiffness of the stretched wire is greater than 3.92 \(\times\) 10\({}^{5}\) N and the length of the stretched wire is less than 30 m, the minimum DLA of the PSM is 0.174, and the difference between the material point and the girder is less than 10.0%. (2) Compression Spring Stiffness  
The section stiffness \(E_{w}A_{w}\) of the stretched wire was assumed to be 1.53 \(\times\) 10\({}^{5}\) N with an initial preload of 20 N, whereas other parameters were not modified. The DLAs of the girder and the SWSS were obtained by changing the stiffness of the compression spring as shown in Figure 11. Under the same stretched-wire length conditions, the DLAs and the difference between the girder and the PSM decrease linearly with the spring stiffness of the compression spring, which means that, the lower the stiffness of the compression spring, the lower the measurement errors. When the stiffness of the compression spring is less than 300 N/m and the length of the stretched wire is 5 m, the DLA obtained by the PSM is 0.180 at the minimum, and the maximum difference between the two DLAs is 6.0%.