## 4 Application to Model Updating of an In-Service Bridge with Extra-Width
### Parameters Selected by Sensitivity Analysis  
Before analysis of model updating, the sensitivity of parameters, material modules (shown in Figure 6a), material densities (shown in Figure 6b), section characteristics (shown in Figure 6c), and secondary dead load (shown in Figure 6d), to frequencies like first, second, and third order vertical bending, first order transverse bending, and first order torsion are all taken into consideration. Parameters with a sensitivity value beyond \(\pm 0.1\) are considered to be sensitive ones in this paper in which the value is computed by following formula,  
\[\frac{\partial\lambda_{j}(a_{i})}{\partial a_{i}}=\frac{\delta\lambda_{j}}{ \lambda_{j}}/\frac{\delta a_{i}}{a_{i}} \tag{12}\]  
where \(\delta\lambda_{j}\) is a small perturbation of the \(j\)th modal frequency of \(\lambda_{j}\), \(\delta a_{i}\) is a small perturbation of the \(i\)th parameter of \(a_{i}\). Negative value of sensitivity value indicates negative influence when parameters increase.  
Figure 5: The first five modal shapes of main girder of Hunan Road Bridge from FE model.  
Figure 6: Sensitivity of parameters to frequencies: (**a**) Sensitivity of material modules to frequencies; (**b**) Sensitivity of material densities to frequencies; (**c**) Sensitivity of section characteristics to frequencies; (**d**) Sensitivity of secondary dead load to frequencies.  
In Figure 6a, the material module of girder is sensitive to first three order vertical bending, transverse bending, and torsion with value of 0.33, 0.39, 0.28, 0.42, and 0.37 respectively; the material module of tower is sensitive to second and third order vertical bending with value of 0.14 and 0.21; the material module of cable is sensitive to first order vertical bending and torsion with values of 0.13 and 0.11 while the material module of hanger shows no sensitivity to frequencies as all values are below 0.02. In Figure 6b, the material density of girder is sensitive to first three order vertical bending, transverse bending, and torsion with value of \(-0.40\), \(-0.37\), \(-0.37\), \(-0.22\), and \(-0.42\) respectively while the other material densities show no sensitivity. In Figure 6c, the moment inertia of girder shows sensitivity to first three order vertical bending and torsion with value of 0.26, 0.29, 0.18, and 0.14 respectively when the moment inertia of tower just shows sensitivity to second and third order vertical bending with value of 0.11 and 0.16; just the section area of girder shows the same sensitivity of moment inertia of girder to frequencies with value of \(-0.24\), \(-0.20\), \(-0.21\), and \(-0.28\), while the area of the tower shows no sensitivity. In Figure 6d, the secondary dead load shows a little sensitivity to all frequencies as all values are around \(-0.1\). To research more details, the moment of inertia of girder is divided into three parameters, namely moment inertia of vertical bending, transverse bending, and torsion. The whole results indicate that the selected sensitive parameters are material module of main girder, main tower and main cable, density of main girder, moment of inertia of vertical bending, transverse bending and torsion, moment of inertia of main tower, area of section of main girder, and secondary dead load, respectively.