# Model Updating of Bridges Using Measured Influence Lines
## 3 Results
### Sensitivity Study
#### 3.1.2 Strain IL  
The \(J_{\text{IL}}\) objective function measures the difference between the modelled and the mean measured strain IL. The lower it is, the more similar the compared strain ILs are. \(J_{\text{IL}}\) is defined separately for E_MG1 and E_MG2. Despite comparing two vectors (strain ILs), it is a scalar value. By definition, it is equal to the root-mean-square error (RMSE):  
\[J_{\text{IL}}=\text{RMSE}=\sqrt{\frac{\sum_{j=1}^{m}\left(\text{IL}_{\text{num },j}-\text{IL}_{\text{exp},j}\right)^{2}}{m}}, \tag{3}\]  
where \(\text{IL}_{j,\text{num}}\) is the \(j\)-th ordinate of the modelled strain IL and \(\text{IL}_{\text{exp},j}\) is the \(j\)-th ordinate of the mean measured strain IL \(\text{IL}_{\text{exp}}\). \(m\) is the number of ordinate values considered and \(\text{IL}_{\text{exp}}\) is the mean measured strain IL (from all passages). \(J_{\text{IL}}\) can be interpreted as an average difference between the modelled and measured strain IL.  
The sensitivity study results for the variation in the boundary conditions (bearings) and structural elements Young's modulus on the relative change in the objective function \(J_{f}\) are shown in Figure 15. Those for the variation of boundary conditions (bearings) and structural elements [PERSON]'s modulus on the relative change of the objective function \(J_{\text{IL},\text{E\_MG1}}\) and \(J_{\text{IL},\text{E\_MG2}}\) are shown in Figure 16a,b.  
Figure 15: The sensitivity study results for the variation in boundary conditions (bearings) and structural elements’ Young’s modulus values on the relative change in the objective function \(J_{f}\).  
Based on the sensitivity study results, it was decided to update the boundary conditions and Young's modulus of structural elements. For the mass, it was decided to estimate it as accurately as possible and keep it fixed to ensure that its increase did not decrease stiffness and vice versa (for the frequency-based FEMU). The Young's modulus values of MG1 and MG2 were updated through \"Young's modulus adjustment factors\" \(\alpha_{\text{MG1}}\) and \(\alpha_{\text{MG2}}\). Those factors were defined as a correction factor, relative to the designed Young's modulus (shown in Table 3). A value of 1.00, for example, denotes no change in the Young's modulus and value of 1.20 denotes a 20% increase in the designed Young's modulus. All other structural elements were grouped into the 'OTHER' group, and their Young's modulus was updated by the same percentage through the \(\alpha_{\text{OTHER}}\) Young's modulus adjustment factor.  
It is crucial to note that the objective of FEMU, as outlined by [28], is not to adjust FE model variables to align the predicted and measured values better. Instead, model-based system identification employs physics-based models to deduce variable values. Consequently, the variables chosen for FEMU do not reflect the actual properties of structural elements, such as Young's modulus (adjustment factor). Instead, they should be viewed to consolidate the structural properties into a single variable. This distinction is important to remember, particularly when analysing the absolute values of the updated variables.