## 4 Quantification of the Extent of Damage
### Explanation of the Precision of the Methodology  
When comparing the reference and examined models, slopes of linearised RNFDs are chosen as the relevant comparative quantities; this methodology introduces a certain degree of error into the system. This error results from the fact that the reference model creates linearised courses of RNFDs, but real RNFDs have a nonlinear nature. In Figure 13, slopes of linearized and real RNFDs \(k_{linj}\) and \(k_{polyj}\) (Equation (16)) are shown. \(k_{polyj}\) is attained using second degree polynomial regression of the real RNFDs (Figure 10--\(\Delta f_{0 Ej}\)) and by consecutive division by \(\Delta d_{r}\). \(k_{linj}\) is attained by division of \(\Delta f_{0 Ej}\) in nominal damage \(\Delta d_{rN}=5\%\) by damage \(\Delta d_{rN}\) itself.  
\[k_{linj}=\frac{\Delta f_{0 Ej}(\Delta d_{rN})}{\Delta d_{rN}}=\frac{\Delta f_{ 0 Ej}}{\Delta d_{rN}}k_{polyj}(\Delta d_{r})=\frac{\Delta f_{0 Ej}}{\Delta d_{r }}=\frac{p_{2}\Delta d_{r}^{2}+p_{1}\Delta d_{r}+p_{0}}{\Delta d_{r}} \tag{16}\]  
Figure 12: RNFDs of a reference model with marked RNFDs related to \(\mathrm{x_{DI}}=0.79936\) m (**upper part of the figure**), RNFDs with slopes \(\mathrm{k_{kj}}\) (full lines) and RNFDs with slopes \(\mathrm{k_{kj}}\) (dashed lines) (**lower part of the figure**).  
Figure 13: Comparison of \(k_{linj}\) and \(k_{polyj}\). The red dashed vertical lines represent the nominal damage extent used for the creation of the reference model.  
In Figure 13, it can be seen that the highest correlation between \(k_{polyy}\) and \(k_{liny}\) is when the damage is close to \(\Delta d_{rN}\). This finding implies that the highest precision of the method is present when the damage is relatively close to \(\Delta d_{rN}\) which is confirmed in Figures 14 and 15.  
Figure 14: Identified extent and relative error of the damage quantification method as a function of \(\Delta d_{r}\).  
Figure 15: _Cont._  
The relative error of the quantification method is defined as:  
\[\Delta d_{re}=\frac{\Delta d_{r1}-\Delta d_{rE}}{\Delta d_{rE}}\times 100\% \tag{17}\]  
For dimensioning purposes, it is important to consider the positive/negative character of the error. If the error is negative, the identified damage is less than the reality, which means that it is necessary to consider the safety coefficient to cover the quantification error. When the error is positive, the identified damage will increase more than the real damage, making the dimensioning more robust. The graphs in Figure 14 show the extent of the identified damage in the first row and the relative quantification error in the second row of the figure as functions of the extent of the real damage for positions \(x_{DE}=0.5\) m and \(x_{DE}=0.8\) m. The lowest error of the method is achieved at \(\Delta d_{rE}(x_{DE}=0.5\) m) \(\cong\) 5% and \(\Delta d_{rE}(x_{DE}=0.8\) m) \(\cong\) 4% which relatively correlates with Figure 13. However, the method is based on a comparison of the examined model with the reference model, which is based on multiple polynomial regressions, which brings an additional system error to the method that explains that the minimal error for \(x_{DE}=0.8\) m \(\Delta d_{re}\cong\) 4% and not 5%.  
To demonstrate precision of the quantification method linearised reference, identified and real RNFDs are shown in Figure 15 and defined by the following formulas:  
* identified RNFDs,
* real RNFDs,  
While the reference RNFDs are defined in Equation (12), the index L in the RNFDs defined above, represents their linear character with respect to \(\Delta d_{r}\).  
The best match between the identified and real RNFDs is achieved in the case of damage with a rate of \(\Delta d_{rE}=5\%\), since the relative error is the smallest for this damage rate for the considered cases. The identified RNFDs are relatively highly correlated with the reference decreases, which is because the reference models are used in a correlation with the examined model in the quantification mechanism.  
Figure 15: Comparison of reference, real and identified RNFDs.