## 4 Severity Curves Derived from the Calculated Damage Deflections
We calculate the crack severities with Equation (1), the data utilized being \(\delta(a,0)\) for the damaged beam and \(\delta_{u}=23.046\) mm for the intact beam. The results obtained for the closed cracks are represented graphically in Figure 5.  
Furthermore, we determine the severity values for open cracks, considering the results presented in Tables 3-5. The severities calculated accordingly are depicted in Figure 6, along with the severities for the closed cracks for comparison. One can observe that an increase in the damage width has as a result an increase in the severity. However, the curves have the same shape. Figure 5: Severity evolution versus crack depth for the closed crack scenario. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Damage** & \(a\) & \(P_{1}\)**(\(a\),100)** & \(P_{1}\)**(\(a\),200)** & \(P_{1}\)**(\(a\),300)** & \(\delta\)**(\(a\),0)** \\
**Scenario** & **[mm]** & **[mm]** & **[mm]** & **[mm]** & **[mm]** \\ \hline
31 & 0.2 & 23.05984 & 23.05340 & 23.05021 & 23.06241 \\
32 & 0.4 & 23.10104 & 23.08376 &  
Testing the Capacity of the Derived Severities to Accurately Predict Frequency Changes Due to Damage  
In this section, we test the accuracy of the developed SHC algorithm and implicitly the analytical relation to calculate the deflection of beams with cracks. To this aim, we perform FEM simulation and laboratory experiments. Because the frequency changes are small, and even big errors can be overseen, we also compare the relative frequency shifts (RFS). The RFSs are frequency changes normalized by the natural frequencies of the intact beam and are calculated using the following mathematical relation [35]:  
\[\Delta\overline{f}_{i-D}(a,x)=\frac{f_{i-U}-f_{i-D}(a,x)}{f_{i-U}}=\gamma(a) \big{[}\overline{\phi}_{i}^{\prime\prime}(x)\big{]}^{2} \tag{11}\]  
By normalization, the changes become more easily comparable, and a better assessment of the method's reliability is possible. Moreover, these RFSs are used in damage detection, so it is important to evaluate if analytically deduced RFSs can be used to construct reliable databases that contain the structural response for a multitude of damage scenarios. We can extract the severity from Equation (11), resulting in  
\[\gamma(a)=\frac{1}{\big{[}\overline{\phi}_{i}^{\prime\prime}(x)\big{]}^{2}} \frac{f_{i-U}-f_{i-D}(a,x)}{f_{i-U}} \tag{12}\]  
Testing is made by comparing the severity obtained from static tests with the RFS obtained from dynamic tests made in the laboratory. From the static tests, made through FEM simulations, we obtain the deflections and calculate the severity with Equation (1). From the dynamic tests, made involving the FEM or laboratory experiments, we obtain the frequencies of the beam in the intact and damaged state. In addition, we can calculate the normalized modal curvature \(\overline{\phi}_{i}^{\prime\prime}(x)\), and eventually the right term in Equation (12), which has also the meaning of the severity. Now, by comparing the two results, we can conclude if these fit and if the prediction of frequency changes can be reliably made with Equation (3). In this mathematical relation, we consider the measured frequency of the intact real beam and the severity deduced from the deflections of the beam under its own weight.