## 4 Severity Curves Derived from the Calculated Damage Deflections  
To determine the deflection caused by a crack that is located at the fixed end, using the described SHC algorithm, we have conducted FEM static simulations considering multiple damage scenarios. The beam considered in this study is like that presented in Section 2, thus it has the following dimensions: the length \(L=1000\) mm, the width \(B=20\) mm, and the thickness \(H=5\) mm. We also used the same material (_Structural steel_) and applied an identical simulation strategy. The simulated transverse cracks are located at distances \(x_{1}=100\) mm, \(x_{2}=200\) mm, and \(x_{3}=300\) mm from the fixed end. The crack depth starts from \(a=0.2\) mm and increases iteratively with a step \(\Delta a=0.2\) mm until the depth of 2 mm is achieved. The applied load was the dead weight, which produces a deflection in the transverse (vertical) direction. For each crack depth, we obtain from the FEM simulations three deflections for the beam's  
Figure 4: The geometry of the open crack. Figure 3: The main window of the PySHC application. free end, which are \(P_{1}(a,100)\), \(P_{2}(a,200)\) and \(P_{3}(a,300)\). Using PySHC, we determine the theoretical deflection at the free end. The input data for the considered damage scenarios and the obtained deflection values are presented in Table 2 for the closed crack and in Tables 3-5 for open cracks with different widths. \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
21 & 0.2 & 23.05194 & 23.04966 & 23.04784 & 23.05331 \\
22 & 0.4 & 23.08282 & 23.07153 & 23.06199 & 23.09481 \\
23 & 0.6 & 23.11673 & 23.09484 & 23.07764 & 23.13992 \\
24 & 0.8 & 23.16138 & 23.12612 & 23.09826 & 23.19984 \\
25 & 1 & 23.21792 & 23.16637 & 23.12452 & 23.27626 \\
26 & 1.2 & 23.28805 & 23.21417 & 23.15746 & 23.37020 \\
27 & 1.4 & 23.37186 & 23.27379 & 23.19689 & 23.48408 \\
28 & 1.6 & 23.47622 & 23.34780 & 23.24623 & 23.62638 \\
29 & 1.8 & 23.60532 & 23.44105 & 23.30606 & 23.80360 \\
30 & 2 & 23.77482 & 23.55553 & 23.38395 & 24.03240 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Deflection at the free end for a cantilever beam with a crack that has the width \(w=1\) mm. \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
1 & 0.2 & 23.05051 & 23.04915 & 23.04807 & 23.05203 \\
2 & 0.4 & 23.06576 & 23.05987 & 23.05525 & 23.07250 \\
3 & 0.6 & 23.09069 & 23.07740 & 23.06701 & 23.10592 \\
4 & 0.8 & 23.12546 & 23.10178 & 23.08331 & 23.15287 \\
5 & 1 & 23.17079 & 23.13365 & 23.10461 & 23.21369 \\
6 & 1.2 & 23.22881 & 23.17437 & 23.13201 & 23.29203 \\
7 & 1.4 & 23.30028 & 23.22457 & 23.16553 & 23.38859 \\
8 & 1.6 & 23.38823 & 23.28624 & 23.20691 & 23.50768 \\
9 & 1.8 & 23.49632 & 23.36178 & 23.25774 & 23.65446 \\
10 & 2 & 23.62840 & 23.45544 & 23.32010 & 23.83521 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Deflection at the free end for a cantilever beam with a crack that has the width \(w=0\) mm. \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
11 & 0.2 & 23.05194 & 23.04741 & 23.04642 & 23.05161 \\
12 & 0.4 & 23.07330 & 23.06490 & 23.05782 & 23.08210 \\
13 & 0.6 & 23.10278 & 23.08532 & 23.07122 & 23.12135 \\
14 The main purpose of determining the theoretical deflections is to calculate the damage severities, which have a direct application in Structural Health Monitoring (SHM).