## 4 Damage Identification of Beam-like Structures Considering Shear Deformation
### Example 2: Simply Supported Beam with Five Elements  
To further validate the proposed method, the simply supported beam depicted in Figure 3 was examined.  
The beam has a length of 1.5 m and consists of five beam elements and six nodes, with each element being 0.3 m in length. According to the boundary conditions of the structure, node 1 restricts horizontal and vertical displacement, while node 6 restricts only vertical displacement. The material properties of this example correspond to that of the beam discussed in Section 4.1, with a rectangular cross-section shape (see Table 1). Moreover, two concentrated vertical forces of \(-100\) kN were applied separately to nodes 2 and 5, and the damage scenarios of the structure are shown in Table 6.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Scenario**} & **Element 1** & **Element 2** & **Element 3** & **Element 4** & **Element 5** \\  & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** \\ \hline
1 & 0.75 & / & / & 0.3 & / \\
2 & / & 0.5 & / & 0.3 & 0.2 \\
3 & 0.75 & 0.5 & 1 & / & 0.2 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Damage scenarios.  
Figure 3: Simply supported beam with five elements.  
Table 6 provides three different scenarios of damage, where Scenario 1 involves damage to two beam elements, Scenario 2 involves damage to three beam elements, and Scenario 3 involves damage to four beam elements.  
The proposed damage identification method, which takes into account shear deformation, is employed to identify damage in the structure. The objective function was formulated based on nodal displacements of DoFs 5, 8, 11, and 14. The boundary conditions of the interior point method range from 0 to 2, and the starting point is one. Figure 3(a)-c illustrate the identification results of corrosion depth for Scenario 1, Scenario 2, and Scenario 3, respectively.  
The horizontal axis of Figure 4 represents the number of iterations, and the vertical axis represents the corrosion depth of the beam elements. Additionally, the dashed lines indicate the actual corrosion depth of each element. As can be seen from Figure 3(a), the objective function of Scenario 1 terminates after 24 iterations, and the final identification results of elements 1 and 4 match the actual corrosion depths perfectly. In Figure 3(b), the objective function for Scenario 2 terminates after 46 iterations, and the damage of elements 2, 4, and 5 are successfully identified. Figure 3(c) shows the damage identification process for Scenario 3, where the objective function terminates after 53 iterations and the damage of all four beam elements is identified.  
From the results presented in Figure 4, it can be concluded that the proposed method can effectively identify the damage of simply supported beams. Furthermore, Scenario 3 achieves the highest number of iterations among the three scenarios.  
To provide a more intuitive display of the accuracy of damage identification, Figure 5 shows the trend of the MRE changes with iterations for Scenario 1, Scenario 2, and Scenario 3.  
Figure 4: Identification results of corrosion depth for different scenarios.  
From Figure 5, it can be observed that the MRE decreases as iterations increase for all scenarios, indicating that the proposed damage identification method converges to a satisfactory solution.