## 4 Damage Identification of Beam-like Structures Considering Shear Deformation  
### Example 1: Simply Supported Beam with Three Elements  
Based on the discussion in the previous section, it is clear that shear deformation induces errors in the calculation of nodal displacements, and the impact of these errors on damage identification needs to be investigated further. This section focuses on investigating damage identification for the simply supported beam structure shown in Figure 1, which has the wide flange cross-section and the rectangular cross-section, as specified in Table 1. These structures are subjected to different damage scenarios, as listed in Table 2. Two different beam theories, the Euler-Bernoulli beam theory and the Timoshenko beam theory, are \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scenario** & \begin{tabular}{c} **Element 1** \\ **(mm)** \\ \end{tabular} & \
\begin{tabular}{c} **Element 2** \\ **(mm)** \\ \end{tabular} &
\begin{tabular}{c} **Element 3** \\ **(mm)** \\ \end{tabular} \\ \hline
1 & 0.8 & / & / \\
2 & / & 1.2 & / \\
3 & / & / & 1 \\
4 & 0.8 & 1.2 & / \\
5 & 0.8 & / & 1 \\
6 & / & 1.2 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Damage scenarios. Figure 2: Relative errors in nodal displacement under different damage scenarios. used for damage identification. In this study, the measured displacements for DoFs 5 and 8 were calculated based on the Timoshenko beam theory. Following the methodology presented in Section 2, the analytical displacements were obtained. Subsequently, the objective function shown in Equation (4) was established to identify the corrosion depth of beam elements. The interior point method was applied to optimize the objective function based on the current damage condition and cross-sectional dimensions of the beam elements, with boundary conditions set to zero and two with a starting point of one. Table 3 presents the identification results for simply supported beams with wide flange cross-sections, including the iterations of objective functions based on both the Timoshenko beam theory and the Euler-Bernoulli beam theory, as well as the optimal value of the identified corrosion depth. From the data in Table 3, it can be observed that the damage identification results obtained based on the Timoshenko beam theory are consistent with the corrosion depths given in Table 2. However, for Scenario 1 to Scenario 6, the damage identification results based on the Euler-Bernoulli beam theory show different degrees of deviation from the actual damage situation. The most significant errors in damage identification results occurred in Elements 1 and 3, with corrosion depths of 0.89 mm and 1.08 mm in Scenarios 1 and 3, respectively. Table 4 presents the identification results for the simply supported beam with a rectangular cross-section. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Scenario**} & \multirow{2}{*}{**Beam Theory**} & \multirow{2}{*}{**Iterations**} & **Element 1** & **Element 2** & **Element 3** \\  & & & **(mm)** & **(mm)** & **(mm)** \\ \hline \multirow{2}{*}{1} & [PERSON] & 8 & 0.80 & / & / \\  & [PERSON]–[PERSON] & 9 & 1.02 & / & / \\  & [PERSON] & 9 & / & 1.20 & / \\  & [PERSON]–[PERSON] & 15 & / & 1.28 & / \\  & [PERSON] & 5 & / & / & 1.00 \\  & [PERSON]–[PERSON] & 9 & / & / & 1.22 \\  & [PERSON] & 18 & 0.80 & 1.20 & / \\  & [PERSON]–[PERSON] & 19 & 0.80 & 1.28 & / \\  & [PERSON] & 13 & 0.80 & / & 1.00 \\  & [PERSON]–[PERSON] & 12 & 0.93 & / & 1.12 \\  & [PERSON] & 21 & / & 1.20 & 1.00 \\  & [PERSON]–[PERSON] & 17 & / & 1.28 & 1.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Identification results of the simply supported beam with a rectangular cross-section.