## 3 Effect of Shear Deformation on Nodal Displacement of a Simply Supported Beam  
Figure 1 shows a three-element simply supported beam model, where the number of elements is denoted within the box, the number of nodes in the circle, and the DoFs of each node are represented by the number next to the arrow.  
The beam under consideration has a length of 4.5 m and is modeled with three beam elements and four nodes, where each element length is set at 1.5 m. The modulus of elasticity (\(E\)) is 206 GPa and the shear modulus of the material (G) is 79.23077 GPa. According to the boundary conditions of the structure, node 1 restricts horizontal and vertical displacement, while node 4 only restricts vertical displacement. The external forces are applied as concentrated vertical forces of \(-50\) kN at nodes 2 and 3. For analysis and comparison, two different cross-sectional shapes were considered, along with six varied damage scenarios. The dimension parameters of the cross-sectional shapes of the structure are given in Table 1.  
The data presented in Table 1 indicate that the wide flange cross-section and the rectangular cross-section have the same depth and width dimensions, resulting in the same depth-to-span ratio (\(h/L\)) of 0.033 for the structure.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Shape**} & **Depth** & **Width** & **Thickness** & **Thickness** \\  & \(h\) & \(b\) & \(t_{f}\) & \(t_{w}\) \\ \cline{2-5}  & **mm** & **mm** & **mm** & **mm** \\ \hline Wide flange cross-section & 150 & 100 & 5 & 4 \\ Rectangular cross-section & 150 & 100 & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 1: The cross-sectional shapes and dimensions.  
Figure 1: Simply supported beam with three elements.  
Assuming that the beam elements are subjected to corrosion in the cross-sectional area, Table 2 presents the corresponding corrosion depths specified for each scenario from Scenario 1 to Scenario 6.  
Table 2 presents six distinct damage scenarios, with Scenarios 1, 2, and 3 involving damage to a single beam element, and Scenarios 4, 5, and 6 involving damage to two beam elements. The numbers in the table represent the corrosion depth of the beam elements.  
The nodal displacements under six different damage scenarios were analyzed using the Euler-Bernoulli beam theory and the Timoshenko beam theory, respectively. Considering the large vertical displacement response of the simply supported beam structure, the relative errors in nodal displacements at DoFs 5 and 8 under different damage scenarios were analyzed. Figure 2 illustrates the relative error of vertical displacement based on two different beam theories. Specifically, Figure 2 displays the relative errors in nodal displacement at DoF 5 for simply supported beams with wide flange cross-section and rectangular cross-section, while Figure 2 displays the relative errors in nodal displacement at DoF 8 for the same structure.  
From Figure 2,b, it can be observed that shear deformation affects the nodal displacements of simply supported beams with wide flange cross-section and rectangular cross-section. For the simply supported beam with a wide flange cross-section, the relative errors in nodal displacements induced by shear deformation exhibit a variation under different damage scenarios. In particular, under Scenario 5, the maximum relative error is observed, with relative errors in DoFs 5 and 8 being 2.04% and 2.12%, respectively.