## 4 Localization of the Damaged Head Studs
### Damage Localization Based on Modal Curvature Difference  
(1) Case 1  
Case 1 refers to the validated FE model in Section 3.3, where the damage was simulated by deleting the headed shear studs at corresponding coordinate positions. The first three modal shapes were used to calculate the modal curvature difference. Figure 7 provides the calculated modal curvature difference distributed along the longitude of the SCCB. The damaged shear studs are located at 1900 mm for Figure 7a and 1210 and 2590 mm for Figure 7b. According to Figure 7, the calculated modal curvature difference manifests the damaged shear studs effectively, for both sole-damage and dual-damage conditions. The maximum error was no more than 1%.  
(2) Case 2  
To guarantee the validity of the proposed flexibility difference curvature for damage location, another steel-concrete composite beam was modeled in ABAQUS. The design of the composite beam was based on the Euler-Bernoulli beam theory and in compliance with the provisions in the Chinese Code for design of steel structure (GB 50017-2017) [25] and the Chinese technical specification for steel structure of tall building (JGJ 99-2015) [26]. Details of the composite beam are described below:  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Flexural Modes** & **1** & **2** & **3** & **4** & **5** \\ \hline FEM frequency (Hz) & 33.08 & 109.20 & 202.22 & 351.52 & 455.50 \\ Test frequency (Hz) & 31.25 & 102.52 & 201.37 & 363.09 & 455.01 \\ Error & 5.52\% & 6.12\% & 0.42\% & 3.29\% & 0.11\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of vibration modes.  
Figure 7: Damage localization based on modal curvature difference in Case 1: (a) damage located at 1900 mm; and (b) damage located at 1210 mm and 2590 mm.  
The simply supported composite beam was designed with a partial shear connection of 65%, a net span of 2500 mm, and a total height of 230 mm. The ML15 headed stud has a diameter of 13 mm and a height of 60 mm. The concrete slab has a cross section of 400 \(\times\) 80 mm. The profiled steel HM 150 \(\times\) 100 was utilized as the steel beam. Detailed dimensions and the distribution of headed studs are shown in Figure 8a.  
Figure 8b provides overviews of the three-dimensional FE model considering the actual geometry, material non-linearity, and boundary constraints. The friction coefficient was assumed to be 0.05 for the interface between the steel flange and concrete slab, and 0.45 for the surfaces of the shear studs and surrounding concrete. A general mesh size of 20 mm was adopted for the overall structure, and a fine mesh size of 5 mm was adopted for head studs and surrounding concrete, as shown in Figure 8b. The constitutive model of all materials was consistent with Section 3.3. Table 4 provides the detailed properties of the materials. The linear perturbation analysis and Lanczos iteration were adopted to obtain the flexural modal shapes and corresponding natural frequencies.  
In Case 2, the damage was also simulated by deleting the headed shear studs at corresponding coordinate positions. The modal curvature differences were calculated based on the first three flexural modal shapes. Figure 9 shows that the distribution of the modal curvature difference manifested the location of damaged studs for both sole-damage and dual-damage conditions successfully.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & **Yield** & **Ultimate** & **Elastic** & **Thickness** & \multirow{2}{*}{**Density** (kg/m\({}^{3}\))**} \\  & **Strength** & **Strength** & **Modulus** & **or Diameter** & **(kg/m\({}^{3}\))** \\  & **(MPa)** & **(MPa)** & **(GPa)** & **(mm)** & \\ \hline Steel beam & 205 & 300 & 206 & 9/6 & 7850 \\ Headed stud & 320 & 400 & 206 & 13 & 7850 \\ Concrete & – & 26.8(C40) & 31.1 & – & 2450 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Material properties of Case 2.  
Figure 8: Design and FE model of SCCB (dimension in mm): (**a**) design drawing; and (**b**) FE model.