# Temperature Effects on the Natural Frequencies of Composite Girders
## 4 Numerical Simulations for Modal Testing
### Simulation Procedure and Results  
To account for thermal effects, the natural frequencies of the specimens were determined through three consecutive analyses: heat transfer, heat expansion, and frequency. Heat transfer analysis was conducted to obtain the temperature field in the specimen. For the heat transfer analysis, the sink condition was assumed to be 20 \({}^{\circ}\)C, simulating the room temperature in the test. The temperature of the top deck was increased from 20 to 50 \({}^{\circ}\)C. The temperature distribution was measured and compared with the test data along the depth of the specimen. As shown in Figure 9, the temperature gradient obtained from the heat transfer analysis matches well with the test data. Evidently, the temperature decreased with depth and converged to room temperature. This phenomenon could be caused by heat conduction and emission processes. The heat flux generated by the PTC film from the deck to the steel beam was measured. Considering the large surface area of the beam compared to its volume, its heat emission was greater than that of the deck. Therefore, the temperature at the bottom of the beam was close to room temperature. A temperature change in the specimen induces thermal expansion. Thermal stress can also occur, depending on the boundary conditions. To account for these factors, heat expansion analysis was conducted using the temperature field obtained from the heat transfer analysis. Hinge\(-\)roller boundary conditions were adopted for the analysis to match with that of the test specimen. Figure 10 presents the displacement variations measured at the bottom of the midspan and the end of the roller support. The analysis results match well with the test and show the same tendency along the top deck temperature. The displacement at the midspan deflected upward, as the top temperature increased. This phenomenon can be attributed to the uneven temperature distribution along the depth and the different thermal expansion coefficients between concrete and steel. As a result of the temperature gradient along the depth, the concrete deck expanded more than the steel beam. Therefore, a convex-shaped displacement of the specimen occurred, which agrees with the displacement results presented in Figure 10. The temperature gradient and discrepancy in the expansion rates resulted in convex deformation of the girder. Such deformation can change the bending stiffness of the specimen, thereby affecting its natural frequencies under bent conditions. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & **Mass Density** & **Poisson’s Ratio** & **Expansion** & **Specific Heat** & **Thermal** \\  & **(kg/m\({}^{3}\))** & **Coefficient (\(V^{\circ}\)C)** & **Capacity (J/kg\(\cdot^{\circ}\)C)** & **Conductivity** \\ \hline Concrete & 2400 & 0.18 & \(1\times 10^{-5}\) & 880 & 1.5 \\ Steel and rebar & 7850 & 0.3 & \(1\times 10^{-5}\) & Table 1 & Table 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material parameters used in the model. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Temperature (\({}^{\circ}\)C)** & \
\begin{tabular}{c} **Specific Heat Capacity (J/kg\(\cdot^{\circ}\)C)** \\ **(Equation (3))** \\ \end{tabular} &
\begin{tabular}{c} **Thermal Conductivity (W/m\(\cdot^{\circ}\)C)** \\ **(Equation (4))** \\ \end{tabular} \\ \hline
20 & 439.8 & 53.33 \\
25 & 443.3 & 53.16 \\
30 & 446.7 & 53.00 \\
35 & 450.1 & 52.83 \\
40 & 453.4 & 52.66 \\
50 & 459.7 & 52.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Specific heat capacity and thermal conductivity values of steel and rebar. temperature on the natural frequency corresponding to the bending mode, the temperature of the top deck was varied from 20 to 50 \({}^{\circ}\)C. Figure 11 presents the natural frequencies and the corresponding mode shapes determined from the FEA. The specimen was based on a girder structure, rendering it vulnerable to torsion. A plate girder bridge consists of multiple girders, and its torsional rigidity is considerably higher than that of the specimen.