# Temperature Effects on the Natural Frequencies of Composite Girders
## 4 Numerical Simulations for Modal Testing
### Simulation Procedure and Results
Therefore, the bending model is typically fundamental; hence, we focused on the natural frequency corresponding to the bending mode. Figure 12 presents the FE results, which were compared to the test results corresponding to the bending mode. The circular markers with solid lines indicate the FEA results, where the temperature dependence of the elastic modulus and thermal properties are incorporated in the FE model. The FE model with the temperature dependencies predicted the natural frequency as 37.78 Hz at 20 \({}^{\circ}\)C. As the temperature rose, the natural frequency exhibited a decreasing tendency conforming to the test results with a predicted value of 37.66 Hz at 50 \({}^{\circ}\)C. The difference between the FE results and the linear regression of the test results varied from \(-\)0.05% to 0.06%, demonstrating that the FE simulation results generally matched well with the test results. The diamond shapes with solid lines represent the FE results without considering the temperature dependency of the elastic modulus. In contrast to the temperature-dependent model, the FE model without the temperature dependence of the elastic modulus yielded an increase in natural frequency with temperature. This can be attributed to the convex deformation at the midspan induced by the temperature gradient. The convex deformation increased with temperature, thereby increasing the geometrical stiffness of bending (Figure 10). Therefore, the frequency increased with temperature. Otherwise, the frequency would be steady along the temperature direction for the FE model without any temperature  
Figure 11: Natural frequencies and corresponding mode shapes at a deck temperature of 20 \({}^{\circ}\)C: (**a**) torsional mode; (**b**) bending mode; (**c**) torsional bending mode. dependency. In Figure 12, the triangular markers with a solid line represent the results of the FEA without considering the temperature dependencies of the specific heat and the thermal conductivity of the steel. This model yielded a decrease in the natural frequency, similar to the model incorporating the temperature dependencies of the specific heat and the thermal conductivity. The maximum frequency difference between the two models was only 0.03%, indicating that the temperature effects of the specific heat and the thermal conductivity were not comparable to those of the elastic modulus. This suggests the importance of considering the temperature-dependent nature of materials, particularly their moduli of elasticity, for achieving a comprehensive understanding of their dynamic responses to temperature variations. Figure 13 presents the variation in the frequency differences between the linear regression, the FE model with an independent modulus, and the FE model with independent thermal properties, and the FE model with a temperature-dependent elastic modulus and thermal properties. Evidently, the maximum difference was 0.03% for the FE model with the independent thermal properties in the temperature range of 20-50 \({}^{\circ}\)C. The maximum difference was 0.06% for the linear regression model and fluctuated with temperature, which could be attributed to the nonlinear relationship between specific heat and temperature, as indicated in Equation (3). As mentioned previously, the natural frequency of the FE model with an independent modulus of elasticity tended to increase with temperature, which caused the frequency difference to increase, as indicated by the square markers in Figure 13. The maximum difference was 0.42% for the FE model with an independent modulus. Figure 12: Relationship between temperature and natural frequency.