# Effect of Temperature on the Modal Variability in Short-Span Concrete Bridges
## 4 Discussion
### Analysis of Modes Considering Temperature Dependency  
PCA (principal component analysis) was carried out to assess the effect of temperature as a major factor influencing the modal variability. Figure 10 plots the regression lines together with the first principal component of PCA. Apart from the fourth mode of Seolmun Bridge and the third and fourth modes of Gajwa Bridge, the regression lines and the first principal components of PCA show similar patterns. This indicates that temperature is a major factor in the modal variability. Based upon such observations, it is now of interest to find the reason why temperature causes the changes in the natural frequencies to occur in the two bridge types. There is certainly the temperature-induced variation of the elastic modulus of concrete as the main material in both bridges. According to [PERSON], et al. [10], the elastic modulus of concrete with compressive strength of 40 MPa decreases by 0.125 GPa  
Figure 10: Temperature-dependent modes: (**a**) Seolmun Bridge; (**b**) Gajwa Bridge. increases by 1 \({}^{\circ}\)C. In other words, the elastic modulus of concrete decreases as much as the temperature increases, which in turn results in the decrease in the natural frequency. Apart from such temperature dependency of the material, the elastomeric bearings of the RC slab bridge also exhibit temperature dependency. Elastomeric bearings are made of rubber and steel plates in which rubber shows loss of its elastic modulus as much as the temperature increases. This also causes the natural frequency of the bridge to decrease with higher temperatures. It follows that the temperature dependency of the RC rahmen bridge is mainly attributable to the change in the elastic modulus of concrete whereas that of the RC slab bridge can be attributed additionally to the change in the elastic modulus of the elastomeric bearings. Equations (1) and (2) express the greater changing rate of the natural frequency with respect to the temperature for the RC slab bridge that can now be understood as the effect of the elastomeric bearings. However, the difference in the temperature dependency appears to not be significant when comparing with the RC rahmen bridge that has no bearings. Accordingly, the effect of the elastomeric bearing on the variation of the natural frequency can be regarded as being relatively lesser than the effect of the elastic modulus of concrete. Model updating [11] was conducted to verify that the temperature dependency of the elastic modulus of concrete is the main factor influencing the temperature dependency of bridge modes. Figure 11 presents the FE (finite element) models of Seolmun Bridge and Gajwa Bridge. The concrete structures are modeled by four-node shell elements. The size of shell element is determined to be 0.5 \(\times\) 0.5 m through the mesh convergence test. For Gajwa Bridge, the fixed boundary condition is imposed on the bottom of the pier. The elastomeric bearings of Seolmun Bridge are modeled by boundary conditions in the vertical direction and springs for rotation. Model updating was performed first for Gajwa Bridge, the RC rahmen bridge without bearings, to identify the elastic modulus of the concrete members per temperature. Figure 12a plots the variation of the elastic modulus of concrete. The elastic modulus is seen to vary from 26.61 GPa to 25.22 GPa when the temperature changes from \(-5\)\({}^{\circ}\)C to 20 \({}^{\circ}\)C. Even if Seolmun Bridge was designed with the same compressive strength as Gajwa Bridge, the damage conditions of both bridges were likely to be different at the time of monitoring. Rather than having the same elastic modulus, Seolmun Bridge and Gajwa Bridge shall be assumed to have the same variation range of the elastic modulus for the same variation range of the temperature. Following, model updating was performed for the rotational stiffness of the elastomeric bearing by maintaining the variation range of the elastic modulus of concrete for Seolmun Bridge identical to that of Gajwa Bridge.