## 2 Methodology
### Selection of Boundary Conditions  
The boundary conditions of the concrete base plate were defined with respect to thermal, moisture, and structural problems, which are illustrated in Figure 6.  
For thermal analysis, the convection can be expressed by the following:  
\[q_{c}=\beta_{c}(T_{sur}-T_{air}) \tag{6}\]  
where \(q_{c}\) is the convective heat flux (W/m\({}^{2}\)); \(\beta_{c}\) is the convective coefficient (W/m\({}^{2,\circ}\)C); and \(T_{sur}\) and \(T_{air}\) are the concrete surface and air temperatures (\({}^{\circ}\)C), respectively. The convection coefficient is largely dependent on wind speed \(v\) (m/s) and it can be estimated as \(\beta_{c}\)= \(2.6\left(\sqrt[4]{|T_{sur}-T_{air}|}+1.54\
u\right)\)[32]. Figure 5: Finite element model of CRTS III base plate on bridge foundation. The heat exchange between the concrete surface and air due to long wave re-radiation is calculated by the following Stefan-Boltzmann equation:  
\[q_{r}=\varepsilon\sigma\Big{(}T_{sur}^{4}-T_{env}^{4}\Big{)} \tag{7}\]  
where \(\sigma\) is the Stefan-Boltzmann constant (\(5.669\times 10^{-8}\cdot\mathrm{W}\cdot\mathrm{m}^{-2}\cdot\mathrm{K}^{-4}\)), and \(\varepsilon\) represents the emissivity of concrete (0.85). In order to simplify the calculation, re-radiation could be accounted for together with convection via a single radiation-convection coefficient, \(\beta_{cr}=\beta_{c}+\beta_{r}\), where the following is the case [27]. \[\beta_{r}=\left\{ \begin{array}{ll}\varepsilon[4.8+0.0075(T_{sur}-278.15)],&T_ {sur}\geq 278.15K\\ 4.8\;\varepsilon,&T_{sur}<278.15K\end{array} \right. \tag{8}\]  
When temporary curing is adopted, an \"equivalent heat transfer coefficient\" \(\beta_{eq}\) can be applied in Equation (6). The material layers between the concrete surface and the air can  
Figure 6: Boundary conditions of the CRTS III base plate. be observed as associated in series, and the equivalent convective coefficient is described as follows [27]:  
\[\beta_{eq}=\left(\frac{1}{\beta_{cr}}+\sum_{1}^{n}\frac{d_{i}}{k_{i}}\right)^{-1} \tag{9}\]  
where \(d_{i}\) is the thickness, and \(k_{i}\) is the convective coefficient of each ith layer. Heat transfer from short-wave solar radiation (SR) to concrete can be computed as follows:  
\[q_{s}=\alpha_{s}\ I_{n}(1-\gamma) \tag{10}\]  
where \(\alpha_{s}\) is the absorptivity of concrete surface (0.65); and \(I_{n}\) is the direct SR intensity (W/m\({}^{2}\)), assumed to follow a sinusoidal function during the daytime. In this study, the data of the daily maximum \(I_{n}\) were obtained from a public meteorological information website. \(\gamma\) is the absorption efficiency factor that relies on cover materials and thicknesses. When the heat flux equations in Equations (6), (7), and (10) are combined, the thermal boundary condition can be expressed as follows:  
\[\alpha_{s}\ I_{n}(1-\gamma)-\beta_{eq}(T_{sur}-T_{air})+k_{T}\bigg{(}\frac{ \partial T}{\partial n}\bigg{)}=0 \tag{11}\]  
where \(\partial T/\partial n\) is the temperature gradient in direction \(n\) of the concrete surface. In order to ensure realistic temperature filed simulation of the base plate, the temperature field of the girder was also included in the thermal analysis. Heat flux at the top boundary of the FE model shown in Figure 5 consisted of combined heat flow from convection, solar radiation, and re-radiation. With respect to the lateral and bottom boundaries of the FE model, heat flux was assumed to be affected only by convection (Figure 6). For the moisture analysis, the boundary condition is described as follows:  
\[\lambda(H-H_{env})+D(H)\frac{\partial H}{\partial n}=0 \tag{12}\]  
where \(\partial H/\partial n\) is the RH gradient in direction \(n\) of the surface, and \(H_{env}\) is the environment RH. The average RH is taken as 45% based on historical weather data. For structural analysis, \"small sliding\" contact was used to simulate the interface between the base plate and bridge foundation. The friction coefficient was chosen as 0.6.