## 4 Heat Transfer Equation and Thermal Boundary Conditions  
The heat transfer partial equation is used to model the heat conduction within the girder. The temperature variation over time, position and material properties, such as the density, specific heat and thermal conductivity coefficient, can be determined using Equation (1). The heat conduction equation expressed in Equation (1) considers the variations in temperature (T) as a function of time (t) at position (x, y, z). The heat conduction within the girder depends on the material density (\(\rho\)), the material specific heat (C) and the thermal conductivity coefficient (k) [38]. \[\centering\thick@vector{\varpi}\frac{\partial\mathrm{T}}{\partial\mathrm{t}}= \mathrm{k}(\frac{\partial^{2}\mathrm{T}}{\partial\mathrm{x}^{2}}+\frac{ \partial^{2}\mathrm{T}}{\partial\mathrm{y}^{2}}+\frac{\partial^{2}\mathrm{T}} {\partial\mathrm{z}^{2}})\@add@centering \tag{1}\]  
The heat energy exchange between the girder surfaces and the surrounding environment has been greatly enhanced due to climate change exposure. The heat transfer mechanisms responsible for the heat energy exchange are (1) conduction heat transfer, (2) convection heat transfer between the bridge surfaces and ambient temperature and (3) solar radiation exposed by the bridge and irradiation reflected from the bridge sur  
Figure 2: Cross-section of the steel I girder with the locations of the thermocouples. face [37]. Moreover, the thermal boundary conditions equation (Equation (2)) has been modified to account for the increased heat fluxes on the girder surfaces, as expressed in Equation (3), where (q) is the sum of the heat fluxes on the girder surfaces and (I) represents the direction cosine (x, y, z) of the unit outward vector perpendicular to the boundary surfaces. \[\mathrm{k}\frac{\partial\mathrm{T}}{\partial\mathrm{n}}\mathrm{l}+\mathrm{q} =0 \tag{2}\]  
\[\mathrm{q}=\mathrm{q}_{\mathrm{s}}+\mathrm{q}_{\mathrm{con}}+\mathrm{q}_{ \mathrm{re}} \tag{3}\]  
where \(\mathrm{q}_{\mathrm{s}}\), \(\mathrm{q}_{\mathrm{con}}\) and \(\mathrm{q}_{\mathrm{re}}\) are the total solar radiation heat flux, heat convection and longwave radiation in W/m\({}^{2}\), respectively. The total solar radiation (I\({}_{\mathrm{s}}\)) striking the girder surfaces from the sun consists of three components, as expressed in Equation (4): beam solar radiation, which is the direct solar radiation from the sun without being diffused; diffuse solar radiation, which is the scattered part of the beam solar radiation due to the surroundings; and ground-reflected radiation. The total solar radiation (I\({}_{\mathrm{s}}\)) is determined by the solar incident angle (\(\theta\)), which is the angle between the sun's rays and the normal direction of the surface; the solar zenith angle (\(\beta\)), which is the angle between the sun's rays and the vertical direction; and the tilted angle between the girder surface and the horizontal plane (\(\delta\)). These three components work together to determine the total solar radiation (Is) striking the girder surfaces of a structure. The reflected radiation from the ground shown in Equation (5) depends on the ground reflection coefficient (albedo) \(\mathrm{(r}_{\mathrm{g}}\)), which is 0.2 for ground reflection [39] and the tilted angle between the girder surface and the horizontal plane (\(\delta\)). The total solar radiation heat flux \(\mathrm{(q}_{\mathrm{s}}\)) is controlled by the girder section's absorptivity (\(\alpha\)) as presented in Equation (6) [37]. This absorptivity is a crucial factor in determining the amount of thermal energy that the girder section is capable of absorbing.