## 2 Finite Element Simulation  
### Box Girder  
One concrete bridge with a three-cell single box girder in Shenzhen, China was chosen as the case study. The dimensions of the box girder cross-section are illustrated in Figure 1. Twenty-one temperature sensors were installed in the box girder during the construction process, as shown in Figure 1. These include five sensors in the top flange (T-1 to T-5), three sensors in the bottom flange (B-1 to B-3), eight sensors in the exterior webs (W-1 to W-4 in the left exterior web and W-7 to W-10 in the right exterior web), two sensors in the interior webs (W-5 in the left interior web and W-6 in the right interior web), and three sensors (C-1 to C-3) in three cells of the box girder. It can be observed from the field test data that the differences between the temperatures inside the cavities and the ambient air temperature were large in the summer because of high solar radiation, while the temperature differences were small in the winter. The daily measured temperature responses obtained at different times of year have been used by researchers to analyze the temperature distributions on cross-sections [13; 22; 24; 25; 39]. Due to space limitations, the data from 13 August 2017 in summer, which is a sunny day with the highest ambient air temperature during the monitoring period, as well as high solar radiation and low wind speed, were selected for analysis. The meteorological data, including ambient air temperature, hourly global solar radiation, hourly diffuse solar radiation, and wind speed, were measured using a movable automatic meteorological station at the bridge site with a measurement time interval of one hour. The midas FEA software was chosen to establish the FEM for the box girder cross-section. Two-dimensional plane strain elements, which can be used for steady-state or transient analyses,  
Figure 1: Layout of the box girder cross-section (unit: cm). were used, as shown in Figure 2. Each node has a single temperature degree of freedom. The measurement time interval in the FEM was set with the same as that for the field test. For the thermal parameters of the concrete, density, heat conductivity, and specific heat values of 2500 kg/m\({}^{3}\), 1.5 W/(m\({}^{\circ}\)C), and 900 J/(kg\({}^{\circ}\)C), respectively, were used [14; 15; 17; 47]. Based on the measured ambient air temperature, wind speed, and solar radiation obtained from the movable automatic meteorological station at the bridge site, the parameters associated with solar radiation, convection, irradiation, and the temperature of the surrounding fluid medium were used as the external thermal boundary conditions for the box girder in the FEM. The influence of solar radiation on different parts of the box girder was considered by using the following rules and assumptions: (a) the external surface of the top flange is influenced by both the beam and diffuse solar radiation; (b) the external surfaces of the two exterior webs in the shadow due to the top flange are influenced by the diffuse solar radiation and ground reflection; (c) the external surfaces of the two exterior webs not in the shadow are influenced by the beam solar radiation, diffuse solar radiation, and ground reflection; and (d) the undersides of the top and bottom flanges are influenced by the ground reflection alone. The environmental and climatic conditions existing on previous days must be considered to accurately determine the initial temperature distribution [13; 14; 17; 25; 39]. A 120-h period with the same environmental conditions imposed cyclically was considered in the analysis. The four simulation methods for the internal thermal boundary conditions summarized in Section 1 were considered in the FEMs of the box girder. The internal thermal boundary condition was not influenced by solar radiation.