## 1 Introduction  
Closed girder cross-sections, such as box girders, small box girders, and adjacent box girders, are typically used worldwide for concrete bridges [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13]. Numerous cases of recorded bridge damages, such as severe cracking, deterioration, or even failure, particularly in bridges with closed girder cross-sections, were caused by temperature-induced stresses and deformations [14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26]. Therefore, it is particularly important to accurately predict the temperature distributions on closed girder cross-sections. Moreover, thermal influence is considered to be a key parameter not only for bridges with expansion joints but also for jointless bridges [7; 13; 22; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39]. Finite element simulation is one of the most commonly used and effective methods to analyze the temperature distributions on concrete bridge cross-sections. The accurate input of external and internal thermal boundary conditions for the closed girder cross-sections in the finite element model (FEM) has a great influence on the accuracy of the calculated results. The Robin boundary condition can be used to simulate the external and internal thermal boundary conditions. For the external thermal boundary condition, the temperatures associated with solar radiation, convection, irradiation, and the temperature of the surrounding fluid medium, as well as the overall heat transfer coefficient (which is the combination of the convection heat transfer coefficient and the radiation heat transfer coefficient), should be taken into account [7; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46]. The internal thermal boundary condition is affected by the temperatures inside the cavities and the convection heat transfer coefficient because it is not influenced by solar radiation [7; 13; 14; 15; 16; 17; 22; 23; 25; 26; 39; 40; 42; 43; 44; 45; 46]. The ambient air temperature can be measured by a meteorological station. The temperature variation inside the cavity is usually less than that of the ambient air temperature and has a time delay [42; 43] because there is no solar radiation inside the cavity and the thermal conductivity of concrete is low. In order to accurately simulate the internal thermal boundary conditions for concrete closed girder cross-sections, researchers have proposed different methods. Temperature sensors were installed inside the cavities by some researchers during the girder construction process to measure the temperature variation curve inside the cavity (hereafter referred to as the Measured Temperature Method) [17; 23; 25; 39; 44]. The Measured Temperature Method can reflect the actual temperature inside the cavity, but measurements on site are cost- and time-prohibitive. Therefore, the application range of the Measured Temperature Method is limited. When the measured temperature inside the cavity can not be obtained, the ambient air temperature was used to simulate the temperature inside the cavity (hereafter referred to as the Ambient Temperature Method) [15; 45]. Numerical methods were used by some researchers to calculate the temperature variation curve inside the cavity, such as the mean temperature over the previous two days (hereafter referred to as the Mean Temperature Method) [27; 28; 41], using the following Equation [40]:  
\[T_{\rm mean}=0.25\times(T_{\rm max,i}+T_{\rm min,i}+T_{\rm max,i-1}+T_{\rm min,i-1}) \tag{1}\]  
where \(T_{\rm mean}\) is the mean temperature over the previous two days, \(T_{\rm max,i}\), \(T_{\rm min,i}\), \(T_{\rm max,i-1}\), and \(T_{\rm min,i-1}\) are the maximum and minimum temperatures of the ambient air temperature curves for the ith and (i.-1)th days. The temperature inside the cavity that is obtained or calculated by using the Measured Temperature Method, Ambient Temperature Method, or Mean Temperature Method can be input as the internal thermal boundary condition in the FEM.