# Fire Behavior of U-shaped Steel Beams Filled with Demolished Concrete Lumps and Fresh Concrete
## 4 Numerical Study
### Thermal and Mechanical Properties  
Variations of the thermal and mechanical properties of the U-shaped steel and longitudinal reinforcements (including density, specific heat, thermal conductivity, thermal strain, and stress-strain relationship) with temperature are determined, according to Eurocode 4 [38]. The elastic modulus of the U-shaped steel and longitudinal reinforcements at room temperature is taken as 2.1 \(\times\) 10\({}^{5}\) MPa. The yield strengths of the U-shaped steel and longitudinal reinforcements listed in Table 1 are employed in the structural analysis. The thermal properties of the demolished concrete are assumed to be the same as those of the fresh concrete and are determined in accordance with Eurocode 2 [39]. The upper limit of concrete thermal conductivity given in Eurocode 2 is adopted and the moisture content of the concrete is taken as 5% by weight. The density of concrete at room temperature is taken as 2400 kg/m\({}^{3}\)  
Figure 21: Finite-element model of the beam for thermal analysis.  
Figure 22: Finite-element model of the beam for structural analysis.  
During the structural analysis, the in-filled concrete composed of DCLs and FC is considered as a whole and the combined compressive strength (\(f_{\text{cu, com}}\)) of the in-filled concrete is calculated based on Equation (1) [24]. The stress-strain relationship of the in-filled concrete at an elevated temperature is determined according to Eurocode 2. The thermal strain of concrete with siliceous aggregates given in Eurocode 2 is employed.  
\[f_{\text{cu, com}}= f_{\text{cu, old}}\times\eta+f_{\text{cu, new}}\times(1-\eta) \tag{1}\]  
Due to a lack of information about the density and specific heat of the fire insulation at a high temperature, the room temperature density is adopted in the numerical analysis and the specific heat is taken as 1047 J/(kg \({}^{\circ}\)C) [40]. The thermal conductivity of the insulation generally increases as the temperature rises and a rate of increase ranging from about 0.0001 W/m K\({}^{2}\) to 0.0003 W/m K\({}^{2}\) has been recommended by other researchers [41; 42]. In this way, based on the aforementioned value pertaining to room temperature, the thermal conductivity of the fire insulation at an elevated temperature (\(T\)) is determined, according to Equation (2).  
\[\lambda_{\text{i}}=0.0806+0.00015(T-20) \tag{2}\]  
[PERSON] and [PERSON] [27] have investigated the influence of thermal resistance at the steel tube-concrete interface on the temperature distribution of the concrete-filled steel tubular columns and constant values of 0.026 m\({}^{2}\) K/W and 0.022 m\({}^{2}\) K/W were, respectively, suggested for the interface thermal resistance of the circular and rectangular columns with a cross-sectional size above 300 mm. The loading pattern of the U-shaped steel beams filled with DCLs and FC is completely different from that of the concrete-filled steel tubes, which means the values of interface thermal resistance for the columns may be not suitable for the beams. Through trial-and-error, the thermal resistance at the interface between the web of the U-shaped steel and the in-filled concrete is estimated as 0.03 m\({}^{2}\) K/W and that at the interface between the bottom flange of the U-shaped steel and the in-filled concrete is estimated to be 0.02 m\({}^{2}\) K/W. It should be noted that the thermal resistance at the steel-concrete interface is expressed as \(t_{\text{i}}/\lambda_{\text{i}}\) where \(t_{\text{i}}\) is the thickness of the putative interface layer between the steel and the in-filled concrete (assumed to be 2 mm) and \(\lambda_{\text{i}}\) is the thermal conductivity of the interface layer. During the thermal analysis, the density and specific heat of the interface layer are ignored [43].