# Performance-Based Analysis of Single-Layer Cylindrical Steel Reticulated Shells in Fire
## 5 Structural Response Analysis in Different Fire Scenarios
### Structural Model
#### 5.1.2 Material Properties  
In order to investigate the behaviors of the cylindrical latticed shell under different temperatures, it is necessary to define the material properties changing with temperature including the coefficient of thermal expansion, elasticity modulus, yield strength, and stress-strain relationship. According to the Chinese Technical Code for Fire Safety of Steel Structure in Buildings (CECS200) [21], the material properties used in this paper are illustrated as follows. * Elasticity modulus  
The elasticity modulus decreases with the increase of temperature. Along with the decrease of the elasticity modulus, the structural deformation will gradually increase until it reaches the limit state and bearing capacity. The reduction coefficient \(\psi_{T}\) of elastic modulus is defined as the ratio of the elasticity modulus at high temperature T \({}^{\circ}\)C to the elasticity modulus at room temperature, which is shown as \(\psi_{T}=E_{T}/E\). In this paper, the reduction coefficient of elastic modulus of steel at high temperature is selected according to CECS200 shown as Equations (9). \[ \begin{cases}\psi_{T}=\frac{7T_{s}-4780}{6T_{s}-4760},&20^{\circ}\text{C}\leq T _{s}<600^{\circ}\text{C}\\ \psi_{T}=\frac{1000-T_{s}}{6T_{s}-2800},&600^{\circ}\text{C}\leq T_{s}\leq 1 000^{\circ}\text{C}\end{cases}  \tag{9}\]  
* Yield strength \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Fire Source** & **\(Q_{max}\)** & **Rise-Span** & **Member Sections** & **Rood Load** & **Thickness** \\
**Location** & **(MW)** & **Ratio** & **(mm)** & **(kN/m2)** & **(mm)** \\ \hline I & & 1/2 & \(\Phi 110\times 4\), \(\Phi 54\times 4\), \(\Phi 110\times 4\), \(\Phi 57\times 4^{\circ}\) & 0.6 & Di = 1 \\ II & 20 & 1/3 & \(\Phi 110\times 4\), \(\Phi 60\times 4\), \(\Phi 108\times 4\), \(\Phi 60\times 4\) & 1.2 & Di = 2 \\ III & & 1/4 & \(\Phi 106\times 4\), \(\Phi 60\times 4\), \(\Phi 146\times 5\), \(\Phi 76\times 4\) & 1.8 & Di = 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters of single-layer cylindrical reticulated shell. Figure 15: Integral points of the steel tubular members. The reduction coefficient \(\eta_{T}\) of yield strength is defined as the ratio of the yield strength at high temperature T \({}^{\circ}\)C to the yield strength at room temperature, which is \(\eta_{T}=f_{yT}\eta_{y}\) as Equation (10). \[\eta_{T}=1.0,\,20\ ^{\circ}\mathrm{C}\leq T_{s}<300\ ^{\circ}\mathrm{C}\] \[\eta_{T}=1.24\times 10^{-8}T_{s}{}^{3}-2.096\times 10^{-5}T_{s}{} ^{2}+9.228\times 10^{-3}T_{s}-0.2168,\,300^{\circ}\mathrm{C}\leq T_{s}<800\ ^{\circ}\mathrm{C} \tag{10}\] \[\eta_{T}=0.5-T_{s}/2000,\,800\ ^{\circ}\mathrm{C}\leq T_{s}\leq 1 000^{\circ}\mathrm{C}\]
* Constitutive model  
The stress-strain relationship of steel at high temperature is investigated by scholars globally. Based on the European standard EC3 [22], the constitutive model at high temperature without considering post yield strengthening is utilized in this paper. The stress-strain relationship model of steel at different temperatures prescribed in EC3 is shown in Figure 16.  
* Physical properties  
Material physical properties including density, heat capacity, coefficient of thermal expansion and coefficient of thermal conductivity are determined according to CECS200. Because the change of temperature has little effect on the density of steel, the density \(\rho_{s}\) is 7580 kg/m\({}^{3}\). The heat capacity \(C_{s}\) is 600 J/(kg.\({}^{\circ}\)C). The coefficient of thermal expansion is used to express the volume change caused by the change of unit temperature. The coefficient of thermal expansion refers to the ratio of the elongation and the original length when the temperature of steel member changes by 1 \({}^{\circ}\)C. Thermal expansion coefficient \(\alpha_{s}\) is \(1.2\times 10^{-5}\). The thermal conductivity of material is one of the important physical properties.