## 2 Materials and Methods
### The Basics of the Fire Design  
The basics of designing the structure under fire conditions are described in Eurocode EN 1991-1-2. If the fire behavior is not modeled by specialists in the special software for a specific building, it is allowed to use predefined fire curves. The basic fire curves are as follows: standard, hydrocarbon, and the curve of external fire. If some details of compartments such as floor and walls, and number and area of openings are known, the parametric fire curve can be used. In the presented article, the standard fire curve was applied. In this case, the temperature of fire gases is defined as follows:  
\[\theta_{g}=20+345 log_{10}(8t+1) \tag{2}\]  
where: t--time of fire duration (min).  
Knowing the temperature of fire gases, it is possible to calculate the temperature of steel members. For a uniform temperature distribution in a cross section, the temperature increase \(\Delta\theta_{a,t}\) of an insulated steel member during the time interval \(\Delta\)t is obtained from the following formula [24]:  
\[\Delta\theta_{a,t}=\frac{\lambda_{p}A_{p}/V\big{(}\theta_{g,t}-\theta_{a,t} \big{)}}{d_{p}c_{a}\rho_{a}(1+\varphi/3)}\Delta t-\Big{(}e^{\varphi/10}-1 \Big{)}\Delta\theta_{g,t}\ \text{but}\ \big{(}\Delta\theta_{a,t}\geq 0\ if\ \ \Delta\theta_{g,t}=0\big{)} \tag{3}\]  
with  
\[\varphi=\frac{c_{p}\rho_{p}}{c_{a}\rho_{a}}d_{p}A_{p}/V \tag{4}\]  
where:  
\(A_{p}\)/\(V\) is the section factor for steel members insulated by fire protection material, for the cross section with contour insulation, heated from each side, it is defined as the ratio of steel perimeter to steel cross-section area;  
Ap is the appropriate area of fire protection material per unit length of the member (m\({}^{2}\)/m);  
\(V\) is the volume of the member per unit length (m\({}^{3}\)/m);  
\(c_{a}\) is the temperature-dependent specific heat of steel (J/kgK);  
\(c_{p}\) is the temperature.independent specific heat of the fire protection material (J/kgK);  
\(d_{p}\) is the thickness of the fire protection material (m);  
\(\Delta t\) is the time interval; it should not be more than 30 s;  
\(\theta_{a,t}\) is the steel temperature at time t (\({}^{\circ}\)C);  
\(\theta_{g,t}\) is the ambient gas temperature at time t (\({}^{\circ}\)C);  
\(\Delta\theta_{g,t}\) is the increase in the temperature of the ambient gas during the time interval \(\Delta t\);  
\(\lambda_{p}\) is the thermal conductivity of the fire protection system (W/mK);  
\(\rho_{a}\) is the unit mass of steel (kg/m\({}^{3}\));  
\(\rho_{p}\) is the unit mass of the fire protection material (kg/m\({}^{3}\));  
The characteristic of insulation material (\(\rho_{p}\), c\({}_{p}\), \(\lambda_{p}\), \(d_{p}\)) and unit mass of steel (\(\rho_{a}\)) according to Eurocode can be treated as temperature-independent. The dependence of specific heat of steel on temperature is described in detail in part 3.4.1. of the Eurocode [24].  
The increase in the temperature of the steel elements results in the decrease of mechanical properties values, including the yield strength and Young's modulus. The appropriate dependencies are presented in Table 1. For the intermediate values of temperature, the linear interpolation can be used.