# Influence of Internal Pressure on Hollow Section Steel Members in Fire
## 3 Gas Pressure Inside the Steel Hollow Profiles
In the case of a circular hollow section, the hoop stresses \(\sigma_{H}\) can be calculated using the established theory as:  
\[\sigma_{H}=\frac{D_{1}}{2t}\Delta P \tag{5}\]  
Mechanical strain \(\varepsilon_{s}\) can be calculated using an available material model. In this paper, a material model from Eurocode [1] is used. A stress-strain diagram for the steel class S355 is presented in Figure 4. The strain \(\varepsilon_{s}\) can be calculated from the hoop stresses \(\sigma_{H}\) using the reversed standard procedure. In case of the material model from EN 1993-1-2 [1], the procedure is as follows [25]: in the case that hoop stresses are lower than the temperature-adjusted proportionality limit stress (\(f_{p,\theta}\)), the strain can be easily calculated using Equation (6); in the opposite case, the strains must be calculated using Equation (7). Figure 1:0  
Figure 1:1  
In Equations (6) and (7), \(\varepsilon_{y,\theta}\) is the yield strain at temperature \(\theta\); \(a\) and \(c\) are the parameters of the EN 1993-1-2 [1] material model; and \(E_{a,\theta}\) is the temperature-adjusted value of the deformation modulus. The model for thermal elongation is also adopted from EN 1993-1-2 [1], as described in Equation (8). \[ \begin{array}{c}\theta<750\ ^{\circ}\mathrm{C}:\ \varepsilon_{th}=1.2*10^{-5 }\theta+0.4*10^{-8}\theta^{2}-2.616*10^{-4}\\ 750\ ^{\circ}\mathrm{C}\ \leq\theta<860\ ^{\circ}\mathrm{C}:\ \varepsilon_{th}=1.1*10^{-2}\\ 860\ ^{\circ}\mathrm{C}\ \leq\theta<1200\ ^{\circ}\mathrm{C}:\ \varepsilon_{th}=2*10^{-5} \theta-6.2*10^{-3}\end{array}  \tag{8}\]  
In the case that \(\varepsilon_{s}\) and \(\varepsilon_{th}\) are defined, the length \(L_{1}\) of the centreline of the profile can be calculated using Equation (9), and the increased volume of the internal space of the circular section can be easily calculated. \[L_{1}=L_{0}(1+\varepsilon_{s}+\varepsilon_{th}) \tag{9}\]  
It is evident that the process is non-linear: the growth of the overpressure partially depends on the mechanical deformation, which itself is a function of overpressure. The mechanical and thermal response models are non-linear functions of temperature. Figure 4: EN 1993-1-2 [1] material model for S355 steel in fire conditions: normalized engineering stress vs. engineering strain. The dependence of the overpressure on the temperature and the impact of the section deformation on the compensation of the overpressure are presented in Figure 5. In Figure 6, the hoop stresses are reported in relation to the yield limit stress \(f_{y,\theta}\), and in Figure 7, hoop stresses are reported in relation to the proportionality limit stress \(f_{p,\theta}\). In Figure 5, the increase in overpressure for conditions where the volume has not changed (i.e., no deformation of the steel material takes place, \(V_{0}=V_{1}\)) is represented by the black line. Lines of other colours represent the pressure growth inside various circular hollow sections. It is evident that the influence of steel material deformation on the overpressure remains quite moderate up to temperatures around 1000 \({}^{\circ}\)C, especially in the practical temperature range of 350 \({}^{\circ}\)C to 850 \({}^{\circ}\)C, where the overpressure is lower by 1.7% to 3.2% compared to that in the case of \(V_{0}=V_{1}\) (black line). The drop in overpressure for the \(\varTheta 300\times 5\) profile in Figure 5 at around 1000 \({}^{\circ}\)C is explained by the patterns of the hoop stresses shown in Figures 6 and 7. For sections of larger diameter, the hoop stresses reach the proportionality limit stress faster than for tubes of smaller diameter. For the \(\varTheta 300\times 5\) section at temperature 1000 \({}^{\circ}\)C, the hoop stresses start to approach the yield limit stress, which, according to the material model, implies a very large mechanical strain \(\varepsilon_{s}\). This work will further focus on rectangular hollow sections, widely used as a standard solution for roof trusses.