# Influence of Internal Pressure on Hollow Section Steel Members in Fire
## 4 Influence of Internal Gas Pressure on the Capacity of Hollow Section Members
If \(\sigma_{x}\) and \(\sigma_{H}\) have the opposite sign, the situation is reversed--hoop stresses will intensify the approach of the plastic state. For example, if \(\sigma_{x}=0.40\,f_{y}\) and \(\sigma_{H}=-0.75f_{y}\), then \(\sigma_{vonMises}\) = 1.00 \(f_{y}\). In the case of SHS, hoop stresses mainly result from the bending moment, which means that hoop stresses vary across the thickness from compression to tension. Consequently, no positive effect should be expected from the hoop stresses on the overall capacity of the SHS and RHS. A certain positive impact can appear for the CHS in tension. Figure 9: Hoop stresses vs. section dimension—Comparison between CHS and SHS sections. The influence of the internal pressure on the axial capacity of the SHS and RHS in fire conditions was investigated using non-linear FEM simulations. Considering the above, it is expected that internal pressure will have a notable negative impact on both the compression and the tensile axial resistance of the section. It is assumed that the bending capacity should show tendencies similar to compression and tension capacities. Simulations were performed using ANSYS 2023 R1 software implementing the \"SHELL181\" finite element types. The geometry of the model was rather simple, as shown in Figure 11a. Steel S355 was applied for the analysis. The material model from EN 1993-1-2 [1] was used as described above (Figure 4, Equation (8)). Calculations were performed for the set of cold-formed hollow sections [26] presented in Table 1, characterizing the practical range of profiles used in construction. Simulations were performed in two stages as follows. At first, the case with no axial load was analyzed: the temperature and the corresponding internal pressure were simultaneously elevated in accordance with Equation (3) until the limit state due to plasticization \begin{table}
\begin{tabular}{c c c} \hline \hline
**Section Type** & **Dimension** & **Wall Thickness** \\ \hline CHS & \(\mathcal{O}\)152.4 mm & 3 mm/5 mm/6 mm \\ CHS & \(\mathcal{O}\)323.9 mm & 5 mm/8 mm/10 mm \\ SHS & 90 mm \(\times\) 90 mm & 3 mm/5 mm/6 mm \\ SHS & 150 mm \(\times\) 150 mm & 4 mm/6 mm/12.5 mm \\ SHS & 300 mm \(\times\) 300 mm & 6 mm/8 mm/12.5 mm \\ RHS & 400 mm \(\times\) 200 mm & 6 mm/8 mm/12.5 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cross sections used for the simulations. Figure 11: (a) FEM model; (b) simulation results/deformed shape. was reached. The achieved maximum temperature and implicitly the corresponding internal pressure was registered and denoted as \(\theta_{max}\). For the second stage, 300 \({}^{\circ}\)C was chosen as the initial temperature, because from 300 \({}^{\circ}\)C upwards, the yield strength of steel starts to degrade. The temperature was increased in steps up to \(\theta_{max}\), obtained in the first stage. For each temperature value \(\theta_{i}\), the internal pressure was generated in accordance with Equation (3), and then axial force was applied and increased until the limit state due to plasticization was reached. For each temperature value \(\theta_{i}\), two cases, compressive and tensile axial force, were simulated. The maximum values of the tensile and compressive axial force achieved at every temperature \(\theta_{i}\) were denoted as \(N_{fl,INP}\). For relatively low internal pressure values and corresponding lower temperature values, the axial force value \(N_{fl,INP}\) was limited by the axial plastic capacity of a section \(N_{fl,INP}\), which was calculated according to the model from EN 1993-1-2 [1] using Equation (12), where \(A\) is the cross-sectional area and \(f_{y,\theta}\) is the temperature-adjusted yield limit stress. It was expected, for higher temperature/internal pressure values, that \(N_{fl,tl,fl}\) would diverge from \(N_{fl,tl,fl}\). The procedure was repeated for all the sections from Table 1. An example of a simulated deformed shape is presented in Figure 11b.