# Influence of Internal Pressure on Hollow Section Steel Members in Fire
## 4 Influence of Internal Gas Pressure on the Capacity of Hollow Section Members  
It is quite easy to analyze the impact of the inner pressure on the stress state of circular profiles using well-established hoop-stress theory. As shown in Figure 6, the hoop stress in circular sections remains relatively moderate up to the gas temperature value of 850 \({}^{\circ}\)C (\(\sigma_{H}\)/\(f_{y,\theta}\approx 0.2\)). The stress state caused by internal pressure is quite different for square or rectangular hollow sections, as is demonstrated in Figure 8.  
In the case of a circular section, the internal pressure causes only axial tension force in the section wall. In rectangular sections, the internal pressure also produces a bending moment (green line). In Figure 8, \(L_{i}\) is the length of the straight segment of the section (the distance between the rounded corners). It is possible to estimate the value of the bending moment using the classical equation for a beam with fixed ends:  
\[M_{max}=\frac{\Delta PL_{i}^{2}}{12} \tag{10}\]  
Equation (10) gives a good estimation if linear material behaviour is assumed. Bending moment values in the middle of the straight segment (\"span\") and on the edge of the straight segment (\"support\") are absolute values equal to those calculated using Equation (10). The internal pressure causes tensile stresses in the outer fibres in the middle  
Figure 8: Internal pressure inside a CHS and SHS. Figure 7: Hoop stresses vs. temperature related to proportionality limit stress \(f_{\bar{r},\theta}\). of the straight segment and on the inner fibres of the rounded corners. In order to compare the maximum stresses for the SHS and CHS sections of the same dimension \(D\) and thickness \(t\), diagrams in Figure 9 are presented. The term \"hoop stresses\" will be conditionally used here analogous to CHS for the normal stresses caused by internal pressure in the walls of the SHS. Hoop stresses in the wall are normal stresses orthogonal to the longitudinal bending or axial stresses of the SHS member. The maximum hoop stresses for a CHS (red line) and an SHS section with the same dimension and thickness (blue line) are compared. The linear material model is adopted. The value of internal pressure \(\Delta P\) = 200 kPa is used, and the thickness is \(t\) = 6 mm (the choice of \(\Delta P\) and \(t\) is trivial for this purpose, as it just scales the maximum stress values). It is evident that the influence of the internal pressure on the value of maximum hoop stresses is much more intense for the SHS than for CHS sections, and the growth of the section dimensions intensifies the influence. The same conclusion can be reached for the RHS sections. In the standard design procedure, the analysis is focused on normal longitudinal stresses from axial force, bending, and shear stresses. The value of hoop stresses is relatively low and is ignored, especially when the design is performed according to the theory of beams. The Von-Mises yield criterion is often used for steel structures in a multidimensional stress state. Longitudinal normal stresses from the main action (axial force and bending moment) are denoted as \(\sigma_{x}\), and the orthogonal normal stresses are represented by the hoop stresses \(\sigma_{H}\). Ignoring shear stresses, the principal plane stress criterion (\(\sigma_{vonMises}\)) is formulated as Equation (11). \[\sigma_{vonMises}=\sqrt{\sigma_{x}^{2}+\sigma_{H}^{2}-\sigma_{x}\sigma_{H}} \tag{11}\]  
The interdependence between the \(\sigma_{vonMises}\), \(\sigma_{x}\), and \(\sigma_{H}\) is presented in Figure 10, where stresses are normalized to yield limit stress \(f_{y}\). In that case that both \(\sigma_{x}\) and \(\sigma_{H}\) are of the same sign (both correspond to tension or compression), hoop stresses will delay the approach of the plastic state. For example, if \(\sigma_{x}=1.00\,f_{y}\) and \(\sigma_{H}=0.50\,f_{y}\), then \(\sigma_{vonMises}=0.90\,f_{y}\).