# Influence of Internal Pressure on Hollow Section Steel Members in Fire
## 3 Gas Pressure Inside the Steel Hollow Profiles  
The issue of internal pressure in tubular members can be approached similarly to that of pressure vessels. The pressure inside the vessel caused by the expanding gas is based on the fundamental ideal gas, Equation (1), where \(P\) is the pressure, \(V\) is the volume of the space occupied by the gas, \(T\) is the temperature of the gas, \(n\) is the number of moles (fundamental unit in chemistry), and \(R\) is the ideal gas constant. \[PV=nRT \tag{1}\]  
It is assumed that the gas is the same in different temperature conditions. The change in the state of the gas can be estimated using the well-known Equation (2), where \(P_{0}\), \(V_{0}\), and \(\theta_{0}\) are the pressure, volume, and temperature in the initial state (in our case, initial inner volume of the profile at normal temperature and atmospheric pressure), and \(P_{1}\), \(V_{1}\), and \(\theta\) are the pressure, volume, and temperature in the modified state (in our case, the  
Figure 1: Designed truss—Fragment of the IFC model. Figure 2: Deformed trusses with oxidized SHS members. event of fire). In order to obtain the overpressure \(\Delta P\), Equation (2) can be modified into Equation (3). \[\frac{P_{0}V_{0}}{\theta_{0}}=\frac{P_{1}V_{1}}{\theta} \tag{2}\]  
\[\Delta P=\frac{P_{0}V_{0}}{\theta_{0}}\frac{\theta}{V_{1}}-P_{0} \tag{3}\]  
The ideal gas law is a good approximation of the real gas behaviour in case that the working temperature is higher than to the critical value \(\theta_{cr}\), while working pressure is lower compared to the critical value \(P_{cr}\). The thermodynamic behaviour of air is considered close to the ideal gas [21]. Critical parameter values of air are as follows: \(\theta_{cr}=-140\)\({}^{\circ}\)C; \(P_{cr}=3.8\) MPa. The working temperature in the case of fire is obviously above \(-140\)\({}^{\circ}\)C. As will be shown further, the practical range of the internal pressure is below 0.4 MPa. Ideal gas is often used in models related to the dynamics of fire [22; 23]. The validity of the ideal gas law is therefore assumed. The increase in temperature within the enclosed section is primarily the result of the elevation in external temperatures. This, in turn, causes the temperature of the steel material to rise, consequently leading to an increase in the temperature of the trapped gas within the tubular steel member. The time lag between the temperature of the steel profile and the internal gas is negligible [24]. The primary factor contributing to the increase in overpressure \(\Delta P\) is linked to the rise in gas temperature \(\theta\) partially counterbalanced by the expansion of volume \(V_{1}\). The change in volume from \(V_{0}\) to \(V_{1}\) is caused by two main factors: thermal expansion of the steel and mechanical deformation under so-called hoop stresses. For simplicity, at first the focus is on circular hollow sections, as shown in Figure 3. On the left, a tubular cross section is in the initial state. On the right, the same section is presented under changed conditions: the temperature of both the steel and the gas inside the inner space of the section has increased; because of the internal pressure, hoop stresses have been activated, causing axial stresses and strains (\(\epsilon_{s}\)) inside the steel material. The increase in temperature of the steel leads to the development of thermal strains (\(\epsilon_{th}\)) in the steel section. Mechanical and thermal strains induce changes in the inner volume up to \(V_{1}\), which partially compensates for the increase in the inner pressure. The following notation is used: \(D_{0}\) is the initial external diameter of the tubular cross section; \(D_{1}\) is the external diameter of the deformed tubular cross section; \(t\) is the thickness of the wall of the tubular section; and \(L_{0}\) is the length of the centreline of the tubular cross section, calculated as follows:  
\[L_{0}=\pi(D_{0}-t) \tag{4}\]  
Figure 3: Internal pressure inside the circular hollow section.