# Ideal Gas in Structural Analysis

### Technical Article

001507

20 February 2018

In theory, an ideal gas consists of freely moving mass particles without extension in a volume space. In this space, each particle moves at a speed in one direction. The collision of one particle with another particle or the volume limitations lead to a deflection and a change in the speed of the particles.

The state of the enclosed gas can be described by means of these assumptions about a thermodynamic equilibrium. This results in the following general gas equation:
p ∙ V = n ∙ R ∙ T
with the state variables
p = compression
V = solid
n = molar amount
R = universal gas constant
T = temperature

#### Properties of Ideal Gases

By keeping certain state variables constant in the general gas equation, special properties of the ideal gas are obtained. Being familiar with these properties helps you to use ideal gases in structural analysis and helps you to simulate certain load states accordingly.

Isothermal state change (Boyle-Mariotte)
If we keep the variables T and n constant and increase the applied pressure p, the volume V of the considered gas unit is reduced.

The following applies:
$\begin{array}{l}\mathrm p\;\sim\;\frac1{\mathrm V}\\\mathrm p\;\cdot\;\mathrm V\;=\;\mathrm{const}\\\frac{{\mathrm p}_1}{{\mathrm p}_2}\;=\;\frac{{\mathrm V}_2}{{\mathrm V}_1}\end{array}$

Isobaric state change (Gay-Lussac)
If the quantities p and n are kept constant and the acting temperature T is increased, the volume V of the considered gas unit is increased.

The following applies:
$\begin{array}{l}\mathrm V\;\sim\;\mathrm T\\\frac{\mathrm V}{\mathrm T}\;=\;\mathrm{const}\\\frac{{\mathrm V}_1}{{\mathrm V}_2}\;=\;\frac{{\mathrm T}_1}{{\mathrm T}_2}\end{array}$

Isochoric change of state (amotons)
If the values V and n are kept constant and the acting temperature T is increased, the pressure p of the relevant gas unit is increased.

The following applies:
$\begin{array}{l}\mathrm p\;\sim\;\mathrm T\\\frac{\mathrm p}{\mathrm T}\;=\;\mathrm{const}\\\frac{{\mathrm p}_1}{{\mathrm p}_2}\;=\;\frac{{\mathrm T}_1}{{\mathrm T}_2}\end{array}$

#### Use in structural analysis

In structural analysis, enclosed gases are usually used for the transfer of external forces. The requirement here is that a locally acting force at a certain location on the solid shell can be transported via the trapped gas to all other sides of the solid shell.

This property is used, for example, for insulating glass panes or inflated membrane cushions. In both cases, the solid shell consisting of structural elements is described and filled with a gas. For insulating glass panes, the volume limitation consists of rigid shell elements and membrane diaphragms made of non-rigid membrane elements. In both cases, however, the wind or snow load attacks on one side of the volume limitation and is transferred via the enclosed gas to the adjacent volume limits.

Since the temperature does not change suddenly in the load situations considered in the construction industry, an ideal gas with isothermal state properties is usually simulated in the solid shell.

#### Implementation in RFEM

Solid definitions are available in RFEM. These volumes are described in relation to the surrounding surfaces. In such a solid cell consisting of surrounding shell and solid components, you can enter a volume definition with the type Gas. The resulting gas volume requires a description of the enclosed gas and a definition of the atmospheric state variables. The atmospheric state variables have no effect on the enclosed solid and describe only an initial situation for the simulation.

In the assigned load cases, a corresponding solid load can be applied for each gas volume. To simulate open or closed solids, it is possible to specify resulting pressures / solids or changes in pressure / volume.

#### References

 [1] Wikipedia: Ideal gas [2] Wikipedia: Thermal equation of state of ideal gases [3] Wagner, R .: Building with cables and membranes. Berlin: Beuth, 2016