# Investigating Load-Bearing Capabilities and Failure Mechanisms of Inflatable Air Ribs
## 2 The Load-Bearing Performance of Air Ribs with Different Rise-Span Ratios
### The Failure Mechanism Analysis of Air Ribs Under Vertical Loading
Initially, air ribs go downward linearly with the increase in the vertical load, which is crucial to evaluate the bearing capacity of ribs. With the increase in the load, the displacement increases disproportionately, the structure approaches a buckled state, and obtains an impending structural failure. For the air rib named 0.333-S, it has a similar pattern of stress distribution, and the critical load is 2.62 \(\text{kN}/\text{m}^{2}\), meaning it provides greater resistance to buckling and instability. It can be seen that the critical bearing load of the 0.5-S/R is the minimum and lower than 1.46 \(\text{kN}/\text{m}^{2}\). When the load is 1.46 \(\text{kN}/\text{m}^{2}\), the stress distributes uniformly initially. With the increase in the load, the stress concentrates around the waist region of the rib locally, and the structure enters an unstable state when the load is 1.47 \(\text{kN}/\text{m}^{2}\). The \(q_{0.5\text{-}S/R}\) is large, but the \(F_{0.5\text{-}S/R}\) is the minimum, and the critical bearing load \(N_{0.5\text{-}S/R}\) is based on the value of \(F_{0.5\text{-}S/R}\). So, the rib named 0.5-S/R fails at the upright region, which is consistent with the structural form of the rib in the failure condition. The \(q_{0.667\text{-}R}\) and \(q_{0.5\text{-}S/R}\) are similar, but \(F_{0.667\text{-}R}\) is larger than \(F_{0.583\text{-}R}\) and \(F_{0.5\text{-}S/R}\). The critical bearing load (\(N_{0.5\text{-}S/R}\)) is based on the value of \(q_{0.667\text{-}R}\), and it is 1.73 \(\text{kN}/\text{m}^{2}\). Then, the rib named 0.667-R fails at the arc-like region, which can be seen in Figure 8. In this paper, the variation of buckling load (\(q\)) or critical failure load (\(F\)) with different structures are studied, but the specific value of buckling load (\(q\)) and critical failure load (\(F\)) are not calculated. The failure mechanism of the ribs named 0.417-S and 0.583-R are confirmed with the deformation results at structural failure. For the rib named 0.417-S, \(q_{0.417\text{-}S}\) and \(F_{0.417\text{-}S}\) are large, and it fails at the middle of the arc-like region; so, \(q_{0.417\text{-}S}\) is smaller than \(F_{0.417\text{-}S}\), and the critical bearing load (\(N_{0.417\text{-}S}\)) is 1.97 \(\text{kN}/\text{m}^{2}\). The \(q_{0.583\text{-}R}\) and \(F_{0.583\text{-}R}\) of the rib named 0.583-R are smaller than those of the rib named  
Figure 7: The parameter schematic of the upright region. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Air Rib** & **\(L\)/(m)** & **\(e\)/(m)** & **\(k\)** & **\(\alpha\)** & **\(M\)** \\ \hline
**0.417-S** & 1.81 & 0.07 & 0.73 & 48.2\({}^{\circ}\) & 0.61 \\
**0.667-R** & 2.09 & 0.05 & 0.83 & 46.9\({}^{\circ}\) & 0.56 \\
**0.583-R** & 1.84 & 0.11 & 0.63 & 27.3\({}^{\circ}\) & 0.38 \\
**0.5-S/R** & 2.04 & 0.12 & 0.6 & 32\({}^{\circ}\) & 0.34 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Shape parameters of the upright region of the air ribs. 0.667-R. The critical bearing load (\(N_{0.583-R}\)) is 1.51 kN/m\({}^{2}\), based on the value of \(F_{0.583-R}\), and fails at the upright region. By the analysis of the five types of ribs, the segmentation failure competition mechanism can be used to evaluate the critical bearing load.