# 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
A lower \(\theta\) and a larger radius of curvature suggest that the air rib possesses a gentler arc segment. This gentler curvature enables a more uniform distribution of stress under external loads, thereby increasing the strength of the arc-like region. Figure 5: Structural form and stress distribution of air ribs under critical load. reducing stress concentrations and delaying local buckling and instability. Conversely, a higher \(\theta\) implies that the air rib has a steeper curvature and a smaller radius of curvature. This sharper curvature is more prone to causing localized stress concentrations under external loads. The stress concentration in the curved sections leads to premature buckling and instability. Therefore, \(\theta\) can be set as an effective method for assessing load-bearing performance. The relationship between the buckling load, radius of curvature, and the covered angle is analyzed by previous studies, and the buckling load decreases with the increase in the \(\theta\)[29]. For the arc-like region, the buckling load \(q\) of ribs is ranked as \(q_{0.417\text{-}5}>q_{0.5\text{-}5/R}>q_{0.667\text{-}R}>q_{0.583\text{-}R}\). For the upright region of air ribs, the critical failure load \(F\) is calculated by Formula (2) as a thin-walled cylinder under axial load [30]:  
\[F_{cr}=A\left(\frac{i}{2L\cos\alpha}\right)\left(\frac{kE\delta}{\Phi}\right) \,,\;k=\frac{\Phi-e}{\Phi} \tag{2}\]  
where \(F_{cr}\) is the critical instability load of the thin-walled cylinder, \(E\) is the Young's modulus of the thin-walled cylindrical material, \(L\) is the length of the thin-walled cylinder, \(\Phi\) is the equivalent diameter, \(\delta\) is the thickness of the thin wall, \(A\) is the cross-sectional area, \(i\) is the inertia radius, \(e\) is the eccentricity of the load, and \(k\) is the eccentricity effect coefficient. The initial geometry of the air ribs plays a significant role in determining the load deformation of the upright region. It can be seen from Formula (2) that the load-bearing capacity of the upright cylindrical structure is influenced by the material, length, and eccentricity of the upright segment and the distribution of external loads. The initial geometry of the air ribs plays a significant role in determining the load deformation of the upright region. As the \(k\) is smaller, the upright region is closer to an ideal cylindrical form and can bear more load effectively in the vertical direction. The parameter schematic of the upright region is shown in Figure 7. The load is added on the top of the arc-like region and transferred to the upright region, so there is an angle (\(\alpha\)) to describe the effective load. The materials of ribs studied in this paper, namely \(A\), \(i\), \(E\), and \(\delta\), are the same. So, \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Air Rib** & **S/(m)** & **r/(m)** & \(N\) & \(\theta\) \\ \hline
**0.417-S** & 6.64 & 5.82 & 2.48\% & 63.8\({}^{\circ}\) \\
**0.667-R** & 6.32 & 4.96 & 3.10\% & 70.8\({}^{\circ}\) \\
**0.583-R** & \(7.13\) & 4.55 & 4.24\% & 86.2\({}^{\circ}\) \\
**0.5-S/R** & \(8.15\) & 5.90 & 3.74\% & 69.6\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Shape parameters of the mid-section of the air ribs. Figure 6: The parameter schematic of the arch structure. \(F\) is proportional to the value of \(k/L\cos\alpha\), which is defined as \(M\). The values of \(M\) are shown in Table 3. The \(M\) of the rib named 0.417-S is the largest and that of the rib named 0.5-S/R is small. For the upright region, the critical failure load (\(F\)) of ribs is ranked as \(F_{0.417\text{-}S}>F_{0.667\text{-}R}>F_{0.583\text{-}R}>F_{0.5\text{-}S/R}\). According to the segmentation failure competition mechanism proposed above, the critical bearing load (\(N\)) of the ribs is based on the \(\min\{q,F\}\). Figure 8 provides the relation between the load and deformation of ribs and the destabilized structural configurations of air ribs under failure loads.