# 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  
As a typical annular flexible body, the air rib deforms differently under external loads. The middle section of the air rib flattens, while the end sections spread outward during the deformation process. The stress concentration occurs in the bending regions. To investigate the mode of air ribs under critical loads, the deformation of five air ribs (0.333-S, 0.417-S, 0.5-S/R, 0.583-R, and 0.667-R) is analyzed. The structural model and loading mode are illustrated in Figure 4. A uniformly distributed vertical load is applied to the upper surface  
Figure 3: The trend of the displacement with the resisted load and stiffness. of the air ribs. The load is applied on the ribs gradually and calculated by the explicit dynamic method [23]. This approach minimizes oscillations and stabilizes the model prior to further analysis. During the compression process, the rib gradually deforms downward until it reaches its lowest point. The load corresponding to this lowest position is defined as the critical load. Figure 5 demonstrates the stress distribution of different air ribs under critical load. The rib named 0.333-S presents a significantly smoother structural shape, and the maximum stress is 18.43 MPa. The stress distributes uniformly and the stress concentrates on a small area. For the other air ribs, there is stress concentration at the top and bending regions. In order to explain the bearing and deformation performance, the air ribs are divided into two parts by the stress concentration regions. The upright region near the root of the air rib is bare of deflection, and the arc-like middle section suffers large deformation. The arc-like region is similar to an arc under radial load, while the upright region near the root behaves like an upright cylinder. With the increase in the load, the arc-like region and upright region can be seen as two independent bearing parts. The upright region is simplified as a thin-walled cylinder, which is subjected to eccentric load. The arc-like region is simplified as a rod that is hinged at both ends. With the increase in the load, there will be two failure mechanisms. If the stiffness of the upright region is larger than that of the arc-like region, the middle of the arc-like region is crushed first and the upright region keeps straight. If the stiffness of the upright region is less than that of the arc-like region, the upright region goes down and the structure collapses from the bending stress concentration regions. The competition between the upright region and arc-like region can be analyzed by the classical theory. The parameter schematic of the arch structure is shown in Figure 6. When the load is added to a circular arch, the buckling load is influenced by its initial radius of curvature and the covered angle. The covered angle \(\theta\), initial radius of curvature \(R\), and the ovality of the arch \(n\) are calculated as follows:  
\[\theta=\frac{180S}{\pi R},\ R=r(1+n)\,\ n=\frac{r_{\max}-r_{\min}}{r_{\rm avg}} \tag{1}\]  
where \(S\) is the length of the fitted circular arc of the mid-section of the rib, \(r\) is the radius of the fitted arc, \(r_{\max}\) is the maximum distance and \(r_{\min}\) is the minimum distance from the rib nodes to the center of the fitted circle, and \(r_{\rm avg}\) is the average distance from the rib nodes to the center of the fitted circle. Figure 4: The simulation model of air ribs to calculate the critical load. The intermediate section is not a perfect circle; the concept of ovality is introduced to represent the actual geometry of the air rib more accurately. The arc length \(S\) and radius \(r\) of the arc-like region are obtained by fitting the node data of the rib under critical load. As shown in Table 2, the \(\theta\) of the 0.417-S configuration is the smallest, and the value is 63.8\({}^{\circ}\). The \(\theta\) of the 0.583-R configuration is the largest, and the value is 86.2\({}^{\circ}\).