## 2 Structural Typology and Arrangement of Circular Silos
### Buckling Types and Analysis  
Like most of the conventional steel structures, buckling under vertical compressive stresses is the critical consideration for the thin-walled steel silos prone to a loss of stability [15; 67]. The main sources of the vertical compressive forces in silos are the frictional traction pressure imposed by the stored material and the horizontal pressure. While the horizontal pressure imposed by the initial filling slightly increases with stored material depth, the frictional traction pressure significantly increases as the depth of the stored material increase, as shown in Figure 4 and according to the [PERSON]'s theory [33]. For this reason, the tall (or slender) silos are built with RC material, where the vertical traction pressure dominates the horizontal one. Instead, steel shells are susceptible to vertical pressure, thus, the shortest (or squat) silos are usually built with steel, especially where the horizontal pressure is dominant with regard to the vertical traction pressure. Historically, extensive knowledge has been developed on the buckling behavior of an empty cylindrical shell under uniform compression even in combination with internal pressure. Different aspects of buckling and post buckling behavior of shell structures under uniform and well-quantified loads were historically addressed by several researchers, as defined in [68] and references therein. In [69], [PERSON] extensively presented the research work performed on shell buckling through the last century. Nevertheless, the buckling strength of silos is dependent on many factors, such as the magnitude and distribution of both the frictional and horizontal pressures, the imperfections amplitude and shape, and the elastic restraint of the stored material against buckling. For instance, the eccentric filling or discharge causes variation in the applied pressure resulting in a worse stress state in the bin wall than even higher uniform pressure [70]. Therefore, some studies were introduced in the literature addressing the buckling behavior of silos under eccentric discharge. One of the earliest comprehensive studies was conducted by [PERSON] [15], in which experiments were performed to investigate the buckling behavior of a cylindrical shell under pressure directly induced by the stored solids. This study took into consideration buckling strength increment derived from the stored solid stiffness. The study treated flat-bottom silos, considering concentric/eccentric filling and discharge. For  
Figure 4: Silo and wall loads: (**a**) normal pressure; (**b**) vertical compression variation. los, according to the results of the experiments, a benign buckle of mostly axisymmetric mode was observed with significant reserves of post-buckling strength compared to the empty pressurized cylinders. For silos under concentric discharge, an accentuated buckling mode was observed. However, larger and different buckling modes were observed under eccentric discharging with a catastrophic collapse. In addition, two main remarks were also recorded as outcomes of this study: the first one is related to the filling depth, where it was found that the critical filling depth under discharges is as low as that one when the silo under filling loads only; the second one is relevant to the channel of flowing, where it was found that it was not critical to the buckling strength [15]. Similar issues were studied by [PERSON] [50], which presented buckling features outlining of thin shells under axial loads. The study addressed moderately slender perfect silos. Within this study, the author tried to come up with practical design recommendations. In this context, the study concluded that the behavior of these shells has many complexities, and to derive design rules out of the observed buckling features is far from being complete. Moreover, the author stated that the effect of unsymmetrical axial distribution among the cylindrical is the most critical point to understand.