## 2 Structural Typology and Arrangement of Circular Silos
### Imperfections' Effects and Modelling
Inspired by SPLA, single perturbation displacement approach, and single boundary perturbation approach were later developed [59]. Particularly, according to SPLA, the geometrical imperfection effect in a cylindrical shell could be included by introducing a radial perturbation load into the middle part of the cylinder. However, it was proved that using SPLA in the analyses of cylindrical shells provides realistic buckling characteristics [32]. SPLA has been used in many studies for different kinds of cylindrical shells as isotropic metallic [62] or composite [32; 63]. Unlike probabilistic approaches, SPLA is independent from the need of measured imperfections, however, some limitations are noticed, as it does not cover all types of imperfections, as the boundary condition imperfection. Therefore, comparing the numerical results with the test buckling loads, the SPLA method can be not enough conservative, as shown in [63]. Using this geometrical imperfection approach for isotropic metallic shells, the obtained results provide superior design loads if compared with the standard of NASA SP-8007 [64], especially for those shells showing an axisymmetric buckling pattern in the pre-buckling range under axial loads [59]. Multiple Perturbation Load Approach (MPLA) was introduced by [PERSON] et al. [65] as an extension of the SPLA for composite cylindrical shell. According to this approach, multiple perturbation loads are considered instead of the single perturbation load. Thus, three parameters were accounted for: the location, the number, and the magnitude of the considered loads. Based on numerical and experimental approaches, [PERSON] et al. [62] stated that MPLA is a more rational design method, especially for metallic cylindrical shell structures, comparing to SPLA. As a matter of fact, the lower-bound of cylindrical shell is sensitive to the number of perturbation loads, especially when these loads are evenly distributed along the circumferential direction. Still, the smaller inhomogeneous degree of the perturbation loads magnitude, the more possibility to acquire a robust lower-bound buckling load. Steel silos are constructed conventionally by rolling separated steel panels and by welding them to form the silo walls, thus, unique imperfection types develop in the silo walls as consequences of the construction process. For instance, [PERSON] and [PERSON] [66] presented a study in which the imperfections caused by the weld-induced residual stresses in circular cylindrical silos was investigated. The study found that the weld material properties contributed to the changes in buckling and post-buckling behavior of the investigated cylinders. Moreover, the study considered the weld-induced residual stresses and found that these latter contribute to an earlier onset of yielding. Another study proposed by [PERSON] et al. [11] indicated that the geometric imperfections in silos are closely associated with the joints of steel panels forming the wall. Using a numerical approach, [PERSON] et al. [7] presented a study investigating the impact of different imperfection forms on the failure behavior of locally supported steel silos, considering different arrangements of stiffening/supporting and different equivalent imperfections shapes, e.g., nonlinear buckling mode, linear bifurcation mode, several post-buckling deformed shapes of the perfect shell, a weld-induced imperfection accounting for varying amplitudes and orientations. The study ranked the relevant imperfection shapes according to their adverse effects on structural behavior and response in terms of nonlinear buckling modes and post-buckling deformed shapes. It is worth noting that circumferential weld depression (also named type A) is a realistic imperfection as it is closely related to the fabrication process of silos. However, it is relatively considered the most deleterious compared to other imperfection forms according to this study. The study concluded that the inward imperfections are more unfavorable than the outward ones [7].