## 2 Structural Typology and Arrangement of Circular Silos
### Imperfections' Effects and Modelling  
In typical cases, steel silos design concerns about avoiding stability failures, as occurs in many different shell structures, whether in the form of local or global buckling failure. However, the buckling capacity is very sensitive to the geometric imperfections as a function of amplitude and form. Geometric imperfections can be defined as the shape deviations from the perfect structure due to the manufacturing process. Hence, in the design of silos, a local high pressure must be imposed on the wall, for accounting the existence of geometrical imperfections. In spite of the extensive experimental and theoretical works in the literature addressing the behavior of the axially compressed cylinders [14; 45; 46], there is still a gap between the predictions obtained by the numerical models and the realistic results provided by the experiments. However, the usual scatter among numerical and experimental predictions, in terms of buckling capacity, could be chiefly traced back to the structural imperfections that are unavoidable in the practical construction [47]. In addition, it is worth considering that this discrepancy is much severe when combining other imperfection types, such as loading imperfections [48; 49]. This issue is considered as one of the main classical problems about homogeneous isotropic structural mechanics, which has not been fully understood [50]. The classical formulation for estimating the buckling load of cylindrical shell was derived more than hundred years ago, without considering any kind of imperfection [51]:  
\[N_{pre}=\frac{2\cdot\pi\cdot E\cdot t^{2}}{\sqrt{3(1-\theta^{2})}} \tag{1}\]  
where \(N_{pre}\) is the buckling load of actual perfect cylindrical shells, \(E\) is the elastic modulus of the construction material, \(t\) is the wall thickness, and \(\theta\) is Poisson's ratio of the construction material. However, it has been long stated that the reduction of the geometrical imperfections decreases the discrepancies between the experimental results and the analytical [52; 53] or numerical estimates [54]. Still, these differences are greater and greater in the case in which the axial compression is more significant than either pressure or torsion [52]. Nevertheless, there are no closed-form solutions to account for imperfection during the design phases [32]. Typically, in the design practice of thin-walled shell structures, the influence of the geometric imperfections could be considered by employing techniques known as artificial substitute imperfections (ASI) [55]. For example, eigen-mode imperfection technique could be adopted where the imperfection is assumed in the form of the bifurcation buckling mode taking into account varying imperfection amplitudes. Thus, the buckling strength could vary depending on the considered buckling mode and amplitude [54; 56]. Eigen-mode imperfection is adopted in preliminary design of cylindrical shell structures; however, it could lead to a very conservative design and it is not easy to define the order and the magnitude of the rational eigen-mode shape imperfections [57]. Techniques of ASI are developed in the literature based on probabilistic methods to represent the geometric imperfection in cylinders. For instance, Monte Carlo simulation and first-order second-moments method were developed by [PERSON] et al. [58], aiming to determine the stochastic distribution of the buckling load. However, imperfection representation based on probabilistic methods are rather complex [59]. In the same context, [PERSON] et al. [60] developed a probabilistic design procedure for cylindrical shells with a reduced computational cost comparing to the conventional ones. Away from probabilistic methods, different perturbation methods were introduced as deterministic approaches, such as the Single Perturbation Load Approach (SPLA) firstly proposed by [PERSON] et al. [61] and dealing with thin-walled cylindrical composite shells.