## 7 Assessment of Steel Silos Vulnerability
[147], identifying a remarkable deviation from linearity in the case of plate structures with relatively significant initial imperfection. Some attempts to exploit the concept of VCTs for cylindrical shells were firstly proposed in 1970s [148], for the purpose of aerospace applications. In this application, VCT was used for determining the actual boundary conditions in numerical calculation of stringer-stiffened shells, as based on laboratory-type and on realistic boundary conditions [149]. Then, it was extended tothe direct prediction of buckling, as just shown in [148]. Currently, several researchers attempt to further exploit VCT for the purpose of erected cylindrical shell assessment. For instance, [PERSON] et al. [145] identified the range of applicability of the VCT for unstiffened cylindrical shells, showing the efficiency of the technique applied to these structures. In addition, the authors demonstrated the advantages given by the results of the FE modelling, considering the realistic boundary conditions obtained by VCT in conjunction with an actual measurement of the initial geometric imperfections. On this basis, in [145] a new methodology to estimate the buckling load of unstiffened cylindrical shells using the VCT was proposed. An experimental verification of this approach was presented by [PERSON] et al. [143], which measured the first natural frequency of vibration and the related mode shape by using a 3D laser scanner on two composite laminated cylindrical shells and two stainless steel cylinders. The authors recommended the monitoring of the first and second vibration modes, as this latter can provide a better prediction when compressive loads increase. In addition, [145] suggested that the maximum load to be adopted in the VCT should be limited to 50% of the buckling load as non-destructive test. Still, the investigated approach presented in [143] returned a very good correlation when the ratio of the test applied load to the experimental buckling capacity is higher than 80%. This fact was also demonstrated by a further experimental work presented by [PERSON] et al. [146], which addressed unstiffened cylindrical shells loaded in axial compression and two laminated composite cylinders loaded repeatedly up to instability point. The study concluded that tests up to 65% of the buckling load can give a 90% fidelity in estimation of buckling load. Moreover, the applicability of the modified VCT, presented in [145], has been investigated by [PERSON] et al. [150] for the thin-walled isotropic cylindrical shells with and without circular cut-outs. The study concluded that VCT provides a reliable estimation of the buckling load of uncut shells and when the global failure mode is governing the collapse. On the other hand, the study stated that using VCT for shells with a cutout is invalid due to developing of local buckling. The study suggested that the global failure mode and the reliability of the VCT estimation could be enabled for these shells by using reinforcement with a ring of the same material, adhesively bonded around the cutout. In addition, using an analytical approach, [PERSON] et al. [151] demonstrated the reliability of the approach suggested in [145] for isotropic unstiffened cylindrical shell. The study defined the basis of numerical modelling for which the second-order relationship between the applied load and the squared natural frequency holds. Recent studies examine the effectiveness of VCT for steel silos. [PERSON] and [PERSON] [152] presented a study in which a steel silo was analyzed. The investigated silo was made through a corrugated wall and stiffened with cold-formed columns. Aiming to evaluate the impact of imperfections on the VCT effectiveness, both imperfect and perfect geometries were taken into account with different imperfection amplitudes. Particularly, the impact of such imperfections on relation between squared natural frequencies and compressive forces was evaluated.