## 7 Assessment of Steel Silos Vulnerability  
As the last part of this paper, it is essential to mention that despite of the fact that robust design methodologies can be improved to avoid the risks mentioned in Section 6, it is necessary to prevent losses by monitoring the conditions of existing silos, aiming to assess the behavior under the realistic circumstances. Consequently, a safe and continuous operation of these structures and the relevant facilities must be guaranteed. In fact, destructive (DTs) and non-destructive tests (NDTs) have been developed as tools to estimate the cylindrical shells efficiency, or as proof-testing of structure, or for the purpose of theoretical analysis validation. However, the common buckling tests have a destructive or terminal nature, as the loaded structure buckles with large plastic deformations. Thus, the same test cannot be repeated and definitely it is not suitable for silos in service. Therefore, the desire for NDTs has grown [30]. However, for more practical solution, several assessment techniques were developed as NDTs applicable for existing silo structures. Generally, non-destructive methods can be classified to be dynamic or static and they could be employed for either direct determination of the buckling load and determination of the actual boundary conditions leading to better numerical determination of the buckling load [30]. One of the first NDT to predict the buckling loads of steel structures is the Southwell approach, which was initially developed for a simple column [51]. Later, this approach was extended to include cylindrical shell [142], but the notable drawback of this approach is represented by the need of applying high loads, in order to come up with reliable prediction and this can threaten the non-destructive feature of the test [143]. However, one of the most common NDT is the vibration correlation technique (VCT) that is being employed currently to predict the buckling capacity of shell structures used for several applications. In the following, this technique and its applicability for silos is presented. For purpose of vulnerability assessment, the vibration correlation method can be used. This method can be defined as an NDT used to estimate the buckling load from the pre-buckling stage of the structure, based on the variation of the natural frequency with the applied loads. Historically, this approach was firstly derived for columns depending on the fact that the buckling modes and vibration modes are similar for a simple structure of a column. In other words, this method takes advantage from the similarity between the buckling behavior and the free vibration behavior of the relevant structures. The relationship between the squared frequency and the compressive load is nearly linear for columns with different boundary conditions [144], while it is exactly linear in the case of simple supports columns, where the vibration mode is identical to the buckling mode and it is presented analytically in [30] and reported below:  
\[\left(\frac{\omega_{n}}{\omega_{n0}}\right)^{2}=1-\left(\frac{P}{P_{n}}\right) \tag{2}\]  
In the equation, \(\omega_{n}\) is the \(n\)th natural frequency of the loaded column, \(\omega_{n0}\) is the \(n\)th natural frequency of the unloaded column, \(P\) is the applied load, and \(P_{n}\) is the Euler buckling load corresponding to \(n\)th vibration mode. Later, VCT was further developed to address plates [145] and shells [146]. The main feature of this method is the ability to estimate the destructive buckling behavior of relevant structure from a simple vibration test, where results are obtained by subjecting the addressed structure (e.g., cylindrical shell) to compressive loads without reaching the instability point. This technique was thus extended for plates by [PERSON] [144], which declared that VCT is reliable only when it is applied on specimens having small initial imperfections. This fact was also confirmed by [PERSON] et al.