PSI - Issue 78

Francesco Martini et al. / Procedia Structural Integrity 78 (2026) 2022–2029 2025 where and are the external and internal radius of the silo, ρ s is the density of steel, is the density of filling material. With the above schematization, if the silo is filled at 60%, the mass on the top will be only the one for empty silo (Equation 1), while the masses in the middle and on the bottom will be the ones for filled silo (Equation 1).

Figure 1. Proposed simplified modelling approach and analogy with the detailed model.

Regarding stiffness, a similar approach to the above one can be employed, by summing the stiffness of the silo structure and the one of the contents. Nevertheless, both stiffness provided by the steel silo and granular material are not only dependent on the characteristics of the material itself and the assumption of a static scheme. In fact, the stiffness of the silo structures is strongly related to the geometry of the silo, while the stiffness of the filling material is strongly related to the geometry of the silo and service condition. Due to the lack of data for analytically deriving stiffness from different geometries of silos, this parameter was estimated by assuming a cantilever scheme for both silo and filling material, modified through two coefficients, i.e., and , respectively, both dependent on the geometry of the silo. In this way, the stiffness of the silo in any condition, , can be expressed as: = 1 + 2 + 3 = � 3 ∙ 3∙ ∙ �∑ 3 =1 � 3 ∙ 1 → 3 ∙ 3∙ ∙ �∑ 3 =1 � 3 ∙ 1 + ∙ 3∙ ∙ �∑ ℎ = 1 � 3 ∙ 1 → (2) where and are the elastic moduli of steel and filling material, respectively, is the moment of inertia of the circular hallow steel section, is the moment of inertia of the circular granular material section (full), i and j represent indexes for the silo and content portions, respectively. Note that while i assumes always values from 1 to 3 (i.e., the entire silo wall), in the case of filled silo the value of j can assume a value from 1 to n , where n is the number of content portions effectively present in the silo (until h mat ). To define values of and , an iterative procedure was employed, in which the stiffness matrix of the simplified 3DOFs model was updated by solving the motion equation and by running eigenvalue analysis to match the analytical frequencies with the numerical solutions obtained by the sample of silos investigated by Khalil et al. (2024). Twenty configurations of silos were accounted for, given by five geometries, i.e., very slender (V), slender (S), boundary (B), intermediate squat (I) and squat (Q), and four filling level, i.e., 0%, 30%, 60%, 90%. The results are reported in Figures 2 and 3, respectively for and . As observed, the graph of reports a trend proportional to the square of the inverse of the slenderness for the silo (i.e., height over diameter), with the coefficient that increases for squat silos and decreases for slender silos. Regarding , the trend is like , but in this case, besides observing more points due to the involvement of the height of the material in the silo, the obtained relationship is proportional to the cube of the inverse of the slenderness. In fact, the