PSI - Issue 78

Francesco Martini et al. / Procedia Structural Integrity 78 (2026) 2022–2029

2024

silo; (ii) diamond-shaped (or similar) failures, which develops at the middle of the silo; (iii) top-of-wall damage, which develops at the top of the silo. To fully predict the above failure modes on silos, both experimental and numerical investigations were performed over the past years. Especially regarding numerical investigations, very refined modelling approaches were proposed, in which both silo structure, filling material and the related interaction were simulated, in order to predict the insights above reported. However, when dealing with seismic actions, the behavior of structures is strongly dependent on the type of input actions, which results in the simulation of the so-called record-to-record variability. Obviously, if considering overdetailed numerical models, the possibility to carry out several nonlinear time history analyses is limited, due to the high computational cost required for achieving the solution and, above all, convergence. An alternative option, aimed at limiting computational burden in running seismic analysis, consists of developing a simplified model, with reduced degrees of freedoms (DOFs), able to reproduce both elastic and nonlinear behavior of the structure with a certain accuracy. Such an approach, representing a trade-off between complexity and accuracy of the simulation, was several times employed for different typologies of structures, as well as for silos and tanks. Just to give a few examples, concerning tanks, Bakalis et al. (2017) proposed the joystick model, characterized by an impulsive mass acting on different springs circularly disposed at the base to simulate both effects of sloshing and soil structure interaction. Regarding silos, Durmus and Livaoglu (2015) proposed a 3-DOFs model aimed at predicting the behavior of silos accounting for soil-structure interaction. The aim of this work is to propose a simplified model able to predict the seismic behavior of cylindrical ground supported steel silos by means of simplified analytical relationships, and able to account for the interaction with the filling material. The idea behind the work is to define analytical rules and equations to derive the elastic dynamic parameters (i.e., period of vibrations and horizontal and static pressures) of different silos geometries (from slender to squat geometries) and different service levels (from empty to fulfilled). The paper reports firstly the definition of the simplified modelling approach, which was calibrated on the sample of silos investigated by Khalil et al. (2024). As above reported, at this stage, only the elastic behaviour was managed, even though the proposed model can serve as a basis for the simulation of the nonlinearities, and then to furtherly run a high number of nonlinear time history analyses. 2. Proposed simplified modelling approach and definition of the elastic features The proposed simplified modelling approach consists of simulating the silos and filling materials through a 3-DOFs model, constituted by 3 masses and 3 springs in series. The logic behind the subdivision in the 3-DOFs is that each spring should be set to simulate a specific failure mode, i.e., first spring simulates the elephant-foot buckling; second spring simulates diamond-shaped (or similar) failures; third spring simulates top-of-wall damage. Being the failure modes beyond the scopes of this work (only the elastic part was accounted for), the other reason related to this schematization is due to the possibility of simulating different service levels, i.e., empty, 30%, 60% and 90%, according to Khalil et al., 2024 and then performing a calibration phase with the detailed models proposed in the reference. The scheme of the simplified model is shown in Figure 1, in which m 1 , m 2 , and m 3 represent the masses of the model, k 1 , k 2 , and k 3 represent the stiffness of the springs. Note that the DOFs of the three masses experience in the horizontal direction (as in a shear-type model), due to horizontal seismic force. Regarding height, three heights, i.e., H 1 , H 2 , and H 3 , are defined and are necessary for the definition of masses and stiffness. In detail, H 1 , H 2 , and H 3 are equal, and their sum is the height of the silo, i.e., H . Instead, the heights of the contents are indicated with h 1 , h 2 , h 3 , they are equal, and their sum is the height of the silo content, i.e., h . Note that the height of the content can be different from the above quantities and can generically be indicated as h mat . Due to the proposed schematization, the masses of the simplified model, m 1 , m 2 , and m 3, can be derived as follows: 1 , 2 , 3 = � ( − ) 2 ∙ 1 , 2 , 3 ∙ ∙ → ( − ) 2 ∙ 1 , 2 , 3 ∙ ∙ + ℎ 1 , 2 , 3 ∙ ∙ 2 ∙ → (1)