A numerical procedure to estimate seismic fragility of cylindrical ground-supported steel silos containing granular-like material

Abstract

The paper presents a study on the evaluation of seismic fragility of cylindrical ground-supported steel silos intended for storing solid material. Silos are a key facility in industrial processes. For example, cylindrical steel silos constitute the main structural component for several industrial activities, such as the ones aimed at the production of food and beverage, and seismic actions can cause high economic losses and long functionality interruptions. Thus, the main goal of this paper is to propose a numerical procedure aimed to assess the seismic fragility of different cylindrical steel silos, accounting for varying geometries and service conditions (i.e., filling level of granular-like material), and observing different failure modes. In detail, a set of smooth steel silos was selected, considering different geometrical configurations (i.e., varying from squattest to slenderest structures). Different service conditions were simulated, with the aim to observe the behaviour of empty and filled silos (30%, 60%, and 90% of filling degree with respect to the maximum capacity). For each configuration, a detailed numerical model was developed under proper boundary conditions, adequately simulating the shell structure, the solid material inside, and the interactions between them. After validating the numerical models against existing literature data, three different failure modes were identified and assessed, accounting for the most recurrent post-elastic buckling type (i.e., elephant foot) and considering the possible occurrence of the elastic ones (i.e., diamond or similar shape failures at the middle and top of the structures). Both static and dynamic analyses were performed to identify the most probable failure modes and evaluate the probability of exceeding each one. As the output of the proposed approach, the seismic performance of each silo under a specific limit state was provided in the form of fragility curves. The results highlight some novel aspects, starting from the role that service conditions assume in the silos seismic performance up to the possible differences in terms of failure modes for different silos geometrical structural configurations.

1 Introduction

The continuous development of industrial activities has highlighted the vulnerability of relevant facilities under hazardous events, both man-made and natural ones (e.g., earthquake, flood, tsunami, blast). Industrial sites inherently exhibit a complex configuration consisting of heterogeneous structural and non-structural components, which can suffer widespread damages under exceptional actions, causing enormous losses on several levels, e.g., human lives, economic, environmental issues, interruptions. Thus, the safety of the different industrial components under seismic events deserves high attention in earthquake-prone countries. For instance, in Italy, about 30% of industrial plants are located in regions with a high seismic hazard (Paolacci et al. 2012). In this framework, typical assets of industrial plants are the storage systems. A main classification of storage systems can be made, by distinguishing tanks storing liquid and silos storing solid materials. While the seismic vulnerability of tanks has been deeply investigated in the scientific literature (Bakalis et al. 2017a, b, 2019; Merino Vela et al. 2019; O’Rourke and So 2000; Virella et al. 2006), fewer contributions are available for silos, despite their relevance in food and chemical industry sectors and the high seismic vulnerability observed in recent severe seismic events (e.g., 2023, Turkish Earthquake; 2012, Emilia Earthquake).

This study aims to investigate the seismic behaviour of flat-bottomed ground-supported silos used for storing granular-like material. However, limitations should be acknowledged regarding the structural typology of the silos under investigation. The silos considered in this study are of smooth walls that are unstiffened neither in the vertical nor in the horizontal direction. Although not the most widespread typology in practice, their behaviour under seismic excitation is of a great interest as they exhibit a higher vulnerability to buckling phenomena comparing to the other silos with corrugated and/or stiffened steel walls. Despite new silos for storing granular-like material could be of different typologies (e.g., corrugated, stiffened), several companies still propose this kind of silos (see Fig. 1) as the most traditional and consolidated ones in the practice. Therefore, the proposed procedure aims to investigate seismic behaviour of new and, above all, existing structures characterized by similar features.

Fig. 1

Examples of new unstiffened smoothed walls steel silos (from left to right, phots are retrieved from the website of the companies Cepi Spa and Technobins, respectively)

In the same context, unstiffened smooth wall silos were investigated in the literature under different loading conditions (e.g.,Butenweg et al. 2017; Mehretehran and Maleki 2018; Sadowski and Rotter 2011a; Silvestri et al. 2016). However, in the light of the state-of-the-art presented in Sect. 2, it is evident that the most influencing parameters on the seismic response of these structures are: (a) the geometry of the structure (i.e., height, diameter, walls thickness); (b) the silo functionality (i.e., filled or empty); (c) the intensity and characteristics of the ground motion. However, additional aspects also need to be investigated in detail. First of all, since silos are thin-walled shell structures, under dynamic excitations steel walls can develop different Buckling Modes (BMs), such as the elephant foot bucking at the base of the structure and the diamond or similar shapes failures at the middle and top of the structures. Secondly, silos could show a different seismic behaviour when different filling levels of the stored granular-like material are considered, situation that aims to reproduce the possible real service conditions when the unexpected seismic event occurs.

With these goals in mind, the paper proposes a numerical procedure (in the framework of the performance-based earthquake engineering, PBEE) to derive fragility curves of steel silos storing solid materials, accounting for different geometrical configurations, different service conditions, and different failure modes. In detail, a set of smooth and unstiffened steel silos was selected, varying two main geometrical aspects, that is, radius-to-thickness and height-to-diameter ratios. A specific granular-like material was considered, according to the detailed information provided by Moya et al. (2013). In the analyses, different levels of filling material were investigated, namely 90%, 60%, and 30% of the maximum capacity level, as well as the empty condition. Moreover, different BMs (also named throughout the paper as failure modes) were considered. For each combination of the involved parameters, detailed numerical models were proposed, and after a phase of modelling validation, static and dynamic analyses were carried out. As the output of the proposed procedure, fragility curves were derived for each specific geometry of the silo storing an assigned level of filling material and looking at a particular BM. The results show in probabilistic terms the likely BM occurring as the seismic intensity varies, providing some insights on the influence of the accounted parameters.

2 Background

2.1 Seismic behaviour of ground-supported steel silos

Steel silos are more commonly adopted types of storage facilities than reinforced concrete (RC) ones, since they provide an efficient solution with limited dimensions, weight, and cost. Among existing silos, a first classification can be made distinguishing the elevated silos resting on a framed structure (Jansseune et al. 2016; Kanyilmaz and Castiglioni 2017), and the ground-supported ones (Butenweg et al. 2017; Sadowski and Rotter 2011a). Referring to ground-supported steel silos, different structural configurations and typologies exist, usually aimed at different functions. Basically, two main categories of flat-bottomed ground-supported steel silos can be identified: (a) smooth silos, composed of thin-walled isotropic plain rolled sheets, with stepped or constant wall-thickness (e.g., Iwicki et al. 2014; Mehretehran and Maleki 2021; Sadowski and Rotter 2011a, b; Song 2004; Song and Teng 2003); (b) corrugated silos, composed of stiffened (e.g., Iwicki et al. 2015, 2016, 2019; Rejowski et al. 2023; Wójcik et al. 2017) or unstiffened (e.g., Kuczyńska et al. 2015) corrugated shell.

Focusing on smooth and unstiffened ground-supported silos containing solids, and on their behaviour under seismic actions, early investigations trace back to the second half of the last century, when the quantification of the seismic loads, the dynamic characteristics, and the distribution of the response acceleration along the height were the key matters. Different approaches, including experimental (Jing et al. 2022; Shimamoto et al. 1984; Silvestri et al. 2016, 2022; Yokota et al. 1983), numerical (Butenweg et al. 2017; Durmuş and Livaoglu 2015; Hardin et al. 1996; Holler and Meskouris 2006; Mehretehran and Maleki 2018, 2021; Rotter and Hull 1989; Sasaki and Yoshimura 1992; Shimamoto et al. 1984; Yokota et al. 1983), and analytical (Lee 1981; Pieraccini et al. 2015; Silvestri et al. 2012; Trahair et al. 1983; Veletsos and Younan 1998; Younan and Veletsos 1998) studies, were proposed and adopted. Turning to more recent literature contributions, the focus was aimed at identifying the most influencing parameters governing the seismic behaviour of ground-supported steel silos, such as the slenderness ratio (Holler and Meskouris 2006; Mehretehran and Maleki 2018; Nateghi and Yakhchalian 2012; Rotter and Hull 1989; Veletsos and Younan 1998; Younan and Veletsos 1998), the properties of stored material (Hardin et al. 1996; Shimamoto et al. 1984; Yokota et al. 1983), the effective mass (Lee 1981; Sasaki and Yoshimura 1992; Silvestri et al. 2012), the additional normal pressure (Holler and Meskouris 2006; Silvestri et al. 2012; Trahair et al. 1983), the compaction of the solids (Silvestri et al. 2022), the soil-structure interaction (SSI) (Butenweg et al. 2017; Holler and Meskouris 2006), and the wall-solids interaction (Holler and Meskouris 2006; Mansour et al. 2022; Nateghi and Yakhchalian 2012).

Discussing the available literature in chronological order, one of the first contributions about the seismic response of ground-supported silos was provided by Rotter and Hull (1989) in which the authors assessed, based on finite element (FE) approach, the critical seismic stresses in squat silo walls and derived the corresponding analytical expressions. A uniform horizontal acceleration was adopted (as quasi-static horizontal body force), and the stored solid material was characterized as an isotropic and homogenous mean with a specific elastic modulus. The results stated that the membrane stress developed in the shell walls are proportional to the height-to-diameter ratio. Sasaki and Yoshimura (1992) evaluated the effective mass pushing on the silo wall under an earthquake based on a numerical model of a tested scaled silo. Hardin et al. (1996) evaluated the seismic response of a real scale steel wheat-silo. The stress–strain distribution, the acceleration history, and the amplification of the horizontal acceleration were recorded. In their work, a composite shear-beam model was employed to simulate the investigated grain-silo system. Veletsos and Younan (1998) and Younan and Veletsos (1998) provided an analytical formulation to describe the seismic response of ground-supported silos considering the effect of the slenderness ratio and walls flexibility, where the ensiled material was simulated as a linear viscoelastic solid.

The above-mentioned works represent milestones about seismic response of ground-supported silos, which were followed by the drafting of the current European standard, EN 1998-4-2006 (2006) and a series of subsequent studies. With this regard, it is worth mentioning that the guidelines in EN 1998-4 (2006) employed a static equivalent approach to provide a practical method for seismically designing the silos structure. In fact, seismic loads are considered by defining an effective mass horizontally applied on the silo walls, determined as a percentage of the total mass. Following this approach, Holler and Meskouris (2006) investigated the behaviour of steel silos containing granular material and, in the study, they considered the effect of the aspect ratio, the grain-wall interaction, the SSI, and the granular material nonlinearity. The results suggested to apply a reduction to the effective mass proposed by the code for squat silos (and not for slender ones) to achieve a more realistic design. Yakhchalian and Nateghi (2012) presented a numerical investigation to assess the influence of the aspect ratio on the seismic behaviour of ground-supported steel silos. The study quantified the additional normal pressures over the silo height and compared the results with those obtained by the application of a constant distribution of accelerations. The authors concluded that this distribution was conservative for squat silos and non-conservative for slender silos. A physically based estimation of the effective mass of grain-silo systems was presented by Silvestri et al. (2012) and later refined by Pieraccini et al. (2015). The works provided a set of analytical formulations aimed at estimating the pressures on the silo walls. Based on the obtained results, authors emphasized that the approach proposed by the code about squat silos is conservative. In the same context, Silvestri et al. (2016) presented an experimental study with a shaking table test for a reduced-scale silo filled with granular material. The result of the experimental campaign revealed a significant effect of the wall-friction coefficient on the overturning moment at the silo base. Further investigation on the simplified approach of the static equivalent load adopted by the code was presented by Butenweg et al. (2017), who compared the code method with a more sophisticated approach, based on the time history analysis and nonlinear FE modelling. The proposed FE model considered the solid nonlinearity, the solid-wall interaction, and the SSI. The study revealed that the application of static equivalent loads on slender silos can be too conservative if a simplified linear acceleration along the wall height is considered. Instead, the use of an acceleration profile based on the multimodal analysis of a simplified beam yields more realistic result. Similarly, the study stated that the simplified static equivalent approach is also conservative in case of squat silos (as stated by Holler and Meskouris 2006; Silvestri et al. 2012). However, the authors attributed this circumstance to the fact that the equivalent static approach does not precisely consider the likely stress reduction caused by the friction of the bulk material in case of squat silos. In the end, the study suggested to opt for nonlinear numerical modelling as it leads to more realistic and economic silo design. Durmus and Livaoglu (2015) introduced an analytical solution based on a simplified model consisting of a single-degree-of-freedom flexural cantilever beam with a lumped mass. The findings based on the proposed approach were compared to those obtained from a more sophisticated numerical FE modelling, showing the soundness of the proposal. The study also concluded that the SSI effect has a negligible contribution to the seismic behaviour of the system and can be ignored in practical applications. Mehretehran and Maleki (2018) investigated the dynamic buckling behaviour of steel silos by implementing incremental dynamic analysis (IDA) with 10 ground motion records. In this case, the investigated silos were assumed filled up to 90% of the maximum capacity and the Elephant Foot Buckling (EFB) was the solely investigated damage pattern. The main findings of this study indicated that slender silos are more vulnerable to buckling failure, while squat silos presented a considerably higher resistance under same conditions. Recently, the same authors extended their findings to include the stepped wall steel silos under seismic conditions (Mehretehran and Maleki 2021) and observing different buckling modes. Particularly, local diagonal shear wrinkles were found to be dominant for squat and intermediate slender silos, while EFB damage was observed at the base of slender ones. Recently, Silvestri et al. (2022) performed an experimental study in which the seismic behaviour of a full-scale flat-bottom steel silo filled with wheat was investigated on a shaking table. The aim was to estimate several parameters of interest, such as the static pressure, the dynamic characteristics, and the dynamic overpressure. The study, moreover, reported the role of the compaction degree of the granular material and its effect on the acceleration profile along the silo wall and within the fillings. Another experimental study was presented by Jing et al. (2022), whose purpose was the assessment of the favourable effect provided by the existence of the granular material on the energy dissipation of the grain-silo system. The effect of different level state on the acceleration vertical profile and on the fundamental frequency was also mentioned. Based on a numerical investigation, Khalil et al. (2023) evaluated the seismic fragility of silo containing wheat considering different silo geometries. For the sake of synthesis, the state-of-the-art herein reported is limited to the ground-supported steel silos, which characterize the subject of this work. More information about silo structures and their behaviour under different conditions is presented in (Khalil et al. 2022).

2.2 Possible failure modes of ground-supported steel silos under earthquake

Failure of structures can be defined as the state beyond which the structure (or a part of it) does not satisfy the design performance requirements. In the case of steel silos, structural failure is governed by specific damage patterns. When talking about earthquake effects on steel silos, failure modes depend on several parameters, such as the geometry, the presence or absence of filling material, and the ground motion intensity. BMs can be generally associated to a specific damage scenario that is observed on the shell body of the silo. Looking at the post-earthquakes site inspections on industrial sites (e.g., Brunesi et al. 2015; Buratti and Tavano 2014; Niwa and Clough 1982), a wide variety of possible failure modes and mechanisms under earthquake can be noted. However, this study focuses on the damage state associated to the instability phenomena in the silo walls, which is commonly associated to the shell buckling damages and that can anticipate any other type of collapse. As a matter of fact, under a seismic event, the silo walls undergo additional normal pressures (EN 1998-4, 2006). However, the increased internal pressure can provoke buckling in the shell wall and, considering the complex grain-wall interaction, different buckling phenomena can develop in different positions of the wall. In this view, three BMs can be identified: (a) the elasto-plastic buckling named Elephant Foot Buckling and known as EFB (Rotter 2006) (Fig. 2a); (b) the elastic Top-of-Wall Damage (TWD) buckling (Malhotra et al. 2018; Mehretehran and Maleki 2021; Virella et al. 2006) (Figure b); (c) the elastic wall deformation occurring near to the middle part of the silo, known as Elastic Buckling (EB) (Virella et al. 2008) (Figure c), and recalling the diamond (or similar) shape buckling. Importantly, although the BMs under consideration are practically more relevant to tanks storing liquids as depicted in Fig. 2, this study consider investigating the possibility to experience each of the three BMs in case of silos storing solids as silos and tanks share similar structural configuration and similar functionality as storage system, especially in the light of the lack of post-earthquake inspection of silos which implies vagueness in the seismic damage patterns for silos in contrast to tanks.

Fig. 2

Possible BMs observed after earthquake events: a EFB (Brunesi et al. 2015); b TWD (Malhotra et al. 2018); c EB (Brunesi et al. 2015)

The EFB, which was well investigated by Rotter (2006), is a BM usually induced by the combination of axial compressive stress exceeding the critical shell stress and the circumferential tensile stress close to yield strength. EFB is described by an outward bulge, near to the base, which extends around the circumference of the wall. In general, EFB is the most likely phenomenon when considering seismic actions on silos. On the other hand, elastic phenomena as TWD and EB must be considered for the case at hand, as observed in several post-earthquake inspections for similar structure, such as tanks (e.g., Buratti and Tavano 2014; Malhotra et al. 2018). In addition, the TWD damage was numerically studied by Mehretehran and Maleki (2021) for silos storing granular material under seismic conditions. Instead, the EB was investigated by Sadowski and Rotter (2011a, b) for silos containing solids and, despite this BM was attributed to the normal pressure induced by discharge loading, it is typical of shell structures and could occur also in case of dynamic excitation.

All three BMs are investigated in this paper, with the twofold aim to provide a specific Engineering Demand Parameter (EDP) for each failure mode and to explore the conditions in which EFB can be anticipated by the elastic BMs (i.e., TWD and EB).

3 Proposed numerical procedure

The proposed numerical procedure is illustrated in Fig. 3, and is composed by four consecutive steps, which are aimed to derive fragility curves for ground-supported steel silos with different geometries and different service conditions, i.e., different filling levels of stored material. Based on PBEE approach, fragility curves are derived, accounting for the silo geometries, the service conditions, and the possible BMs.

Fig. 3

Flowchart outlining the proposed numerical procedure

The first step consists in the selection of a set of representative cases of steel silos, which can cover a wide range of possible conditions, according to a combination of the main influencing parameters the seismic behaviour. Assuming as focus of this paper only the ground supported smooth steel silos, different geometries must be considered, ranging from very slender (low radius-to-thickness and high height-to-diameter ratios) to squat cases (high radius-to-thickness and low height-to-diameter ratios). To adequately consider the influence of the stored material and the occurrence of seismic events in any condition, different filling levels must be considered in the investigation ranging from the near-full level to the empty case.

The second step consists in the detailed modelling of the considered samples. Given the boundary conditions and the applied loads, complex models are strongly suggested to reproduce the peculiar structural behaviour that is mainly ruled by the steel walls (as shells elements), the stored material, and the interaction between the two above elements. For the scope of the proposed procedure, the focus is on the steel walls, whereas adequate modelling simplification can be applied on the filling material, which can be simulated through an elastic approach. Obviously, dealing with shell structure, an adequate mesh must be defined, in order to accurately simulate the failure modes. For such a numerical approach, a validation process is required to assess the basic parameters against real-life (or experimental) case studies. As minimum requirements, models should be assessed in terms of static conditions (e.g., controlling the pressure on the walls and comparing it to the analytical solutions) and dynamic conditions (e.g., computing the eigen frequencies and comparing them to existing or experimental benchmark cases). In relation to steps 1 and 2, it is worth noting that, to account for uncertainties in the generation of the sample, the variation of several parameters should be considered, such as the mechanical parameters of the structural steel and the physical properties of filling material. Nevertheless, it is worth remarking that to perform a more comprehensive parametric analysis on this type of structures, a less detailed modelling approach than the one proposed here should be considered. Therefore, for the proposed procedure, specific geometrical configurations and physical properties of the stored material are considered, reserving more intensive analyses for further developments.

Once numerical models are available, the third step consists in the analysis campaign. To probabilistically characterize the seismic performance of the silos, nonlinear time history analyses (NTHAs) are required, and for this purpose, several parameters must be defined: (a) selection of ground motion records; (b) EDP and intensity measure (IM) to characterize the demand-intensity relationship; (c) possible failure modes and thresholds identifying their achievement; (d) type of analysis approach to employ. Concerning the ground motion selection, specific hazard-consistent records can be considered, selected according to a specific hazard curve or to a response spectrum for the site of interest (Kohrangi et al. 2017). In a PBEE-based approach, record selection should be performed by selecting a range of ground motion wide enough to cover any type of IM and to ensure sufficiency (Luco and Bazzurro 2007). Nevertheless, given the complexity of the proposed modelling approach and the absence of specific code prescriptions, a set of 11 records (analogously to the idea developed in Ruggieri et al. 2021) spectrum-compatible with the reference site was employed (record selection is to be processed according to the Eurocode 8 prescriptions (EN 1998-1 2004), while a higher number of records than Eurocode provisions is suggested). The evident simplification in terms of record selection (i.e., possible high dispersion in the EDP-IM relationship, with probable issues in terms of regression analysis and high epistemic uncertainties) is balanced by an important advantage: a reduction of the computational analysis cost, which could be very expensive when a very detailed modelling approach is employed.

In respect of the EDP selection, several parameters could be employed, such as displacements or an index of the stress/strain relationship. Nevertheless, considering that the aim of the study is to characterize the failure of steel silos looking at different buckling phenomena, which can occur in different zones along the height, a typical representative parameter can be the rotation of the wall with respect to the base, θ_W,BM (the subscript BM oversees all failure modes and then, according to the specific failure mode, it is replaced throughout the text with the proper acronym, i.e., EFB, TWD, EB) expressed as the ratio between the horizontal displacement in the likely buckling zone (at the top: δ_W,TWD; in the middle: δ_W,EB; close to the base:, δ_W,EFB) and the height of the identified zone from the silo base (at the top: h_W,TWD; in the middle: h_W,EB; close to the base: h_W,EFB):

$${\theta }_{W,BM}\to {\theta }_{W,EFB}=\frac{{\delta }_{W,EFB}}{{h}_{W,EFB}};\,{\theta }_{W,TWD}=\frac{{\delta }_{W,TWD}}{{h}_{W,TWD}};\,{\theta }_{W,EB}=\frac{{\delta }_{W,EB}}{{h}_{W,EB}}$$

(1)

The use of θ_W,BM is suitable for all BMs, as it is an unbiased parameter neutralizing the position of the buckling zone and then, is effective for the purpose of the procedure. Figure 4 graphically illustrates the EDP calculation with reference to the EFB case as an example. With respect to the IM selection, two options are eligible: peak ground acceleration (PGA), which is adopted in several research works (Buratti and Tavano 2014; Kildashti et al. 2018; Mehretehran and Maleki 2021; O’Rourke and So 2000; Salzano et al. 2003; Sobhan et al. 2017; Virella et al. 2006), or spectral acceleration of the first vibration mode (Sa(T₁)). It is worth noting that, unlike tanks, silos storing granular-like materials do not have a convective response, which represents a limit in the use of Sa(T₁) because the first vibration period (T₁) cannot accurately describe the effective acceleration. For the case at hand, instead, if compaction phenomena of the stored granular material are neglected, T₁ tends to be invariant for a given condition of functionality and Sa(T₁) can provide a better quantification of the seismic effect on the couple grain-silo system than PGA. Still, considering that the solid filling material represents an additional mass for the system, the values of T₁ are expected to fall in the spectral range of linear-constant acceleration (plateau).

Fig. 4

EDP calculation for the sample of silos (e.g., Q silo, 60% filling level for EFB)

Once EDP-IM quantities are defined and given the failure modes defined in Sect. 2.2, BMs thresholds (EDP_C,EFB, EDP_C,TWD, EDP_C,EB, where the subscript C stands for the capacity) can be quantified by using pushover analysis. In detail, all numerical models can be investigated under a specific load profile defined as a vector of forces proportional to the nodal masses of the silo wall. This load profile is preferred to the one proportional to the first vibration mode, considering that in some cases the participating mass for the main vibration mode could be lower than a reasonable value, and therefore not representative of the dynamic behaviour of the structure. The results of the pushover analysis are used to capture the θ_W,BM values characteristic of the zones where the BMs occur (i.e., base, middle, and top of the wall). These values can be directly observed on the pushover curves expressed in terms of base shear (V_b) vs. θ_W,BM. In particular, in the case of EFB mode, which is an elasto-plastic failure mode, the threshold can be identified by the point identifying the transition from the elastic to the post-elastic zone (i.e., yielding-like point of the capacity curve). When observing EB and TWD modes, which are elastic failure modes, two possibilities are available: (i) identify the change of stiffness in the capacity curve (i.e., change of slope); (ii) record the final point of the curve, which corresponds to the final step of the pushover and indicates the collapse of the structure. The latter could also merely correspond to a numerical convergence problem, which can however be easily discarded by observing if a real failure mechanism is activated at the end of the analysis. At this point, is worth specifying two aspects. Firstly, by considering different service conditions, at a first glance, different thresholds values could be recorded. Nevertheless, considering that the focus of the study is the structural behaviour of the wall, it is necessary that the threshold value, which is the lower limit behind which the relevant BM is observed, is established independently of the filling level. Secondly, the pushover analysis can clarify the occurrence of different failure modes that anticipate the EFB occurrence (considering EFB the most likely failure mode in silos under earthquakes).

Once defined the thresholds values, NTHAs can be performed according to a proper methodology capable of characterizing the distribution of EDP|IM (or IM|EDP). One can optimally choose among IDAs (Vamvatsikos and Cornell 2002), multi stripe (Bazzurro et al. 1998; Jalayer and Cornell 2009) or cloud analysis (Bazzurro et al. 1998; Jalayer 2003) and in line with the necessity to reduce the computational demands, cloud analysis is considered. Undeniable are the uncertainties related to the regression analysis, which could even require the implementation of logistic regression, especially in the case of elevated number of collapses. Nevertheless, some advantages over IDAs can be mentioned, such as the computational efficiency (that is fundamental for the proposed numerical modelling) and the possibility to employ a local fit in the IM-band of interest. In the end, it is worth noting that, given the axial-symmetry of the proposed models, NTHAs can be run on half models, provided that proper boundary conditions are imposed on the symmetry plan, so to ensure realistic mechanical conditions of the structure, i.e., displacements in the perpendicular direction to the symmetry plan and rotations in the main directions on the symmetry plan are not allowed. Results in the EDP-IM plane can be processed through the probabilistic seismic demand model proposed by Cornell et al. (2002). Thus, for each BM, it can be expressed as:

$${EDP}_{BM}={\alpha }_{BM}{IM}^{{\beta }_{BM}}$$

(2)

where ln(α_BM) is the intercept and β_BM is the slope of the regression line in the log-space, both evaluated using the least square method.

The last step of the proposed procedure consists in the definition of the fragility curves, aimed to define the probability for the silos of exceeding specific BMs. Fragility functions are mathematically defined as the cumulative distribution function (CDF) that describes the relation between the IM and the probability of collapse (P_f) according to a certain BM:

$${P}_{f}\left(EDP\ge {EDP}_{C}|IM\right)=\Phi \left(\frac{\mathrm{ln}\left(x/{\mu }_{BM}\right)}{{\upsigma }_{BM}}\right)$$

(3)

where x is the generic IM value, and μ_BM and σ_BM are the median and the dispersion of the CDF, respectively. For the case at hand, the fragility of the silos has been estimated for all geometries and all service conditions (i.e., filling level), and the governing fragility function is the one corresponding to the BM having the higher P_f.

$${P}_{f}=max\left({P}_{f}\left(EFB\right), {P}_{f}\left(TWD\right),{P}_{f}(EB)\right)$$

(4)

4 Selection of the sample of ground-supported steel silos and description

To reflect the variety in the geometry of the existing silos used in the practice, a set of five different structural configurations with different geometries were considered. The set was the same designed by Sadowski and Rotter (2011a) according to the Eurocode prescriptions [12], which was investigated under different conditions of discharge loading. Figure 5 graphically depicts the variation in terms of geometry among the considered cylindrical silos, which differ in terms of 3 main characterizing parameters: height, diameter, and thickness.

Fig. 5

Schematic representation of the five considered silos, drawn to scale

In the design of a typical silo, the relevant geometrical parameters are usually expressed through two ratios: (i) height-to-diameter ratio (h/D) and radius-to-thickness ratio (R/t). The choice of the sample set of case studies aims to cover a wide range of typologies, including very slender (V), slender (S), boundary (B), intermediate squat (I) and squat (Q) silos. Regarding the design roles, h/D reasonably assumes a value ranging from 0.65 to 5.2, while R/t ranges from 357 to 1667 (corresponding to the most economic design values). In the proposed sample set, the storage capacity is constant and equal to 510 m3. The geometric features of the set are reported in Table 1, describing the slenderness type, the silo acronym, the height (h), the diameter (D), the thickness (t) and the abovementioned design ratios.

Table 1 Geometric characteristics of the considered set of silos

With respect to the service conditions, the presence of the filling material has a favourable impact on the lateral stiffness and imperfection amplitude (Buratti and Tavano 2014), and then on the structural strength. The existence of the granular material, in fact, represents an elastic lateral support for the shell wall, and prevents buckling. The reduction of the filling load could be an efficient approach to reduce the seismic demand in earthquake-prone zones since the stored solids contribute to the greater part of the seismic mass of the silo. From this first discussion, it can be deduced that the filling level is a crucial factor governing the failure mechanism of silos under earthquakes. In order to investigate and quantify the effect of the filling level on the dynamic capacity of the silo under earthquake excitation, each silo of the sample set was investigated considering different possible filling levels, expressed in percentage terms of the maximum storage capacity. The considered values are 90%, 60%, and 30%, while also the empty condition was considered for the sake of completeness. Combining the different silos typologies and the different filling levels, 20 specific cases were investigated.

5 Numerical modelling

For all the silos of the set, three-dimensional (3D) numerical models were developed by means of the FE software ABAQUS (Simulia 2012). According to the software philosophy, three numerical parts were identified to form and assemble each FE model: (a) the cylindrical steel silo (modelled through Four-nodes shell elements S4R); (b) the bulk material stored in the silo (modelled through 8 node linear brick elements C3D8R); (c) the RC base (modelled through quadrilateral shell elements S4R). Fixed boundary conditions were applied to the steel wall bottom, by considering the radial, the circumferential, and the meridian displacements fixed to the RC base. Same considerations were employed for the rotations (constraints were applied around the circumference, at uniformly spaced points, in order to reproduce a fully fixed base). The detail of the roof structure was neglected in the model since the primary interest of the study is the behaviour of the shell wall. However, a rigid body constraint, connecting the nodes of the upper edge of the silo wall to a master point, was considered to reproduce the restricting effect of the roof on the out-of-roundness displacement at the upper boundary of the silo shell wall. On this subject, it is worth noting that a ring beam placed at the very top of the silo wall is often used in the practice to avoid the out-of-roundness displacements at the silo top. 4-node shell elements with reduced integration (S4R) were adopted to model the steel silo wall. S4R is classified as a general-purpose and three-dimensional 4-nodded shell element that considers the finite membrane strains and uses both displacement and rotational degrees of freedom (Simulia 2012). However, with S4R elements, the change of the shell thickness as a function of in-plane deformation is accounted for. This element can be employed to model the behaviour of thin and thick shell structures under different loading conditions. The filling material was modelled by using eight-node classical brick element with reduced integration (C3D8R element), and the RC base foundation was considered as a fully stiff element, restrained to the ground. It is worth nothing that the shell failure is very sensitive to the imperfection amplitude, when considering under axial compression (Jansseune et al. 2016). Since the focus of this work is the behaviour of silo shell wall under horizontal loads, it is more convenient to override the effect of the imperfection with reasonable simplification. In fact, for the case at hand the horizontal pressure is the most predominant action to consider and the not investigation of other vertical actions (e.g., vertical component of the seismic motion) reduces the impact that imperfection amplitude can provoke on the silos behaviour (Buratti and Tavano 2014).

Concerning the mesh, a specific investigation was performed for each silo, under different analysis conditions. For the case at hand, as modelling strategy, a higher number of elements was considered in the zones in which the variation of stress rate was high (i.e., bottom part of the silos). According to this modelling strategy, a mesh size ranging from 125 to 1000 mm was considered for the shell of the silo wall. For the mesh of the filling material, a stepwise graded approach was considered in the radial direction. Particularly, a finer mesh size of 250 mm was used at the interface near the wall to ensure a proper developing of the friction interaction between the filling material and the wall, while a coarser mesh size of 1000 mm was adopted for the inner part of the filling solid. A mesh size ranging from 250 to 500 mm was used for circumferential and meridian directions. An example of the employed mesh strategy is reported in Figure 6, which shows the FE models of the silos’ walls and the filling material.

Fig. 6

Modelling and meshing approaches for the considered sample of silos. From left to right, steel wall, granular material and complete model with base plate are reported

Regarding the constitutive laws of employed materials, the steel of shell walls was assumed with an elastic-hardening behaviour and the Von Mises yield criterion was accounted for. The features of the steel are: Elastic Modulus, E_s, equal to 210 GPa; Poisson’s ratio, ν_s, equal to 0.3; yield strength, f_s,y, equal to 275 MPa; ultimate strength, f_s,u, equal to 430 MPa; strain hardening modulus, E_t, equal to 3880 MPa; density, ρ equal to 7850 kg/m3. The considered silos are filled with ‘Camacho’ wheat having a density of 8.36 kN/m³; the coefficient of friction was assumed equal to 0.19 for the steel wall and 0.42 for the RC base. Other specific properties of the used filling material are reported in Table 2 according to Moya et al. (2013), where relevant information derived by experimental investigations are provided. It is worth noting also that only the elastic properties of the filling material were considered, neglecting the insight about the nonlinear behaviour, which is not the objective of the study. However, seeking more realistic modelling approach, the effect of the granular material nonlinearity could be accounted for, by considering more sophisticated constitutive material models (e.g., the hypoplastic material law as adopted by Butenweg et al. (2017) or the extended Drucker-Prager law as used by Mehretehran and Maleki (2018)) that can more accurately capture the solids behaviour and its impacts on the silo walls, in terms of additional pressures. Obviously, this would strongly increase the computational demand to achieve convergence in the numerical model and, for this reason, any type of granular material nonlinearity was neglected in the numerical model.

Table 2 Mechanical properties of the stored material (‘Camacho’ wheat) (Moya et al. 2013)

Concerning the interaction between the filling material and the structure, a Coulomb frictional model was assumed, by defining the critical shear stress (threshold) between the two contact surfaces. According to this model, the slip between the two surfaces occurs after that the shear stress between them exceeds the threshold. The critical shear stress is mainly proportional to the normal pressure on the contact surface and the friction coefficient between the two materials. While the normal pressure on the contact surface is calculated based on the numerical analysis, the coefficient of friction, as it was derived from Moya et al., (2013), was manually inserted by the authors as input of the FE model.

6 Modelling approach assessment and validation

In order to assess the validity of the proposed modelling approach, a specific section is devoted to the aspects related to the appropriateness of the proposed mesh size and the type of FE selected from the software library (Sect. 6.1) and the validation with the existing literature (Sect. 6.2).

6.1 Mesh resolution and assessment of the FE type selection

The software ABAQUS accounts for the use of different FE elements from the available libraries, which could be used for detailing the numerical models. For the case at hand, it is relevant the modelling technique adopted for the structural part of the silos (i.e., steel walls) and for this reason some additional basic analyses were performed. However, for the granular-like material and the surface interaction, which are not the central focus, the used modelling approach can be assumed to be adequate. For modelling the steel of shell walls, ABAQUS allows to select more complex FE types than the used one, such as the eight-node continuum shell element (SC8R element) or the eight-node continuum solid shell element (CSS8 element). Comparing to the S4R element, which is a 4-node quadrilateral shell element with reduced integration that allows transverse shear deformation, SC8R element is 8-node hexadreon continuum shell element with reduced integration that allows transverse shear deformation. Differently from the S4R, the SC8R is used to discretize an entire three-dimensional body, and from the modelling point of view, it is like a 3D continuum solid presenting kinematic and constitutive behaviour like conventional shell elements and with the thickness determined from the element nodal geometry. Still, CSS8 element is an 8-node linear solid shell brick, that is typically suggested for thin-walled structures, and it can fill the gap between incompatible mode elements, which use 3D constitutive laws but tend to exhibit locking in bending for large aspect ratios, and continuum shell elements, which present good bending response for large aspect ratios but are limited to two-dimensional plane stress constitutive behaviours.

To understand which of the above FEs (i.e., S4R, SC8R, CSS8) provides the better performance, higher accuracy and shorter time of analysis, a preliminary linear analysis was performed on an axially compressed cylindrical shell. In particular, the purpose was to estimate the buckling capacity (N_numerical) under axial meridian stresses by employing the three FE types and compare it to a classical analytical solution (N_analytical). For this new numerical experiment, the shell was assumed to be fixed at the bottom and with reduced degrees of freedom at the top, i.e., with possibility of free vertical displacements. The geometry of the numerical specimen presents D equal to 800 mm, h equal to 1200 mm, and t equal to 1 mm. For the steel, it was assumed having E_s equal to 210 GPa, and Poisson ratio, ν, equal to 0.3. The solution N_analytical was estimated through the classical formulation of the elastic critical force applied to a shell under axial compression, that is, the Timoshenko formulation:

$${N}_{analytical}=\frac{2 \cdot \pi \cdot {E}_{s} \cdot {t}^{2}}{\sqrt{\left.3(1-{\nu }^{2}\right)}}$$

(5)

Using the same values of the numerical benchmark, N_analytical results in 798.58 kN. By excluding the geometrical imperfections from the evaluation of the numerical solution, the results for N_numerical are shown in Table 3, which also reports the percentage difference from N_analytical. Table 3 reports also the CPU time required for estimating N_numerical through a workstation equipped with a Core i5 CPU, 4 GBs of RAM, and an INTEL HD Graphics 4000. Basing on the obtained results, the differences among the analytical and numerical solutions are quasi-negligible (element S4R provides a lower percentage difference, but also other models provide good results). Nevertheless, the main discriminant parameter is the time of analysis, which was sensibly lower for S4R and SC8R elements than CSS8. This first assessment demonstrated the reliability of the proposed modelling technique compared to the more complex FE types.

Table 3 Comparison among different FEs types for steel shell element, in terms of numerical results and CPU time

6.2 Validation

For purpose of modelling approach validation, two different steps were performed: (a) a comparison among the numerical results in terms of static pressure imposed by the filling material on the silo walls and those obtained by analytical solutions based on Janssen’s theory (Janssen 1895), and modified Reimbert’s theory (Reimbert and Reimbert 1987); (b) a comparison among the fundamental frequencies provided by the proposed numerical models, the ones obtained by experimental investigations on a reduced-scale silo reported in (Silvestri et al. 2016), and the numeral investigations of a full-scale silo reported in Mehretehran and Maleki (2018).

Concerning to the first validation step, horizontal (or normal) pressure (P_h) and frictional (or vertical) traction (P_w) imposed by the filling on the walls were numerically computed for all models with grain material inside, i.e., 15 cases over 20, excluding the empty cases. For the cases V, S and B, Janssen’s theory (see Eq. 6) was applied, while for I and Q cases, modified Reimbert’s theory (see Eq. 7) was employed, according to the following expressions:

$$P_{h} \left( z \right) = P_{h0} \left( {1 - e^{{{{ - z} \mathord{\left/ {\vphantom {{ - z} {z_{0} }}} \right. \kern-0pt} {z_{0} }}}} } \right)$$

(6)

$${P}_{h}\left(z\right)={P}_{h0}\left({1-\left\{\left(\frac{z-{h}_{0}}{{z}_{0}-{h}_{0}}\right)+1\right\}}^{n}\right)$$

(7)

where P_h0 is a pressure proportional to the lateral pressure ratio, the unit weight of the granular solid and z₀; h₀ is the height between the equivalent surface of the solid and the highest solid to wall contact; z is the height of a generic point of the silo; z₀ is the ratio between the silo radius and a quantity proportional to the wall friction coefficient between the granular solid and the wall and the lateral pressure value. Then, the values of P_w are defined as a fraction of P_h, as:

$${P}_{w}={\mu }_{F}{P}_{h}$$

(8)

where μ_F is the coefficient of friction.

It is worth remembering that both theories are currently adopted by most of the international standards such as the European ones (EN 1991-4, 2006). Figure 7 shows the comparison between numerical and analytical results in terms of P_h and P_w against the normalized height, h_norm, for the 5 different geometries of silos with 90% filling ratio. The static pressures numerically computed (red lines) are in good agreement with those provided by the theoretical solutions (blue lines), even though some discrepancies can be observed in the bottom parts of the silos, as also reported in Ayuga et al. (2001) and in Mehretehran and Maleki (2018).

Fig. 7

Comparison among numerical and analytical values of P_h and P_w for all silos geometries with 90% filling level

In order to verify the dynamic characteristics of the FE models, the second validation step was performed by considering the experimental results on a reduced-scale benchmark silo manufactured in laboratory and subjected to dynamic excitations (Silvestri et al. 2016). The benchmark silo was numerically reproduced by employing the proposed modelling approach, and modal analysis was performed to compare outputs with the experimental results. The numerical results showed a good agreement with the experimental ones, where the first and second natural frequencies calculated by the FE model were 15.65 Hz and 43.79 Hz, respectively, while the ones identified in Silvestri et al. (2016) after shaking table tests were declared in the ranges of 12.7–14.1 Hz and 43.9–44.9 Hz, respectively. It is also worth mentioning that due to the axial-symmetry, the model showed duplicated mode shapes in two main directions. Figure 8 reports the two lowest-frequency modes of vibration corresponding to a global cantilever flexural mode in the translational directions.

Fig. 8

Model of the benchmark silo in (Silvestri et al. 2016) and vibration modes a the first vibration mode 15.65 Hz; b the second vibration mode 43.79 Hz

To complete the second validation step, a second FE model was developed, for reproducing the outcomes of the numerical model of a full-scale silo presented in Mehretehran and Maleki (2018) in terms of modal analysis and static pressures. This silo presents t equal to 10 mm, h equal to 25 m, D equal to 10 m, while steel material was modelled with a f_y,s equal to 250 MPa, E_s equal to 210 GPa, ν equal to 0.3, and density equal to 7850 kg/m³. Using these data, modal analysis was carried out on the FE model and the frequency of the first vibration mode was found to be equal to 2.8 Hz, which was identical to the one obtained by Mehretehran and Maleki (2018).

7 Seismic analysis of the selected set of silos

7.1 Modal analysis

For the selected set of silo models, modal analysis was performed, aimed at calculating T₁ and the related participating mass (M[%]) as the geometry changes from V to Q, and as the percentage of filling level varies from 90% to the empty case. The results are reported in Table 4, in which it can be observed that the two varied parameters provide different results. Obviously, duplicated T₁ values were obtained in the two main orthogonal directions, due to the axial-symmetry of the structure. For a same filling level, going from the V to Q silo, the value of T₁ reduces at the same filling level, which is due to an increment of stiffness among silos (higher values of R/t provide a decrease in T₁, as well as occurs for lower values of h/D). For the same geometry, going from the 90% filling level to the empty case, the value of T₁ reduces, which is due to a gradual decrease of the total mass.

Table 4 Variation of the fundamental periods (T₁) and the related participating mass (M[%]), according to the silos’ geometry and percentage of filling level

Looking at the dispersion of T₁ for a same geometry, as the filling level varies, a high dispersion can be observed for the cases V and S, while a lower dispersion was obtained for the cases I and Q. This observation reveals that the squattest geometries are less sensitive to the filling level parameter, while the slenderest geometries have an opposite behavior. Different considerations can be provided for the values of M[%]. In general, values of M[%] range from 55 to 75% and for all service conditions, the M[%] increases going from the slenderest to the squattest configurations. Observing the service conditions, different outcomes can be derived. In fact, going from 90 to 30% of filling level, the values of M[%] decrease for the same geometry. Instead, different trend from the filled silos is observed for empty silos, where the values of M[%] are higher than the ones obtained in 30% and 60% filled silos, and similar to the ones obtained in 90% filled silo.

7.2 Pushover analysis and thresholds definition

Once modal analysis results have been derived, pushover analysis was performed. As stated in Sect. 3, this step has two objectives: (a) predefine the most vulnerable buckling zones, which should be monitored when running NTHAs; (b) derive the values of proper thresholds, EDP_C,BM, beyond which the structure is considered to be failed exhibiting a specific BM. With these goals in mind, all silos were investigated, and the evolution of the deformed configurations was recorded. Pushing the structures, accumulation of stress was observed in different zones of the silo wall. In the squattest configurations, EFB anticipated EB and TWD, confirming the available literature. Also the structures with 90% and 60% filling ratio showed a similar outcome. When coming to the slenderest configurations and to the empty (or near-empty) cases, it was observed that the first occurring buckling mode is not always the EFB but can be the EB or the TWD. As a physical interpretation of the obtained results, in the case of the empty or 30% filling level, the development of elastic failures such as EB can be attributed to the absence of the stored solids, which could be imagined like an elastic lateral support against buckling. A summary of these outcomes is graphically outlined in Figs. 9 and 10, where the evolution of the deformation under horizontal static loads is shown in terms of displacement for the Q silo, 0% filling level, and for the Q silo, 60% filling level, respectively. In particular, for the case in Fig. 9, the first BM developed is the EB in the middle zone of the silo, while EFB does not appear. Instead, in Fig. 10, despite a hint of EB occurs, the main developed BM is the EFB. In both cases, due the application of the forces, TWD can be observed, and they are more or less accentuated depending on the analysis step. These observations assume high relevance, because they confirm that the combination between geometry and service condition governs the BM development under horizontal actions. Still, the failure modes assumed at the base of this study are confirmed.

Fig. 9

Evolution of the deformation in terms of displacements (from left to right) under pushover analysis of Q silo, 0% filling level. Values in the colored legend are expressed in mm

Fig. 10

Evolution of the deformation in terms of displacements (from left to right) under pushover analysis of Q silo, 60% filling level. Values in the colored legend are expressed in mm

The second step of the pushover analysis is the definition of the deformation limit beyond which the buckling phenomena develops (i.e., capture the thresholds) and the identification of the specific zone to monitor during NTHAs. After running the analyses, it is worth noting that while for EFB and TWD at the bottom and upper parts of the structures can be well identified, for the EB failure mode there is not a specific position to consider (e.g., a position that, for all silos, could be representative if normalized over the height). Thus, EB failure needs to be observed case-by-case, to define a specific height of the zone. Table 5 reports the obtained thresholds values, θ_W,BM, and the related quantities for estimating the EDP.

Table 5 Thresholds values for the considered silos, in terms of θ_W,BM, δ_W,BM, h_W,BM

As observed, for the values of θ_W,BM it is difficult to derive a trend among thresholds, considering that similar values occur for TWD and EB failure modes, while higher values were recorded for EFB. Looking at the other parameters, such as the δ_W,BM, it can be observed that going from V to Q silos, the achievement of the EFB occurs at an increasing displacement, while the achievement of TWD and EB occurs at a decreasing displacement. Still, looking at the parameter h_W,BM, it was observed that the EFB occurs at the same height from the ground, while the TWD and EB failures occur at different heights, according to the type of considered silo.

Finally, it is useful to show how the values in Table 5 were estimated. For this scope, capacity curves in terms of V_b vs. θ_W,BM are reported in Fig. 11 for all silos, accounting for specific filling levels and monitoring different zones, that is to say, the base of the walls for the EFB and the middle of the walls for the EB. Thresholds are reported as black dots on each curve, according to the values in Table 5 and the criteria reported in Sect. 3. Regarding these latter, no cases can be mentioned in which convergence problems occurred and black dots reported in the final point of the curve. By varying the geometry of the silos and going from the slenderest to the squattest silos, the value of V_b increases and the value of θ_W,BM decreases.

Fig. 11

Pushover curves obtained by monitoring EFB and EB zones, for all silos with 90% and 0% filling level

It is worth remarking that pushover analyses, here, are not employed with the objective of characterizing the seismic performance of silos, which is instead achieved by employing more reliable NTHA approach, as reported in Sect. 8.

8 Record selection and cloud analysis

To run NTHAs on the silos, a proper record selection was firstly performed. For the case at hand, a set of 11 records was selected from the European-Strong-Motion-Database (Ambraseys et al. 2004), by referring to a target spectrum obtained from the municipality presenting the highest value of PGA in Italy (Ferla, Sicily), and by amplifying the spectrum according to a soil category of type C. The choice was performed, on one hand, in order to ensure high acceleration values on silos (and then, induce failures) and, on the other hand, to characterize the analysis in the country of the authors’ universities. Following the prescriptions of Eurocode 8, records were selected in order to limit the difference between mean and target spectra to + 30% and − 10% in a period range of interest for the investigated structures, that is, from 0 to about 1 s (low–medium range of periods) (Ruggieri and Vukobratović 2023). The latter value was selected because it was twice larger than the maximum obtained T₁ value, which occurred for the V silo with 90% filling level. All elastic ground motion records (5% damping) are reported in Fig. 12, with the indication of target and mean spectra.

Fig. 12

Set of selected records

To reduce the effort of the analysis, considering the axial-symmetry of the structures, the numerical models developed were modified, opting for running NTHAs on half of the systems and by applying the adequate boundary conditions described in Sect. 3. It is worth noting that this approach was also employed by Mehretehran and Maleki (2018), with the purpose of running IDAs on complex numerical models. In the NTHAs, a damping ratio of 2% was considered for the models and the implicit time-integration technique provided by ABAQUS was used. Coherently with the modelling simplifications, accelerograms were applied in one direction, that is, the one in the symmetry plan. Nevertheless, it is worth mentioning that the vertical component of the seismic action could be crucial for the silo safety (Butenweg et al. 2017), considering an additional acceleration on the silos mass, which yields additional dynamic pressures. This could be important especially for the slender or elevated silos (since it particularly relevant for the vertical pressure on the hopper).

From the observation of the first outputs, silos behaved as expected, exhibiting an EFB failure mode, especially when 90% and 60% of filling material was considered. This effect is graphically shown in Fig. 13, which reports the development of EFB for the Q (from left to right, the first two images) and S silos (from left to right, the last two images), under a randomly chosen record among the above-mentioned set.

Fig. 13

Numerically predicted EFB for Q (from left to right, the first two images) and S (from left to right, the last two images) silos, subjected to a randomly chosen ground motion record from the selected set. Values of stresses are indicated in a colored scale in terms of MPa (although no specific values are indicated)

The NTHAs campaign was performed in the form of cloud analysis, without performing record scaling. Figure 14 shows a couple of examples reporting the distribution of cloud points in the EDP-IM space and the global power law fit. Some comments can be provided about the obtained results. Firstly, as expected, the points are dispersed, which means that on the base of the performed regression, the values at higher IM could deviate from the near reality. Thus, other approaches could provide substantial improvements, such as by opting for IDAs or by scaling the ground motions to better capture the effective BMs. On the other hand, it is worth considering manifold aspects that justify our analysis simplifications. First of all, an undeniable computational effort reduction can be mentioned, considering that for running 220 analyses (20 silos with 11 records) on the detailed models, using a more powerful workstation than the one described in Sect. 6.1 (Core i9-13900HK CPU, 64 GBs of RAM, and an NVIDIA GeForge 3090 GPU equipped with 24 GBs or VRAM), more than 1 month was spent. Anyway, with the selected suits of input, no collapses occurred, which means that all the planned EDP-IM points were obtained for the global fitting. Moreover, taking into account the combination between the input records and the selected IM, the analyses performed have covered a band from about 0.5 g to about 2–2.2 g in the low–medium range of periods, providing an EDP-IM relationship in an IM-band consistent with the existing literature and accounting for a seismic intensity able to bring the considered structures to failure (Mehretehran and Maleki 2018). An important consideration should be provided about the obtained results in Fig. 14. Considering silos as short period structures, the obtained exponent of the power low is extremely lower than 1, which is not properly correct if considering an expected value close or greater than 1 (see for example the outcomes in O’Reilly and Monteiro 2019). This numerical aspect, due to the extremely scattered results, confirms once again the necessity to increase the number of analyses to run. Finally, although the results obtained could be not so accurate for some parts of the EDP-IM space, our investigation provides clear information on the most likely (dominant) failure mode at the variation of the geometry and service conditions.

Fig. 14

Cloud analysis and power law regression for the Q silo with 90% filling level (left) and I silo with 60% filling level. IM is Sa(T₁), expressed in unit of g, while EDP is θ_W,EFB

9 Estimation of seismic fragility and discussion of results

From the previous results, fragility curves can be derived according to the approach described in Bakalis and Vamvatsikos (2018) for all silos, accounting for the combination of all geometries, all service conditions, and all BMs. Results in terms of fragility curves are reported in Fig. 15, while Table 6 reports the detailed values of μ_BM and σ_BM.

Fig. 15

Fragility curves accounting for all silos geometries, all conditions of functionality and all BMs

Table 6 Values of μ_BM and σ_BM for fragility curves, accounting for all geometries, all conditions of functionality and all BMs

As expected, the EFB is the most probable BM occurring on silos when dynamic excitation is applied, while the other BMs, i.e., TWD and EB, could occur in specific cases (quasi-empty and empty silos) and for high values of the seismic intensity, anticipating the EFB. Discussing in detail the results obtained from the point of view of service conditions, it can be observed that going from the filled to the empty cases, the EFB fragility curves reduce their P_f (μ_EFB increases). The physical interpretation of this result can be given by considering that a reduction of the stored material induces a reduction of the mass excited under the earthquake, inducing a benign effect on the seismic performance of the silo. Looking at the EFB fragility curves for a given filling level, varying the geometry from the slenderest to the squattest silo the μ_EFB increase (an increment of about 40% was recorded, with an almost doubled median), which implies a better performance of squat silos under earthquake actions. As a matter of fact, silos with lower aspect ratio present higher performance than those with high aspect ratio under seismic actions, given the same service conditions. Moving to silos with 30% and 0% of filling level, EFB fragility curves shift to right, and the EB and TWD fragility curves can anticipate the previous one. Anyway, this estimate cannot be accurate for the reasons repeatedly discussed throughout the paper: for high levels of seismic intensity, different BMs could occur and anticipate the usually expected EFB. Also this result assumes a physical sense, by considering that a low total mass characterizing an empty system implies a low fundamental period of vibration and then, a reduction of the seismic demand. The very high values of μ_BM for quasi-empty or empty cases suggest that a low seismic risk is connected to these kinds of structures (they can difficultly collapse under a natural seismic event) and that the most influent parameter in the overall fragility estimate are the stored material type, its level and its interaction with the structure.

10 Conclusions

The paper presents a numerical procedure to derive seismic fragility of cylindrical ground-supported steel silos storing granular-like material. The procedure aims to investigate the influence of three main aspects on the seismic behaviour of these kinds of structures: (a) the geometry of the silos wall; (b) the service conditions, that is, the filling level of the stored material; (c) the type of failure mode. With this goal in mind, the procedure has been articulated in four consecutive steps. The first step consists in the selection of a set of smooth steel silos, which cover a large range of possibilities. The set was selected by assuming different geometries (e.g., varying from slender to squat silos) and different filling level of stored materials (from filled to empty case), obtaining a number of 20 cases. The second step consisted in the detailed modelling of the generated set of silos, which was performed through the software ABAQUS. Three-dimensional numerical models were created for simulating the behaviour of steel shell walls, the physical properties of a granular like material and the interaction between the above components. Some validations with data from the existing literature and the available analytical solutions were carried out, by assessing the numerical results in terms of static pressures and dynamic features. Considerations were provided about the mesh of the detailed numerical models, in order to ensure effective solutions of the successive analysis. The third step consists in the analysis campaign, which was based on the combination of static and dynamic nonlinear analyses. The role of static analyses is double: (a) the assessment of the possible failure modes occurring on steel silos under seismic actions, i.e., Elephant Foot Buckling in the bottom part of the wall, Elastic (diamond or similar shape) Buckling in the middle part of the wall, Top Wall Damage in the upper part of the wall; (b) the definition of likely thresholds to define the achievement of all the considered buckling modes. Instead, nonlinear time history analyses have been developed in order to define the probabilistic relationship between demand and capacity, assuming specific parameters for this purpose. With this regard, a set of 11 records was suggested to excite structures through a cloud analysis, and a regression analysis was performed through the power law fit. Although the number of analyses performed was reduced (to limit the computational cost for analysing the models), the fourth step consists in the derivation of fragility curves for all silos geometries, all filling levels and all possible failure modes. The obtained results highlight some novel aspects. Accounting for the influence of the silo geometry, the results have revealed that silos storing solids are more vulnerable to the elephant foot buckling, and this is emphasized when structures are filled with 90% and 60% of the maximum capacity. Looking at the median of fragility curves, going from squattest to slenderest silos, the probability of failure for same seismic intensity increases, which means that squattest silos show better performance than the slenderest ones under seismic actions (a difference of about 40% was recorded). With regard to the effect of the ensiled material, results indicated that Elephant Foot Buckling is the governing failure mode for filled silos (90% and 60%) while, in case of quasi-empty or empty (i.e., 30% and 0%), all three investigated failure modes present comparable probability of failure, despite high median values are obtained. Still, empty (or quasi-empty) silos exhibit very high seismic capacity, regardless of the failure mode, which means that in these service conditions there is a low probability of failure for silos, especially under ordinary natural seismic events.

Further developments of the work will be aimed to extend the investigation to different kinds of silos (e.g., stiffened, corrugated), and to consider different types of stored materials. In addition, a refinement of the numerical models could be performed, by considering additional physical factors for the granular solid material, such as the compaction of it under earthquakes actions. From the fragility curves derivation point of view, different approaches from the simple one herein used should be employed, by opting for using more records and by involving other analysis techniques (e.g., IDA).