SIMULATING CYLINDRICAL STEEL SILO FILLING AND DISCHARGE PROCESSES: FINITE ELEMENT MODELING AND VALIDATION WITH EXPERIMENTAL DATA

ABSTRACT

Cylindrical steel silos with flat bottoms are widely used in agriculture and industry for storing granular materials. While research has advanced our understanding of pressure on silo walls, accurate prediction, especially during the dynamic filling and discharge phases, remains a challenge. This study presents a finite element (FE) analysis of pressure distribution in a model cylindrical steel silo with a flat bottom, investigating the influence of the height-to-diameter (H/D) ratio. The numerical results were validated against experimental data from a pilot-scale test facility storing corn. Material properties were determined through laboratory experiments, with mechanical properties obtained from literature. An arbitrary Lagrangian formulation was employed for the FE calculations. The FE results showed good agreement with experimental data for static pressure distribution on the silo wall across all H/D ratios analyzed. While the patterns of dynamic pressure curves were similar, the FE-predicted magnitudes were lower than those observed experimentally. Notably, the simulations captured significant pressure fluctuations during silo discharge.

cylindrical steel silos; pressure distribution; filling and discharge processes; free-flowing product; numerical model; stored material properties

INTRODUCTION

Grain storage structures are typically thin-walled metal cylindrical silos (Cao & Zhao, 2017; 2018; Maleki & Mehretehran, 2019; Trung et al., 2020; Tian & Jiao, 2021; Khalil et al., 2023). The rapid growth of the food industry has dramatically increased the demand for grain storage, making the structural integrity and safety of silos increasingly important (Ranjan Debbarma et al., 2022; Khalil et al., 2023).

The slenderness of commercially produced silos, defined as the height-to-diameter ratio of the cylindrical part, ranges from 0.4 to 3.5 (Sondej et al., 2018; Khalil et al., 2023). For large-diameter silos, this ratio generally does not exceed H/D = 1.0. International codes and guidelines describe silos with low height-to-diameter ratios (H/D ≤ 1.5) as squat and those with high height-to-diameter ratios (H/D ≥ 1.5) as slender (Hussein & Risan, 2022). The predominant configuration of silos used on farms features a low height-to-diameter ratio and a flat bottom, while in the industrial sector, slender silos are more common.

The essential starting point in the structural and flow design of silos is determining the physical properties of stored products. This should be conducted under the most severe conditions expected to occur in a silo, enabling accurate prediction of pressures in both the static and dynamic phases (Maleki & Mehretehran, 2019).

Wall pressure distribution depends on the type of silo. Based on loading characteristics, silos are generally classified as slender silos or low silos. For slender silos, most standards are based on Janssen's theory, though each standard considers different wall pressure coefficients. The classical wall pressure theories for low silos include Rankine theory and modified Coulomb theory (Sun et al., 2018; Jayachandran et al., 2019).

Although many studies have been conducted on these structures, significant uncertainties remain, as the laws governing the behavior of products stored in silos are not fully understood (Tian & Jiao, 2021). Analytical methods do not always accurately predict these pressures (González-Montellano et al., 2011) The large number of variables involved complicates the development of a mathematical formalism that can adequately express the phenomenon while ensuring an acceptable margin of safety and economy.

Goodey et al. (2017) found that the existing design guidelines in EN 1991-4:2006 result in considerably more expensive sizing of metal silos than necessary. Sondej et al. (2018) stated that the analytical sizing of corrugated sheet silos based on current normative silo codes is overly simplified and uneconomical, particularly for silos with relatively large distances between uprights.

The failure rate in silos is much higher than in other industrial constructions (Sondej et al., 2018; Chavez Sagarnaga & Kulkarni, 2020; Hussein & Risan, 2022; Ranjan Debbarma et al., 2022; Gabbianelli et al., 2023). The pressures exerted by stored materials are a key factor in these failures and collapses (Sondej et al., 2018; Bywalski & Kamiński, 2019; Gandia et al., 2022). While the pressure on silo walls during filling is well understood, predicting the pressure during discharge remains a significant challenge in silo design (Wang et al., 2014; Sondej et al., 2018; Jayachandran et al., 2019).

Numerical techniques have opened new avenues for understanding the complex interactions between stored solids and silo walls (Zhao & Jofriet, 1992; Holst et al., 1999; Ayuga et al., 2006; Gallego et al., 2010; Gallego et al., 2015; Pardikar et al., 2020; Gandia et al., 2021a; Moya et al., 2022). Significant progress has been made in computational approaches for granular solids and silo design worldwide in recent decades (Hilal et al., 2022). Among numerical methods, the finite element method (FEM), the discrete element method (DEM), and their combinations (Vidal et al., 2008) are particularly important. Studies have shown that FEM is more accurate for predicting silo stresses (Gallego et al., 2015; Jayachandran et al., 2019). EN 1993-4-1:2007 (Eurocode 3 - Design of steel structures - Silos) identifies FEM as the most accurate and reliable tool for structural stress analysis, recommending it as the validated numerical method for silos in Action Reliability Classes 3 (Tian & Jiao, 2021).

Numerical methods are invaluable for understanding these complex structures (Ooi & Rotter, 1990; Chen et al., 1999; Martinez et al., 2002; Wang et al., 2014; Wang et al., 2015, , Horabik et al., 2016), as they allow for simultaneous simulation of a flexible wall and materials stored in the silo, or consideration of nonlinear material models for the granular solid, among other factors (Gallego et al., 2015). Consequently, using numerical methods can improve silo quality and durability and generate economic benefits (Sun et al., 2018). According to Sondej et al. (2018), silos designed using FEM offer greater opportunities for stability evaluation compared to normative standards-based calculations, leading to significant material savings and allowing for detailed analysis of important phenomena.

The FEM model has vast usability and is suitable for predicting pressures in static and dynamic conditions in cylindrical or rectangular silos (Ooi & Rotter, 1990; Gallego et al., 2010; Gallego et al., 2015; Goodey et al., 2017; Gandia et al., 2021a; Hilal et al., 2022; Abdelbarr et al., 2024). It can also predict wall pressure in silos with inserts (Askif et al., 2021), wall pressures in cylindrical silos with conical hoppers (Wang et al., 2014; Han et al., 2019; Sadowski et al., 2020), eccentric hoppers (Guaita et al., 2003; Vidal et al., 2008), multiple bottom hoppers (Ranjan Debbarma et al., 2022), or flat-bottomed silos (Sanad et al., 2001; Wang et al., 2015). Additionally, it can analyze phenomena such as buckling behaviors of steel silos (Zhao and Teng, 2004; Iwicki et al., 2011; Cao & Zhao, 2017; Wang et al., 2017; Maleki & Mehretehran, 2019; Mehretehran & Maleki, 2022), seismic activity on silo systems (Durmuş & Livaoglu, 2015; Hussein & Risan, 2022), shear localization in granular specimens (Wójcik & Tejchman, 2009), and geometric imperfections (Wójcik & Tejchman, 2016; Mehretehran & Maleki, 2022).

However, numerical models must be validated by experimental tests (Gallego et al., 2015; Gandia et al., 2021a; Khalil et al., 2023), which provide essential parameters for simulations (Couto et al., 2013; Moya et al., 2022) and allow proper adjustment of the models (Moya et al., 2022). Validated numerical simulations can overcome many limitations, provide data that closely approximate real values, and enhance understanding of pressures in silos.

Due to the high investment required for construction, instrumentation, and operations, there are few full-scale experimental silo facilities worldwide (Schwab et al., 1994; Brown et al., 2000; Teng et al., 2001; Zhao & Teng, 2004; Teng & Lin, 2005; Härtl et al., 2008; Ramírez et al., 2010a; Couto et al., 2012; Jayachandran et al., 2019; Sun et al., 2020; Gandia et al., 2021b). Additionally, the scale factor is crucial for confidence in the data (Brown & Nielsen, 1998).

A full-scale experimental silo model enables the acquisition of approximate real values, facilitating the understanding of pressure effects in silos (Gandia et al., 2021a). In this study, we used the pilot-scale test station proposed by Pieper & Schütz in 1980 (Pieper and Schütz, 1980) and supported the guidelines of DIN 1055-6: Basis of design and actions on structures - Part 6 (DIN, 2005). This setup allows measuring numerous variables directly influencing silo pressure behavior. The pilot silo was described and validated in detail in two previous publications (Gandia et al., 2021b and 2021c).

To advance scientific knowledge in granular product storage, we aimed to develop an experimental and numerical study of static and dynamic pressures due to corn storage in vertical silos with concentric filling and discharge. For analysis, smooth walls, flat bottoms, and three height-to-diameter ratios were considered. A two-dimensional and axisymmetric analysis was performed using the finite element method (FEM) with the ANSYS Student R1 2022 software package (ANSYS STUDENT, 2022). The numerically obtained results were compared with the maximum pressures normal to the silo walls obtained experimentally. Our results were also compared to the analytical solution of Janssen (1895), which is commonly used in standards for computing the initial filling pressures of silos.

MATERIAL AND METHODS

Physical and mechanical properties of the product

The reliability of finite element method (FEM) results is highly dependent on the physical and mechanical properties of analyzed products for validating and implementing constitutive models (Jayachandran & Rao, 2018; Moya et al., 2022). Yet, these properties are not commonly included in traditional theories and are found in a limited number of scientific studies for only a few products (Moya et al., 2002, 20, 2006, 2013, 2022; Costa et al., 2014).

As granular material, we used corn (Zea mays) with a minimum purity of 97%. Its flow properties were determined experimentally at the Federal University of Campina Grande, Brazil, following British standards (British Materials Handling Board, 1985; Jenike, 1964) and using the Jenike Shear tester, which conforms to Eurocode 1 (EN 1991-4, 2006). Table 1 shows these properties.

TABLE 1
Corn properties used in the FEM simulation.

Property	Value	Reference
Grain specific weight, γ (kN/m3)	7.75	Obtained experimentally
Grain modulus of elasticity, E (kPa)	34.8	Moya et al. (2022)
Effective angle of internal friction of bulk material, ϕi	38°	Obtained experimentally
Wall friction angle, ϕw	9.95	Obtained experimentally
Wall friction coefficient, μ	0.17	Obtained experimentally
Cohesion, c (kPa)	0.78	Obtained experimentally
Grain Poisson’s ratio, ν	0.31	Gallego et al. (2015)
Angle of dilatancy of bulk material, ψ	33.6º	Moya et al. (2022)

The mechanical properties adopted in numerical simulations were based on Moya et al. (2022), who suggested reference values for the studied mechanical properties of corn to be used in silo load calculations involving numerical methods. Table 1 presents these respective input parameters associated with the stored product.

Experimental tests

Description of the Experimental Installation

The experimental setup consists of a pilot silo equipped with instrumentation for measuring pressures and an auxiliary silo for product storage. The tested product is transferred using a bucket lift to perform automated filling and discharge operations (Fig. 1).

FIGURE 1
Diagram of the experimental station.

The pilot silo comprises 12 independent rings, each 49.5 cm high, separated by a 5 mm gap to restrict vertical influence between the rings, resulting in a total height of 6 m. The silo walls are made of 10 mm thick galvanized steel,

making them rigid. According to Eurocode 1, Part 4 classification (EN 1991-4, 2006), the silo can be classified as low, medium slender, or slender, depending on the height-to-diameter ratio adopted during testing (due to the independence of each ring).

The internal diameter of the silo is 0.691 m, and the discharge orifice has a diameter of 0.20 m. A transition box is located immediately below the discharge opening of the pilot silo to feed the bucket lift continuously, allowing maximum discharge flow into the pilot silo and neutralizing any influence from the bucket lift transport (Fig. 1).

Normal wall pressures are measured using extensometer-type tensile load cells positioned horizontally at the opening of each calendar ring. In each ring of the pilot silo, pairs of load cells are pre-tensioned by a set of three springs (Fig. 2).

FIGURE 2
Wall pressure and friction measurement system (8-kN load cells).

The rings were suspended by vertical measuring clamps (8 kN load cells) to determine the frictional force on the walls (Fig. 2). The total weight of the silo was measured by another group of load cells located under the two silo columns (Fig. 3). More information about this experimental station was provided in two previous papers published by the team responsible for the calibration and validation of the structure (Gandia et al., 2021b; Gandia et al., 2021c).

FIGURE 3
Beam-type load cells at silo pillar bases (50-kN capacity) to measure total frictional force.

Test description

The experimental tests were conducted with the following configurations: 1) three H/D ratios, as described in Table 2; 2) flat bottom; 3) smooth wall for the pilot silo; and 4) five replicates for each configuration (filling and discharge cycles).

TABLE 2
Configuration of the H/D ratios evaluated in the pilot silo.

Silo slenderness classification*	Ratio H/D*	Silo height (m)	Inner diameter (m)	Experimental H/D ratio	Storage volume (m3)
Squat	0.4≤H/D≤1.0	0.75	0.691	≅ 1.0	0.34
Intermediate slenderness	1.0<H/D≤2.0	1.50	0.691	≅ 2.00	0.64
Slender	H/D>2.0	2.00	0.691	≅ 3.00	0.83

• * Slenderness classification according to the EN 1991-4:2006 guidelines.

For each combination evaluated in the pilot silo, filling and discharge pressures were measured separately. The silo was loaded until the grain mass reached the desired height, and after a pause of nearly 10 minutes (static condition), it was unloaded. The filling was carried out through a vertical tube, allowing grains to fall centrally by gravity. Discharge was also concentric, with the discharge orifice being fully opened, producing a free-flow discharge during the initial moments until the transition box was full (Fig. 4).

FIGURE 4
Description of the test. a) H/D ratios analyzed in the pilot silo; b) Test stages: filling, static, and discharge conditions.

Normal wall pressure values (p_h[1,12]) were obtained directly from the sum of readings from load cells installed in each ring of the pilot silo. This was done by dividing the obtained force value by the height of the ring [h_r=0.495 m] and the value of the constant of the rings (C_r[1,12]), as shown in [eq. (1)].

$p_{h(1,12)}=\frac{F_{h(1,12)u}+F_{h(1,12)d}}{h_r\cdot c_{r(1,12)}}$

The constant of each ring was obtained experimentally by calibrating the pilot silo. Only the coefficients of the rings corresponding to the maximum filling height used in the experimental tests (2.0 m) are shown (Table 3).

TABLE 3
Constants of the rings used for the experiment in the pilot silo.

Constant	Value obtained experimentally
Cr1	0.275
Cr2	0.288
Cr3	0.253
Cr4	0.280

Numerical Analysis - Finite Elements

Silo geometries

For analysis purposes, the experimentally tested silo was geometrically modeled using finite elements to create a numerical model for comparison with the experimental results. The simulated silo models were constructed with the same dimensions as the pilot silo (Fig. 5), featuring an

FIGURE 5
Height-to-diameter ratios of the silos in the developed numerical model.

internal diameter of 0.691 m and a discharge orifice of 0.20 m. Three H/D ratios were adopted for flat-bottom cylindrical silos, varying only the product height and, consequently, the slenderness. The ANSYS Student R1 2022 software package (ANSYS STUDENT, 2022) was used to develop the 2-D numerical axisymmetric model, accurately reproducing the dimensions and shape of the pilot silo.

Elements used and contact simulation

The isoparametric four-node element (PLANE42) was used to represent the grain in the two-dimensional analyses in the ANSYS program (ANSYS STUDENT, 2022). This Lagrangian-type element admits large deformations, different plasticity models, death and birth analyses, and allows for various behavioral laws of the stored product to be adopted, including parameters such as sliding or rupture criteria considering the dilatancy and high deformation capacity of the product. The degrees of freedom for this element are translations in the x and y axes only (Fig. 6). PLANE42 can be used as a plane element (plane stresses or plane strains) or as an axisymmetric element.

FIGURE 6
Simulation of product-wall contact.

To represent the interaction between the stored product and the silo wall, the "Surface-to-Surface" contact element was used in ANSYS (ANSYS STUDENT, 2022). This method is based on defining two surfaces: the "Target Surface" and the "Contact Surface." A more detailed description of the grain-wall contact simulation in silos can be found in Gallego et al. (2010).

The wall is represented by the target surface, as it represents the stiffer material, while the contact surface represents the deformable material, i.e., the granular product. The TARGET element169 was used to represent the target surface, and the CONTACT element172 was used to represent the contact surface (Fig. 6).

The TARGET169 finite element contact target surface is discretized by a set of segmented finite elements that pair with each other from the other contact surface. Contact occurs when the finite element CONTACT172 penetrates one of the target contact finite elements, TARGET169. By determining which contact target surface and which contact surface, it is possible to model a deformable and a rigid surface, respectively.

The Mohr-Coulomb model was used in this study to simulate friction. Therefore, the grain-wall friction coefficient is the only parameter necessary to simulate this interaction, as adherence between corn grains and the smooth wall of the metallic silo is negligible.

Loads and restrictions

The only load considered in the analysis was the material weight, as the effect of gravity on the stored product mass. The numerical model had the following constraints:

- In the static condition, all nodes at the bottom of the silo were fully constrained, while nodes located at x=0 had their displacements constrained only in the x-direction to create the symmetry condition (Fig. 7);

FIGURE 7
Loads and constraints considered in the numerical model.

- For discharge simulations, the constraints on the nodes at the discharge hole were removed to allow free flow of the granular product during the transient analysis. However, the constraints on the wall nodes at the bottom of the silo were maintained.

The Newton-Raphson procedure was applied to solve the nonlinear equations for the filling process, while the direct and implicit integration method of Newmark was used to solve the discharge process.

Material model

An elastoplastic material model was used to simulate the behavior of the bulk solid stored inside the silo. The ordinary isotropic-linear model was employed to represent the elastic behavior of the product in the static phase, while the perfect plasticity criterion of Drucker & Prager (1952) was used to define the behavior of the grain in the plastic (discharge) phase.

In the elastic phase, only two material parameters are required: the Poisson coefficient of the product (ν), and the modulus of elasticity (E). The stress limit beyond which the product plasticizes is represented by the plasticization criterion or creep surface.

For the plastic phase, three parameters of the stored product are necessary: the angle of internal friction (ϕi), the cohesion (c), and the dilatancy angle (ψ).

Analysis description

The results and discussion present the maximum horizontal pressures obtained experimentally, predicted numerically by FEM, and predicted by the theory of Janssen (1895). Horizontal loading and unloading pressures were obtained using the FE model with Lagrangian formulation. To do so, the same configurations tested in a pilot silo designed by Pieper & Schütz (1980), enabling comparison with experimental observations.

Theoretical pressures were determined using Janssen's model (1895) and the experimentally determined physical properties of the product. Janssen's model was chosen as the reference because it forms the basis for predicting static pressures during loading in current regulatory codes. Due to limited understanding of the unloading process, most standards define dynamic pressure by applying a coefficient to the filling pressure based on Janssen's theory and its modifications (Wang et al., 2015). These coefficients vary between standards and can be fixed values or use empirical equations, leading to different pressure results for the same silo (Lopes Neto et al., 2014).

For the loading condition, experimental maximum horizontal pressures were compared with those predicted by FEM and Janssen's model (Figs. 8, 9, and 10). For the unloading condition (Figs. 11, 12, and 13), comparisons were made between experimental results from the cylindrical silo and those from the FEM developed with the ANSYS software package.

RESULTS AND DISCUSSION

Normal silo wall pressures - static condition

Figure 8 presents the maximum static normal pressure distribution along the height of the silo with a H/D ratio of 1 and a flat bottom. The FEM numerical results for static pressures were compared with experimental results from the pilot silo and Janssen's theory (1895).

FIGURE 8
Comparison of maximum normal static pressures on the silo wall with a height-to-diameter ratio of 1.

Figure 8 shows that the pressure distribution curve obtained by FEM (0.53 kPa) is higher at the top of the silo compared to the maximum experimental pressure (0.41 kPa) and decreases by 6% near the base. At the same height, FEM recorded 2.03 kPa while the experimental sensor in ring 1 recorded 2.15 kPa, due to the embedded base consideration in the numerical model. Overall, the FEM results trend closely matches the experimental averages along the silo height for an H/D ratio of 1.

The static pressure curve from FEM (Fig. 8) exhibited a linear distribution along the height for the silo with a H/D ratio of 1, similar to observations by Ooi & Rotter (1990), Freitas & Calil Junior (2005), Sun et al. (2018, 2020), 2020) in low height-to-diameter ratio silos. In the static phase, the stored product behaved elastically and linearly, as described by Hooke's law (Pardikar et al., 2020). Gallego et al. (2010) noted that filling pressures in flat-bottomed silos with this slenderness are insensitive to the constitutive model used.

Janssen's (1895) theory predicts significantly higher normal pressures than experimental or numerical results for silos with low height/diameter ratios, as it does not account for the boundary conditions of such storage units (Freitas & Calil Júnior, 2005).

Figure 9 presents the normal pressure distribution for the silo with a H/D ratio of 2 during filling. The FEM results at 0.75 m and 1.25 m heights are close to and slightly higher than the experimental values, indicating an accurate model that ensures project safety. At 0.25 m height, the FEM value (4.3 kPa) differed by approximately 4% from the experimental value (4.13 kPa), with discrepancies increasing near the bottom due to the section being considered rigid in the numerical model, a behavior also

FIGURE 9
Comparison of maximum normal static pressures on the silo wall with a height-to-diameter ratio of 2.

noted by Hilal et al. (2022) and Hussein & Risan (2022). Thus, the numerical method reliably predicts static pressures for this silo configuration.

For the H/D = 2 ratio, the horizontal filling pressure obtained by FEM (Fig. 9) shows an exponential trend, consistent with Janssen's (1895) model (Madrona & Calil Junior, 2009; Wang et al., 2014; Gallego et al., 2015; Hilal et al., 2022; Horabik et al., 2022). Therefore, when appropriate parameters are considered, silo-filling pressures closely match Janssen's theory (Rotter, 2001).

Madrona & Calil Júnior (2009) found that the pressure curve obtained by FEM was nearly identical to the pressure curve for silos in the first reliability class of EN 1991-4:2006, similar to the present work. This is because the Poisson coefficient and wall friction coefficient used in the numerical model were based on average values of K and μ. Therefore, excluding the variability in the physical properties of the products, FEM pressures closely match those predicted by Janssen's equation, except at the base, where the silo was considered embedded in the numerical model, while Janssen's theory assumes uniform vertical pressures in the section (infinite cylinder).

Figure 10 shows the maximum static normal pressure distribution along the height for the silo with an H/D ratio of 3. Comparing experimental values with FEM model predictions, we observed that FEM-predicted normal pressures are generally very close to the experimental values during filling. The largest percentage difference was 17%, occurring at the third ring (height of 1.25 m).

FIGURE 10
Comparison of maximum normal static pressures on the silo wall with a height-to-diameter ratio of 3.

As noted by researchers (Gallego, 2006; Sielamowicz et al., 2015) and reflected in EN 1991-4:2006, eccentricity during filling can cause overpressures in silo walls. Likely, a filling eccentricity occurred due to the 6 m height of free fall, causing overpressure at the 1.25 m height (third ring), as shown in Figure 10. The slope formed at the end of the filling reached the silo wall around this height.

In practice, avoiding eccentric filling is difficult due to factors such as inlet eccentricities, opening type, product free fall height, segregation, and mechanical limitations (Cao & Zhao, 2018). Small eccentricities during filling can lead to unexpected silo behavior (Sielamowicz et al., 2015).

As with the H/D ratio of 2 (Fig. 9), for the H/D ratio of 3, the pressure distribution determined by FEM exhibits an exponential trend, similar to Janssen's theory (1895).

Normal pressures on the silo wall - dynamic condition

Figure 11 shows the maximum normal pressure distribution along the height for the silo with an H/D ratio of 1 during corn discharge. The maximum pressure values were obtained experimentally and through FEM models. In the dynamic phase, the FEM-predicted normal pressure at a height of 0.25 m (ring 1) was 17.8% lower than the experimental value, and at 0.75 m (ring 2), it was 13.6% lower than the experimental value.

FIGURE 11
Comparison of maximum normal dynamic pressures on the silo wall with a height-to-diameter ratio of 1.

The highest pressure recorded by the sensor at 0.25 m (Fig. 11) likely corresponds to the pressure peak caused by the V-shaped grain layer formed during discharge, a characteristic of funnel flow (Han et al., 2019). Appendix A provides the formulations used (BMHB, 1985) to determine the effective transition position of the flow along the silo height for all H/D ratios analyzed. Similar behavior was observed for the other H/D ratios in this study (Figs. 12 and 13).

In silos with funnel flow, pressure peaks during discharge can result from the volumetric increase of the grain mass due to product dilatancy. Wang et al. (2014) found similar behavior when simulating filling and discharge pressures in silos. While their FEM model reproduced static pressure accurately, it could not predict a 15-20% increase during discharge, raising questions about potential changes in input parameters at the onset of discharge. One plausible explanation is the transition of wall friction from static to dynamic values (Wang et al., 2014).

Figure 12 presents the maximum dynamic normal pressure distribution along the height for the silo with an H/D ratio of 2. Discontinuities and peaks in the dynamic normal pressure curve obtained with FEM (Fig. 12) are attributed to product shear zones, as reported by Wójcik & Tejchman (2009) and Gallego et al. (2015).

FIGURE 12
Comparison of maximum normal dynamic pressures on the silo wall with a height-to-diameter ratio of 2.

The internal friction of the product (ϕi) defines the rupture surface slope, while the dilatancy angle (ψ) corresponds to the angle between the sample's sliding line and the actual direction of sliding, relating to the product's volumetric dilation under shear. This dilation is believed to increase pressures during silo discharge (Madrona & Calil Junior, 2009; Ramírez et al., 2009, 2010a), 2010a).

Traditional theories, such as Janssen's (1895), used for silo sizing, are based on common product properties like internal friction (ϕi), the grain-to-wall friction coefficient (μw) and the specific weight (γ), but do not include Poisson's ratio (ʋ), the modulus of elasticity (E), and dilatancy angle (ψ), which are crucial for FEM parameterization (Lopes Neto et al., 2016; Moya et al., 2013, 2022), 2022).

The robustness of FEM results depends on the mechanical properties of the material in the constitutive model used to describe the grain's nonlinear behavior (Jayachandran et al., 2019). However, these values are rarely found in the literature (Moya et al., 2013; Lopes Neto et al., 2016; Pardikar et al., 2020), leading researchers to use values from other studies (Vidal et al., 2005, 20, 2008; Gallego et al., 2015; Wang et al., 2016; Wójcik & Tejchman, 2016; Gao et al., 2018; Jayachandran et al., 2019; Pardikar et al., 2020), particularly the work of Ramírez et al. (2010b), Moya et al. (2002, 2006, , 2006, 2013, 2022), and Lopes Neto et al. (2016).

Further research is needed to understand the influence of adopting mechanical parameters from other studies on FEM predictions. Future developments should consider how different methods or assumed values of other properties impact FEM pressure predictions (Lopes Neto et al., 2016).

Comparing the pressure values, the normal pressure calculated with FEM was 25%, 10%, and 19% lower than the experimental pressure for heights corresponding to rings 1, 2, and 3, respectively (Fig. 12).

The experimental pressure recorded at 0.25 m height is considerably higher (Fig. 12), as seen with a H/D ratio of 1, due to stagnant zones in the lower corners of flat-bottom silos during discharge (Zhang et al., 2022). This overpressure results from contact with the stagnant product zone.

Zhang et al. (2022) used discrete element method simulations to show that flat silos with funnel flow exhibit pressure fluctuations at the highest position, where the converging flow zone meets the stagnant zone, characterized by significant oscillations of contact force. Their findings suggest no intrinsic difference between oscillatory fluctuations during funnel flow and mixed flow discharges.

Figure 13 shows the maximum normal pressure distribution for the silo with an H/D ratio of 3 during discharge. Similar to the H/D ratio of 2, discontinuities and peaks were observed (Fig. 13). The greatest discrepancy between experimental and numerical data occurred at 1.25 m (ring 3), with a 48% difference. The mixed (plug-flow) discharge caused a pressure peak near ring 3, where the flow touched the silo wall (Han et al., 2019).

FIGURE 13
Comparison of maximum normal dynamic pressures on the silo wall with a height-to-diameter ratio of 3.

The pressure peak between mass flow and funnel flow was not detected by FEM (Fig. 13) because the finite element formulation does not predict the inversion of principal stresses in transition zones. The discrete element method is more suitable for these observations. Sanad et al. (2001) found that while FEM reasonably predicts internal stresses and pressures, it struggles with modeling silo discharge, particularly localized phenomena like shear zone formation and product flow through the orifice. Conversely, DEM satisfactorily predicts various dynamic phenomena in silos, such as flow patterns, arc formation, and shear zones.

Although the dynamic normal pressures obtained by FEM were lower than the experimental values at all analyzed points (Fig. 13), the curve followed the same trend as the experimental data. In the lower part of the silo, the experimental maximum dynamic pressures in rings 1 and 2 were 5.58 and 4.27 kPa, respectively, while FEM values were 5.27 and 4.25 kPa. In the upper part, FEM recorded 1.47 kPa compared to the experimental 2.12 kPa at 1.75 m. The peak pressure at ring 3 (1.5 m) was 4.78 kPa experimentally, and 2.92 kPa by FEM.

Han et al. (2019) highlighted the lack of quantitative and systematic investigations on flow pattern transitions in slender silos. For the three H/D ratios analyzed, FEM showed dynamic pressure curves with a peak near the bottom (Figs. 11, 12, and 13) due to the discontinuity in the two-dimensional axisymmetric model at this location, causing difficulty in accurately determining the pressure (Madrona & Calil Júnior, 2009).

Pressure fluctuations recorded both experimentally and by FEM in the dynamic condition for slender silos (H/D ratios of 2 and 3) can be attributed to the change in the orientation of the principal stress from vertical to convergent. This can cause tremor and vibration issues due to intermittent macro-slippage of the granular product against the silo wall (stick-slip) (Wang et al., 2014). These macro-slippings and wave propagation are associated with significant pressure fluctuations in the wall (Wang et al., 2014).

The stick-slip mechanism involves shear zone formation near the silo wall, with intense shear in this region causing significant stress fluctuations.

CONCLUSIONS

An experimental setup was used to determine the maximum lateral pressures exerted by stored material in a silo with three height-to-diameter (H/D) ratios (1, 2, and 3). A finite element model (FEM) was developed using the ANSYS software to simulate experimental conditions.

For the filling process, the normal pressures predicted by the FEM closely matched the experimental data for all three H/D ratios. This approach can be applied to model the overall process of filling flat-bottom silos.

The dynamic pressure curves from the numerical model showed a similar trend to the experimental data for all H/D ratios but with lower magnitudes. The numerical values correlated qualitatively well with the experimental values but showed weaknesses in quantitative comparison. This discrepancy is due to stagnant zones in the lower corners of flat-bottom silos, which cause experimental overpressure during discharge that FEM did not detect. The finite element formulation does not predict the inversion of principal stresses in these zones, and the discrete element method is more suitable for such observations.

The FEM model accurately predicted the static condition pressures but underestimated the experimental pressures in the dynamic condition along the depth. Future investigations should refine numerical models by considering the variability of mechanical parameters in granular solid material.

https://minio.scielo.br/documentstore/1809-4430/pJKHMyqzvLhLd8tbcGjkGjr/5573803c1109a836c6103f04f286dcff5c5fb653.pdf

ACKNOWLEDGMENTS

The authors thank CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) for funding the scholarship associated with this project. They also appreciate the partnership with the Federal University of Lavras and the support of the ICAA (Institute of Agricultural and Environmental Sciences - Federal University of Mato Grosso, Sinop Campus).

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Edited by

• Area Editor: Gizele Ingrid Gadotti

Publication Dates

• Publication in this collection
  20 Sept 2024
• Date of issue
  Sept 2024

History

• Received
  27 Jan 2024
• Accepted
  10 June 2024