INFLUENCE OF SPECIFIC WEIGHT AND WALL FRICTION COEFFICIENT ON NORMAL PRESSURES IN SILOS USING THE FINITE ELEMENT METHOD

The objective of this work was to develop models using the Finite Element Method (FEM) to assess the maximum normal pressures in the static condition in silos using different wall friction coefficient and specific weight of the stored product compared to the pressures obtained by the Eurocode 1, part 4. The geometries of the silos models were developed based on the dimensions of the experimental station at the Universidad de Leon (Spain). The material properties were obtained by Jenike shear cell tests and were used to generate the models by the MEF. 3D models were generated varying the friction coefficient (0.2, 0.4, and 0.6) and the specific weight (6; 7.5 and 9 KN m-3). It was verified that the models by FEM follow the theory of pressures in silos: normal pressures increase due to the increase in specific weight and decrease due to the increase in the friction coefficient. Moreover, the maximum normal pressure occurs at the hopper silo transition. The experimental pressures (FEM models) compared with Eurocode 1, part 4 allowed to validate the models developed, presenting trends of similar values to those found by the MEF. The experimental models demonstrated that the influence of the wall friction coefficient and specific weight significantly interferes with the pressures in slender silos.


INTRODUCTION
In 2020, the estimated grain production in Brazil was 268.7 million tons (CONAB, 2020). In less than 39 years (2019), Brazil has more than quadrupled its static capacity (177.7 million tons of grain). Out of this total, 86.6 million tons of grain (49%) were stored in silos (DPE, 2019).
Since 1895, Janssen (JANSSEN, 1895) has been studying the flow and pressure of products stored in silos. Ever since then, other theories have been developed (WALKER, 1967) (WALTERS, 1973a, 1973b) (JENIKE et al., 1973 supporting international standards (CEN, 2006;DIN, 2005). It is necessary to understand the different theories that support the standards to comprehend the actions (flow and pressures) in silos. However, many factors are still not conclusive due to the randomness of the physical properties of the stored products (differently from liquids) (CALIL; CHEUNG, 2007).
The specific weight of the main agricultural products in Brazil (corn, soybeans, rice, wheat, and beans) (CONAB, 2020) and products derived from the feed and flour industry vary widely. For example, flour can have bulk unit weight from 6.5 to 7 KN m -3 , while corn and soybean from 7 to 8 KN m -3 and wheat from 7.5 to 9 KN m -3 (CEN, 2006). The wall friction coefficient and the specific mass of the main agricultural products grown in Brazil varies considerably and is directly influenced by the roughness of the wall of the silo. The wall friction coefficient of this stored products are: flour (0.24 to 0.48); corn (0.22 to 0.53); soybean 0.24 to 0.48 and wheat 0.24 to 0.57 (CEN, 2006).
Due to the advancement of engineering combined with computational development, modeling is supported by a tripod: theoretical model, experimental model, and numerical model. These three models (working together) optimize and advance studies in most areas of engineering. For example, the finite element method (FEM) is a numerical procedure to determine approximate solutions of value problems on the contour of differential equations. The FEM subdivides the problem domain into smaller parts, denominated finite elements. Therefore, in recent years, studies using the FEM in silos have significantly increased (JOFRIET, 1992;HOLST et al., 1999;GALLEGO et al., 2010;AGUADO, 2015;PARDIKAR et al., 2020;WASSGREN, 2020;ZHAO, 2004), mainly for the quality of the answers and for the relative time and cost-saving in comparison to the experimental models in real scale.
Therefore, the objective of this paper is to simulate through the Finite Element Method, the interference of the specific mass and the wall friction coefficient using representative values of the leading Brazilian agricultural products to obtain the normal static pressures in slender silos compared to the Eurocode 1, part 4 (CEN, 2006).

MATERIAL AND METHODS
The numerical simulation was performed using elements and the software Ansys 2020R2 student version. The model was developed by simulating the static condition (after filling the silo). The dimensions of the silo, hopper inclination were based on the silo of the experimental station designed to evaluate the pressure in silos at the University of León (Spain) designed and built by the research group of the Department of Agricultural Engineering and Sciences (COUTO; RUIZ; AGUADO, 2012; RUIZ; COUTO; AGUADO, 2012).
The silo was 2.00 meters high and had one meter in diameter with a concentric hopper angle of 34.3 and 0.48 meters in height. The silo is made of polished steel metal. According to Eurocode 1, part 4 (CEN, 2006), this silo is classified as slender (height/diameter ratio = 2) and can be seen in Figure 1. For the numerical simulation, three different wall friction coefficients and the product (u) were used: 0.2; 0.4, and 0.6 (values that include the limits of wall friction coefficient of most of the leading national agricultural products (corn, soybeans, rice, wheat, and beans) (CEN, 2006) and also products characterized as feed and flour. In addition, as variables, the specific weight of the stored product (γ) (6; 7.5 and 9 KN m -3 ) was also analyzed. Those are values that include the upper and lower specific weight of most agricultural products, such as corn, wheat, soybeans, rice, beans, and flour products from the animal feed industry.
The parameters of the stored product and the steel sheet of the silo were obtained in the literature ( Table 1).
The proposed model was performed using elastoplastic material (DRUCKER; PRAGER, 1952) to simulate the pressure of the stored product (solid) inside the silo. The standard isotropic and linear model was used to represent the elastic behavior of the silo, while Drucker and Prager's perfect plasticity criterion (DRUCKER; PRAGER, 1952) was used to define the plastic part.
The model was created in 3D geometry, aiming at further studies to understand the volume of the GANDIA, R. M. et al. Thickness (m) 0.02 -model. The modeling was done initially using a small number of elements (mesh simplification). After the model was defined and generated, the mesh was refined, mainly in the areas of most significant interest (silo-hopper transition) ( Figure  2). The size of the elements was 0.15 m for the solid and 0.075 m for the silo, totaling 14688 elements.
The generated mesh was composed of prismatic elements with 8 and 4 knots, as shown in Figure  2. This type of mesh is stable and, when possible, allows analysis with better results. For the stored product (solid), an element with eight nodes (Solid 45, Lagrangian type) was used, which is an element that supports large deformations (GALLEGO et al., 2015). The silo (shell) wall was modeled using the 8-node shell element (Shell 63), ideal for analyzing non-linear applications. For the contact (friction) between the stored product (solid) and the silo wall (shell), 4-node elements were used (Conta 173 and Target 170) (GALLEGO et al., 2010). The model was vertically (z-axis) constrained in the contour of the silo-hopper transition. The gravitational force was inserted towards the z-axis. The models were subsequently compared with Eurocode 1, part 4 (CEN, 2006). The normal static pressure curves were calculated using the same parameters proposed in the development of the model using FEM: silo dimensions (height: 2 meters; internal diameter: 1 meter; β: 34.3 °); work variables (wall friction coefficient: 0.2; 0.4 and 0.6); specific weight: 6; 7.5 and 9 KN m -3 ); other values extracted from Table E.1 -Properties of Disaggregated Solids used for the product (CEN, 2006).

RESULTS AND DISCUSSION
Information on Z displacement (m), Z stress ( KN m -2 ), and main stress ( KN m -2 ) of the mass of the product inside the silo was extracted according to the variation of the wall friction coefficient ( Figure 3) and a specific weight (Figure 4).
The main tension increased as the wall friction coefficient (Figure 3a) decreased. The maximum values (SMN) are negative because they are in the opposite direction to the Z-axis, which corroborates the statements (CEN, 2006;JENIKE et al., 1973). It is observed that the lowest stresses (red) occur at the top of the silo and the highest (blue) near the silo-hopper transition.
The maximum stresses in Z showed the opposite behavior, increasing according to the increase of the wall friction coefficient, mainly in the silo-hopper transition. The reason is that with the increase in Figure 2. Geometry and mesh of the developed model the wall friction coefficient, the friction pressure increases, resulting in a reduction in the normal pressure in the silo's wall and the occurrence of a slight increase in the vertical tension in the silohopper transition.
A decrease was found in the displacement of the grain mass (Figure 3c) caused by the vertical displacement (z) due to the increase in the wall friction coefficient, which was visually observed through the smaller area of the ellipse (blue) approximately in the middle of the silo and numerically confirmed by the displacement value (SMN) plotted in each simulation: 1.118; 0.940 and 0.838 mm respectively for the wall friction coefficient 0.2; 0.4 and 0.6.
The main tension increased according to the increase in the specific weight of the stored product (CEN, 2006;JENIKE et al., 1973) (Figure 4a). The lowest stresses (red) occur at the top of the silo and the highest (blue) near the silo-hopper transition. However, for the specific weight of 9 KN m -3 , the minimum main stress occurs at the outlet gate because the increase in the specific weight of the material and the pressure of the material column above the hopper provides a mechanical arc above the outlet opening.
The maximum stresses in Z have the same dynamic, increasing according to the increase in the specific weight of the stored product, mainly in the silo-hopper transition. Thus, the maximum and minimum principal stresses occur at the hopper silo transition and the top of the silo. Except for the specific weight of 9 KN m -3 , where the minimum occurs at the outlet gate due to the formation of the mechanical arc provided by the increase in weight of the product, the hopper's inclination, and the wall friction coefficient of the hopper. The displacement in the z (vertical) direction increased according to the specific weight of the stored material. The greater the weight of the product, the greater the displacement of the grain mass, as compression occurs according to the upper layers of the product, forcing the more significant displacement in the lower positions of the silo.
For a better comparison among the developed simulations, the normal static pressure curves were measured throughout the silo. The curves comparing the three models varying the wall friction coefficients ( Figure 5) and the specific weight of the stored product ( Figure 6) are shown below.
The natural behavior among the three curves varying the wall friction coefficient (u) is expected in view of the theories and experiments (CEN, 2006;INTERNACIONAL ORGANIZATION FOR STANDARDIZATION, 2012;JENIKE et al., 1973;WALKER, 1967). The reason is that normal pressures have maximum values close to the hopper transition. Also, as expected in the face of theories (JENIKE; JOHANSON; CARSON, 1973;WALKER, 1967) and standards (CEN, 2006;INTERNACIONAL ORGANIZATION FOR STANDARDIZATION, 2012), the higher the wall friction coefficient, the lower the normal pressures will be due to the increased friction pressure. Therefore, the model with a wall friction coefficient (u) 0.2 was the one with the highest normal pressures and 0.6 showed the lowest normal pressure values. The standard behavior among the three curves varying the specific weight of the stored product (γ) was expected given the theories and codes (CEN, 2006;INTERNACIONAL ORGANIZATION FOR STANDARDIZATION, 2012;JENIKE et al., 1973;WALKER, 1967). Thus, normal pressures have maximum values close to the hopper transition. Also, as expected given theories and standards, the greater the specific weight of the stored product, the greater the normal pressures on the wall and hopper. Therefore, it is observed that the model with specific weight (γ) 9 KN m -3 was the one with the highest pressure values and 6 KN m -3 showed the lowest values.
Aiming to validate the models and analyze Eurocode 1, part 4 (CEN, 2006), all the developed models were compared to the standard. The curves comparing the three models varying the wall friction coefficients and the pressures obtained by Eurocode 1, part 4 are shown below (Figure 7). The curves comparing the three models varying the specific weight of the stored product and the pressures obtained by Eurocode 1, part 4 are shown below (Figure 8).
The analysis of errors relating to the pressures obtained through the simulation using FEM and those calculated with Eurocode 1, part 4 was shown in Table 2. In comparison with Eurocode 1, part 4, it is possible to observe that the initial normal static pressures obtained by the FEM at the top of the silo (height 0 to 0.75 m) were higher than the standard ones. Furthermore, in relation to the increase in the wall friction coefficient and consequently a decrease in normal pressures, it is observed that the greater tendency of the pressures calculated using FEM to exceed the pressures obtained from the standard, also in the case of the wall friction coefficient u = 0.6.
Although the pressures in the static phase were generally lower than the discharge, it is possible to assume that the standard should increase the pressures when the wall friction coefficient is greater than 0.5. In addition, concerning the decrease in the specific weight of the stored product and consequently the decrease in normal pressures, it is observed a greater tendency of the pressures calculated through the MEF to exceed the pressures obtained from Eurocode 1, part 4, even exceeding in the case of the specific weight of the product (γ) 6.0 KN m -3 . Therefore, despite being pressures in the static phase, generally lower than the discharge, it is possible to assume that the standard should increase the pressures for products with lower specific weights. It can be seen that Eurocode safely increases the normal static pressures for silo projects in extreme conditions (products with higher specific weights and lower wall friction coefficient), however, under conditions of low normal pressures the trend of pressures by FEM exceeds those of Eurocode 1, part 4.
According to Gallego (GALLEGO et al., 2015) in evaluating wheat efforts experimentally, numerically, and by comparing with Eurocode 1, part 4 (CEN, 2006), pressures with magnitudes close to that of this work are found for similar properties of the stored material. Ruiz and collaborators (COUTO et al., 2012), also using wheat, found that the maximum vertical pressure obtained experimentally in loading is close to the model proposed in the present work. The experimental study using corn (COUTO et al., 2013) also showed a tendency for pressure values in the silo during loading close to those presented in this work.
The variation of the wall friction coefficient considerably influences the study and silo projects, changing the coefficient by only 0.4; an increase is found in the maximum normal static pressure of 58%. The same importance can be considered for the specific weight of the stored product, with an increase of 62% due to the increase in weight by 3 KN m -3 . The specific weight of the stored product has an increase of 62% due to the increase in weight by 3 KN m -3 .
For all the generated models, the location was GANDIA, R. M. et al. exactly 4.8 cm below the transition (both for the variation of the wall friction coefficient and the specific weight of the stored product).

CONCLUSIONS
• The use of FEM for the calculation of pressures in silos is recommended. The two variables (specific weight and wall friction coefficient) influence the static pressures in the silo, which were observed in the pressures obtained by Eurocode and also in the results of the simulated models.
• Products with similar particular characteristics are considered. In this experiment, the highest normal static pressures will occur in the products with the highest specific weight and the lowest wall friction coefficient.
• Compared with Eurocode 1, part 4 (CEN, 2006), it could be seen that all models generated by the MEF show consistent errors, presenting normal static pressures below the standard. However, two issues were observed: although the normal pressures at the top of the silo are the lowest, the model created by the MEF presented a divergence from the standard; the second issue is related to the higher coefficient used in the standards for products that tend to have higher pressures and lower coefficients for products that tend to generate lower pressures, in other words, for products stored with lower specific weight and higher wall friction coefficient, the standard presents lower safety coefficients that could be observed in the comparisons with MEF.
• The variation of the wall friction coefficient and specific weight considerably influences the study and silo projects.
• The location of the maximum normal static pressure in the silo is located in the silo-hopper transition zone.