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This work consists in the analytical derivation and numerical solution of the equation which determines the shape of the section of a silo bag: a long horizontal cylindrical plastic bag filled with either a liquid or a granular material which behaves similarly, exerting normal, but not frictional forces against the wall. The bag is considered inextensible and completely flexible, capable of supporting only tensile loads. These suppositions lead to a secondorder differential equation for the membrane shape, which is normalised and solved, in a way that allows, for any bag with any amount of filling, a simple computation of its geometry, enclosed area and tensile loads. A discussion is included about the effects on the theoretical results of the silo bag tensile deformation. Introduction Silo bags are cylindrical plastic bags, hermetically sealed, extensively used for grain storage (Figure 1). They are made of low-density extruded polyethylene, with additives to increase its resistance to UV radiation (INTA, 2009). A bag consists of three layers: an outer white one, which reflects the sunlight, especially ultra-violet radiation, a middle one, and an interior black layer, which minimises the penetration of solar radiation. Only in Argentina, about 45 million tons per year are stockpiled in silo bags (Rozadilla et al., 2018). They present advantages as a low cost per ton of stored grain, variable storage capacity, no need of freight during harvest, and good quality of storage, retaining grain moisture, grade and colour (Bartosik et al., 2013). The motivation for this work is to determine the shape adopted by a silo bag when filled with grain and the forces exerted by the grain on the bag. Although granular materials do not strictly follow the law of hydrostatic pressure P = Po + ρgh due to friction effects, being their normal stresses usually modelled by Janssen’s equation (Janssen, 1895; Schulze, 2019), the hydrostatic pressure model gives a good approximation to their asymptotic behaviour for small storage heights. The pressure distribution of a variety of grains commonly stored in silo bags, as soybean, wheat, corn, rice, barley or sunflower seed, is well modelled by the hydrostatic law at these heights to diameter ratios. Plastic cylindrical bags could also be employed for storage of liquids or could be filled with either water or sand as inflatable dams, for which the present hypothesis is also valid. Although there is increasing bibliography on the physical processes taking part inside the silo bag (Bispo dos Santos et al., 2007; Gaston et al., 2008; Bratsk, 2012), most studies consider an approximate shape, without any geometrical analysis. The closest approaches to derive the true shape of closed horizontal membranes when filled came from the geometries of inflatable dams, as the theoretical analysis of Watson (1985), or the numerical solutions of Abdullah et al. (2005), and others. Some theoretical investigation of analogue problems can also be found in Vassilev et al. (2008). This work is an attempt to describe the shape, tensile forces, and storage volume per unit length of a flexible horizontal cylindrical membrane filled with a material, which exerts inner hydrostatic pressure. Unlike the mathematical transformations used in Watson (1985) and Vassilev et al. (2008), this analysis will be restrained to Cartesian coordinates, and efforts will be made in order to keep the physical meaning of each parameter clear. Theoretical analysis The hypotheses for this analysis are: i) the bag is completely flexible and inextensible and can only support tensile loads. Its weight is neglected; ii) the bag’s longitudinal dimension is considerably larger than its diameter, so the analysis will be two-dimensional for a typical section far from the ends; iii) the filling substance is homogeneous, of constant density (bulk density for grains) and without static friction, so that the hydrostatic pressure gradient holds; iv) the bag shape is symmetrical on a vertical plane, so only one half will be studied. The bag thickness is around 235 μm. Typical silo bags diameters range from 6 to 12 ft (approx. 1.83 to 3.66 m), and lengths, from 60 to 100 m (INTA, 2009). Low density polyethylene average mechanical properties in sheet extrusions, reported in the materials database Matweb (MATWEB, 2019), are: density 0.922 kg/dm3, yield tensile strength 11.2 MPa, elongation at break 533% Correspondence: Ana Scarabino, Computational Fluid Dynamics Group, School of Engineering, National University of La Plata, Argentina. E-mail: scarabino@ing.unlp.edu.ar

Geometric shape and tensile forces on silo bags for grain storage