# Three-Dimensional Finite Element Analysis for Pressure on Flexible Wall Silos
## 1 Introduction
### Literature Review  
Over the last few decades, significant progress has been made in the computational approaches to granular solids and the design of silos around the world. Many researchers [5; 6; 7; 8; 9] have worked on silo design and implemented various principles and suggestions, but only a few have investigated rectangular silos, and even fewer have investigated flexible walled silos compared to the most widely used circular silo due to their thin walls and simplified analysis methods [10]. Rectangular silos have the advantage of the ease of manufacture and rapid construction, justifying their use when manufacturing processes are simplified or construction space is limited [11]. Moreover, more advanced analysis methods for rectangular silos still need to be established, taking into account the stiffness of walls and hoppers, as well as their interactions with the bulk solid.  
Figure 1: The main components of the rectangular planform silo.  
Current design approaches estimate the horizontal and vertical pressures acting on the wall and hopper, then use the equilibrium equations to calculate the bending moment, shear, and normal forces acting on each structural member. The final results of equilibrium depend on the postulated boundary conditions. A rigid wall boundary condition gives fundamentally different values compared to a very flexible wall. In the latter case, the walls can bend away from the solids, moving the loads from the flexible sections to the rigid sections, and changing the load distribution across the wall.  
Because of the huge number of interconnected elements, including the main supporting system, bulk solid/structure interaction, wall deformability, the failure mechanism, and wall imperfection, the silo response becomes more sophisticated to predict. To estimate vertical and horizontal pressures in circular silos, many theories are commonly used.  
The majority of classical theories developed by different authors are based on the same model of equilibrium of a horizontal slice of the silo, not distinguishing between them by defining certain parameters such as the angle of internal friction in the stored material, the angle of wall friction, and the ratio of horizontal and vertical pressures [12]. Historically, the wall pressures were considered to be similar to hydrostatic pressures [13]. According to [PERSON] and [PERSON] [14], this assumption was not valid, since the wall carries part of the weight of the bulk solid due to the friction between the bulk materials and the walls. [PERSON] [15] confirmed this phenomenon and developed his widely used method for calculating pressure along a wall's height. Most codes and standards [16] have adopted the [PERSON] method, which expressed the wall pressure by Equation (1).  
\[P_{x}=\frac{A}{U}\cdot\frac{\omega}{\mu}\,\left(1-\mathrm{e}^{-\frac{\omega}{ \lambda}\cdot\lambda\cdot\mu\cdot Z}\right) \tag{1}\]  
where \(P_{x}\) is the horizontal pressure to the wall; \(A\) and \(U\) are the area and perimeter of the horizontal section, respectively; \(\omega\) is the specific weight of the material; \(\mu\) is the wall friction coefficient; \(\lambda\) is the lateral pressure ratio; and \(Z\) is the depth from the free surface.  
[PERSON] [17] subsequently proposed another theory to calculate the pressure acting on the circular silo wall. The theory provides the pressure of the grains on the surface of the silo wall; it involves the angle of internal friction of the stored material and the coefficient of wall friction. Recently, [PERSON] and [PERSON] [18] created other standard theories to estimate horizontal and vertical pressures based on actual data and soil mechanics theories. According to these approaches, the silo wall is rigid, and the pressure in the silo is constant at any level. However, computer models developed in the late 1980s revealed that wall flexibility influences wall pressure distribution [19].