# Three-Dimensional Finite Element Analysis for Pressure on Flexible Wall Silos
## 2 Numerical Approach
### Numerical Result and Validation
#### 2.5.2 Comparison with Janssen Equation  
The Janssen equation is the oldest and most widely used approach in silo design. It was regarded as the first equation in driving the differential equation for the equilibrium of a solid layer in a silo. This equation was derived for an axis-symmetrical layer of the circular silo and assumes a constant lateral pressure ratio.  
In fact, most current design codes employ the Janssen method to calculate the pressure acting on silo walls, which has been generalized to cover non-circular silos by considering square silos as equivalent circular silos with the same hydraulic radius [28]. Alternatively, a silo of the same volume may be used to evaluate maximum pressure [12]. It was assumed that the wall pressure at any level on a horizontal section is uniform. In contrast, in the behavior of a flexible wall, the middle section has the lowest value while the corners experience the most lateral pressure, resulting in an arching mechanism inside the bulk materials across the diagonals of the silo [16].  
A comparison was made between the finite element findings and [PERSON]'s equation in the flexible wall silo for the same studied silo (flexible wall with dimensions 1.5 \(\times\) 1.5 \(\times\) 2.5 m) and filled with Leighton Buzzard sand, as illustrated in Figure 9.  
As shown in Figure 9c, this discrepancy disappears when reaching section 3-3, where the values of the finite element became very close to [PERSON]'s. The reason for the difference in values between finite elements and [PERSON]'s is the effect of wall flexibility where the rigid zone of walls (corner) attracts more load and hence produces higher pressure, while the flexible part of the wall (middle) releases its pressure and deflects outward. The full redistribution of load and pressure in the horizontal direction is completely ignored in the [PERSON] approach, in which the values of [PERSON] predictions remain constant over the three sections with no wall deformation.  
There is another divergence between [PERSON]'s predictions and finite element calculations towards the bottom of the three sections (1-1, 2-2, and 3-3). This is due to the influence of the base's connection to the wall and the respective boundary condition. The [PERSON] equation ignores this effect and does not integrate boundary conditions [29].  
The finite element distribution of wall pressure in section 1-1 has a similar trend as the [PERSON] pressure distribution with smaller values. A difference of 2 kPa can be noticed between the two curves along the full height. Figure 9b confirms the same distribution in section 2-2.  
To prove that [PERSON]'s prediction is close to rigid wall silo outcomes, another rigid wall model was developed for the same silo dimensions (1.5 \(\times\) 1.5 \(\times\) 2.5 m) with the same filling granular material (sand), where the walls were prevented from lateral displacements using the proper boundary conditions to fulfill that target.  
As illustrated in Figure 10, the finite element results are compared to [PERSON]'s predictions at the mid-span of the wall (section 1-1). In the case of rigid walls, finite element results show good agreement with [PERSON]'s calculations. In the lower part, the FE trend exceeds [PERSON]'s prediction due to the influence of the junction between the bulk and the wall. Many researchers have formerly reported such end effects [16; 29].  
Figure 9: Comparison between lateral filling pressure of the finite element and [PERSON] predictions along the silo height in case of the flexible walled silo. (**a**) section 1-1: at mid-side of the wall; (**b**) section 2-2: at 0.35 m from silo center; (**c**) section 3-3: at 0.55 m from silo center.  
From the above comparisons, two main conclusions can be drawn:  
1. Finite element results are in great agreement with [PERSON]'s experimental data for flexible silo walls.
2. Finite element results closely match [PERSON]'s predictions for rigid silo walls, where the [PERSON] method is applicable.