# Dynamic Response of Non-Yielding Wall Supporting Over-Consolidated Sand
## 3 Methodology
### Numerical Modeling  
Before constructing the prototype-scale numerical model, a FLAC model was first developed with the dimensions of the reduced-scale physical model. The depth, length, and wall thickness of the model were the same as the physical model. The width of the box was not modeled since plane strain conditions were assumed, and a 2D analysis was carried out. The back side of the box was modeled as a rigid boundary similar to the rigid  
Figure 1: Configuration and dimensions of prototype non-yielding wall. Figure 2: Set-up of an experimental model in the laboratory. (**a**) Photo, (**b**) Schematic diagram [6]. back of the physical model. The boundary condition at the toe of the wall was modeled using one-dimensional two-node beam elements with 3 hinges, as shown in Figure 3. This boundary condition facilitates the output of vertical and horizontal forces developed at the base. The boundaries around the soil, namely, the front non-yielding wall, the rigid foundation, and the rigid back wall were defined using 4-noded linear elastic zones. The facing panel was modeled as a linear and elastic material with a unit weight of 17.24 kN/m\({}^{3}\). Panel shear (G\({}_{\mathrm{w}}\)) and bulk (k\({}_{\mathrm{w}}\)) moduli of 1000 MPa and 1100 MPa, respectively, were adopted and the failure stress of the panel was set to be unlimited. These properties were used to ensure the high stiffness of the wall. The interface friction angle (\(\delta\)) between the uniform wall was back-calculated from the experimental results. An average interface friction angle of 0 was obtained; however, a value of 2\({}^{\circ}\) was adopted for the FLAC model to maintain numerical stability. A rigid interface was defined at the base to prevent the slipping of soil particles along the foundation. This was performed to simulate the effect of gluing a thin sand layer to the base of the container. A mesh convergence study was carried out to determine the optimum number of mesh elements for the model. Figure 4 shows the variation in vertical and horizontal force on the wall with a changing number of mesh elements, indicating that a minimum of 1288 soil elements is adequate. The soil was modeled as an elastoplastic material following the Mohr-Coulomb constitutive model. The cohesion of the soil was assumed 0, while the friction angle was taken as the \"true\" peak plane strain friction angle. The peak plane strain friction angle of the sand in the experimental study was back-calculated by [PERSON] et al. [39] using experimental and numerical direct shear tests. The friction angle in the numerical model  
Figure 4: Mesh convergence study outcome. Figure 3: FLAC numerical model used for validation by [PERSON]’s experiments. was taken as 58\({}^{\circ}\), according to their findings. This friction angle was found to be consistent with [PERSON]'s [40] equation for predicting the plane strain friction angle of soil. The shear modulus (G) and bulk modulus (K) of the sand was also back-calculated by [PERSON] et al. [39] using the same procedure and found to be equal to 7 MPa and 6 MPa, respectively. Material properties of the numerical model are summarized in Table 1.  
The numerical model was built following the compaction sequence of the experimental model, i.e., compaction was carried out for each soil layer for 8 layers. Compacting soil layers using vibration in the numerical model proved to be time-consuming; thus, a different approach was utilized. After the placement of each soil layer, the soil was compacted by applying a horizontal stress equivalent to the stress calculated using K\({}_{\text{o}}\), as defined by [PERSON]'s equation [41]:  
\[K_{o}=\big{(}1-sin\phi^{\prime}\big{)}OCR^{sin\phi^{\prime}} \tag{1}\]  
where '\(\phi\)' is the soil friction angle and OCR is the soil over-consolidation ratio. The soil is then taken to equilibrium prior to the placement of the next soil layer. This method of compaction was successfully employed by [PERSON] and [PERSON] [5].