# Dynamic Response of Non-Yielding Wall Supporting Over-Consolidated Sand
## 3 Methodology
### Analysis of Prototype Model  
After validation, the numerical model was scaled up to the prototype dimensions, as shown in Figure 1, following Iai's scale factors for 1 g models [36]. The same modeling procedure used in the validation study was utilized to create the prototype model. The geometry of the control FLAC numerical model is shown in Figure 6, while the adopted control material properties are presented in Table 2, together with Iai's scale factors \(\lambda=4\). A parametric study was conducted on this model by subjecting it to the 14 s of real seismic base excitation shown in Figure 7. This real ground acceleration record was chosen due to its predominant high-frequency content, which can be critical for the short-period structure considered in this study. The selected ground motion record, Figure 7, mainly includes high-frequency content, which aligns with the expected natural frequencies of stiff, short-period structures, such as the non-yielding wall modeled in this study. The validation of the numerical model developed based on the experimental model contains the relatively shallow and dense backfill soil with an OCR as high as 4 (Section 3.2), resulting in higher shear wave velocities and, thus, a higher natural frequency of the soil deposit. It is expected that the coupled system (wall + backfill) will have a natural frequency within a short-period range, making the chosen seismic input suitable for capturing the dynamic behavior of both the wall and the surrounding soil. The values of the investigated parameters that include the backfill friction angle, \(\phi\), over-consolidation ratio, OCR, and wall elastic modulus, E\({}_{\text{w}}\), are summarized in Table 3. The seismic performance of each run is examined by considering the output parameters shown in Figure 8: (a) the incremental lateral deformation of the wall at mid-height (\(\Delta\)x), (b) the seismic pressure distribution (c), the total lateral earth force (R\({}_{\text{H}}\)), and (d) the elevation of the resultant earth force (y\({}_{\text{R}}\)). Note that the incremental deflection refers to the difference between the deflection due to seismic action only, after subtracting the static deflection. The displacement of the base due to excitation is subtracted from the total displacement at the wall's mid-height to obtain the deflection of the wall with respect to its bottom support. The elevation of the resultant lateral earth force was calculated using the recorded forces at the top and bottom of the wall (R\({}_{\text{HT}}\) and R\({}_{\text{HB}}\), respectively). Both peak and residual values of these outputs are considered, and the base excitation is applied after the wall has been subjected to the compaction effort and its own weight. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Material Property** & **Iai Scale Factor (\(\lambda\) = 4)** & **Value** \\ \hline Peak friction angle, \(\phi_{\text{P}}\) (\({}^{\circ}\)) & 1 & 58 \\ Residual angle, \(\phi_{\text{Res}}\) (\({}^{\circ}\)) & 1 & 46 \\ Cohesion, \(c\) (kPa) & 4 & 0 \\ Dilation angle, \(\psi\) (\({}^{\circ}\)) & 1 & 14.5 \\ Soil unit weight, \(\gamma\) (kN/m\({}^{3}\)) & 1 & 15.7 \\ Soil over-consolidation ratio, OCR & 1 & 4 \\ Soil shear modulus, G (MPa) & 4 & 28 \\ Soil bulk modulus, K (MPa) & 4 & 24 \\ Wall unit weight, \(\gamma_{\text{w}}\) (kN/m\({}^{3}\)) & 4 & 17.24 \\ Wall elastic modulus, E\({}_{\text{w}}\) (MPa) & 4 & 2100 \\ Wall shear modulus, G\({}_{\text{w}}\) (MPa) & 4 & 4400 \\ Wall bulk modulus, K\({}_{\text{w}}\) (MPa) & 4 & 4000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material properties of the prototype-scale model. Figure 6: FLAC numerical model of the prototype-scale wall. Figure 7: Seismic base excitation used in the study. The initial conditions of the dynamic analysis are based on the state of the wall after static deflections and stresses have been developed.