# Settlement Prediction of Footings Using \(V_{s}\)
## 3 Centrifuge Study
### Results and Discussion  
#### 3.2.1 Shear Wave Velocity (\(V_{s}\))  
The \(V_{S}\) was calculated after determining the first arrival time of received shear wave signals and the tip-to-tip distance between the two corresponding bender elements [24]. Interpolation based on a regression analysis was conducted to produce continuous \(V_{S}\) profiles for precise predictions because the shear wave was detected by point-based measurements. The measured \(V_{S}\) values and theoretical confining stress were used for the regression analysis as shown below [25]:  
\[V_{s}=C_{s}(\sigma^{\prime}/P_{a})^{n/2} \tag{12}\]  
where \(C_{S}\) is material constant, \(\sigma^{\prime}\) is effective confining stress, and \(P_{a}\) is atmosphere pressure to normalize the stress term, which was generally assumed to be 100 kPa. In this study, the theoretically calculated vertical effective stress was adopted as the effective confining stress in Equation (12).  
Figure 3 shows the measured and interpolated \(V_{S}\) profiles with respect to depth, which is normalized by footing breadth (\(B\)) for the three experiments. As shown in the figure, three \(V_{S}\) profiles before the 1 st loading are almost identical, which means the model grounds were constructed homogeneously and uniformly. It is interesting to note that the \(V_{S}\) measured before 2 nd loading becomes greater than the 1 st loading and the depth at which \(V_{S}\) increased was deeper with increasing  
\begin{table}
\begin{tabular}{c c} \hline
**Properties** & **Silica Sand** \\ \hline Unified soil classification (USCS) & SP \\ Median particle size (\(D_{50}\)), mm & 0.22 \\ Curvature coefficient (\(C_{C}\)) & 1.11 \\ Uniformity coefficient (\(C_{U}\)) & 1.96 \\ Specific gravity (\(G_{S}\)) & 2.65 \\ Plasticity index (\(PI\)) & NP \\ Maximum void ratio (\(e_{max}\)) & 1.130 \\ Minimum void ratio (\(e_{min}\)) & 0.611 \\ \hline \end{tabular}
\end{table}
Table 1: Physical properties of the silica sand.  
\(L/B\) ratio. This could be attributed to residual horizontal stress in the subsurface after the 1 st loading, and the area influenced by the residual stress deepened with increasing \(L/B\) ratio. Given that the residual stress is evidence of load transfer during the 1 st loading, the residual stress can be correlated with the \(I_{z}\). The \(I_{z}\) profiles from [PERSON] et al. (2016) and elasticity theory reflect the aforementioned trend that the stress increment transmitted from the surface is deeper with increasing \(L/B\) ratio. The influenced depth depending on the \(L/B\) ratio converges to zero at 2.0 \(B\) depth for square footing and at 4.0 \(B\) depth for strip footing as shown in Figure 1. Figure 4 shows the discrepancies between the \(V_{S}\) measured before 2 nd loading and that before 1 st loading with respect to depth, which is normalized by footing breadth (\(B\)). In this figure, it is possible to confirm the phenomena mentioned before more clearly. The influenced depth and depth to peak, which indicates the maximum \(V_{S}\) difference, increase with increasing \(L/B\) ratio as presented by shading.  
Figure 3: Shear wave velocity profiles: (a) \(L/B=1.0\); (b) \(L/B=2.5\); (c) \(L/B=5.0\).