## 4 Peak Penetration Resistance in Sand Overlying Clay
The calculated \(q_{peak}\) are compared with those estimated by theoretical models of [PERSON] et al. (2011) and [PERSON] et al. (2012) in Figure 9.  
It should be noted that both the models of [PERSON] et al. (2011) and [PERSON] et al. (2012) require an iterative procedure to determine the operative friction and dilation angles, as summarized in the previous section. Additionally, the relative density of the sand layer needs to be defined. In this evaluation, the relative densities for \(\phi^{\prime}\) = 31\({}^{\circ}\) and 35\({}^{\circ}\) were defined as 44% and 60%, respectively, based on the empirical correlation of [PERSON] et al. (2019). The operative friction angles determined from this iterative procedure vary from 33.5\({}^{\circ}\) to 36.1\({}^{\circ}\) and 32.1\({}^{\circ}\) to 34.1\({}^{\circ}\) for \(\phi^{\prime}\) = 31\({}^{\circ}\) and 35\({}^{\circ}\) cases, respectively. The comparisons show that \(s_{u}\) has an important influence on the predicted \(q_{peak}\). It is also demonstrated that the theoretical models provide a reasonable estimate up to \(s_{u}\) = 20 kPa, which fall within the range of data from which the models were developed. This overestimation is caused by the empirical correlations for \(D_{F}\), which are only dependent on \(H_{s}\)/D. \(q_{peak}\) increases linearly with \(s_{u}\), thus not capturing the reduction in the mobilized area of the bottom clay layer which produces a lower capacity. Following the approaches of [11; 12], an empirical correlation for \(D_{F}\) was developed through a trial-and-error process. \(D_{F}\) was varied until \(d_{peak}\) predicted by the theoretical model of [PERSON] et al. [12] most favorably fit the calculated output. It should be noted that the iterative procedure to determine the operative friction and dilation angles were not used, because it requires an additional assumption for the relative density, which is most often not provided. Instead, the friction and dilation angles used in the simulations were applied as operative angles, because comparisons demonstrate that this additional step has marginal influence on the calculated response for the cases used in this study. The back-calculated \(D_{F}\) values are plotted against \(H_{\text{s}}\)/\(D\) for the range of selected \(s_{u}\) in Figure 10. The \(D_{F}\) relationship proposed by [PERSON] et al. [12] is also plotted in Figure 10 for comparison purposes. It is  
Figure 8: Peak bearing resistance calculated results for various spudcan diameters: (**a–d**) \(\phi^{\prime}=35^{\circ}\), (**e–h**) \(\phi^{\prime}=31^{\circ}\). demonstrated that \(D_{F}\) is strongly dependent on both \(H_{s}\)/\(D\) and \(s_{u}\). \(D_{F}\) is shown to be negatively correlated to both \(H_{s}\)/\(D\) and \(s_{u}\). For all \(s_{u}\) values, the variation in \(D_{F}\) is more significant for \(H_{s}\)/\(D\) < 0.5. It is also illustrated that the relationships are non-linear. A new relationship for \(D_{F}\) accounting for the influence of \(s_{u}\) in addition to \(H_{s}\)/\(D\) is as follows:  
\[D_{F}=\left(0.74\ -\ 0.53\frac{s_{u}}{p_{a}}\right)*\left(\frac{H_{s}}{D} \right)^{-(0.57\ -\ 1.07\frac{s_{u}}{p_{a}})} \tag{12}\]  
where \(p_{a}\) is the atmospheric pressure. The equation is applicable for the range applied in this study, which is \(0.28\leq H_{s}\)/\(D\leq 1.0\). The proposed equation is compatible with the \(q_{peak}\) model of [PERSON] et al. (2012). Figure 11 compares the predicted peak penetration resistances with the numerically calculated outputs. The residuals between the numerically calculated and the predicted peak resistances are within 20%. The comparisons highlight that the proposed equation provides reliable estimates of peak penetration resistance even for stiff clays with \(s_{u}\) up to 60 kPa. The proposed equation is also compared with the [PERSON] et al. (2010)\(D_{F}\) relationship. [PERSON] et al. (2010) proposed an analytical model for calculation of the post-peak penetration resistance in sand over clay soils using measurements from centrifuge tests.