## 2 Peak Penetration Resistance Prediction Models in Sand Overlying Clay  
Among various penetration resistance models developed specifically for spudcans, two \(q_{peak}\) models widely used in practice are introduced in this section. [PERSON] et al. [11] proposed an equation to predict \(q_{peak}\) of spudcan foundation in dense sand over NC clay using a simplified conceptual model, illustrated in Figure 1a. An inverted truncated cone of sand is assumed to be pushed into the underlying clay. \(q_{peak}\) is mobilized by the shearing resistance along the sliding surface and the bearing capacity of the clay layer. It was assumed \(q_{peak}\) is mobilized at the surface, based on the centrifuge tests of [PERSON] et al. [22]. The predictive equation is as follows:  
\[q_{peak} \tag{1}\]  
where \(N_{c0}\) is the bearing capacity factor for clay at the base of a circular foundation, \(s_{u}\) is the undrained strength of clay at the sand-clay interface, \(q_{0}\) is the overburden pressure, \(H_{s}\) is the height of the sand layer, \(D\) is the diameter of spudcan, \(\psi\) is the dilation angle of sand, and \(E^{*}\) is an empirical parameter, and \(\gamma^{\prime}s\) is the unit weight of sand. The following equation proposed by [PERSON] and [PERSON] [23] for isotropic clays was used for \(N_{c0}\):  
\[N_{c0}=6.34\ +0.56\frac{k(D\ +\ 2H_{s}tany)}{s_{u}} \tag{2}\]  
where \(k\) is the strength gradient of NC clay. \(E^{*}\) is defined as follows:  
\[E^{*}=2\left[1\ +\ D_{F}\left(\frac{tamb^{*}}{tamb\psi}-1\right)\right] \tag{3}\]  
where \(tamb^{*}\) is calculated as:  
\[tamb^{*}=\frac{sin\phi^{\prime}cos\psi}{1-sin\phi^{\prime}sin\psi} \tag{4}\]  
\(D_{F}\) is defined as:  
\[D_{F}=\frac{\sigma^{\prime}_{n}}{\overline{\sigma^{\prime}_{z}}} \tag{5}\]  
where \(\sigma^{\prime}_{n}\) is the normal stress along the failure surface and \(\overline{\sigma^{\prime}_{z}}\) is the average of the vertical stress within the failure surface (frustrum). Because of difficulties in determining the failure surface and normal stress acting on it, the following empirical correlation for \(D_{F}\) was presented from a linear regression to fit the centrifuge measurements as follows:  
\[D_{F}=1.333\ -\ 0.889\frac{H_{s}}{D}\left(\frac{H_{s}}{D}\ \leq\ 0.9\right) \tag{6}\]  
The operative friction and dilation angle were recommended to be selected through an iterative process, because they are dependent on the stress level [PERSON] [24]. The following equation was presented to determine \(q_{peak}\) dependent operative friction and dilation angles:  
\[\phi^{\prime} =\phi_{cv}+2.65\left[\ I_{D}\left(Q-\ lnq_{peak}\right)-1\ \right] \tag{7}\] \[\psi\ =3.3\left[\ I_{D}\left(Q-\ lnq_{peak}\right)-1\ \right] \tag{8}\]where \(I_{D}=\) relative density, \(Q\) is the particle crushing strength of sand, and \(\phi_{cv}\) is the critical state friction angle of sand. It was reported that \(Q=10\) for silica sand. [PERSON] et al. [12] performed spudcan penetration tests on medium-loose sand overlying clay and compared \(q_{peak}\). It was reported that \(q_{peak}\) is not mobilized at the surface, but at a depth of \(d_{peak}\). To account for \(d_{peak}\), the model of [PERSON] et al. [11] was modified as illustrated in Figure 1b. The following modified equation was proposed:  
\[ \begin{array}{c}q_{peak}=\Big{(}N_{co}s_{u}+\ q_{o}+0.12\gamma_{s}^{\prime}H_ {s}\Big{)}\Big{(}1+\frac{1.76H_{s}}{D}tan\psi\Big{)}^{E^{*}}+\frac{\gamma_{s}^ {\prime}D}{2 tan\psi(E^{*}+1)}\\ \left[1-\Big{(}1-\frac{1.76H_{s}}{D}E^{*}tan\psi\Big{)}\Big{(}1+\frac{1.76H_{s} }{D}tan\psi\Big{)}^{E^{*}}\right]\end{array}  \tag{9}\]  
Additionally, the relationship of \(D_{F}\) was recalibrated and optimized for a wide range of spudcan geometries and sand stiffness.