## 4 Failure Envelopes in the \(V\)-\(H\)-\(M\) Domain
### Empirical Design Procedure and Validation  
Equations (2)-(4) can be applied based on geometry parameters as \(D_{\rm w}D_{\rm p}/L^{2}\) = 0.05-0.1 and soil properties with \(s_{\rm um}\) = 20-40 kPa, \(\varphi\) = 30\({}^{\circ}\)-40\({}^{\circ}\), which is in line with engineering practice [47; 48; 49]. The equations are calculated with \(T_{\rm s}/L\) = 0.1, 0.3, 0.5 and 0.7, which can also be used for other values of \(T_{\rm s}/L\) through interpolating. The failure envelopes calculated by FEM and Equations (2)-(4) are compared in Figure 10b for an example, and good agreement is obtained. The maximum relative error of about 7% occurs when \(T_{\rm s}/L\) = 0.7 and \(V/(2\pi R^{3}\gamma^{\prime}s)\) is relatively small, which means that the prediction of the equations can be considered conservative. Based on the above geometric dimensions and soil properties, a suggested design procedure is proposed: (i) determining the shape of the failure envelopes in the \(V\)-\(H\)-\(M\) domain with the assistance of Equation (2) and Table 2; (ii) calculating the absolute values of uniaxial lateral bearing capacity and vertical capacity using Equations (3) and (4) along with Tables 3 and 4; (iii) obtaining the lateral and moment bearing capacities under a certain vertical load of the composite foundation. To validate the suitability and accuracy of the proposed empirical design procedure, it is used to estimate the bearing capacity of composite foundations reported by other researchers. Firstly, the centrifuge test data reported by [PERSON] et al. (2014) [27] on a composite foundation in pure sand are compared with the predicted results using the proposed method in this study (\(D_{\rm w}D_{\rm p}/L^{2}\) = 0.048, \(e/D_{\rm p}\) = 8, \(V/V_{\rm ult}\) = 0). The lateral bearing capacities from the empirical formula and test results are 59.23 and 63.14 in normalized form, respectively, which indicates an error of only 6.61%. Then, to further validate the accuracy, the laboratory testing results from the centrifuge tests carried out by [PERSON] et al. (2018) [36] are also compared with the prediction results using the proposed empirical formula in this work (\(D_{\rm w}D_{\rm p}/L^{2}\) = 0.094, \(e/D_{\rm p}\) = 8, \(V/V_{\rm ult}\) = 0). The normalized lateral bearing capacities are 143.64 and 181.89 for the empirical design procedure and the centrifuge test, respectively. It reflects an error of 21.03%. The difference is probably because in pure sand, the shape of the failure envelope is slightly different to that of \(T_{\rm s}/L\) = 0.7 in certain situations, and the empirical design procedure for the bearing capacities of the composite foundation in sand-over-clay deposit may not be perfectly suitable for that in single soil layer conditions. Since very limited results are available for composite \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(T_{\rm s}/L\) & \(a_{\rm 1}\) & \(b_{\rm 1}\) & \(c_{\rm 1}\) & \(d_{\rm 1}\) & \(e_{\rm 1}\) & \(f_{\rm 1}\) & \(g_{\rm 1}\) & \(h_{\rm 1}\) \\ \hline
0.1 & 1.61 & 230 & \(-\)0.21 & \(-\)2.78 & \(-\)1.22 & 399.44 & 0.19 & \(-\)12.22 \\
0.3 & \(-\)15.99 & 457.22 & 0.53 & \(-\)11.67 & \(-\)16.99 & 957.22 & 1.33 & \(-\)37.78 \\
0.5 & \(-\)2.25 & 164.44 & 0.07 & \(-\)2.22 & 0.09 & 948.33 & \(-\)0.035 & \(-\)25.56 \\
0.7 & 7.52 & \(-\)260.56 & \(-\)0.6 & 15 & \(-\)9.02 & 1506.67 & 0.54 & \(-\)45 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values of parameters related to \(T_{\rm s}/L\) in Equation (3) (\(h_{0}\)). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(T_{\rm s}/L\) & \(tan(\varphi)\gamma^{\prime}sD_{\rm p}/s_{\rm um}\) & \(A\) & \(B\) & \(C\) & \(D\) & \(E\) \\ \hline
0.7 & 0.754 & 0.302 & 0.015 & 0.203 & 3.422 & 4.655 \\
0.7 & 0.848 & 0.602 & 0.321 & 0.161 & 1.794 & 4.572 \\
0.7 & 0.949 & 0.874 & 0.869 & 0.314 & 1.448 & 3.357 \\
0.7 & 1.096 & 1.063 & 1.709 & 0.889 & 1.862 & 3.471 \\ \hline \hline \end{tabular}
\end{table}