## 6 Numerical Simulation of Pile Group Subjected to Cyclic Torsional Loads  
Three-dimensional finite element modelling is a convenient and reliable approach to accounting for the continuity of the geomaterial mass and the nonlinearity of the geomaterial pile group interaction, which act as a valuable mechanism of load transfer from the pile to the surrounding geomaterial, especially for pile groups. Therefore, several trial models have been executed with initial geomaterial moduli and varying dilations of 0-12\({}^{\circ}\) (Bolton [33]) to capture the cyclic torque-twist relationship for the pile group. The analysis has been performed using the computational algorithm (PGCYT). The pile group (2,2) input parameters have been considered as shown in Table 1.  
The geomaterial continuum has been divided into several volume elements. The tangential contact has been defined to model the interaction behavior between the pile and the geomaterial using penalty friction formulation with a friction coefficient of 0.36 (Figure 3a,b). The interaction uses the allowable elastic slip, which is the fraction of the characteristic surface dimension of 0.005. The eight-node linear brick hexahedral element (C3D8R), with reduced integration, has been considered for modelling pile group (2,2), as shown in Figure 3c,d, which could capture the interlocking effects in terms of strength parameters. The master surface is represented by the exterior surface of the pile and the slave surface by the interior surface of the geomaterial. The geomaterial has been selected such that the flow potential for the yield surface is defined by the formulation used by [PERSON] and [PERSON] [20; 21]. The yield surface in the front cycle of loading is isotropic and has an onset of nonlinearity at \(\varepsilon_{ij}^{p}>0\). In the reverse cycle, it follows a mixed hardening model necessarily consistent with the hardening rule of a flow-controlled geomaterial. A relatively fine mesh was adopted for the pile group, and a coarser mesh was adopted for the geomaterial. The mesh sensitivity analysis has been performed using the single seeding bias technique consisting of a number of elements (8). The bias ratio (0.5) has been selected to achieve convergence behavior. It further considers a flip bias from a near field to a far field with increasing mesh size. The computational program considers a peak value of the angle of internal friction, \(\mathcal{O}_{p}\) (\(\mathcal{O}_{p}=\mathcal{O}_{c}+\mathcal{O}_{d}\)), which is a function of constant volume friction angle, \(\mathcal{O}_{c}\), and angle of dilation, \(\mathcal{O}_{d}\). The input parameters for the geomaterial medium have been defined in (Table 1).  
\begin{table}
\begin{tabular}{l c} \hline
**Parameters** & **Description** \\ \hline Young modulus of pile material, (E\({}_{\text{p}}\), Gpa) & 210 \\ Poisson’s ratio (\(\mu_{\text{p}}\)) & 0.3 \\ Unit weight, (\(\gamma_{\text{p}}\), kN/m\({}^{3}\)) & 78 \\ Diameter, (D\({}_{\text{p}}\), mm) & 40 \\ Length (L\({}_{\text{p}}\), mm) & 600 \\ Pile-to-pile spacing (S\({}_{\text{p}}\)) & 3D\({}_{\text{p}}\) \\ Equivalent perimeter (mm) & 50.3 \\ \hline Initial elastic modulus of geomaterial (Mpa) & 20 \\ Effective unit weight of geomaterial (\(\gamma_{\text{s}}\), kN/m\({}^{3}\)) & 16 \\ Initial Poisson’s ratio of geomaterial (\(\mu_{\text{s}}\)) & 0.3 \\ Constant volume friction angle (\(\mathcal{O}_{\text{c}}\), degrees) & 30 \\ Dilation angle (\(\mathcal{O}_{d}\), degrees) & 2–12 \\ \hline \end{tabular}
\end{table}
Table 1: Pile group and geomaterial parameters.