## 2 Cyclic Torsional Load in Relation to Twist and Displacements  
A nonlinear numerical scheme is presented to illustrate the behavior of an individual pile and geomaterial due to the application of a cyclic torsional load at the head of a pile group. In this scheme, the pile is assumed to be elastic and embedded in cohesionless geomaterial. The modified torque-twist relation of a pile under cyclic torsional loading ([PERSON] and [PERSON] [28]) is expressed as follows:  
\[\frac{d^{2}\theta(z,t)}{dz^{2}}=\frac{\pi D^{2}}{2J_{P}G_{P}}\tau(z,t) \tag{1}\]  
where \(\theta(z,t)\) is the twist at the pile head; \(G_{P}\) is the shear modulus of the pile shaft; \(J_{P}\) is the polar second moment of the area of the pile section; \(z\) is the depth; \(D\) is the diameter of the pile; \(G_{P}J_{P}\) is the torsional rigidity of a pile; \(\tau(z,t)\) is the shear stress along the shaft. The basic equation of motion for multi-degree freedom systems is expressed as follows:  
\[m\ddot{y}+c_{h}\dot{y}+k_{h}y=0 \tag{2}\]  
\[m\ddot{\theta}+c_{\theta}\dot{\theta}+k_{\theta}\theta=0 \tag{3}\]  
where \(c_{h}\) and \(c_{\theta}\) are the coefficients of damping for lateral and torsional cyclic loading, respectively; \(k_{h}\) and \(k_{\theta}\) are the modulus of subgrade reaction for lateral and torsional cyclic loading, respectively. The modified second and fourth-order differential equilibrium equations for lateral displacement (\(y\)) and twist angle (\(\theta\)) of the pile in the geomaterial, assuming lateral displacement and twist as a function of time are expressed as follows:  
\[\rho_{s}\mathrm{A}\frac{\partial^{2}y(z,t)}{\partial t^{2}}+c_{h}\frac{ \partial y(z,t)}{\partial t}+k_{hN}y(z,t)=-E_{P}I_{P}\frac{\partial^{4}y(z,t) }{\partial z^{4}} \tag{4}\]  
\[\rho_{s}\mathrm{A}\frac{\partial^{2}\theta(z,t)}{\partial t^{2}}+c_{\theta} \frac{\partial\theta(z,t)}{\partial t}+k_{0N}\theta(z,t)=G_{P}J_{P}\frac{ \partial^{2}\theta(z,t)}{\partial z^{2}} \tag{5}\]  
where \(E_{P}\) is the elastic modulus of the pile shaft; \(I_{P}\) is the second-moment area of the pile section; \(\rho_{s}\) is the mass density of geomaterial and \(N\) is the number of cycles of torsional loading. Therefore, the differential equation of the geomaterial-pile system due to damped cyclic torsional loads assuming the harmonic motion induced through the pile head is expressed as follows:  
\[m\frac{\partial^{2}\delta(z,t)}{\partial t^{2}}+c_{h}\frac{\partial\delta(z,t )}{\partial t}+c_{\theta}\frac{\partial\delta(\theta,t)}{\partial t}+k_{hN} h(z,t)+k_{hN}\ \theta(z,t)+E_{P}I_{P}\frac{\partial^{4}\delta(z,t)}{\partial z^{4}}+G_{P}J_{P} \frac{\partial^{2}\theta(\theta,t)}{\partial z^{2}}=T_{0}e^{int} \tag{6}\]where \(T_{0}\) is the torsional load applied to the center of the pile group and \(\omega\) is the natural frequency. Figure 1a demonstrates the numerical model of the pile group employed in the present investigation. The loading has been applied as a variation of two-way symmetrical cyclic torsional loading with time (Figure 1b). The lateral and torsional resistance in clockwise and anticlockwise directions subjected to cyclic torsional loading for the pile group (2,2) is shown (Figure 1c). We consider the finite element in bending (Figure 1d). The approximation of the time-dependent displacement field \(\delta(z,t)\) using the [PERSON] method ([PERSON] [29]) is expressed as follows:  
\[ \begin{split}\delta(z,t)=N_{1}(z)\delta_{1}(t)+N_{2}(z)\delta_{2 }(t)+N_{3}(z)\delta_{3}(t)+N_{4}(z)\delta_{4}(t)+N_{5}(z)\Psi_{5}(t)+N_{6}(z) \Psi_{6}(t)+N_{7}(z)\Psi_{7}(t)\\ +N_{8}(z)\Psi_{8}(t)+N_{9}(z)\theta_{0}(t)+N_{10}(z)\theta_{10}(t) =[N]\{\delta(t)\}\end{split}  \tag{7}\]  
where \([N]=[N_{1}\ N_{2}\ N_{3}\ N_{4}\ N_{5}\ N_{6}\ N_{7}\ N_{8}\ N_{9}\ N_{10}]\)  
\[\{\delta(t)\}=\{\delta_{1}\delta_{2}\ \delta_{3}\delta_{4}\ \Psi_{5}\ \Psi_{6}\ \Psi_{7}\Psi_{8}\theta_{9}\ \theta_{10}\}\{t\}\]  
where, \(N_{m}\) are the shape functions for interpolation \(\delta(z,t)\) using its nodal values. \[