Cyclic Degradation Parameters of Flow-Controlled Geomaterial for Pile Group Subjected to Torsional Loads

Abstract

The pile groups support large foundation structures which often experience cyclic torsional loads. Several theoretical and experimental investigations were carried out on pile groups under axial and lateral cyclic loads; however, the study of the response of cyclic torsional loads on pile foundation groups remains elusive. This paper proposes a numerical scheme to capture the behavior of the symmetrical pile group in the flow-controlled geomaterial under cyclic torsional loading. Based on the numerical scheme, three-dimensional finite element analysis was performed to capture the nonlinear response of the pile group using a computational program. The results from the numerical analysis have been compared with the experimental observations. The peak twist and peak shear stress logarithmically decrease and then become asymptotic after a number of cycles of loading for a pile group (2,2) in a flow-controlled geomaterial. A set of cyclic degradation parameters of flow-controlled geomaterial is identified for the pile group and presented as a function of dilation.

1. Introduction

The dynamics of pile groups are important due to their applications in machine foundations and structures exposed to cyclic loads such as wind or earthquakes. The response of structures supported by pile groups depends upon the cyclic stiffness and damping generated by geomaterial–pile interaction. Due to complex geomaterial–pile interaction, the dynamics of pile groups are not well understood. There are no readily available methods that could evaluate the response of geomaterial–pile interaction hence the geomaterial stiffness is ignored and only the pile stiffness is considered. The omission of geomaterial–pile interaction makes the analysis of pile groups quite unrealistic (Novak [1] and Novak Sharnouby [2,3]). Several analytical and numerical studies on the piles subjected to static lateral and torsional load (Poulos [4,5]; Randolph [6,7]; Chow [8]; Georgiadis [9]; Anagnostopoulos and Georgiadis [10]; Guo and Randolph [11]; Kong [12]; Kong and Zhang [13,14]; Chen, Kong and Zhang [15]; Kong, Zhang and Chen [16]) are extended to the pile groups in the present work. A series of experiments was reported on single piles and pile groups subjected to torsional loads (Mehra and Trivedi [17,18,19,20,21]). The numerical model was developed for the combined axial and torsional loads in flow-controlled geomaterial for the pile groups (Mehra and Trivedi [20]). The progressive twist, displacements and torsional energy around the pile groups have been presented (Mehra and Trivedi [21]). The settlements of sand due to the cyclic twisting of the tube were investigated by Cudmani and Gudehus [22]. The solutions for dynamic analysis of piles (Novak and Howell [23]; Novak and Sharnouby [3]; Long and Vanneste [24]; Ladhane and Sawant [25]; Basack and Nimbalkar [26]) including the literature reported mainly focused on the generic behavior of pile groups subjected to torsion while the cyclic torsion effect have not been considered. The investigators Zhu and Huang [27] have used the finite element -finite difference method to evaluate the safety against the liquefaction of a reservoir subjected to cyclic loading.

The design of offshore pile groups requires consideration of the effects of cyclic torsional loading. Therefore, the present paper considers a numerical scheme that provides a framework for the estimation of the behavior of a pile group in the sand under cyclic torsional loading. Based on a numerical scheme, the three-dimensional finite element analysis has been used to clarify the mechanism of load transfer from the pile to the surrounding geomaterial for the pile groups. The cyclic analysis of the torsionally loaded pile group in the sand is carried out by idealizing the individual pile as beam elements and the geomaterial as nonlinear spring elements. An iterative procedure is adopted and the effect of lateral and torsional load on deflection and twist, respectively, for the pile groups are investigated. Based on the lateral deflection and twist at the end of the first cycle, the degradation factor is evaluated and p-y and τ–θ curves are modified. The results from the numerical analysis work well with the published experimental results on the pile group subjected to two-way cyclic loading. Therefore, regarding the effect of torsional loads on the pile group, the following mechanisms have been investigated, namely, (a) cyclic twist behavior of the pile group (2,2) in the flow-controlled geomaterial as a result of the torsional loading and (b) the plastic strain and peak shear stress in a flow-controlled geomaterial captured for cyclic torsional loads, which, in turn, is a function of the number of cycles and dilation.

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 (Timoshenko and Goodier [28]) is expressed as follows:

$$\frac{d^2\mathsf{\theta }(z,t)}{{dz}^2}=\frac{\pi D^2}{2J_P{G}_P}\tau (z,t)$$

(1)

where $\mathsf{\theta }(z,t)$ 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:

$$m\ddot{y}+c_h\dot{y}+k_{h}y=0$$

(2)

$$m\ddot{\mathsf{\theta }}+c_{\mathsf{\theta }}\dot{\mathsf{\theta }}+k_{\mathsf{\theta }}\mathsf{\theta }=0$$

(3)

where $c_h$ and $c_{\mathsf{\theta }}$ are the coefficients of damping for lateral and torsional cyclic loading, respectively; $k_h$ and $k_{\mathsf{\theta }}$ 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 ($\mathsf{\theta })$ of the pile in the geomaterial, assuming lateral displacement and twist as a function of time are expressed as follows:

$${\mathsf{\rho }}_s\mathrm{A}\frac{{\partial }^2\mathrm{y}(\mathrm{z},\mathrm{t})}{\partial {\mathrm{t}}^2}+c_h\frac{\partial \mathrm{y}(\mathrm{z},\mathrm{t})}{\partial \mathrm{t}}+k_{hN}\mathrm{y}\left(z,t\right)=-E_P{I}_P\frac{{\partial }^4\mathrm{y}(\mathrm{z},t)}{\partial z^4}$$

(4)

$${\mathsf{\rho }}_s\mathrm{A}\frac{{\partial }^2\mathsf{\theta }(\mathrm{z},\mathrm{t})}{\partial t^2}+c_{\mathsf{\theta }}\frac{\partial \mathsf{\theta }(\mathrm{z},\mathrm{t})}{\partial \mathrm{t}}{+k}_{\mathsf{\theta }N}\mathsf{\theta }\left(z,t\right){=G}_P{J}_P\frac{{\partial }^2\mathsf{\theta }(\mathrm{z},t)}{\partial z^2}$$

(5)

where $E_p$ is the elastic modulus of the pile shaft; $I_p$ is the second-moment area of the pile section; ${\mathsf{\rho }}_s$ 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:

$$m\frac{{\partial }^2\mathsf{\delta }(\mathrm{z},\mathrm{t})}{\partial {\mathrm{t}}^2}+c_h \frac{\partial \mathsf{\delta }(\mathrm{z},\mathrm{t})}{\partial \mathrm{t}}+c_{\theta } \frac{\partial \mathsf{\delta }(\mathsf{\theta },\mathrm{t})}{\partial \mathrm{t}}+k_{hN }h\left(\mathrm{z},\mathrm{t}\right)+k_{\theta N }\mathsf{\theta }\left(z,\mathrm{t}\right){+E}_P{I}_P\frac{{\partial }^4\mathsf{\delta }(\mathrm{z},\mathrm{t})}{\partial {\mathrm{z}}^4}+G_P{J}_P\frac{{\partial }^2\mathsf{\theta }(\mathsf{\theta },\mathrm{t})}{\partial {\mathrm{z}}^2}=T_0{e}^{iwt}$$

(6)

where $T_0$ is the torsional load applied to the center of the pile group and ω 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 $\mathsf{\delta }\left(z,t\right)$ using the Galerkin method (Hutton [29]) is expressed as follows:

$$\mathsf{\delta }\left(z,t\right)=N_1\left(z\right){\mathsf{\delta }}_1\left(t\right)+N_2\left(z\right){\mathsf{\delta }}_2\left(t\right)+N_3\left(z\right){\mathsf{\delta }}_3\left(t\right)+N_4\left(z\right){\mathsf{\delta }}_4{\left(t\right)+N}_5\left(z\right){\mathsf{\Psi }}_5\left(t\right)+N_6\left(z\right){\mathsf{\Psi }}_6\left(t\right)+N_7\left(z\right){\mathsf{\Psi }}_7\left(t\right)+N_8\left(z\right){\mathsf{\Psi }}_8\left(t\right)+N_9\left(z\right){\mathsf{\theta }}_9\left(t\right)+N_{10}\left(z\right){\mathsf{\theta }}_{10}\left(t\right)=\left[N\right]\left\{\mathsf{\delta }\left(\mathrm{t}\right)\right\}$$

(7)

where $\left[N\right]=\left[N_1 N_2{ N}_3 N_4{ N}_5 N_6 N_7{ N}_{8 }{N}_9{ N}_{10}\right]$

$$\left\{\mathsf{\delta }\left(\mathrm{t}\right)\right\}=\left\{{\mathsf{\delta }}_1{\mathsf{\delta }}_{2 }{\mathsf{\delta }}_3{\mathsf{\delta }}_{4 }{\mathsf{\Psi }}_5 {\mathsf{\Psi }}_6 {\mathsf{\Psi }}_7{\mathsf{\Psi }}_8{\mathsf{\theta }}_9 {\mathsf{\theta }}_{10}\right\}\left\{t\right\}$$

where, $N_m$ are the shape functions for interpolation $\mathsf{\delta }\left(z,t\right)$ using its nodal values.

$$\begin{gathered} N_1=N_2=1-3{\mathsf{\xi }}^2+2{\mathsf{\xi }}^3;N_3=N_4=3{\mathsf{\xi }}^2-2{\mathsf{\xi }}^3; \\ {N}_5=N_6=L\left(\mathsf{\xi }-2{\mathsf{\xi }}^2+{\mathsf{\xi }}^3\right);N_7=N_8=L{\mathsf{\xi }}^2\left(\mathsf{\xi }-1\right) \end{gathered}$$

The Galerkin residual equation for the finite element of length, L, assuming displacements as a function of time is expressed as follows:

$$N_m\left(z\right){\int }_0^L\left(\mathrm{\rho }A\frac{{\partial }^2\mathsf{\delta }(\mathrm{z},\mathrm{t})}{{\partial \mathrm{t}}^2}+EI\frac{d^4\mathsf{\delta }(\mathrm{z},\mathrm{t})}{d{z}^4}\right)dz=0$$

(8)

Therefore, the pile element matrix in bending assuming displacement as a function of time is expressed as follows:

$$\left[{\Lambda }_{\mathrm{P}\mathrm{P}}\right]=\mathrm{\rho }A{\int }_0^L\left(N_m\frac{{\partial }^2\mathsf{\delta }}{{\partial \mathrm{t}}^2}+N_n\frac{{\partial }^2\mathsf{\delta }}{{\partial \mathrm{t}}^2}\right)dz+EI{\int }_0^L\frac{{d^{2}N}_m}{{dz}^2}\frac{{d^{2}N}_n}{{dz}^2}dz$$

(9)

where $m=1,2,5,6; n=3,4,7,8$.

Assuming twist as a function of time (Figure 1d), the pile element matrix in torsion is expressed as follows:

$$\left[{\Lambda }_{\mathrm{P}\mathrm{T}}\right]=\mathrm{\rho }A{\int }_0^L\left(N_m\frac{{\partial }^2\mathsf{\delta }}{{\partial \mathrm{t}}^2}+N_n\frac{{\partial }^2\mathsf{\delta }}{{\partial \mathrm{t}}^2}\right)dz+c{\int }_0^L\left(N_m\frac{\partial \mathsf{\delta }}{\partial \mathrm{t}}+N_n\frac{\partial \mathsf{\delta }}{\partial \mathrm{t}}\right)dz+G_{P}J{\int }_0^L\frac{{dN}_m}{dz}\frac{{dN}_n}{dz}dz$$

(10)

where $m=9; n=10$

$$\begin{gathered} \left[N\right]=\left[N_9 N_{10}\right]; \left\{\mathsf{\theta }\right\}=\left\{{\mathsf{\theta }}_9{\mathsf{\theta }}_{10}\right\}\left\{t\right\}; \\ {N}_9=1-\frac{Z}{L},N_{10}=\frac{Z}{L} \end{gathered}$$

3. Response of the Geomaterial to Cyclic Torsional Loading

The pile group is placed into a homogenous isotropic granular geomaterial medium. The application of cyclic torsional loading on the pile group is resisted by the flow-controlled geomaterial. The stress–strain response of geomaterial is highly nonlinear in all phases of loading. The hyperbolic stress–strain curve (Fahey and Carter [30]) is expressed as follows:

$$\frac{G_s}{G_{max}}=1-f_s{\left(\frac{\tau }{{\tau }_{max}}\right)}^{g_s}$$

(11)

where $G_s$ and $G_{max}$ are the initial and maximum shear modulus of geomaterial, respectively; $\tau$ and ${\tau }_{max}$ are the current and maximum shear stress, respectively; $f_s$ and $g_s$ are the curve fitting parameters. The parameter $f_s$ controls the magnitude or the extent of degradation whereas the parameter $g_s$ controls the rate of degradation and the curvature of the curve. A two-way torsional load applied at the centre of the pile group varies sinusoidally with time. The torsional and lateral resistances concurrently resist a portion of the applied torsional load (Kong [12]). Therefore, the initial modulus of subgrade reaction for the lateral load, $k_h$, and torsional load, $k_{\mathsf{\theta }}$, using the nonlinear $p-y$ and $\tau -\theta$ curves are expressed as follows:

$$k_h=\frac{P}{y} ; k_{\mathsf{\theta }}=\frac{2T_s}{\mathsf{\pi }{D}^{2}L}$$

(12)

where $P$ and $T_s$ are the lateral and torsional geomaterial reactions per unit length of the pile ($L)$, respectively. In the cyclic torsional analysis of pile groups, the geomaterial degradation has been captured by modifying the geomaterial spring stiffness (lateral translational spring and a torsional spring) after the completion of a number (N) of cycles. Little and Briaud [31] model calculates the deterioration of the geomaterial reaction modulus, $k_h$, due to cyclic lateral loading and is expressed as follows:

$$k_{hN}=k_{h1}{N}^{-\alpha }$$

(13)

where $k_{hN}$ takes the value of $k_h$ at the Nth cycle of load; $k_{h1}$ is the value of the geomaterial lateral reaction modulus for the first cycle of load, and α is the degradation parameter for geomaterial lateral modulus. Similarly, the deterioration of the geomaterial shear modulus,${ k}_{\theta }$, due to cyclic torsional loading is expressed as follows:

$$k_{\mathsf{\theta }N}=k_{\mathsf{\theta }1}{N}^{-\beta }$$

(14)

where $k_{\mathsf{\theta }N}$ takes the value of $k_{\theta }$ at the Nth cycle of load; $k_{\mathsf{\theta }1}$ is the value of the geomaterial torsional reaction modulus for the first cycle of load; and β is the degradation parameter for geomaterial torsional modulus. The degradation parameters for geomaterial lateral reaction modulus, α and geomaterial torsional modulus, β, are expressed as follows:

$$\alpha =\frac{E_c}{E_s};\beta =\frac{G_c}{G_s}$$

(15)

where $E_c$ and $E_s$ are the geomaterial modulus for cyclic and static loading, respectively; $G_c$ and $G_s$ are the geomaterial shear modulus for cyclic and static loading, respectively. The degradation is insignificant unless pile–geomaterial slippage takes place at the interface (Poulos [32]). As the angle of the pile twist increases, the mobilized shear stress at the pile shaft will reach the limiting value, and a slip will occur between the pile and the geomaterial. The limiting shaft friction ${\tau }_u$ (Guo and Randolph [11]) can be expressed as follows:

$${\tau }_u=A_t{z}^t$$

(16)

where $A_t$ is the constant that determines the magnitude of the shaft friction and t is the corresponding non-homogeneity factor. A correlation between ${\theta }_i$ and τ for an initial no-slip condition (Randolph [7]) is expressed as follows:

$${\theta }_i=\frac{\tau }{2G_s}$$

(17)

where ${\theta }_i$ is the angle of twist for ith element; τ is the pile–geomaterial interfacial shear stress; and $G_s$ is the initial geomaterial modulus.

The lateral spring stiffness at a node has been related to parameters inside the yield surface assuming displacement as a function of time and is expressed as follows:

$$\left[s{\Lambda }_{He}\right]={\mathrm{\rho }A{\int }_0^L\left(N_m\frac{{\partial }^2\mathsf{\delta }}{{\partial \mathrm{t}}^2}+N_n\frac{{\partial }^2\mathsf{\delta }}{{\partial \mathrm{t}}^2}\right)dz+c_h{\int }_0^L\left(N_m\frac{\partial \mathsf{\delta }}{\partial \mathrm{t}}+N_n\frac{\partial \mathsf{\delta }}{\partial \mathrm{t}}\right)dz+k}_{hN }{\int }_0^L\frac{{dN}_m}{dz}\frac{{dN}_n}{dz}dz$$

(18)

Similarly, the lateral translational spring stiffness at a node has been related to yield flow parameters assuming displacement as a function of time and is expressed as follows:

$$\left[s{\Lambda }_{Hf}\right]=f\left({\mathrm{s}\Lambda }_{\mathrm{H}\mathrm{e}},{\mathrm{s}\Lambda }_{\mathrm{H}\mathrm{p}}\right)=\left(k_{hN }f\left(G_{se}\right)\right){\int }_0^L\frac{{dN}_m}{dz}\frac{{dN}_n}{dz}dz$$

(19)

where $k_{hN}=\frac{P_N}{y_N}$, $P_N \mathrm{and} y_N$ are the lateral geomaterial reaction per unit length of the pile and lateral displacement of the pile for Nth cycles, respectively; $\left({s\Lambda }_{H_e}\right)$ is the stiffness as long as $f\to 0$,$\left({s\Lambda }_{H_p}\right)$ is the flow-controlled stiffness, and $G_{se}$ is the equivalent shear modulus of the geomaterial (Mehra and Trivedi [20]) at a node and is expressed as follows:

$$G_{se}=f\left(f,g\right)=f\left(\frac{\frac{\partial f}{\partial {\sigma }_{kl}}\frac{\partial g}{\partial {\sigma }_{ij}}}{\frac{\partial f}{\partial {\epsilon }_{pq}^p}\frac{\partial g}{\partial {\sigma }_{pq}}}\right)$$

(20)

The torsional spring stiffness at a node has been related to parameters inside the yield surface assuming twist as a function of time and is expressed as follows:

$$\left[s{\Lambda }_{\mathsf{\theta }e}\right]=\mathrm{\rho }A{\int }_0^L\left(N_m\frac{{\partial }^2\mathsf{\delta }}{{\partial \mathrm{t}}^2}+N_n\frac{{\partial }^2\mathsf{\delta }}{{\partial \mathrm{t}}^2}\right)dz+c_{\mathsf{\theta }}{\int }_0^L\left(N_m\frac{\partial \mathsf{\delta }}{\partial \mathrm{t}}+N_n\frac{\partial \mathsf{\delta }}{\partial \mathrm{t}}\right)dz+k_{\theta N }{\int }_0^L\frac{{dN}_m}{dz}\frac{{dN}_n}{dz}dz$$

(21)

The torsional spring stiffness at each node has been related to the yield flow parameters and is expressed as follows:

$$\left[s{\Lambda }_{\mathsf{\theta }f}\right]=f\left({\mathrm{s}\Lambda }_{\mathrm{T}\mathrm{e}},{\mathrm{s}\Lambda }_{\mathrm{T}\mathrm{p}}\right)=\left(k_{\theta N }f\left(G_{se}\right)\right){\int }_0^L\frac{{dN}_m}{dz}\frac{{dN}_n}{dz}dz$$

(22)

4. Equation of Equilibrium and Group Assembly of Stiffness Matrix

The equation of equilibrium for pile–geomaterial element considering Equations (1)–(22) is expressed as follows:

$$\begin{gathered} \left\{F\left(t\right)\right\}=\left[EI{\int }_0^L\frac{{d^{2}N}_m}{{dz}^2}\frac{{d^{2}N}_n}{{dz}^2}dz+GJ{\int }_0^L\frac{{dN}_m}{dz}\frac{{dN}_n}{dz}dz+\mathrm{\rho }A{\int }_0^L\left(N_m\frac{{\partial }^2\mathsf{\delta }}{{\partial \mathrm{t}}^2}+N_n\frac{{\partial }^2\mathsf{\delta }}{{\partial \mathrm{t}}^2}\right)dz \\ +c_h{\int }_0^L\left(N_m\frac{\partial \mathsf{\delta }}{\partial \mathrm{t}}+N_n\frac{\partial \mathsf{\delta }}{\partial \mathrm{t}}\right)dz+c_{\mathsf{\theta }}{\int }_0^L\left(N_m\frac{\partial \mathsf{\delta }}{\partial \mathrm{t}}+N_n\frac{\partial \mathsf{\delta }}{\partial \mathrm{t}}\right)dz {+k}_{hN }{\int }_0^L\frac{{dN}_m}{dz}\frac{{dN}_n}{dz}dz \\ +\left(k_{hN }f\left(G_{se}\right)\right){\int }_0^L\frac{{dN}_m}{dz}\frac{{dN}_n}{dz}dz+k_{\theta N }{\int }_0^L\frac{{dN}_m}{dz}\frac{{dN}_n}{dz}dz+\left(k_{\theta N }f\left(G_{se}\right)\right){\int }_0^L\frac{{dN}_m}{dz}\frac{{dN}_n}{dz}dz \right]\left\{\mathsf{\delta }\right\} \end{gathered}$$

(23)

where, $\left\{\mathsf{\delta },\mathsf{\Psi },\mathsf{\theta }\right\}=\left\{{\mathsf{\delta }}_1{\mathsf{\delta }}_2{\mathsf{\delta }}_3{\mathsf{\delta }}_4{\mathsf{\Psi }}_5{\mathsf{\Psi }}_6{\mathsf{\Psi }}_7{\mathsf{\Psi }}_8{\mathsf{\theta }}_9{\mathsf{\theta }}_{10}\right\}\left\{t\right\}$,

$$\left\{F\left(t\right)\right\}=\left\{{ P}_{mx} P_{my} P_{nx} P_{ny} M_{mx} M_{my} M_{nx} M_{ny} T_m { T}_n\right\}\{t\}$$

The overall equilibrium equations have been formulated based on the assembled group stiffness matrix and the known load vector is expressed as follows:

$$\left\{T\left(t\right)\right\}=\left[{\Lambda }_P\right]\left\{d_p\left(t\right)\right\}+\left[s\Lambda \right]\left\{{\mathrm{d}}_{\mathrm{s}}\left(t\right)\right\}$$

(24)

where [${\Lambda }_p)$ and [$s\Lambda )$ are the global stiffness matrix of all the elements of the pile groups and geomaterial, respectively; $\left\{d_p\left(t\right)\right\}$ and ${\{d}_s\left(t\right)\}$ are the vector of deformations at pile nodes and geomaterial, respectively; {T(t)} is the cyclic torsional load applied on the pile group. The Gauss elimination has been used to solve Equation (24) for unknown nodal displacements. The 3D FE modelling of the geomaterial–pile group has been performed based on the aforesaid idealizations.

5. Computational Algorithm

The computation is carried out using a pile group cyclic torque computational program (PGCYT). The flowchart is shown in Figure 2. The step-by-step algorithm for obtaining the cyclic degradation parameter of flow-controlled geomaterial is as follows:

(a)

Using the input parameters for geomaterial medium, pile group and cyclic loading, the analysis is carried out under cyclic torsional loads.

(b)

The torsional and lateral spring stiffnesses of the pile group are determined using Equations (9)–(10), assuming displacement and twists as the function of time for the first cycle (N = 1).

(c)

The stress–strain response of geomaterial is obtained from the hyperbolic model using Equation (11).

(d)

The lateral and torsional spring stiffnesses for geomaterial at a node related to parameters inside the yield surface are determined using Equation (18) and Equation (21), respectively.

(e)

The lateral and torsional spring stiffnesses for geomaterial at node-related yield flow parameters are determined using Equation (19) and Equation (22), respectively.

(f)

The nodal displacements and twists are evaluated using Gauss elimination for all elements using Equation (24).

(g)

The values of $\tau$ and ${\tau }_u$ are evaluated for all elements using Equations (16) and (17).

(h)

The degradation parameter for lateral and torsional geomaterial stiffnesses has been computed.

(i)

Using reduced values of the geomaterial stiffness, analysis is again carried out following the repeating steps (a–h), and the procedure is repeated until the desired convergence is achieved.

(j)

The distributions of twist, plastic strain, and peak shear stress are computed for the pile group.

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° (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. Pile group and geomaterial parameters.

Parameters	Description
Young modulus of pile material, (Ep, Gpa)	210
Poisson’s ratio (µp)	0.3
Unit weight, (γp, kN/m3)	78
Diameter, (Dp, mm)	40
Length (Lp, mm)	600
Pile-to-pile spacing (Sp)	3Dp
Equivalent perimeter (mm)	50.3
Initial elastic modulus of geomaterial (Mpa)	20
Effective unit weight of geomaterial (γs, kN/m3)	16
Initial Poisson’s ratio of geomaterial (µs)	0.3
$\mathrm{Constant} \mathrm{volume} \mathrm{friction} \mathrm{angle} ({\varnothing }_{\mathit{C}}$ , degrees)	30
$\mathrm{Dilation} \mathrm{angle} ({\varnothing }_{\mathit{d}}$ , degrees)	2–12

.

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 Mehra and Trivedi [20,21]. The yield surface in the front cycle of loading is isotropic and has an onset of nonlinearity at ${\epsilon }_{ij}^p>0$. 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, ${\mathrm{\varnothing }}_p$ $\left({\varnothing }_p={\varnothing }_c+{\varnothing }_d\right)$, which is a function of constant volume friction angle, ${\mathrm{\varnothing }}_c$, and angle of dilation, ${\mathrm{\varnothing }}_d$. The input parameters for the geomaterial medium have been defined in (Table 1).

7. Results, Discussion and Validation

The response of footings and structures supported by pile groups depends upon the cyclic stiffness and damping generated by geomaterial–pile group interaction. The three-dimensional sectional view of the geomaterial–pile group (2,2) for 2°, 6°, and 12° dilation, respectively, is shown in Figure 4a–c.

Figure 4d–f show the geomaterial–pile group (2,2) sectional view for plastic strain values for 2°, 6°, and 12° dilation, respectively. The peak plastic strain decreases with an increase in dilation. The peak plastic strain of the geomaterial has been computed as 43.44, 27.34, and 14.19% for 2°, 6°, and 12° dilation, respectively (

Table 2. Peak values of plastic strain (%) of the geomaterial.

Cycle	∅D = 12°	∅D = 6°	∅D = 2°
1	0.92	1.34	1.67
10	7.08	12.74	18.49
20	11.02	20.76	32.35
30	14.19	27.34	43.44

). The color scheme is selected to represent the intensity of plastic strain, where the green color scheme is for maximum and minimum plastic strain for 2° and 12° dilation, respectively.

Figure 5 shows the variation of twist angle measured at the pile cap with the number of cycles of normalized cyclic torque for pile group (2,2) evaluated for 2°, 6°, and 12° dilation. The twist angle decreases with an increase in the number of cycles. The peak twist angle decreases with an increase in dilation. The peak twist logarithmically decreases with an increase in the number of cycles and dilation and then becomes asymptotic after 30 cycles of loading in a flow-controlled geomaterial.

The relationship between the peak twist angle and the number of cycles, N is expressed in matrix form as follows:

$$\left[{\theta }_t\right]=\left[\begin{array}{cc}{\log }_{e}N& 1\end{array}\right]\left[\begin{array}{c}{a}_t\\ b_t\end{array}\right]$$

(25)

$${\theta }_t=a_t{\log }_{e}N+b_t$$

(26)

$$N=e^{\frac{{\theta }_t-b_t}{a_t}}; N=1; {\theta }_t=b_t$$

(27)

where $a_t$ and $b_t$ are the logarithmic degradation parameters for the twist (

Table 3. Flow-controlled cyclic degradation parameters.

$\mathbf{Dilation}, {\varnothing }_{\mathit{D}}$	${\mathit{D}}_{\mathit{f}}^{″}$	${\mathit{D}}_{\mathit{f}}^{\prime }$	${\mathit{D}}_{\mathit{f}}$	${\mathit{a}}_{\mathit{t}}$	${\mathit{b}}_{\mathit{t}}$	${\mathit{a}}_{\mathit{s}}$	${\mathit{b}}_{\mathit{s}}$
2°	−0.019	2.006	0.1445	−0.007	0.0357	−1.079	8.289
6°	−0.0156	1.3342	0.6356	−0.003	0.0244	−1.298	8.246
12°	−0.009	0.7087	0.6871	−0.002	0.0189	−1.364	8.118

), which depend upon the number of cycles, dilation angle, and plastic strain. The logarithmic parameter $b_t$ is a measure of the residual angle of twist (${\theta }_0$) required in excess to mobilize the torque in the pile group over and above the maximum dilation at the end of the first cycle (N = 1).

Figure 6 shows the variation of plastic strain (%) with the number of cycles for pile group (2,2) evaluated for 2°, 6°, and 12° dilation, with enlarged views magnifying the convexities of plastic strain. The plastic strain increases with an increase in the number of cycles. The variation in the peak shear stress with the number of cycles for pile group (2,2) evaluated for 2°, 6°, and 12° dilation, respectively, has been shown in Figure 7.

The relationship between the peak values of plastic strain and the number of cycles has been expressed in matrix form as follows:

$$\left[{\epsilon }_p\right]=\left[\begin{array}{ccc}{N}^2& N& 1\end{array}\right]\left[\begin{array}{c}{D}_f^{″}\\ D_f^{\prime }\\ D_f\end{array}\right]$$

(28)

where $D_f^{″}$ is the power law cyclic degradation parameter, which reduces the magnitude of plastic strain at a decrement rate for increasing dilation angle; $D_f^{\prime }$ represents the linear cyclic degradation parameter, which decreases with the increasing dilation angle; and $D_f$ represents the constant cyclic degradation parameter for a cyclic strain, which increases with the dilation angle (Figure 8a and Table 3).

The peak shear stress for pile group (2,2) and a single pile for normalized cyclic torque is found in the range of (5–8.2) kPa at dilations of 2° to 12°, respectively, in the present analysis. The peak shear stress for the pile group (2,2) logarithmically decreases and then asymptotically converges after 60 cycles of loading in a flow-controlled geomaterial, which, in turn, is a function of the number of cycles and dilation (Figure 7).

The relationship between the peak shear stress and the number of cycles, N is expressed in matrix form as follows:

$$\left[{\tau }_p\right]=\left[\begin{array}{cc}{\log }_{e}N& 1\end{array}\right]\left[\begin{array}{c}{a}_s\\ b_s\end{array}\right]$$

(29)

$${\tau }_p=a_s{\log }_{e}N+b_s$$

(30)

$$N=e^{\frac{{\tau }_p-b_s}{a_s}}; N=1; {\tau }_p=b_s$$

(31)

where, $a_s$ and $b_s$ are the logarithmic degradation parameters for shear stress. The experimental data of Cudmani and Gudehas (2001) [22] have the values of $a_s$= $-$0.782 and $b_s=2.9964$, and the same for the present analysis for a single pile are $-$1.438 and 6.3183, respectively. Figure 8b shows the variation of a set of logarithmic degradation parameters with the angle of dilation (Table 3).

8. Conclusions

This study considers the degradation of the geomaterial medium associated with the plastic flow, reflecting the actual response of the interaction of pile groups and the geomaterial, which is normally ignored by the classical design of pile foundations. The main conclusions of the study are as follows:

• Due to the application of the cyclic torsional loading, there is a progressive deterioration of strength and stiffness of the surrounding geomaterial, resulting in significant degradation of the pile group capacity.
• A cyclic degradation parameter for plastic strain and a logarithmic degradation parameter for twist and shear have been identified as a function of frictional characteristics of flow-controlled geomaterial which in turn depend upon the number of cycles, plastic strain, and dilation. The flow-controlled cyclic degradation parameters for plastic strain ($D_f^{″},D_f^{\prime },D_f)$ and logarithmic degradation parameters for the twist $(a_t$, $b_t)$ and shear $(a_s$, $b_s)$ have been classified and recommended for designing pile groups.
• The twist decreases with an increase in the number of cycles and dilation captured for the pile group (2,2). The peak twist logarithmically decreases and then becomes asymptotic after 30 cycles of loading in a flow-controlled geomaterial.
• The plastic strain increases with an increase in the number of cycles for the pile group (2,2). The peak plastic stain has been obtained as 43.44, 27.34, and 14.19% for 2°, 6°, and 12° dilation, respectively.
• The 3D numerical model on the pile group (2,2) and the single pile have been compared to the result of the experimental data of the single pile. The peak shear stress for pile group (2,2) and a single pile for normalized cyclic torque are found in the range of (5–8.2) kPa at dilations of 2° to 12° in the present analysis.
• The peak shear stress for the pile group (2,2) logarithmically decreases and then asymptotically converges after 60 cycles of loading in a flow-controlled geomaterial which in turn is a function of the number of cycles and dilation.