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Edited by: Sakdirat Kaewunruen, University of Birmingham, United Kingdom

Reviewed by: Rims Janeliukstis, Riga Technical University, Latvia; Zhang Qian, Qingdao University, China

This article was submitted to Transportation and Transit Systems, a section of the journal Frontiers in Built Environment

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Pile foundations supporting large structures (such as high-rise buildings, oil drilling platforms, bridges etc). are often subjected to eccentric lateral load (in addition to the vertical loads) due to the action of wind, waves, high speed traffic, and ship impacts etc. The eccentric lateral load, which is usually cyclic (repetitive) in nature, induces torsion in the pile foundation. This paper presents a numerical model based on boundary element approach to study the performance of a single pile subjected to the torsional cyclic load. The model is initially validated by comparing it with the experimental data available from the literature. Thereafter, the model has been utilized to conduct a parametric study to understand the influence of the torsional cyclic loading parameters on the axial pile capacity. The results indicated that the model is able to capture the degradation in the axial pile capacity due to the torsional cyclic loading with a reasonable accuracy. Moreover, the parametric study showed that the frequency, amplitude and number of cycles play a significant role in the torsional cyclic response of the pile. The present study is essential for the development of design guidelines for pile foundations subjected to torsional cyclic load.

The construction of large structures such as high-rise buildings, oil drilling platforms, electrical transmission towers, wind turbines, bridges, and railway embankments over a soft compressible clay poses a serious challenge to the design engineers. Therefore, a cost-effective foundation system with an acceptable degree of safety is required and consequently, these structures are usually supported by pile foundations. These piles are often subjected to cyclic (repetitive) lateral loads in addition to the vertical loads, during their service life due to wind, sea waves, high speed train traffic and ship impacts etc. (Arshad and O'Kelly,

The cyclic load initiates the reversal of soil-pile interface shear stress, thus producing a progressive deterioration in strength and stiffness of the surrounding soil and consequently, pile-soil interactive performance undergoes significant degradation (Basack,

Several theoretical, laboratory and field-based investigations on the pile foundations subjected to vertical and lateral cyclic loads have been conducted in the past (El Naggar et al.,

The behavior of pile foundation subjected to monotonic and cyclic axial loading can be investigated by using several numerical and analytical techniques available in the literature. These include the dynamic response analysis, cyclic stability analysis, finite element and boundary element analysis etc. to name a few (Poulos,

The paper is presented in the following sequence: first, a model is formulated using BEM approach followed by its validation by comparing the BEM computed results with the available field data. Such comparison indicates reasonable accuracy of the proposed numerical solutions. Thereafter, a prototype parametric study is conducted using the developed model to analyse the influence of cyclic loading parameters on the soil-pile interactive performance. Finally, the normalized soil-pile interface shear stress profiles (predict using the proposed solution) are presented to show the distribution of stresses along the length of pile for static and post-cyclic condition.

_{t} and a two way symmetrical cyclic torsional load (given by Equation 1):

Numerical modeling:

Where, _{b} and σ_{b} at the base of the pile, and shear stresses _{b}are primarily induced due to the axial load and are static, whereas, the stresses _{b} are induced due to the torsional load and are cyclic. The primary aim of this study is to evaluate these unknown interface shear stress components and subsequently, determine the axial pile capacity after completion of a certain number of load cycles (

The pile material in the present study has been idealized as elastic-perfectly plastic. The stress-strain behavior of the soil in shear is assumed to be non-linear up to the peak shear stress (τ_{u}), followed by a perfectly plastic post-peak response (Basack and Sen, _{i} and a reduction factor of _{f} (Duncan and Chang, _{f} usually varies in the range of 0.8–1.0 (Randolph,

Where, τ andγ are the shear stress and shear strain, respectively. The post peak response can be mathematically represented as:

The pile-soil interaction in the present study has been analyzed using the Boundary Element Modeling (BEM) following the methodology of (Basack and Sen, ^{th} element have been denoted as _{i} and θ_{i} corresponding to the vertical and torsional modes, respectively.

Initially, the analysis has been performed for static loading with subsequent extension for the cyclic loading by using appropriate parameters for simulating the degradation of soil strength and stiffness (under cyclic loading), and the influence of the cyclic loading parameters. First, the static axial and torsional loadings are analyzed separately, followed by a coupled analysis to arrive at specific solutions. The governing differential equations for the static torsion (Equation 4) and static axial load (Equation 5) are given as (Basack and Sen,

Where, θ is the angle of twist; ρ is the vertical displacement; _{p} is the polar moment of inertia of pile cross section; _{p} is the modulus of rigidity of the pile; _{p} is the Young's modulus of the pile; _{p}, _{p} and pile head torque (_{t}). The correlations are then compiled together (in a matrix form) and the resultant matrix is given by:

Where, [CM] is a coefficient matrix of order (_{i} and _{i} and

Where, _{s} is the soil secant modulus. Using the provided correlation (Equation 7), the Equation (6) is modified to:

Where, [_{p}, _{s}, _{p}, and _{t}. The initial values of the unknown horizontal torsional interface shear stress (_{u}) for each element. The elements are assumed to have slipped if the value of _{u}. The initial value of _{u} and appropriate adjustments are made to the initial values of _{s} to incorporate the soil non-linearity. The entire computation procedure is then repeated for the rest of the non-slipped elements until the desired convergence is achieved.

The governing differential equation for the static axial load is solved by using a similar procedure as described above i.e., by utilizing the finite difference technique and a correlation between ρ and

Addition of interface stresses in coupled analysis.

The resultant interface shear stress is then compared with the ultimate shear stress and the values of

The analysis for the torsional cyclic loading has been performed by using a quasi-static method with a peak torsion of

Where, _{r} is a datum loading rate, γ_{c}is the peak nodal shear strain and

The post cyclic axial pile capacity is then evaluated, based on the degraded values of the shaft and end bearing resistance, using the following expression:

Where, _{ui} is the elemental ultimate soil strength, σ_{bu} is the ultimate base restraint and _{p} is the self-weight of pile.

Finally, the pile degradation factor (_{p}) is evaluated. It is defined as the ratio of the post-cyclic to static axial pile capacities (Equation 12):

where,

Flowchart for the FORTRAN program PTCYC.

The proposed solutions have been validated using the field test results available in literature. The field investigation on piles under composite torsional cyclic and axial static loading is extremely difficult and expensive. In absence of such research work, the authors have used the available laboratory and field test data to validate their numerical model. Stoll (

Comparison of the results computed using BEM with Guo and Randolph (

Stuedlein et al. (

Comparison of the results computed using BEM with field test data of Stuedlein et al. (

The present boundary element model has been used to predict the response of a prototype vertical concrete floating pile embedded in soft clay subjected to a combined axial and cyclic torsional loading. _{t}_{u0}) is 0.4. The number of pile elements are fixed at 100 after conducting a sensitivity check (Basack and Nimbalkar,

Input parameters of soil and pile for the parametric study.

Soil | Unit weight (_{s}^{3} |
18 |

Unit cohesion at ground surface (_{u0} |
30 | |

Adhesion factor (α) | 0.8 | |

Friction angle ( |
0 | |

Poisson's ratio (_{s} |
0 | |

Initial tangent shear modulus at ground surface (_{st0} |
300 | |

Pile | Length ( |
15 |

Diameter (D), m | 0.5 | |

Unit weight (_{p}^{3} |
25 | |

Young's modulus (_{p} |
20 | |

Shear modulus (_{p} |
8.3 |

In the present study, the variation of the pile degradation factor (_{p}) with cyclic loading parameters namely, number of cycles (_{c}) has been investigated. The cyclic loading level _{c} is defined as the ratio of peak cyclic torsion to the static ultimate torsional pile capacity. The number of cycles, frequency and cyclic loading level have been varied in the range of 10–1,000 cycles, 5–30 c.p.m., and 15–30%, respectively. Moreover, the analysis has been conducted for two values of reduction factor (0.85 and 0.95) to investigate its influence on the soil-pile interactive performance.

_{p} with _{c}, respectively. It can be observed that the parameter _{p} decreases with an increase in the number of loading cycles. However, the trend shows an asymptotically stabilizing tendency, i.e., the parameter _{p} becomes almost constant after a certain number of loading cycles. This may be attributed to the exponential degradation of the soil strength and stiffness with _{p} increases with an increase in the loading frequency. This is because the strength and stiffness of the soil increases logarithmically with _{p}) decreases with an increase in _{c} following a curvilinear pattern with an increasing slope. This observation is reasonable because an increase in the value of the torsional cyclic amplitude or cyclic loading level is likely to cause the failure of more number of elements (which is initiated by the rapid yielding of soil adjacent to the soil-pile interface) (Basack, _{f} shows insignificant effect on the soil-pile interactive performance.

Variation of pile degradation factor with: _{C}.

Normalized interface shear stress profile of pile for static and post-cyclic condition.

The present study attempts to investigate the effect of the cyclic loading parameters namely, frequency, number of cycles and the amplitude, on the axial load carrying capacity of the pile foundation subjected to axial and torsional cyclic load. The results from the present study show that the axial loading capacity of the pile decreases with an increase in the number of loading cycles up to a particular number of cycles, beyond which the capacity becomes constant. Moreover, the pile capacity decreases with an increase in the amplitude of the cyclic torsional loading or the cyclic loading level. These findings can be used to predict the

Another practical aspect of the present study is the proper assessment of the

In the present study, a numerical solution based on the boundary element modeling has been developed for a single floating pile subjected to combined axial and torsional cyclic loads. The numerical model is successfully calibrated using appropriate values of the key input parameters and is validated against the field data published in the literature. The validation of the computed results with available field data exhibits the accuracy of the proposed solution. The results of numerical analysis indicate that the cyclic loading parameters, viz. number of load cycles, frequency, and cyclic loading level, significantly influence the degradation of the axial pile capacity due to torsional cyclic loading. Moreover, the interface shear stress has been found to decrease in a curvilinear pattern from a maximum value at the ground surface to a minimum value at the pile base. Furthermore, the proposed numerical solution can be used to evaluate the post-cyclic factor of safety relevant to the ultimate pile capacity. Thus, the results of the present parametric studies (conducted to investigate influence of key design parameters) can be utilized for formulating the design criteria for pile subjected to axial and torsional cyclic loads.

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The Authors acknowledge the in-kind support from the School of Civil and Environmental Engineering, University of Technology, Sydney, Australia. The assistance received from Mr. Sankhasubhra Sen, former postgraduate student of Bengal Engineering and Science University, India in developing the computer program is acknowledged as well.

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δ Height of pile elements (m)

θ_{i} Twist on the i^{th} element (

λ_{r} Datum loading rate (Dimensionless)

ρ_{i} Vertical displacement at the i^{th} element (m)

τ Shear stress (N/m^{2})

τ_{b} Shear stress at the base of the pile (N/m^{2})

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τ_{u} Peak shear stress (N/m^{2})

σ_{b} Normal stress at the base of the pile (N/m^{2})

σ_{bu} Ultimate base restraint (N/m^{2})

γ Shear strain (Dimensionless)

γ_{c} Peak nodal shear strain (Dimensionless)

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