Response of pile groups subjected to lateral loads

An approach is presented for the prediction of linear and nonlinear load–deformation behaviour of laterally loaded pile groups. The individual pile response is obtained by the conventional p–y curve technique, while group interaction is modelled using the Mindlin's solution. Good agreement is observed when comparing the present method of analysis with the commonly used interaction factor approach for the computation of response of pile groups embedded in a homogenous, isotropic elastic half-space. The predicted pile group behaviour also compares favourably with existing field data on laterally loaded pile groups in soft clay.     

Three‐dimensional nonlinear finite element analysis of pile groups in saturated porous media using a new transmitting boundary

The dynamic behaviour of pile groups subjected to an earthquake base shaking is analysed. An analysis is formulated in the time domain and the effects of material nonlinearity of soil, pile–soil–pile kinematic interaction and the superstructure–foundation inertial interaction on seismic response are investigated. Prediction of response of pile group–soil system during a large earthquake requires consideration of various aspects such as the nonlinear and elasto‐plastic behaviour of soil, pore water pressure generation in soil, radiation of energy away from the pile, etc. A fully explicit dynamic finite element scheme is developed for saturated porous media, based on the extension of the original formulation by Biot having solid displacement (u) and relative fluid displacement (w) as primary variables (u–w formulation). All linear relative fluid acceleration terms are included in this formulation. A new three‐dimensional transmitting boundary that was developed in cartesian co‐ordinate system for dynamic response analysis of fluid‐saturated porous media is implemented to avoid wave reflections towards the structure. In contrast to traditional methods, this boundary is able to absorb surface waves as well as body waves. The pile–soil interaction problem is analysed and it is shown that the results from the fully coupled procedure, using the advanced transmitting boundary, compare reasonably well with centrifuge data. Copyright © 2007 John Wiley & Sons, Ltd.     

Selection of p–y curves for the design of single laterally loaded piles

Two-dimensional finite element analysis has been used to find load–transfer relationships for translation of an infinitely long pile through undrained soil for a variety of soil-constitutive models. It has been shown that these load–transfer curves can be used as p–y curves in the analysis of single piles undergoing lateral pile head loading in undrained soils with non-linear stress–strain laws. Lateral pile response deduced from 2-D analysis input to the subgrade reaction method has been compared to the behaviour of a single pile analysed using three-dimensional finite element analysis. Good agreement between the two methods for non-linear soils suggests that the 2-D analysis may form a useful design method for calculation of p–y curves. Copyright © 1999 John Wiley & Sons, Ltd.     

Seismic Response Analysis of Pile Foundations

A quasi-3D continuum method is presented for the dynamic nonlinear effective stress analysis of pile foundation under earthquake excitation. The method was validated using data from centrifuge tests on single piles and pile groups in liquefiable soils conducted at the University of California at Davis. Some results from this validation studies are presented. The API approach to pile response using p–y curves was evaluated using the quasi-3D method and the results from simulated earthquake tests on a model pile in a centrifuge. The recommended API stiffnesses appear to be much too high for seismic response analysis under strong shaking, but give very good estimates of elastic response.     

Nonlinear three-dimensional analysis of pile group supported columns considering pile cap flexibility

A numerical method that takes into account the coupling between the rigidities of the piles, the cap, and the column has been developed for analyzing the response of pile group supported columns. Special attention is given to consideration of pile cap flexibility. A load transfer approach using t–z/q–z and p–y curves is used for the analysis of single piles. The finite element technique is used to combine the pile stiffness with the stiffness of the cap and column. The numerical method developed has been verified by comparing the results with other numerical methods for pile groups. Through comparative studies, it has been found that the maximum load on the individual piles in a group is highly influenced by pile cap flexibility. The prediction of the lateral loads and bending moments in the pile cap is much more conservative in the present analysis than in FBPier 3.0 and shows a definitely larger lateral load and bending moment for various cap thicknesses.     

A modulus‐multiplier approach for non‐linear analysis of laterally loaded pile groups

A modulus‐multiplier approach, which applies a reduction factor to the modulus of single pile p–y curves to account for the group effect, is presented for analysing the response of each individual pile in a laterally loaded pile group with any geometric arrangement based on non‐linear pile–soil–pile interaction. The pile–soil–pile interaction is conducted using a 3D non‐linear finite element approach. The interaction effect between piles under various loading directions is investigated in this paper. Group effects can be neglected at a pile spacing of 9 times the pile diameter for piles along the direction of the lateral load and at a pile spacing of 6 times the pile diameter for piles normal to the direction of loading. The modulus multipliers for a pair of piles are developed as a function of pile spacing for departure angle of 0, 90, and 180sup>/sup> with respect to the loading direction. The procedure proposed for computing the response of any individual pile within a pile group is verified using two well‐documented full‐scale pile load tests. Copyright © 2006 John Wiley & Sons, Ltd.     

Transverse seismic kinematics of single piles by a modified Vlasov model

Within the framework of soil–pile interaction, a novel displacement scheme for the transverse kinematic response of single piles to vertically propagating S waves is proposed on the basis of the modified Vlasov foundation model. The displacement model contains a displacement function along the pile axis and an attenuation function along the radial direction. The governing equations and boundary conditions of the two undetermined functions are obtained in a coupled form by using Hamilton's principle. An iterative algorithm is adopted to decouple and solve the two unknown functions. In light of the governing equation of the pile kinematics, a mechanical model is proposed to evaluate the present method on a physical basis considering material damping. The coefficient of the equivalent Winkler spring is derived explicitly as function of the displacement decay parameter γ and soil Poisson's ratio. A parametric study is performed to investigate the effects of the soil–pile system properties on the kinematic response of single piles. The results show that the dimensionless pile length controls the transverse kinematics of piles. In terms of the theory of beams on elastic foundation, the classification limits of the dimensionless pile length may be π ∕ 4 and π, respectively. Copyright © 2014 John Wiley & Sons, Ltd.     

Explicit extension of the p–y method to pile groups in cohesive soils

Although simplified numerical methods are reliable for evaluating the response of a single pile under horizontal load, their application is questionable for assessing the response of pile groups. The notion of “p–y” curves has been considered with the aim of establishing a transformation relationship able to provide the “p_G–y_G” curves of soil resistance around a pile in a group from the well-known curves of soil resistance around the single pile.This transformation extends the applicability of the “p–y” method to pile groups, without the need for time consuming numerical computations, rendering the proposed method efficient and attractive. Comparative examples demonstrated the applicability and the effectiveness of the proposed method. In addition, the method can be straightforwardly extended to account for varying soil resistance, according to the particular location of a pile in a group. It can therefore be used to estimate accurately force and bending moment distributions along the characteristic piles of a group, which are required for the efficient design of foundations.     

Analysis of vertically loaded pile groups

An approach is presented for the analysis of linear and non-linear responses of vertically loaded pile groups. The soil behaviour of individual piles in modelled using load-transfer curves and the pile-soil-pile interaction is determined based on Mindlin's solution. Good agreement between the present method of analysis and the rigorous boundary integral method is observed for the computation of the response of pile groups embedded in a homogeneous, isotropic elastic half-space. The computed non-linear response of pile groups compares favourably with measured results from field load tests.     

Response evaluation of axially loaded fixed‐head pile groups in clayey soils

The aim of this paper is to investigate the interaction between the piles in a group with a rigid head and correlate the response of a group of piles to that of a single pile. For this purpose, a computationally intensive study using 3‐D nonlinear numerical analysis was carried out for different pile group arrangements in clayey soils. The responses of the groups of piles were compared with that of a single pile and the variation of the settlement amplification factor R_a was then quantified. The influence of the number of piles, the spacing, and the settlement level on the group response is discussed. A previously proposed relationship for predicting the response of a pile group, based on its configuration and the response of a single pile, has been modified to extend its applicability for any pile spacing. The modified relationship provides a reasonable prediction for various group configurations in clayey soils. Copyright © 2009 John Wiley & Sons, Ltd.     

Formulation of the axisymmetric CPDI with application to pile driving in sand

In foundation engineering practice, pile driving is often used as an efficient method to install piles. While large distortions take place along the pile shaft during the installation, the zone around the pile toe experiences compression. In an attempt to fully understand the build up of resistance when driving piles, it is desirable to model the driving process and the corresponding soil behaviour. The non-linear dynamic analysis of this problem is challenging, given the large deformation that develops together with the associated changes in soil properties. Some numerical methods offer the possibility of handling large material movements by utilising Lagrangian and Eulerian frames of references. However, few of these methods are capable of tracing the material displacement, such as the Material Point Method (MPM). Early implementation of MPM assumes that the mass is concentrated at the material points, which causes noise in the solution. Later implementations assign a spatial domain to the material points to mitigate the grid crossing error. The Convected Particle Domain Interpolation (CPDI) is one such implementation.This paper extends the two-dimensional CPDI formulation for an axisymmetric problem where a pile is driven into sand that is modelled as a hypoplastic model. The extended formulation is tested, validated and compared to that for the case of the two-dimensional plane-strain within the framework of the method of manufactured solution. The hammer blows on the pile are represented by a periodic forcing function. In contrast to earlier studies on pile installation using advanced models, deep penetration is achieved in the present analysis. A non-regular distribution for the particle domains is suggested to avoid unnecessary computation. A frictional contact algorithm is introduced to describe the pile–soil interaction.     

Explicit extension of the p–y method to pile groups in sandy soils

The method of “p–y” curves has been extensively used, in conjunction with simplified numerical methods, for the design and response evaluation of single piles. However, a straightforward application of the method to assess the response of pile groups is questionable when the group effect is disregarded. For this reason, the notion of p-multipliers has been therefore introduced to modify the “p–y” curves and account for pile group effect. The values proposed for p-multipliers result from pile group tests and are limited to the commonly applied spacing of 3.0 D and layout less than 3 × 3, restricting the applicability of the method to specific cases. With the aim of extending the applicability of the “p–y” method to pile groups, the authors have already proposed a methodology for estimating the “p _G–y _G” curves of soil resistance around a pile in a group for clayey soils. A complementary research allowing for the estimation of the “p _G–y _G” curves for sandy soils is presented in this paper. The well-known curves of soil resistance around the single pile in sandy soils are appropriately transformed to allow for the interaction effect between the piles in a group. Comparative examples validate the applicability and the effectiveness of the proposed method. In addition, the method can be straightforwardly extended to account for varying soil resistance, according to the particular location of a pile in a group. It can therefore be used in a most accurate manner in estimating the distribution of forces and bending moments along the characteristic piles of a group and therefore to design a pile foundation more accurately.     

Response of rigid piles during passive dragging

This paper develops a three‐layer model and elastic solutions to capture nonlinear response of rigid, passive piles in sliding soil. Elastic solutions are obtained for an equivalent force per unit length p_s of the soil movement. They are repeated for a series of linearly increasing p_s (with depth) to yield the nonlinear response. The parameters underpinning the model are determined against pertinent numerical solutions and model tests on passive free‐head and capped piles. The solutions are presented in non‐dimensional charts and elaborated through three examples. The study reveals the following:

• On‐pile pressure in rotationally restrained, sliding layer reduces by a factor α, which resembles the p‐multiplier for a laterally loaded, capped pile, but for its increase with vertical loading (embankment surcharge), and stiffness of underlying stiff layer: α = 0.25 and 0.6 for a shallow, translating and rotating piles, respectively; α = 0.33–0.5 and 0.8–1.3 for a slide overlying a stiff layer concerning a uniform and a linearly increasing pressure, respectively; and α = 0.5–0.72 for moving clay under embankment loading.
• Ultimate state is well defined using the ratio of passive earth pressure coefficient over that of active earth pressure. The subgrade modulus for a large soil movement may be scaled from model tests.
• The normalised rotational stiffness is equal to 0.1–0.15 for the capped piles, which increases the pile displacement with depth.

The three‐layer model solutions well predict nonlinear response of capped piles subjected to passive loading, which may be used for pertinent design. Copyright © 2016 John Wiley & Sons, Ltd.     

Numerical study of group effects for pile groups in sands

The OpenSees finite element framework was used to simulate the response of 3×3 and 4×3 pile groups founded in loose and medium dense sands. Several numerical static pushover tests were conducted to investigate the interaction effects for pile groups. The results were then compared with those from centrifuge study. It is shown that our simulations can predict the behaviour of pile groups with good accuracy. Special attention was given to the three dimensional distribution of bending moment. It was found that bending moment develops in the plane perpendicular to the loading direction. In addition, bending moment data from simulations was used to derive p–y curves for individual piles, which were used to illustrate different behaviour of individual piles in the same group. Copyright © 2003 John Wiley & Sons, Ltd.     

Visco‐elastic load transfer models for axially loaded piles

Viscoelastic or creep behaviour can have a significant influence on the load transfer (t–z) response at the pile–soil interface, and thus on the pile load settlement relationship. Many experimental and theoretical models for pile load transfer behaviour have been presented. However, none of these has led to a closed‐form expression which captures both non‐linearity and viscoelastic behaviour of the soil. In this paper, non‐linear viscoelastic shaft and base load transfer (t–z) models are presented, based on integration of a generalized viscoelastic stress–strain model for the soil. The resulting shaft model is verified through published field and laboratory test data. With these models, the previous closed‐form solutions evolved for a pile in a non‐homogeneous media have been readily extended to account for visco‐elastic response. For 1‐step loading case, the closed‐form predictions have been verified extensively with previous more rigorous numerical analysis, and with the new GASPILE program analysis. Parametric studies on two kinds of commonly encountered loading: step loading, ramp (linear increase followed by sustained) loading have been performed. Two examples of the prediction of the effects of creep on the load settlement relationship by the solutions and the program GASPILE, have been presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Nonlinear response of laterally loaded piles and pile groups

In spite of extensive studies on laterally loaded piles carried out over years, none of them offers an expedite approach as to gaining the nonlinear response and its associated depth of mobilization of limiting force along each pile in a group. To serve such a need, elastic–plastic solutions for free‐head, laterally loaded piles were developed recently by the author. They allow the response to be readily computed from elastic state right up to failure, by assigning a series of slip depths, and a limiting force profile. In this paper, equivalent solutions for fixed‐head (FixH) single piles were developed. They are subsequently extended to cater for response of pile groups by incorporating p‐multipliers. The newly established solutions were substantiated by existing numerical solutions for piles and pile groups. They offer satisfactory prediction of the nonlinear response of all the 6 single piles and 24 pile groups investigated so far after properly considering the impact of semi‐FixH restraints. They also offer the extent to ultimate state of pile groups via the evaluated slip depths. The study allows ad hoc guidelines to be established for determining input parameters for the solutions. The solutions are tailored for routine prediction of the nonlinear interaction of laterally loaded FixH piles and capped pile groups. Copyright © 2008 John Wiley & Sons, Ltd.     

Load–Deflection response of transversely isotropic piles under lateral loads

In general, pile materials are assumed to be isotropic during the analysis of the load–deflection response of piles under lateral loads. However, commonly used materials such as reinforced concrete and timber as well as potentially promising new pile materials such as fiber reinforced polymers are typically transversely isotropic materials. Experimental studies have shown that transversely isotropic materials have a high ratio of section longitudinal modulus to the section in‐plane shear modulus (E_zz/G_xz) compared to the value for isotropic materials. The high modulus ratio leads to a more significant shear deformation effect in beam bending. To account for the shear deformation effect, the Timoshenko Beam Theory has been adopted in deriving the solutions for the load–deflection response of transversely isotropic piles under lateral loads instead of the Classical (Euler–Bernoulli) Beam Theory. The load–deflection responses depend on the shear effect coefficient, the lateral soil resistance, the embedment ratio, and the boundary conditions. The deflection of the pile, if the shear deformation effect is considered, is always larger than if it is neglected. Copyright © 2000 John Wiley & Sons, Ltd.     

A simplified analysis method for piled raft foundations in non‐homogeneous soils

A simplified method of numerical analysis based on elasticity theory has been developed for the analysis of axially and laterally loaded piled raft foundations embedded in non‐homogeneous soils and incorporated into a computer program “PRAB”. In this method, a hybrid model is employed in which the flexible raft is modelled as thin plates and the piles as elastic beams and the soil is treated as springs. The interactions between structural members, pile–soil–pile, pile–soil–raft and raft–soil–raft interactions, are approximated based on Mindlin's solutions for both vertical and lateral forces with consideration of non‐homogeneous soils. The validity of the proposed method is verified through comparisons with some published solutions for single piles, pile groups and capped pile groups in non‐homogeneous soils. Thereafter, the solutions from this approach for the analysis of axially and laterally loaded 4‐pile pile groups and 4‐pile piled rafts embedded in finite homogeneous and non‐homogeneous soil layers are compared with those from three‐dimensional finite element analysis. Good agreement between the present approach and the more rigorous finite element approach is demonstrated. Copyright © 2002 John Wiley & Sons, Ltd.     

On the response prediction of horizontally loaded fixed-head pile groups in sands

The aim of this paper is to investigate the behavior of laterally loaded pile groups in sands with a rigid head and correlate the response of a pile group it to that of a single pile. For this purpose, a computationally intensive study using 3-D nonlinear numerical analysis was carried out for different pile group arrangements in sandy soils. The responses of the pile groups were compared to that of the single pile and the variation of the displacement amplification factor R_a was then quantified. The influence of the number of piles, the spacing, and the deflection level on the group response is discussed. A relationship for predicting the response of a pile group, based on its configuration and the response of a single pile, has been formulated allowing also for soil shear strength which was found to affect the group response. The relationship provides a reasonable prediction for various group configurations in sandy soils.     

The P-y Response of Laterally Loaded Flexible Piles in Residual Soil

This paper describes the main features related to lateral displacements with depth after successive lateral loading–unloading cycles applied to the top of reinforced-concrete flexible bored piles embedded in naturally bonded residual soil. The bored piles under study have a cylindrical shape, with 0.40-m in diameter and 8.0-m in length. Both bored piles types (P1 and P2) include an embedded steel pipe section in their center as longitudinal steel reinforcements: pile type P1 has another 16 steel rods as steel reinforcement to concrete while pile type P2 has no further steel reinforcement. Pile type P1 has three times as much stiffness (EI) and four and a half times the plastic moment (M_y) than pile type P2. A similar load–displacement performance was observed at initial loads as for small displacements of both piles. At this initial loading stage, the response of the reinforced concrete piles is a function of the soil characteristics and of a linear elastic pile deformation. During this stage, piles can even be understood as probes for evaluating soil reactions. For larger horizontal displacements, after the concrete section starts undergoing large deformations, approaching the ultimate bending moment, pile behavior and consequently the load–displacement relation starts to diverge for both piles. For pile P1 the values of relevant lateral displacements are extended to about 2.5-m in depth, while for pile P2 lateral displacements are mostly constrained to about 2.0-m in depth. Measurements of horizontal displacements of pile P1 against depth recorded with a slope indicator show that, after unloading, lateral loads at distinct stages (small and near failure loads), exhibits a much higher elastic phase of the system response. An analytical fitting model of soil reaction is proposed based on the measured displacements from slope indicator. The integration of a continuous model proposed for the soil reaction agrees fairly well with the measured displacements up to moments close to plastic limit. Results of load–displacement show that the stiffer pile (P1) was able to mobilize twice as much lateral load compared to pile P2 for a service limit displacement of about 20 mm. The paper shows results that enable the isolation of the structural variable through real scale pile load tests, thus granting understanding of its importance and enabling its quantitative visualization in examples of piles embedded in residual soil sites.