Dynamic p-y curves for vertical and batter pile groups in liquefied sand

Zhang Jian, Li Yurun, Rong Xian and Liang Yan

1.School of Civil and Transportation Engineering, Hebei University of Technology, Tianjin 300401, China

2.Campus Construction and Management, Hebei University of Technology, Tianjin 300401, China

Abstract: In this study, centrifuge model tests of vertical and batter pile groups in liquefied sand were conducted on a centrifuge shaking table.The dynamic p-y curves for these pile groups before and during sand liquefaction were obtained from calculations based on test data.The results confirm that liquefaction contributes to a reduction in the energy consumption of pile foundations, with the degradation effect being more pronounced for batter pile groups.At shallow depths, the difference in the backbone gradients of the p-y curves after liquefaction for vertical and batter pile groups indicates that the lateral stiffness of a batter pile group is greater than that of a vertical pile group.As shaking intensity increases, the lateral stiffness of a vertical pile group increases with depth during the late stage of sand liquefaction.However, the lateral stiffness of a batter pile group during liquefaction does not vary with depth.The results of this study provide a reference for the seismic design of vertical and batter pile groups in liquefied soil.

Keywords: soil liquefaction; pile-soil interaction; p-y curve; centrifuge shaking table

1 Introduction

Pile foundations are deep foundations that are used in numerous engineering projects.Many pile foundations have been used in areas prone to liquefaction.Postearthquake site investigations have revealed that soil liquefaction is an important cause of the structural damage induced by an earthquake.In particular, soil liquefaction damages the pile foundation of a structure(Bhattacharyaet al., 2011; Hamada and O′Rourke,1992; Lombardi and Bhattacharya, 2014; Mylonakiset al., 2006; TOKIMATSU and ASAKA, 1998).Therefore,the dynamic response of liquefied and non-liquefied soils has been studied by researchers (Lianget al., 2017;Zhuanget al., 2018).The behavior of pile foundations during liquefaction has been extensively investigated(Abdoun and Dobry, 2002; Brandenberget al., 2007;Dashet al., 2010; Haigh and Gopal Madabhushi, 2011;Liuet al., 2018; Suet al., 2016; Tanget al., 2014; Wanget al., 2017; Yasudaet al., 2000).

Thep-ycurve method has often been used to analyze and design piles that are subjected to lateral loads.The method itself has been developed based on the beamon-non-linear-Winkler-foundation (BNWF) model,which is a simplified model used to analyze pile-soil interaction.In the BNWF model, pile-soil contact is simplified as ap-yspring, whereprepresents the pilesoil interaction force (per unit length of the pile) andyrepresents pile-soil relative displacement.In the 1970s and 1980s, the empirical relationship betweenpandywas obtained through a series of tests on small-diameter steel pipes and expressed in the form ofp-ycurves(Matlock, 1970; Murchison and O′Neill, 1986; Reeseet al., 1974).According to this empirical relationship, the American Petroleum Institute (API) proposed a variety of construction methods for thep-ycurve for various soil types (API, 2000).However, the API specifications do not provide a clear explanation for thep-ycurves of liquefied soil.Therefore, researchers have begun to conduct detailed research on the lateral behavior of pile foundations under non-liquefied and liquefied soil conditions, through static tests under gravity and centrifugal gravity.Subsequently, variousp-ycurve correction methods have been proposed (Elkasabgy and El Naggar, 2019; Franke and Rollins, 2013; Jianget al.,2020; Konget al., 2019; Yanget al., 2019).

A review of existing literature suggests that research onp-ycurves for different soil types under static reciprocating loads has received more attention, whereas research on thep-ycurve of pile foundations under dynamic loads has been limited.In particular, studieson thep-ycurves for pile foundations in liquefied soils are rare.Lianget al.(2018) proposed a new method for constructing quasi-staticp-ycurves, which can be used to analyze pile-soil interaction under cyclic loads.Lombardiet al.(2017) proposed a method to calculate thep-ycurves based on a stress-strain model.This method was identified as a reasonable one for constructingp-ycurves for liquefied soils.Elsawyet al.(2019)comprehensively studied dynamicp-ycurves for helical piles in dry sand.They discussed the applicability of the curve fitting method and the interpretation of dynamicp-ycurves.In their work, thep-ycurves for liquefied soils were derived from the stress-strain relationship of soils.To date, few studies have examined the backcalculation of relevant data through dynamic tests to obtainp-ycurves, especially for batter piles.

Batter piles are used when the lateral bearing capacity of vertical piles does not meet project requirements.However, in seismically active regions,batter piles are not recommended according to certain building codes (AFPS, 1990; Eurocode 8, 2004).The main disadvantages of batter piles are the following:stress on the pile increases at the junction of the pile cap and the pile, the residual bending moment caused by soil settlement increases, and bending capacity decreases(Gerolymoset al., 2008).However, batter piles are still commonly used in certain projects, especially in ports or on offshore platforms (Medinaet al., 2015; Mondal and Rai, 2008; Poulos and Harry, 2006; Rajeswari and Sarkar, 2020).Moreover, some studies have reported on the positive performances of batter piles (Berrillet al., 2001; Harn, 2004; Liet al., 2016).Caoet al.(2018)proposed a newp-ycurve for laterally loaded batter piles,as well as a method to determine parameters in thep-ycurve.Ashouret al.(2020) usedp-ycurves to predict the dynamic response of batter piles in sand.Their technique focuses on the magnitude and direction of the inclination angle of the pile.We can understand, therefore, how the study of thep-ycurve of batter piles in dry sand has begun to attract attention.However, studies have not been conducted on the dynamicp-ycurve of batter pile groups during sand liquefaction.Therefore, this study aims to explore the differences in thep-ycurve laws of vertical and batter pile groups in liquefied soil, and to reveal the differences in the pile-soil interaction between different pile types in liquefied soil.The results from this study are expected to provide references for the seismic design of different pile types for future projects.

2 Experimental design

To study the pile-soil interaction of vertical and batter pile groups in liquefied sand under horizontal dynamic loads, we conducted two sets of centrifuge shaking table tests, which took place at Zhejiang University(Fig.1).The centrifuge shaking table included a onedimensional hydraulic shaker, jointly developed by Zhejiang University and Japan′s Solution Company.A laminar box with an inner length, width, and depth of 730 mm, 330 mm, and 420 mm, respectively, was used in this test.The laminar shear box included a stack of 12 metal laminar frames.Miniature rollers were placed between each pair of layers of frames.The influence of the boundary effect on the test results was thus reduced.For detailed information on the parameters of the testing equipment , the reader is referred to Zhouet al.(2018).

Fig.1 Geotechnical centrifuge at Zhejiang University.Photograph taken by J.Zhang

Two sinusoidal waves with the same frequency (i.e.,1 Hz at the prototype scale) but different amplitudes (i.e.,0.05 g and 0.1 g) were selected as the input motions for the tests.The accelerations recorded on the shaking table are presented in Fig.2.The air pluviation method was adopted to prepare cohesionless soil samples; pluviation height was controlled to achieve a designed relative density of 50%.The basic procedure involved in the air pluviation method starts with calculating the weight of each layer of sand and placing the sand into an air pluviation container.Next, the air pluviation container was raised to a specified height and sand spreading was initiated.A uniform speed was maintained during the air pluviation process.After the spreading of each layer of sand was completed, sensors were placed at designated locations.These steps were repeated until the model was completed.Table 1 represents the physical and mechanical properties of the soil used as the ground in the experiments.A detailed layout of sensors in the foundation soil model is shown in Fig.3.

Fig.2 Shaking table input motions (A1)

Fig.3 Schematic diagram of the experimental setup for the vertical and batter pile groups (unit: mm)

Centrifugal acceleration was 50 g.Table 2 enumerates the related scaling factors.Considering the influence of pore fluid permeability on the tests, the ground was saturated with silicone oil, which has a dynamic viscosity 50 times greater than that of water.A 2×2 arrangement was used for the vertical and batter pilegroup models, and the piles and caps were created using a 6061 aluminum alloy, with an elastic modulus of 68.9 GPa.The pile models had a total length of 500 mm,a thickness of 2 mm, and an outer diameter of 20 mm.The angle of inclination for the batter pile group was 10°.Nine pairs of equally spaced strain gauges were glued to the surfaces of two piles on the same side in each pile group.The model pile was installed at the bottom of the laminar box using the wish-in-place approach.The pile toes were rigidly attached to the base.The pile group and pile cap were then rigidly connected.One pile each from the vertical pile group (labelled VP) and batter pile group (labelled BP) was selected for analysis.

Table 1 Physical and mechanical properties of sand

Table 2 Scaling laws for the model to the prototype

3 Calculation principle of dynamic p-y curves

3.1 Interaction force between the piles and soil

Bending moments are obtained through strain values recorded by the equally spaced strain gauges glued to pile surfaces.According to basic beam theory, the strain is proportional to the curvature and changes linearly with the distance from the neutral axis.The relationship between strain and curvature is as follows:

whereεis the strain,yis the distance from the surface of the pile to the neutral axis, andρis the curvature.

Assuming that the beam undergoes bendingdeformation, the relationship between the bending moment and the curvature is as follows:

whereMis the bending moment,Eis the elastic modulus,andIzis the moment of inertia.

By combining Eqs.(1) and (2), the relationship between the bending moment and strain can be obtained as follows:

In this study, the soil-pile interaction force and pile deflection were obtained by differentiating and integrating the bending moment, respectively.A similarmethod was used for the construction of thep-ycurves(Jeanjean, 2009; Wilson, 1998).To obtain the function of the bending moment with respect to depth, we fitted the discrete bending moment values at different depths.Considering the accuracy of the fitting curve and the influence of high-frequency noise on these fitting results, a cubic polynomial function was used to obtain a continuous bending curve.Figure 4 represents the continuous bending curves for the vertical and batter pile groups at different shaking intensities.It can be seen that the fitted curves are ideal.The bending moment of the pile at a certain depthzcan be calculated as follows:

whereMis the bending moment of the pile;zis the soil depth; anda1,a2,a3, anda4are the fitting parameters.

According to classic elastic foundation beam theory,the lateral interaction force can be obtained as follows:

wherepis the lateral interaction force,zis soil depth,andMis the bending moment of the pile.

3.2 Relative pile-soil horizontal displacement

Pile deflection is obtained from double integration of the bending moment.Two new parameters would be generated after the double integration, and the function can be solved once the boundary conditions are introduced.Pile deflection at different depths could be solved using Eq.(3), with the bending moment expressed in Eq.(2).

whereyis the lateral deflection of the pile,zis soil depth,Mis the bending moment, andEIrepresents the flexural rigidity of the pile.

Lateral soil displacement is the result of double integration of accelerations recorded by the accelerometers.The difference between pile deflection and soil displacement is considered as relative pile-soil horizontal displacement.

The relationship between the soil-pile interaction force and relative pile-soil horizontal displacement at any given time can be calculated using the abovementioned process.Owing to the large dataset obtained during the experiments, a program was designed to efficiently calculate the dynamicp-ycurves.Figure 5 presents a flowchart of the calculation process that was utilized by this program.

4 Analyses of dynamic p-y curves for pile groups

Dynamicp-ycurves were investigated for vertical and batter pile groups before and after sand liquefaction.However, when the pile foundation is subjected to dynamic loads, as is the case during earthquakes,the overall stress and deformation of the pile will be considerably influenced by the surrounding soil.Especially when sand is saturated, the soil is liquefied due to the action of seismic load, which leads to a considerable change in deformation of the soil and the constraint imposed by the soil on the pile.Therefore,before discussing thep-ycurves, we must discuss the dynamic response of the soil.

4.1 Soil response

Figure 6 demonstrates soil response at various depths.Based on this figure, when the shaking intensity is 0.05 g, peak acceleration of the shallow soil is slightly higher than that of deep soil.The excess pore pressureratio (EPPR) at a depth of 3.5 m increases significantly,but its maximum value only reaches about 0.5, which indicates that the soil does not undergo liquefaction.As shaking intensity increases, the EPPR at various depths increases to a certain extent.In particular, the EPPR at a depth of 3.5 m reaches about 0.8, which suggests that shallow soil undergoes liquefaction after shaking begins.After the EPPR value reaches its maximum, the peak acceleration of the shallow soil (depth=3.5 m) decreases significantly.This also shows that soil liquefaction can alter the stiffness of the soil.In addition, regardless of shaking intensity, as the depth increases from 11 m to 16 m, the EPPR values gradually decrease.Furthermore,the EPPR values are between 0 and 0.5, an insufficient number to cause soil liquefaction.Therefore, there is no obvious change in the peak acceleration.The main reason for this phenomenon is that the pore water in deep soil is continuously discharged upward during the dynamic process, which causes excess pore pressure in the shallow soil to rapidly increase.When excess pore pressure equals the effective stress of the soil, soil liquefaction occurs.However, deep soil does not readily undergo liquefaction, because the pore water in it is continuously released upward, and the effective stress of soil increases with depth.

Fig.4 Fitted curve of the bending moment along the instrumented pile at different shaking intensities

Fig.5 Flow chart for calculating dynamic p-y curves

The change in soil stiffness was analyzed based on the time history of acceleration and EPPR.The effect of soil deformation, caused by dynamic loads on pilesoil interaction needs to be investigated.Figure 7 shows the distribution of peak soil displacement at depth at different shaking intensities.When the shaking intensity was 0.05 g, the peak displacement of shallow soil (0.009 m) was slightly higher than that of deep soil(0.008 m), although the difference was small.When the shaking intensity was 0.1 g, peak soil displacement notably increased as burial depth decreased.Peak soil displacement near the ground surface reached 0.027 m,which represented an increase of approximately 70%when compared to the peak displacement (0.016 m)of deep soil.This trend, combined with the EPPR distribution trend shown in Fig.6, indicates that soil displacement will significantly increase after liquefaction.The foregoing analysis, which is based on the distribution of stiffness and the deformation of saturated soil before and after liquefaction, provides thebasis for the analysis of soil-pile interaction, which is reported in the following section.

Fig.6 Time history of acceleration and the excess pore water pressure ratio at various depths and for different shaking intensities

4.2 Dynamic p-y curves for vertical and batter pile groups in different cases

4.2.1 Vertical pile group

Figure 8 presents the dynamicp-ycurves for the vertical pile group in saturated sand at various depths when the shaking intensity was 0.05 g.The same relative pile-soil displacement was observed at all depths, except at a depth of 1 m, for which the relative displacement was the largest.Furthermore, a positive relationship was observed between the loop area of thep-ycurve and depth, indicating that the vertical pile group exhibited good energy dissipation.Moreover, thep-ycurves obtained at different depths during the tests exhibited different angles of inclination, which could reflect the occurrence of the “piles pushing soil” or “soil pushing piles” phenomena at different depths.At depths of 1 m,8.5 m, and 11 m, the “soil pushing piles” phenomenon was dominant, whereas the “piles pushing soil”phenomenon was dominant at depths of 3 m and 6 m.At a depth of 13.5 m, the “piles pushing soil” phenomenon was partially visible, although it was not obvious.This phenomenon occurred mainly because the deformation of soil and piles in deep ground was smaller than that in shallow ground.

Figure 9 presents the dynamicp-ycurves for the vertical pile group in saturated sand at various depths when the shaking intensity was 0.1 g.The relative pilesoil horizontal displacement was large in the shallow ground, but it decreased as soil depth increased.A part of the shallow saturated sand layer was liquefied during shaking, which resulted in significant relative displacement in shallow ground.However, relative displacement decreased as depth increased, owing to the absence of liquefaction in deeper ground.The loop area of thep-ycurves increased with depth, and good energy dissipation was observed in the deeper ground.

Another notable observation was that the peak value of the pile-soil interaction force in the saturated sand gradually increased with depth during shaking.This can be attributed to the fact that liquefaction caused by the shaking led to significant influence by the inertial force from the upper structure on the pile in deeper ground.

4.2.2 Batter pile group

Figure 10 presents the dynamicp-ycurves for the batter pile group in saturated sand at various depths when the shaking intensity was 0.05 g.During the experiments, as depth increased the area of thep-ycurve continuously decreased until it reached a minimum value at a depth of 6 m.Subsequently, the area of thep-ycurve began to increase with depth.The main reason for this was the variation of the lateral stiffness of the batter pile at different depths.As depth increased, the horizontal displacement of the soil gradually decreased.However,at a depth of 6 m, the relative pile-soil displacement appeared to be very large.It can be shown that the constraint imposed by the soil on the pile was greatest at a depth of 6 m.This increased lateral pile-soil stiffness as a whole.The area of thep-ycurves increased with soil depth, indicating that the batter pile group exhibited good energy dissipation in the deeper ground when the shaking intensity was 0.05 g.

Figure 11 presents the dynamicp-ycurves for the batter pile group at various depths when the shaking intensity was 0.1 g.Both the interaction force and relative pile-soil horizontal displacement were notably lower than those observed for the vertical pile group.This implies that the lateral dynamic response of the batter pile group was considerably reduced when the shaking intensity was 0.1 g.

4.3 Dynamic p-y curves for vertical and batter pile groups at different times

The discussion in the previous section focused on thep-ycurves during the entire dynamic process.To thoroughly analyze the interaction between the pile and soil, thep-ycurves at various specific times are discussed in this section.

To study the distribution and change trend of the dynamicp-ycurves for the vertical and batter pile groups at the same depth, EPPRs in two ground locations at specific time points during shaking were selected for further analysis.

Figure 12 presents the dynamicp-ycurves for the vertical and batter pile groups at a depth of 3.5 m under different EPPRs at two different shaking intensities.At a low shaking intensity (0.05 g), thep-ycurves for all pile groups did not show obvious changes as the EPPR increased, because the saturated sand was not yet liquefied.However, at a high shaking intensity (0.1 g),the EPPR increased faster with time and the saturated sand also underwent liquefaction within a shorter period.Furthermore, the areas of thep-ycurves for the vertical pile group were much larger than those of the batter pile group.This indicates that the energy dissipation performance of the vertical pile group was superior to that of the batter pile group during rapid soil liquefaction.

Fig.7 Soil peak displacement at various depths and different shaking intensities

Fig.8 Dynamic p-y curves for the vertical pile group at various depths when the shaking intensity was 0.05 g

Fig.9 Dynamic p-y curves for the vertical pile group at various depths when the shaking intensity was 0.1 g

Fig.10 Dynamic p-y curves for the batter pile group at various depths when the shaking intensity was 0.05 g

Fig.11 Dynamic p-y curves for the batter pile group at various depths when the shaking intensity was 0.1 g

Fig.12 Dynamic p-y curves for vertical and batter pile groups at a depth of 3.5 m at different shaking intensities

The analyses of the backbones of the dynamicp-ycurves (Fig.13) for both pile groups were included in the analyses of the EPPR.

Fig.13 Backbones of p-y curves for vertical and batter pile groups at a depth of 3.5 m

When the shaking intensity was 0.05 g, the backbone gradients of thep-ycurves for the vertical and batter pile groups did not present obvious differences.At EPPR values of 0.4 and 0.6, the backbones of thep-ycurves were almost identical, whereas they exhibited differences when the shaking intensity was 0.1 g.At a high shaking intensity (0.1 g), the slope of the backbones of thep-ycurves for both types of pile groups gradually decreased;this was especially evident for the vertical pile group.Thus, the significant development of the EPPR led to sand liquefaction.When sand liquefaction occurs, the constraint imposed by the sand on the pile declines sharply.However, the reduction in stiffness of the soil around the batter pile group was not as significant as that around the vertical pile group.

The results of our experiments indicate that the pile type and the EPPR had major influences on thep-ycurves, and that the EPPR at a certain time fluctuated considerably, depending on depth in the saturated sand.Thus, it is necessary to discuss thep-ycurves at different depths.

Figure 14 presents thep-ycurves for both types of pile groups at different depths at different shaking intensities (0.05 g and 0.1 g).In our experiments, when the shaking intensity was low (0.05 g), thep-ycurves at different depths exhibited similar patterns.Moreover,thep-ycurves for both types of pile groups demonstrated an increasing loop area with depth, indicating that theenergy dissipation of both types of pile groups improved as depth increased.When the shaking intensity was high(0.1 g), the EPPR exhibited variations with depth, which produced distinct differences between thep-ycurves for various depths.For instance, the lateral reaction force on the vertical pile group was far larger than that on the batter pile group, and this effect was more evident in the deeper ground.Moreover, the variation in the pile stiffness was reflected in the dynamicp-ycurve.A comparison of thep-ycurves for both types of pile groups, when the shaking intensity was 0.1 g yielded the conclusion that depth had a major impact on pile stiffness in the vertical pile group, especially in the final phase of shaking.The pile stiffness in the vertical pile group rose rapidly with depth, whereas it did not vary significantly with depth in the batter pile group, except at a depth of 3.5 m.

Fig.14 Dynamic p-y curves for vertical and batter pile groups at different depths and at different shaking intensities

4.4 Pile-soil interaction mechanism

Based on the foregoing analyses ofp-ycurves,we can assume that sand liquefaction has a significant influence on the lateral stiffness of different piles.To better reveal the impact of sand liquefaction on the pilesoil interaction mechanism, we first analyze the stress characteristics of different pile types.As shown in Fig.15,part of the horizontal load on the top of the batter pile will be decomposed into the pile′s axial force, which reduces the influence of the horizontal load on the bending of the pile.Moreover, the stress characteristics of the pile are closely related to the soil around the pile.According to the BNWF model (see Fig.15 (b)), the soil-pile interaction is simplified using spring elements.When the soil undergoes liquefaction, the spring rate is significantly weakened.This leads to an increase in therelative pile-soil horizontal displacement.

Fig.15 Pile-soil interaction model when the pile top is subjected to a lateral load: (a) vertical and batter piles; (b) the beam-onnon-linear-Winkler-foundation model

According to the aforementioned analysis, after liquefaction occurs in saturated sand, the lateral stiffness of batter piles is significantly higher than that of vertical piles.As the degree of liquefaction increases,the advantages of the lateral stiffness of the batter piles become more apparent.Therefore, during the seismic design of pile foundations in liquefied soil, appropriate adjustments should be made for the different stress characteristics of vertical and batter pile groups.

5 Conclusions

Two sets of dynamic centrifuge tests were conducted on pile groups, subjected to different shaking intensities in saturated sand.The lateral forces acting on the piles and the lateral pile displacement caused by the increments in excess pore water pressure at various depths were evaluated.The distributions of thep-ycurves at different times and depths were analyzed to evaluate the performance of the piles in liquefied soil under dynamic lateral loading.The main findings of the study are as follows:

1.The back-calculation program that was designed to obtain thep-ycurves using the bending moments and accelerations at different soil depths considerably improved data processing efficiency.

2.The energy dissipation in both the vertical and batter pile groups exhibited an overall increasing trend as soil depth increased.Energy dissipation was reduced as shaking intensity increased, suggesting that soil liquefaction had a significant effect on the energy dissipation of the pile foundation; this effect was more pronounced for the batter pile group.

3.At a depth of 3.5 m, thep-ycurves for the vertical and batter pile groups changed dramatically as shaking intensity increased.The backbone gradients of thep-ycurves for the vertical pile group exhibited an obvious reduction during shaking (a shaking intensity of 0.1 g),which was closely related to the change in the EPPR of the sand.However, the backbone gradients of thep-ycurves for the batter pile group did not exhibit this trend during shaking, indicating that the lateral pile stiffness of the batter pile group was higher than that of the vertical pile group during sand liquefaction.

4.Thep-ycurves for the pile groups at different depths displayed interesting features, as the EPPR of the sand increased during shaking.Small changes were observed in thep-ycurves for both types of pile groups when the shaking intensity was 0.05 g.However, when the shaking intensity was 0.1 g, the saturated sand gradually approached the liquefied state, owing to a sharp increase in the EPPR.Based on the correspondingp-ycurves, it can be concluded that the stiffness of the vertical pile group significantly increased with soil depth, whereas the stiffness of the batter pile group demonstrated only minor differences as depth increased.

5.The pile type had a notable influence on pile-soil interaction in liquefied soil.This influence was mainly determined by the stress characteristics of the different types of piles.Therefore, the seismic design process for pile foundations in liquefied soils should account for the different pile types.

This study provides a useful reference for the analysis of the dynamicp-ycurve laws for vertical and batter pile groups in liquefied soil.It should be noted here that the above findings are based on the conditions in effect during the tests.If the pile or soil parameters of the models change, these conclusions may not hold true.We have also conducted centrifuge model tests on the dynamic response characteristics of the same modelin dry sand.The dynamicp-ycurves in dry sand will be analyzed in detail in future research.

Acknowledgement

This work was supported by the National Natural Science Foundation of China (Grant No.51778207), the Project of Graduate Students′ Innovative Ability Training of Hebei Province (Grant No.CXZZBS2019041), and the Natural Science Foundation of Hebei Province(Grant No.E2018202107).