Effects of cement-enhanced soil on the ultimate lateral resistance of composite pile in clayey soil

Zhijun Yng ,Kexin Chen ,Xudong Fu ,Zhiyn Zou

a School of Civil Engineering,Wuhan University,Wuhan,430000,China

b Changjiang Design Group Co.,Ltd.,Wuhan,430000,China

c Institute of Geotechnical Engineering,College of Civil Engineering and Architecture,MOE Key Laboratory of Soft Soils and Geoenvironmental Engineering,Zhejiang University,Hangzhou,310000,China

Keywords: Composite pile Ultimate soil resistance Finite element limit analysis (FELA)Plasticity theory Failure mode

ABSTRACT The composite pile consisting of core-pile and surrounding cement-enhanced soil is a promising pile foundation in recent years.However,how and to what extent the cement-enhanced soil influences the ultimate lateral resistance has not been fully investigated.In this paper,the ultimate lateral resistance of the composite pile was studied by finite element limit analysis (FELA) and theoretical upper-bound analysis.The results of FELA and theoretical analysis revealed three failure modes of laterally loaded composite piles.The effects of the enhanced soil thickness,strength,and pile-enhanced soil interface characteristics on the ultimate lateral resistance were studied.The results show that increasing the enhanced soil thickness leads to a significant improvement on ultimate lateral resistance factor(NP),and there is a critical thickness beyond which the thickness no longer affects the NP.Increasing the enhanced soil strength induced 6.2%-232.6% increase of NP.However,no noticeable impact was detected when the enhanced soil strength was eight times higher than that of the natural soil.The maximum increment of NP is only 30.5% caused by the increase of interface adhesion factor (α).An empirical model was developed to calculate the NP of the composite pile,and the results show excellent agreement with the analytical results.

1.Introduction

In recent years,several composite piles consisting of Prestressed High-strength Concrete (PHC) core-pile and surrounding cementenhanced soil have been proposed to undertake lateral loads,such as stiffened deep cement mixing piles (Wang et al.,2018;Eltaweila et al.,2020;Yu et al.,2021),jet-grouting-reinforced piles(Wang et al.,2015;He et al.,2016),piles with cement-treated soil(Faro et al.,2015;Tariq and Maki,2019),pre-bored grouted planted piles(Qian and Wang,2015;Zhou et al.,2021),and drilled and postgrouted concrete pipe piles(Yang et al.,2020).The composite piles are constructed by inserting a PHC pipe pile into a cement mixing pile or post-grouting cement slurry around the PHC pipe piles.In other words,the PHC pipe pile is surrounded with a layer of cement-enhanced soil in which the shear strength is improved by the cement slurry.The combination of PHC core-pile and surrounding enhanced soil not only ensures the strength of the pile section but also improves the lateral bearing capacity of piles.These advantages endow the composite piles with a good prospect for application(Du et al.,2018;Li et al.,2018;Voottipruex et al.,2011).

The ultimate lateral resistance factor can be written asNP=Pu/suD,wherePudenotes the ultimate force per unit length on the pile;suis the undrained strength of natural soil;andDis the pile diameter.The ultimate lateral resistance factor is an important parameter in the analysis of laterally loaded piles and is often incorporated in thep-ycurves method (Klar,2008;Li et al.,2015;Yu et al.,2021).The studies by Murff and Hamilton (1993) and Yu et al.(2015) showed that the value ofNPincreased with depth until reaching a limiting value at a critical depth.Below the critical depth,the correspondingNPis independent of depth due to the plane strain flow-around failure mode.For monopile in cohesive soil,an empirical value ofNP=9 for the depth below 3Dwas firstly proposed by Broms(1964).Although this value is widely employed in thep-ymethod,it is derived mainly from empirical analysis withnot theoretical justification.

A rigorous theoretical analysis of theNPproblem was performed using the plasticity theorems,in which the upper-bound (UB) and lower-bound (LB) methods were used to estimate the upper-and lower-bounds of the true value ofNP,and the possible values ofNPlying in between (Randolph and Houlsby,1984;Martin and Randolph,2006;Klar and Randolph,2008).TheNPvalues of monopiles and pile groups embedded in undrained natural soil have been extensively studied(e.g.Georgiadis et al.,2013a,b;Zhao et al.,2017a,b;Zhou et al.,2020;Sheil,2021).However,theNPof the composite pile has not yet been fully investigated,nor has the effect of cement-enhanced soil.

In this paper,the finite element limit analysis(FELA)and upperbound plasticity theory were used to investigate the ultimate lateral bearing resistance and the failure mode of the composite pile.This study focuses on the effects of enhanced soil on the ultimate lateral bearing resistance and failure mode below the critical depth.The parameters,such as the thickness and strength of enhanced soil as well as the pile-enhanced soil interface characteristic,were analyzed.An empirical model for estimating the value ofNPwas proposed based on the analytical upper-bound solutions.

2.Finite element limit analysis (FELA)

2.1.Problem statement

As shown in Fig.1a,a composite pile consisting of a PHC corepile and a layer of cement-enhanced soil is embedded in the surrounding natural soil.In Fig.1b,the laterally loaded composite pile below the critical depth is simplified as a two-dimensional (2D)plane strain model to investigate theNPvalue.

In the analytical model,the core-pile radius is denoted asr,and the enhanced soil thickness is δr.The parameter δ is the ratio between enhanced soil thickness and core-pile radius.The undrained strength of enhanced soil is denoted as ηsu,where η is the strength ratio between enhanced and natural soils.The shear strength along the core pile-enhanced soil interface is αηsu,where α denotes the interface adhesion factor.In practice,the enhanced soil layer is usually formed by grouting or rotary jetting,and the cement slurry penetrates the natural soil to a certain extent before hardening.Therefore,no strength reduction is considered at the interface between enhanced and natural soils.

2.2.Numerical model and material properties

The numerical FELAs were performed using the OptumG2,which is a convenient software for addressing 2D limit analysis problems(Zhou et al.,2020).Fig.2 shows a typical numerical model of the laterally loaded composite pile.A half model was used due to the geometry and loading symmetry.The left mesh boundary is a symmetrical face on which the normal displacement is zero.The other three boundaries are fixed in both normal and tangent directions.The core-pile diameter was taken asD=1 m,and the thickness of enhanced soil as δr=0.5δ.Furthermore,the outer boundaries were positioned at a 10(1+δ)Ddistance away from the pile center.

Fig.2.Numerical model of the composite pile:(a)Geometry dimension,boundary and load condition,and (b) Representative mesh after auto-adaptive iterations.

The deformation and failure of saturated clay under undrained conditions are governed by the maximum shear stress.Besides,the limit analysis only focuses on the state and load at failure and ignores the deformation prior to failure.Therefore,the enhanced and natural soil were simulated by rigid-plastic Tresca material with an associated flow rule.The undrained shear strengths of the natural and enhanced soil were taken as 100 kPa and 100η kPa,respectively.The enhanced soil of different strengths was considered by varying the parameter η.The strength of the core-pile was much higher than that of the enhanced and nature soils,so the core-pile section was assumed as a rigid body.The geogrid elements were used to simulate the interface between core-pile and enhanced soil.At the enhanced soil side of the geogrid elements,the shear strength of this interface was 100αη kPa.

As shown in Fig.2a,a uniformed pressure load was applied to the core-pile section.Through the optimization program provided by OptumG2,the minimum upper-bound and maximum lowerbound of the pressure load,which respectively represent the upper-and lower-bound ofPucould be obtained.Then theNPwould be calculated byNP=Pu/(suD).

The OptumG2 provides an auto-adaptive meshing which can automatically adjust the element size according to the intensity of energy dissipation (Fig.2b) to obtain more accurate results.To evaluate the convergence and accuracy of the numerical model and determine a suitable mesh quantity,several numerical models with mesh quantity ranges from 1000 to 20,000 were performed.

The influence of mesh quantity onNPis illustrated in Fig.3.Higher mesh quantity brings in lower UB and higher LB ofNP,leading to a narrower possible range of exact solution.Therefore,the convergence and accuracy of the numerical model are verified.Moreover,when the mesh quantity reaches 15,000,the relative error between UB and LB decreases to about 1.1%,which is acceptable in numerical analysis.Thus,the mesh quantity of 15,000 is suitable and used in the following numerical models.

Fig.3.The influence of mesh quantity on the value of NP and the accuracy of numerical model.

2.3.Failure modes for different enhanced soil parameters

As shown in Fig.4,there are three failure modes for laterally loaded composite piles according to the energy dissipation distribution patterns.

Fig.4.The energy dissipation density and adapted mesh of three kinds of failure modes: Modes (a) I,(b) II,and (c) III.

Fig.5.Proposed kinematic mechanism for the composite pile.

In failure mode I (see Fig.4a),the energy dissipation concentrates on the natural soil,which indicates that the failure occurs in the natural soil without damaging the enhanced soil.In failure mode II(Fig.4b),the energy is concentrated on both enhanced and natural soils,which means that the failure of enhanced and natural soils occurs simultaneously.In failure mode III(Fig.4c),failure only occurs in the enhanced soil where the energy dissipation dominates.

3.Theoretical analysis and analytical upper-bound calculation

3.1.Analytical upper-bound theorem

The upper-bound theorem can be stated as follows: For a constructed kinematic mechanism,the rate of internal plastic energy dissipation could not be greater than the rate of work determined by the external force.Thus,the lowest upper-bound ofPuandNPcan be calculated by minimizing the rate of internal plastic energy dissipation in the proposed kinematic mechanism.Note that the force acting on the piles is the only external force.Thus,the upperbound of ultimate lateral resistancePuand factorNPcan be calculated using

whereZdenotes the velocity discontinuities,τfis the shear stress atZ,Δνzis the velocity difference atZ,Lis the length ofZ,Adenotes the shear zone,and ˙γ is the shear strain rate atA.

It is known from Eqs.(1)and(2)that the essential processes in the analytical upper-bound calculation are constructing a kinematic mechanism and calculating the rate of internal plastic energy dissipation.

3.2.Kinematic mechanism for composite pile

The geometric features of this kinematic mechanism are controlled by parameters λ,λ1,λ2,λ3and β,which can be optimized freely to obtain the lowest internal energy dissipation rate and upper-bound solution.The parameter λ4is related to λ1,λ2,and λ3.The normal velocity difference atmust be zero due to the kinematical admissibility.Therefore,the following equation should be satisfied for arbitrary pointPon(see Fig.6).

Fig.6.Kinematical admissibility of the discontinuity in the kinematic mechanism.

The relationship between λ4and λ1,λ2,λ3could derive from simplifying Eq.(3)

3.3.Analytical upper-bound calculations

The proposed mechanism consists of rigid bodies and velocity discontinuities.The internal plastic energy dissipation rate can be calculated by summing up the energy dissipation rate in each discontinuity because the internal plastic energy dissipation rate in rigid body is zero.It can be proved that the velocity differences on,,andare respectively constant for any combinations of λ,λ1,λ2,λ3and β.Thus,the energy dissipation rate in each discontinuity can be obtained by multiplying its velocity difference with length and shear strength.The detailed calculations of the energy dissipation are presented in Appendix A.

A genetic algorithm-based optimization program was developed to optimize the proposed kinematic mechanism.As shown in Fig.7a,the genetic algorithm is an optimization algorithm based on genetic evolution and natural selection.It first encoded the combination of optimization parameters λ,λ1,λ2,λ3and β into chromosomes (i.e.individuals),then evaluated them with an objective function(see Eq.(A18))to select and retain the“better”individuals.After that,the retained individuals were crossed over and mutated to produce the next generation of individuals.Repeat the evaluation,selection,cross-over,and mutation until the maximum difference between all individuals of a generation is less than the error tolerance.In Fig.7b,the initial individuals are randomly generated and then optimized to the global optimum with the genetic algorithm.

Fig.7.The genetic algorithm: (a) Genetic optimization process,and (b) Schematic diagram of the genetic optimization.

Analyses were performed on 10,780 cases with various combinations of δ,η and α.The results indicate that β would converge to either 1 or (1+δ) for all cases,which is rational and reasonable.When β converges to 1,failure occurs at the pile-enhanced soil interface,which is consistent with failure modes II and III in FELA(Fig.4b and c).When β converges to (1+δ),failure occurs at the interface between enhanced and natural soils,corresponding to failure modes I in FELA (Fig.4a).

4.Results and discussion

The cases in which η varies from 2 to 9,δ from 0 to 7,and α from 0 to 1 were analyzed with upper-bound FELA,lower-bound FELA,and analytical upper-bound theorem.The accuracy of the numerical and analytical results was firstly verified by comparing the analysis results when δ=0,which corresponds to the ordinary circle pile,with the existing results (Randolph and Houlsby,1984;Martin and Randolph,2006).The effects of the enhanced soil strength,enhanced soil thickness,and pile-enhanced soil interface adhesion factor α on theNPwere then analyzed.

4.1.Verification of numerical and analytical results

Table 1 tabulates the analysis results when δ=0 and their comparison with the existing LB solution (Randolph and Houlsby,1984) and UB solution (Martin and Randolph,2006).It shows that the solutions in this context yield great consistency with existing LB and UB solutions,and the maximum relative error is less than 2.37%.The accuracy of the FELA results by OptumG2 and the analytical results by the proposed kinematic mechanism is therefore verified.

Table 1Comparisons of FELA,analytical,and existing solutions for ordinary circle pile (δ=0).

4.2.Effect of the thickness of enhanced soil δ on NP

In Fig.8,theNPincreases with δ until it reaches the value of 55.24 when δ=δ3.When δ ≥δ3,theNPis independent of δ.Interestingly,55.24 is equal to 9.21η for η=6,and the 9.21 is approximately equal to theNPvalue of the ordinary pile with α=0(see Table 1).It means that when δ ≥δ3,the failure mode III illustrated in Fig.4c occurs,and the enhanced soil zone covers the whole failure zone.In that condition,theNPvalue of the composite pile exclusively depends on η and α.

Fig.8.Relationship between NP and δ for the condition of η=6 and α=0.

When δ ＜δ3,the curves consist of three phases which are divided by δ1and δ2,respectively.The failure mode II (Fig.4b)emerges at a small value of δ(δ ＜δ1)because the energy dissipation rate increment due to enhanced soil failure is not large enough to prevent the damage from occurring at the pile-enhanced soil interface.Therefore,theNPincreases nonlinearly with δ when δ ＜δ1.However,the energy dissipation rate corresponding to the enhanced soil damage grows to so great that the failure would not occur in the enhanced soil when δ1＜δ ＜δ2.In that scenario,the damages occur in the natural soil only,which means that the failure mode I illustrated in Fig.4a occurs withNPrising linearly.Meanwhile,the damaged area and energy dissipation rate increase over δ.When δ2＜δ ＜δ3,the energy dissipation rate corresponding to failure mode I is so high that the failure mode II(Fig.4b)emerges,and the failure occurs in both enhanced and natural soils for minimum energy dissipation rate.In that condition,theNPof the composite pile increases nonlinearly with δ.

It is worth noting that failure mode III only occurs when the enhanced soil zone is large enough to contain the whole plastic zone.That requires the thickness of enhanced soil to be at least four times greater than the diameter of the core-pile (see Fig.8).It is necessary to point out that such a thick layer of enhanced soil is rarely,if ever,possible in practical engineering.The failure mode III was discussed only for the integrity of the research logic.

4.3.Effect of the pile-enhanced soil interface adhesion factor α on NP

In Fig.9a,the effect of α on theNPvaries under different values of δ.TheNPis independent of α when δ is equal to 1 or 2.However,when δ ＞2,theNPvalue increases slightly with the maximum increment of 30.5%.

Fig.9.Effect of α on the analytical UB NP and failure mode:(a)NP versus α for different values of δ;and (b) NP versus δ for different values of α.

It can be seen from Fig.9b that the variation of α affects the values of δ1,δ2and δ3,which correspond to the transitions of different failure modes(see Fig.8).The δ1firstly decreases and then keeps zero when α ＞0.2,suggesting that failure mode I occurs in a thinner enhanced soil layer.It is because the α increment leads to a higher shear strength at the pile-enhanced soil interface,and the failure is more likely to occur in natural soil.And when α ＞0.2,the shear strength of pile-enhanced soil interface is greater than that of natural soil (αηsu＞su),therefore the failure mode I no longer domains at small δ (δ1=0).As the α increases,the δ2increases from 2.15 to 2.9,and δ3from 4.35 to 6.2.It indicates failure modes II and III would occur at smaller value of δ when the value of α is smaller.

4.4.Effect of the strength ratio η between enhanced and natural soils on NP

In Fig.10a,theNPfirstly increases with η until reaching a plateau.Additionally,the effects of η onNPbecome more significant as the value of δ increases.Augment ofNPvalue surges from 6.2%(from η=2 to η=3)to 232.6%(from η=2 to η=8)when δ value varies from 0.1 to 5.

Fig.10.Effect of η on the analytical UB NP and failure mode: (a) NP versus η for different values of δ;and (b) Influence of η and δ on the failure mode.

From Fig.10b,it can be seen that as η value increases,the failure pattern shifts from mode II to mode I when δ ＜3,and then from mode III to mode II and mode I successively when 3 ＜δ ＜6.Regardless of δ value,the damage mode I occurs when η exceeds a specified value.That can explain why,as stated previously,NPremains constant when η reaches a specified value,since the enhanced soil keeps intact in failure mode I.When δ ＜6,which is in the most cases in practice,further increasing value of η makes no difference toNPwhen η ＞8.

4.5.Empirical model for estimating NP

The calculation ofNPis not always convenient in practice because an optimization program is needed.Therefore,it is necessary to establish an empirical model to estimate the value ofNP.According to the energy minimization principle,theNPshould be the lowest value for the three failure modes:

wherei=1,2,3 correspond to failure modes I,II,and III,respectively.

For failure modes I and III,damage occurs merely in natural or enhanced soils.Therefore,the composite pile can be simplified as an ordinary pile with radius of(1+δ)rin natural soil or radius ofrin enhanced soil.According to theNPequation proposed by Randolph and Houlsby(1984),Npfor the composite pile in failure modes I and III can be expressed as

For failure mode II,damage occurs in both enhanced and natural soils,andNPrelates to η,δ,and α.According to the spatial relationship between the velocity discontinuities and the interface between enhanced and natural soils,the failure mode II is divided into four subclasses (Fig.11) and then theNPof mode II can be estimated by

Fig.11.Subclasses of mode II: (a) Subclasses-mode II-I;(b) Subclasses-mode II-II;(c) Subclasses-mode II-III;and (d) Subclasses-mode II-IV.

wherei(=1,2,3,4)indicates the four subclasses of failure mode II,respectively;ai,bi,ci,di,ei,andfiare fitting parameters,which can be determined by equations in Table 2.

Table 2The fitting parameters in Eq.(8).

To verify the accuracy of the proposed empirical model,the error between the empirical model and the analytical upper-bound solution is defined as

whereandareNPcalculated by empirical model and analytical upper-bound,respectively.In Fig.12,the maximum errors are only 3.6%,5.5%,and 5.1%,respectively,when η=2,4,6.That means theNPof the composite pile can be estimated effectively and accurately by the established empirical model.

Fig.12.Errors between the empirical model and the analytical upper-bound:(a)η=2,(b) η=4,and (c) η=6.

4.6.Discussion

In this paper,the effects of enhanced soil layer on the ultimate lateral resistance factorNPof composite piles below the critical depth were studied,and a method for estimating theNPof composite piles was proposed.The results can not only apply to determining the lateral bearing capacity of composite piles subjected to lateral loads but also have reference for studying the bearing capacity of composite piles subjected to horizontal soil displacement.However,it should be noted that in practical engineering,the interaction between composite piles and soil is a threedimensional (3D) problem.The plane strain assumption in this paper only applies to the part beyond the ground influence range(i.e.below the critical depth).The studies of Murff and Hamilton(1993) and Yu et al.(2015) show that theNPbelow the critical depth is only related to factors such as pile diameter and soil parameters,but not depth.Therefore,the effect of depth on theNPis not considered in this paper.For shallow soils,the soil deformation will be influenced by the ground,resulting in vertical displacements,such as bulging or depression.In that case,the pile-soil interaction is more complex,and the effect of depth onNPmust be addressed,which may be another topic in future study.

The soil is usually simplified to a series of independent springs during the analysis of laterally loaded piles.The relationship between the pile displacement and the lateral load can be obtained by substituting the deformation and resistance relation of the soil springs (i.e.p-ycurves) into the differential equation of piles.The ultimate soil resistance factorNPis an essential parameter in thepycurves.Its physical meaning is the maximum lateral resistance the soil springs can impose on per unit length pile.This paper focuses on the enhanced soil’s effects on theNP.Thus,the pile body is assumed to be a rigid body.In actual projects,the piles would be damaged under the bending moments caused by lateral loads.Therefore,when applying the results of this context to the laterally loaded pile analysis,the possible pile damage caused by the pile bending moment needs to be further investigated.

5.Conclusions

The effects of cement-enhanced soil on theNPof the composite pile are studied by FELA and analytical upper-bound analysis.The following conclusions can be drawn:

(1) Three distinct failure modes are identified for the laterally loaded composite pile.In failure mode I,the enhanced soil keeps intact and bonds to the core-pile,resulting inNPproportional to(1+δ).In failure mode II,damage occurs in both enhanced and natural soils,meanwhile value ofNPgoes up at higher values of δ,η and α.Failure mode III is rarely observed,if ever,possible in practical engineering because it only occurs when the η is large enough to let the enhanced soil zone contain the whole plastic zone.

(2) The thickness of enhanced soil significantly influences both the failure mode and the value ofNP.As the value of δ increases,with transitions of δ1,δ2and δ3,failure modes II,I,II and III occur successively.When δ ＜δ3,the increase in the value of δ leads to theNPincreasing by times.However,when δ ＞δ3,the value of δ no longer affects the value ofNPbecause the enhanced soil zone is large enough to let the failure mode III occurs.

(3) The strength of enhanced soil possesses a significant impact onNPand failure mode,depending on the thickness of enhanced soil.The value ofNPincreases at first and then keeps constant after η exceeds a specified value.Additionally,the maximum effect of η onNPincreases from 6.2% to 232.6% as δ value varies from 0.1 to 5 when failure mode II occurs.When η ＞8,theNPvalue stays constant due to failure mode I and intact enhanced soil layer.

(4) The interface adhesion factor α has slight influence onNPthan δ and η.The α affectsNPonly when failure mode II or mode III occurs,with maximum augment of only 30.5%.

(5) The proposed empirical model can accurately predict theNPvalue of the composite pile.The maximum relative error between the empirical model and rigorous analytical upperbound solutions is only 4.37%.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The work was supported by the National Natural Science Foundation of China (Grant No.51978540).

Appendix A.Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.jrmge.2023.03.010.