A theoretical solution for pile-supportedembankments on soft soils under one-dimensionalcompression

R.P. Chen, Y.M. Chen, J. Han, and Z.Z. Xu

Abstract: Pile-supported embankments are increasingly being used for highways, railways, storage tanks, etc. over softsoil because of their effectiveness in accelerating construction and minimizing deformation. The stress transfer mechanismsamong all of the components in a piled embankment, including the embankment fill, the piles and (or) caps, and the foun-dation soils, are complicated. In this study, a closed-form solution for one-dimensional loading was obtained taking intoconsideration the soil arching in the embankment fill, the negative skin friction along the pile shaft, and the settlement ofthe foundation soil. In the derivations, the piles, the embankment fill, and the foundation soil were assumed to deformone-dimensionally. This study investigated the stress concentration on top of the pile, the axial load and skin friction distri-butions along the pile, and the settlement of the embankment. Comparisons demonstrate that the results from this solutionare in good agreement with those obtained using a finite element method. It is worth pointing out that this solution shouldbe applied to the piles close to the centerline of the embankment and not to those near the toe of the embankment becauseof the two-dimensional loading condition near the toe.

Key words: embankment, pile, settlement, soil arching, soil–pile interaction, soft soil.

Resume : Les remblais sur pieux sont de plus en plus utilises pour les routes, les chemins de fer, les reservoirs d’entre-posage, etc. sur des sols mous a cause de leur efficacite pour accelerer la construction et minimiser la deformation. Lesmecanismes de transfert des contraintes a toutes les composantes du remblai sur pieux incluant le remblai, les pieux et(ou) les chapeaux des pieux, et les sols de fondation, sont compliques. Dans cette etude, une solution exacte pour un char-gement unidimensionnel a ete obtenu en tenant compte de l’effet de voute dans le remblai, du frottement superficiel nega-tif le long du fut du pieu, et du tassement du sol de la fondation. Dans les derivations, on a suppose que les pieux, lemateriau du remblai, et le sol de fondation se deformaient dans une seule dimension. Cette recherche a etudie la concentra-tion de contraintes sur le sommet du pieu, les distributions de la charge axiale et du frottement superficiel le long du pieu,et le tassement du remblai. Des comparaisons demontrent que les resultats de cette solution sont en bonne concordanceavec ceux obtenus avec la methode d’elements finis. Il est bon de noter que cette solution devrait etre appliquee aux pieuxpres de la ligne de centre du remblai plutot qu’a ceux pres du pied du remblai a cause de leur condition de chargement bi-dimensionnel.

Mots-cles : remblai, pieu, tassement, effet de voute de sol, interaction sol–pieu, sol mou.

[Traduit par la Redaction]

Introduction

Pile-supported embankments, which contain foundationsoil, piles, pile caps, geosynthetics if needed, and embank-ment fill, have been increasingly used for embankmentsover soft soils as shown in Fig. 1. When compared withother ground improvement methods for soft soils, the pile-supported embankments have the advantages of rapid con-

struction, small vertical and lateral deformations, and globalstability. This technology has been used in a number of ap-plications worldwide to solve many geotechnical problems,for example, the railway widening project at the Stanstedairport in London (Jones et al. 1990), the retaining wallproject in the north of St. Paul in Brazil (Alzamora et al.2000), the highway construction project in the Netherlands(AASHTO/FHWA 2002), and the Hangzhou–Ningbo ex-pressway widening project in China (Chen et al. 2007).

Han and Gabr (2002) indicated that the mechanisms ofload transfer in pile-supported embankments are a combina-tion of the soil arching effect in the embankment fill, the re-inforcement effect of the geosynthetics, and the load transferbetween piles and soils as a result of their stiffness differ-ence. Hewlett and Randolph (1988) and Low et al. (1993)investigated soil arching in the piled embankments usingmodel tests and then developed theoretical solutions for esti-mating the degree of the load transfer between piles and soilby assuming that the embankment fill was in an equilibriumlimit state. British Standard (1995) used Marston’s formula

Received 2 November 2006. Accepted 12 December 2007.Published on the NRC Research Press Web site at cgj.nrc.ca on14 May 2008.

R.P. Chen, Y.M. Chen,1 and Z.Z. Xu. Key Laboratory of SoftSoils and Geoenvironmental Engineering of Ministry ofEducation, Department of Civil Engineering, ZhejiangUniversity, 38 Zheda Road, Hangzhou 310027, China.J. Han. Department of Civil, Environmental, and ArchitecturalEngineering, The University of Kansas, 1530 West 15th Street,Lawrence, KS 66045, USA.

1Corresponding author (e-mail: chenyunmin@zju.edu.cn).

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to estimate the vertical stress on top of the piles. These stud-ies were mainly focused on the soil arching in the embank-ment. In reality, the mechanisms of load transfer are muchmore complicated than just the soil arching. A number ofnumerical analyses were conducted to investigate the loadtransfer mechanisms, for example, Han and Gabr (2002),Pham et al. (2004), and Huang et al. (2006). Han and Gabr(2002) investigated the effect of geosynthetics on soil arch-ing and the pile–soil stress ratio using an axisymmetric nu-merical model. Pham et al. (2004) used a two-dimensional(2D), plane-strain finite element method (FEM) code tostudy the aggregate pier–soil–geogrid interaction. Huang etal. (2006) concluded that 2D and 3D numerical modeling ofdeep mixed column-supported embankments yield close set-tlement results. In most studies so far, piles are seated on afirm soil or bedrock layer. In realty, however, soft soils mayexist under the pile toe, and the settlement of the substrataunder the pile toe may play an important role in the mecha-nisms of load transfer. In addition, the effects of pile capshave not been well investigated. Numerical results also indi-cated that the geosynthetic reinforcement has an insignifi-cant effect on the maximum settlement of the embankmentand the stress concentration when the tensile stiffness of thereinforcement is less than 860 kN/m (Han and Gabr 2002;Pham et al. 2004) and the spacing of the piles is relativelyclose. Even though numerical methods are effective for in-vestigating the interactions in the pile-supported embank-ments, however, they are generally time-consuming anddifficult to be adopted for routine use in practice. Poulos(2007) provided simple design charts for the preliminary de-sign of piled embankments.

In this study, a closed-form solution for one-dimensional

(1D) embankment loading was developed for the evaluationof the interactions among the rigid pile cap, the pile, thesoil, and the embankment fill. This solution is verified bythe results from a FEM. In the development of this solution,geosynthetic reinforcement was not considered and the em-bankment fill, the foundation soil, and the piles were all as-sumed to deform one-dimensionally. Hence, the proposedsolution in this study should be used for the piles close tothe centerline of the embankment and not for those nearthe toes of the embankment.

Development of the theoretical solution

Analytical modelFigure 2 illustrates an axisymmetric unit cell model,

which includes a pile and a rigid cap, the surrounding foun-dation soil, and the embankment fill. It is assumed that the zaxis starts at the top of the pile cap and is positive in thedownward direction. The embankment fill is homogeneous,isotropic, and cohesionless with internal friction angle 4c,unit weight �c, and Young’s modulus Ec. The piles, the soil,and the embankment fill only compress one-dimensionally.The diameter, the cross-sectional area, and the length of thepile are expressed as Dp, Ap, and L, respectively. The area ofthe pile cap is Ai. In this analysis, the fill above the eleva-tion of the pile cap is simplified as an inner column with adiameter of Di and a hollow cylinder with an inner diameterof Di and an outer diameter of Do. The cross-sectional areaof the hollow cylinder is Ao.

It is expected that, at the top elevation of the pile cap, thesettlement of the foundation soil is larger than that on thepile cap. The differential settlement Se between the pile capand the foundation soil at z = 0 induces the movement of theembankment fill downward so that shear stress would de-velop within the embankment fill. As a result, the pile capcarries a higher load while the foundation soil between thepile caps carries less stress than the overburden stress. Thisphenomenon is called soil arching. As Terzaghi (1943) pro-posed, the slip plane between the stationary portion and themoving portion is assumed to be vertical. As a result of thesoil arching effect, the differential settlement at the same el-evation in the embankment decreases from the base of theembankment towards the top surface of the embankmentand reaches zero differential settlement at an elevation ofz = –he if the embankment is thick enough (Fig. 2). Terzaghi(1943) referred to this plane as the plane of equal settle-ment. Above this plane of equal settlement, the settlementand the vertical stress are evenly distributed.

If the compression of the fill above the pile cap andwithin the range of 0 ‡ z ‡ –he is Si and the compression ofthe fill above the foundation soil and within the range of 0 ‡z ‡ –he is So, then the differential settlement is Se = Si – So.If the settlements of the pile and the foundation soil withinthe range of z = 0 to L are defined as Wp(z) and Ws(z), re-spectively, at the depth of the pile toe (z = L), then the soilhas the settlement Ws(L) because of the compression of theunderlying substrata, and the pile has the settlement Wp(L).The penetration of the pile at the toe is Wp(L) – Ws(L),which causes positive skin friction along the lower part ofthe pile. Below the pile cap, however, the settlement of thesoil is much greater than that of the pile. This differential

Fig. 1. Types of pile-supported embankments. (a) Piles floating insoft soil. (b) Piles embedded in firm soil.

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settlement would induce negative skin friction along the topportion of the pile. Therefore, the skin friction changes fromnegative to positive from the top to the toe of the pile. Aneutral location with zero skin friction occurs at a certaindepth. The differential settlement Se at z = 0 can also be ex-pressed as Wp(0) – Ws(0).

Soil arching in the embankmentAs shown in Fig. 2, at the top elevation of the pile cap,

the settlement of the pile is less than that of the soil. As aresult, the outer cylinder fill would move downwards rela-tive to the inner column. This relative movement inducesshear stresses acting downwards on the surface of the innercolumn but upwards on the surface of the outer cylinder.

Taking a unit element with thickness of dz in the inner

column (Fig. 2b) and assuming no lateral deformation inthe embankment, the force equilibrium of the unit elementin the inner column in the vertical direction can be ex-pressed as

½1� AidPi ¼ ð�cAi þ �DiFÞdz

where Pi is the vertical stress in the inner column; F is thefriction on the slip surface and F = PiK0 tan 4c, K0 is thecoefficient of the earth pressure at rest, and K0 = 1 –sin 4c.

When z = –he, Pi = �c(h – he). The integration of eq. [1]from z = –he to z = 0 yields

½2� PiðzÞ ¼�cDi

4K0tan’c

1þ 4K0tan’c

Di

ðh� heÞ

24

35exp 4K0tan’c

zþ he

Di

0@

1A� 1

8<:

9=; ð�he � z � 0Þ

In eq. [2], he is the only unknown and can be determinedbased on the differential settlement Se between the pile andthe soil at z = 0.

The equilibrium of the vertical force in the embankmentrequires that

½3� mPiðzÞ þ ð1� mÞPoðzÞ ¼ �cðhþ zÞð�he � z � 0Þ

where m is the ratio of the cap area to the total area and m =Ai/(Ai + Ao), and Po is the vertical stress in the outer cylin-der. The proportion of the load carried by the pile n is de-fined as the load on the cap divided by the total weight ofthe embankment fill over the total area

½4� n ¼ Pið0ÞAi

�chðAo þ AiÞ¼ mPið0Þ

�ch

Since no differential settlement between the inner columnand the outer column occurs at the plane of equal settle-ment, the settlements of the outer cylinder and the inner col-umn are the same, namely

½5� Si þWpð0Þ ¼ So þWsð0Þ

where Si and So are the compression of the inner columnand the outer cylinder, respectively, within the range of z =0 to –he, respectively.

Equation [5] can be rewritten as

Fig. 2. Analytical model for the pile-supported embankment. (a) Embankment before settlement. (b) Embankment after settlement.

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½6� Se ¼ Si � So ¼ Wsð0Þ �Wpð0Þ ¼Z 0

�he

PiðzÞ � PoðzÞEc

dz ¼ 1þ Ai

Ao

0@

1A �cD

2i

16K0tan2’cEc

1þ 4K0tan’c

Di

ðh� heÞ

24

35

0@

� exp 4K0tan’c

he

Di

0@

1A� 1

24

35� he

8<:

9=;1A þ �c

Ec

1þ Ai

Ao

0@

1A h2e

2� hhe

0@

1A ð�he � z � 0Þ

Equation [6] presents the relationship between the heightof the equal settlement plane he and the differential settle-ment Se. If he is determined using eq. [6], Pi(z), Po(z), and ncan be calculated using eqs. [2], [3], and [4].

The value of n reflects the effect of the soil arching.For two extreme cases, n ¼ 0 represents the complete soilarching, whereas n ¼ 1 represents no soil arching. In addi-tion, n depends on the differential settlement between thepile and the soil, the properties of the embankment fill,the height of the embankment, the pile spacing, and thecap size. The model tests by Hewlett and Randolph(1988) and Low et al. (1993) demonstrated the effects ofthese factors.

The above-mentioned equations are only valid for thecase when h ‡ he. To ensure the limited depression on theroad surface, the height of the embankment should be atleast 1.2 times the center-to-center spacing of the piles, orthe maximum center-to-center spacing of the piles shouldnot be greater than 0.8h (NGG 2002). In practice, he is gen-erally less than the spacing of piles so that the condition ofh ‡ he is mostly satisfied.

Load transfer in pile and soilThe equilibrium equations of the pile and the soil can be

written as

½7� d2WpðzÞdz2

¼ �U

EpAp

½8� d2WsðzÞdz2

¼ �U

EsAs

where U is the perimeter of the pile; Ep is Young’s modulusof the pile; Es is the constrained modulus of the soil, � is theskin friction along the pile shaft, and Ap and As are thecross-sectional areas of the pile and the soil in the unit cell,respectively.

The toe resistance of the pile depends upon the penetra-tion at the pile toe Wp(L) – Ws(L) and not on the total set-tlement of the substrata under the pile toe. It is assumedherein that the soil has elastoplastic behavior. As shown inFig. 3, the toe resistance and the skin friction can be ex-pressed as:

½9� qp ¼kb½WpðLÞ �WsðLÞ� ½WpðLÞ �WsðLÞ� � qu=kbqu ½WpðLÞ �WsðLÞ� > qu=kb

�

½10� � ¼ ks½WpðzÞ �WsðzÞ� ½WpðzÞ �WsðzÞ� � �u=ks�u ½WpðzÞ �WsðzÞ� > �u=ks

�

where kb is the soil stiffness at the pile toe; qu is the ulti-mate toe resistance; ks is the shear stiffness of soil, which isstrongly dependent on the method of pile installation; and �uis the ultimate skin friction of the soil around the pile. Un-der a drained or long-term condition, the ultimate skin fric-tion can be calculated using the � method, i.e.,

½11� �u ¼ ��0v ¼ ��0z

where � is the coefficient of the skin friction, �0v is the ef-

fective overburden stress at the depth z for calculation, and�’ is the effective unit weight of the soil.

The skin friction along the pile shaft can be divided intothree sections as shown in Fig. 4. In the first section (0 £z < z1), the negative skin friction is assumed to reach itsultimate value. In the second section (z1 £ z £ z2), the skinfriction is proportional to the relative pile–soil displace-ment. In the third section (z2 < z £ L), the positive skinfriction is assumed to reach its ultimate value. The shearstiffness in the second section is assumed to be equal tothat at a depth of (z1 + z2)/2, therefore, the distribution ofthe skin friction along the pile in these three sections canbe given by:

½12� � ¼���0z 0 � z � z1

� �ksðWp �WsÞ z1 � z � z2

� ���0z z2 � z � L

� �8<:

Substituting eq. [12] into eqs. [7] and [8] and then solvingthese two differential equations yields the expressions forWs(z) and Wp(z) as follows:

½13a� z ¼ 0 to z1:

Wp1ðzÞ ¼ � 1

6�pz

3 þ F1zþ F2

Ws1ðzÞ ¼1

6�sz

3 þ F3zþ F4

8>>><>>>:

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½13b� z ¼ z1 to z2:

Wp2ðzÞ ¼�pp

�3F5ðe�z � e��zÞ þ

�pp

�2F6ðe�z þ e��zÞ þ F7zþ F8

Ws2ðzÞ ¼ � �ss

�3F5ðe�z � e��zÞ � �ss

�2F6ðe�z þ e��zÞ þ F7zþ F8

8>>><>>>:

½13c� z ¼ z2 to L:

Wp3ðzÞ ¼1

6�pz

3 þ F9zþ F10

Ws3ðzÞ ¼ � 1

6�sz

3 þ F11zþ F12

8>>><>>>:

in which �p ¼ �0�UEpAp

, �s ¼ �0�UEsAs

, �pp ¼ ksUEpAp

, �ss ¼ ksUEsAs

,

� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�pp þ �ssÞ

p, and F1... F12 are the integral constants.

The subscripts 1, 2, and 3 denote the three skin friction sec-tions as described previously.

Boundary and compatibility conditionsAt the top of the pile cap (z = 0)

½14a� Ep

dWp1ðzÞdz

jz¼0 ¼ �Pið0ÞAi

Ap

½14b� Es

dWs1ðzÞdz

jz¼0 ¼ �Poð0ÞAo

As

½14c� Ws1ð0Þ �Wp1ð0Þ ¼ Se

At the pile toe (z = L)

½15a� Ep

dWp3ðzÞdz

jz¼L ¼(�kb½Wp3ðLÞ �Ws3ðLÞ� ½Wp3ðLÞ �Ws3ðLÞ� � qu=kb�qu ½Wp3ðLÞ �Ws3ðLÞ� > qu=kb

½15b� Ws3ðLÞ ¼ Sb

where Sb is the settlement of the substrata and can be deter-mined by an approximate method that will be discussed in alater section. As it is assumed that the compression of thesubstrata is uniform and does not have any effect on thepile–soil interaction, Sb = 0 can be assigned for now tomake the equations simpler. After all of the equations aresolved, the vertical displacements above the substrata shouldbe added to by a value of Sb, which will be determined later.

Considering the continuity of the displacements and thestresses in the pile and the soil, the compatibility conditionscan be obtained as follows:

For z = z1:

½16a� Wp1ðz�1 Þ ¼ Wp2ðzþ1 Þ

½16b� Ep

dWp1ðzÞdz

����z¼z�

1

¼ Ep

dWp2ðzÞdz

����z¼zþ

1

½16c� Ws1ðz�1 Þ ¼ Ws2ðzþ1 Þ

½16d� Es

dWs1ðzÞdz

����z¼z�

1

¼ Es

dWs2ðzÞdz

����z¼zþ

1

For z = z2:

½17a� Wp3ðzþ2 Þ ¼ Wp2ðz�2 Þ

½17b� Ep

dWp3ðzÞdz

����z¼zþ

1

¼ Ep

dWp2ðzÞdz

����z¼z�

1

½17c� Ws3ðzþ2 Þ ¼ Ws2ðz�2 Þ

½17d� Es

dWs3ðzÞdz

����z¼zþ

1

¼ Es

dWs2ðzÞdz

����z¼z�

1

In addition, the magnitudes of the relative displacements be-tween the pile and the soil at the depths of z1 and z2 are thesame and equal to the shear deformation of the soil, i.e.,

½18a� Wp2ðz1Þ �Ws2ðz1Þ ¼ ��u=ks

½18b� Wp2ðz2Þ �Ws2ðz2Þ ¼ �u=ks

SolutionThe substitution of eq. [13] into eqs. [14]–[18] and the

combination of eqs. [3] and [6] yield 17 equations, in whichthe variables F1 to F12, Pi(0), Po(0), z1, z2, and he are un-known; because the number of unknowns is equal to thenumber of equations, all of these unknowns can be solved.

As these equations are highly nonlinear, no simple expres-sion of the unknowns can be obtained directly. These equa-tions have to be solved numerically. Expressing F1 to F12,Pi(0), and Po(0) in terms of z1, z2, and he, reduces the numberof equations from 17 to 3. The three equations including threeunknowns z1, z2, and he can be solved using the Newton–

Chen et al. 615

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Raphson method, and then all of the other unknowns andvariables can be obtained accordingly. The details on thederivations of these solutions can be found in Appendix Aof this paper.

The settlement of the substrata Sb below the elevation ofthe pile toe can be solved based on the following proce-dures. Detailed presentation of these procedures would un-duly lengthen this paper, therefore references are providedinstead for readers to follow. The additional stresses in thesubstrata can be considered to be induced by the verticalstress on the top of the foundation soil between piles, thepositive and negative skin friction along the pile shaft, andthe stress under the pile toe transferred from the pile cap asshown in Fig. 5. The portions of the additional stresses in-duced by the negative and positive skin friction along thepile shaft and the stress under the pile toe can be estimatedusing Geddes’s formula (Geddes 1966). At the same time,the portion of the additional stresses induced by the verticalstress on the top of the foundation soil between piles can beestimated using Boussinesq’s equation. Finally, the settle-ment of the substrata can be estimated by summing the com-pression of each sublayer.

Determination of soil parameters

ks and koRandolph and Wroth (1978) defined the shear stiffness of

a pile in an isotropic elastic medium as follows:

½19� ks ¼ 2�G=lnðrm=r0Þ

in which G is the shear modulus of the soils, ro is the pileradius, and rm is the influence radius of the pile. Randolphand Wroth (1978) suggested rm ¼ 2�ð1� �0sÞL, where � isthe ratio of the soil shear modulus in the middle of the pileto that at the pile toe, L is the pile length, and �0s is Pois-son’s ratio of the soil at the pile toe. A parametric studyshows that ks changes slightly with rm but it changes signif-icantly with the shear modulus of the soil. Wong and Teh(1995) proposed a method for determining the shear modu-lus of soil based on triaxial tests. Both G and ks may bechanged as a result of the disturbance or densification ofsoil due to the pile installation.

The normal stiffness kb of the soil under the pile toe canexpressed as Randolph and Wroth (1978) suggested

½20� kb ¼ 4Gr0=ð1� �0sÞ

�u and quAs previously described, the ultimate skin friction under a

drained or long-term condition can be determined using the� method. The parameter � depends on the soil type, theroughness of the pile shaft, and the pile installation method,which can be estimated using the following formulae (Wongand Teh 1995):

½21� � ¼ �NCOCR0:5 for saturated clay

Ktan for sand

�

where is the interface friction angle between the pile and

Fig. 5. Stresses applied on and inside the foundation soil.

Fig. 4. Distribution of skin friction along the pile.Fig. 3. Relationships between soil reaction and pile–soil relativedisplacement for (a) skin friction and (b) toe resistance.

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the soil, OCR is the overconsolidation ratio, and K is thecoefficient of lateral earth pressure, which can be estimatedby K0 (K/K0). The K/K0 ratio depends on the pile type andthe method for construction and can be found in the litera-ture (e.g., Coduto 2001). The subscript ‘‘NC’’ stands fornormally consolidated soils. Wong and Teh (1995) sug-gested �NC = 0.22 if no test data is available.

The ultimate toe resistance under a drained or long-termcondition can be determined by the following equation:

½22� qu ¼ c0Nc þ �0vNq

where Nc and Nq are the bearing capacity factors of the soilat the pile toe, c’ is the effective cohesion of the soil at thetoe, and �0

v is the effective overburden stress at the toe.

Comparison with the finite element method

A FEM incorporated in the PLAXIS code (Brinkgreve

Fig. 6. Settlement profiles at different elevations of the embankment above the pile cap. (a) Pile toe in soft soil. (b) Pile toe in firm soil.

Table 1. Material properties used in the numerical modeling and the proposed analytical solution.

Embankment Soft soil Firm soil Pile Cap

Height (m) 4 — — — —Friction angle (8) 30 — — — —Effective friction angle (8) — 9 22 — —Cohesion (kPa) 0 — — — —Effective cohesion (kPa) — 15 30 — —Young’s modulus (GPa) 0.03 — — 35 35Constrained modulus (MPa) — 2.2 15 — —Poisson’s ratio 0.25 0.35 0.35 0.15 0.15Unit weight of fill (kN/m3) 20 — — — —Saturated unit weight (kN/m3) — 17.5 18 — —Coefficient of skin friction, � — 0.25 0.25 — —Pile spacing (m) — — — 2.5 1Diameter (m) — — — 0.40 1.13Thickness (m) — 0.25 3 — 0.35

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Fig. 8. Distribution of axial force in the pile with depth when em-bedded in soft soil.

Fig. 9. Distribution of axial force in the pile with depth when em-bedded in firm soil.

Fig. 7. The settlement contours within the embankment fill. (a) Pile floating in soft soil. (b) Pile embedded in firm soil.

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and Vermeer 2000) was used to verify the method proposedin this paper. The numerical analysis was conducted usingaxisymmetric models, which simulate the unit-cell conceptused for the development of the analytical solution. Eachsingle pile has a surrounding cylindrical soil column. Thevertical boundary is allowed for vertical displacement only.The foundation soils and the embankment fill were modeledas linearly elastic – perfectly plastic materials with theMohr–Coulomb failure criteria. The piles and caps weremodeled as linear elastic materials. The interface betweenthe pile and the soil was modeled using interface elements.The interface shear strength was estimated based on thestrength reduction method suggested by PLAXIS (Brink-greve and Vermeer 2000) as follows:

½23� ci ¼ Rintercs

½24� tan’i ¼ Rintertan’s � tan’s

where ci and 4i are the interface cohesion and friction angle,respectively; cs and 4s are the cohesion and friction angle ofthe soil, respectively; and Rinter is the strength reduction fac-tor. In general, Rinter is on the order of 2/3 for a sand–steelinterface and 1/2 for a clay–steel interface, and a rough in-terface usually has a higher Rinter value (Brinkgreve andVermeer 2000). In this paper, Rinter = 0.7 was used. In addi-

tion, the embankment was assumed to be placed instantly.All parameters used in the analysis are listed in Table 1.

For the proposed analytical method, the shear modulus ofthe soil was calculated from the constrained modulus andPoisson’s ratio. The normal stiffness of the soil at the piletoe and the shear stiffness around the pile were calculatedusing eqs. [19] and [20].

Two cases were selected and analyzed using the numeri-cal and analytical methods. In the first case, the pile is20 m long and installed in a 25 m thick soft layer, therefore,the pile toe is in the soft soil and has 5 m underlying softsubstratum. In the second example, the pile is 27 m longand penetrates through the 25 m thick soft soil into a firmsoil for 2 m. These two cases were selected for evaluatingthe effects of end-bearing conditions.

The settlement profiles analyzed using the FEM, at differ-ent elevations of the embankment, are presented in Fig. 6. Itis shown that the settlement above the cap is less than thatabove the foundation soil and there is obvious differentialsettlement at the cap elevation. These settlement profilesjustify the assumption of the embankment fill above thepile cap being approximately treated as the inner columnand the outer cylinder. Figure 7 presents the numerical re-sults of the settlement contours within the embankment fill,

Fig. 10. Distribution of skin friction along the pile with depth whenfloating in soft soil.

Fig. 11. Distribution of skin friction along the pile with depth whenembedded in firm soil.

Chen et al. 619

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which clearly demonstrate the existence of the equal settle-ment plane at the height of 2.0 m for these two specificcases. The height of the equal settlement plane calculatedby the proposed method is 1.76 m if the pile is installed inuniform soft soil or 1.67 m if the pile is installed throughthe soft soil into the firm layer.

Figures 8 and 9 present the distributions of the axial forcein the pile with depth from FEM and the proposed method.It is shown that the results obtained by these two methodsagree reasonably well. Figures 8 and 9 both indicate thatthe axial forces increase first and then decrease with thedepth. The increase of the axial forces results from negativeskin friction while the decrease of the axial forces resultsfrom positive skin friction. Figures 10 and 11 present thedistributions of skin friction along the pile shaft, whichshow three distinguished sections of skin friction from thepile top to the pile toe including the ultimate negative skinfriction section, the transitional section from the negative topositive skin friction, and the ultimate positive skin frictionsection. Since the pile has less settlement when embedded inthe firm layer, it has a wider negative skin friction regionand a deeper neutral point than when floating in the softlayer.

The proportions of the load carried by the piles are pre-sented in Table 2. It is shown that the proportions of theload carried by the piles for these two cases are close.

The settlements at different elevations of the embankmentfill are also listed in Table 2. The settlement of the embank-ment is mainly influenced by the compressibility of the sub-stratum especially when a soft substratum exists below thepile toe. A thick soft substratum induces a large embank-ment settlement. At the pile top elevation, the differentialsettlements computed by these two methods are very close,that is, 20–24 mm for these two cases.

Summary of design procedureThe method proposed in this paper can calculate the load

transfer and the embankment deformation of the piled em-bankment. A brief description of the design procedure issummarized as follows:

(1) Input the design parameters of the embankment fill, pilesand caps, and foundation soils. According to the formu-lae provided in Appendix A, calculate the deformationsof the pile and the embankment, the vertical pressureson the pile head and the soil at the pile top elevation,and the stress distribution along the pile shaft and onthe pile toe.

(2) Check if the penetration deformation of the pile toe(Wp(L) – Ws(L)) exceeds the maximum elastic deforma-tion of the soil under the pile toe (qu/kb). If it does,change the pile length, the pile spacing, and (or) the capsize to ensure the penetration deformation at the pile toeis less than qu/kb.

(3) Calculate the maximum axial force in the pile, PT, whichis the summation of the vertical force on the pile capPi(0) and the total downdrag along the pile, Rg, that is,

½25� PT ¼ Pið0Þ þ Rg

(4) The required ultimate capacity of the pile Pu at a speci-fied factor of safety FS should be

½26� Pu ¼ FS � PT

where Pu is the summation of the ultimate positive sideresistance and the toe bearing capacity of the pile.

(5) Calculate the settlement of the substrata, Sb. The totalsettlement of the embankment should be the summationof the settlement of the substrata and the vertical displa-cement above the substrata.

ConclusionsThe interactions among the embankment fill, the piles and

caps, and the foundation soil in pile-supported embankmentsare complicated. This paper proposed a closed-form solutionbased on a simplified unit-cell concept considering soil arch-ing in the embankment and the distribution of skin frictionalong the pile. In the development of this solution, the pile,the soil, and the embankment fill were assumed to deformone-dimensionally. The proposed method can determine theplane of equal settlement, the proportion of load carried bythe pile, the distribution of skin friction along the pile, andthe settlement of the embankment. The results from the pro-posed method compared reasonably well with the numericalmethod incorporated in the PLAXIS software.

AcknowledgmentThis work was supported by the National Natural Science

Foundation of China (Project No. 50308026).

ReferencesAASHTO/FHWA. 2002. Innovative technology for accelerated

construction bridge and embankment foundations. AASHTO/FHWA Preliminary Summary Report. Federal Highway Admin-istration, United States Department of Transportation, Wash.

Alzamora, D.E., Wayne, M.H., and Han, J. 2000. Performance of

Table 2. Settlement and proportion of load carried by pile obtained by the finite element method (FEM) and theproposed method.

Piles embedded in firm soil Piles embedded in soft soil

FEM Proposed method FEM Proposed methodProportion of load carried by pile (%) 65.6 68.7 66.7 74.0Settlement at crest of embankment (mm) 32 34 166 156

At pile cap elevationSettlement on cap (mm) 12 21 148 141Settlement on soil (mm) 32 43 168 165Differential settlement between cap and soil (mm) 20 22 20 24

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SRW supported by geogrids and jet grout columns. In Proceed-ings of Sessions of ASCE Specialty Conference of PerformanceConfirmation of Constructed Geotechnical Facilities. Edited byA.J. Lutenegger and D.J. DeGroot. ASCE Geotechnical SpecialPublication No. 94, pp. 456–466.

Brinkgreve, R.B.J., and Vermeer, P.A. 2000. Plaxis Manual, Ver-sion 7.2. A.A. Balkema, Rotterdam. Delft, The Netherlands.pp. 5.1–5.18.

British Standard. 1995. Code of practice for strengthened/rein-forced soils and other fills. BS 8006. British Standard Institu-tion, London. p. 162.

Chen, R.P., Xu, Z.Z., and Chen, Y.M. 2007. Research on key pro-blems of pile-supported reinforced embankment. China Journalof Highway and Transport, 20: 7–12. [In Chinese.]

Coduto, D.P. 2001. Foundation design. 2nd ed. Prentice-Hall, Inc.,Upper Saddle River, N.J.

Geddes, J.D. 1966. Stress in foundation soil due to vertical subsur-face loading. Geotechnique, 16: 231–255.

Han, J., and Gabr, M.A. 2002. Numerical analysis of geosynthetic-reinforced and pile-supported earth platforms over soft soil.Journal of Geotechnical and Geoenvironmental Engineering,128: 44–53. doi:10.1061/(ASCE)1090-0241(2002)128:1(44).

Hewlett, W.J., and Randolph, M.F. 1988. Analysis of piled em-bankments. Ground Engineering, 21: 12–18.

Huang, J., Han, J., and Porbaha, A. 2006. Two and three-dimen-sional modeling of deep mixed columns under embankments. InProceedings of the ASCE GeoCongress, Atlanta, Ga., 27 Febru-ary – 1 March 2006. American Society of Civil Engineers, NewYork. pp. 1–5.

Jones, C.J.F.P., Lawson, C.R., and Ayres, D.J. 1990. Geotextile re-inforced piled embankments. In Proceedings of 4th InternationalConference on Geotextiles, Geomembranes and Related Pro-ducts, The Hague, The Netherlands, 28 May – 1 June 1990. In-ternational Geosynthetics Society. pp. 155–160.

Low, B.K., Tang, S.K., and Choa, V. 1993. Arching in piled em-bankments. Journal of Geotechnical and Geoenvironmental En-gineering, 120: 1917–1938.

NGG. 2002. Nordic handbook – Reinforced soils and fills. NordicGeotechnical Society, Stockholm. Available from www.nordicinnovation.net/_img/nordisk_handbok_armerad_jord-engelsk.pdf.

Pham, H., Suleiman, M.T., and White, D.J. 2004. Numerical analy-sis of geosynthetic-rammed aggregate pier supported embank-ments. In Proceedings of Geo-Trans 2004 Conference:Geotechnical Engineering for Transportation Projects, Edited byM.K. Yegian and E. Kavazanjian. ASCE Geotechnical SpecialPublication No. 126. Vol. 1, pp. 657–664.

Poulos, H.G. 2007. Design charts for piles supporting embankmentson soft clay. Journal of Geotechnical and Geoenvironmental En-gineering, 133: 493–501. doi:10.1061/(ASCE)1090-0241(2007)133:5(493).

Randolph, M.F., and Wroth, C.P. 1978. Analysis of deformation ofvertically loaded piles. Journal of the Geotechnical EngineeringDivision, ASCE, 104: 1465–1488.

Terzaghi, K. 1943. Theoretical soil mechanics. John Wiley & Sons,New York.

Wong, K.S., and Teh, C.T. 1995. Negative skin friction on piles inlayered soil deposits. Journal of Geotechnical Engineering ASCE,121: 457–465. doi:10.1061/(ASCE)0733-9410(1995)121:6(457).

List of symbols

Ai plane area of inner column (and pile cap)Ao plane area of outer cylinder

Ap area of pileAs area of cross section of soilci interface cohesioncs cohesion of the soilcu unconsolidated–undrained triaxial compress strengthc’ effective cohesion of the soil at the toe

Di equivalent diameter of capDo outer diameter of cylinderDp pile diameterEc Young’s modulus of embankment fillEs constrained modulus of soilEp Young’s modulus of pileF friction on slide interface of embankment

FS specified factor of safety of pile capacityG shear modulus of soilh embankment height

he elevation of equal settlementK coefficient of earth pressure between pile and soil

Ko coefficient of earth pressure at rest of the embank-ment fill

kb soil stiffness at pile toeks shear stiffness of soil along pile shaftL pile lengthm degree of cap coverage

NC normally consolidated and saturated soilNc, Nq coefficients of bearing capacity at pile toe

OCR overconsolidation ratioPi vertical pressure in inner columnPo vertical pressure in outer cylinderPT maximum normal force in pilePu summation of ultimate positive friction along pile

and end bearing capacityqu ultimate toe resistanceRg total downdrag along pile

Rinter strength reduction factorro pile radiusrm influence radius of pileSb settlement of substratumSe differential settlement between pile and soil at z = 0Si settlement of pile (cap) at z = 0So settlement of soil at z = 0U pile perimeter

Ws displacement of soil at z = 0 ~ LWp displacement of pile at z = 0 ~ L� coefficient of skin resistance

�NC coefficient of skin resistance of normally consoli-dated and saturated soil

�c unit weight of embankment fill�’ submerged unit weight of soil angle of external friction between pile and soil 0s Poisson’s ratio of soil� ratio of shear modulus in middle of pile to that at

pile toe�0v effective overburden pressure� skin resistance of soil�u ultimate skin resistance of soil4c internal friction angle of embankment fill4i interface friction angle4s friction angle of soil

Appendix A

Substituting eq. [13a] into eq. [14a], taking eq. [6] intoconsideration yields

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½A1� F1 ¼ � Ai

EpAp

Pið0Þ

½A2� F3 ¼ � 1

Es

Poð0Þ

½A3� F4 � F2 ¼ Se

Considering eqs. [2], [3], and [6], eqs. [A1], [A2] and [A3] can be expressed by he as

½A4� F1 ¼ � Ai�cDi

4EpApK0tan’c

1þ 4K0tan’c

Di

ðh� heÞ� �

exp 4K0tan’c

he

Di

� � 1

�

½A5� F3 ¼�ch

Esð1� mÞ �mAi�cDi

4Esð1� mÞEpApK0tan’c

1þ 4K0tan’c

Di

ðh� heÞ� �

exp4K0he

Di

tan’c

� � 1

�

½A6� F4 � F2 ¼�cD

2i ðAo þ AiÞ

16K0EcAotan2’c

1þ 4K0tan’c

Di

ðh� heÞ� �

exp4K0he

Di

tan’c

� � 1

� �� he

� þ ðAi þ AoÞ�c

AoEc

h2e2� hhe

�

Substituting eq. [13b] into eqs. [18a] and [18b] results in

½A7� F5

�e�z1 � e��z1� �

þ F6 e�z1 � e��z1� �

¼ � �u

ks

½A8� F5

�e�z2 � e��z2� �

þ F6 e�z2 � e��z2� �

¼ �u

ks

By solving eqs. [A7] and [A8], F5 and F6 can be written as

½A9� F5 ¼�u

ks

�ð1þ e�z1þ�z2Þ2ð�e�z1 þ e�z2Þ

½A10� F6 ¼�u

ks

ð�1þ e�z1þ�z2Þ2ðe�z1 � e�z2Þ

Substituting eqs. [13a], [13b], and [13c] into Eqs. [16], [17], and [15b] yields

½A11� � 1

6�pz

31 þ z1F1 þ F2 ¼

�pp

�3ðe�z1 � e��z1ÞF5 þ

�pp

�2ðe�z1 þ e��z1ÞF6 þ z1F7 þ F8

½A12� 1

6�sz

31 þ z1F3 þ F4 ¼ ��ss

�3ðe�z1 � e��z1ÞF5 �

�ss

�2ðe�z1 þ e��z1ÞF6 þ z1F7 þ F8

½A13� � 1

2�pz

21 þ F1 ¼

�pp

�2ðe�z1 þ e��z1ÞF5 þ

�pp

�ðe�z1 � e��z1ÞF6 þ F7

½A14� 1

2�sz

21 þ F3 ¼ � �ss

�2ðe�z1 þ e��z1ÞF5 �

�ss

�ðe�z1 � e��z1ÞF6 þ F7

½A15��pp

�3ðe�z2 � e��z2ÞF5 þ

�pp

�2ðe�z2 þ e��z2ÞF6 þ F7z2 þ F8 ¼

1

6�pz2 þ z2F9 þ F10

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½A16� � �ss

�3ðe�z2 � e��z2ÞF5 �

�ss

�2ðe�z2 þ e��z2ÞF6 þ z2F7 þ F8 ¼ � 1

6�sz

32 þ z2F11 þ F12

½A17��pp

�2ðe�z2 þ e��z2ÞF5 þ

�pp

�ðe�z2 � e��z2ÞF6 þ F7 ¼

1

2�pz

22 þ F9

½A18� � �ss

�2ðe�z2 þ e��z2ÞF5 �

�ss

�ðe�z2 � e��z2ÞF6 þ F7 ¼ � 1

2�sz

22 þ F11

½A19� � 1

6�sL

3 þ F11Lþ F12 ¼ Sb

From eqs. [A13], [A17], [A18], [A19], [A16], [A15], [A11], and [A12], F7, F9, F11, F12, F8, F10, F2, and F4 can be written as

½A20� F7 ¼ � 1

2�pz

21 þ F1 �

�pp

�2ðe�z1 þ e��z1ÞF5 �

�pp

�ðe�z1 � e��z1ÞF6

½A21� F9 ¼�pp

�2ðe�z2 þ e��z2ÞF5 þ

�pp

�ðe�z2 � e��z2ÞF6 þ F7 �

1

2�pz

22

½A22� F11 ¼ � �ss

�2ðe�z2 þ e��z2ÞF5 �

�ss

�ðe�z2 � e��z2ÞF6 þ F7 þ

1

2�sz

22

½A23� F12 ¼ Sb þ1

6�sL

3 � LF11

½A24� F8 ¼ � 1

6�sz

32 þ z2F11 þ F12 �

�ss

�3ðe�z2 � e��z2ÞF5 þ

�ss

�2ðe�z2 þ e��z2ÞF6 � z2F7

½A25� F10 ¼1

6�pz

32 þ z2F9 �

�pp

�3ðe�z2 � e��z2ÞF5 �

�pp

�2ðe�z2 þ e��z2ÞF6 � F7z2 � F8

½A26� F2 ¼�pp

�3ðe�z1 � e��z1ÞF5 þ

�pp

�2ðe�z1 þ e��z1ÞF6 þ z1F7 þ F8 þ

1

6�pz

31 � z1F1

½A27� F4 ¼ � �ss

�3ðe�z1 � e��z1ÞF5 �

�ss

�2ðe�z1 þ e��z1ÞF6 þ z1F7 þ F8þ

1

6�sz

31 � z1F3

As a result, F1 to F12 are expressed in terms of z1, z2, and he. Substituting eq. [13c] into eq. [15a] leads to

½A28� 1

2�pL

2 þ F9 ¼� kb

Ep

1

6�pL

3 þ F9Lþ F10

0@

1A

� qu

Ep

8>>>><>>>>:

The three eqs. [A6], [A14], and [A28], including three unknowns z1, z2, and he, can be solved using the Newton–Raphsonmethod included in the Mathematics software Mathematica (Wolfram Research Inc., Champaign, Il.)and then all the otherunknowns and variables can be obtained accordingly. The formula for these solutions are not included in this paper becausethey are too large.

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