piles subjected to lateral soil movement

7/28/2019 piles subjected to lateral soil movement

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On limiting force profile, slip depth and response of lateral piles

Wei Dong Guo *

School of Engineering, Griffith University, Gold Coast 9726, Qld, Australia

Received 11 February 2005; received in revised form 30 January 2006; accepted 7 February 2006Available online 31 March 2006

Abstract

A wealth of numerical analyses have been performed for lateral piles using finite difference, boundary element, and finite elementmethods, etc. Their essence, especially for a free-head pile, is to simulate the mobilisation of the maximum (limiting) force along the pile.This paper presents new closed-form solutions for a free-head pile embedded in an elasticplastic, non-homogeneous soil by simulatingpilesoil interaction using a series of springs distributed along the pile shaft. The stiffness of each spring is theoretically related to soilmodulus, pilesoil relative stiffness, and loading properties; and the limiting force is catered for by a new generic expression. The solu-tions permit non-linear response of the pile to be readily estimated right up to failure. They compare well with continuum-based 3D finiteelement analysis of a pile embedded in stratified soils, irrespective of whether the limiting force (or pilesoil relative slip) is progressivelymobilised downwards as the solutions assume or not. Presented in explicit forms, the solutions allow the dominant limiting force profile(LFP) to be back-estimated against measured data, and may be used as a boundary element for simulating beamsoil interaction underlateral resistance. Ranges of input parameters are provided, in light of predictions carried out to date against 62 tested piles in clay andsand. Finally, study on three typical piles is presented to elaborate the calculation procedure. 2006 Elsevier Ltd. All rights reserved.

Keywords: Piles; Closed-form solutions; Non-linear response; Lateral loading; Soilstructure interaction

1. Introduction

A wealth of theoretical studies on response of laterallyloaded piles have been undertaken using a load transfer(py curves) model [1]. In the model, the pilesoil interac-tion is simulated using a series of independent (uncoupled)springs distributed along the pile shaft [2]. With the pycurves at any depths, solution of the response of the piles

is normally obtained using numerical approaches such asfinite difference method (e.g., COM624 [3]). This model,however, often offers notable different predictions againstcontinuum-based finite element analysis (FEA) (e.g. [4]).Derived from a displacement mode for soil around the pile,a new coupled load transfer model has been recently devel-oped [5] that allows the interaction among the springs to becaptured. Characterised by modulus of subgrade reaction

(k), and a fictitious tension (Np), the coupled model com-pares well with finite element analysis. However, it is con-fined to elastic state.

With increase in lateral loads, maximum (limiting) resis-tance along a pile is gradually mobilised. This limiting forceis the sum of the passive soil resistance acting on the face ofthe pile in the direction of soil movement, and sliding resis-tance on the side of the pile, less any force due to active

earth pressure on the rear face of the pile. Variation ofthe net limiting force per unit length pu with depth (referredto as LFP) is critical to pile design. Significant researcheffort has been made to date to construct the LFP, usingmainly the following three techniques of: force equilibriumon a passive soil wedge [6,7]; upper-bound method of plas-ticity on a conical soil wedge [8]; and a strain wedge modeof soil failure [9]. Around a pile, the wedge was assumed tohave developed near ground level; below which, lateralflow was assumed. The LFP is generally non-uniform withdepth even along piles embedded in a homogeneous soil, as

0266-352X/$ - see front matter 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compgeo.2006.02.001

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Computers and Geotechnics 33 (2006) 4767
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it was taken for granted [10]. The most popularly used pu(thus the LFP) was that derived from the force equilibriumon a wedge [7,11]. However, it is noted that the scope of the

wedge should associate with rotation center for a rigid pile

at one extreme, while the wedge may never occur around abatter (or capped) pile at the other extreme. A universalexpression of LFP should be applicable to any mode of soil

failure.

Nomenclature

AL coefficient for the LFP;su undrained shear strength of soil;

su average undrained shear strength over pile

embedment;d diameter of an equivalent solid cylinder pile;Ep Youngs modulus of an equivalent solid cylinder

pile;e eccentricity, i.e., the height from the loading

location to the mudline;Gs average shear modulus of the soil;G* average soil shear modulus, G* = (1 + 0.75ms)Gs;Ip moment of inertia of an equivalent solid cylin-

der pile;J empirical factor lying between 0.5 and 3 for esti-

mating Ng;k modulus of subgrade reaction;

k1 parameter for estimating the load transfer fac-tor, c;

Ki(c) modified Bessel function of second kind of ithorder;

Kp tan2(45 + / 0/2), the passive earth pressure coef-

ficient;L embedded pile length;Lc critical embedded pile length, beyond which the

pile is classified as infinitely long;LFP net limiting force profile;M(x) moment induced in a pile element at a depth of

x, also written as M;

MA(x), MB(z) moment induced in a pile element, atdepth x and z, respectively;Mx Mxkn2=AL, normalised bending moment at

depth x;Mmax maximum bending moment within a pile;n power for the LFP;N blow count of the SPT;Nco lateral capacity factor correlated soil undrained

strength with the limiting pilesoil pressure atmudline;

Nc equivalent lateral capacity factor correlatedaverage soil undrained strength with averagelimiting force per unit length on pilesoil inter-

face by pu suNcd;Ng gradient correlated soil undrained strength with

the limiting pilesoil pressure;Np fictitious tension for a stretched membrane used

to tied together the springs around the pile shaft;P lateral load applied at a distance of e above

mudline;

Pe value of the lateral load applied P when the slipdepth is just initiated at mudline;

p force per unit length;

pu limiting force per unit length;P Pkn1=AL, normalised pile-head load;pu average limiting force per unit length over the

pile embedment;r0 radius of an equivalent solid pile;x depth measured from mudline;xmax depth at which maximum bending moment

occurs (xmax = xp + zmax when wxpP 0;xmax xmax when wxp < 0 );

xp slip depth from the elastic to the plastic state;w lateral deflection;wA lateral deflection in the upper plastic zone;wIVA , w

000A, w

00A, w

0A fourth, third, second and first deriva-

tives, respectively, of deflection w with respectto the depth x;

wB lateral deflection in the lower elastic zone;wIVB , w

000B , w

00B, w

0B fourth, third, second and first deriva-

tives, respectively, of deflection w with respectto the depth x;

wg, w0g lateral pile deflection, and rotation angle (in

radian) at mudline, respectively;wg wgkk

n=AL, normalised mudline deflection;wp pu/k, lateral deflection at the slip depth of xp;wIVp , w

000p , w

00p, w

0p values of fourth, third, second and first

derivatives, respectively, of deflection w with

respect to the depth x at depth xp;wt lateral deflection;z depth measured from the slip depth;zmax depth below the slip depth, where maximum

bending moment occurs;a, b stiffness factors for elastic solutions using load

transfer model;ac ratio of the shear modulus over the undrained

strength, Gs/su;aN, bN a/k, and b/k, normalised a and b by k, respec-

tively;a0 equivalent depth to account for mudline limiting

force;

c load transfer factor;c0s effective density of the overburden soil;k the reciprocal of characteristic length;ms Poissons ratio of soil;/ 0 effective frictional angle of soil.

48 W.D. Guo / Computers and Geotechnics 33 (2006) 4767


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With the kdeduced from the coupled model, and the pufrom LFP, an ideal elasticplastic load transfer (py) curveat any depth can be constructed. Transition from the initialelastic to the ultimate plastic state may be assumed in otherforms of non-linear curves, such as those proposed for softclay, stiff clay, and sand, etc. [6,7,11]. In terms of instru-

mented piles embedded in uniform soils [7], these formsof py curves are proved to be very useful. However, tosynthesise the curves, a number of parameters are requiredto be properly determined [12], which is often time-con-suming, wearisome, and only warranted for large projects.In contrast, a simplified elasticplastic py curve is suffi-ciently accurate [13] and normally expedite.

A rigorous closed-form expression should be developedeven though there are numerical solutions, as the formercan then be used as a new boundary element to advancethe latter. Elastic, perfectly plastic solutions (typically forfree-head piles) have been proposed [14,15], using (uncou-pled) Winkler model. Valid for either a uniform or a line-

arly increase limiting force profile, they were neverthelessnot rigorously linked to properties of a continuum mediumsuch as shear modulus etc. Thus, a new approach isrequired to render response of a pile (e.g., loaddeflection,and loadmaximum bending moment relationships) to besimulated, which should be consistent with continuum-based analysis. It should also permit a new LFP to beback-estimated using measured responses of a pile regard-less of mode of soil failure.

This paper addresses the elasticplastic solutions of lat-erally loaded, free-head piles that are embedded in non-homogeneous medium, for which limiting force per unit

length varies monotonically with depth. The following tar-gets are set:

Propose a generic expression to describe the distributionof limiting force per unit length with depth, and find itsvalid depth (maximum slip depth).

Establish new closed-form solutions for piles in elasticplastic, non-homogenous soil.

Examine the effect of each input parameter by conduct-ing parametric analysis and case studies.

The study uses the 3D FE analysis for a pile in two strat-ified soils to verify the closed-form solutions. The solu-tions are implemented into a spreadsheet program tofacilitate the prediction of response of a number of testedpiles, and parametric analysis. Selection of input parame-ters is discussed at length. Three typical examples areelaborated.

2. Solutions for the pilesoil system

The problem addressed here is schematically depicted inFig. 1, where a laterally loaded pile is embedded in a non-homogenous elastoplastic medium. No constraint isapplied at the pile-head and along the effective pile length,

except for the soil resistance. The free length (eccentricity)

measured from the point of applied load, P to the groundsurface is written as e. The pilesoil interaction is simulatedby the model shown in Fig. 2(a). The uncoupled modelindicated by the pu is utilised to represent the plastic inter-action, and the coupled load transfer model indicated bythe k, and Np to portray the elastic pilesoil interaction,

respectively. The two interactions occur, respectively, inregions above and below the slip depth, xp. In otherwords, the following hypotheses are adopted:

(i) Each spring is described by an idealised elasticplas-tic py curve (y being written as w in this paper,Fig. 2(b)).

(ii) In elastic state, equivalent, homogenous and isotropicelastic properties (modulus and Poissons ratio) areused to estimate the k and the Np.

(iii) In plastic state, the interaction among the springs isignored by taking the Np as zero.

(iv) Pilesoil relative slip occurs down to a depth where

the displacement, wp is just equal to pu/k and netresistance per unit length pu is fully mobilised.

(v) The slip (or yield) can only occur from ground level,and progress downwards.

All of the five assumptions are adopted to establishclosed-form solutions presented in this paper. Influence

e

L

xp

x, or z

Plastic zone

Elastic zone

P

Pressure distribution

(L > Lc)

Fig. 1. The problem addressed.

(a) Coupled load transfer model (b) Normalised p -y (w) curve

P

Plastic

zone,xp

Elastic

zone

Membrane,Np

Spring, k

pu 0

0.5

1

0 1 2 3

Normalised deflection, w/d (%)

Normalisedforce,p/pu

Plastic zone

Elastic zone

Transition zone

Fig. 2. Coupled load transfer analysis.

W.D. Guo / Computers and Geotechnics 33 (2006) 4767 49


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of deviation from these assumptions are assessed and com-mented upon in the later sections.

2.1. Coupled load transfer model (elastic state)

Applying a variational approach to simulate elastic

response of the coupled pilesoil system shown inFig. 2(a) with xp = 0, the following conclusions drawn pre-viously [5] are directly adopted herein.

Infinitely long. A free-head pile is defined as infinitelylong, should the pilesoil relative stiffness, Ep/G* be lessthan a critical ratio, (Ep/G*)c [16], where (Ep/G*)c =0.052(L/r0)

4/(1 + 0.75ms), G* = (1 + 0.75ms)Gs, Gs, ms isthe shear modulus, and Poissons ratio of an equivalent,homogenous and isotropic elastic soil, respectively; L isthe embedded pile length; Ep and r0 are the Youngsmodulus, and radius of an equivalent solid cylinder pile,respectively. In other words, unless a slenderness ratio

L/r0 is less than Lc/r0, the laterally loaded pile can betreated as if it were infinitely long.

Modulus of subgrade reaction k. The modulus k [FL2]may be estimated by Eq. (1)

k 1:5pGsf2cK1c=K0c c2K1c=K0c

2 1g

1

where Ki(c) is the modified Bessel function of secondkind of ith order (i= 0, 1). This k is not the custom-ary one [5,17]. The factor c for a free-head, infinitelylong pile is a power function of the relative stiffness,i.e.,

c k1Ep=G

0:252

where k1 = 1.0, and 2.0, respectively, for a lateral loadapplied at ground surface, and at an infinitely higheccentricity (i.e., a pure moment loading). k1 lies in be-tween 1.0 and 2.0.

Fictitious tension Np. The tension, Np [F] of the mem-brane linking the springs is determined by

Np pr20GsK1c=K0c

2 1 3

2.2. Generic net LFP (plastic state)

A monotonic variation in the net limiting force per unitlength, pu with depth x (LFP) may be expressed by a gen-eric expression of

pu ALx a0n 4

where AL is the coefficient for the LFP [FL1n]; x is the

depth below mudline [L]; a0 is an equivalent depth to coverthe force at x = 0 [L]; pu is the net limiting force per unitlength [FL1]; n is a constant ( Lc(infinitely long pile) means that the pu is generally fullymobilised from mudline to the slip depth, xp that increaseswith lateral loads, as indicated in Fig. 1. Thus, the LFPmay be conveniently plotted using Eq. (4), but it is effectiveonly to the maximum xp.

In determining the parameters Ng, Nco, and n for Eq. (4),

six options may be considered.

Option 1. For piles in a cohesionless soil, should the LFPbe taken as that suggested by [10], referred to as BromsLFP later on, the parameters should be adopted as

Ng 3Kp; Nco 0; n 1 5b

where Kp = tan2 (45 + / 0/2), the passive earth pressure

coefficient; / 0 is the effective friction angle of the soil. Option 2. For piles in a cohesive soil, if the LFP pro-

posed by [6,7] is adopted, then the parameters shouldbe given by

Ng c0sd=su J; Nco 2; n 1 5c

where J is a factor lying between 0.5 and 3 [6]. TheLFP obtained using Eqs. (4) and (5c) will be referredto as Matlock using J= 0.5; and as Reese (C) ifJ= 2.8 [7].

Option 3. An available LFP [19] may be expressed bychoosing a set of parameters Ng, Nco and n. Forinstance, the LFP for a pile in sand employed inCOM624P, which is referred to as Reese (S), was wellfitted using Ng K

2p; Nco 0, and n = 1.7 [20].

Option 4. Should a set of py curves be available,numeric value of pu may be acquired from the curves

for each depth, hence the LFP is resulted.



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Option 5. For a layered soil profile, the LFP may be con-structed using the interim procedure as follows. Firstly,assuming that the entire soil consists of the clay or thesand only, LFP of Reese (C) or (S) is obtained, respec-tively. Secondly, the pu within a zone of 2d above orbelow an interface should be increased in average by

$40% for a weak (clay) layer adjoining a stiff (sand)layer; and decreased by $30% for a stiff layer adjoininga weaker layer [21]. Thirdly, the increased and thedecreased pu of the two adjacent layers is representedaveragely by Eq. (4), with the n being gauged visually,as an exact shape of the LFP (thus n) makes little differ-ence to the final predictions (shown later in Case stud-ies). And finally, a LFP is created for a two-layeredsoil. For a multiple layered soil, the same principlesapply, but the n (thus the LFP) should permit the overalllimiting force to be represented. Any layer located morethan 2d below the maximum slip depth is excluded inthis process. Lastly,

Option 6. Should measured pile response be available,the parameters Ng, Nco, and n may be back-figuredthrough matching predicted with measured responsesof a pile as elaborated in the section of back-estimationof LFP.

Use of options 15, and 6 are illustrated in sections ofValidation, and Case studies, respectively.

2.3. Elasticplastic solutions

For convenience, response of the pile is denoted by the

subscripts A and B for the upper plastic, and the lowerelastic zones, respectively (see Fig. 2(a)). Above the depthxp, using the uncoupled model for the plastic zone (x 6 xp),the governing equation for the pile is

EpIpwIVA ALx a0

n 0 6 x 6 xp 6

where wA is the deflection of the pile at depth x that ismeasured from ground level; wIVA is the fourth derivativeof wA with respect to depth x; Ip is the moment of inertiaof an equivalent solid cylinder pile. Below the depth xp,using the coupled model for the elastic zone (xp 6 x 6 L),the governing equation for the pile may be written as[5,22]

EpIpwIVB Npw

00B kwB 0 0 6 z6 L xp 7

where wB is the deflection of the pile at depth z(=x xp) that is measured from the slip depth. w

IVB ,

w00B is the fourth, and second derivatives of wB with re-spect to depth z. Taking Np = 0, Eq. (7) reduces to theone based on Winkler model. By invoking the deflectionand slope (rotation) compatibility restrictions at x = xp(z = 0) for the infinitely long pile, Eqs. (6) and (7) weresolved as elaborated in Appendix I. The solutions allowresponse of the pile at any depth to be predicted readily.In particular, three key responses were recast in dimen-

sionless forms, which are:

Normalised load, P (=Pkn+1/AL), where P is a lateralload applied at a distance of e above mudline; k is thereciprocal of characteristic length, k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik=4EpIp

4p

. Normalised mudline deflection, wg (=wgkk

n/AL), wherewg is the pile deflection at mudline.

Normalised bending moment, Mx (=M(x)kn+2/AL),

where M(x) is the bending moment in the pile at depthx.

The key responses are described as follows.

2.3.1. Lateral load

Using Eq. (A-26), an expression for the normalised lat-eral load is derived as

P F1; 0xp aN

xp aN e

F2;xp F2; 0 aNF1;xp F0;xp=2

xp aN e 8

where xp kxP, normalised slip depth using k; e ke, nor-malised eccentricity; and

aN a=k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 Np=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi4EpIpk

pq

a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik=4EpIp

q Np=4EpIp

r9a

and

Fm;xp xp a0knm

Ym

j1

n j

,10

whereQm

j1n j n m n 2n 1, m is theinteger (6 4). For instance, when m = 2, Fm;xp becomes

F2;xp, andQ2

j1n j n 2n 1. In particular, atm = 0,

Q0j1n j is taken as 1.

2.3.2. Groundline deflection

Substituting Eqs. (A-21) and (8) into Eq. (A-23), thenormalised pile deflection at ground level is obtained as

wg 4F4;xp xpF3;xp F4; 0 Cx2F2;xp

Cx1F1;xp Cx0F0;xp Co2F2; 0

Co1F1; 0 11

where

Cx2 41 a2

N Cm; Cx1 2 4a2

Nxp aNCm;

Cx0 1 2aNxp Cm=2; Co2 2x2p Cx2;

Co1 4x3p=3 2xp xp aNCm;

Cm 4

3x3p 2xp 2x

2p 4a

2N 4e

1

xp aN e

12

At a relative small eccentricity, e, pile-head deflection, wtmay be approximately taken as wg e w

0g, where w

0g is

the mudline rotation angle (in radian) obtained using Eq.

(A-22).

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2.3.3. Maximum bending moment

The maximum bending moment, Mmax occurs at a depthxmax (or zmax) at which shear force is equal to zero. Thedepth could locate in plastic or elastic zone, thus it needsto be determined by using either QA(xmax) = 0 in Eq. (A-1), or QB(zmax) = 0 in Eq. (A-30). Which zone the Mmax lies

in depends on a function, wxp that itself is derived fromEq. (A-30)

wxp bN=aN 2kw00P=w

000P 13

where

bN b=k

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 Np=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi4EpIpk

pq

b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik=4EpIp

q Np=4EpIp

r9b

w000P kkn=AL 4k

3F1;xp F1;0 P 14

w00Pkkn=AL 4k

2F2;xp F2;0 F1;0xp xp eP

15

The depth zmax may be obtained from Eq. (A-30) as

zmax tan1wxp=b 16

which expression to be used for determining the Mmax de-pends on the wxp.

(a) If wxp > 0, then zmax > 0. The maximum momentMmax should occur in elastic zone, thus it locates at adepth (measured from mudline) that is equal to xp +zmax. The value of Mmax may be estimated using Eq.(17) by replacing z with zmax, i.e., Mmax = MB(zmax).

MBz eaz

C1xp cosbz C2xp sin bz 17where C1xp EpIpw

00Pk

n2=AL, C2xp EpIpw000P

aw00Pkn2=bAL. MB(z) is the bending moment at

depth z that is derived from Eq. (A-29).(b) Ifwxp < 0, then zmax < 0, which is expected at a rel-

atively high value of xp. The Mmax should locate inplastic zone, and at depth xmax. The normalised depth

xmax (=xmaxk) is obtained, using Eq. (A-1), as

xmaxk a0kn1 n 1P1

=n1 a0k 18

The Mmax may be calculated using Eq. (19), derivedfrom Eq. (A-2) by replacing x with xmax.

Mmax

AL

1

n 2an10 n 1

P

AL

n2=n1

an20

n 2a0P

AL

Pe

AL19

The Mmax of Eq. (19) may be recast in form of thenormalised slip depth as well by replacing the P withthat given by Eq. (8). The form of Eq. (19) is to stressits independence of the characteristic length but AL,a0, n (hence the LFP), and the load P. However, thenormalised form, Mmaxk

n+2/AL will be used later toprovide a consistent presentation (e.g., Figs. 7 and

8) from elastic through to plastic state.

In summary, responses of the laterally loaded piles arepresented in explicit expressions of the slip depth, at whichthe maximum pu normally occurs. The expressions arevalid for infinitely long (L > Lc) piles that are embeddedin a soil of a constant modulus (k) with depth. This con-stant may be regarded as an average value over the maxi-

mum slip depth expected (initially taken as 8d) belowground level. Derived from Eq. (4), the new solutions arenot based on mode of soil failure. Influence of the failuremode is catered for by selecting different parameters and/or types of solutions, such as the current solutions for afree-head pile, and those for a fixed-head pile [23] etc.

2.4. Some extreme cases

The normalised slip depth under lateral loads may beestimated using Eq. (8) that associates with the LFP (viaAL, a0, n), and the pilesoil relative stiffness (via k). The

minimum load, Pe required to initiate the slip at mudline(xp 0) is given by

Pekn1=AL a0k

n=2aN e 20

The current solutions can be reduced to available solutionsfor some special cases.

(a) Imposing n = e = 0, aN = 1, and a0 = 0, the norma-lised load by Eq. (8) reduces to

Pk=AL 1 xp=2 21

And the normalised deflection by Eq. (11) reduces to

wgk=AL x4p=6 2x3p=3 x2p xp 1 22

Eqs. (21) and (22) are essentially identical to thosebrought forwarded previously [24] using Winklermodel (Np = 0) for a pipeline that is embedded in ahomogenous soil, and has a constant (n = 0) limitingforce (resistance) along its length.

(b) Setting a0 = 0, and e = 0, Mmax obtained from Eq.(19) then reduces to P2/(2AL), and

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8P3=9AL

p, respec-

tively, for n = 0, and 1; and the corresponding depthxmax derived from Eq. (18) becomes P/AL, and

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2P=AL

p. These results are what were put forwarded

previously [25].(c) Introducing xp 0 (elastic state), and at e = 0, Eq.

(15) offers that w00P 0. Furthermore, using Np = 0in Eqs. (16) and (17), zmax and Mmax obtained, respec-tively, are virtually identical to the results obtainedusing Winkler model [24].

2.5. Numerical calculation and back-estimation of LFP

In spite of the complicated appearance the expressionssuch as Eqs. (8), (11), (13) and (17) can be readily estimatedusing modern mathematical packages. In this paper, they

have been implemented into a spreadsheet operating in

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EXCEL called GASLFP, along with Eqs. (1)(5) and theexpressions shown in Appendix I. At a0 = 0, and aN = 1,simplified forms of Eqs. (8), (11), (13) and C1xp; C2xpfor Eq. (17) have also been obtained and provided inAppendix II.

Using Eq. (8), the slip depth, xp may be obtained itera-

tively using a purpose- designed macro under a lateral load,P, as is adopted in GASLFP. The slip depth, xp may alsobe assumed to estimate the load, P when the simplifiedexpression is employed. In either approach, the normalisedslip depth, xp may be calculated with the k, which allowsthe ground-level deflection, wg to be computed, the wxpto be calculated. The latter in turn permits the maximumbending moment, Mmax and its depth, xmax to be evaluated.The calculation is repeated for a series of P or xp, eachoffers a set of load, deflection, bending moment and itsdepth, thus the P wg, Mmax wg, and xmax Mmaxcurves etc are determined. All of the numerical values ofthe current solutions presented subsequently are obtained

using GASLFP, except where specified. For comparison,the predictions using simplified expressions are sometimesprovided as well.

Using GASLFP, the three input parameters Ng, Nco andn (or AL, a0, and n) may also be adjusted until the predictedwg, Mmax and xmax across all load levels agree well with thecorresponding measured data (if available), respectively.The values thus obtained should reflect the overall pilesoilinteraction rather than a real limiting force profile in thecase of a layered soil. They are unique, so is the corre-sponding LFP (option 6). This back-estimation may alsobe carried out using other expressions given in Appendix

I, for instance, Eq. (A-22) for rotation. If only one mea-sured (normally wg) response is available, Ng may beback-estimated by taking Nco as 04, and n as 0.7 and1.7 for clay and sand, respectively. Should two measured(say wg and Mmax) responses be available, both Ng and nmay be back-estimated by an assumed value of Nco. Mea-sured responses should encompass the integral effect of allintrinsic factors on piles in a particular site, so should thecorresponding back-figured LFP. In this manner, LFPcan be gradually updated to reflect the effect for futuredesign.

3. Validation against FEA

Yang and Jeremic [4] reported a 3D finite element anal-ysis (FEA) of a laterally loaded pile. The square aluminiumpile was 0.429 m in width, and 13.28 m in length. It had aflexural stiffness EpIp of 188.581 MN m

2. The pile wasinstalled to a depth of 11.28 m in a deposit of claysandclay profile, and sandclaysand profile, respectively. Lat-eral loads were applied at 2 m above ground level. Theclaysandclay profile refers to a uniform clay layer thathas a uniform interlayer of sand between a depth of1.72 m (4d, d= width of the pile) and 3.44 m (i.e. 8d). Acontrasting case was a clay layer sandwiched between

two sand layers and is refereed to as the sandclaysand

profile. In either profile, the clay has a undrained shearstrength of 21.7 kPa, Youngs modulus of 11 MPa, Pois-sons ratio of 0.45, and unit weight of 13.7 kN/m3. Themedium dense sand has an internal friction angle of37.1, shear modulus of 8.96 MPa at the level of pile base,Poissons ratio of 0.35, and unit weight of 14.5 kN/m3.

The FE analysis of the pile offered the following resultsfor either soil profile: (1) py curves at depths up to 2.68 m;(2) pile-head load and mudline displacement relationship;(3) mudline displacement and maximum bending momentcurve; and (4) profiles of bending moment under 10selected load levels. In this study, with the py curves,the limiting p (i.e., the pu) at each depth was approximatelyevaluated, thus the variation of pu with depth (i.e., LFP)was obtained (option 4), and is shown in Figs. 3 and 4(a)as FEA, respectively. Using the bending moment profiles,the depths of maximum bending moment were estimated,and are shown in Figs. 3 and 4(d), respectively.

In the current predictions of the pile embedded in the

claysandclay profile, an average shear modulus of thesoil Gs was calculated as 759.5 kPa (=35su [26]). This ren-ders k, and Np to be estimated as 2.01 MPa, and892.9 kN using Eqs. (1) and (3), respectively. Followingoption 5 for constructing LFP for a stratified soil, LFPof Reese (C) or (S) was obtained, respectively, using prop-erties of the clay or sand layer. The pu at depth (24)d ofthe clay layer was approximately increased in average by30% due to the underlying stiff sand layer, while at depth(46)d, it was approximately reduced by 20%, due to theoverlying weak clay layer [21]. A smooth transition fromthe upper to the low layer allows the n to be gauged visually

as 0.8 (Fig. 3). The bottom clay layer is ignored as justifiedlater. Thus, the current LFP for the pile in the stratified soilis described by n = 0.8, Ng = 6, and a0 = 0.

The predictions using the current LFP compare wellwith the FEA results, in terms of mudline displacement(see Fig. 3(b)), and maximum bending moment (seeFig. 3(c)). Nevertheless, the depth of the Mmax, xmax isoverestimated by up to 20%, because the current predic-tions use an equivalent homogenous medium through theconstant k [2], and the stratified profile reduces the depthxmax. The maximum slip depth xp was calculated as 4.63dunder the maximum pile-head load. Thus, the bottom claylayer located at 3.37d (=8d 4.63d) below the maximumxp can be ignored. The current pu between 1.5d and 8dslightly exceeds what would be obtained using the previousinstruction [27] for a layered soil that is not shown herein.However, it is quite compatible with the overall trend ofFEA result within the maximum slip depth. Interestingly,replacing the Current LFP with the Reese (C), the currentsolutions still offer good predictions to 30% of the maxi-mum load, to which the response is dominated by theupper layer.

For the pile in the sandclaysand profile, shear mod-ulus, Gs was obtained as 1.206 MPa by averaging that ofthe sand at mid-depth (=0.5 8.96 MPa) and of the clay

(=0.759 MPa). This leaded to that k= 3.33 MPa and

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Np = 5.731 MN. Using option 5, LFPs of both Reese (S)and Broms were ascertained using the sand properties;and that of Reese (C) using the clay ones. Importantly,the effect of the bottom sand layer on LFP is consideredin this case, as the interface (located at depth of 8d) is lessthan 2d from the maximum slip depth xp of 6.75dobtained subsequently. In comparison with the BromsLFP, the pu was maintained unchanged in depth of (04)d, and it was reduced approximately by 20% in thedepth of (68)d. For instance, pu=c

0sd

2 at depth of 8dwas found to be 70.0. An overall fit to the LFP of thetop layer and the point (70, 8) allows the Current LFPto be described by using n = 0.8, Ng = 16.32[=14.5 tan4(45+37.1/2)], and a0 = 0. The current pre-dictions compare well with the FE analysis in terms ofthe mudline displacement, wg (see Fig. 4(b)), and maxi-mum bending moment, Mmax (see Fig. 4(c)). Only thedepth of the Mmax was overestimated at a load less than270 kN, probably due to the same reason as explainedpreviously for claysandclay profile. The maximum slip

depth xp was found to be 6.75d at a load of 400 kN.

The current LFP is close to the overall trend of thatobtained from FEA within the xp.

For the pile in sandclaysand profile, it is noted thatat the depth of 4d, the wp is calculated as 25.2 mm(=9 21.7 0.429/3.33) using the clay strength; but it is39.7 mm (=85.71 (4 0.429)0.8/3330) using the sandproperties and AL = 85.71 kN/m

1.8. This implies thataround the depth of the interface, pilesoil relative slipoccurred in the clay before it did in the overlying sand.This sequence is unlike the assumption of slipprogressing downwards. The very good comparisonsnoted above indicate that (1) response of the pile ismainly affected by the overall trend of the LFP withinthe maximum slip depth xp; and (2) Current solutionsare sufficiently accurate for the stratified soil by employ-ing n = 0.8 to describe the LFP; and using xp to repre-sent a transition zone that should occur if non-linearpy curves [4] are employed. To decide whether a layercomes into effect, initial xp may be based on an educatedguess, say, 8d. These conclusions are further enhanced by

the facts that:

(a) (b)

(c) (d)

Fig. 3. Comparison between the current predictions and FEA results [4] for a pile in claysandclay layers.



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Should the LFP be replaced with that of Broms or Reese(S), excellent predictions up to a load of 400 kN wouldalso be achieved. Beyond the load level, the displace-ment, wg and the moment, Mmax would be graduallyunderestimated.

Should the gradient of the LFP reduce by 10%, betterpredictions against the FEA results would be achieved.

Should the Georgiadis proposal be adopted, an overes-timation ofwg and Mmax would be expected, as a muchlower gradient LFP than the current one would beresulted.

The later two analyses are not shown herein.

4. Pile response versus slip depth

The previous section indicates that responses of a later-ally loaded pile are generally dominated by the LFP withinthe maximum slip depth. Taking e = 0 (thus wt = wg), nor-malised responses such as P, wg, Mmax, and so forth are pri-

marily affected by the three non-dimensional parameters n,

a0k and xp. The ranges of these parameters were identifiedlater in Comments on use of current solutions as: n = 0.52.0, a0k < 0.3, and xp < 2:0, for which parametric analysiswas carried out using GASLFP and presented below.

Fig. 5(a), and (b) indicate that:

(1) The normalised load, P is generally below 2.(2) With n > 0, and xp < 2:0, P normally increases with

decrease in n, and/or increase in a0k.(3) As P increases, the slip at n > 0 expands at a gradually

slowrate[Therate refersto net increasein the slip depthover corresponding increase in the load], as implied bythe increasing gradients of the concave curves.

Fig. 6(a), and (b) indicate that

(1) wg is1020timesthevalueofP,andnormallyshouldbebelow 50. For instance, atxp 2:0 (n = 0.5, a0k = 0),itis noted thatP 1:38 and wg 21:1, respectively.

(2) wg follows an opposite trend to P in response to n and

a0k.

(b)

(d)(c)

(a)

Fig. 4. Comparison between the current predictions and the FEA results [4] for a pile in sandclaysand layers.



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In comparison with Fig. 6, Fig. 7 demonstrates thatresponse of the normalised moment, Mmax to the load,

P resembles that of the deflection wg to the load P. Fur-thermore, the figure together with Fig. 8 demonstratesthat:

(1) Mmax and xmax are below 5.0, and 3.0, respectively.(2) As xp increases from 2 to 3 (n = 1.0), xmax increases by

35%, and Mmax increases by 250%.(3) At xp 2, and n = 1, as a0k increases from 0 to 0.2, P

increases by 20% (from 1.47 to 1.77), Fig. 7(b), and

Mmax by 13% (from 1.68 to 1.90), Fig. 8(b). It seems

that wg and Mmax are more susceptible to xp than Pand xmax are, which are also demonstrated in Table4 discussed next.

5. Case studies

Predictions using the current solutions are presented forthree typical piles embedded in a two-layered silt, a sand-silt layer, and a stiff clay, respectively. Highlighted in bold,solid lines in the subsequent figures, the cases of best pre-

dictions or match with measured data are obtained using

a b

Fig. 6. Normalised load and groundline deflection (e = 0).

a b

Fig. 5. Normalised load versus slip depth (e = 0).



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the parameters Ng, Nco, and n (thus the LFPs) that areprovided in Table 1, along with six other input parameters.These predictions are elaborated individually in thissection.

5.1. Piles tested in two-layered silt, and sand-silt layer

Single piles A, and C were driven into two differenttypes of two-layered soils, respectively. They were tested

a b

Fig. 7. Normalised load versus maximum bending moment (e = 0).

a b

Fig. 8. Normalised moment versus depth xmax (e = 0).

Table 1Parameters for the predictions of bold lines in Figs. 911

Figures su (kPa) or (N)* ac or (Gs/N)* EpIp (MN m2) e (m) L (m) r0 (m) Ng/Nco Nc=a0k n

9 15 121 298.2 0.1 17.4 0.305 4.53/3.28 5.8/.08 0.510 (12)* (640)* 169.26 0.2 23.3 0.305 8.3/4.67 1.011 153 545.1 493.7 0.31 14.9 0.32 0.85/0.0 1.67/0 1.5

NB, indicated by *, the values refer to N (blow count of SPT); and Gs/N (ratio of shear modulus and blow count) in kPa, respectively.



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individually and instrumented to measure the bending straindown the depth [28]. Pile A was 17.5 m in length, and 0.61 min diameter. It had a flexural stiffness EpIp of 298.2 MN m

2.The pile was driven into a two-layered silt, with a uniformundrained shear strength, su of 15 kPa to a depth of 4.8 m,and 22.5 kPa below the depth. Gs was taken as 121su (su =

15 kPa) [28]. Thus, critical slenderness ratio, Lc/r0 was esti-mated as 26.06, and the pile was classified as infinitely long.With the shear modulus, Gs, pilesoil relative stiffness,

Ep/G* was calculated; factor c was obtained using Eq.(2); modulus of subgrade reaction, k using Eq. (1); norma-lised fictitious tension, Np/(2EpIp) using Eq. (3); stiffness

factors, a and b, using Eqs. (9a) and (9b); and k using itsdefinition. All of these values for the elastic state are sum-marised in Table 2(a). The LFP for the upper layer(su = 15 kPa) was determined using the profile of Reese(C). Similarly, for the lower layer (xP 7.87d= 4.8 m),the pu was found to be a constant of 9d 22.5 kN/m, which

gives a pu/(sud) of 13.5 using the upper layer su of 15 kPa asthe normaliser. Using option 5, the overall LFP for thetwo-layered soil should be close to the Reese (C) near theground level, and pass through point (13.5, 7.87), as indi-cated by n = 0.5 in Fig. 9(a). The LFP may be expressedby using n = 0.5, AL = 53.03 kPa/m

0.5, and a0 = 0.32 m in

Table 2(a)Parameters for the bold lines, Case (I) elastic state

Piles EP/G* c k (MPa) NP/(2EpIp) a (m1) b (m1) k (m1)

A 18587.7 0.0856 5.378 0.0169 0.2750 0.2422 0.2591C 2739.2 0.1382 26.400 0.0674 0.4808 0.4047 0.4444

Notes. Gs = 1.82 and 7.68 MPa for piles A and C, respectively. If Np = 0, then a = b = k.

(a) (b)

(d)

(c)

Fig. 9. Comparison between the calculated and measured [28] response of Pile A.



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Eq. (4). The three parameters are thereby obtained usingEq. (5a) as Ng = 4.53, Nco = 3.28, and n = 0.5. The higherstrength su of 22.5 kPa of the lower layer renders 050%(an average of 25%) increase in the pu in the depth of (47.87)d, which is similar to the effect of an interlayer of sandon its overlying clay deposit [8,21].

The above-mentioned parameters offer excellent predic-tions of the pile response as shown in Fig. 9. They allow theresponses shown in Table 2(b) to be obtained as well, whichencompass:

(a) Minimum lateral load, Pe for initiating the slip.(b) Lateral load, P** for the slip depth shown in Fig. 9.(c) Maximum imposed lateral load, Pmax.(d) Slip depth, xp under the Pmax.

Those critical loads (Pe, P**, and Pmax) and the slipdepth are useful to examine the depth of influence of eachsoil layer. Pilesoil relative slip occurred along the pile A at

a low load, Pe of 54.6 kN, but it touched the second layer(i.e., xp = 4.8 m) at a rather high load, P** of 376.9 kN.Influence of the lower layer on the pile A seems to be wellcatered for by the LFP. The above study is referred to asCase (I).

To examine the effect of the parameters on the predic-tions, against Case (I) the following investigations (seeTable 3) are made:

Case (II). Taking Np as 0, the responses obtained, andshown in Fig. 9 as n = 0.5 (Np = 0), are slightly higherthan those obtained otherwise, indicating the elasticcoupled interaction is limited.

Case (III). Assuming a0 = 0, the LFP reduces to the tri-angular dots shown in Fig. 9(a). For this LFP, using the

simplified expressions (Np = 0, Appendix II), the pre-dicted responses are indicated by (S. Eqs) in Fig. 9.They are close to those bold lines obtained earlier(a0 6 0, Np 6 0). Furthermore, assuming e = 0, theresponses for xp = 1, and 5 are hand-calculated as illus-trated in Appendix II. The results are summarised inTable 4 along with those obtained for xp = 3, and 8.

Case (IV). A new LFP of n = 0.4 described byNg = 4.79, Nco = 3.03 and n = 0.4, is utilised, whichoffers an equivalent resistance force to that estimatedusing the Reese (C) profile along the pile within the max-imum xp of 8.36d. This LFP offers very good predictionsagainst the measured data as well. Thus, the n value

has limited effect on the predictions of response of thepile, and use of visual assessment of n value is deemedsufficiently accurate.

Pile C was 23.3 m in length, and 0.61 m in diameter, andhad an EpIp of 169.26 MN m

2. It was driven through asand layer that extended to 15.4 m from the ground level,and subsequently into a underlying silt layer with a su of

Table 2(b)Parameters and predictions for the bold lines, Case (I) plastic state

Piles Input parameters Predictions

Ng or c0s (kN/m

3), / (deg) AL (kN/mn+1) a0 (m) Pe (kN) P** (kN) Pmax (kN) xpxp at Pmax

A Ng = 4.53 53.03 0.320 54.57 376.9 393.0 8.36d(1.32)C c0s 16:5 /

0 = 28 83.58 0.563 48.90 244.4 440.0 4.72d(1.28)

Notes. Pe = P at xp = 0; P P at the slip depth xp shown in Figs. 9 and 10.

Table 3Sensitivity of current solutions to k(Np), LFP and e (see Fig. 9)

Cases Limiting force profiles Remarks References

(I) Ng, Nco and n provided in Table 1 Using the current LFP n = 0.5(II) Ng, Nco and n provided in Table 1 Np = 0 n = 0.5(Np = 0)(III) Ng = 4.53, Nco = 0, and n = 0.5 Using e = Nco = Np = 0 n = 0.5(S. Eqs)(IV) Ng = 4.79, Nco = 3.03, and n = 0.4 Total force is equal to that obtained using Reese(C)s LFP over the max. xp n = 0.4

Table 4Response of pile A using simplified expressions and e = 0 (see Fig. 9)

xp (m) xp Pkn1AL

wg kkn

ALMmaxk

n2

ALP (kN) wg (mm) Mmax (kN m) xmax (m)

1 0.2591 0.2792 0.6797 0.1079 112.3 13.2 167.4 3.093 0.7774 0.5851 2.3435 0.3218 235.2 45.4 501.1 3.655 1.2957a 0.8982 6.1526 0.6574 361.1 119.2 1020.0 4.718 2.0730a 1.4187 20.327 1.3716 570.4 393.8 2185.1 6.38

a When xp exceeds 1.118, Mmax occurs at the upper plastic zone (i.e., xmax < xp).



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55 kPa. The blow count of SPT, N of the sand layer wasfound as: 12 (in depth of 011.0 m), 8 (11.013.8 m), and16 (13.815.4 m), respectively. The effective unit weight c0swas about 16.5 kN/m3. The shear modulus, Gs, and angleof friction of the sand, / 0 were correlated with the blowcount by the expression of [28] Gs = 640NkPa, and

/

0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

8N 4p

20, respectively. Thus, with N= 12 forthe top 11 m, Gs and /0 were estimated to be 7.68 MPa,

and 28, respectively. This allowed Lc/r0 to be estimatedas 15.8, and the pile to be classified as infinitely long. Theeffective pile length Lc is within the top layer, thus the prob-lem becomes a pile in a single layer.

Similar to pile A, relevant parameters for elastic statewere estimated and are shown in Table 2(a). An apparentcohesion was reported in the wet sand near the groundlevel around the driven pile. This is represented by aNco of 4.67 (of similar magnitude to Nc). Ng was obtainedas 8.31 using Eq. (5b). Thus, with n = 1.0, AL was com-puted as 83.58 kPa/m, and a0 as 0.563 m. The LFP is then

plotted as n = 1 in Fig. 10(a). The above-mentioned

parameters offer close predictions of the pile responsesto the measured data. The responses at a typical slipdepth of 2.5d (=1.52 m) are also highlighted. Table 2(b)tabulates the critical values explained before. The Pewas 48.9 kN, the xp extended to 2.88 m at a Pmax of440.1 kN.

In the study, the n = 1 and Nco > 0 are dissimilar ton = 1.31.7 and Nco = 0 normally used for sand. Thus theireffects are examined (Table 5) as follows:

Case (II). Should the apparent cohesion be ignored,the n = 1.0 LFP then reduces to the Broms profile(Fig. 10(a)). This new profile leads to overestimationof displacement (Fig. 10(b)) against measured data.The Reese (S) LFP after depth corrections was foundnearly identical to the Broms one for this case,thereby the over-estimation reported previously [28]using the former LFP is anticipated, so are the over-estimations of maximum bending moment, Mmax,

and depth of the Mmax (not shown herein).

(c)

(d)

(b)

(a)

Fig. 10. Comparison between the calculated and measured [28] response of Pile C.



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Case (III). Taking a0 (Nco) = 0 and n = 2, a lower lim-iting force than that derived from the n = 1 LFP is seenin Fig. 10(a) above a depth of 1.8d, and vice versa. Con-sequently, deflection, Fig. 10(b) and bending moment,Fig. 10(c) are slightly overestimated up to a load levelof 380 kN, to which the total limiting force in the slipzone reaches that for n = 1 case, Fig. 10(a). Thereafter,an underestimation of each response is observed.

Case (IV). Employing an identical LFP to that of Case(III), but taking e = 0, the predictions using the simpleexpressions are slightly higher than those obtained

earlier.

5.2. A pipe pile tested in stiff clay

A steel pipe pile was tested in a stiff clay near Manor [7].The pile was 14.9 m in length, and 0.641 m in diameter. Themoment of inertia, Ip was 2.335 10

3 m4, and the flexuralstiffness, EpIp was equal to 493.7 MN m

2. The undrainedshear strength, su of the clay increases linearly from25 kPa at the ground level to 333 kPa at depth 4.11 m.The submerged unit weight, c0s was 10.2 kN/m

3. No infor-

mation was available about the ac, so it was back-estimatedas 545.1 by substituting k of 331.3 MPa [7] into Eq. (1).Lc/r0 was estimated as 10.8, thus the pile was infinitelylong. The pu profile was estimated by using Eq. (5c) withJ= 0.92, which was then modified using the depth factorprovided by Reese [7]. The LFP thus obtained can bedescribed by using Eqs. (4) and (5a) with Ng = 0.961,Nco = 0.352, and n = 1.0, which is plotted in Fig. 11(a) asStiff clay (J= 0.92). In presenting this pu/(dsu), however,an average su of 153 kPa was used as the normalisationvalue. Using this LFP, good predictions of the pileresponses to a load level of 450 kN are achieved againstthe measured data (see Fig. 11).

To improve the overall predictions, an excellent back-estimation was undertaken which rendered the three param-eters Ng, Nco, and n to be adjusted to 0.854, 0, and 1.5,respectively. The n = 1.5 unfolded is quite close to n =1.7 used for piles in sand, as is the strength (su) profile.The slip was initiated upon loading, thus the back-figuredLFP is rational. Bending moment profiles were computedfor lateral loads of 179.7, 317.7, 485.6, and 606.2 kN, respec-tively, using Eqs. (A-2) and (A-9) provided in Appendix I.As depicted in Fig. 11(d), there is an excellent agreementbetween the calculated and the measured profiles, althoughthe transferring depths of the bending moments of 57 m

predicted are up to 1 m higher than the measured ones.

Using a constant k in the current solutions, the limitingdeflection wp should increase with depth at a power, n of1.5, starting with zero at mudline. In contrast, a linearlyincreasing k is adopted in the COM624P. The effect ofthese differences was examined by assuming a conservativek of 150x (MPa) [7] so that the average k over the maxi-mum slip depth of 2.71 m is 203.3 MPa (ac = 348.4). Usingthis k value, the predicted P wg is shown in Fig. 11(b) asDifferent ac. Only slight overestimation of mudline deflec-tion is noted in comparison with those obtained from then = 1.5 case utilising k= 331.3 MPa (see Table 6). Thus,

the effect of k on the predictions is generally not obvious.The pronounced overestimation [29] using Characteristicload method (based on COM624P) for this case may thusbe attributed to the LFP.

6. Comments on use of the current solutions

The current solutions were used to predict response of12 infinitely long single piles tested in clay, and sand dueto lateral loads, or soil movements. In particular, usingthe Matlocks LFP via Eq. (5c), good predictions weremade for three laterally loaded single piles tested [30] in

Shanghai clay and two single piles due to lateral soil move-ment [31,32]. Good comparison with measured responsefor a pile embedded in sand due to soil movement was alsonoticed previously using Bartons LFP [32,33]. In particu-lar, the analyses of 10 piles in clay showed that:

(a) Ng = 0.34.79 (clay).(b) Nco = 04.67; [or a0k < 0.8 (all cases), and a0k < 0.3

(full-scale piles)].(c) n = 0.52.0 with n = 0.50.7 for a uniform strength

profile, n = 1.31.7 for a linearly increase strengthprofile (similar to sand).

(d) xp 0:51:69 at maximum loads [or xp = (48.4)d].(e) ac = 50340 (clay), and 556 (stiff clay).

These magnitudes are consistent with those summarisedbelow for piles in clay:

(a) Ng = 24 for n = 0 [34].(b) Nco = 2 for a smooth shaft [35], 3.57 for a rough shaft

[36], and 0.0 for a pile in sand.(c) The values of n tentatively obtained by fitting the

reported pu profiles [19] with Eq. (4). Particularly,the case of n > 1 is consistent with the theoreticalsolution [37], and the upper bound method under-

taken for layered soil profiles [8].

Table 5Sensitivity of current solutions to k(Np), LFP and e (Fig.10)

Cases Limiting force profiles Remarks References

(I) Ng, Nco and n provided in Table 1 Using the current LFP n = 1(II) Ng = 3Kp, Nco = 0, and n = 1.0 Using Broms LFP Broms LFP(III) Ng = 8.3, Nco = 0, and n = 2 e = 0.31 m, & Gs = 7.68 MPa n = 2(VI) Ng = 8.3, Nco = 0, and n = 2 identical to (III) Using e = Nco = Np = 0 n = 2(S. Eqs)



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(d) Finally, ac being equal to 80140 [26]; 210280 [13];175360 [38], 330550 [39], and those summarisedpreviously [40].

The analyses also showed that each combination of n,

Ng, and Nco can produce a special LFP. The existing LFPs

such as those ofMatlock, Reese (C) and (S) may work wellfor relevant predictions. However, the special factors suchas a layered soil profile (see Fig. 9), an apparent cohesionaround a driven pile in sand (see Fig. 10) and so forthcan only be readily accommodated through the current

LFP. By means of an equivalent, homogeneous modulus

(b)(a)

(c)

(d)

Fig. 11. Comparison between the calculated and measured [7] response of the Manor test.

Table 6Effect of elastic parameter ac on the Manor test (Fig. 11)

Cases Input parameters Calculated elastic parameters

n a0 (m) AL(kN/mn+1) ac Ep/G* k (MPa) NP/(2EpI p)

Stiff claya 1.0 0.234b 147.0 545.1 551.20 331.28 0.1683

n = 1.5c 1.5 0 163.3 545.1 551.16 331.31 0.1683Different ac 1.5 0 163.3 348.4 862.44 203.30 0.1234

Notes for all cases: c0s 10:2 kN=m3.

a LFP for stiff clay (J= 0.92)0.b Corresponding Pe is 24.5 kN.c Maximum xp = 4.23d, and xp 1.73.



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for elastic state, and the generic LFP for plastic state, thecurrent solutions are sufficiently accurate for analysingoverall response of lateral piles in layered soil if a specificdistribution profile of limiting force is not a major concern.The above-mentioned analysis, together with our recentstudy on another 22 piles in clay indicates that

(a) k= (2.73.92)Gs with an average of 3.04Gs; Gs = (25340)su with an average of 92.3su.

(b) Given an equivalent, uniform strength profile, theLFP may simply be obtained using n = 0.7, a0 =0.050.2 m (average of 0.11 m) and Ng = 0.64.79(1.6).

Study so far on 20 piles installed in sand shows that

(a) k= (2.43.7)Gs with an average of 3.2Gs; Gs = (0.250.62)N (MPa) with an average of 0.5N (MPa).

(b) Given an equivalent uniform sand, it follows that

n = 1.7, a0 = 0, and Ng 0:4

2:5K2p.

The LFP can be more rigorously deduced from soilstrength parameters. All of these results are ready forpublication.

Finally, the current (CF) solutions are compared withthe numerical program COM624P, as can be seen fromTable 7. Although both are capitalised on load transfermodel, only the CF solutions are linked to soil modulusvia Eq. (1), which allow continuum-based pilesoil interac-tion to be simulated. COM624P can incorporate variousforms of non-linear py curves, but the resulting overall

pile response is negligibly different from that obtainedusing the current solutions. COM624P and the CF solu-tions actually offer predictions for a linearly increasing,and a uniform profile of k, respectively, which should

bracket non-homogeneous k normally encountered.Reflecting overall pilesoil interaction, only the parametersfor the LFP is dependent of mode of soil failure. To date, itoffers very good to excellent prediction of pile response incomparison with measured data of 62 tested piles. Thusthe current solutions may be used as a boundary element

to simulate beamsoil interaction due to lateral forcewithin a complicated reinforced soil structure.The current solutions are based on five assumptions

mentioned previously.

Contradictory to the assumption (i), the py curve maybe a parabola [6] or a hyperbola [41]. A transition zone(not depicted in Fig. 2(a)) may thus form in between theupper plastic zone and the lower elastic zone. Neverthe-less, adopting the idealised elasticplastic py(w) curveinstead, response of the pile is negligibly affected [13,42].

Assumption (ii) may lead to a 20% overestimation ofxmax.

Assumption (iii) has negligible influence, as slip gener-ally occurs under a very low load level, but it extendsto a limited depth under a maximum load level.

Assumption (iv) is automatically satisfied, with theintroduction of assumption (i), as the latter renders thetransition zone to be reduced to a single slip depth(see Fig. 2(a)).

Assumption (v) is ensured by two conditions: firstly, thepile should be infinitely long with L > Lc, otherwise,another slip may be initiated from a short, rigid pile baseat a rather high load level [43]. And secondly, Eq. (4) isused.

Along with assumption (ii), Eq. (4) allows a gradualincrease in wp with depth. This is not always true in a strat-ified soil, as a deeply embedded weak layer may have a

Table 7Salient features of COM624P and the current CF solutions

Item COM624P [48] GASLFP (CF solutions)

Pilesoil interaction model Uncoupled, inconsistent with continuum-based numericalanalysis

Coupled, consistent with continuum-based numericalanalysis

Subgrade modulus, k Increase linearly with depth at a gradient of nh(k= nhx). A constant calculated from an average modulus, Gs overthe maximum slip depth, xp and using Eq. (1)

Empirically related to soil properties Theoretically related to soil and pile properties, pile-head,and base conditions

Limiting force per unitlength (LFP)

1. Many parameters, different expressions and proceduresfor different soils

2. Parameters are mainly derived from soil failure modesof wedge type and lateral plastic flow

1. Three parameters n, Nc (or a0), and Ng, a unifiedexpression of Eq. (4), and procedure for all kinds ofsoils

2. Parameters are deduced from overall pile response,regardless of mode of failure

py curve Consisting of four piecewise curves, dashed l ine inFig. 2(b)

An elasticperfectly plastic curve, solid line in Fig. 2(b)

Computation Finite difference method Explicit expressions of the xp using spreadsheet programGASLFP or by hand

Advanced use In form of numerical program; Other use has not beenspecified

In form of explicit expressions; Reflecting overall pilesoilinteraction by LFP, and indicating the effective depth byxp. May be used as a boundary element for advanced

numerical simulation



18/21

lower wp than a shallow, stiff layer may. However, the useof an overall LFP is sufficiently accurate as demonstratedpreviously against 3D FEA results. This is also valid fora very stiff, upper layer, since in such a case, pilesoil rela-tive slip may never extend to an underlying weak layer. Pileresponse would be mainly affected by the stiff layer. In

brief, assumption (v) is generally acceptable. Deriving fromthe normal and shear stresses, respectively, on the pilesoilinterface [44,45], the resistance in elastic state may be suffi-ciently accurately evaluated using elastic theory [5], and inthe plastic (slip) state by Eq. (4).

In rare cases, the non-homogeneous modulus may affectmarkedly the pile response, for which the previous numer-ical results [46,47] may be consulted along with the currentpredictions.

7. Conclusions

This paper put forwards new elasticplastic solutions for

laterally loaded, infinitely long, free-head piles. They havebeen calibrated against FEA results for a pile in two differ-ent types of stratified soils. The solutions permit non-linearresponse of the piles to be readily estimated right up to fail-ure. Presented in explicit expressions of slip depth andLFP, the current solutions may be used as a boundary ele-ment to represent beamsoil interaction in the context ofanalysing a complicated soilstructure interaction. In termsof analysis of 62 pile tests to date, the ranges of inputparameters are provided. In particular the following con-clusions are drawn:

The generic expression of Eq. (4) is applicable to alltypes of soils. It can generally accommodate existingLFPs through selecting a suitable set of parameters.

Response of free-head piles is dominated by the LFP andthe maximum slip depth. Thus, non-linear predictionsmay be made by selecting a series of slip depth xp, usingGASLEP or the simplified expressions provided.

By maintaining total resistance within a maximum slipdepth xp, pile response is insensitive to the shape ofthe LFP. Available, verified procedures may be usedto construct LFP for current solutions.

To generate LFP, the new procedure proposed hereinalong with n = 0.52.0 may be used. A low value of nmay correspond to a uniform strength profile, and ahigh one to a sharply changed strength profile. For alayered soil, the generated LFP may not necessarilyreflect a detailed distribution profile of limiting forcealong a pile but an overall trend.

LFP should be back-estimated using current solutionsalong with measured data, as it then can account foroverall pilesoil interaction rather than sole soil failuremechanism. In this manner, LFP may be updated tocater for various influence factors.

The current study has been limited to static loading, and

linear elastic, free-head piles, but it can be extended to

complicated loading. The current solutions have been usedsuccessfully for non-linear piles. They have been extendedto fixed-head piles as well. All of these results will be pub-lished in due course.

Acknowledgments

The work reported herein was sponsored by AustralianResearch Council research fellowship (F00103704) andDiscovery Grant (DP0209027). It was initiated in Singa-pore through the sponsor of the (Singapore) National Sci-ence and Technology Board. This financial assistance isgratefully acknowledged. The author also would like toacknowledge the reviewers comments.

Appendix I. Development of elasticplastic solutions

In this appendix, derivation of the elasticplastic solu-tions for the pile mentioned in the paper is elaborated.All of the symbols used are of identical meanings to thosedefined in the paper.

Integrating Eq. (6) for plastic state yields expressions forshear force, QA(x), bending moment, MA(x), rotation,w0Ax, and deflection, wA(x) of the pile at depth x, asdetailed below

QAx EpIpw000Ax

ALx a0

n1 an10n 1

P

AL

!A-1

MAx EpIpw00

Ax

ALx a0

n2 an20Q2j1n j

an10

n 1

P

AL

x

Pe

AL

" #

A-2

w0Ax AL

EpIp

x a0n3Q3

j1n j

an10n 1

P

AL

x2

2

"

an20Q2

j1n j

Pe

AL

!x

# C3 A-3

wAx AL

EpIp

x a0n4

Q4j1n j

an10

n 1

P

AL x

3

6"

an20Q2

j1n j

Pe

AL

!x2

2

# C3x C4 A-4

where C3, C4 = constants. In the integration, the condi-tions for a free-head pile (at x = 0) are adopted as follows

QA0 EpIpw00A0 P;MA0 EpIpw

00A0

Pe A-5

Eq. (7) for the elastic state may be solved as(Np < 2 ffiffiffiffiffiffiffiffiffiffiffiffi

kEpIpp

wBz eazC5 cosbz C6 sin bz A-6



19/21

whereas C5, C6 = constants; a, b are given by Eqs.(9a),(9b). From Eq. (A-6), it follows that

wIVB z eaza4 6a2b2 b4C5 4aba

2 b2C6 cosbz

eaz4aba2 b2C5 a4 6a2b2 b4C6 sinbz

A-7

w000B z eazaa2 3b2C5 b3a2 b2C6 cosbz

eaz3a2 b2bC5 aa2 3b2C6 sinbz

A-8

w00Bz eazfa2 b2C5 2abC6 cosbz

2abC5 a2 b2C6 sinbzg A-9

w0Bz eazaC5 bC6 cosbz bC5 aC6 sinbz

A-10

The constants Ci (i= 36) are determined using the com-patibility conditions at the slip depth x = xp (z = 0) fromelastic to plastic state, which require

wIVA x xp wIVB z 0 w

IVP A-11

w000Ax xp w000B z 0 w

000P A-12

w00Ax xp w00Bz 0 w

00P A-13

w0Ax xp w0Bz 0 w

0P A-14

wAx xp wBz 0 wP A-15

With Eq. (A-8), Eq. (A-12) may be written as an expressionof unknown constants C5 and C6. In terms of Eq. (A-9),Eq. (A-13) offers another expression for the two constants.Solution of these two expressions offers

C5 EpIp

k 2aw000P 3a2

b2

w00P A-16

C6 EpIp

kba2 b2w000P aa

2 3b2w00P A-17

Utilising Eqs. (A-3) and (A-10), Eq. (A-14) can be ex-panded, thus C3 is determined as

C3 4ALk

1n

kF3;xp F2;0xp F1;0 P

x2p

2Pexp

" #

aC5 bC6 A-18

In the same manner, Eqs. (A-4) and (A-6) allowEq. (A-15) to be expanded, which gives C4 as

C4 4ALkk

n F4;xp F1; 0 Px3p

6 F2; 0 Pe

x2p

2

" #

C3xp

k C5 A-19

Substituting Eqs. (A-16) and (A-17) into Eq. (A-18), anormalised C3 is derived as

C3kkn1

AL 4F3;xp xpF2; 0 4aNF2;xp F2; 0

2F1;xp 21 2aNxp x2pF1; 0

2x2p 1 2aNxp 2aN xpeP A-20

In light of Eqs. (A-17) and (A-20), the normalised form ofC4 is obtained

C4kkn

AL 4F4;xp xpF3;xp 41 a

2NF2;xp

F2; 0 2x2pF2; 0 2xp1 2a2

NF1;xp

1 2aNxpF0;xp 4x2p=3 2xpF1; 0 4x3p=3 2xp 2x

2p 4 4a

2NeP A-21

At ground level, Eqs. (A-3) and (A-4) permit the rotation,w0g and deflection, wg to be expressed, respectively, as

w0gkkn1

AL 4F3; 0

C3kkn1

ALA-22

wgkkn

AL 4F4; 0

C4kkn

ALA-23

Eq. (A-23) has been rewritten as Eq. (11). SubstitutingEq. (A-6) into Eqs. (A-14), (A-15), the normalised rota-tion and deflection at the slip depth are written, respec-tively, as,

w0pkkn

AL a

C5kkn

AL b

C6kkn

ALA-24

wpkkn

AL

C5kkn

ALA-25

Eqs. (A-11)(A-13) render the following relationship at theslip depth to be established

0:5wIVP aw000P k

2w00P 0 A-26

In terms of Eqs. (6), (14) and (15), Eq. (A-26) can be re-written in the explicit form of Eq. (8), correlating the loadto the normalised slip depth.

The slip depth under a given load may be computedusing Eq. (8), which is then used to calculate the pileresponses. Particularly, the mudline rotation of the pilemay be predicted using Eq. (A-22), as it becomes impor-tant to predictions of pile response due to soil movement.Responses of the pile along its length are predicted sepa-rately using elastic and plastic solutions. Within plasticstate of x < xp, Eqs. (A-4), (A-3), (A-2), and (A-1) areused for deflection, rotation, moment, and shear force,respectively. Otherwise, in the elastic state of xP xp,

Eqs. (A-27)(A-30) should be employed, which werederived from Eqs. (A-7)(A-10) using Eqs. (A-16) and(A-17).

wBz eazEpIp

k

2aw000P 3a

2 b2w00P cos bz:

a2 b2

bw000P

a

ba2 3b2w00P

sin bz

A-27

w0Bz eaz

2k2

w000P 2aw

00P cosbz

aw000P a

2 b2w00P

b sin bz A-28

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20/21

MBz EpIpw00Bz

EpIpeaz w00P cos bz

w000P aw00P

b

sin bz

A-29

QBz EpIpw000B z

EpIpeaz w000P cos bz

aw000P 2k2w00Pb

sin bz

A-30

At depth zmax, the shear force, QB(zmax) is zero, and themaximum bending moment should occur. Therefore, Eq.(A-30) allows the function wxp to be defined as Eq.(13), and zmax to be derived as Eq. (16), respectively.

Appendix II. Simplified expressions (at a0 = 0, and aN = 1)

Provided that a0 = 0, and aN = 1 (i.e., bN = 1), Eq. (*)provided in the paper may be simplified to Eq. (*s) givenbelow. Thus, for instance, Eq. (8) may be replaced withEq. (8s).

Pkn1

AL

0:5xnpn 1n 2 2xp2 n xp

xp 1 en 1n 28s

wgkkn

AL

2

3xn3p

2x2p 2n 10xp n2 9n 20

xp 1 en 2n 4

2x2p 2xp 1x

np

xp 1 e

2x4p n 4xp 12x

2p n 1xp 1

xp 1 en 1n 4exnp

11swxp

2xp2 ne 21 nx

2p n 2n 1

2n 2xp 1 ne 2x2p 22 nxp n 2n 1

13s

Under the above-mentioned conditions, the constantsC1xp, and C2xp in Eq. (17) may be replaced with

C1xp 0:5xnp

xp 1 e

1 n 2xp1 n

e 2 n 2xp

2 nxp

C2xp 0:5xn

p A-31

From Eq. (13s), the normalised slip depth, xp (rewritten asx0) at wx0 0 is obtained as

x0 0:52 n

1 ne 0:5

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 n

1 ne2 2n 2

rA-32

The condition of xp > x0 will lead to wxp < 0, thus maxi-

mum bending moment should occur above the slip depth.The estimation using above-mentioned expressions is

referred to as S. Eqs. It can be readily undertaken in aspreadsheet, similar to the form shown in Table 4 for pileA. Here provides the calculation for two typical values ofxp. From Table 2(a), k = 0.2591/m, given xp = 1 m, xp iscomputed to be 0.2591. Using n = 0.5 (Table 1), ande = 0, x0 is computed as 1.118, thus

As xp < x0wxp > 0, the maximum bending moment

should occur below the slip depth. Substitutingwxp 0:602, and b = k into Eq. (16), zmax is computedto be 2.091 m, thus, tan(k zmax) = 0.602, and cos(kzmax) =0.8567. Also from Eq. (A-31)

C1xp 0:5 2:5 2 0:2591 0:25911:5

2:5 1:2591 0:06323;

C2xp 0:5 0:25910:5 0:25451

These values ofC1 and C2 allow the normalised moment tobe estimated using Eq. (17) as

MBzmaxkn2=AL e

0:541880:06323 0:25451 0:602

0:8567 0:10785

Assuming xp = 5 m, and e = 0, xp is found to be 1.2957.Following the above-mentioned procedures, it follows thatPk1.5/AL = 0.8982, and wgkk

0.5/AL = 6.1526. As xp > x0,

the depth ofxmax may be obtained directly from Eq. (18) as

xmax 1:5 0:89821=1:5

=0:2591 4:71m

The moment, M0 (=P e) is zero. Thus, the normalisedmaximum moment may be estimated from Eq. (19) as

Mmaxkn2=AL 1:5 0:8982

2:5=1:5=2:5 0:6574

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wxp 2 1:5 0:25912 2:5 1:5

2 0:25912 2 2:5 0:2591 2:5 1:5 0:6020

Pk1:5

AL 0:5 0:2591

0:51:5 2:5 2 0:2591 2:5 0:25911:2591 1:5 2:5

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piles subjected to lateral soil movement

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