Abstract Lateral load-deflection behaviour of

single piles is often analysed in practice on the

basis of methods of load-transfer P–Y curves.

The paper is aimed at presenting the results of

the interpretation of five full-scale horizontal

loading tests of single instrumented piles in two

sandy soils, in order to define the parameters of

P–Y curves, namely the initial lateral reaction

modulus and the lateral soil resistance, in correla-

tion with the pressuremeter test parameters. P–Y

curve parameters were found varying as a power of

lateral pile/soil stiffness, on the basis of which

hyperbolic P–Y curves in sand were proposed. The

predictive capabilities of the proposed P–Y curves

were assessed by predicting the soil/pile response

in full-scale tests as well as in centrifuge tests and a

very good agreement was found between the

computed deflections and bending moments, and

the measured ones. Small-sized database of full-

scale pile loading tests in sand was built and a

comparative study of some commonly used P–Y

curve methods was undertaken. Moreover, it was

shown that the load-deflection curves of these test

piles may be normalised in a practical form for an

approximate evaluation of pile deflection in a

preliminary stage of pile design. At last, a

parametric study undertaken on the basis of the

proposed P–Y curves showed the significant influ-

ence of the lateral pile/soil stiffness on the non-

linear load-deflection response.

Keywords Lateral loading test Æ Lateral reaction

modulus Æ P–Y curves Æ Pressuremeter test Æ Sand ÆSingle pile

List of symbols and unitsB diameter or frontal width of the pile (m)

D embedded length of the pile (m)

De effective pile length (m)

E elastic soil modulus (MPa)

e excentricity of lateral load (m)

Ec characteristic soil modulus (MPa)

Em first load pressuremeter modulus (MPa)

Er reload pressuremeter modulus (MPa)

Eti initial lateral reaction modulus (MPa)

EpIp flexural pile stiffness (MN m2)

F tangential lateral reaction (kN/m)

Fl limit tangential lateral reaction or

tangential lateral resistance (kN/m)

Gr pressuremeter shear modulus (Gr = Er/

[2(1 + m)]) (MPa)

H lateral load applied on the pile top (kN)

Id density index (%)

K pile/soil compressibility

Kr lateral pile/soil stiffness

L tangential dimension of the pile section

(parallel to H) (m)

A. Bouafia (&)Department of civil engineering, University of Blida,P.O. Box 270, R. P Blida 09000, Algeriae-mail: bouafia1@yahoo.fr

Geotech Geol Eng (2007) 25:283–301

DOI 10.1007/s10706-006-9110-7

123

ORIGINAL PAPER

Single piles under horizontal loads in sand: determinationof P–Y curves from the prebored pressuremeter test

Ali Bouafia

Received: 11 June 2005 / Accepted: 20 September 2006 / Published online: 27 October 2006� Springer Science+Business Media B.V. 2006

L0 transfer length or elastic length (m)

M bending moment at a given depth (kN m)

M0 bending moment applied to the pile top

(kN m)

NH rate of increase of Eti with depth in

Gibson’s soil (MPa/m)

Nspt N value of the SPT (blow counts/30 cm)

P lateral soil reaction at a given depth

(kN/m)

Pu lateral soil resistance or limit lateral

reaction (kN/m)

pf pressuremeter creep pressure (kPa)

pl limit pressuremeter pressure (kPa)

p* net limit pressuremeter pressure (kPa)

P�le net equivalent limit pressuremeter

pressure (kPa)

p0 at-rest lateral earth pressure (kPa)

Q frontal lateral reaction (kN/m)

qc cone penetration resistance (MPa)

qs limit skin friction along the pile shaft

(kPa)

R least-squares regression coefficient (%)

R0 initial radius of pressuremeter borehole

(m)

DR increase in PMT borehole radius (mm)

Sf, St shape factors

Y lateral displacement or deflection at a

given depth (mm)

Y0 pile deflection at ground level (mm)

Yref. reference deflection or threshold of lateral

soil resistance Pu(mm)

z depth with respect to the ground level (m)

zc critical depth (m)

k rate of linear increase of Em with depth in

Gibson’s soil (MPa/m)

l rate of linear increase of P�l with depth in

Gibson’s soil (kPa/m)

g lateral resistance factor

m Poisson’s ratio

w ratio Eti to Em

n ratio Pu to P�LB

1 Introduction

Pile foundations were initially designed in order

to transmit vertical loads to the soil. When these

foundations were besides horizontally loaded,

inclined piles, often difficult to achieve were to

be added. Due to the progress done in the

knowledge of piles foundation behaviour, it is

nowadays recognised that vertical piles can sus-

tain horizontal loads. Earth pressures on a bridge

abutment piles, lateral displacement of soft clayey

layer underlying an access embankment to a

motorway and wind pressures on slender struc-

tures built on piles are usual examples of hori-

zontal loading of piles. Behaviour of piles under

horizontal loads is a complex soil/pile interaction

problem because of the tridimensional nature of

the phenomenon and its dependence on a multi-

tude of key parameters. This fertile domain of

research was investigated since more than a half

century.

Geotechnical literature contains a wealth of

methods of analysis mainly based on elasticity,

finite/boundary elements or on subgrade reaction

theory. However, It should be emphasised that

the theoretical approaches offer simplistic

schemes of soil/pile response and therefore do

not take into consideration many pile/soil inter-

action parameters such as the pile installation, the

soil/pile interface roughness and the soil com-

pressibility. Furthermore, some particular aspects

of the problem of laterally loaded piles such as

the proximity of a slope, the group effects, and

piles undergoing lateral soil movement are diffi-

cult to be analysed by theoretical methods.

Experimental research may then be considered

as the most adapted way to investigate such a

problem. The last four decades were marked by a

considerable progress in the understanding of the

response of a pile to bending forces by means of

several experimental studies in full-scale as well

as in centrifuge.

Prior to the development of numerical meth-

ods in geotechnical engineering, piles were usu-

ally designed by evaluating the deflections under

working loads on the basis of small displacement

methods such as the elasticity. Subgrade reaction

theory was also used for the linear analysis of pile

deflection by modelling the pile as a beam on

elastic foundations. These approaches were

adapted for simple pile/soil configuration and do

not account for the soil properties heterogeneity

and the non-linear lateral response of the pile/soil

284 Geotech Geol Eng (2007) 25:283–301

123

system. Moreover, foundations of some structures

working under severe lateral loading conditions

are designed on the basis of limit equilibrium

methods. These latter ones are based on approx-

imate mechanisms of soil resistance derived from

the lateral earth pressures theory (Bouafia 1990,

1998; Bouafia et al. 1991).

It is nowadays recognised that the design

methods based on P–Y curves are the most

reliable to the analysis of the behaviour of

laterally loaded single piles with possibility of

taking account of the non-homogeneity of soil

properties as well as of the material non linearity

in lateral pile/soil response. Soil/pile interface is

modelled by infinity of non-linear springs in

which the soil reaction P at a given depth is

undertaken by the spring for a lateral pile

displacement Y.

Full-scale tests on instrumented piles are often

used to investigate the soil/pile response in the

light of load-transfer theory. P–Y curves are

derived from bending moment profiles measured

by strain gauges along the pile. However, a few

full-scale tests on instrumented piles in sand

were reported in the literature with successful

derivation of P–Y curves from double differen-

tiation and integration of the bending moment

profile. The main difficulty in deriving these

curves is due to the high sensitivity of the lateral

soil reaction P to the experimental conditions as

well as to the method of fitting and differenti-

ation of bending moments (Bouafia and Garnier

1991).

This paper is aimed at presenting the results of

an extensive analysis of full-scale horizontal piles

loading tests in quite homogeneous sandy soils.

Test piles were well instrumented, and P–Y

curves were derived from the interpretation of

bending moment distribution along the experi-

mental pile. The experimental results presented

herein are part of an important research pro-

gramme carried out by the LCPC (Laboratoire

Central des Ponts & Chaussees, France) during

more than three decades.

It was shown the existence of fundamental

relationships between the P and Y curves param-

eters namely the lateral soil modulus and the

lateral soil resistance, the parameters of pressure-

meter test (PMT) and the lateral pile/soil stiffness

Kr. Based on these relationships, hyperbolic

functions were proposed to describe P–Y curves.

Validation process was undertaken by comput-

ing the tests piles used to derive such a method as

well as other test piles in sandy soils. Comparative

study showed the good prediction capability of

the proposed soil/pile stiffness dependant P–Y

curve methods compared to the current

approaches based on the PMT test.

2 Brief review of the methods of construction

of P–Y curves

To the knowledge of the author, the first study on

the basis of P–Y curves was due to Reese and

Matlock (1977) by introducing the concept of the

lateral reaction modulus previously defined by

Winkler (1867). The first generation of P–Y

curves was bilinear describing an elastic plastic

behaviour at the pile/soil interface.

The in-situ tests such as the PMT become

usual tools for pile foundations analysis and

design. The PMT test provides an experimental

stress–strain curve describing the borehole

response under radial loads. Some similitude

exists between the expansion of the PMT beor-

ehole and the mobilisation of the frontal lateral

reaction of the soil around a pile (Menard et al.

1969).

Geotechnical literature contains a diversity of

methods for deriving P–Y curves from PMT

parameters, namely the PMT deformation mod-

ulus Em and the limit PMT pressure pl. For

brevity, only the commonly used methods will be

presented hereafter.

2.1 Method of Menard et al. (1969)

This method was initially suggested by Menard

et al. (1969), and then improved by Gambin

(1979). As illustrated in Fig. 1, the curve 1 is tri-

linear shaped. The first portion has a slope equal

to the initial lateral reaction modulus Eti, the

second one has a slope equal to the Eti/2 and the

third one corresponds to the lateral soil resistance

taken equal to net limit pressure multiplied by the

diameter (or the frontal width) B.

Geotech Geol Eng (2007) 25:283–301 285

123

Lateral reaction modulus Eti was evaluated by

Menard et al. (1969) on the basis of the formula

of settlement of strip foundation, by considering

the pile as a infinitely long rigid foundation whose

settlement is horizontal and equal to the pile

deflection Y. Modulus Eti was derived as a

function of Em, B and a as follows:

Eti

Em¼ 18

4ð2:65Þa þ 3afor B � B0 ¼ 0:60 m ð1Þ

Eti

Em¼ 18B

4B0ð2:65 BB0Þa þ 3Ba

for B[B0 ð2Þ

a is a rheological factor called ‘‘coefficient of soil

structure’’ depending on the nature of the soil and

its compressibility. It is equal to 1/3 for loose and

medium dense sands and 1/2 for very dense sands.

For small diameter piles (B £ 0.60 m) ratio Eti/

Em ranges between 2.24 and 2.75.

Pressuremeter parameters defining the P–Y

curves above a critical depth zc, should be

reduced to take into consideration a reduction

in soil resistance due to soil heave and a probable

reduction in soil confinement (Baguelin et al.

1978; Briaud 1986). According to Menard (1971),

the critical depth zc is equal to 4 diameters in

granular soils and to 2 diameters in cohesive soils

(Frank and Jezequel 1989).

It should be emphasised that the model of

lateral reaction proposed by Menard (1971) is

limited to a rigid pile section and therefore

neglects the effects of the pile flexural rigidity.

Moreover, analogy assumed by Menard (1971)

between the pile and an infinitely long beam leads

to neglect the effect of the slenderness ratio D/B

of the pile, D being the embedded length of the

pile.

Evidences from full-scale lateral loading tests

showed that in the same site the response of piles

characterised by different pile/soil stiffnesses Kr

could not be characterised by a unique lateral

reaction modulus (Bouafia 1990, 1997, 2002a).

Tests on instrumented pile models in centrifuge

showed rather a variation of the modulus Eti as a

power of Kr (Bouafia 2002b).

Many investigators have confirmed from the

analysis of full-scale pile loading tests that this

method is rather pessimistic in predicting small

deflections behaviour (Frank 1984; Briaud 1986;

Baguelin and Jezequel 1972; Baguelin et al. 1990)

and optimistic in the domain of large deflections

(Baguelin et al. 1990; Bouafia and Bouguerra

1995, 1996).

According to this method, for a small diameter

pile (B < 0.60 m) the deflection noted Yref. in

Fig. 1, beyond which the lateral reaction reaches

the soil resistance ranges between 5 and 10% of

the diameter whatever the pile/soil stiffness.

2.2 Method of the French code Fascicule-62

(MELT 1993)

The previous method was integrated in the French

geotechnical code with reduction of the lateral soil

resistance to the net creep pressure pf* multiplied

by B, as illustrated by curve 2 in Fig. 1. This

adaptation was dictated by the necessity to obtain

conservative prediction of the pile response at

large deflections (Baguelin et al. 1978).

For non-circular pile section, in addition to the

lateral reaction defined by curve 2 in Fig. 1,

tangential lateral reaction F is mobilised along the

tangential sides. Lateral tangential F–Y curve is

defined as a bilinear curve. The first linear portion

has a slope equal to that of P–Y curve and the

second one is horizontal and represents the

tangential lateral resistance Fl given by

Fl ¼ 2qsðL� BÞ ð3Þ

qs is the limit skin friction equal to that mobilised

under vertical loads and L is the tangential

dimension of the pile section. The overall P–Y

Yref.

3 Yref.

2

Eti /2

P*

f.B

P (

kN/m

)

Y (mm)

1: Ménard at al2: Fascicule-623: Proposed P-Y

P*

l.B

Pu=P*

l.B.ξ(K

r)

1

2

3

Eti=E

m.ψ(K

r)

Eti

Yref.

1

Fig. 1 Schematisation of some typical P–Y curves

286 Geotech Geol Eng (2007) 25:283–301

123

curve is the superposition of the two lateral

reaction curves.

2.3 Method of Dunand (1981)

This method is based on a bi-linear P–Y curve as

illustrated by curve 2 in Fig. 1. Lateral reaction

modulus was correlated to Em on the basis of an

elastic method whereas the limit lateral reaction

is equal to plB. The concept of critical depth is

introduced as in the previous methods. According

to the author, this method is recommended to the

design of drilled piers supporting transmission

line structures.

2.4 Method of Briaud et al. (1985)

The total lateral reaction P to the deflection Y at

a given depth is the sum of the frontal reaction Q

and the tangential reaction F. As a result, the P–Y

is the addition of the Q–Y curve and the F–Y

curve. Carayannacon et al. (1979) showed by

finite element analysis that the contribution of

the tangential reaction increases with slenderness

of the pile section.

The main assumption in this method is that

radial displacements of a PMT borehole and the

pile deflections are homothetic. Q–Y is directly

built from the expansion curve of the PMT test as

follows:

Q ¼ Sf p�B ð4Þ

Y ¼ BDR

2R0ð5Þ

The shape factor Sf is equal to 1 for square

piles and to p/4 for circular section. R0 and DR are

respectively the initial PMT borehole radius and

the increase in borehole radius under the net

pressure p*.

F–Y curves have a bilinear shape composed of

an initial portion with a slope equal to 2Gr, Gr

being the PMT shear modulus, and a horizontal

asymptote equal to the limit frontal reaction Fl as

follows:

F l ¼ StqsB ð6Þ

The shape factor St is equal to 2 for square piles

and to 1 for circular pile section. Limit skin

friction qs slightly differs from the one mobilised

along the pile shaft under vertical loads and then

may be evaluated with usual bearing capacity

formulae (Smith 1987).

According to the authors, the assessment of

this method with respect to the experimental

evidence of 27 pile loading tests carried out in a

variety of piles and soils showed a good predictive

capability of pile deflections (Briaud 1986).

2.5 Method of Baguelin et al. (1978)

In this method, based on the self-boring pres-

suremeter test (SBPMT), the total P–Y curve is

constructed point by point from the experimental

PMT expansion curve as follows:

F ¼ g p�B ð7Þ

Y ¼ BDV

4V0ð8Þ

V0 and DV are respectively the initial PMT

borehole volume and the increase in borehole

volume under the net pressure p* g is called

lateral resistance factor taking into account the

surface effect and varies form 0.33 to 3 (Baguelin

1982).

2.6 Method of Robertson et al. (1984)

P–Y curve is constructed for a bored pile from a

prebored PMT or a self-boring PMT, and for a

driven pile from a pushed-in PMT test. Formulae

5 and 7 should be used with factor g equal for

sandy soils to 0 at surface and increasing linearly

with depth to 1.5 at the critical depth and below.

The critical depth was estimated to 4 diameters

(Robertson et al. 1985).

Atukoralla and Byrne (1984) analysed by finite

element modelling the lateral displacements of a

rigid disk within an elastic plastic material as well

as those due the cylindrical cavity expansion

within the same material. It was shown that the

ratio of lateral pressures surrounding the disk to

those around the PMT cavity varies between 1.4

and 1.7 with an average value of 1.50, which is in

Geotech Geol Eng (2007) 25:283–301 287

123

accordance with the factor g of this method.

However, results of this study do not account for

the tridimensional response of the pile under

lateral loads.

As summarised in Table 1, the ratio of lateral

soil resistance Pu to PlB proposed by the methods

mentioned above ranges in a wide margin betw-

een 0.3 and 3, which shows some uncertainty in

predicting the soil resistance. As an alternative,

the Experimental analysis of instrumented test

piles will be used in the next section to evaluate

the lateral soil resistance.

3 Presentation of full-scale tests in sand

3.1 Sites and geotechnical conditions

The first site, noted S1, is located in Chatenay-sur-

Seine, 70 km south east of Paris (France). A big

pit whose volume is 424 m3 was previously dug to

a depth of 3.20 m in a chalky soil. It was

waterproofed by plastic sheets, and then filled in

by Fontainebleau sand into two medium dense

layers. The underlying layer is 1.40 m thick with a

density index Id = 37% whereas the upper layer

has a thickness of 1.80 m and Id = 57%. Fon-

tainebleau sand is poorly graded sand. In-situ

tests, notably PMT (Menard pre-bored pressure-

meter test), CPT (static cone resistance test) and

DPT (dynamic penetration test) were carried out

and typical profiles are shown in Fig. 2.

The second site, noted S2, is located in Le-Rheu,

5 km south west of Rennes (France). The soil is

composed of reddish poorly graded clean sand of

marine origin from the Pliocene era. Ground

water table was found at 10 m of depth. The sand

above water table has average water content of

8% corresponding to a saturation degree of 31%.

It was possible to recover some samples with a

150 mm diameter auger sampler up to 4.0 m of

depth. The density index Id measured according

to ASTM standard is 66%. Profiles of PMT, CPT

and DPT tests are illustrated in Fig. 3.

3.2 Test piles

Test piles are steel pipes instrumented by strain

gauges distributed by pairs along two diametri-

cally opposite axes. Table 2 summarises the

main geometrical and mechanical characteristics

of the piles. Three tubes, noted T5, T10 and T15

were tested in site S1 and two piles P1 and P2

tested in site S2. The slenderness ratio (embed-

ded length D/diameter B) varies between 5.5

and 15.3.

Piles in site S1 are externally instrumented by

seven pairs of strain gauges irregularly distributed

along two diametrically opposite axes and pro-

tected by an adhesive papers of aluminium.

Figure 4 illustrates a general view of piles in site

S1 with the scheme of loading device. Each pile in

this site was connected at its tip by a 90� jacking

cone in order to facilitate the jacking process into

soil. As shown in Fig. 4, the cone has same

diameter as that of the pile. Each pile was jacked

by means of a hydraulic jack in contact with a

reaction beam (Canepa et al. 1987).

Piles P1 and P2 in site S2 are instrumented by

20 and 22 pairs of strain gauges, respectively.

These latter are externally placed along the pile

P1 and protected by steel valley for each axis. For

pile P2, they are internally stuck along two axes

along the pile. For both the piles strain gauges are

regularly spaced of 25 cm and the first one

corresponds to the ground surface. Each pile

was placed into a borehole previously made by a

helical drilling engine 6 m high. Some irregularity

in diameter of borehole within 5 cm was noticed.

It was likely due to a default of verticality of

drilling axis (Jezequel 1988). Each pile in site S1

was filled in with bentonite-cement grout. The

diameter of pile was directly measured at surface

as well as estimated from the volumes of steel and

bentonite-cement. Uniaxial compression tests of

bentonite samples have given a Young’s modulus

of 3,500 MPa at 28 days.

Table 1 Comparison of theoretical ratios Pu/plB

Method Pu/plB Remarks

Menard et al. (1969) 1.00Fascicule-62 0.50 Usual correlation

pf � pl/2Dunand (1981) 1.00Briaud et al. (1982) 0.83 Bored pile in sandBaguelin et al. (1977) 0.3–3.0Robertson et al. (1984, 1985) 1.50 Beyond 4B

288 Geotech Geol Eng (2007) 25:283–301

123

3.3 Other experimental devices

Horizontal load was applied at 1 m and 0.07 m

above the ground surface in sites S2 and S1

respectively. It was given by a hydraulic jack in

contact with a concrete block of reaction and

measured by a high precision electric load cell.

Top deflections were measured in site S1 by

two pairs of LVDT located above the axis of

lateral loading of pile. In site S2, measurement

was made by an LVDT located at load level as

well as manually by a distance-meter (invar wire)

with a precision of 1/10 mm.

Rotation was measured by an inclinometer

located at piles top in site S1, whereas it was

measured along the piles in site S2 by means of

electro-levels BRE with a precision of 1/100 mRad.

One pipe access was fixed at the central axis of pile

P1 and two ones were internally fixed at two

diametrically opposite axes into pile P2. Each pipe

access allows the installation of an inclinometer or

the electro-levels BRE every 50 cm depth.

Figure 5 shows the experimental configuration

for piles P1 and P2.

4 Programme of tests

Each pile was subjected to a series of static

horizontal loads at pile head. Each load incre-

ment was maintained 15 min in site S1 and 2 h in

site S2. Some troubles in the hydraulic jack

performance in site S2 led to carry out a series

of three loading-unloading sequences for pile P2

and 2 ones for pile P1. Response of strain gauges

above or at ground surface were used to check to

actual load applied to the pile.

5 Analysis of P–Y curves

5.1 Methodology

Measurement of the axial deformation by strain

gauges along the pile allows the determination of

Fig. 2 Typical profiles ofin-situ tests in site S1

Geotech Geol Eng (2007) 25:283–301 289

123

bending moments curve for a given load at pile

top. Two successive integrations of this curve lead

to easily determine lateral displacement Y along

the pile. Moreover, two successive differentia-

tions of this curve allow the determination of

horizontal soil reaction P and then to define P–Y

curve at any depth.

Since the soil reaction geometrically represents

the curvature of bending moment distribution, it is

therefore very sensitive to any fluctuation of

bending moment at a given depth and strongly

depends on the choice of the fitting curve of

bending moment (Bouafia 1990; King 1994).

Quintic spline functions or polynomial functions

were used to fit the bending moment distribution.

The fitting function was chosen according to the

criterion of static equilibrium of the test pile under

lateral reaction profile P(z) and the loads on the

pile top within a given tolerance (Bouafia and

Fig. 3 Typical profiles ofin-situ tests in site S2

Table 2 Characteristics of test piles

Site Pile B (m) D/B EpIp(kN m2)

S1 T5 0.050 14.2 59.74T10 0.100 15.3 868.9T15 0.150 15.3 4,331.6

S2 P1 0.500 10.0 56,370P2 0.900 5.50 743,600

Fig. 4 General view of test pile in site S1

290 Geotech Geol Eng (2007) 25:283–301

123

Garnier 1991). This criterion was subsequently

adopted in other studies in LCPC (Mezazigh 1995;

Remaud 1999).

5.2 Interpretation of results

Figure 6 illustrates an example of P–Y curves

obtained according to this methodology. It can be

seen that P–Y curves at different depths are non-

linear shaped with an increase in soil stiffness

with depth. It is to be noticed that deflections and

soil reaction change in sign at almost the same

depth, say 10 diameters, which is in accordance

with Winkler’s hypothesis regarding the soil

reaction modulus (Bouafia 1998). Furthermore,

it can be seen that beyond a deflection of about

3% of B, limit lateral reaction is reached with

exhibition avec asymptotic values in the P–Y

curves along the pile.

The procedure of construction of P–Y curves

was validated by back-computation of all the test

piles. P–Y curves were introduced in the P–Y

curve based computer program single pile under

lateral loads (SPULL) developed in the university

of Blida. As shown in Fig. 7, computed deflec-

tions were found in very good agreement with the

experimental results. It is possible to accurately

describe the lateral load-deflection of the test

piles by means of these experimental P–Y curves.

5.3 Lateral reaction modulus

Hyperbolic formulation is often used to describe

the elastic plastic constitutive laws of soils (Dun-

can and Chang 1970) as well as the P–Y curves

(Reese 1971; Garassino 1976; Georgiadis et al.

1992). Experimental P–Y curves were fitted by

the following hyperbolic function:

P ¼ Y1

Etiþ Y

Pu

ð9Þ

Least squares regression coefficient was found

greater than 95% for curves corresponding to

depths above the zero displacement depth.

Beyond this depth values of Eti seem to be

inaccurate, since P and Y become small and the

ratio P/Y has no significance regarding the

uncertainties due to experiments as well as to

the procedure of interpretation of bending

moment curves.

For all the piles, the modulus Eti varies

linearly with depth. This fact is in accordance

with the distribution of soil modulus in homo-

geneous granular soils called Gibson’s soils.

Figure 8 shows a typical profile, which may be

described by

Eti ¼ NHZ ð10Þ

5.4 Influence of lateral pile/soil stiffness

on P–Y curves

It has been already stated that the influence of

lateral pile/soil stiffness on the P–Y curves was

not accounted for by the current methods. Most

of these methods simply correlate the parameters

of P–Y curves to those measured in PMT test.

Lateral pile/stiffness may be defined as follows:

Fig. 5 General view of piles P1 and P2 in site S2

Geotech Geol Eng (2007) 25:283–301 291

123

Kr ¼EpIp

EcD4ð11Þ

where Ec is a characteristic soil modulus

evaluated as an average value of PMT modulus

along the pile

Ec ¼1

D

ZD

0

Emdz ð12Þ

The ratio

w ¼ Eti

Emð13Þ

was studied for all the P–Y curves constructed. It

wad found that the average values of w vary as a

power of Kr as illustrated in Fig. 9. It is to be

noticed that pile P2 whose slenderness ratio D/B

is equal to 5.5 should be considered rather as a

pier. The proposed correlation is valid only for

long pile with D/B greater than 10 and may be

computed as follows:

EtiðzÞ ¼ EmðzÞw ¼ EmðzÞ � 0:28�K�0:55r ð14Þ

Within the interval 10–3 – 10–2 of pile/soil

stiffness studied, ratio w ranges between 3 and 9,

whereas Menard et al. (1969) recommended 2.75.

According to Eq. 11, lateral reaction modulus

-20-450

-400

-350

-300

-250

-200

-150

-100

-50

0

50

100

150

200

Pile P1 site Le-Rheu B=0.5 m D/B=10 Ep.Ip=56.37 MN.m2

Z/B= 0.5 = 1.0 = 1.5 = 2.0 = 2.5 = 3.0 = 3.5 = 4.0 = 4.5 = 5.0 = 5.5 = 6.0 = 6.5 = 7.0 = 7.5 = 8.0 = 8.5 = 9.0 = 9.5 = 10.

Lat

eral

rea

ctio

n P

(kN

/m)

Deflection Y (mm)-10 0 10 20 30 40 50 60

Fig. 6 Typical P–Ycurves for pile P1

00

5

10

15

20

25

30

35

40Pile T15

Pile T10

Pile T5

measured computed from

P-Y curves

Lat

eral

load

(kN

)

Top deflection (mm)20 40 60 80 100 120 140 160 180

Fig. 7 Comparison ofcomputed and measureddeflections in site S1

292 Geotech Geol Eng (2007) 25:283–301

123

approximately decreases with the square root of

the flexural pile stiffness and decreases with the

embedded length. Such dependence was not taken

account by the method of Menard et al. (1969)

and Fascicule-62.

Moreover, limit lateral reaction Pu was corre-

lated to the net limit pressure by defining the ratio

n ¼ Pu

p�l Bð15Þ

Figure 10 shows that for all the test piles with D/B

‡ 10, n increases as a power of Kr:

Pu ¼ 3P�LBffiffiffiffiffiffiKr

pð16Þ

Accordingly, lateral soil resistance around a rigid

pile is greater that around a flexible pile. Accord-

ing to Eq. 12, Pu increases with the square root of

the stiffness EpIp and decreases with the embed-

ded length. For the test piles, n ranges between

0.1 and 0.3, which is less than the recommended

values of Table 1.

In case of a solid circular pile, Eqs. 14 and 16

lead to the following simplified formulae

Eti

Em� 5

4ðD=BÞ2 1ffiffiffiffi

Kp ð17Þ

Pu

p�LB¼ 2

3

ffiffiffiffiKp

ðD=BÞ2ð18Þ

where K = Ep/Ec is the pile/soil compressibility.

These formulae show the important influence of

the slenderness ratio D/B on the parameters of

P–Y curve.

Curve 3 in Fig. 1 illustrates the proposed

hyperbolic P–Y curve. The reference displace-

ment Yref. corresponds to the intercept of the

initial linear portion with a slope equal to Eti, and

the horizontal asymptote corresponding to the

lateral resistance PuYref. is therefore defined as

the threshold of large lateral deflections of the

pile section and of full mobilisation of the lateral

soil resistance according to the elastic plastic

scheme of P–Y curve. Based on Eqs. 14 and 16,

ratio yref./B is simply expressed by the following

function of Kr and PMT characteristics

Fig. 8 Typical lateral reaction modulus profile

0,011

10

Eti/Em=a.(Kr)b

a=0.28b=-0.55R=94 %

D/B >=10 D/B=5.5

ψ

Kr

0,1

Fig. 9 Variation of the ratio Eti/Em with lateral pile/soilstiffness Kr

1E-30,01

0,1

1

Pu/(P*

L.B)=a.(Kr)b

a=3.0b=0.50R=94%

D/B >=10 D/B=5.5

ξ

Kr

0,01 0,1 1

Fig. 10 Variation of the ratio Pu/P�LB with lateral pile/soilstiffness Kr

Geotech Geol Eng (2007) 25:283–301 293

123

Yref:

B� 11Kr

p�LEm

ð19Þ

It is to be noticed from Eq. 9 that Yref. corre-

sponds to half the limit lateral reaction in the

hyperpoblic formulation and to all the limit

lateral reaction in the elastic plastic formulation.

Equation 19 provides a simple and useful tool to

estimate the threshold of large lateral load-

deflection behaviour. Figure 11 drawn for typical

values of Em/pL* of sandy soils gives for a flexible

pile characterised by Kr equal to 10–3 a reference

deflection of 0.22, 2.2 and 22 % of B in loose sand,

medium dense sand and very dense sand respec-

tively.

Hyperbolic P–Y curves proposed on the basis

of the interpretation of full-scale instrumented

piles in sandy soils provides a simple approach to

construct P–Y curves taking into consideration

some physical parameters of pile/soil interaction.

Lateral reaction modulus and lateral soil resis-

tance are defined as function of PMT character-

istics and the pile/soil stiffness according to Eq. 14

and 16.

The embedded length D is a key factor in

lateral response of piles. The effective embedded

length De is the relevant length of pile mobilised

in load-deflection behaviour and beyond which,

pile sections are at rest. It is usually expressed in

the function of the elastic length (or transfer

length) L0.

Length L0 of a pile embedded in a soil charac-

terised by a lateral reaction modulus increasing as a

linear function of depth

Eti ¼ bþNHz ð20Þ

or as a power function of depth

Eti ¼ NHzn ð21Þ

is given by (Matlock and Reese 1960)

L0 ¼

ffiffiffiffiffiffiffiffiffiffiEpIp

NH

ðnþ4Þ

sð22Þ

The effective length De is equal to 5L0 for

n ‡ 1 and to 3L0 for n = 0. A pile is classified as a

long one if D is greater than De and as a rigid one if

D is less than 2L0 for n ‡ 1 and to L0/2 for n = 0.

Lateral pile/soil stiffness should be evaluated in

Eq. 11 by introducing the minimum of the effec-

tive length and the total embedded length. In case

of a long pile, simultaneous use of Eqs. 11 and 22

necessitates an iterative process.

6 Validation of the proposed P–Y curve

The proposed P–Y curves method was assessed

by predicting the lateral response of full-scale as

well as in centrifuge test piles in sandy soils.

Tables 3 and 4 summarise the main geotechnical

and physical characteristics of soil/pile configura-

tions used in this regard. Piles are identified as

mentioned in the references.

The Lock & Dam 26 site is composed of

alluvial deposits (poorly graded sand) 3 m thick

and overlying glacial deposits (medium to coarse

sand with gravel) 17 m thick. The bedrock is a

hard limestone from the Mississipian age. Lateral

load tests were performed on two identical -

HP-14 · 73 piles socketed in the limestone

bedrock, jacked apart, and the lateral displace-

ment of each pile were measured.

The Longjuemau site is located near Paris and

composed of tertiary silty fine sand, rather

Table 3 Characteristics of soil/pile in full-scale tests

Site Pile D/B EpIp(MN m2) Installation Ec (MPa) Kr D/L0 Reference

Lock & Dam-26 3–12 57.3 61.00 Driven 20.6 1.9 · 10–3 16.2 (Briaud et al. 1989)Lock & Dam-26 3–13 57.3 61.00 Driven 20.6 1.9 · 10–3 16.2 (Briaud et al. 1989)Longjumeau TG 10.0 7.31 Driven 6.65 3.3 · 10–2 2.44 (Gambin 1979)Longjumeau TD 10.0 7.31 Driven 5.33 4.1 · 10–2 2.50 (Gambin 1979)Roosevelt bridge 16 18.4 958.50 Driven 61.3 1.1 · 10–2 6.80 (Townsend 1997)Lock & Dam-26 T3 44.5 61.70 Driven 15.6 2.3 · 10–3 12.3 (O’Neill 1983)Arkansas River 2 39.7 700.50 Driven 10.6 2.8 · 10–3 7.30 (Meyer 1979)Arkansas River 16 39.7 688.80 Driven 10.6 2.8 · 10–3 7.30 (Meyer 1979)

294 Geotech Geol Eng (2007) 25:283–301

123

uniformly graded. Piles TG and TD are installed

and loaded as in the above site.

The Roosevelt bridge site is composed of loose

layer of sand thick of 4 m, overlying a thick layer

of very dense partially cemented sand. The site

with submerged by water up to 2 m above the

ground level. Square prestressed concrete pile

was driven and tested up to cracking under a load

of 200 kN and concrete failure occurred under a

load of 320 kN.

Pile T3 was tested in lock & Dam site 7 years

prior to tests on piles 3–12 and 3–13, PMT data

were not available. Prediction of the pile T3 was

made with the PMT data of piles 3–12 and 3–13.

In the Arkansas site, the soil is a saturated SP/

SM sand and only the SPT test was carried out.

PMT data were estimated by usual correlation

with the SPT.

It should be emphasised that the reliability of

the predictions of piles T3 in Lock & Dam and

piles 2 and 16 in Arkansas site will decrease

because of the scatter in the estimation of the

PMT data for pile T3 or in the correlation with the

SPT test for piles 2 and 16. In this regard,

predictions of these piles will interpreted sepa-

rately.

For each pile, lateral pile/soil stiffness was

evaluated and hyperbolic P–Y curves according

to the Eqs. 14 and 16 were defined. In most of

cases, piles were sufficiently long to be considered

as restrained at their tips. SPULL programme was

used to predict the load-deflection curve of each

pile. As shown in Fig. 12, very good agreement

between predicted and measured deflections, with

remarkable fluctuation of the 35 points of com-

parison around the ratio predicted to measured

deflection of 1.11. Moreover, Y0pred./Y0

meas. Varied

between 0.81 and 1.84 with a mean value of 1.22

and a coefficient of variation of 21%. The results

of predictions are encouraging seeing the multi-

tude of approximations made during the process

of definition of this method.

Lateral response of test pile T3 in Lock & dam

26 and piles 2 and 16 in the Arkansas River was

predicted. As illustrated by Fig. 13, the ratio

Y0pred./Y0

meas. was found fluctuating around 1.33

within an interval 094–2.80 and a coefficient of

variation of 36%.

Sandy materials of sites S1 and S2 were used in

the LCPC centrifuge to study the lateral behaviour

of centrifuged models in sand within the scope of an

important programme of research undertaken by

the LCPC since two decades. Reduction scales of

piles were 1/40 and 1/18 for models in Fontaine-

bleau and Le-Rheu sands respectively. Character-

istics summarised in Table 4 correspond to the

prototype ones. Sandy mass was characterised by

cone penetration tests (CPT) carried out by min-

iature cone during the centrifugation. CPT tests in

centrifuge were used to estimate the PMT data by

adopting the same correlation CPT/PMT found

in-situ. This assumption leads to a rough estimation

0,000

5

10

15

20

251 : Loose sand E

m/P

L*=5

2 : Medium dense sand Em/P

L*=10

3 : Very dense sand Em/P

L*=20

Yre

f. /B

(%

)

Kr

1

2

3

0,02 0,04 0,06 0,08 0,10

Fig. 11 Variation of reference displacement with Kr

Table 4 Characteristics of prototype soil/piles in centrifuge

Soil Pile D/B EpIp (MN m2) Installation Ec (MPa) Kr D/L0 Reference

Fontainebleau – 16.7 473.60 Driven 34.1 3.0 · 10–3 7.33 (Remaud 1999)Le-Rheu ENSM-1 15.6 44.70 Bored 7.60 9.4 · 10–3 3.80 (Bouafia 1987)Le-Rheu ENSM-2 15.6 44.70 Bored 10.3 6.9 · 10–3 4.16 (Bouafia 1987)Le-Rheu P1–2 10.0 56.60 Bored 3.60 2.5 · 10–2 2.15 (Bouafia 1990)Le-Rheu P1–4 10.0 56.60 Bored 11.6 7.8 · 10–3 4.20 (Bouafia 1990)Le-Rheu P1–t 10.0 56.60 Bored 18.7 4.8 · 10–3 4.30 (Bouafia 1990)

Geotech Geol Eng (2007) 25:283–301 295

123

of the PMT data of sand in centrifuge and then to

an approximate prediction of the piles behaviour.

As shown in Fig. 14, good prediction is to be

noticed for small deflections up to 10% of B.

The ratio Y0pred./Y0

meas. of the 27 points of

comparison varied within the interval 0.56–2.40

with a mean value of 1.43 and a coefficient of

variation of 30%.

7 Comparative study

The predictive capability of the proposed method

is to be compared with that of the current

methods of construction of P–Y curves. Due to

the non-availability of the PMT expansion curves,

the comparison was limited to the method of

Fascicule-62. For all the piles where the PMT

data was available, P–Y curves illustrated by

curve 2 in Fig. 1 were defined and input in

SPULL. Figure 15 summarises the comparison

between predicted and measured deflections. It

can be seen that the ratio Y0pred./Y0

meas. of the 53

points of comparison fluctuates around 0.81

within a margin of 0.31–3.30 and a coefficient of

variation of 40%. The proposed method slightly

overpredicted the pile deflections whereas the

method of Fascicule-62 underpredicted them.

0 5 10 15 20 25 30 35 40 45 50

0

5

10

15

20

25

30

35

40

45

50

Y0/

B p

redi

cted

%

Y0/B measured %

Lock & Dam, Pile 3-12 Lock & Dam, Pile 3-13 Roosevelt bridge, Pile 16 Longjumeau, Pile TG Longjumeau, Pile TD

Full-scale pile loading tests

(PMT data available)

Y0pred./Y0

meas.=1,11

(R2=94%)

Fig. 12 Comparison ofpredicted and measureddeflections (PMT dataavailable)

0 10 20 30 40 50 60 70 80 90 100

0

10

20

30

40

50

60

70

80

90

100

Y0 pr

edic

ted

(mm

)

Y0 measured (mm)

Lock & Dam, 1978 Pile T-3 Arknsas river, 1970 Pile 2 Arkansas river, 1970 Pile 16

Full-scale pile loading tests

(PMT data estimated)

Y0pred./Y0

meas.=1,33

(R2=94%)

Fig. 13 Comparison ofpredicted and measureddeflections (PMT dataestimated)

296 Geotech Geol Eng (2007) 25:283–301

123

8 Normalised load-deflection curves

Test piles in Lock & Dam 26 and Roosevelt

bridge were added to those in sites S1 and S2 to

build a small database of nine lateral loading tests

in four sandy sites. As shown in Fig. 16, load-

deflection curves were normalised by dividing the

lateral load by an equivalent net limit pressure P�leand the frontal area De · B, and by dividing the

deflection Y0 at ground level by the diameter or

the width B.

Net equivalent limit pressure is an average net

value along the effective length De of the pile,

evaluated as follows:

p�Le ¼1

De

ZDe

0

p�Ldz ð23Þ

P�Le is to be computed along the total embedded

length D if it is shorter than the effective length.

Normalised curves were located in a rather

narrow band, which allowed fitting all the exper-

imental data by unique fitting curve. In a

preliminary stage of pile design, tabulated values

of this function in Table 5 provides a simple

approach to estimate the pile deflections under

working loads.

0 5 10 15 20 25 30 35 40 45 50

0

5

10

15

20

25

30

35

40

45

50

Centrifuge tests

Y0/

B p

redi

cted

%

Y0/B measured %

Fontainebleau sand, 1999 Le-Rheu sand, 1987 (test 1) Pile ENSM-1 Le-Rheu sand, 1987 (test 2) Pile ENSM-2 Le-Rheu sand, 1988 (test 2) Pile P1-2 Le-Rheu sand, 1988 (test 4) Pile P1-4 Le-Rheu sand, 1991 (test t) Pile P1-t

Fig. 14 Comparison ofpredicted and measureddeflections (centrifugetests)

00

10

20

30

40

50

60

70

80

90

100

Y0 pr

edic

ted

(mm

)

Y0 measured (mm)

Lock &Dam Pile3-12 Lock &Dam Pile3-13 Roosevelt bridge Pile 16 Longjumeau Pile TG Longjumeau Pile TD Chatenay Pile T5 Chatenay Pile T10 Chatenay Pile T15 Le-Rheu Pile P1 Le Rheu Pile P2

Full-Scale Pile Loading testsMethod: Fascicule -62

Y0pred,/Y0

meas,=0,81

R2=84%

10 20 30 40 50 60 70 80 90 100

Fig. 15 Comparison ofpredicted and measureddeflections (Fascicule-62)

Geotech Geol Eng (2007) 25:283–301 297

123

9 Parametric study of piles in Gibson’s soils

The influence of the lateral pile/soil stiffness on

the non-linear behaviour of piles embedded in a

soil characterised by linear profiles of PMT

characteristics was investigated.

Dimensional analysis according to Bucking-

ham’s theorem of the following general equation

f ðY0;H;M0;EpIp;B;D; k; lÞ ¼ 0 ð24Þ

led to the following dimensionless equation:

gðH=ðp�LeBDeÞ;Y0=B;Kr; k=lÞ ¼ 0 ð25Þ

where M0 is the bending moment applied at the

ground level and taken equal to 0 in this study.

k and l are respectively the rate of linear

increase of Em and P�L with depth. The ratio k/l is

equal to Em/P�L.

Two extreme cases of pile/soil stiffness were

studied: rigid pile characterised by Kr equal to

1.15 and a flexible pile characterised by Kr equal

to 0.002. the piles were embedded in two extreme

types of soils: loose sand and very dense sand with

Em/pL* equal to 5 and 20 respectively. Hyperbolic

P–Y curves were defined by Eqs. 14 and 16 and

introduced in SPULL.

As shown in Figs. 17 and 18, load-deflection

curves in both the two soils are very sensitive to

Kr, particularly in the domain of large deflections.

This result shows that the lateral pile capacity

should be analysed in relation with the pile/soil

stiffness, contrarily to the current methods of

estimation of the lateral pile capacity which

simply neglect this important factor by necessity

of simplification of the analysis.

10 Conclusions

The lateral response of single piles in sand was

investigated on the basis of interpretation of full-

scale lateral loading tests of single instrumented

piles in quite homogeneous sandy soils, in order

to define the parameters of P–Y curves, namely

the lateral reaction modulus and the lateral

soil resistance, in correlation with the PMT

parameters.

After a brief description of some current

methods of construction of P–Y curves, experi-

mental P–Y curves of the test piles were derived

and the corresponding parameters were found

varying as a power of lateral pile/soil stiffness, on

the basis of which hyperbolic P–Y curves in sand

were proposed. The predictive capabilities of the

proposed P–Y curves were assessed by predicting

the soil/pile response in full-scale tests as well as

in centrifuge tests and a very good agreement was

found between the computed deflections and the

measured ones.

Simple linear relationship was proposed

between the reference deflection, which is the

threshold of large deflections, and the PMT

characteristics and the pile/soil stiffness.

Small sized database of full-scale pile loading

tests in sand was built. It was shown the load-

deflection curves of these test piles may be

normalised in a practical form for a rough

estimation of the pile deflection under working

loads in a preliminary stage of pile design.

1E-41E-3

0,01

0,1

1

Le Rheu Pile P1 Le Rheu Pile P2 Chatenay Pile T5 Chatenay Pile T10 Chatenay Pile T15 Lock & Dam Pile 3-12 Lock & Dam Pile 3-13 Lock & Dam Pile Pile T3 Roosevelt Bridge Pile 16

H/(

P* le

.D.B

)

Y0/B

R2=93%

1E-3 0,01 0,1 1

Fig. 16 Normalised load-deflection curves

Table 5 Tabulated values of the normalised load-deflection curve

Y0/B 10–4 10–3 5 · 10–3 10–2 5 · 10–2 10–1 2 · 10–1

H/(P�LBDe) 2 · 10–3 7.06 · 10–3 1.70 · 10–2 2.50 · 10–2 6.0 · 10–2 8.8 · 10–2 12.8 · 10–2

298 Geotech Geol Eng (2007) 25:283–301

123

At last, a parametric study of the influence of

lateral pile/soil stiffness on the non linear response

of piles embedded in Gibson’s soils was under-

taken on the basis of the proposed P–Y curves. It

was shown the significant influence of the lateral

pile/soil stiffness on the load-deflection curves

particularly in the domain of large deflections.

Acknowledgments The study reported herein is supportedby the Algerian ministry of higher education and researchMESRS within the scope of the project COPIFOR(COmportement des PIeux FORes) under the code J0901/04/02/2000.

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