review of p-y relationships in cohesionless soil

Review of p-y relationships in cohesionless soil

K.T. BrødbækM. Møller

S.P.H. SørensenA.H. Augustesen

Department of Civil EngineeringISSN 1901-726X DCE Technical Report No. 57

DCE Technical Report No. 57

Aalborg University Department of Civil Engineering

Water & Soil

Review of p-y relationships in cohesionless soil

by

K.T. Brødbæk M. Møller

S.P.H. Sørensen A.H. Augustesen

January 2009

© Aalborg University

Scientific Publications at the Department of Civil Engineering Technical Reports are published for timely dissemination of research results and scientific workcarried out at the Department of Civil Engineering (DCE) at Aalborg University. This medium allows publication of more detailed explanations and results than typically allowed in scientificjournals. Technical Memoranda are produced to enable the preliminary dissemination of scientific work bythe personnel of the DCE where such release is deemed to be appropriate. Documents of this kindmay be incomplete or temporary versions of papers—or part of continuing work. This should be kept in mind when references are given to publications of this kind. Contract Reports are produced to report scientific work carried out under contract. Publications ofthis kind contain confidential matter and are reserved for the sponsors and the DCE. Therefore,Contract Reports are generally not available for public circulation. Lecture Notes contain material produced by the lecturers at the DCE for educational purposes. Thismay be scientific notes, lecture books, example problems or manuals for laboratory work, orcomputer programs developed at the DCE. Theses are monograms or collections of papers published to report the scientific work carried out atthe DCE to obtain a degree as either PhD or Doctor of Technology. The thesis is publicly availableafter the defence of the degree. Latest News is published to enable rapid communication of information about scientific workcarried out at the DCE. This includes the status of research projects, developments in thelaboratories, information about collaborative work and recent research results.

Published 2009 by Aalborg University Department of Civil Engineering Sohngaardsholmsvej 57, DK-9000 Aalborg, Denmark Printed in Aalborg at Aalborg University ISSN 1901-726X DCE Technical Report No. 57

Review of p�y relationships in cohesionless soilK. T. Brødbæk1; M. Møller2; S. P. H. Sørensen3; and A. H. Augustesen4

Aalborg University, January 2009

AbstractMonopiles are an often used foundation concept for o�shore wind turbine converters.These piles are highly subjected to lateral loads and thereby bending moments due towind and wave forces. To ensure enough sti�ness of the foundation and an acceptablepile-head de�ection, monopiles with diameters at 4 to 6 m are typically necessary. Incurrent practice these piles are normally designed by use of the p�y curve methodalthough the method is developed and veri�ed for small diameter, slender piles with adiameter up to approximately 2 m. In the present paper a review of existing p�y curveformulations for piles in sand under static loading is presented. Based on numericaland experimental studies presented in the literature, advances and limitations ofcurrent p�y curve formulations are outlined.

1 Introduction

It is a predominating opinion that theglobal warming is caused by the emissionof greenhouse gasses. According to UnitedNations (1998) there is a strong politicalinterest to raise the percentage of renew-able energy, and reduce the use of fossilfuels, in the years to come. Wind energyplays a major role in attaining these goalsboth onshore and o�shore which is why afurther development is of interest.

Several concepts for o�shore wind turbinefoundations exist. The choice of foun-dation concept primarily depends on site

1Graduate Student, Dept. of Civil Engineer-ing, Aalborg University, Sohngaardsholmsvej 57,9000 Aalborg, Denmark.



4Assistant Professor, Dept. of Civil Engineer-ing, Aalborg University, Sohngaardsholmsvej 57,9000 Aalborg, Denmark.

conditions and the dominant type of load-ing. At great water depths the most com-mon foundation principle is monopiles,which are single steel pipe piles. The foun-dation should be designed to have enoughultimate resistance against vertical andlateral loads. Moreover, the deformationcriteria and sti�ness of the foundationsshould be acceptable under lateral load-ing which is normally the primary designcriterion for this type of foundation. Inorder to avoid resonance the �rst naturalfrequency of the structure needs to be be-tween 1P and 3P , where P denotes thefrequency corresponding to one rotor rota-tion. According to LeBlanc et al. (2007)monopiles installed recently have diame-ters around 4 to 6 m and a pile slender-ness ratio (L/D) around 5 where L is theembedded length and D is the outer pilediameter.

In current design of laterally loaded o�-shore monopiles, p�y curves are normallyused. A p�y curve describes the non-linear relationship between the soil resis-tance acting against the pile wall, p, and

1

2 Review of p�y relationships in cohesionless soil

the lateral de�ection of the pile, y. Notethat there in present paper is distinguishedbetween soil resistance, p, and ultimatesoil resistance, pu. The soil resistance isgiven as the reaction force per unit lengthacting on the pile until reaching the ulti-mate soil resistance. The ultimate soil re-sistance is given as the point of maximumsoil resistance.

Several formulations of p�y curves existdepending on the type of soil. These for-mulations are originally formulated to beemployed in the o�shore gas and oil sec-tor. However they are also used for o�-shore wind turbine foundations, althoughpiles with signi�cantly larger diameter andsigni�cantly smaller slenderness ratio areemployed for this type of foundation.

In this paper the formulation and imple-mentation of p�y curves proposed by Reeseet al. (1974) and API (1993) for piles insands due to static loading will be anal-ysed. However, alternative methods fordesigning laterally loaded piles have beenproposed in the literature. According toFan and Long (2005) these alternative ap-proaches can generally be classi�ed as fol-lows:

� The limit state method.

� The subgrade reaction method.

� The elasticity method.

� The �nite element method.

Simplest of all the methods are the limitstate methods, e.g. Broms (1964), consid-ering only the ultimate soil resistance.

The simplest method for predicting thesoil resistance due to a given applied hor-izontal de�ection is the subgrade reactionmethod, e.g. Reese and Matlock (1956)and Matlock and Reese (1960). In thiscase the soil resistance is assumed lin-early dependent on the pile de�ection.

Full-scale tests though substantiate a non-linear relationship between soil resistanceand pile de�ection. The subgrade reac-tion method must therefore be consideredtoo simple and highly inaccurate. In addi-tion the subgrade reaction method is notable to predict the ultimate lateral resis-tance. The p�y curve method assumesa non-linear dependency between soil re-sistance and pile de�ection and is there-fore able to produce a more accurate so-lution. Furthermore the ultimate lateralresistance can be estimated by using thep�y curve method.

In both the p�y curve method and thesubgrade reaction method the winkler ap-proach, cf. section 2, is employed to cal-culate the lateral de�ection of the pile andinternal forces in the pile. When employ-ing the Winkler approach the pile is con-sidered as a beam on an elastic foundation.The beam is supported by a number of un-coupled springs with spring sti�ness givenby p�y curves. When using the Winklerapproach the soil continuity is not takeninto account as the springs are considereduncoupled.

The elasticity method, e.g. Banerjee andDavis (1978), Poulos (1971), and Pou-los and Davis (1980), includes the soilcontinuity. However, the response is as-sumed to be elastic. As soil is more likelyto behave elasto-plastically, this elasticitymethod is not to be preferred unless onlysmall strains are considered. Hence, themethod is only valid for small strains andthereby not valid for calculating the ulti-mate lateral resistance.

Another way to deal with the soil con-tinuity and the non-linear behaviour isto apply a three-dimensional �nite ele-ment model, e.g. Abdel-Rahman andAchmus (2005). When applying a three-dimensional �nite element model both de-formations and the ultimate lateral re-sistance can be calculated. Due to thecomplexity of a three-dimensional model,

K. T. Brødbæk, M. Møller and S. P. H. Sørensen 3

substantial computational power is neededand calculations are often very time con-suming. Phenomena such as liquifaction,due to non-appropriate kinematic mod-els, and gaps between soil and pile areat present hard to handle in the models.Hence, a �nite element approach is a use-ful method but the accuracy of the resultsis highly dependent on the applied consti-tutive soil models as well as the calibrationof these models.

2 p�y curves and Winklerapproach

As a consequence of the oil and gas indus-try's expansion in o�shore platforms in the1950s, models for design of laterally loadedpiles were required. The key problem isthe soil-structure interaction as the sti�-ness parameters of the pile, Ep, and thesoil, Es, may be well known but at thesoil-pile interface the combined parameterEpy is governing and unknown. In order toinvestigate this soil-pile interaction a num-ber of full-scale tests on fully instrumentedpiles have been conducted and various ex-pressions depending on the soil conditionshave been derived to predict the soil pres-sure on a pile subjected to lateral loading.

Historically, the derivation of the p�ycurve method for piles in sand is as fol-lows:

� Analysing the response of beams onan elastic foundation. The soil ischaracterised by a series of linear-elastic uncoupled springs, introducedby Winkler (1867).

� Hetenyi (1946) presents a solution tothe beam on elastic foundation prob-lem.

� McClelland and Focht (1958) as wellas Reese and Matlock (1956) suggestthe basic principles in the p�y curvemethod.

� Investigations by Matlock (1970) in-dicates that the soil resistance in onepoint is independent of the pile defor-mation above and below that exactpoint.

� Tests on fully instrumented test pilesin sand installed at Mustang Islandare carried out in 1966 and reportedby Cox et al. (1974).

� A semi-empirical p�y curve expressionis derived based on the Mustang Is-land tests, cf. Reese et al. (1974).The expression becomes the state-of-the-art in the following years.

� Murchison and O'Neill (1984) com-pare the p�y curve formulation pro-posed by Reese et al. (1974)with three simpli�ed expressions(also based on the Mustang Islandtests) by testing the formulationsagainst a database of relatively well-documented lateral pile load tests. Ahyperbolic form is found to providebetter results compared to the origi-nal expressions formulated by Reeseet al. (1974).

Research has been concentrated on deriv-ing empirical (e.g. Reese et al. 1974)and analytical (e.g. Ashour et al. 1998)p�y curve formulations for di�erent typesof soil giving the soil resistance, p, as afunction of pile displacement, y, at a givenpoint along the pile. The soil pressure ata given depth, xt, before and during a sta-tic excitation is sketched in �g. 1b. Thepassive pressure on the front of the pile isincreased as the pile is de�ected a distanceyt while the active pressure at the back isdecreased.

An example of a typical p�y curve is shownin �g 2a. The curve has an upper horizon-tal limit denoted by the ultimate soil resis-tance, pu. The horizontal line implies thatthe soil behaves plastically meaning that


(a) Pile bending dueto lateral loading.

(b) Stresses on a pile before and dur-ing lateral excitation.

Figure 1: Distribution of stresses before and dur-ing lateral excitation of a circular pile. pt denotesthe net force acting on the pile at the depth xt,after Reese and Van Impe (2001).

no loss of shear strength occurs for increas-ing strain. The subgrade reaction modu-lus, Epy, at a given depth, x, is de�ned asthe secant modulus p/y. Epy is therebya function of both lateral pile de�ection,y, and depth, x, as well as the physicalproperties and load conditions. Epy doesnot uniquely represent a soil property, butis simply a convenient parameter that de-scribes the soil-pile interaction. Epy is con-stant for small de�ections for a particulardepth, but decreases with increased de�ec-tion, cf. �g. 2b. A further examination ofthe shape of p�y curves is to be found insection 3, and an overview of the used pa-rameters are given in tab. 1.

Since the pile de�ection is non-linear aconvenient way to obtain the soil resis-tance along the pile is to apply the Winklerapproach where the soil resistance is mod-elled as uncoupled non-linear springs withsti�ness Ki acting on an elastic beam asshown in �g. 3. Ki is a non-linear loadtransfer function corresponding to Epy.By employing uncoupled springs layeredsoils can conveniently be modelled.

The governing equation for beam de�ec-

(a) p�y curve.

(b) Variation of subgrade re-action modulus.

Figure 2: Typical p�y curve and variation ofthe modulus of subgrade reaction at a given pointalong the pile, after Reese and Van Impe (2001).

Figure 3: The Winkler approach with the pilemodelled as an elastic beam element supportedby non-linear uncoupled springs. K is the sti�nesscorresponding to Epy.

tion was stated by Timoshenko (1941).The equation for an in�nitesimal small ele-ment, dx, located at depth x, subjected tolateral loading, can be derived from staticequilibrium. The sign convention in �g.4 is employed. N , V , and M de�nes theaxial force, shear force and bending mo-ment, respectively. The axial force, N , isassumed to act in the cross-section's centreof gravity.

Equilibrium of moments and di�erentiat-ing with respect to x leads to the follow-ing equation where second order terms are


Table 1: De�nition of parameters and dimensions used in the present paper.Description Symbol De�nition DimensionPile diameter D LPile length L LSoil resistance p F/LUltimate resistance pu F/LSoil pressure P P = p/D F/L2

Pile de�ection y LDepth below soil surface x LSecond moment of inertia Ip L4

Young's modulus of elasticity of the pile Ep F/L2

Modulus of subgrade reaction Epy Epy = p/y F/L2

Initial sti�ness E∗py E∗

py = dpdy , y = 0 F/L2

Initial modulus of subgrade reaction k F/L3

neglected:

d2M

dx2+

dV

dx−N

d2y

dx2= 0 (1)

Following relations are used:

M = EpIpκ (2)dV

dx= −p (3)

p(y) = −Epyy (4)

Ep and Ip are the Young's modulus of elas-ticity of the pile and the second momentof inertia of the pile, respectively. κ is thecurvature strain of the beam element.

Figure 4: Sign convention for in�nitesimal beamelement.

Use of (2)�(4) and the kinematic assump-tion κ = d2y

dx2 which is valid in Bernoulli-Euler beam theory the governing fourth-

order di�erential equation for determina-tion of de�ection is obtained:

EpIpd4y

dx4−N

d2y

dx2+ Epyy = 0 (5)

In (5) the shear strain, γ, in the beam isneglected. This assumption is only validfor relatively slender beams. For short andrigid beams the Timoshenko beam theory,that takes the shear strain into account,is preferable. The following relations areused:

V = GpAvγ (6)

γ =dy

dx− ω (7)

κ =dω

dx(8)

Gp and Av are the shear modulus and thee�ective shear area of the beam, respec-tively. ω is the cross-sectional rotation de-�ned in �g. 5. In Timoshenko beam the-ory the shear strain and hereby the shearstress is assumed to be constant over thecross section. However, in reality the shearstress varies parabolic over the cross sec-tion. The e�ective shear area is de�nedso the two stress variations give the sameshear force. For a pipe the e�ective sheararea can be calculated as:

Av = 2(D − t)t (9)


Figure 5: Shear and curvature deformation of abeam element.

where t is the wall thickness of the pipe.

By combining (1) � (4) and (6) � (8) twocoupled di�erential equations can be for-mulated to describe the de�ection of abeam:

GAvd

dx

(dy

dx− ω

)−Epyy = 0 (10)

EpIpd3ω

dx3−N

d2y

dx2+ Epyy = 0 (11)

In the derivation of the di�erential equa-tions the following assumptions have beenused:

� The beam is straight and has a uni-form cross section.

� The beam has a longitudinal plane ofsymmetry, in which loads and reac-tions lie.

� The beam material is homogeneous,isotopic, and elastic. Furthermore,plastic hinges do not occur in thebeam.

� Young's modulus of the beam mate-rial is similar in tension and compres-sion.

� Beam de�ections are small.

� The beam is not subjected to dynamicloading.

3 Formulations of p�ycurves for sand

p�y curves describing the static behaviourof piles in cohesionless soils are describedfollowed by a discussion of their valid-ity and limitations. Only the formulationmade by Reese et al. (1974), hereafterdenoted Method A, and the formulationproposed by API (1993), Method B, willbe described. Both p�y curve formulationsare empirically derived based on full-scaletests on free ended piles at Mustang Is-land.

3.1 Full-scale tests at MustangIsland

Tests on two fully instrumented, identicalpiles located at Mustang Island, Texas asdescribed by Cox et al. (1974), are thestarting point for the formulation of p�ycurves for piles in sand. The test set up isshown in �g. 6.

To install the test- and reaction piles aDelmag-12 diesel hammer was used. Thetest piles were steel pipe piles with a dia-meter of 0.61 m (24 in) and a wall thick-ness of 9.5 mm (3/8 in). The embed-ded length of the piles was 21.0 m (69 ft)which corresponds to a slenderness ratio ofL/D = 34.4. The piles were instrumentedwith a total of 34 active strain gauges from0.3 m above the mudline to 9.5 m (32 ft)below the mudline. The strain gauges werebonded directly to the inside of the pilein 17 levels with highest concentration ofgauges near the mudline. The horizontaldistance between the centre of the two testpiles was 7.5 m (24 ft and 8 in). Betweenthe piles the load cell was installed on fourreaction piles. The minimum horizontaldistance from the centre of a reaction pileto the centre of a test pile was 2.8 m (9 ftand 4 in). The water table was located atthe soil surface, hereby the soil was fullysaturated.


Elev

ation

[m

]

2.7

0.90.30.0

8.89.5

21.0

Instr

umen

ted

secti

on

2.8 1.9 2.8Horizontal distance [m]

Load arrangement

Pile 1 Pile 2

Reaction piles

Diaphragm

Steel Pipe Pile :Outside diameter = 0.61 mWall thickness = 9.525 mmEmbedment length = 21 mEpIp = 1.7144 x 105 kNm2

Figure 6: Set-up for Mustang Island tests, after Cox et al. (1974).

Prior to pile installation, two soil boringswere made, each in a range of 3.0 m (10ft) from a test pile. The soil samplesshowed a slight di�erence between the twoareas where the piles were installed, asone boring contained �ne sand in the top12 m and the other contained silty �nesand. The strength parameters were de-rived from standard penetration tests ac-cording to Peck et al. (1953). The stan-dard penetration tests showed large vari-ations in the number of blows per ft. Es-pecially in the top 40 ft of both boringsthe number of blows per ft varied from 10to 80. From 40 to 50 ft beneath the mud-line clay was encountered. Beneath theclay layer the strength increased from 40to 110 blows per ft. From 60 ft beneaththe mudline to the total depth the num-ber of blows per ft decreased from 110 to15.

The piles were in total subjected to sevenhorizontal load cases consisting of two sta-tic and �ve cyclic. Pile 1 was at �rst sub-jected to a static load test 16 days after in-stallation. The load was applied in stepsuntil a maximum load of 267 kN (60000lb) was reached. The maximum load was

determined as no failure occurred in thepile. After the static load test on pile 1 twocyclic load tests were conducted. 52 daysafter installation a pull-out test was con-ducted on pile 2. A maximum of 1780 kN(400000 lb) was applied causing the pileto move 25 mm (1 inch). After anotherweek pile 2 was subjected to three casesof cyclic loading and �nally a static loadtest. The static load case on pile 2 was per-formed immediately after the third cyclicload case which might a�ect the results.Reese et al. (1974) do not clarify whetherthis e�ect is considered in the analyses.

3.2 Method A

Method A is the original method based onthe Mustang Island tests, cf. Reese et al.(1974). The p�y curve formulation con-sists of three curves: an initial straightline, p1, a parabola, p2, and a straight line,p3, all assembled to one continuous piece-wise di�erentiable curve, cf. �g. 7. Thelast straight line from (ym,pm) to (yu,pu)is bounded by an upper limit characterisedby the ultimate resistance, pu.


Figure 7: p�y curve for static loading usingmethod A, after Reese et al. (1974).

Ultimate resistance

The total ultimate lateral resistance, Fpt,is equal to the passive force, Fp, minus theactive force, Fa, on the pile. The ultimateresistance can be estimated analyticallyby means of either statically or kinemat-ically admissible failure modes. At shal-low depths a wedge will form in front ofthe pile assuming that the Mohr-Coulombfailure theory is valid. Reese et al. (1974)uses the wedge shown in �g. 8 to analyt-ically calculate the passive ultimate resis-tance at shallow depths, pcs. By using thisfailure mode a smooth pile is assumed, andtherefore no tangential forces occur at thepile surface. The active force is also com-puted from Rankine's failure mode, usingthe minimum coe�cient of active earthpressure.

At deep depths the sand will, in contrastto shallow depths, �ow around the pile anda statical failure mode as sketched in �g. 9is used to calculate the ultimate resistance.The transition depth between these failuremodes occurs, at the depth where the ul-timate resistances calculated based on thetwo failure modes are identical.

The ultimate resistance per unit length ofthe pile can for the two failure modes be

Fp

D

h

Direction of

movement

x

Figure 8: Failure mode for shallow depths, afterReese et al. (1974).

Pile

Pile movementShear failure surfaceMovement of block

Figure 9: Failure mode for deep depths, afterReese et al. (1974).

calculated according to (12) and (13):

pcs = γ′xK0x tanϕtr sinβ

tan(β − ϕtr) cos α(12)

+ γ′xtanβ

tan(β − ϕtr)(D − x tanβ tanα)

+ γ′x(K0x tanϕtr(tanϕtr sinβ − tanα)−KaD)

pcd = KaDγ′x(tan8 β − 1) (13)+ K0Dγ′x tanϕtr tan4 β

pcs is valid for shallow depths and pcd fordeep depths, γ′ is the e�ective unit weight,and ϕtr is the internal angle of frictionbased on triaxial tests. The factors α andβ measured in degrees can be estimated bythe following relations:

α =ϕtr

2(14)

β =45◦ +ϕtr

2(15)

Hence, the angle β is estimated accordingto Rankine's theory which is valid if the


pile surface is assumed smooth. The factorα depends on the friction angle and loadtype. However, the e�ect of load type isneglected in (14). Ka and K0 are the coef-�cients of active horizontal earth pressureand horizontal earth pressure at rest, re-spectively:

Ka = tan2(45− ϕtr

2) (16)

K0 = 0.4 (17)

The value of K0 depends on several fac-tors, e.g. the friction angle, but (17) doesnot re�ect that.

The theoretical ultimate resistance, pc, asfunction of depth is shown in �g. 10.As shown, the transition depth increaseswith diameter and angle of internal fric-tion. Hence, for piles with a low slender-ness ratio the transition depth might ap-pear far beneath the pile toe.

By comparing the theoretical ultimate re-sistance, pc, with the full-scale tests atMustang Island, Cox et al. (1974) founda poor agreement. Therefore, a coe�cientA is introduced when calculating the ac-tual ultimate resistance, pu, employed inthe p�y curve formulations:

pu = Apc (18)

The variation of the coe�cient A withnon-dimensional depth, x/D, is shown in�g. 11a. The deformation causing the ul-timate resistance, yu, cf. �g. 7, is de�nedas 3D/80.

p�y curve formulation

The soil resistance per unit length, pm, atym = D/60, cf. �g. 7, can be calculatedas:

pm = Bpc (19)

B is a coe�cient depending on the non-dimensional depth x/D, as plotted in �g.11b.

0 1 2 3 4 5 6

x 104

−30

−25

−20

−15

−10

−5

0

pc [kN/m]

x [m

]

φtr=30°

φtr=40°

(a) D = 1.0 m

0 1 2 3 4 5 6

x 105

−120

−100

−80

−60

−40

−20

0

pc [kN/m]

x [m

]

φtr=30°

φtr=40°

(b) D = 4.0 m

Figure 10: Theoretical ultimate resistance, pc,as function of the depth. γ′ = 10 kN/m3 has beenused to plot the �gure. The transition depths aremarked with circles.

The slope of the initial straight line, p1

as shown in �g. 7, depends on the initialmodulus of subgrade reaction, k, cf. tab.1, and the depth x. This is due to the factthat the in-situ soil modulus of elasticityalso increases with depth. Further, it is as-sumed that k increases linearly with depthsince laboratory test shows, that the initialslope of the stress-strain curve for sand isa linear function of the con�ning pressureTerzaghi (1955). The initial straight lineis given by:

p1(y) = kxy (20)

Reese et al. (1974) suggest that the valueof k only depends on the relative den-sity/internal friction angle for the sand.On basis of full-scale experiments values


A1 2 300

1

2

3

4

5

6

Ac (cyclic)

As (static)

>5, A=0.88xD

xD

(a) Non-dimensional coe�cient A for de-termining the ultimate soil response.

B1 2 300

1

2

3

4

5

6

Bc (cyclic)

Bs (static)

>5, Bc=0.55Bs=0.5

xxD D

(b) Non-dimensional coe�cient B for de-termining the soil response, pm.

Figure 11: Non-dimensional variation of A andB, after Reese et al. (1974).

of k for loose sands, for medium sands,and for dense sands are 5.4 MN/m3 (20lbs/in3), 16.3 MN/m3 (60 lbs/in3), and 34MN/m3 (125 lbs/in3), respectively. Thevalues are valid for sands below the wa-ter table. Earlier estimations of k hasalso been made, for example by Terza-ghi (1955), but according to Reese andVan Impe (2001) these methods have beenbased on intuition and insight. Design reg-ulations (e.g. API 1993 and DNV 1992)recommend the use of the curve shown in�g. 12. The curve only shows data for re-lative densities up to 80 %, which causeslarge uncertainties in the estimation of kfor very dense sands.

Relative density [%]

Below thewater table

Above thewater table

V. Loose

28

0 20 40 60 80 100

80000

70000

60000

50000

40000

30000

20000

10000

0

29 30 36 40 45

Loose Medium dense Dense V. dense

Friction angle, [degrees]

k[kPa

/m]

Figure 12: Variation of initial modulus of sub-grade reaction k as function of relative density,after API (1993).

The equation for the parabola, p2, cf. �g.7, is described by:

p2(y) = Cy1/n (21)

where C and n are constants. Theconstants and the parabola's start point(yk,pk) are determined by the followingcriteria:

p1(yk) = p2(yk) (22)p2(ym) = p3(ym) (23)

∂p2(ym)∂y

=∂p3(ym)

∂y(24)

The constants can then be estimated:

n =pm

mym(25)

C =pm

y1/nm

(26)

yk = (C

kx)n/(n−1) (27)

where m is the slope of the line, p3.

3.3 Method B

Design regulations (e.g. API 1993 andDNV 1992) suggest a modi�ed formulation


of the p�y curves, in which the analyti-cal expressions for the ultimate resistance,(12) and (13), are approximated using thedimensionless parameters C1, C2 and C3:

pu = min(

pus = (C1x + C2D)γ′xpud = C3Dγ′x

)

(28)

The constants C1, C2 and C3 can be de-termined from �g. 13.

Angle of internal friction, , [degrees]j

C 1

C 2

C 3

Cand C

[-]

12

C[-

]3

1

2

3

4

5

6

0

25 30 35 40 45

0

20

40

60

80

100

120

Figure 13: Variation of the parameters C1, C2

and C3 as function of angle of internal friction,after API (1993).

A hyperbolic formula is used to describethe relationship between soil resistanceand pile de�ection instead of a piecewiseformulation as proposed by method A:

p(y) = Apu tanh(kx

Apuy) (29)

The coe�cient A could either be deter-mined from �g. 11a or by:

A =(

3.0− 0.8H

D

)≥ 0.9 (30)

Since:

dp

dy|y=0= Apu

kxApu

cosh2( kxyApu

)|y=0= kx (31)

the p�y curve's initial slope is then simi-lar using the two methods, cf. (20). Alsothe upper bound of soil resistance will ap-proximately be the same. However, there

is a considerable di�erence in soil resis-tance predicted by the two methods whenconsidering the section between the points(yk,pk) and (yu,pu) as shown in �g. 14.The soil parameters from tab. 2 has beenused to construct the p�y curves shown in�g. 14.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

500

1000

1500

2000

2500

3000

3500

y [m]

p [k

N/m

]Method AMethod Bkmu

x = 5 m

x = 10 m

Figure 14: Example of p�y curves based onmethod A and B. The points k, m, and u refers tothe points (yk,pk), (ym,pm), and (yu,pu), respec-tively, cf. �g. 7.

Table 2: Soil parameters used for plotting thep�y curves in �g. 14.

γ′ φtr D k[kN/m3] [◦] [m] [kN/m3]

10 30 4.2 8000

3.4 Comparison of methods

A comparison of both static and cyclic p�ycurves has been made by Murchison andO'Neill (1984) based on a database of 14full-scale tests on 10 di�erent sites. Thepile diameters varied from 51 mm (2 in.)to 1.22 m (48 in.). Both timber, concreteand steel piles were considered. The soilfriction angle ranged from 23◦ to 42◦. Thetest piles' slenderness ratio's were not pro-vided.

Murchison and O'Neill (1984) comparedthe di�erent p�y curve formulations withthe full-scale tests using the Winkler ap-proach. The predicted head de�ection,maximum moment, Mmax, and the depth


of maximum moment were compared ac-cording to the error, E:

E =|predicted value - measured value|

measured value(32)

In the analysis it was desired to assessthe formulations ability to predict the be-haviour of steel pipe monopiles. Multi-plication factors were therefore employed.The error, E, was multiplied by a factor oftwo for pipe piles, 1.5 for non-pipe drivenpiles and a factor of one for drilled piers.When predicted values were lower than themeasured values the error was multipliedby a factor of two. By using these factorsunconservative results are penalised andpipe piles are valued higher in the com-parison. In tab. 3 the average value of Efor static p�y curves are shown for the twomethods. As shown, method B results ina lower average value of E for all the cri-teria considered in the comparison. Thestandard deviation of E was not providedin the comparison.

Table 3: Average values of the error, E. Themethods are compared for pile-head de�ection,maximum moment and depth to maximum mo-ment.

Pile-head Mmax Depth tode�ection Mmax

Method A 2.08 0.75 0.58Method B 1.44 0.44 0.40

Murchison and O'Neill (1984) analysedthe sensitivity to parameter variation formethod B. The initial modulus of sub-grade reaction, k, the internal friction an-gle, ϕ, and the e�ective unit weight, γ′,were varied. They found that a 10 % in-crease in either ϕ or γ′ resulted in an in-crease in pile-head de�ection of up to 15and 10 %, respectively. For an increaseof 25 % in k an increase of up to 10% of the pile-head de�ection was found.The sensitivity analysis also shows thatk has the greatest in�uence on pile-headde�ection at small de�ections and that ϕhas a great in�uence at large de�ections.

Murchison and O'Neill (1984) state thatthe sizes of the errors in tab. 3 can-not beexplained by parameter uncertainty. Theamount of data included in the databasewas very small due to the unavailability ofappropriately documented full-scale testsand Murchison and O'Neill (1984) there-fore concluded that a further study of thesoil-pile interaction was needed.

4 Limitations of p�y curves

The p�y curve formulations for piles in co-hesionless soils are, as described, devel-oped for piles with diameters much lessthan 4 to 6 m which is often necessaryfor nowadays monopiles. Today, thereis no approved method for dealing withthese large diameter o�shore piles, whichis probably why the design regulationsare still adopting the original p�y curves,cf. Reese et al. (1974), API(1993), andDNV(1992).

The p�y curve formulations are, as de-scribed, derived based on the Mustang Is-land tests which included only two pilesand a total of seven load cases. Further-more, the tests were conducted for onlyone pile diameter, one type of sand, onlycircular piles etc. Taken into account thenumber of factors that might a�ect thebehaviour of a laterally loaded pile andthe very limited number of full-scale testsperformed to validate the method, the in-�uence of a broad spectre of parameterson the p�y curves are still to be clari�ed.Especially when considering o�shore windturbine foundations a validation of sti�piles with a slenderness ratio of L/D < 10is needed as the Mustang Island test pileshad a slenderness ratio of L/D = 34.4. Itis desirable to investigate this as it mighthave a signi�cant e�ect on the initial sti�-ness which is not accounted for in the p�ycurve method. Briaud et al. (1984) postu-late that the soil response depends on the�exibility of the pile. Criteria for sti� ver-


sus �exible behaviour of piles have beenproposed by various authors, for exampleDobry et al. (1982), Budhu and Davies(1987), and Poulus and Hull (1989). Thedi�erence in deformation behaviour of asti� and a �exible pile is shown in �g.15. A pile behaves rigidly according tothe following criterion, cf. Poulus and Hull(1989):

L < 1.48(

EpIp

Es

)0.25

(33)

Es is Young's modulus of elasticity of thesoil. The criterion for a �exible pile be-haviour is:

L > 4.44(

EpIp

Es

)0.25

(34)

According to (33) a monopile with anouter diameter of 4 m, an embedded lengthof 20 m and a wall thickness of 0.05 m be-haves rigidly if Es < 7.6 MPa. In con-trast, the pile exhibits a �exible behaviourif Es > 617 MPa. Even dense sands haveEs < 100 MPa, so the recently installedmonopiles behave, more like a rigid pilethan a �exible.

Figure 15: Rigid versus �exible pile behaviour.

For modern wind turbine foundations onlysmall pile head rotations are acceptable.Furthermore, the strict demands to thetotal sti�ness of the system due to reso-nance in the serviceability mode increasethe signi�cance of the p�y curve's initial

slope and hereby the initial sti�ness of thesoil-pile system.

When using the p�y curve method the pilebending sti�ness is employed when solvingthe beam on an elastic foundation prob-lem. However, no importance is attachedto the pile bending sti�ness in the formu-lation of the p�y curves, hence Epy is inde-pendent of the pile properties. The valid-ity of this assumption can be questioned asEpy is a parameter describing the soil-pileinteraction.

When decoupling the non-linear springsassociated with the Winkler approach an-other error is introduced since the soil inreality acts as a continuum.

In the following a number of assumptionsand not clari�ed parameters related to thep�y curve method are treated separately.The treated assumptions and parametersare:

� Shearing force between soil layers.

� The ultimate soil resistance.

� The in�uence of vertical pile load onlateral soil response.

� E�ect of soil-pile interaction.

� Diameter e�ect on initial sti�ness ofp�y curves.

� Choice of horizontal earth pressurecoe�cient.

� Shearing force at the pile toe.

4.1 Shearing force between soillayers

Employing the Winkler approach the soilresponse is divided into layers each rep-resented by a non-linear spring. As thesprings are uncoupled the layers are con-sidered to be independent of the lateral


pile de�ection above and below that spe-ci�c layer, i.e. the soil layers are consid-ered as smooth layers able to move rel-atively to each other without loss of en-ergy to friction. Pasternak (1954) modi-�ed the Winkler approach by taking theshear stress between soil layers into ac-count. The horizontal load per length ofthe pile is given by:

p(y) = −Eppyy −Gs

dy

dx(35)

where Gs is the soil shear modulus. Thesubgrade reaction modulus Epy given intab. 1 may indirectly contain the soil shearsti�ness as the p�y curve formulation is�tted to full-scale tests. Ep

py is a modulusof subgrade reaction without contributionfrom the soil shear sti�ness.

Belkhir (1999) examines the signi�canceof shear between soil layers by compar-ing the CAPELA design code, which takesthe shear between soil layers into account,with the French PILATE design code,which deals with smooth boundaries. Thetwo design codes are compared with theresults of 59 centrifuge tests conductedon long and slender piles. Analyses showconcordance between the two design codeswhen shear between soil layers is not takeninto account. Furthermore, the analysesshows a reduction from 14 % to 5 % in thedi�erence between the maximum momentsdetermined from the centrifuge tests andthe numerical simulations when taking theshear between the soil layers into account.However, it is not clear from the paperwhether or not the shear between soil lay-ers is dependent on pile diameter, slender-ness ratio etc. Furthermore it is not clar-i�ed whether the authors distinguish be-tween Epy and Ep

py.

4.2 The ultimate soil resistance

The p�y curve formulations according toMethod A and Method B, cf. (12) and(29), are both dependent on the ultimate

soil resistance. The method for estimatingpu is therefore evaluated in the following.

Failure modes

When designing large diameter monopilesin sand, the transition depth will most of-ten occur beneath the pile toe, cf. �g.10b. There are however several uncertain-ties concerning the ultimate resistance atshallow depths.

The prescribed method for calculating theultimate resistance at shallow depths as-sumes that the pile is smooth, i.e. noskin friction appears and a Rankine fail-ure mode will form. However, in real-ity a pile is neither perfectly rough norperfectly smooth, and the assumed failuremechanism is therefore not exactly cor-rect. According to Harremoës et al. (1984)a Rankine failure takes place for a per-fectly smooth wall and a Prandtl failurefor a perfectly rough wall, cf. �g. 16a and�g. 16b, respectively. Due to the fact thatthe pile is neither smooth nor rough a com-bination of a Rankine and Prandtl failurewill occur. Furthermore, the failure modesare derived for a two-dimensional case.

In (12) the angle α, which determines thehorizontal spread of the wedge, appears.Through experiments Reese et al. (1974)postulate that α depends on both the voidratio, friction angle, and the type of load-ing. However, the in�uence of void ratioand type of loading is neglected in the ex-pression of α.

Nowadays monopiles are non-slender pileswith high bending sti�ness. The piles willtherefore de�ect as almost rigid piles andrather large deformations will occur be-neath the point of zero de�ection. How-ever, when calculating the ultimate resis-tance according to method A and B thepoint of zero de�ection is disregarded. Fornon-slender piles a failure mode as shownin �g. 17 could form. This failure mode


(a) Failure mode proposed by Rankine fora smooth surface at shallow depth.

(b) Failure mode proposed by Prandtl for a roughsurface at shallow depth.

Figure 16: Rankine and Prandtl failure modes.

is derived for a two-dimensional case andconsists of sti� elastic zones and Rankinefailures.

Figure 17: Failure mode for non-slender pile atshallow depth.

Soil dilatancy

The e�ect of soil dilatancy is not includedin method A and B, and thereby the e�ectsof volume changes during pile de�ectionare ignored.

Fan and Long (2005) investigated the in-�uence of soil dilatancy on the ultimateresistance by use of a three-dimensional,

non-linear �nite element model. Theconstitutive model proposed by Desai etal. (1991) incorporating a non-associative�ow rule was employed in the analyses.The �nite element model was calibratedbased on the full-scale tests at MustangIsland. The magnitudes of ultimate resis-tance were calculated for two compactionsof one sandtype with similar friction an-gles but di�erent angles of dilatancy. Thedilatancy angles are not directly speci-�ed by Fan and Long (2005). Estimateshave therefore been made by interpreta-tion of the relation between volumetricstrains and axial strains. Dilatancy anglesof approximately 22◦ and 29◦ were found.An increase in ultimate resistance of ap-proximately 50 % were found with the in-crease in dilatancy angle. In agreementwith laboratory tests, where the dilatancyin dense sands contributes to strength, thismakes good sense. However, as the dila-tancy is increased and the friction angleis held constant extra strength is put intothe material.

Alternative methods

Besides the prescribed method for calcu-lating the ultimate resistance other formu-lations exist, see for example Broms (1964)and Hansen (1961). Fan and Long (2005)compared these methods with a �nite ele-ment solution for various diameters, fric-tion angles, and coe�cients of horizontalearth pressure. Hansen's method showedthe best correlation with the �nite elementmodel, whereas Broms' method resulted inconservative values of the ultimate resis-tance. Further, a signi�cant di�erence be-tween the �nite element solution and theAPI method was found. The API methodproduces conservative results for shallowdepths and unconservative results for deepdepths. The results of the comparison areshown in �g. 18.

When calculating the ultimate soil resis-tance according to method A and B, the


API method

Dept

h [m

]

0

1

2

30.0 1.0 2.0

Broms� method

0.0 1.0 2.0

Dep

th [m

]

0

1

2

3pult/pult(fem) [-]

Hansen�s method

0.0 1.0 2.0

Dept

h [m

]

0

1

2

3

D=0.3D=0.61D=1.2

K=0.8K=1.2K=1.6

=33°'=38°'=45°'

Legend:

pult/pult(fem) [-]

pult/pult(fem) [-]

Figure 18: Comparison of the ultimate resistance estimated by Broms' method, Hansen's method andAPI's method with a �nite element model, after Fan and Long (2005). pult/pult(fem) de�nes the ratio ofthe ultimate resistance calculated by the analytical methods and the ultimate resistance calculated bythe �nite element model.

side friction as illustrated in �g. 19 is ne-glected. To take this into account Briaudand Smith (1984) has proposed a modelwhere the ultimate resistance is calculatedas the sum of the net ultimate frontal re-sistance and the net ultimate side friction.In the model both the net ultimate frontalresistance and the net ultimate side fric-tion are taken to vary linearly with pilediameter. Zhang et al. (2005) refer toa comparison made by Barton and Finn(1983) of the model made by Briaud andSmith (1984) and lateral load tests per-formed in a centrifuge. The circular piles,

associated with the tests, have diametersof 9, 12, and 16 mm and a slenderness ra-tio larger than 20. The magnitude of theacceleration in the centrifuge is not given.The ultimate resistance is compared forfour depths and the error between mea-sured and computed values is found to beless than 10 %. The methods proposed byBroms (1964) and Reese et al. (1974) arealso compared with the model tests. Con-clusions similar to Fan and Long (2005)are reached.

Side friction is due to the model proposed


Figure 19: Side friction and soil pressure on thefront and the back of the pile due to lateral de-�ection.

by Briaud and Smith (1984) una�ectedby the diameter since both the ultimatefrontal resistance and the net ultimate sidefriction vary linearly with diameter. How-ever, the ultimate frontal resistance variesnon-linearly with diameter in both the mo-del proposed by Hansen (1961) and Reeseet al. (1974). The importance of side fric-tion might therefore be more signi�cant forlarge diameter monopiles. Furthermore,it should be emphasized that the normalrestistance at the back of the pile is ne-glected in the analysis.

Summary

Several assumptions are employed whencalculating the ultimate resistance accord-ing to Reese et al. (1974) and the de-sign regulations (e.g. API 1993 and DNV1992). These methods do not account forfriction between pile and soil as the pilesurface is assumed smooth. Furthermore,the failure modes do not consider de�ec-tions beneath the point of zero de�ection.Thus, the assumed failure modes might beinaccurate.

The dilatancy of the soil a�ects the soil re-sponse, but it is neglected in the p�y curveformulations.

Several methods for determining the ulti-mate resistance exist. The method pro-posed by Hansen (1961) were found to cor-

relate better with a �nite element modelthan the methods proposed by Reese etal. (1974) and Broms (1964). In orderto take the e�ect of side friction into ac-count a model was proposed by Briaudand Smith (1984). Predictions regardingthe ultimate resistance correlate well withcentrifuge tests.

4.3 The in�uence of vertical loadon lateral soil response

In current practice, piles are analysedseparately for vertical and horizontal be-haviour. Karthigeyan et al. (2006) inves-tigate the in�uence of vertical load on thelateral response in sand through a three-dimensional numerical model. In the mo-del they adopt a Drucker-Prager constitu-tive model with a non-associated �ow rule.

Karthigeyan et al. (2006) calibrate the nu-merical model against two di�erent kindsof �eld data carried out by Karasev et al.(1977) and Comodromos (2003). Concretepiles with diameters of 0.6 m and a slen-derness ratio of 5 were tested, cf. Karasevet al. (1977). The soil strata consisted ofsti� sandy loam in the top 6 m underlainby sandy clay. Comodromos (2003) per-formed the tests in Greece. The soil pro�leconsisted of silty clay near the surface withthin sublayers of loose sand. Beneath amedium sti� clay layer a very dense sandygravel layer was encountered. Piles with adiameter of 1 m and a slenderness ratio of52 were tested.

To investigate the in�uence of verticalload on the lateral response in sandKarthigeyan et al. (2006) made a modelwith a squared concrete pile (1200 × 1200mm) with a length of 10 m. Two typesof sand were tested, a loose and a densesand with a friction angle of 30◦ and 36◦,respectively. The vertical load was appliedin two di�erent ways, simultaneously withthe lateral load, SAVL, and prior to thelateral load, VPL. Various values of verti-


cal load were applied. The conclusion ofthe analyses were that the lateral capac-ity of piles in sand increases under verticalload. The increase in lateral capacity de-pended on how the vertical load was ap-plied as the highest increase was in thecase of VPL. For the dense sand with alateral de�ection of 5 % of the side lengththe increase in lateral capacity was, in thecase of SAVL, of up to 6.8 %. The samesituation in the case of VPL resulted inan increase of up to 39.3 %. Furthermore,the analyses showed a large di�erence inthe increase of lateral capacity between thetwo types of sand as the dense sand re-sulted in the highest increase. Due to ver-tical loads higher vertical soil stresses andthereby higher horizontal stresses occur,which also mobilise larger friction forcesalong the length of the pile. Therefore, thelateral capacity increases under the in�u-ence of vertical loading.

Although the analyses made byKarthigeyan et al. (2006) indicateda considerable increase in the lateralcapacity at a relative high de�ection, theimprovement at small displacements arenot as signi�cant. Therefore it mightnot be of importance for wind turbinefoundations.

4.4 E�ect of soil-pile interaction

No importance is attached to the pilebending sti�ness, EpIp, in the formulationof the p�y curves. Hereby, Epy is indepen-dent of the pile properties, which seemsquestionable as Epy is a soil-pile interac-tion parameter. Another approach to pre-dict the response of a �exible pile underlateral loading is the strain wedge (SW)model developed by Norris (1986), whichincludes the pile properties. The conceptof the SW model is that the traditional pa-rameters in the one-dimensional Winklerapproach can be characterised in termsof a three-dimensional soil-pile interactionbehaviour.

The SW model parameters are related to athree-dimensional passive wedge develop-ing in front of the pile subjected to lateralloading. The wedge has a form similar tothe wedge associated with method A, asshown in �g. 8. However the angles α andβ are given by:

α = ϕm (36)β = 45◦ +

ϕm

2(37)

where ϕm is the angle of mobilised internalfriction.

The purpose of the method is to relatethe stresses and strains of the soil in thewedge to the subgrade reaction modulus,Epy. The SW model described by Ashouret al. (1998) assumes a linear de�ectionpattern of the pile over the passive wedgedepth, h, as shown in �g. 20. The dimen-sion of the passive wedge depends on twotypes of stability, local and global, respec-tively. To obtain local stability the SWmodel should satisfy equilibrium and com-patibility between pile de�ection, strainsin the soil and soil resistance. This is ob-tained by an iterative procedure where aninitial horizontal strain in the wedge is as-sumed.

Figure 20: Linear de�ection assumed in the SW-model, shown by the solid line. The dashed lineshows the real de�ection of a �exible pile. AfterAshour et al. (1998).

After assuming a passive wedge depth thesubgrade reaction modulus can be calcu-lated along the pile. Based on the calcu-lated subgrade reaction modulus the pile


head de�ection can be calculated fromthe one-dimensional Winkler approach.Global stability is obtained when concor-dance between the pile head de�ection cal-culated by the Winkler approach and theSW-model is achieved. The passive wedgedepth is varied until global stability is ob-tained.

The pile bending sti�ness in�uence the de-�ection calculated by the one-dimensionalWinkler approach and hereby also thewedge depth. Hence, the pile bending sti�-ness in�uence the p�y curves calculated bythe SW-model.

The equations associated with the SWmo-del are based on the results of isotropicdrained triaxial tests. Hereby an isotropicsoil behaviour is assumed at the site. TheSW model takes the real stresses into ac-count by dealing with a stress level, de-�ned as:

SL =∆σh

∆σhf(38)

where ∆σh and ∆σhf are the mobilisedhorizontal stress change and the horizontalstress change at failure, respectively. Thespread of the wedge is de�ned by the mo-bilised friction angle, cf. (36) and (37).Hence the dimensions of the wedge de-pends on the mobilised friction.

Although the SW model is based on thethree-dimensional soil-pile interaction andit is dependent on both soil and pile prop-erties there still are some points of crit-icism or doubt about the model. Themodel does not take the active soil pres-sure that occurs at the back of the pileinto account. This seems to be a non-conservative consideration. Furthermore,the wedge only accounts for the passivesoil pressure at the top front of the pile butneglects the passive soil pressure beneaththe zero crossing point which will occur fora rigid pile, cf. section 4.2. The assump-tion of an isotropic behaviour of the soil inthe wedge seems unrealistic in most casesfor sand. To obtain isotropic behaviour

the coe�cient of horizontal earth pressure,K, needs to be 1, which is not the case formost sands. E�ects of cyclic loading arenot implemented in the SW model whichis a large disadvantage seen in the lightof the strict demands for the foundationdesign.

Ashour et al. (2002) criticise the p�y curvemethod as it is based and veri�ed througha small number of tests. However theSW model, has according to Lesny et al.(2007) been veri�ed only for conventionalpile diameters.

Ashour et al. (2000) investigate by meansof the SW model, the in�uence of pile sti�-ness on the lateral response for conditionssimilar to the Mustang Island tests. A p�ycurve at a depth of 1.83 m is shown in�g. 21 for di�erent values of EpIp. Thep�y curve proposed by Reese et al. (1974)is also presented in the �gure. It is seenthat there is a good concordance betweenthe experimental test and the SW modelfor similar pile properties. Furthermore,the soil resistance increases with increas-ing values of EpIp.

0

Reese et al . (1974a)SW Model

500

400

300

200

100

0 40 80 120y [mm]

p[kN

/m]

EpIp

10 EpIp

0.1 EpIp

0.01 EpIp

Figure 21: The in�uence of pile bending sti�-ness, after Ashour et al. (2000).

Changing the pile sti�ness a�ects the p�ycurves drastically according to the SWmodel. Fan and Long (2005) investigatedthe problem by changing the Young's mod-ulus of elasticity of the pile while keepingthe diameter and the second moment ofinertia constant in their three-dimensional


�nite element model. The results areshown in �g. 22. The investigation showedthat the pile sti�ness has neither signi�-cant in�uence on the ultimate bearing ca-pacity nor the initial sti�ness of the soil-pile system. The e�ect of pile sti�nessshown in �g. 21 and 22 has not been ver-i�ed trough experimental work.

Depth = 1 m

Depth = 2 m

Depth = 3 m

0.00 0.04 0.06

0.00 0.04 0.06

0.00 0.02 0.04 0.06

0.02

0.02

y [m]

y [m]

y [m]

p[k

N/m

]p

[kN

/m]

p[k

N/m

]

200

400

600

200

400

600

200

400

600

1.689x106 kNm2

1.689x105 kNm2

1.689x104 kNm2

Rigid pile

Figure 22: E�ect of pile bending sti�ness, afterFan and Long (2005).

4.5 Diameter e�ect on initialsti�ness of p�y curves

The initial modulus of subgrade reac-tion, k, is according to API (1993), DNV(1992), and Reese et al. (1974) only de-pendent on the relative density of the soilas shown in �g. 12. Hence, the methodsA and B do not include EpIp and D in

the determination of k, which might seemsurprisingly. Di�erent studies on the con-sequences of neglecting the pile parame-ters have been conducted over time withcontradictory conclusions. Ashford andJuirnarongrit (2003) point out the follow-ing three conclusions.

The �rst signi�cant investigation was theanalysis of stress bulbs conducted byTerzaghi (1955). Terzaghi concluded thatby increasing the pile diameter the stressbulb formed in front of the pile is stretcheddeeper into the soil. This results in agreater deformation due to the same soilpressure at the pile. Terzaghi found thatthe soil pressure at the pile is linearly pro-portional to the inverse of the diametergiving that the modulus of subgrade reac-tion, Epy is independent on the diameter,cf. de�nitions in tab. 1.

Secondly, Vesic (1961) proposed a relationbetween the modulus of subgrade reactionused in the Winkler approach and the soiland pile properties. This relation showedthat Epy is independent of the diameterfor circular and squared piles.

Thirdly, Pender (1993) refers to two re-ports conducted by Carter (1984) and Ling(1988). Using a simple hyperbolic soil mo-del they concluded that Epy is linear pro-portional to the pile diameter.

The conclusions made by Terzaghi (1955),Vesic (1961), and Pender (1993) con-cerns the subgrade reaction modulus, Epy.Their conclusions might also be applicablefor the initial modulus of subgrade reac-tion, k, and the initial sti�ness, E∗

py.

Based on the investigations presented byTerzaghi (1955), Vesic (1961), and Pen-der (1993), it must be concluded that noclear correlation between the initial mod-ulus of subgrade reaction and the pile dia-meter has been realised. Ashford andJuirnarongrit (2005) contributed to thediscussion with their extensive study ofthe problem which was divided into three


steps:

� Employing a simple �nite elementmodel.

� Analyses of vibration tests on large-scale concrete piles.

� Back-calculation of p�y curves fromstatic load tests on the concrete piles.

The �nite element analysis was very sim-ple and did not account very well for thesoil-pile interaction since friction along thepile, con�ning pressure in the sand, andgaps on the back of the pile were not in-cluded in the model. In order to isolate thee�ect of the diameter on the magnitude ofEpy, the bending sti�ness of the pile waskept constant when varying the diameter.The conclusion of the �nite element anal-ysis were that the diameter has some ef-fect on the pile head de�ection as well asthe moment distribution. An increase indiameter leads to a decreasing pile headde�ection and a decreasing depth to thepoint of maximum moment. However, itis concluded that the e�ect of increasingthe diameter appears to be relatively smallcompared to the e�ect of increasing thebending sti�ness, EpIp, which was not fur-ther investigated in the present case.

The second part of the work by Ashfordand Juirnarongrit (2005) dealt with vibra-tion tests on large-scale monopiles. Thetests included three instrumented pileswith diameters of 0.6, 0.9, and 1.2 m (12m in length) and one pile with a diameterof 0.4 m and a length of 4.5 m. All pileswere cast-in-drilled-hole and made up ofreinforced concrete. They were installedat the same site consisting of slightly ho-mogenous medium to very dense weaklycemented clayey to silty sand. The pileswere instrumented with several types ofgauges, i.e. accelerometers, strain gauges,tiltmeters, load cells, and linear poten-tiometers. The concept of the tests werethat by subjecting the piles to small lateral

vibrations the soil-pile interaction at smallstrains could be investigated by collectingdata from the gauges.

Based on measured accelerations the natu-ral frequencies of the soil-pile system weredetermined. These frequencies were in thefollowing compared to the natural frequen-cies of the system determined by means ofa numerical model. Two di�erent expres-sions for the modulus of subgrade reaction,Epy, were used, one that is linearly depen-dent and one that is independent on thediameter. The strongest correlation wasobtained between the measured frequen-cies and the frequencies computed by us-ing the relation independent of the diame-ter. Hence, the vibration tests substanti-ate Terzaghi and Vesic's conclusions. It isnoticed that the piles were only subjectedto small de�ections, hence Epy ≈ E∗

py.

Finally, Ashford and Juirnarongrit (2005)performed a back-calculation of p�y curvesfrom static load cases. From the back-calculation a soil resistance was found atthe ground surface. This is in contraireto the p�y curves for sand given by Reeseet al. (1974) and the recommendations inAPI (1993) and DNV (1992) in which theinitial sti�ness, E∗

py, at the ground surfaceis zero. The resistance at the ground sur-face might be a consequence of cohesion inthe slightly cemented sand.

Furthermore, a comparison of the resultsfrom the back-calculations for the variouspile diameters indicated that the e�ectsof pile diameter on E∗

py were insigni�cant.The three types of analyses conducted byAshford and Juirnarongrit (2005) indicatethe same: the e�ect of the diameter on E∗

py

is, insigni�cant.

Fan and Long (2005) investigated the in-�uence of the pile diameter on the soil re-sponse by varying the diameter and keep-ing the bending sti�ness, EpIp, constant intheir �nite element model. The results aregiven as curves normalised by the diame-ter and vertical e�ective stress as shown in


�g. 23. No signi�cant correlation betweendiameter and initial sti�ness is observed.It must be emphasised that the investiga-tion considers only slender piles.

Depth = 1 m

Depth = 2 m

Depth = 3 m

0.00 0.04 0.06

0.00 0.04 0.06

0.00 0.02 0.04 0.06

0.02

0.02

y/D

y/D

y/D

p/(

v� D)

10

30

50

D = 1.2 m D = 0.61 mD = 0.3 m

10

30

50

10

30

50

p/(

v� D)

p/(

v� D)

Figure 23: E�ect of changing the diameter, afterFan and Long (2005).

For non-slender piles the bending sti�nessmight cause the pile to de�ect almost as arigid object. Therefore, the de�ection atthe pile toe might be signi�cant. Thus acorrect prediction of the initial sti�ness isimportant in order to determine the cor-rect pile de�ection.

Based upon a design criterion demand-ing the pile to be �xed at the toe, LesnyandWiemann (2006) investigated by back-calculation the validity of the assumptionof a linearly increasing E∗

py with depth.The investigation indicated that E∗

py isoverestimated for large diameter piles atgreat depths. Therefore, they suggested a

parabolic relation, to be used instead of alinear relation, cf. �g. 24. A �nite elementmodel was made in order to validate theparabolic assumption. The investigationsshowed that employing the parabolic ap-proach gave more similar de�ections to thenumerical modelling than by using the tra-ditional linear approach in the p�y curvemethod. However, it was emphasised thatthe method should only be used for deter-mination of pile length. The p�y curvesstill underestimates the pile head de�ec-tions even though the parabolic approachis used.

* *a

parabolic refpy py

ref

xE Ex

æ ö= ×ç ÷ç ÷è ø

*linearpyE k x= ×

refx

*refpyE *

pyE

Dep

th

Initial stiffness

Figure 24: Variation of initial sti�ness, E∗py,

as function of depth, after Lesny and Wiemann(2006). The linear approach is employed in Reeseet al. (1974) and the design codes (e.g. API 1993,and DNV 1992). The exponent a can be set to0.5 and 0.6 for dense and medium dense sands,respectively.

The above mentioned investigations aresummarised in tab. 4. From this tabularit is obvious that more research is needed.

Looking at cohesion materials the testsare also few. According to Ashfordand Juirnarongrit (2005) the most signif-icant �ndings are presented by Reese etal. (1975), Stevens and Audibert (1979),O'Neill and Dunnavant (1984), and Dun-navant and O'Neill (1985).

Reese et al. (1975) back-calculated p�y


Table 4: Chronological list of investigations concerning the diameter e�ect on the initial sti�ness of the p�y curveformulations.Author Method ConclusionTerzaghi (1955) Analytical IndependentVesic (1961) Analytical IndependentCarter (1984) Analytical expression calibrated

against full-scale tests Linearly dependentLing (1988) Validation of the method proposed

by Carter (1984) Linearly dependentAshford and Juirnarongrit (2005) Numerical and large-scale tests Insigni�cant in�uenceFan and Long (2005) Numerical Insigni�cant in�uenceLesny and Wiemann (2006) Numerical Initial sti�ness is

non-linear for longand large diameter piles

curves for a 0.65 m diameter pile in orderto predict the response of a 0.15 m pile.The calculations showed a good approx-imation of the moment distribution, butthe de�ections however were considerablyunderestimated compared to the measuredvalues associated with the 0.15 m test pile.

Based on published lateral pile load testsStevens and Audibert (1979) found thatde�ections computed by the method pro-posed by Matlock (1970) and API (1987)were overestimated. The overestimationincreases with increasing diameter leadingto the conclusion that an increase in mod-ulus of subgrade reaction, Epy, occurs withan increase in diameter.

By testing laterally loaded piles with di-ameters of 0.27 m, 1.22 m, and 1.83 m inan overconsolidated clay, O'Neill and Dun-navant (1984) and Dunnavant and O'Neill(1985) found that there were a non-linearrelation between de�ection and diameter.They found that the de�ection at 50 %of the ultimate soil resistance generallydecreased with an increase in diameter.Hence, Epy increases with increasing pilediameter.

4.6 Choice of horizontal earthpressure coe�cient

When calculating the ultimate resistanceby method A the coe�cient of horizon-tal earth pressure at rest, K0, equals 0.4even though it is well-known that the rela-tive density/the internal friction angle in-�uence the value of K0. In addition, piledriving may increase the coe�cient of hor-izontal earth pressure K.

The in�uence of the coe�cient of horizon-tal earth pressure, K, are evaluated byFan and Long (2005) for three values ofK and an increase in ultimate resistancewere found for increasing values of K. Theincrease in ultimate resistance is due tothe fact, that the ultimate resistance isprimarily provided by shear resistance inthe sand, which depends on the horizontalstress.

Reese et al. (1974) and thereby API(1993) and DNV (1992) consider the initialmodulus of subgrade reaction k to be in-dependent of K. Fan and Long (2005) in-vestigated this assumption. An increase inK results in an increase in con�ning pres-sure implying a higher sti�ness. Hence, kis highly a�ected by a change in K and kincreases with increasing values of K.


4.7 Shearing force at the pile toe

Recently installed monopiles have diame-ters around 4 to 6 m and a pile slender-ness ratio around 5. Therefore, the bend-ing sti�ness, EpIp, is quite large comparedto the pile length. The bending deforma-tions of the pile will therefore be small andthe pile will almost move as a rigid objectas shown in �g. 25.

Figure 25: De�ection curve for non-slender pile.

As shown in �g. 25 there is a de�ection atthe pile toe. This de�ection causes shear-ing stresses at the pile toe to occur, whichincrease the total lateral resistance. Ac-cording to Reese and Van Impe (2001) anumber of tests have been made in orderto determine the shearing force at the piletoe, but currently no results from thesetests have been published and no meth-ods for calculating the shearing force as afunction of the de�ection have been pro-posed.

5 Conclusion

Monopiles are the most used foundationconcept for o�shore wind energy convert-ers and they are usually designed by useof the p�y curve method. The p�y curvemethod is a versatile and practical designmethod. Furthermore, the method has along history of approximately 50 years ofexperience.

The p�y curve method was originally de-veloped to be used in the o�shore oiland gas sector and has been veri�ed withpile diameters up to approximately 2 m.Nowadays monopiles with diameters of 4to 6 m and a slenderness ratio around 5are not unusual.

In the present review a number of the as-sumptions and not clari�ed parameters as-sociated with the p�y curve method havebeen described. The analyses consideredin the review state various conclusions,some rather contradictory. However mostof the analyses are based on numericalmodels and concentrates on piles with aslenderness ratio much higher than 5. Inorder to calibrate the p�y method to nowa-days monopiles further numerical and ex-perimental work is needed. Important�ndings of this paper are summarised asfollows:

� When employing the Winkler ap-proach the soil response is assumedindependent of the de�ections aboveand below any given point. The e�ectof involving the shear stress betweensoil layers seems to be rather small,and from the analysis it is not clearwhether the results are dependent onpile properties.

� The failure modes assumed whendealing with the ultimate soil resis-tance at shallow depth seems ratherunrealistic. In the employed meth-ods the surface of the pile is assumedsmooth. Furthermore, the methoddoes not take the pile de�ection intoaccount, which seems critical for rigidpiles.

� Soil dilatancy a�ects the soil re-sponse, but it is neglected in the p�ycurve formulations.

� Determining the ultimate soil resis-tance by the method proposed byHansen's (1961), seems to give more


reasonable results than the methodassociated with the design codes.Moreover, side friction is neglected inthe design codes.

� In current practice piles are anal-ysed separately for vertical and hor-izontal behaviour. Taking into ac-count the e�ect of vertical load seemsto increase the lateral soil resistance.However the e�ect is minor at smalllateral de�ections.

� Analyses of p�y curves sensitivity topile bending sti�ness, EpIp, givesrather contradictory conclusions. Ac-cording to the Strain Wedge modelthe formulations of p�y curves arehighly a�ected by pile bending sti�-ness. This is in contradiction to theexisting p�y curves and a numericalanalysis performed by Fan and Long(2005).

� The initial sti�ness is independent ofpile diameter according to the exist-ing p�y curves. This agrees withanalytical investigations by Terzaghi(1955), and Vesic (1961). Similarly,Ashford and Juirnarongrit (2005)concluded that initial sti�ness is in-dependent of the pile diameter basedupon an analysis of a �nite elementmodel and tests on large scale con-crete piles. Carter (1984) and Ling(1988) however, found that the initialsti�ness is linear proportional to pilediameter. Based upon a numericalmodel, Lesny and Wiemann (2006)found that the initial sti�ness is over-predicted at the bottom of the pilewhen considering large diameter piles.

� The initial sti�ness of the p�y curveas well as the ultimate resistance in-creases with an increase in the co-e�cient of horizontal earth pressure.This e�ect is not taking into consid-eration in the existing p�y curve for-mulations.

� A pile which behaves rigidly will havea de�ection at the pile toe. Tak-ing this de�ection into considerationmight give an increase in net soil re-sistance.

6 References

Abdel-Rahman K., and Achmus M., 2005.Finite element modelling of horizontallyloaded Monopile Foundations for O�shoreWind Energy Converters in Germany. In-ternational Symposium on Frontiers inO�shore Geotechnics, Perth, Australia,Sept. 2005. Balkerna.

API, 1993. Recommended practicefor planning, designing, and construct-ing �xed o�shore platforms - Workingstress design, API RP2A-WSD, AmericanPetrolium Institute, Washington, D.C.,21. edition.

Ashford S. A., and Juirnarongrit T.,March 2003. Evaluation of Pile Dia-meter E�ect on Initial Modulus of Sub-grade Reaction. Journal of Geotech-nical and Geoenvironmental Engineering,129(3), pp. 234-242.

Ashford S. A., and Juirnarongrit T., 2005.E�ect of Pile Diameter on the Modulus ofSubgrade Reaction. Report No. SSRP-2001/22, Department of Structural En-gineering, University of California, SanDiego.

Ashour M., Norris G., and Pilling P., April1998. Lateral loading of a pile in layeredsoil using the strain wedge model. Jour-nal of Geotechnical and GeoenvironmentalEngineering, 124(4), paper no. 16004, pp.303-315.

Ashour M., and Norris G., May 2000.Modeling lateral soil-pile response basedon soil-pile interaction. Journal ofGeotechnical and Geoenvironmental Engi-neering, 126(5), paper no. 19113, pp.420-428.


Ashour M., Norris G., and Pilling P., Au-gust 2002. Strain wedge model capa-bility of analyzing behavior of laterallyloaded isolated piles, drilled shafts and pilegroups. Journal of Bridge Engineering,7(4), pp. 245-254.

Banerjee P. K., and Davis T. G., 1978.The behavior of axially and laterallyloaded single piles embedded in non-homogeneous soils. Geotechnique, 28(3),pp. 309-326.

Barton Y. O., and Finn W. D. L., 1983.Lateral pile response and p�y curves fromcentrifuge tests. Proceedings of the 15thAnnual O�shore Technology Conference ,Houston, Texas, paper no. OTC 4502, pp.503-508.

Belkhir S., Mezazigh S., and Levacher, D.,December 1999. Non-Linear Behavior ofLateral-Loaded Pile Taking into Accountthe Shear Stress at the Sand. Geotechni-cal Testing Journal, GTJODJ, 22(4), pp.308-316.

Briaud J. L., Smith T. D., and Meyer, B.J., 1984. Using pressuremeter curve todesign laterally loaded piles. In Proceed-ings of the 15th Annual O�shore Technol-ogy Conference, Houston, Texas, paper no.OTC 4501.

Broms B. B., 1964. Lateral resistance ofpiles in cohesionless soils. Journal of SoilMechanics and Foundation Div., 90(3),pp. 123-156.

Budhu M., and Davies T., 1987. Nonlin-ear Analysis of Laterally loaded piles incohesionless Soils. Canadian geotechnicaljournal, 24(2), pp. 289-296.

Carter D. P., 1984. A Non-Linear Soil Mo-del for Predicting Lateral Pile Response.Rep. No. 359, Civil Engineering Dept,.Univ. of Auckland, New Zealand.

Comodromos E. M., 2003. Response pre-diction for horizontally loaded pile groups.

Journal of the Southeast Asian Geotechni-cal Society, 34(2), pp. 123-133.

Cox W. R., Reese L. C., and Grubbs B. R.,1974. Field Testing of Laterally LoadedPiles in Sand. Proceedings of the SixthAnnual O�shore Technology Conference,Houston, Texas, paper no. OTC 2079.

Desai C. S., Sharma K. G., Wathugala G.W., and Rigby D. B., 1991. Implementa-tion of hierarchical single surface δ0 and δ1

models in �nite element procedure. Inter-national Journal for Numerical & Analyt-ical Methods in Geomechanics, 15(9), pp.649-680.

DNV, 1992. Foundations - Classi�cationNotes No 30.4, Det Norske Veritas, DetNorske Veritas Classi�cation A/S.

Dobry R., Vincente E., O'Rourke M., andRoesset J., 1982. Sti�ness and Damping ofSingle Piles. Journal of geotechnical engi-neering, 108(3), pp. 439-458.

Dunnavant T. W., and O'Neill M. W.,1985. Performance analysis and inter-pretation of a lateral load test of a 72-inch-diameter bored pile in overconsoli-dated clay. Report UHCE 85-4, Dept. ofCivil Engineering., University of Houston,Texas, p. 57.

Fan C. C., and Long J. H., 2005. As-sessment of existing methods for predict-ing soil response of laterally loaded pilesin sand. Computers and Geotechnics 32,pp. 274-289.

Hansen B. J., 1961. The ultimate re-sistance of rigid piles against transver-sal forces. Danish Geotechnical Institute,Bull. No. 12, Copenhagen, Denmark, 5-9.

Harremoës P., Ovesen N. K., and Jacob-sen H. M., 1984. Lærebog i geoteknik 2, 4.edition, 7. printing. Polyteknisk Forlag,Lyngby, Danmark.

Hetenyi M., 1946. Beams on Elastic Foun-dation. Ann Arbor: The University ofMichigan Press.


Karasev O. V., Talanov G. P., and BendaS. F., 1977. Investigation of the work ofsingle situ-cast piles under di�erent loadcombinations. Journal of Soil Mechanicsand Foundation Engineering, translatedfrom Russian, 14(3), pp. 173-177.

Karthigeyan S., Ramakrishna V. V. G. S.T., and Rajagopal K., May 2006. In�u-ence of vertical load on the lateral responseof piles in sand. Computers and Geotech-nics 33, pp. 121-131.

LeBlanc C., Houlsby G. T., and Byrne B.W., 2007. Response of Sti� Piles to Long-term Cyclic Loading.

Lesny K., and Wiemann J., 2006. Finite-Element-Modelling of Large DiameterMonopiles for O�shore Wind Energy Con-verters. Geo Congress 2006, February 26to March 1, Atlanta, GA, USA.

Lesny K., Paikowsky S. G., and GurbuzA., 2007. Scale e�ects in lateral load re-sponse of large diameter monopiles. Geo-Denver 2007 February 18 to 21, Denver,USA.

Ling L. F., 1988. Back Analysis of Lat-eral Load Test on Piles. Rep. No. 460,Civil Engineering Dept., Univ. of Auck-land, New Zealand.

Matlock H., and Reese L. C., 1960. Gener-alized Solutions for Laterally Loaded Piles.Journal of the Soil Mechanics and Foun-dations Division, 86(5), pp. 63-91.

Matlock H., 1970. Correlations for De-sign of Laterally Loaded Piles in SoftClay. Proceedings, Second Annual O�-shore Technology Conference, Houston,Texas, paper no. OTC 1204, pp. 577-594.

McClelland B., and Focht J. A. Jr., 1958.Soil Modulus for Laterally Loaded Piles.Transactions. ASCE 123: pp. 1049-1086.

Murchison J. M., and O'Neill M. W, 1984.Evaluation of p�y relationships in cohe-sionless soils. Analysis and Design of Pile

Foundations. Proceedings of a Symposiumin conjunction with the ASCE NationalConvention, pp. 174-191.

Norris G. M., 1986. Theoretically basedBEF laterally loaded pile analysis. Pro-ceedings, Third Int. Conf. on NumericalMethods in o�shore piling, Editions Tech-nip, Paris, France, pp. 361-386.

O'Neill M. W., and Dunnavant T. W.,1984. A study of e�ect of scale, ve-locity, and cyclic degradability on later-ally loaded single piles in overconsolidatedclay. Report UHCE 84-7, Dept. of Civ.Engrg., University of Houston, Texas, p.368.

Pasternak P. L., 1954. On a New Methodof Analysis of an Elastic Foundation byMeans of Two Foundation Constants, Go-sudarstvennoe Izadatelstro Liberatuni poStroitelstvui Arkhitekture, Moskow, pp.355-421.

Peck R. B., Hanson W. E., and Thorn-burn T. H., 1953. Foundation Engineer-ing, John Wiley and Sons, Inc. New York.

Pender M. J., 1993. Aseismic Pile Foun-dation Design Analysis. Bull. NZ Nat.Soc. Earthquake Engineering., 26(1), pp.49-160.

Poulos H. G., 1971. Behavior of laterallyloaded piles: I - Single piles. Journal of theSoil Mechanics and Foundations Division,97(5), pp. 711-731.

Poulos H. G., and Davis E. H., 1980. Pilefoundation analysis and design, John Wi-ley & Sons.

Poulus H., and Hull T., 1989. The Roleof Analytical Geomechanics in FoundationEngineering. in Foundation Eng.: Currentprinciples and Practices, 2, pp. 1578-1606.

Reese L. C., and Matlock H., 1956.Non-dimensional Solutions for LaterallyLoaded Piles with Soil Modulus AssumedProportional to Depth. Proceedings of the


eighth Texas conference on soil mechanicsand foundation engineering. Special pub-lication no. 29.

Reese L. C., Cox W. R., and Koop, F. D.,1974. Analysis of Laterally Loaded Pilesin Sand. Proceedings of the Sixth AnnualO�shore Technology Conference, Houston,Texas, 2, paper no. OTC 2080.

Reese L. C., and Welch R. C., 1975. Lat-eral loading of deep foundation in sti� clay.Journal of Geotechnic Engineering., Div.,101(7), pp. 633-649.

Reese L. C., and Van Impe W. F., 2001.Single Piles and Pile Groups Under Lat-eral Loading, Taylor & Francis Group plc,London.

Stevens J. B., and Audibert J. M. E.,1979. Re-examination of p-y curve for-mulation. Proceedings Of the XI AnnualO�shore Technology Conference, Houston,Texas, OTC 3402, pp. 397-403.

Terzaghi K., 1955. Evaluation of coe�-cients of subgrade reaction. Geotechnique,5(4), pp. 297-326.

Timoshenko S. P., 1941. Strength of ma-terials, part II, advanced theory and prob-lems, 2nd edition, 10th printing. NewYork: D. Van Nostrand.

United Nations, 1998. Kyoto protocol.United Nations framework convention onclimate change.

Vesic A. S., 1961. Beam on Elastic Sub-grade and the Winkler's Hypothesis. Pro-ceedings of the 5th International Confer-ence on Soil Mechanics and FoundationEngineering, Paris, 1, pp. 845-850.

Winkler E., 1867. Die lehre von elasticzi-tat and festigkeit (on elasticity and �xity),Prague, 182 p.

Zhang L., Silva F., and Grismala R., Jan-uary 2005. Ultimate Lateral Resistanceto Piles in Cohesionless Soils. Journal of

Geotechnical and Geoenvironmental Engi-neering, 131(1), pp. 78-83.

ISSN 1901-726X DCE Technical Report No. 57

review of p-y relationships in cohesionless soil

Documents

eective unit

horizontal

nite element

nite element

eective shear

typical py

theoretical

laterally