RESEARCH PAPER
A numerical approach to estimate shaft friction of bored pilesin sands
Ylenia Mascarucci
•
Salvatore Miliziano
•
Alessandro Mandolini
Received: 10 August 2013/Accepted: 14 January 2014
SpringerVerlag Berlin Heidelberg 2014
Abstract
A new approach to estimate shaft capacity of bored piles in sandy soils, based on numerical analysis, ispresented. The topic is relevant as current design methodsoften largely underestimate the shaft capacity of piles insands, thus resulting in an overconservative design. Theproposed approach is based on explicitly modelling the thincylinder of soil surrounding the pile, where strain localization concentrates (shear band), and on the fundamentalmechanic behaviour of sandy soils (e.g. dilatancy, softening). This approach is both simple and easy to apply.Results of a broad parametric study involving axially loaded single piles embedded in different sandy soils arepresented, highlighting that relative density and grain sizedistribution mainly affect the shaft capacity. The capabilityof the procedure to predict shaft friction is checked againstdata from a welldocumented fullscale axial load test oninstrumented pile. Some suggestions for calibration andapplication of the method are also reported.
Keywords
Bored piles
Dilatancy
Sand
Shaft friction
Shear band
Soil grading
1 Introduction
Signiﬁcant advances have been made in identifying processes that occur in the soil zone immediately adjacent tothe pile during loading phase. Nevertheless, design methods for bored piles currently used in practice do notexplicitly take into account the fundamental aspects of soilbehaviour (i.e. void ratio and state of stress). This particularly applies to sandy soils, for which the wellknowndifﬁculties in retrieving undisturbed samples have led to awidespread use of in situ testbased methods (often referredto as
empiric
or
direct methods
). The available methodsoften result in very large scattered predictions, clearlyhighlighted by socalled prediction events (e.g. [45]). Thisreinforces the belief that predictive reliability is generallyfar poorer than many practitioners recognize [17].The need for a calculation methodology that can yieldaccurate results is relevant as current design methods oftenlargely underestimate the shaft capacity of piles in sands,thus resulting in an overconservative and more expensivedesign. These methods are very simple to apply, but are toosimplistic as they do not properly take into account thefundamental aspects of the behaviour of sandy soil nor thecomplex phenomena occurring in the thin cylinder of soilsurrounding the pile, where strain localization occurs(shear band). Consequently, their ability to predict pilebehaviour is quite poor.An attempt to overcome this lack is made here, withparticular reference to the shaft capacity of bored castin situ piles embedded in sands. Based on numericalmodelling, the approach presented is relatively practicaland simple to be used routinely and, at the same time, it cantake into account the main factors on which the phenomenadepend. The shear band, which forms close to the shaftduring axial loading, is explicitly considered by means of
Y. Mascarucci (
&
)
A. MandoliniDepartment of Civil Engineering, Design, Building andEnvironment, Second University of Napoli, Via Roma 29,81031 Aversa, CE, Italyemail: ylenia.mascarucci@unina2.itA. Mandoliniemail: alessandro.mandolini@unina2.itS. MilizianoDepartment of Structural and Geotechnical Engineering,Sapienza University of Roma, Via Eudossiana 18,00184 Rome, Italyemail: salvatore.miliziano@uniroma1.it
1 3
Acta GeotechnicaDOI 10.1007/s1144001403054
interface elements, whose constitutive law has been conveniently selected to reproduce the main aspects of themechanical behaviour of sands.A brief summary regarding some of the most commonmethods currently used for the evaluation of pile shaftfriction is given. The main aspects of the interactionmechanisms between pile and surrounding soil are thenillustrated, and the role played by the dilative behaviour of sand in the shear band, partially restrained by the surrounding soil, is highlighted.Numerical analyses performed by means of FLAC 2Dare focused on. Following the description of the numericalmodelling procedure, results obtained for an axially loadedsingle pile are presented in some detail. The results of abroad parametric study are then discussed, with the aim of detecting which parameters play a major role on shaftfriction (relative density, geostatic stresses, strength andshear band thickness, etc.).The predictive capability of the proposed approach ischecked against a selected fullscale axial load test oninstrumented pile [5, 46].
2 Background
The evaluation of the ultimate shaft friction,
q
s
, at a givendepth,
z
, along a vertical pile embedded into a sandy soil, isusually made by methods based on soil properties (socalled
theoretical methods
) or directly on in situ test results(socalled
empirical methods
). For both approaches, several indications are given in Recommendations, Guides orCodes; a detailed overview is currently available in manytextbooks (e.g. [13, 48]).
Regarding theoretical methods, the starting point forestimating values of shaft friction
q
s
for a vertical pile insandy soil is the expression:
q
s
¼
r
0
hf
tan
d
0
¼
K
r
0
v0
tan
d
0
¼
b
r
0
v0
ð
1
Þ
where
r
0
hf
is the effective horizontal stress at failure and
d
0
represents the soil–pile friction angle. The normal effectivestress may be taken as some ratio
K
of the vertical effectivestress
r
0
v0
, thus resulting in the second form of theexpression in Eq. 1. Usually, the appropriate value of
K
depends on (1) the in situ earth pressure coefﬁcient,
K
0
,(2) the method of installation of the pile and (3) the initialrelative density of the sand,
I
D
.Instead of separately evaluating
K
and tan
d
0
, in routinedesign practice it is often suggested to refer to a lumpedparameter,
b
=
K
tan
d
0
[30], giving rise to the wellknown
b
methods (third form of the expression in Eq. 1). Thesrcinal
b
method for piles embedded in granular soils, ﬁrstintroduced by Reese and O’Neill [37] on the basis of 41load tests on bored piles, typically underestimates shaftresistance since it was developed as the lower bound of experimental data. The conservatism of such an approachwas demonstrated by Rollins et al. [38] (Fig. 1), who back
ﬁgured more than one hundred tensile load tests on boredpiles in sands and gravels and suggested different depthdepending
b
curves.Irrespective of the speciﬁc suggestions given by Rollinset al. [38], data in Fig. 1 show that (1)
b
decreases forincreasing depth and (2) at a given depth larger values for
b
are expected passing from sand to gravel.Recently, FHWA [11] proposed the socalled
rational
b
method
for bored cast in situ piles, where the coefﬁcient
K
is set equal to the earth coefﬁcient at rest
K
0
(hence
b
=
b
0
=
K
0
tan
d
0
). The latter can be evaluated by theexpression of Mayne and Kulhawy [28]:
K
0
;
OC
¼
K
0
;
NC
OCR
sin
u
0
ð
2
Þ
where OCR is the overconsolidation ratio,
K
0,NC
=
1

sin
u
0
following Jaky [15] and
u
0
is the friction angle of thesoil; at shallow depth (
z
B
2.3 m), a constant value
K
=
K
0,OC
(calculated at
z
=
2.3 m) is suggested. Thesoil–pile interface friction angle
d
0
is assumed to be coincident with
u
0
.Figure 2 reports the comparison between
b
values suggested by FHWA [11] and those experimentally measuredby more than 100 load tests on piles [7]. When normallyconsolidated soils (NC, OCR
=
1) are assumed, typicalvalues for
u
0
lead to
b
in the range 0.25–0.30, hence largelyunderestimating experimental data. To obtain higher valuesof
b
, very large
u
0
and OCR have to be considered. InFig. 2, the curve ‘‘OC’’ refers to overconsolidated soils,having
u
0
=
45
and OCR decreasing with depth, from
Fig. 1
Experimental
b
values from load tests of bored piles in ﬁneand coarse sands, compared with Reese and O’Neill [37] and Rollinset al. [38] design curves (modiﬁed from [38])
Acta Geotechnica
1 3
more than 30 close to the ground surface to about 2 at
z
=
40 m, thus corresponding to very large
K
0
valuesranging between over 3 to about 0.4. It should be noted thatseveral measures are still larger than the predicted values.Moreover, this approach yields results in terms of
K
0
,which are in evident contrast with some experimental data:Jamiolkowski et al. [16], on the basis of calibrationchamber tests results, found that also in heavily OC sandysoils (OCR
max
=
15)
K
0,OC
is not greater than 1.Therefore, it clearly follows that the estimation of
q
s
inEq. 1 is challenging.Pile installation and loading are a process that causescomplex stress changes in the soil around the pile from thein situ conditions to failure. According to Randolph andGourvenec [34], ‘‘a design method is more robust if it hassome basis in the underlying mechanics of the process(rather than being wholly empirical)’’.Concerning the aforementioned, complexities arise fromthe need to estimate the net result of installation andloading effects, based merely on knowledge of the in situconditions prior to pile installation, as identiﬁed by geotechnical site and laboratory investigations.To tackle the problem, it is useful to separate singlecontributions to shaft resistance, rewriting Eq. 1 as:
q
s
¼
r
0
hf
tan
d
0
¼ ð
r
0
h0
þ
D
r
0
hc
þ
D
r
0
hl
Þ
tan
d
0
ð
3
Þ
where
D
r
0
hc
and
D
r
0
hl
represent the stress changes inducedby pile installation and loading, respectively.The stress change
D
r
0
hc
depends on the drilling operation (e.g. with or without casing or mud), as well as on theconcrete casting and properties (e.g. water/cement ratio).According to Fleming et al. [13], if excavation is properlyexecuted and high ﬂuidity concrete is then placed, it isreasonable to assume
D
r
0
hc
=
0, implying that concretingcan reinstate the effective horizontal stresses existingbefore drilling. It is worth noting that, even under thiscircumstance, the ensuing concrete curing could alter thestate of stress in the soil [10, 26].
The stress change
D
r
0
hl
can be attributed to Poisson’sratio strains in the pile and to dilation of the soil close tothe pile where strains localize, both causing outwardexpansion towards the surrounding soil for piles undercompressive load.With reference to Poisson’s effect, De Nicola andRandolph [9] carried out a number of numerical analysesand showed that this depends on different parameters (pilegeometry, soil and pile properties). Overall, for compressive axial loading, a stress increase of about 10–30 % alongthe entire pile length can be expected.Recent research has led to an improved understanding of effects of soil dilation.Pile response to axial loading is mainly governed by thebehaviour of a thin cylinder of soil (shear band) surrounding the pile itself. Lehane et al. [19] highlighted thatthe increase,
u
r
, of the thickness,
t
s
, of such a shear bandinduced by soil dilatancy, determines an increase of theeffective horizontal stress acting on the pile shaft at failure(Fig. 3), as this is partially restrained by the outer soil.A ﬁrst attempt to quantify
D
r
0
hl
was made by Wernick [49]. Starting from an initial state (Fig. 4a), when the soil
adjacent to the pile shaft dilates of a quantity
u
r
the outersoil, schematically represented by springs with a stiffness
k
1
, reacts with stresses depending on the geometry of theproblem and on soil properties (Fig. 4b). If the theory of expansion of the cylindrical cavity in a linear elasticmedium is considered [4]:
D
r
0
hl
¼
4
G D
u
r
¼
k
1
u
r
ð
4
Þ
where
G
is the soil shear modulus and
D
is the pilediameter. The term
k
1
=
4
G
/
D
represents the conﬁning
50454035302520151050
z ( m )
0 3 41 2 5 6 7
β
values ()
β
m
measuredOCNC
Fig. 2
Experimental mean
b
m
values from axial load tests of boredpiles in sand [7], compared with
b
0
values for NC and OC soils(modiﬁed from [11])
Fig. 3
Typical measured
q
s
versus
r
0
h
acting on pile shaft duringloading (modiﬁed from [19])Acta Geotechnica
1 3
stiffness imposed by the surrounding soil, which decreasesfor increasing pile diameter.From Eq. 4, it clearly derives that a reliable estimate of
D
r
0
hl
strictly depends on the assessment of
G
and
u
r
.To take into account the nonlinearity of soil behaviour,the use of linear elasticity theory requires operative valuesfor
G
to be selected. For instance, Fioravante [12] suggests
G
=
(0.05–0.10)
G
0
,
G
0
being the small strain shearmodulus.The outward displacement is typically evaluated byrelating it to the pile roughness,
R
a
, and/or to the meanparticle grain size,
D
50
(
R
a
=
R
t
/2, where
R
t
is the verticaldistance between the highest peak and the lowest troughmeasured on a reference length of surface proﬁle, whichdepends on
D
50
[44]). For instance, Schneider [42] suggests
u
r
=
2.5
D
500.4
R
a0.6
, while Lehane et al. [19] report
u
r
=
0.7
D
50
. It is evident that these suggestions do notexplicitly take into account the fundamental mechanicalbehaviour of sandy soil and, in particular, the dependenceof soil dilation/contraction on the relative density,
I
D
, andthe mean effective conﬁning stress,
p
0
(e.g. [1]).Improvements have been provided by Lehane et al. [20],who derived an analytical relationship between dilation andthe stiffness of the soil surrounding the shear band. Such anexpression has only been calibrated against centrifuge testsand its use requires (1) constant normal load interface sheartests and (2) the stiffness
k
1
under cavity expansion of thesand mass surrounding the pile. Based on distinct elementnumerical modelling of monogranular sand, Peng et al.[33] recently suggested a similar expression.Loukidis and Salgado [22] carried out broad parametricﬁnite element numerical analyses using an advanced constitutive model [24] capable of considering the role playedby a number of factors affecting
q
s
(
K
0
,
r
0
v0
,
I
D
and shearband thickness). The results were summarized in an analytical expression for directly estimating
D
r
0
hl
for pileembedded in clean sand, although some criticism can bemade.From the above, it can be concluded that although thebasic mechanisms of pile–soil interaction are quite clear, adesign methodology based on the mechanical behaviour of sandy soils and, at the same time, simple to use in practice,is not yet available.
3 Numerical modelling
Numerical modelling of shaft friction of a bored pileshould reproduce the main phenomena that occur at pile–soil contact. Which constitutive law to choose for the soiland which strategies to deal with shear band modelling arerather crucial.Among the several constitutive models available for thesoil, in the authors’ opinion the ﬁnal choice should be thatof selecting a model able to combine simplicity and completeness (the former aimed to be effective for practicalpurposes, the latter aimed to simulate, as best as possible,the shear strain evolution at the pile shaft). In particular, itshould be able to describe (1) nonlinear soil behaviour,from small to large strains, (2) soil softening and evolutionof plastic strains and (3) the achievement of critical stateconditions at large strains.When dealing with a boundary value problem in theCauchy continuum, at collapse it is well known that thesolutions obtained through numerical methods, such asﬁnite different or ﬁnite element, are mesh dependent.However, in the case of pile–soil contact, since the exactlocation of shear band is known a priori (parallel andadjacent to sand–concrete contact), appropriate results canbe obtained by modelling the shear band using continuumelements close to the pile, whose width is set to be equal tothe real shear band thickness (e.g. [22]). Alternatively, theshear band can be simulated by means of interface elements, thus shear band behaviour is described in terms of relative displacements between soil and pile, rather thanstrains (e.g. [3]).3.1 Soil constitutive modelThe strain softening constitutive model (SS) was adoptedsince it can reproduce the essential features of soilbehaviour as, for instance, shown in Fig. 5 for a constantnormal load (CNL) simple shear test. It is a relativelysimple linear elastoplastic model with Mohr–Coulombfailure criterion and a nonassociated ﬂow rule. Theresponse is initially linear elastic; after yielding, isotropicsoil softening is assumed, regulated by plastic shear strains[50].For a cohesionless soil (like sand typically is), if it isassumed that both angles of friction,
u
0
, and dilation,
w
,linearly reduce with plastic shear strains from peak tocritical state (CS) condition (Fig. 5b), only six parameters(physically based and obtainable by routine geotechnical
(a) (b)
Fig. 4
Wernick [49] model: restraining effect on shear band dilation(modiﬁed from [49])Acta Geotechnica
1 3
tests) are needed (Table 1): shear modulus,
G
; Poisson’sratio,
m
0
; friction angle at peak and at CS conditions,
u
0
p
and
u
0
cs
, respectively; angle of dilation at peak,
w
p
; plasticshear strain to achieve CS condition,
c
cs
p
.Apart from the nonlinear experimental behaviourobserved before the peak, the SS model can be calibrated tobetter reproduce peak and CS strength as well as the volumetric expansion due to dilatancy. In other words, despiteelastic linearity, the model can account for the overallnonlinear behaviour of soil by means of the Mohr–Coulomb failure criterion, whose position in the effective stressspace changes according to shear plastic strains (softening).It is worth noting that the SS model does not includevoids ratio but rather its derivatives, such as the tendencyto dilate and contract. In detail, the volumetric plasticstrains development is directly controlled by means of theangle of dilation. The latter has to be properly deﬁned toquantify the change in volume that occurs when movingfrom the current to the critical state. The reduction of angleof dilation with plastic shear strains can account for soilsoftening. The angle of dilation tends to zero as the stresspath in the compression plane tends to the critical state line.This approach corresponds to the use of a ﬂow ruledepending on the ‘‘state parameters’’ deﬁned by Been andJefferies [1] but, at the same time, simpliﬁes the SS calibration procedure.To obtain the six model parameters, backﬁguringexperimental laboratory data from simple shear tests, carried out on a specimen reconstituted at the in situ relativedensity, are recommended. Direct shear test results can alsobe used interpreting data, according to Boulon [3].For a preliminary evaluation, it is possible to refer to theliterature (Bolton [2] and Rowe [39] theories for
u
0
p
and
w
p
, respectively; Randolph et al. [36] and Stroud [43] for
u
0
cs
and
c
cs
p
, respectively).Particular attention must be paid to the choice of soilshear stiffness as its value very much depends on the stresslevel. An appropriate operative value of
G
\
G
0
(secantstiffness) must be selected to take into account soilnonlinearity.3.2 Shear bandTo simulate the shear band, interface elements were used.Such a choice reﬂects the need to achieve a convenientlysimple procedure for practical purposes.The concrete–soil interface has been considered as‘‘rough’’ (e.g. [21, 44]) giving rise to a shear band fully
developed within the sand mass at the shaft surface.According to this premise, it has been suggested (e.g. [14,44, 47, 51]) that soil grading mainly inﬂuences shear band
thickness. Typically,
t
s
ranges between 5 and 20 times
D
50
,thus ranging from a few millimetres to a few centimetresfor the soil under consideration. It follows that, if
(b)(a)(d)(c)
Fig. 5
SS prediction and soil response for CNL simple shear test:
a
stress path; evolution of
c
shear stresses and
d
vertical strains;
b
linearreduction for angle of friction and dilation
Table 1
SS parametersShear modulus
G
Poisson’s ratio
m
0
Angle of friction at peak condition
u
0
p
Angle of friction at critical state condition
u
0
cs
Angle of dilation at peak condition
w
p
Plastic shear strains to get CS condition
c
cs
p
Acta Geotechnica
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