Improved Relationships for the Pile Base

Response in Sandy Soils

Emilios M. Comodromos 1 and Mark F. Randolph 2

Abstract: Design codes vary in their recommendations for the end-bearing response for bored piles founded in sand, both for the ultimate
Downloaded from ascelibrary.org by University of Central Florida on 09/29/23. Copyright ASCE. For personal use only; all rights reserved.

design value and the response at small settlements. The ultimate end-bearing resistance may be expressed either in terms of bearing factors N q
relative to the in situ vertical effective stress, with the N q value varying with the friction angle of the sand, or as a factor applied to in situ test
data, such as the standard penetration test blow count or the tip resistance qc . Numerical studies have led to proposed ratios of design end-
bearing pressure to qc at specific settlement ratios, such as 5% and 10% of the pile diameter. The work presented here used numerical analysis
to evaluate the full pile base response from initial stiffness to ultimate end-bearing resistance at a settlement ratio of 10% of the pile diameter.
The resulting base responses are suitable for implementation in beam column analyses and have been validated by comparison with published
design guidelines and with data from a full-scale instrumented pile load test. DOI: 10.1061/JGGEFK.GTENG-11035. © 2023 American
Society of Civil Engineers.
Author keywords: Pile base resistance; Axial pile response; Piled foundations in sand; Analysis and design of piled foundations.

Introduction demonstrated that relative density I d, stress conditions, dilatancy,

and soil shear modulus G each had a significant effect. Lee and
The base bearing capacity qb; design of piles in sandy soils is generally Salgado (1999) used ABAQUS to incorporate a hyperbolic model
given in terms of a dimensionless bearing capacity factor N q times providing a variation of shear modulus as a function of the initial
the effective overburden pressure (Vesic 1967), or as a function of and current values of the first invariant of stress tenson I 1 and the
penetration resistance measured in situ, such as from the tip resis- second invariant of deviatoric stress tensor J 2. The failure surface
tance qc of a cone penetration test (CPT) (Lee and Salgado 1999) or and dilation angle were also updated from the values of current
in terms of the blow count N from a standard penetration test (SPT) relative density and mean effective stress. Valuable conclusions
(Reese and O’Neill 1989, cited in FHWA 2018). The main difficulty were drawn from their analysis that allowed them to express the
in relating the design end-bearing resistance for piles to data from end-bearing mobilized at displacements of 5% and 10% of the pile
penetrometer tests is the much lower settlement ratio relevant for pile diameter as a fraction of the cone resistance qc estimated using the
design compared to continuous penetration, such as in a CPT. software CONEPOINT (Salgado et al. 1997).
Indeed, for bored piles, it is important to establish the gradual devel- The foregoing literature has focused almost exclusively on suit-
opment of end-bearing resistance with increasing pile displacement. able design values of end-bearing, expressed either in terms of
A range of design values for N q are given in different design 0 or as a fraction of the cone tip resistance q . In a design
N q σvo c
codes, as summarized in Table 1 for nondisplacement piles. under ultimate limit state (ULS) conditions, the ultimate resistance
Empirical values are proposed to relate the pile base resistance to is applied, corresponding to the commonly accepted displacement
the pile tip displacement, i.e., a parabolic form with normalized value of 0.10D. It should be underlined that ULS analysis is asso-
displacement to the pile dimeter D for API (2003), or a tri-linear ciated with the application of augmentative partial loading factors
form for displacement levels 0.02D, 0.03D and 0.10D, for DIN [1.35 to 1.50 according to EN 1997-1 (CEN 2004b)] and reductive
1054 (DIN 2005). Even though the analogy between spherical cav- partial factors to resistances (1.1 to 1.45 according to EN 1997-1).
ity expansion and bearing capacity failure (Gibson 1950) has been The combination of these partial factors leads to an overall safety
adapted to CPT geometry, allowing for the effects of soil density factor on the order of 2, as was usually applied as a global safety
and frictional characteristics (Carter et al. 1986; Yu and Houlsby factor by older design codes. Therefore, for serviceability limit state
1991), there have been few attempts made to define a relationship (SLS) analysis, the operational load level will typically be about
with the ability to provide a continuous response of pile base 50% of the ultimate pile base resistance. In turn, this corresponds
resistance as a function of pile tip displacement. An investigation to a pile base displacement on the order of 2%D (1.3%D by API,
on the design of driven piles in sand (Randolph et al. 1994) 2003; 3.0%D by DIN 1054, 2005).
This paper endeavors to develop complete curves, up to settle-
1
Professor, Dept. of Civil Engineering, Univ. of Thessaly, Pedion Areos, ments of 15% of the pile diameter, for the end-bearing response
Volos 383 34, Greece (corresponding author). ORCID: https://orcid.org of bored piles bearing on sand, covering both SLS and ULS
/0000-0003-2661-867X. Email: ecomo@civ.uth.gr conditions. The curves are provided in generic form as a function
2
Professor Emeritus, Centre of Offshore Foundation Systems, Univ. of 0 , and the relative density of
of the in situ vertical effective stress σv0
Western Australia, 35 Stirling Hwy., Crawly, Perth WA 6009, Australia.
the sand. The curves may be used directly in beam column software
Email: mark.randolph@uwa.edu.au
Note. This manuscript was submitted on May 13, 2022; approved on
for axial pile response (noting the increased use of such software by
March 21, 2023; published online on May 26, 2023. Discussion period industry), or appropriate values of end-bearing mobilized at a given
open until October 26, 2023; separate discussions must be submitted for displacement extracted for use in limit state design calculations.
individual papers. This paper is part of the Journal of Geotechnical As stated previously, for a conventional piled foundation design,
and Geoenvironmental Engineering, © ASCE, ISSN 1090-0241. the operational pile base settlements may be less than 2%D,

© ASCE 04023058-1 J. Geotech. Geoenviron. Eng.

J. Geotech. Geoenviron. Eng., 2023, 149(8): 04023058

 Table 1. Recommendations for pile base bearing factor N q and bearing capacity qb;design from design codes
End bearing End bearing Limiting
capacity factor, capacity factor, value
Design manual or code Soil density Nq qb;design (MPa) (MPa)
API (2003) Very loose to loose sand 8 — 1.9
Loose to medium dense sands 12 — 2.9
Medium to dense sands 20 — 4.8
Dense to very dense sands 40 — 9.6
Dense to very dense sands with gravel 50 — 12.0
Canadian Foundation Silts 10–30 — —
Engineering Manual Loose sands 20–30 — —
(CGS 2006) Medium dense sands 30–60 — —
— —
Downloaded from ascelibrary.org by University of Central Florida on 09/29/23. Copyright ASCE. For personal use only; all rights reserved.

Dense sands 50–100

Gravel 80–150 — —
Deutsches Institut für qc ¼ 10 MPa — 2.0 —
Normung e.V. DIN 1054 (DIN 2005) qc ¼ 15 MPa — 3.0 —
qc ¼ 20 MPa — 3.5 —
qc ¼ 25 MPa — 4.0 —

ignoring the effects of interaction within large groups of piles. the variation of the shear strength (failure surface) and deformation
However, pile base displacements relative to the local soil may ex- moduli (bulk modulus K and shear modulus G) as a function of
ceed 10%D in the case of combined piled raft foundations (CPRF), effective stress level and relative density. Substantial variation of
where much higher mobilisation of the nominal bearing capacity both values is observed as load at the pile toe increases and therefore
of piles is usually exploited. As demonstrated by measurements at the constitutive law must be updated during the solution process.
the Torhaus and Messeturm skyscrapers (Frankfurt, Germany), As stated by Randolph et al. (1994), the ultimate base resistance
a much higher proportion (exceeding 10%) of the applied load is is a function of both the strength and the rigidity index G=p 0
transferred to the pile bases in a CPRF, as a result of reduction in (where p 0 is the mean effective stress) of the material, both of
mobilisation of the shaft resistance because of interaction effects which depend on the effective stress level and the relative density
within the pile group (see Katzenbach et al. 2000; Reul and of the soil. The strength of sandy soils is directly related to the peak
Randolph 2003; Comodromos et al. 2016). effective friction angle, φ 0 , which may be estimated from the mean
There is increasing use of numerical methods by industry to as- effective stress p 0 and relative density I d (Bolton 1986; Kleven
sess the response of piled foundations, in many cases replacing the et al. 1986). From evaluation of the behavior of North Sea sand
soil resistance with the pile-soil interaction captured by nonlinear (Kleven et al. 1986), Toyoura sand (Ichihara and Matsuzawa 1973),
springs, so-called load transfer curves, in the interests of computa- Loch Alines sand (Sutherland and Mesdary 1969), Ham River sand
tional efficiency. The purpose of this study, therefore, is to provide (Green and Reades 1975), and Brasted sand (Cornforth 1973), the
a sound basis for the complete end-bearing response of a pile, relat- following correlation is suggested between the peak effective fric-
ing the end-bearing resistance at a selected displacement level in tion, the relative density and the mean effective stress:
terms of appropriate bearing factors. To that end, a series of numeri- 8  0
cal analyses were carried out, covering a range of combinations of < 32 − 3 ln p þ 13I ∀ p 0 ∶p 0 > p
d a
relevant soil parameters. Based on the evaluation of the numerical φ0 ¼ pa ð1Þ
: 0 0
results a parabolic relationship has been established for providing the 32 þ 13I d ∀ p ∶p ≤ pa
response of pile base resistance as a function of pile toe settlement.
In the proposed method initial geostatic stress conditions are where pa = atmospheric pressure; and I d = relative density.
considered, ignoring any disturbance arising from the pile installa- A comparison between the laboratory results of North Sea sand
tion process (i.e., modeling wished in place conditions) and there- for mean effective stress varying from 200 kPa to 800 kPa, and
fore the validity of the method is limited to non-displacement friction angles from Eq. (1), is given in Fig. 1. Fig. 2 shows cor-
(bored) piles. responding comparisons for Loch Aline sand, Ham River sand,
Brasted sand, and Toyoura sand. It should be noted that the con-
solidation pressure for the foregoing triaxial tests ranged from 175
Objectives and Methodology to 282 kPa, while the correlation from Eq. (1) corresponds to a
mean effective stress of 200 kPa.
Numerical analysis of the pile base response can be carried out to The correlation for the dilation angle ψ, originally provided by
cover a wide range of sandy soils, from loose to very dense, and Bolton (1986) and updated by Bolton (1987) and implemented in
various initial stress conditions. Sufficient numerical experiments the numerical analysis discussed later, is
can reveal the factors that most affect the response and that need
to be incorporated in nonlinear springs to simulate mobilization of ψ ¼ 1.875I r ð2Þ
the soil resistance at the pile base. The validity of this process relies 8
on the quality of data from the numerical experiments, while a val- < 5I
d−1 ∀ p 0 ∶p 0 ≤ 150 kPa
0

idation process against experimental data from in situ pile tests is Ir ¼ p ð3Þ
: I d 5.4 − ln − 1 ∀ p 0 ∶p 0 > 150 kPa
required before incorporating the approach in a structural finite pa
element code or beam column model. Obviously, the constitutive
law used in the numerical experiments is critical and must reflect where I r = dilatancy index.

© ASCE 04023058-2 J. Geotech. Geoenviron. Eng.

J. Geotech. Geoenviron. Eng., 2023, 149(8): 04023058

Downloaded from ascelibrary.org by University of Central Florida on 09/29/23. Copyright ASCE. For personal use only; all rights reserved.

Fig. 1. Relationships of effective angle of friction values arising from

Fig. 3. Nomogram for estimating the effective angle of friction as a
Eq. (1) compared to experimental data.
function of vertical effective stress and CPT cone resistance, Eq. (8).

Based on the results from tests on around 20 different types of

sands it was decided to adopt values of emin ¼ 0.57 and emax ¼
0.92 for the numerical analyses presented here.
The angle of internal friction (or the relative density) of a sandy
layer is usually obtained from the results of in situ tests. Empirical
relationships and correlations in tabular form have been given for
SPT N values (Peck et al. 1974; Bowles 1996). For cone penetra-
tion test (CPT) data, correlations have been developed between the
0
tip resistance qc , vertical effective stress σvo , and relative density I d
(Lancellotta 1983; Jamiolkowski et al. 1985). The proposed equa-
tion, based on calibration chamber tests on Ticino sand, Ottawa
sand, Edgar sand, Hokksund sand, and Hilton sand, is
 
qc
I d ¼ −0.98 þ 0.66 log pffiffiffiffiffiffiffiffiffi 0
ffi ð6Þ
½σvo 

where qc = cone penetration resistance (in t=m2 ); and σvo0

= initial
2
Fig. 2. Comparison of effective angle of friction values arising from effective vertical stress below the cone (in t=m ). The equation can
Eq. (1) and experimental data for Loch Aline sand, Ham River sand, be rewritten in nondimensional form as
Brasted sand, and Toyoura sand.
qc =pa
I d ¼ −0.65 þ 0.287 ln pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0
ffi ð7Þ
σvo =pa
The relative density, required in the foregoing equations, is
where pa is atmospheric pressure.
expressed as
Substitution of the foregoing equation into Eq. (1) yields
e −e
I d ¼ max ð4Þ 8
emax − emin > qc =pa
> 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi 0 0
< 23.6 − 3 lnðp =pa Þ þ 3.73 ln σ 0 =p ∀ p ∶p > pa
>
vo a
where emax and emin are the maximum and minimum void ratios, φ0 ¼
respectively; and e is the current void ratio that determines the rel- > c =pa
> 23.6 þ 3.73 ln pqffiffiffiffiffiffiffiffiffiffiffiffiffiffi
>
: ffi ∀ p 0 ∶p 0 ≤ pa
ative density. The values of emax and emin are characteristic for each 0 =p
σvo a
granular material and can be estimated by performing compaction
tests in the laboratory. According to Bolton (1986), Santamarina ð8Þ
and Cho (2001), and Fioravante and Giretti (2016) the values
for emin vary from 0.49 to 0.63 and for emax from 0.75 to 1.18. A useful nomogram, Fig. 3, may arise from Eq. (8), very similar
Evaluation of an extensive program of triaxial tests on Ticino sand to that given by Robertson and Campanella (1983), providing the
and Toyoura sand allowed Fioravante and Giretti (2016) to define a variability of the effective angle of friction φ 0 as a function of
relationship for the critical state void ratio as a function of the mean the mean effective stress p 0 , the vertical effective stress σvo
0 , and the

effective pressure: cone penetration resistance qc .

The foregoing equations, which provide the current effective
sffiffiffiffiffiffi
angle of friction and the dilation angle as a function of evolving
p0
ecs ¼ 0.923 − 0.046 ð5Þ relative density and mean effective stress, have been incorporated
pa in the constitutive law used in the analyses.

© ASCE 04023058-3 J. Geotech. Geoenviron. Eng.

J. Geotech. Geoenviron. Eng., 2023, 149(8): 04023058

 The correlation of the shear modulus with the relative density
can be obtained from the relationship proposed by Lo Presti (1987):
 0 n
p
G ¼ pa S expðc1 I d Þ ð9Þ
pa

where S ¼ 600; c1 ¼ 0.7; and n ¼ 0.43.

The foregoing correlation arose from tests on Ticino sand, while
for in situ silica sands Randolph et al. (1994) suggested reducing S
to 400 to account for more compressible materials, such as silts or
calcareous sands; this lower value is applied in the present study.
The foregoing equation has been introduced in the constitutive law
and, together with Eq. (10), they update the deformation moduli
Downloaded from ascelibrary.org by University of Central Florida on 09/29/23. Copyright ASCE. For personal use only; all rights reserved.

during the solution process, using the relationship

2ð1 þ νÞG
K¼ ð10Þ
3ð1 − 2νÞ

where ν = Poisson’s ratio.

The formulation given by Eqs. (1)–(8) links the sandy soil re-
sistance to the fundamental soil input parameters of relative density
I d , the mean effective stress p 0 , and the initial effective vertical
0
stress σvo . As discussed, these quantities affect the friction angle
φ 0 , dilation angle ψ, and the deformation moduli K and G. During
the numerical solution process the variation of the void ration can
be estimated from the following equation as a function of volume
change:
 
ΔV
e¼ 1− e ð11Þ
Vo o

where ΔV = volume change of a finite element at a given step; V o =

initial volume of the element at the initial stage; and eo = initial
void ratio of the element.
The initial stress condition and the initial failure envelope for
every finite element was introduced as a function of the initial rel-
ative density I d;o. More specifically, the angle of internal friction φ 0
and the dilation angle ψ of each particular finite element were
attributed using Eqs. (1) and (2). The initial vertical effective
0
stresses σvo were evaluated from the overburden stress, while
the initial horizontal effective stresses were taken as
0 0
σho ¼ K o σvo ð12Þ
Fig. 4. Flowchart demonstrating the numerical procedure of the pro-
0 posed method.
K o ¼ 1 − sin φ ð13Þ

The initial shear modulus Gini was estimated from Eq. (9) in
terms of the initial relative density I d;o and mean effective stress Because the analyses were limited intentionally to a base dis-
po0 . The same equation was applied to update the current shear placement of 15%D, remeshing was not required but updated
modulus based on the current values of I d and p 0 during the sol- Lagrangian (large strain) formulations were adopted to ensure the
ution process. best approximation of volume change in each particular finite
The sand model adopted was based on the Mohr-Coulomb fail- element, which in turn affects the evolving soil parameters, such as
ure criterion, but with the failure envelope and the stiffness of the friction and dilation angles.
soil allowed to evolve according to the stress and deformation state As the focus of this work was on the base resistance, smooth
and the stiffness of the soil. The flowchart of the numerical process interface behavior was attributed to shaft resistance along the pile
is outlined in Fig. 4, as implemented in the finite difference code shaft. Comodromos et al. (2021) found that the effect of the shaft
FLAC. The mesh applied in the parametric axisymmetric analysis, friction on the base resistance of piles in clay was minor and limited
shown in Fig. 5, consisted of 4,624 soil elements, 117 pile ele- to a slight concavity in the load-settlement response of the pile base
ments, and interface elements along the shaft and the pile tip. A before full mobilization of the shaft resistance. Preliminary analy-
refined mesh was applied at the pile tip (i.e., below 20 m depth) ses demonstrated that this effect for piles in sand was also minor, as
with element size less than D=10, to capture the detailed response can be observed in Fig. 6. It can be seen that the pile tip response for
at the pile base. The zone along the pile shaft was found to be rel- a smooth shaft (δ ¼ 0o ) and friction shaft (δ ¼ 30o ) are practically
atively insensitive to the element size, due to the interface elements identical in both loose and very dense soil conditions and different
allowing slippage once the ultimate shaft resistance was reached. initial stress conditions. It is worth noting that this effect is consid-
As a result, the distortion in the zone was limited and a rather coarse erably higher in the case of closely spaced pile groups in sands,
mesh was applied. particularly for the interior piles, due to augmentation of the stress

© ASCE 04023058-4 J. Geotech. Geoenviron. Eng.

J. Geotech. Geoenviron. Eng., 2023, 149(8): 04023058

 Table 2. Input data for parametric study
Parameter A B C D
Initial relative density, I d;ini 0.30 0.50 0.70 0.90
emax 0.92 0.92 0.92 0.92
emin 0.57 0.57 0.57 0.57
eo 0.815 0.745 0.675 0.605
Poisson’s ratio, v 0.29 0.27 0.26 0.24
Initial angle of internal friction, φo0 Assigned from Eq. (1)
Coefficient of earth pressure at rest, K o 0.414 0.377 0.343 0.309
Relative density, I d (%) Being updated by Eqs. (4) and
(11)
Angle of internal friction, φ 0 Being updated by Eq. (1)
Dilation angle, ψ 0 Being updated by Eq. (2)
Downloaded from ascelibrary.org by University of Central Florida on 09/29/23. Copyright ASCE. For personal use only; all rights reserved.

Shear modulus, G (kPa) Being updated by Eq. (9)

Bulk modulus, K (kPa) Being updated by Eq. (10)
Gravity, g ðm=s2 Þ 5, 10, 20
Soil weight, γ 0 ðkN=m3 Þ 10

where Fs;max = limiting shear force at the pile-soil interface; δ =

angle of friction between pile and soil; and Fn = normal force
at the interface. As discussed, here the value of δ was set to zero
at the pile shaft.
The normal and shear forces at the interface nodes during the
iterative solution process are determined from the forces of the pre-
vious step (corresponding to time t) and the incremental relative
displacement vector (corresponding to time t þ 0.5Δt) by

ðtþΔtÞ ðtÞ ðtþ0.5ΔtÞ

Fn ¼ Fn − kn Δun L ð15Þ

ðtþΔtÞ ðtÞ ðtþ0.5ΔtÞ

Fs ¼ Fs − ks Δus L with magnitude limited to Fs;max
ð16Þ
Fig. 5. Axisymmetric FE mesh in FLAC.
where kn = normal stiffness; Δun = incremental relative normal
displacement; ks = shear stiffness; Δus = incremental relative shear
displacement; and L = length of the interface segment. The inter-
face elements behave essentially as a slider with a rigid-plastic
behavior (St Venant model), Comodromos and Papadopoulou
(2012). To satisfy this requirement, the normal and shear stiffness
values for the interface elements were taken equal to 10 GPa=m
(at least 102 times the equivalent stiffness of neighboring zones).

Parametric Study

Results of FLAC Analyses

The parametric study consisted of 12 different cases covering the
range from loose to very dense sand, namely cases A, B, C, and D
for I d;o ¼ 0.30, 0.50, 0.70, and 0.90, respectively (see Table 2). To
investigate the effect of vertical and mean effective stress levels,
Fig. 6. Response comparison of pile base resistance with smooth and
different values for gravity g were applied. Specifically, values
friction shaft behavior.
of g ¼ 5; 10, and 20 m=s2 were applied for the same mesh and ef-
fective soil unit weight of 10 kN=m3 , giving initial vertical effec-
0 of 100, 200, and 400 kPa at the pile base. As
tive stresses σvo
level at the pile base level from load transferred along the pile indicated in Fig. 5, the pile embedment was Lp ¼ 20 m, with a
shafts. diameter of D ¼ 1 m, and elastic properties of Ep ¼ 32 GPa
The constitutive model of the interface elements was defined by and ν p ¼ 0.2 were assumed [corresponding to concrete grade
a linear Coulomb shear-strength criterion that limits the shear force C30/37 according to Eurocode EN 1992-1-1 (CEN 2004a)]. Note
acting at an interface node according to that the effect of changing the g level may alternatively be consid-
ered as equivalent to scaling the pile dimensions to 0.5 m diameter
by 10 m long for g ¼ 5 m=s2 and 2 m diameter by 40 m long
Fs;max ¼ Fn tan δ ð14Þ for g ¼ 20 m=s2.

© ASCE 04023058-5 J. Geotech. Geoenviron. Eng.

J. Geotech. Geoenviron. Eng., 2023, 149(8): 04023058

 In all analyses a ramped vertical displacement (constant veloc-
ity) was applied at the pile head and the response of the base re-
sistance was established from the reaction forces and the vertical
movements at the pile toe. The data used in the analyses are sum-
marized in Table 2.
The pile base responses computed from the numerical analysis
are shown in Figs. 7–10, where, for comparison purposes, the same
scale has been applied for the base resistance qb . The significant
0 and the initial
effects of both the initial vertical effective stress σvo
relative density I d;o are clearly demonstrated. As expected, the form
of the base resistance-settlement relationship is convex downwards
and all curves appear to have rather similar shapes, but with the
magnitude of resistance affected by the key parameters of I d;o
Downloaded from ascelibrary.org by University of Central Florida on 09/29/23. Copyright ASCE. For personal use only; all rights reserved.

and σvo0 .
0
A summary of relevant data from the numerical analyses, Fig. 9. Pile base response for I d ¼ 0.70 and σvo ¼ 100, 200, and
concentrating on values of qb mobilized at displacements of 400 kPa.
2%Dðqb;02 Þ and 10%Dðqb;10 Þ, is provided in Table 3. The first
value can be considered to reflect anticipated responses in the case
of SLS analysis, where the applied loads are not increased by the
application of partial loading factors. To help in that respect, the
base secant stiffness K b;02 at that displacement level is also tabu-
lated, normalized by the ultimate resistance qb;10 . The second dis-
placement value of 10%D corresponds to the commonly accepted
ultimate resistance, although such resistance may be activated at

0 ¼ 100, 200, and

Fig. 10. Pile base response for I d ¼ 0.90 and σvo
400 kPa.

Table 3. Numerical and fitted bearing factors

0
Fig. 7. Pile base response for I d ¼ 0.30 and σvo ¼ 100, 200, and Initial vertical I d;ini
400 kPa. effective stress,
σo0 (kPa) Parameter Source 0.3 0.5 0.7 0.9
100 qb;02 (MPa) FLAC 0.59 0.74 0.97 1.17
K b;02 =qb;10 20.1 18.9 18.3 17.4
qb;02 (MPa) P.M. 0.63 0.74 1.00 1.27
K b;02 =qb;10 19.0 19.0 19.0 19.0
qb;10 (MPa) FLAC 1.47 1.95 2.65 3.37
P.M. 1.67 1.95 2.62 3.33
200 qb;02 (MPa) FLAC 0.79 0.96 1.22 1.63
K b;02 =qb;10 18.9 18.0 17.6 17.1
qb;02 (MPa) P.M. 0.82 1.04 1.41 1.79
K b;02 =qb;10 19.0 19.0 19.0 19.0
qb;10 (MPa) FLAC 2.09 2.66 3.48 4.75
P.M. 2.15 2.74 3.70 4.71
400 qb;02 (MPa) FLAC 1.15 1.39 1.75 2.31
K b;02 =qb;10 18.7 18.0 17.4 17.0
qb;02 (MPa) P.M. 1.08 1.46 1.99 2.53
K b;02 =qb;10 19.0 19.0 19.0 19.0
qb;10 (MPa) FLAC 3.07 3.86 5.03 6.76
0 ¼ 100, 200, and
Fig. 8. Pile base response for I d ¼ 0.50 and σvo P.M. 2.84 3.85 5.23 6.65
400 kPa.
Note: P.M. = proposed method.

© ASCE 04023058-6 J. Geotech. Geoenviron. Eng.

J. Geotech. Geoenviron. Eng., 2023, 149(8): 04023058

Downloaded from ascelibrary.org by University of Central Florida on 09/29/23. Copyright ASCE. For personal use only; all rights reserved.

Fig. 11. Comparison of pile base resistance qb;10 values arisen from the
parametric numerical analysis and those provided by Eq. (17).

working loads in the case of CPRF designs where piles may be Fig. 13. Comparison of parabolic fits to base responses arisen from
loaded close to their ultimate capacity. the parametric numerical analysis for initial vertical effective stress
0 ¼ 200 kPa.
σvo
Power Law Fits to Base Response
In order to allow general application of the numerical analysis re-
sults presented previously, an attempt has been made to fit the set- Lee and Salgado (1999), whose values lie between 1% and 10%
tlement response with a power law function, relating the magnitude higher than the current FLAC results.
of resistance to the key input quantities of in situ vertical effective The base responses were then been fitted by the power law
0 and initial relative density I .
stress σvo d;o
The first step was to quantify the ultimate base resistance qb;10 ,  
qb;10 Sns 0.6
which was found to be matched well by the relationship qb ¼ ð18Þ
4 0.01
sffiffiffiffiffiffiffi
σvo0 0.004
qb;10 ¼ ð0.1 þ 3.6I d;o Þ þ 4 ð17Þ where Sns = settlement normalized by the pile base diameter.
pa I d;o Reasonable overall fits to the finite element results are provided,
as shown in Figs. 12–14, with the exemption of rather marginal
0
The accuracy of this expression is illustrated in Fig. 11, where cases, i.e., low values of I d;o in conjunction with high values of σvo .
the only significant discrepancies are for σvo 0 ¼ 100 kPa and The various parameters and resulting mobilized base resistance
I d;o ¼ 0.3, for which the FLAC value is 16% greater than the with the corresponding base stiffnesses are summarized in Table 3,
fitted result. Also shown on Fig. 11 are qb;10 values reported by where P.M. refers to the proposed method described previously.

Fig. 12. Comparison of parabolic fits to base responses arisen from Fig. 14. Comparison of parabolic fits to base responses arisen from the
the parametric numerical analysis for initial vertical effective stress parametric numerical analysis for initial vertical effective stress
0 ¼ 100 kPa.
σvo 0 ¼ 400 kPa.
σvo

© ASCE 04023058-7 J. Geotech. Geoenviron. Eng.

J. Geotech. Geoenviron. Eng., 2023, 149(8): 04023058

Downloaded from ascelibrary.org by University of Central Florida on 09/29/23. Copyright ASCE. For personal use only; all rights reserved.

Fig. 15. Geotechnical soil profile at the site of Everrich II project in Ho Chi Minh City, Vietnam. (Data from Nguyen and Fellenius 2015.)

Verification mobilization at the relative pile-soil displacement of 2.5 mm,

and the guidelines of DIN 1054 (DIN 2005) that provide an upper
The proposed simplified parabolic model for the base response was limit of 30 mm.
implemented in the general purpose FE code SOFiSTiK, modelling Static pile load tests were carried out during the construction of
a single pile as a 1-dimensional (beam column) member with pile- the Everrich II project in Ho Chi Minh City, Vietnam, consisting
soil interaction springs distributed along the shaft and at the base. of a series of 37-story apartment buildings. A static load test on a
It was afterwards used to simulate a full-scale pile test performed, pile with diameter D ¼ 2.0 m, length Lp ¼ 80.6 m, was per-
using an Osterberg cell instrumented suitably to define the pile base formed using an Osterberg cell installed at a depth of 60.0 m.
resistance. The numerically established base response is also com- According to Nguyen and Fellenius (2015), the soil profile at
pared with recommendations from different design codes. The re- the location of the O-cell tested pile consisted of 28 m of soft
sponse of pile base resistance, as measured in two other static pile clay overlying 16 m of silty sand to sandy silt. A layer of medium
tests, were used for further validation of the proposed method. dense to dense sand was encountered from 44 to 100 m. The
groundwater table was at 1.0 m below ground surface. The geo-
technical soil profile at the site is given in Fig. 15; further infor-
Validation from Results of a Full Scale Test
mation can be retrieved from the personal website of Prof. B.
In the SOFiSTiK beam column analyses, the interaction springs Fellenius, (Nguyen and Fellenius 2015).
along the pile shaft were modelled as inverted parabolas, with The maximum applied load at the O-cell was 27.1 MN. At that
the mobilized shear stress τ expressed as a function of the local load the bottom plate of the O-cell was moved downward by
axial displacement w of the pile shaft according to 105.7 mm, while a settlement of 100.2 mm was observed at the
  2  pile base. The shaft resistance across the pile, from the O-Cell
w w to the pile base, was found from the strain gages SG-1 to SG-4
τ s ¼ fs 2 − ð19Þ
wu wu to vary from 100 to 210 kPa, as shown in Fig. 15. The initial relative
density at the pile base was estimated from the standard blow count
0
where wu = displacement at which the limiting interface resistance N 70 ¼ CN N, where the adjustment factor CN is given by Bowles
occurs; and f s = mobilized and, according to Randolph and Wroth (1996) as
(1978), can be estimated from sffiffiffiffiffiffiffiffiffiffiffi
fs D 95.76
wu ¼ ζ ð20Þ CN ¼ 0 ð21Þ
2G σvo

The value of ζ may be taken as about 5 according to Baguelin where σvo0 is given in kPa.

and Frank (1979) for a normalized pile length Lp =rp > 20. For σvo0 ¼ 730 kPa, C takes the value of 0.362 and N 0 ¼ 18.
N 70
The response using this approach lies between the stiffer rela- For this value and using Table 3.4 in Bowles (1996), the layer is
tionship provided by API (2003) associated with a full shaft considered as medium dense with I d ¼ 0.50. The response of the

© ASCE 04023058-8 J. Geotech. Geoenviron. Eng.

J. Geotech. Geoenviron. Eng., 2023, 149(8): 04023058

 according to DIN 1054 (DIN 2005), was estimated by (Lunne
et al. 1997)
1 qc
Id ¼ ln ð22Þ
2.61 181ðp 0 Þ0.55

For I d ¼ 0.5 and mean effective stress at the pile toe p 0 ¼

423 kPa, the value of qc was found equal to 23 MPa, while Eq. (7)
provided qc ¼ 15 MPa. Application of the mean value to DIN
1054 (DIN 2005) provided base resistance values of 1.33, 1.71,
and 3.4 MPa for normalized settlements Sns ¼ 0.02, 0.03, and
0.10, respectively.
DIN 1054 (DIN 2005) provides a rather conservative approxi-
Downloaded from ascelibrary.org by University of Central Florida on 09/29/23. Copyright ASCE. For personal use only; all rights reserved.

mation, particularly for Sns > 0.03. This tendency is in line with
previous research associated with both experimental and numerical
results (Comodromos et al. 2003, 2009, 2021). In contrast, API
Fig. 16. Comparison of test data with the total and pile base resistance (2003) provides a good approximation for the ultimate resistance,
responses provided by the proposed method. although it overestimates the base resistance for lower settlement
levels. The relationship provided by the proposed method lies be-
tween the curves provided by the two design codes and is very
close to the response shown by the pile test.
shaft resistance τ s across the pile was defined from Eqs. (19) and
(20), where f s and G are given in Fig. 15. These data were intro-
Validation over Pile Base Resistance from Pile Tests
duced to SOFiSTiK and the segment of the test from the level of the
O-Cell to the pile toe (i.e., Lp ≈ 20.0 m) was simulated by apply- Further validation was carried out on pile load tests provided by
ing a ramp vertical displacement. Yusufuku et al. (2001). The first case corresponds to a static load
Corresponding predictions from SOFiSTiK are shown in Fig. 16 test on a pile with diameter D ¼ 1.20 m, length Lp ¼ 38.0 m
by the continuous lines (thick gray, thin gray, and dotted line for the Ogura et al. (1992). The pile toe was embedded in fine sand with
total, base, and shaft resistances, respectively). Close agreement blow count from standard penetration test N ¼ 55 and an effective
0 0
may be observed between the prediction of the proposed method vertical stress σvo ¼ 300 kPa. From Eq. (21) N 70 is calculated as
and the test data for the response of the base resistance. A slight 31 and using Table 3.4 in Bowles (1996) the layer is considered as
difference is observed between the total predicted resistance and dense with I d slightly higher than 0.65. Adopting I d ¼ 0.67in
the test data at the initial part, while a close agreement is noted Eq. (17) yields qb;10 ¼ 4.37 MPa with the response prediction
in the case of base resistance. The shaft resistance, as quantified for the pile base resistance shown in Fig. 18 together with the pile
by the results of the simulation, are also shown in the same figure. test results.
Valuable conclusions can be drawn from comparing the predic- The second case corresponds to the load test 1B carried out by
tions of some design codes with the results of the full-scale test, the Committee of Bearing Capacity of Piles (1971) as a part of a
focusing on the pile base response. The pile base resistance, as a series of load tests in sand. The pile diameter was D ¼ 0.20 m with
function of pile base settlement, as proposed by API (2003) and length Lp ¼ 4.0 m. The pile toe was embedded in a fine sand of
DIN 1054 (DIN 2005), are plotted in Fig. 17. The bearing resis- medium density with void ratio e ¼ 0.89 and qc varying from 1 to
12 MPa. The effective vertical stress at the pile toe was σvo 0 ¼
tance qb;10 for the API approach was calculated using the data
in Table 1, where the upper limiting value of 4.8 MPa was adopted. 60 kPa. A relative density I d ¼ 0.35–0.65 is proposed for medium
The CPT value qc , required to establish the pile base resistance dense sands (Table 3.4 in Bowles) while a median cone resistance

Fig. 18. Comparison of pile base resistance of a pile test and the re-
Fig. 17. Comparison of pile base resistance field data with different sponse provided by the proposed method. (Data from Ogura et al.
codes and proposed method data. 1992.)

© ASCE 04023058-9 J. Geotech. Geoenviron. Eng.

J. Geotech. Geoenviron. Eng., 2023, 149(8): 04023058

 using an Osterberg cell at 60 m depth, with the lowest instrument
level 2 m above the pile base. In spite of the in situ vertical effective
stress being much greater than the range investigated numerically,
the proposed method gave a very good fit to the measured data. The
experimental and proposed base responses were also compared
with those recommended by two different design codes, which
were found to lie well above the experimental data in one case,
and well below the data in the other case. Close agreement was
also observed between the experimental data and the responses pro-
vided by the proposed method and two other static pile tests.
The constitutive law used in the numerical analysis to produce
data takes into account the current relative density, mean effective
stress during loading, and updates the failure envelope and the de-
Downloaded from ascelibrary.org by University of Central Florida on 09/29/23. Copyright ASCE. For personal use only; all rights reserved.

formation moduli (K and G) accordingly. Despite the encouraging

results from the validation process, it is recommended that the
validity of the proposed methodology should be examined thor-
Fig. 19. Comparison of pile base resistance of a pile test carried out by oughly in further case studies through the combined efforts of
the Committee of Bearing Capacity of Piles (1971) and the response the international research community.
provided by the proposed method. (Data from Committee of Bearing
Capacity of Piles 1971.)
Data Availability Statement

Some or all data, models, or code that support the findings of this
of qc ¼ 5.0 MPa suggests I d ¼ 0.55. Adopting I d ¼ 0.50 in
study are available from the corresponding author upon reasonable
Eq. (17) yields qb;10 ¼ 1.54 MPa, with the predicted response
request.
for the pile base resistance together with the pile test results shown
in Fig. 19. Close agreement can be observed in both cases between
the values provided by the proposed method and the pile test
results. Acknowledgments

The authors are grateful to the reviewers and the Associated Editor
Summary and Conclusions for the valuable comments and constructive feedback that improved
this work.
This paper has presented the results of numerical analysis aimed at
quantifying the base response of bored piles founded in sand. The
objective has been to establish reliable relationships for ultimate Notation
design value of end-bearing resistance in sand of any given relative
density between 0.3 and 0.9 and in situ vertical effective stresses in The following symbols are used in this paper:
the range 100 to 400 kPa. In addition, a simple power law relation- D = pile diameter;
ship has been proposed for the complete end-bearing response. Ep = pile Young’s modulus;
The numerical analysis was carried out in the finite difference e = void ratio;
code FLAC, using a Mohr Coulomb model that was extended to ecs = critical state void ratio;
allow the friction and dilation angles, and also the sand modulus, emax = maximum void ratio;
to reflect the current in situ effective stress level and relative density
emin = minimum void ratio;
as they evolved during the analysis.
eo = initial void ratio;
The end-bearing resistance qb;10 mobilized at a base settlement
of 10% of the pile diameter D was found to vary linearly with the Fn = normal force at pile interface;
square root of the in situ vertical effective stress, with the propor- Fs;max = limiting shear force at pile-soil interface;
tionality constant a linear function of relative density. The mobili- fs = unit shaft resistance;
zation of end-bearing resistance qb =qb;10 was found to vary in G = soil shear modulus;
proportion to the relative settlement to a power 0.60. Even for Go = initial soil shear modulus;
the extreme combinations of relative density and in situ vertical I d = soil relative density;
effective stress investigated, the maximum divergence between I d;o = soil initial relative density;
the curve fits and numerical results was less than 13% (qb;02 , case I r = dilatancy index;
I d;ini ¼ 0.70 and σvo0 ¼ 200 kPa), while the mean divergence is
K o = coefficient of earth pressure at rest;
about 5%. kn = interface normal stiffness;
The numerical results and fitted expressions were validated
ks = interface shear stiffness;
against results from Lee and Salgado (1999), who tabulated values
Lp = pile length;
of qb for normalized settlements of 5% and 10% of the pile
diameter. N q = dimensionless base bearing capacity factor;
In addition, the proposed end-bearing response was imple- p 0 = mean effective stress;
mented in 1-dimensional beam column software in order to deter- pa = atmospheric pressure;
mine the response of piles under axial loading. The software was qb = pile base resistance;
then used to simulate a field test on an instrumented 2 m diameter qb;02 = pile base resistance mobilized at base displacement
pile, embedded to a depth of 81 m in sand. The pile was loaded 2%D;

© ASCE 04023058-10 J. Geotech. Geoenviron. Eng.

J. Geotech. Geoenviron. Eng., 2023, 149(8): 04023058

 qb;10 = pile base resistance mobilized at base displacement Comodromos, E. M., M. C. Papadopoulou, and M. F. Randolph. 2021.
10%D; “Improved relationships for the pile base response in clayey soils.”
J. Geotech. Geoenviron. Eng. 147 (10): 04021095. https://doi.org/10
qb:design = pile base bearing capacity;
.1061/(ASCE)GT.1943-5606.0002606.
qc = cone penetration resistance; Comodromos, E. M., M. C. Papadopoulou, and I. K. Rentzeperis. 2009.
rp = pile radius; “Pile foundation analysis and design using experimental data and 3-D
Sns = pile base settlement normalized to the pile diameter; numerical analysis.” Comput. Geotech. 36 (5): 819–836. https://doi.org
s = settlement; /10.1016/j.compgeo.2009.01.011.
V o = initial volume; Cornforth, D. 1973. “Prediction of drained strength of sands from relative
density measurements.” In STP523-EB evaluation of relative density
w = interface shear displacement;
and its role in geotechnical projects involving cohesionless soils,
wu = interface shear displacement at ultimate shaft edited by E. Selig and R. Ladd, 281–303. West Conshohocken, PA:
resistance; ASTM.
γ 0 = effective unit weight of soil; DIN (Deutsches Institut für Normung). 2005. Ground—Verification of the
Downloaded from ascelibrary.org by University of Central Florida on 09/29/23. Copyright ASCE. For personal use only; all rights reserved.

γ sat = saturated unit weight of soil; safety of earthworks and foundations. DIN 1054. Berlin: DIN.
δ = angle of friction across the pile shaft; FHWA (Federal Highway Administration). 2018. Drilled shafts: Construc-
ν = Poisson’s ratio; tion procedures and design methods. Publication FHWA-NHI 18-024,
FHWA GEC 010. Washington, DC: FHWA, DOT.
ν p = Poisson’s ratio of a concrete pile;
0 = initial effective horizontal stress;
Fioravante, V., and D. Giretti. 2016. “Unidirectional cyclic resistance of
σho Ticino and Toyoura sands from centrifuge cone penetration tests.”
0
σvo = initial effective vertical stress; Acta Geotech. 11 (4): 953–968. https://doi.org/10.1007/s11440-015
φ 0 = effective angle of friction; -0419-3.
φo0 = initial effective angle of friction; Gibson, R. E. 1950. “Correspondence.” J. Inst. Civ. Eng. 34 (382): 383.
φp = peak angle of friction; Green, G. E., and D. W. Reades. 1975. “Boundary conditions, anisotropy
and sample shape effects on the stress-strain behaviour of sand in a tri-
ψ = dilation angle; and
axial compression and plane strain.” Géotechnique 25 (2): 333–356.
ψp = peak dilatancy angle. https://doi.org/10.1680/geot.1975.25.2.333.
Ichihara, M., and H. Matsuzawa. 1973. “Application of plane strain tests to
earth pressure.” In Proc., 8th Int. Conf. on Soil Mechanics and Foun-
References dation Engineering, 185–190. Moscow: ISSMGE.
Jamiolkowski, M. C., C. C. Ladd, J. T. Germaine, and R. Lancellotta. 1985.
API (American Petroleum Institute). 2003. Planning, designing and “New developments in field and laboratory testing of soils.” In Proc.,
constructing fixed offshore platforms—Working stress design. API 11th Int. Conf. on Soil Mechanics and Foundation Engineering,
Recommended Practice 2A-WSD (RP 2A-WSD). Washington, DC: 57–153. San Francisco: CRC Press.
API. Katzenbach, R., U. Arslan, and C. Moormann. 2000. “Piled raft foundation
Baguelin, F., and R. Frank. 1979. “Theoretical studies of piles using the projects in Germany.” In Design applications of raft foundations,
finite element method.” In Proc., Conf. Numerical Methods in Offshore 323–391. London: Thomas Telford.
Piling, 83–91. London: Institution of Civil Engineers. Kleven, A., S. Lacasse, and K. H. Andersen. 1986. Soil parameters for
Bolton, M. D. 1986. “The strength and dilatancy of sands.” Géotechnique offshore foundation design. Rep. No. 40013-34. Oslo, NO: Norwegian
36 (1): 65–78. https://doi.org/10.1680/geot.1986.36.1.65. Geotechnical Institute.
Bolton, M. D. 1987. “Discussion: The strength and dilatancy of sands.” Lancellotta, R. 1983. “Analisi di affidabilitá in ingegneria geotecnica.”
Géotechnique 37 (2): 219–226. https://doi.org/10.1680/geot.1987.37 In Atti dell’Istituto di Scienza delle Costruzioni, 625. Turin, Italy:
.2.219. Politecnico di Torino.
Bowles, J. E. 1996. Foundation analysis and design. 5th ed. New York: Lee, J. H., and R. Salgado. 1999. “Determination of pile base resistance in
McGraw-Hill. sands.” J. Geotech. Geoenviron. Eng. 125 (8): 673–683. https://doi.org
Carter, J. P., J. R. Booker, and S. K. Yeung. 1986. “Cavity expansion in /10.1061/(ASCE)1090-0241(1999)125:8(673).
cohesive frictional soils.” Géotechnique 36 (3): 349–358. https://doi Lo Presti, D. 1987. “Mechanical behaviour of Ticino sand from resonant
.org/10.1680/geot.1986.36.3.349. column tests.” Ph.D. thesis, Dip. Di Ingegneria Civile, Politecnico di
CEN. 2004a. Eurocode 2. Design of concrete structures—Part 1: General Torino.
rules and rules for buildings. EN 1992-1-1. Brussels, Belgium: CEN. Lunne, T., P. K. Robertson, and J. J. M. Powell. 1997. Cone penetration
CEN (European Committee for Standardization). 2004b. Eurocode 7. testing. New York: E & FN Spon.
Geotechnical design—Part 1-1: General actions. EN 1997-1. Brussels, Nguyen, H. M., and B. H. Fellenius. 2015. “Bidirectional cell tests on
Belgium: CEN. non-grouted and grouted large-diameter bored piles.” J. Geo-Eng. Sci.
CGS (Canadian Geotechnical Society). 2006. Canadian foundation 2 (3–4): 105–117. https://doi.org/10.3233/JGS-140025.
engineering manual. 4th ed. Alberta, Canada: CGS. Ogura, H., M. Sumi, T. Suzuki, H. Kawamura, and H. Kishida. 1992.
Committee of Bearing Capacity of Piles. 1971. “Field tests on piles in “Simplified load testing method for cast-in-place pile.” In Proc., 27th An-
sand.” Soils Found. 11 (2): 29–49. https://doi.org/10.3208/sandf1960 nual Meeting of JGS, 1531–1532. Tokyo: Japanese Geotechnical Society.
.11.2_29. Peck, R. B., W. E. Hanson, and T. H. Thornburn. 1974. Foundation engi-
Comodromos, E. M., C. Anagnostopoulos, and M. Georgiadis. 2003. neering. 2nd ed. New York: Wiley.
“Numerical assessment of axial pile group response based on load test.” Randolph, M. F., J. Dolwin, and R. Beck. 1994. “Design of driven piles in
Comput. Geotech. 30 (6): 505–515. https://doi.org/10.1016/S0266 sand.” Géotechnique 44 (3): 427–448. https://doi.org/10.1680/geot
-352X(03)00017-X. .1994.44.3.427.
Comodromos, E. M., and M. C. Papadopoulou. 2012. “Response evalu- Randolph, M. F., and C. P. Wroth. 1978. “Analysis of deformation of
ation of horizontally loaded pile groups in clayey soils.” Géotechnique vertically loaded piles.” J. Geotech. Eng. Div. 104 (12): 1465–1488.
62 (4): 329–339. https://doi.org/10.1680/geot.10.P.045. https://doi.org/10.1061/AJGEB6.0000729.
Comodromos, E. M., M. C. Papadopoulou, and L. Laloui. 2016. “Contri- Reese, L. C., and M. W. O’Neill. 1989. “New design method for drilled
bution to the design methodologies of piled raft foundations under com- shafts from common soil and rock tests.” In Vol. 2 of Foundation en-
bined loadings.” Can. Geotech. J. 53 (4): 559–577. https://doi.org/10 gineering: Current principles and practices, edited by F. H. Kulhawy,
.1139/cgj-2015-0251. 1026–1039. New York: ASCE.

© ASCE 04023058-11 J. Geotech. Geoenviron. Eng.

J. Geotech. Geoenviron. Eng., 2023, 149(8): 04023058

 Reul, O., and M. F. Randolph. 2003. “Piled rafts in overconsolidated clay— Sutherland, H., and M. Mesdary. 1969. “The influence of the intermediate
Comparison of in situ measurements and numerical analyses.” Géotech- principal stress on the strength of sand.” In Proc., 7th Int. Conf. on Soil
nique 53 (3): 301–315. https://doi.org/10.1680/geot.2003.53.3.301. Mechanics and Foundation Engineering, 391–399. Mexico: Sociedad
Robertson, P. K., and R. G. Campanella. 1983. “Interpretation of cone pen- Mexicana de Mecanica.
etration tests. Part II: Clay.” Can. Geotech. J. 20 (4): 734–745. https:// Vesic, A. S. 1967. A study of bearing capacity of deep foundations. Final
doi.org/10.1139/t83-079.
Rep. No. Project B-189. Atlanta: Georgia Institute of Technology.
Salgado, R., J. K. Mitchell, and M. Jamiolkowski. 1997. “Cavity expansion
Yu, H. S., and G. T. Houlsby. 1991. “Finite cavity expansion in dilatant
and penetration resistance in sand.” J. Geotech. Geoenviron. Eng.
123 (4): 344–354. https://doi.org/10.1061/(ASCE)1090-0241(1997) soils: Loading analysis.” Géotechnique 41 (2): 173–183. https://doi
123:4(344). .org/10.1680/geot.1991.41.2.173.
Santamarina, J. C., and G. C. Cho. 2001. “Determination of critical parameters Yusufuku, N., H. Ochiai, and S. Ohno. 2001. “Pile end-bearing capacity of
in sandy soils—Simple procedure.” Geotech. Test. J. 24 (2): 185–192. the sand related to soil compressibility.” Soils Found. 41 (4): 59–71.
https://doi.org/10.1520/GTJ11338J. https://doi.org/10.3208/sandf.41.4_59.
Downloaded from ascelibrary.org by University of Central Florida on 09/29/23. Copyright ASCE. For personal use only; all rights reserved.

© ASCE 04023058-12 J. Geotech. Geoenviron. Eng.

J. Geotech. Geoenviron. Eng., 2023, 149(8): 04023058