Design Load of Rigid Footings on Sand

Design load of rigid footings on sand

Edgar G. Diaz and Fernando Rodrıguez-Roa

Abstract: Experimental evidence has shown that most current methods are not able to predict design loads of footings oncohesionless soil with an acceptable degree of accuracy. In the present study, a simple and realistic settlement-basedmethod is proposed to estimate the design load of rigid footings on sand subjected to static vertical loading. The designcriterion based on restricting the end-of-construction settlement to 16 mm because of the inherent variability of the realsoil deposits is herein adopted. A series of finite-element analyses based on an advanced constitutive model were carriedout to study the load–settlement response of footings supported on 14 sandy soils. Routine design charts were developedto predict the net allowable soil pressure of footings on normally consolidated and overconsolidated sands. These chartsconsider footing shape, embedment depth, grain diameters D10 and D60, particle shape, unit weight (or submerged unitweight for saturated sands), and indirect measurements of the shear strength derived from in situ tests, such as relativedensity, standard penetration test (SPT) or cone penetration test (CPT). As shown, the proposed charts match well withavailable experimental data.

Key words: footing design, Lade model, finite-element method, sand, footing settlement, shallow foundations.

Resume : Selon des evidences experimentales, les methodes les plus courantes de prediction ne peuvent pas predire avecun degre de certitude acceptable les chargements de conception sur des semelles placees sur un sol sans cohesion. Danscette etude, une methode simple basee sur des tassements realistes est proposee pour estimer le chargement de conceptionde semelles rigides sur du sable, soumises a un chargement vertical statique. Le critere de conception adopte est base surun tassement maximal de 16 mm a la fin de la construction en raison de la variabilite inherente des sols reels. Une seried’analyses par elements finis, basees sur un modele constitutif avance, ont ete effectuees dans le but d’etudier le comporte-ment en chargement-tassement de semelles supportees sur 14 sols sablonneux. Des chartes de conception ont ete develop-pees afin de predire la pression nette acceptable du sol sur des semelles placees sur des sables consolides normalement etsurconsolides. Ces chartes considerent la forme de la semelle, la profondeur d’enfouissement, les diametres des grains D10

et D60, la forme des particules, le poids unitaire (ou le poids unitaire submerge pour les sables satures), ainsi que des me-sures indirectes de la resistance au cisaillement derivees d’essais in situ, tels que la densite relative, l’essai de penetrationstandard (SPT) ou l’essai de penetration d’un cone (CPT). L’article demontre que les chartes proposees correspondent bienavec les donnees experimentales disponibles.

Mots-cles : conception de semelles, modele Lade, methode d’elements finis, sable, tassement de semelle, fondations peuprofondes.

[Traduit par la Redaction]

Introduction

The design load of footings on sand has been based onlimiting the maximum settlement to a value of 25 mm, ex-cept in narrow and shallow footings, in which case thebearing capacity of the sand may lead to a lower allowablefooting unit load. Several methods have been proposed topredict settlements of footings on sand based on in situtests, such as the standard penetration test (SPT) and thecone penetration test (CPT) (Terzaghi and Peck 1948;Meyerhof 1965; Peck and Bazaraa 1969; Schmertmann1970; Peck et al. 1974; Burland and Burbidge 1985). How-

ever, footing load tests performed on sand at Texas A&MUniversity (Briaud and Gibbens 1999) and the field dataobtained from a research project on 30 bridge footings inOhio (Sargand et al. 1999) showed that most current meth-ods are unable to predict design loads of footings on cohe-sionless soil with an acceptable degree of accuracy. In thepresent study, a simple and realistic settlement-basedmethod is proposed to estimate the design load of rigidfootings on sand.

There are several advanced constitutive models availabletoday. A comparison of their capabilities and limitationswas reported by Lade (2005a). In this research, the singlehardening (SH) model proposed by Lade and Kim (1995)was selected. This is an elastoplastic model that representsthe real nonlinear stress–strain behavior of the soil beforefailure, at failure, and after failure. In addition, the SHmodel only involves 11 parameters for cohesionless soilsthat can be derived from results of isotropic compressionand conventional drained triaxial tests.

The commercial finite element (FE) program ABAQUSwas selected to perform the numerical analyses. This

Received 24 January 2009. Accepted 15 December 2009.Published on the NRC Research Press Web site at cgj.nrc.ca on19 July 2010.

E.G. Diaz and F. Rodrıguez-Roa.1 Department of Structuraland Geotechnical Engineering, Pontificia Universidad Catolicade Chile, Av. Vicuna Mackenna 4860, Codigo Postal 782-0436,Santiago, Chile.

1Corresponding author (e-mail: [email protected]).

872

Can. Geotech. J. 47: 872–884 (2010) doi:10.1139/T09-145 Published by NRC Research Press

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program provides a subroutine for users to add anyconstitutive law not included in the standard library. Thus, athree-dimensional (3D) SH-model interface was coded bythe authors, using the incremental form of the SH modeldeveloped by Lade and Jakobsen (2002) and Jakobsen andLade (2002) and a subroutine for two-dimensional (2D)problems coded by Jakobsen (1999).

The capability of the SH model to capture the real soilresponse was examined by comparing the predicted responsewith experimental data derived from drained triaxialcompression tests performed on loose and dense MaipoRiver (MR) sand (Rodrıguez-Roa 2003).

As mentioned above, the design load of footings onsand has usually been based on limiting the settlementto a value of 25 mm, on the premise that if the maxi-mum settlement is restricted to this amount the angulardistortion among the footings of a typical building wouldbe within tolerable values (Terzaghi and Peck 1948). Onthe other hand, field observations of footings on sandsindicate quite clearly an increase of settlements withtime because of the creep phenomenon (Schmertman1970; Burland and Burbidge 1985; Briaud and Gibbens1999). A settlement found 30 years after the end of con-struction may range between 1.5 and 2.5 times the im-mediate settlement for static and heavily fluctuatingloads, respectively (Burland and Burbidge 1985). Tolimit the settlement of the largest footing to a value of25 mm, Terzaghi et al. (1996) suggested a design crite-rion based on restricting the end-of-constructionsettlement to 16 mm because of the inherent variabilityof real soil deposits. This criterion, which is hereinadopted, may also be regarded as an additional safetymargin to compensate for the anticipated long-term set-tlement of buildings on sand.

A series of FE analyses were carried out to study theload–settlement response of square and strip footings onsands subjected to a static central vertical load. Fourteendry and saturated sands were analyzed to develop charts topredict the net allowable soil pressure of footings on nor-mally consolidated (NC) and overconsolidated (OC) sands.These charts consider footing shape, embedment depth,grain diameters D10 and D60 (grain size at which 10% and60%, respectively, of the sample is finer), particle shape,unit weight (or submerged unit weight for saturated sands),and indirect measurements of the shear strength of soils de-rived from in situ tests as relative density (Dr), SPT or CPT.It is shown that the proposed design charts match well withavailable experimental data derived from several footingload tests.

For footing widths smaller than *1.0 m, the allowablesoil pressure should also be checked against a bearing-capacity failure to ensure a factor of safety of at least 3.0(Peck et al. 1974). This verification can be done by usingstandard bearing-capacity methods, such as those proposedby Brinch Hansen (1970) or Vesic (1973), in which thepeak friction angle may be estimated in terms of thecritical-state friction angle, relative density, and footingshape (Rodrıguez-Roa 2006). The critical-state frictionangle is mainly a function of mineralogy and can bedetermined experimentally (Bolton 1986).

SH model

The SH model is an elastoplastic hardening constitutivemodel with a single isotropic yield surface. It can be appliedto sand, clay or any cemented soil. This model uses a non-associated plastic flow rule and can include both a work-hardening and a work-softening law. Anisotropic and time-dependent behavior cannot be incorporated in the model.

The total stress increments are divided into elastic andplastic components, which are computed separately. Theelastic strain increments are calculated by using Hooke’slaw with a Young’s modulus, E, expressed as

½1� E ¼ MI1

pa

� �2

þ 61þ y

1� 2y

� �J2

p2a

" #lpa

where n is Poisson’s ratio; pa is atmospheric pressure; M andl are dimensionless constants; and I1 and J2 are the first in-variant of the stress tensor and the second invariant of thedeviatoric stress tensor, respectively, given by

½2� I1 ¼ s1 þ s2 þ s3

½3� J2 ¼1

6½ðs1 � s2Þ2 þ ðs2 � s3Þ2 þ ðs1 � s3Þ2�

where s1, s2, and s3 are the principal stresses. The failurecriterion is defined as

½4� I31

I3

� 27

� �I1

pa

� �m

¼ h1

where m and h1 are dimensionless constants, and I3 is thethird invariant of the stress tensor

½5� I3 ¼ s1s2s3

A typical failure surface in the principal stress space forcohesionless soils is shown in Fig. 1. An additional parame-ter must be incorporated in the model to allow the inclusionof the soil cohesion. For this purpose, a translation of theprincipal stress space along the hydrostatic axis is per-formed, i.e., a constant stress apa is added to the normalstresses before substitution in eq. [4], where a is a dimen-sionless constant. The stress apa is slightly greater than theuniaxial tensile strength of the material (Lade 1982).

The plastic strain increments are computed by applyingthe normality rule to the following potential function:

½6� g ¼ j1

I31

I3

� I21

I2

þ j2

� �I1

pa

� �m

where m and j2 are dimensionless constants, I2 is the secondstress invariant of the stress tensor

½7� I2 ¼ �ðs1s2 þ s2s3 þ s1s3Þ

and the dimensionless parameter j1 is computed in terms ofm as follows:

½8� j1 ¼ 0:00155 m�1:27

A typical plastic potential surface in the principal stressspace is shown in Fig. 2. It is seen that the pointed apex of

Diaz and Rodrıguez-Roa 873

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this surface is located behind the origin of the coordinatesfor cohesionless soils.

The yield criterion, fp, is associated with surfaces of con-stant plastic work, and it is expressed as

½9� fp ¼ f 0pðsÞ � f 00p ðWpÞ ¼ 0

where s is the stress state, f 00p is a function defined below,Wp is plastic work, and the function f 0p is defined as

½10� f 0p ¼ j1

I31

I3

� I21

I2

� �I1

pa

� �h

eq

where h is a dimensionless constant and q varies from zeroto unity. If the stress level, S, is expressed as

½11� S ¼ 1

h1

I31

I3

� 27

� �I1

pa

� �m

the value of q in eq. [10] is given by the relationship

½12� q ¼ a S

1� ð1� aÞ S

where a is a dimensionless constant.For isotropic-hardening behavior, the function f 00p in

eq. [9] increases with plastic work, Wp, as follows:

½13� f 00p ¼ ð27j1 þ 3Þ Wp

Cpa

� �hp

where the parameters C and p are dimensionless constants.These parameters are related to the plastic work done duringan isotropic compression test, according to the following re-lationship:

½14� Wp

pa

¼ CI1

pa

� �p

Thus, both parameters C and p can be computed by plottingWp/pa versus (I1/pa) on a log–log scale.

For isotropic-softening behavior, the function f 00p in eq. [9]is given by an exponential decay expression in terms of theplastic work. The transition from hardening to softeningtakes place abruptly when the current stress point reachesthe failure surface, i.e., when S = 1.

A typical yield surface in the principal stress space isshown in Fig. 3. For a more detailed description of the SHmodel, the reader is referred to Lade (2005b). A summaryof the parameters required by this model for a cohesionlesssoil is given in Table 1.

Fig. 1. Failure surface in the principal stress space for a cohesion-less soil.

Fig. 2. Plastic potential surface in the principal stress space.

Table 1. SH-model parameters required for acohesionless soil.

Components of the model ParametersElastic strains M, l, nFailure criterion m, h1

Plastic potential j2, mYield criterion h, aHardening function C, p

Fig. 3. Yield surface in the principal stress space.

874 Can. Geotech. J. Vol. 47, 2010


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Numerical modeling of drained triaxialcompression tests

The measurements derived from isotropic compression andconventional drained triaxial tests performed on MR sand(Rodrıguez-Roa 2003) were used to evaluate the accuracy ofthe model on the basis of laboratory tests. The SH-model pa-rameters fitted to the experimental data by following themethod proposed by Lade (2005b) are included in Table 2for relative densities equal to 0.35, 0.55, and 0.75. The valueof Poisson’s ratio is usually close to 0.2 in cohesionless soils(Lade 2005b). Therefore, this Poisson’s ratio was adopted forall the numerical calculations performed in the present study.

An axisymmetric finite-element mesh composed of four-node elements was selected for modeling the triaxial tests(Fig. 4). After applying an equal all-around confining pres-sure to the specimen (initial stress state), the top cap wassubjected to successive incremental vertical displacementsfor modeling the loading process. Stress, strain, and volumechanges, computed at the centroid of the marked element inFig. 4, did not vary significantly when smooth and rough in-terface elements were inserted between the soil and the cap.

As shown in Figs. 5a and 5b, the observed stress–strainand volume-change curves under an effective confiningpressure of 80 kPa are accurately predicted by the SH modelfor both loose and dense MR sand.

Footings on normally consolidated sands

Relative density approach to net allowable soil pressureA series of FE analyses were carried out to examine the

load–settlement response of square and strip footings onNC sands. Three-dimensional FE fine meshes composed ofeight-node elements were used for modeling rigid squarefootings. Because of symmetrical conditions, only one-quarter of the footing was modeled. A typical 3D FEmesh is illustrated in Fig. 6. After preliminary analyses,the final extension adopted for the mesh was 10 times thefooting width in both the vertical and horizontal directions.

Two-dimensional FE meshes subdivided into four-noderectangular elements were employed for the analyses ofrigid strip footings under plane strain conditions. The 2Dmeshes kept the geometry and extension that were used forthe 3D meshes.T

able

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0.58

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–3.1

62.

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0.26

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eigh

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unit

wei

ght

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hepa

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sand

pres

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ade

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e Dak

oula

san

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n(1

992)

.

Fig. 4. Axisymmetric FE mesh for triaxial compression tests.



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Fourteen dry and saturated NC sands, whose geotechnicalproperties are given in Tables 2 and 3, were considered inthe numerical analyses. In the cases of footings on saturatedsands, the submerged unit weight was used instead of thedry unit weight, and it was assumed that loading was ap-plied under drainage conditions. For most NC sands, the co-efficient of earth pressure at rest, K0, ranges from 0.35 to0.45. Thus, a mean value of 0.40 was assumed.

Several square and strip footings supported on the studysands were modeled by considering different widths and em-

bedment depths (Table 4) to involve usual cases of footingsencountered in practice.

Because of the usual construction method of pouring theconcrete directly against the surrounding soil, no interfaceelement was inserted between the footings and the surround-ing soil. All footings were subjected to successive incremen-tal vertical displacements up to a maximum settlement of25 mm for modeling the load sequence.

Experimental evidence has shown that the load–settlementresponse of footings on sand depends on several variablesrelated to the stress–strain behavior of the soil as well asthe foundation type. Therefore, if the net soil pressure at agiven settlement is expressed as a function of a unique rela-tive density factor, FDr, for the sake of simplicity, such afactor should depend on both soil properties and footingcharacteristics. Thus, different expressions for FDr were triedfor including not only Dr, but also the effects of other pa-rameters, such as grain diameter; particle shape, Ps; unitweight of the sand; footing geometry; and embedmentdepth. As shown in Table 5, from this trial and error processbased on the numerical analyses of 168 square footing cases(six foundation types on 14 dry and saturated sands), a highvalue of the correlation coefficient (R2 = 0.93) was obtainedby expressing FDr as

½15� FDr ¼ Dr

g

gw

� �0:5

Ps logð500ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD10D60

pÞ s

B

� �0:3

� 1þ 0:6Df

B

� �0:2

where Dr is the relative density in decimal form, D10 and D60 aregrain diameters in millimetres, s is the settlement (= 16 mm), Bis the footing width, Df is the embedment depth, g is the unitweight of the sand (or submerged unit weight for saturatedsands), gw is unit weight of water, and Ps is the dimensionless

Fig. 5. Stress–strain and volume-change curves for triaxial compression tests performed with s3 = 80 kPa: (a) MR loose sand (Dr = 0.35);(b) MR dense sand (Dr = 0.75).

Fig. 6. Typical 3D FE mesh for square footings.



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constant fitted to particle-shape effect, as illustrated in Table 6.Units used for s, B, and Df, must be consistent.

The numerical analyses of the strip-footing cases also ledto a correlation coefficient of R2 = 0.93.

The charts obtained for the net soil pressure at 16 mm set-tlement are given in Figs. 7a and 7b for square and stripfootings, respectively, on NC sands. An extension of thismethod to footings on unsaturated sand deposits is a conser-vative approach, because no apparent cohesion was includedin the SH model.

It can be noted from Figs. 7a and 7b that the net allow-able soil pressure for a given square footing on NC sandsmay vary from 1.21 to 1.66 times the predicted value for astrip footing with the same width. These results are consis-tent with the value 1.57, reported by Burland and Burbidge(1985), for the ratio of strip-footing settlement to square-footing settlement when both types of shallow foundationshave the same width and are subjected to the same unit load.

The proposed charts were developed on the assumption ofa very rough sand–concrete contact. To evaluate the effectof this simplification on the predicted net allowable soilpressures, a conservative smooth sand–concrete interfacewas assumed between the foundation walls and the sur-rounding soil. Then, two square prisms with B = 1.0 m andDf/B ratios equal to 1.0 and 2.0, on a medium-dense dry MRsand (Dr = 55%), were analyzed. A decrease was found inthe net allowable soil pressure, ranging from 2% to 7%when the obtained results were compared with those givenby the charts. Despite the fact that these differences are notconsidered large, it is suggested to limit the use of the pro-posed charts to a maximum Df/B value of 2.0 for the mostgeneral case.

To verify the proposed charts on the basis of the experi-mental evidence, load tests performed on six square footingsat Labenne were studied (Lehane et al. 1993; Mestat andBerthelon 2001). The tested soil was composed of a dunesand about 10 m thick. An overburden of 1.5 m was re-moved previous to the execution of the load tests. At thesite, the water table ranged from 3.0 to 4.3 m deep.Geotechnical properties of this sand and a summary of thefooting sizes and the measured unit loads corresponding toa settlement of 16 mm are given in Tables 7 and 8,respectively. It can be seen in Fig. 8 that the predicted soilpressures for square footings on NC sands match well withthe response of the tested footings on the Labenne slightlyOC sand.T

able

3.In

dex

prop

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sof

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Sand

Gs

D60

(mm

)D

50(m

m)

D10

(mm

)Pa

rtic

lesh

ape

Ref

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MR

2.70

0.39

00.

340.

205

Suba

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arR

odrı

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(200

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Bea

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660.

284a

0.27

0.18

aSu

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Lad

ean

dB

oony

achu

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2.65

0.50

0a0.

430.

32a

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ngul

ar–s

ubro

unde

dL

ade

and

Dun

can

(197

3)Sa

cram

ento

Riv

er2.

680.

220

0.21

0.15

Suba

ngul

ar–s

ubro

unde

dL

eean

dSe

ed(1

967)

Eas

tern

Sche

ldt

2.65

0.18

0a0.

170.

12a

Subr

ound

ed–r

ound

edJa

kobs

enet

al.

(199

9)F-

Sand

2.65

0.24

00.

220.

13R

ound

edA

lshi

bli

and

Stur

e(2

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Nev

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2.68

0.18

00.

160.

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and

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e(1

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Otta

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0.09

20.

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bD

akou

las

and

Sun

(199

2)

Not

e:G

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aver

age

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nsi

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a Est

imat

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from

repo

rted

data

.b A

ssum

edpa

rtic

lesh

ape.

Table 4. Width, B, and embedmentdepth, Df , of the square and stripfootings analyzed.

B (m) Df (m)1.0 0.251.0 0.501.0 1.002.0 0.502.0 1.004.0 1.00



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Standard penetration approach to net allowable soilpressure

The SPT is one of the most widely used in situ tests onsandy soils in engineering practice. Although the SPT doesnot directly measure the shear strength of soils, it does pro-vide a good correlation with it. Thus, several empirical ex-pressions have been suggested for correlating the SPT N-value and the relative density (Skempton 1986; Tokimatsuand Seed 1987; Kulhawy and Mayne 1991). The expressionproposed for sands by Kulhawy and Mayne (1991), based onextensive field data, is given by

½16� D2r ¼

ðN1Þ60

CPCAOCR0:18

where Dr is the relative density in decimal form; (N1)60 isthe normalized SPT N-value; Cp = 60 + 25 logD50; D50 isthe grain diameter in millimetres; CA = 1.2 + 0.05 log(t/100); t is the age of the soil in years; and OCR isthe overconsolidation ratio. The expression for the agingfactor, CA, is not very sensitive to the chosen value for t.Thus, it would be sufficient to use a value of t = 1000 yearsfor most practical cases (Coduto 2001), which is the valueof t used in the present study.

The (N1)60 value can be obtained from the expression(Skempton 1986; Youd et al. 2001)

½17� ðN1Þ60 ¼ N CNCECSCBCR

where N is the measured value; CN is the overburden pres-sure correction; CE is the hammer energy ratio correction;CS is the correction for a sampler without liner; CB isthe correction factor for a borehole diameter larger than115 mm; and CR is the rod length correction. According toASTM D4633-05 (ASTM 2005), it must be pointed out thatthe correction for short rods of less than 10 m should be dis-continued, i.e., it is suggested to use CR = 1. In Japan, thecorrection for short rods is not used (Ishihara 1996). Equa-tion [16] can be used for analyses of unsaturated or satu-rated sands (P.W. Mayne, personal e-mail communication,

2008). Therefore, in the case of saturated sands no water ta-ble correction for the SPT N-value is required.

On the basis of eqs. [15] and [16] and considering OCR =1, the following factor FSPT-NC is proposed to provide astandard penetration approach to net allowable soil pressure

½18� FSPT-NC ¼ðN1Þ60

Cp

� �0:5g

gw

� �0:5

Ps log ð500ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD10D60

pÞ

� s

B

� �0:3

1þ 0:6Df

B

� �0:2

By considering the FE results used for developing Figs. 7aand 7b and assuming that CA = 1.25 in eq. [16], charts in termsof the standard-penetration factor, FSPT-NC, were obtained forthe design of square and strip footings on NC sands, as shownin Fig. 9. Footing load tests performed at six sites in Kuwait(Ismael 1985; Ismael and Jeragh 1986) were studied for com-parison purposes. The tested soil was composed of fine to me-dium sand with little nonplastic silt. Mean values of indexproperties of Kuwait sand and representative geotechnicalproperties of the test sites are given in Tables 9 and 10, respec-tively. Values of N and cone tip resistance, qc, included in Ta-ble 10 correspond to reported average values between depthsof 0.5 and 2.0 m. Standard penetration tests were performedby means of a donut hammer, using two turns of the ropearound the cathead system. Thus, an energy-correction factorof CE = 0.75 was adopted to normalize the standard-penetra-tion resistance (Skempton 1986). On the other hand, the fol-lowing expression proposed by Liao and Whitman (1986) wasused for the overburden pressure correction

½19� CN ¼pa

s 0v

� �0:5

where s 0v is the effective vertical overburden pressure. Thiscorrection factor, CN, should not exceed a value of 1.7(Youd et al. 2001).

The effective vertical overburden pressure, s 0v, was calcu-lated at a depth equal to B/2 below the level of the base ofthe footings tested in Kuwait. The water table at the site re-mained below a depth (Df + B) beneath the ground surfacesurrounding the tested square footings (Tables 10, 11).

There is no information with regard to the type of SPT sam-pler used for the subsoil exploration performed in Kuwait inthe early 1980s. According to Kovacs (1994), SPT samplerswith liners were used until approximately 1980. Thus, it wasassumed that a sampler with liner was used in Kuwait. Thecomputed values of (N1)60 and the measured unit loads corre-

Table 5. Some examples of expressions tried for the relative density factor FDr.

Relative density factor FDr = Best-fit set of parameters R2

Dn1r n1 = 1 0.663

Drðs=BÞn2 n2 = 0.3 0.833

Dr logðn3 þ n4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD10D60

pÞðs=BÞ0:3 n3 = 0; n4 = 500 0.906

Dr logð500ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD10D60

pÞðs=BÞ0:3½1þ n5ðDf =BÞ�n6 n5 = 0.6; n6 = 0.2 0.915

Drðg=gwÞn7 logð500ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD10D60

pÞðs=BÞ0:3½1þ 0:6ðDf =BÞ�0:2 n7 = 0.5 0.925

Drðg=gwÞ0:5Ps logð500ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD10D60

pÞðs=BÞ0:3½1þ 0:6ðDf =BÞ�0:2 — 0.932

Note: R2, correlation coefficient.

Table 6. Particle-shape effect, Ps.

Particle shapea Ps

Angular 1.00Subangular 0.95Subrounded 0.90Rounded 0.85

aTypical shapes of coarse particles (Peck et al. 1974).



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sponding to a settlement of 16 mm are given in Table 11. Asshown in Fig. 10, there is good agreement between the pro-posed chart for square footings on NC sands and the experi-mental data obtained from the Kuwait tests. The Burland andBurbidge (1985) predictions, also included in this figure, tendin general to underestimate the measured footing unit loads.

Cone tip resistance approach to net allowable soilpressure

The CPT is another common in situ test. It can also beused to estimate the engineering properties of soils throughthe use of empirical correlations. For sands, Kulhawy andMayne (1991) proposed the following expression betweenDr and the normalized cone tip-resistance:

½20� D2r ¼

qc1

305 CAQc OCR0:18pa

where Dr is the relative density in decimal form; qc1 isthe normalized cone tip resistance; CA = 1.2 + 0.05 log(t/100); t is the age of soil in years (as mentioned above, a va-lue of t = 1000 years was used in the present study); andQc is a dimensionless constant equal to 0.91 for highly com-pressible sands, 1.0 for moderately compressible sands, and1.09 for slightly compressible sands. Equation [20] can be

Fig. 7. Net allowable soil pressure as a function of factor FDr for (a) square footings on NC sands and (b) strip footings on NC sands.

Table 7. Geotechnical properties of Labenne sand.

Dr gd /gw

Moisture content(%)

D60

(mm)D50

(mm)D10

(mm) Particle shape0.43 1.60 5 0.34 0.32 0.18 Subangular–subrounded

Table 8. Unit load at 16 mm settle-ment for footings tested at Labenne.

B (m) Df (m) Unit load/pa

1.00 0.20 4.101.00 1.00 4.150.71 0.10 4.120.71 0.20 4.240.71 0.80 4.950.71 1.10 5.60

Fig. 8. Predicted and measured unit load at 16 mm settlement forsquare footings tested at Labenne.

Fig. 9. Net allowable soil pressure as a function of factor FSPT-NC

for square and strip footings.



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used for analyses of unsaturated or saturated sands (P.W.Mayne, personal e-mail communication, 2008).

The normalized cone tip resistance, qc1, can be calculated as

½21� qc1 ¼ qc

pa

s 0v

� �n

where qc is the measured value, n is a nondimensional expo-nent, typically equal to 0.5 for sands (Robertson and Wride

1998; Ishihara 1996). The overburden correction factor (pa/s 0v)n in eq. [21] should not exceed a value of 1.7 (Youd etal. 2001).

On the basis of eqs. [15] and [20] and consideringOCR = 1, the following factor FCPT-NC is proposed to providea cone tip–resistance approach to net allowable soil pressure

½22� FCPT-NC ¼qc1

pa

� �0:5g

gw

� �0:5

Ps logð500ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD10D60

pÞ

� s

B

� �0:3

1þ 0:6Df

B

� �0:2

By considering the FE results used for developing Figs. 7aand 7b and assuming CA = 1.25 and Qc = 1.0 in eq. [20],charts in terms of the cone resistance factor FCPT-NC wereobtained for the design of square and strip footings on NCsands, as shown in Fig. 11. To support these results, the re-presentative values of qc1 and the measured unit loads corre-sponding to a settlement of 16 mm at Kuwait footing loadtests, included in Table 11, were considered. As shown inFig. 12, the predicted soil pressures closely match the mea-sured values. The predicted soil pressures obtained by theSchmertmann’s method (Schmertmann 1970; Schmertmannet al. 1978) for the Kuwait tests have also been included inFig. 12. As noted, this method tends to slightly underesti-mate the observed footing unit loads.

Table 9. Mean values of index properties of Kuwait sand.

D60 (mm) D50 (mm) D10 (mm) Particle shape0.33 0.31 0.08 Subroundeda

aAssumed particle shape.

Table 10. Reported geotechnical properties for test sites at Kuwait.

Test siteDepth of watertable (m) gd /gw

Moisture content(%) N-value qc/pa

KISR 2.8 1.577 3.0 20 94.0Sabah Hospital 10 1.588 2.6 25 118.0Ahmadi — 1.429 3.5 15 71.0Abra Kheitan — 1.485 2.4 12 56.0Rumaithiyah 2.6 1.553 4.9 10 47.0Shuwaikh industrial area 2.0 1.406 6.3 10 47.0

Table 11. Normalized SPT and CPT resistances and unit load measured at 16 mm settlement forfootings tested at Kuwait.

Test site B (m) Df (m) N60 (N1)60 qc1/pa Unit load/pa

KISR 1.00 0.5 15.0 25.5 159.8 4.1KISR 1.00 1.0 15.0 25.5 159.8 5.0KISR 0.75 1.0 15.0 25.5 159.8 5.9KISR 0.50 1.0 15.0 25.5 159.8 7.7Sabah Hospital 0.50 1.0 18.7 31.9 200.6 9.4Ahmadi 0.50 1.0 11.2 19.1 120.7 6.9Abra Kheitan 0.50 1.0 9.0 15.3 95.2 4.8Rumaithiyah 0.50 1.0 7.5 12.8 79.9 5.6Shuwaikh industrial area 0.50 1.0 7.5 12.8 79.9 5.3

Note: qc1, normalized cone tip resistance.

Fig. 10. SPT-based prediction and measured unit load at 16 mmsettlement for nine square footings tested at Kuwait.



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Footings on overconsolidated sandsThe coefficient of lateral earth pressure must be estimated

previously to analyze the behavior of footings on OC sands.Available experimental evidence has shown that K0 in-creases with an increase in OCR. Several empirical correla-tions have been proposed to predict K0 as a function ofOCR. The well-known K0–OCR relationship proposed byMayne and Kulhawy (1982) was used herein. This relation-ship is expressed as

½23� K0 ¼ ð1� sinfÞOCR sinf

where f is the peak friction angle obtained from triaxialcompression tests.

Values of OCR corresponding to K0 = 1.0 were computedfrom eq. [23] for typical friction angles for dry sands. As shownin Table 12, OCR varied between 4.1 and 5.7, with an averageOCR equal to *5. This average OCR may be regarded as a me-dium degree of overconsolidation (Kulhawy and Mayne 1991).

By using a mean value of K0 = 1.0 in the present study for sim-plicity, design charts were developed for the net allowable soilpressure for footings supported on the study sands. The FE re-sults obtained were plotted in Figs. 13a and 13b for square andstrip footings, respectively, in terms of the relative-density factor,FDr. These charts were developed considering both dry and satu-rated sands. In the case of footings on saturated sands, the sub-merged unit weight was used instead of the unit weight, and itwas assumed that loading was applied under drainage conditions.

From these charts it can be seen, as expected, that for anygiven value of the relative-density factor, the net allowablesoil pressures for footings on OC sands are higher than thosepredicted for identical footings on NC sands. The overcon-solidation effect produces an increase of 13%–32% over thepredicted soil pressure for footings on NC sands.

It was also found that the predicted net allowable soilpressure for a given square footing on OC sands may varybetween 1.30 and 1.78 times the predicted value for a stripfooting with the same width.

By considering the FE results used for developingFigs. 13a and 13b and assuming CA = 1.25 and Qc = 1.0 ineqs. [16] and [20], design charts were developed for the netallowable soil pressure for square and strip footings in termsof results of SPT and CPT, as shown in Figs. 14 and 15, re-spectively. In these figures, the standard penetration andcone resistance factors for OC sands with K0 = 1.0 and aver-age OCR = 5 are given by

½24� FSPT-OC ¼ðN1Þ60

1:34 Cp

� �0:5g

gw

� �0:5

� Ps logð500ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD10D60

pÞ s

B

� �0:3

1þ 0:6Df

B

� �0:2

½25� FCPT-OC ¼0:75qc1

pa

� �0:5g

gw

� �0:5

� Ps logð500ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD10D60

pÞ s

B

� �0:3

1þ 0:6Df

B

� �0:2

Four footing load tests performed at Texas A&M Univer-sity were examined (Briaud and Gibbens 1997). The soil is amedium-dense fine silty silica sand and the water table wasobserved at a depth of 4.9 m. Mean values of index proper-ties of this sand are presented in Table 13. According toseveral authors, the sand is apparently OC (Altaee and Fell-enius1994; Deschamps and Ludlow 1994; Mayne 1994).

Energy measurements conducted during the execution of thestandard penetration tests led to an average energy efficiencyof 53%. Consequently, corrections for energy (CE = 0.88) andfor overburden pressure were used to obtain the (N1)60 values.The sizes and embedment depths of the tested square footings,the average values of (N1)60, qc1 between the level of the baseof the footing and a depth B below that level, and the measuredfooting unit loads corresponding to a settlement of 16 mm areillustrated in Table 14. It can be noted that the normalized cone

Fig. 12. CPT-based prediction and measured unit load at 16 mmsettlement for nine square footings tested at Kuwait.

Table 12. Values of OCRcorresponding to K0 = 1 fordifferent values of f (eq. [23]).

f (8) OCR31 4.133 4.235 4.437 4.639 4.841 5.143 5.445 5.7

Fig. 11. Net allowable soil pressure as a function of factor FCPT-NC

for square and strip footings.



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tip resistance related to the 1.5 m footing is apparently low andnot consistent with the rest of the available experimental data.As shown in Figs. 16a and 16b, the proposed chart for squarefootings on OC sands underestimates the measured soil pres-sures in a mean value of 24%, without considering the 1.5 mfooting in the CPT-based predictions (unfilled point inFig. 16b). These results suggest that the tested sand at TexasA&M University is overconsolidated with an average OCRsomewhat higher than 5, which is a conclusion that is consis-tent with the average OCR & 6 estimated by Mayne (1994).

ConclusionsTo limit the settlement of the largest footing to a value of

25 mm, Terzaghi et al. (1996) suggested a design criterionbased on restricting the end-of-construction settlement to16 mm because of the inherent variability of the real soil de-posits. This criterion, which was herein adopted, may also beregarded as an additional safety margin to compensate for theanticipated long-term settlement of buildings on sand.

Design charts based on the SH model were developed topredict the net allowable soil pressure for rigid footings onNC and OC sands subjected to static vertical loading. Thesupporting soils can be unsaturated or saturated sands. Thesecharts consider footing shape, embedment depth, grain di-ameters D10 and D60, particle shape, unit weight (or sub-merged unit weight for saturated sands), and indirectmeasurements of the shear strength derived from in situtests, such as relative density, SPT or CPT.

As shown, the proposed design charts match well with avail-able experimental data. It is suggested to limit the Df/B ratio toa maximum value of 2.0 because the charts were developed onthe assumption of a very rough concrete–sand contact.

The SPT-based approach proposed by Burland andBurbidge (1985) gives acceptable predictions for theresponse of square footings on NC sands, but it tends tounderestimate the soil pressures.

The CPT-based procedure proposed by Schmertmann(1970) and Schmertmann et al. (1978) also gives acceptablepredictions for the response of square footings on NC sands,and it tends to slightly underestimate the soil pressures. Forroutine design, the proposed charts are easier to use andmore accurate than Schmertmann’s method (Schmertmann1970; Schmertmann et al. 1978).

It was found that the predicted net allowable soil pressurefor a given square footing on NC sands varies between 1.21and 1.66 times the predicted value for a strip footing withthe same width. Such a variation was found to be in therange 1.30–1.78 in the case of footings on OC sands.

Footings on OC sands with K0 = 1 show an increase inthe net allowable soil pressure ranging from 13% to 32%over the value predicted for identical footings on NC sands.

The analysis performed for OC sands, based on both SPTand CPT data, suggests that the tested sand at Texas A&MUniversity is overconsolidated with an average OCR some-what higher than 5.

Fig. 13. Net allowable soil pressure as a function of factor FDr for (a) square footings on OC sands and (b) strip footings on OC sands.

Fig. 14. Net allowable soil pressure as a function of factor FSPT-OC

for square and strip footings (K0 = 1.0, OCR = 5.0).

Fig. 15. Net allowable soil pressure as a function of factor FCPT-OC

for square and strip footings (K0 = 1.0, OCR = 5.0).



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AcknowledgmentsThe authors thank the Chilean National Council for Sci-

ence and Technology Research (CONICYT) and the Schoolof Engineering of the Pontificia Universidad Catolica deChile, Santiago, Chile, for the financial support given to thefirst author during the development of his doctoral thesis.

The authors also thank the anonymous reviewers for theiruseful comments. The license for using the program ABAQUSwas purchased from Dassault Systemes Simulia Corporation.

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Table 13. Mean values of index properties of Texas A&M sand (Briaud and Gibbens1997).

gd/gw

Moisturecontent (%)

D60

(mm)D50

(mm)D10

(mm)Fine content(%) Particle shape

1.472 5 0.25 0.22 0.11 11 Subroundeda

aAssumed particle shape.

Fig. 16. Predicted and measured unit load at 16 mm settlement forsquare footings tested at Texas A&M: (a) SPT-based predictionand (b) CPT-based prediction.

Table 14. Normalized SPT and CPT resistances and unitload measured at 16 mm settlement for footings tested atTexas A&M University (Briaud and Gibbens 1997).

B (m) Df (m) (N1)60 qc1/pa Unit load/pa

1.0 0.71 27.77 145.88 7.001.5 0.76 25.24 89.03 5.402.5 0.76 25.73 121.96 4.603.0a 0.76 27.20 162.03 4.50

Note: SPT and CPT resistances shown in this table were ob-tained from the SPT-boring and CPT-sounding close to the loca-tion of each footing.

aNorth footing.



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Design Load of Rigid Footings on Sand

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