NOTE / NOTE

Reliability analysis of the bearing capacity of ashallow foundation resting on cohesive soil

G.L. Sivakumar Babu, Amit Srivastava, and D.S.N. Murthy

Abstract: In recent years, there have been considerable advances in the characterization of soil variability and its ap-plication in geotechnical designs. It is recognized that in using reliability-based design it is necessary to consider allsources of uncertainty in the analysis and incorporate them in the geotechnical design. It is also necessary to examinethe reliability-based approach in relation to deterministic approaches. In this study, cone tip resistance (qc) data ob-tained from a static cone penetration test on a stiff clay deposit are analyzed by using random field theory; and statisti-cal parameters, such as the mean, variance, and autocorrelation function, are estimated in evaluating the reliability ofthe allowable bearing capacity of a strip footing founded on the above deposit.

Key words: reliability-based design, soil variability, random field, variance, autocorrelation function, bearing capacity.

Résumé : Au cours des dernières années, on a accompli des progrès considérables dans la caractérisation de la variabi-lité des sols et dans son application aux conceptions géotechniques. On réalise que l’utilisation d’une conception baséesur la fiabilité (RBD) est nécessaire pour prendre en considération toutes les sources d’incertitude dans l’analyse et deles incorporer dans la conception en géotechnique. Il est aussi nécessaire d’examiner l’approche basée sur la fiabilitéen relation avec les approches déterministiques. Dans cet article, les données de résistance à la pointe du cône qc obte-nues par l’essai de pénétration au cône statique (SCPT) sur un dépôt d’argile raide sont analysées au moyen de lathéorie aléatoire de terrain; et les paramètres statiques tels que la médiane, la variance et la fonction d’autocorrélationsont estimés pour évaluer la fiabilité de la capacité portante d’une semelle filante reposant sur le dépôt ci-haut.

Mots clés : conception basée sur la fiabilité, variabilité des sols, champ aléatoire, variance, fonction d’autocorrélation,capacité portante.

[Traduit par la Rédaction] Sivakumar Babu et al. 223

Introduction

In geotechnical engineering, the bearing capacity of ashallow foundation can be evaluated by using a deterministicapproach in conjunction with a probabilistic approach. In thedeterministic approach, available equations and charts areused to assess the allowable bearing capacity of the shallowfoundation. Soil parameters required for the analyses are ob-tained from field and (or) laboratory tests. The number ofsoil samples for testing depends on several factors, such asthe importance of the structure, subsoil conditions, and eco-nomic considerations. The factor of safety used in the deter-ministic approach accounts for natural variability, statisticaluncertainty, measurement errors, and limitations of analyti-cal models and is an indirect way of limiting deformation; in

general, a factor of safety of 2.5–3.0 is adopted to accountfor this variability (Bowles 1996). Whitman (2000) andDuncan (2000) indicated that both approaches are comple-mentary and can be used to assess the safety of foundationsby taking into account different sources of uncertainty andspatial variation in soil properties in a rational way.

Quantification of uncertainty in soilproperties

There are three major sources of uncertainty associatedwith geotechnical engineering practice: natural heterogene-ity, measurement, and transformation uncertainty (Phoonand Kulhawy 1999a). Figure 1 shows typical variation in asoil property, characterized by the vertical scale of fluctua-tion, the trend, and the deviation from the trend, which con-stitute important parameters for site characterization andreliability-based design. Analysis of the sources of uncer-tainty in soil properties and its influence on design decisionsand implications have been studied extensively (Meyerhof1982; Phoon and Kulhawy 1999a, 1999b; Cherubini 2000).The use of the stochastic finite element method for bearingcapacity analysis has received considerable attention re-

Can. Geotech. J. 43: 217–223 (2006) doi:10.1139/T05-099 © 2006 NRC Canada

217

Received 24 June 2005. Accepted 26 September 2005.Published on the NRC Research Press Web site athttp://cgj.nrc.ca on 28 January 2006.

G.L. Sivakumar Babu,1 A. Srivastava, and D.S.N. Murthy.Department of Civil Engineering, Indian Institute of Science,Bangalore 560 012, India.

1Corresponding author (e-mail: gls@civil.iisc.ernet.in).

cently (Griffiths and Fenton 2001; Nobahar and Popescu2001; Fenton and Griffiths 2003). Griffiths and Fenton(2001) conducted probabilistic study on bearing capacity offootings using nonlinear finite element analysis, combinedwith random field theory and Monte Carlo simulations. Theyindicated that the mean bearing capacity of the footing on aspatially varying soil is always lower than the deterministicbearing capacity based on the mean value. They also indi-cated that a factor of safety of 3.0–4.0 reduces the probabil-ity of failure to negligible values for soils with a coefficientof variation (COV) for undrained shear strength up to 50%,although in the analysis they considered only inherent vari-ability of the soil properties. Nobahar and Popescu (2001)observed that the failure mechanism of the bearing capacityof shallow foundations is affected by natural variability ofthe foundation soil.

Spatial variability is an important factor affecting design.The spatial averaging of soil properties reduces its pointvariance. A variance reduction factor is derived in terms ofscale of fluctuation (δ); and averaging distance (L), the dis-tance over which the geotechnical properties are averaged.In the analysis, statistically homogeneous data are used toevaluate the experimental autocorrelation coefficient for dif-ferent lag distances, and then analytical expressions are fit-ted by regression analysis based on a least-squares errorapproach to obtain the theoretical autocorrelation function,ρx(r). An autocorrelation distance is defined as the distance(r0) within which the soil property exhibits a fairly strongcorrelation. The numeric value of r0 is taken to be the dis-tance at which ρx(r) decays to 1/e. In a physical sense, it isthe same as the scale of fluctuation, although the methodolo-gies for obtaining the scale of fluctuation (Vanmarcke 1977)and the autocorrelation distance are different. A numericalrelationship can be established between the scale of fluctua-tion (δ) and the autocorrelation distance (r0) for the type ofautocorrelation function used in the analysis (Vanmarcke1983).

The variability of soil property ui from point to point ismeasured by the standard deviation σi, and the standard de-viation of the spatially averaged property uL is given by σL.With an increase in the averaging distance, more fluctuationsin the soil property ui get canceled out, and subsequently thevariance in the soil property value is reduced in the processof spatial averaging. The σL/σi ratio was defined as the vari-ance reduction factor Γu(L) by Vanmarcke (1983):

[1] ΓuL

i

L( ) = σσ

Vanmarcke (1983) derived analytical expressions for dif-ferent theoretical fits to the experimental variance reductionfunction. For an exponential autocorrelation function, the ex-pression is given by

[2] Γ2 2 22 1( ) ( / ) [ /( / ) ]/( / )L L LLL= − + >−0.5 e 0.0δ δδ

δ

Γ2( )LL= =1.0 0.0δ

For any value of L/δ, the numerical value of the variancereduction function can be obtained by using eq. [2]. Thevalue of the standard deviation for spatially averaged soilproperties (σL) can be obtained by using eq. [1] and know-ing the values of the variance reduction factor Γ(L) and thestandard deviation for point properties (σi). The approximaterelationship between the variance reduction function interms of the averaging distance and the scale of fluctuationis as follows (Vanmarcke 1977):

[3] Γ2( )LL

L= >δδ

1.0

Γ2( )LL= ≤1.0 1.0δ

Vanmarcke (1983) modified the above equation as fol-lows:

[4] Γ2 14

( )LL L

L= −⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥ >δ δ

δ0.5

Γ2 5( ) .LL= ≤1.0 0δ

If the soil properties are perfectly correlated, theautocorrelation function can be represented by the straightline AB shown in Fig. 2. If a decaying autocorrelation func-tion in the form of an exponential function (AC) is consid-ered, the variance reduction function Γ2(L) can be defined interms of the ratio of the area under the autocorrelation func-

© 2006 NRC Canada

218 Can. Geotech. J. Vol. 43, 2006

Fig. 1. Statistical description of soil variability (Phoon andKulhawy 1999a).

Fig. 2. Derivation of theoretical variance reduction function forexponential autocorrelation function.

tion (AC) up to the averaging distance L to the area underAB for the same averaging distance.

The area under the curve AC is given by

A y xx

xL

= = −⎛⎝⎜

⎞⎠⎟∫ ∫d d

0

expδ

After transformation, –x/δ = t, and hence, dx = –δ dt. For x =0, t = 0; and for x = L, t = –L/δ. Hence,

A t tL

= −−

∫δδ

exp/

d0

= –δ (exp t)0–L/δ

= δ[1 – exp (–L/δ)]

The variance reduction function Γ2(L) = (δ/L)[1 – exp(–L/δ)]. Therefore,

[5] Γ LL

LL2 1= − − >δ δδ

[ exp( / )] 0

Γ LL2 = =1 0δ

From Fig. 3, it can be seen that with a decrease in thescale of fluctuation and an increase in the averaging dis-tance, the value of the variance reduction function is re-duced, which in turn reduces the variance in the spatiallyaveraged soil property value. In other words, the more er-ratic the variation (i.e., the less the correlation or the lowerthe scale of fluctuation) in the soil property value and thelarger the soil domain considered, the larger the reduction inthe variability of the average property value will be. Thisphenomenon is a result of the increasing likelihood that un-usually high property values at some points are balanced bylow values at other points (NRC 1995). Figure 3 shows a

comparison of different variance reduction functions. It canbe noted that the proposed variance reduction function(curve 4) shows the same trend as and is quite close toVanmarcke’s (1983) equations (curve 1).

The objective of this note is to model the spatial variabil-ity in soil properties by using the autocorrelation functionsdescribed above in the probabilistic analysis of the bearingcapacity of a shallow foundation. A simple case of a rigidshallow strip foundation of 1 m width, resting on a spatiallyvarying cohesive soil deposit at a depth of 0.5 m at the siteof a 300 MW power project in northern India, is used in theanalysis. The averaging distance (L) is taken as the zone ofinfluence beneath the footing. The in situ soil in general is astiff clay, and a few cone tip resistance (qc) profiles were ob-tained at different locations. The undrained shear strength(cu) values from cone tip resistance (qc) data obtained fromstatic cone penetration tests (SCPTs) are evaluated throughthe transformation model given by Phoon and Kulhawy(1999b):

[6]c

Dq

mq

k Dku

vo

c vo

vo

c vo

voσσ

σε σ

σ′= −

′= + −

′( )

( )( )

where cu is the undrained shear strength; qc is the correctedcone tip resistance; σvo′ and σvo are the effective and totaloverburden stresses, respectively; Dk is the uncertain modelterm; mDk

is the mean of Dk; and ε is the zero mean randomvariable representing transformation uncertainty. The abovecorrelation model can be expressed in the following form:

[7] c m t w eDku vo= + + + −( ) ( )ε σ

where t + w + e = qc includes all the three major sources ofuncertainty: that is, uncertainty associated with the deter-ministic trend function (t), inherent soil variability (w), andmeasurement error (e). Phoon and Kulhawy (1999b) alsogave expressions for obtaining the mean of and variance inundrained shear strength values from corrected cone tip re-

© 2006 NRC Canada

Sivakumar Babu et al. 219

Fig. 3. Comparison of variance reduction function developed in the present study (curve 4) with those proposed by Vanmarcke (1977,1983).

sistance data with, without, and with spatial averaging(eqs. [8], [9], and [10], respectively).

[8] mcu= mDk

(t – σvo)

[9] COVCOV COV

COVd

vo

ξ εσ

22 2

22

1

= +

−⎛⎝⎜

⎞⎠⎟

+( )w e

t

[10] COV(L)COV COV

COVa

va

ξ εσ

22 2 2

22

1

= +

−⎛⎝⎜

⎞⎠⎟

+[ ]Γ w e

t

where COVξd and COVξa are the coefficients of variation forthe design property considering point estimates and spatialaveraging, respectively; and σva is the average total overbur-den stress over the averaging distance (L). In the presentstudy, the above equations are used to obtain the probabilis-tic parameters (mean and variance) for undrained shear co-hesion (cu) with and without consideration of spatialvariability in the vertical direction. It is assumed that the av-erage undrained shear strength governs the design in the lon-gitudinal direction. Data from four cone tip resistance (qc)profiles were used in the analysis. The COV values for trans-formation model uncertainty (COVε) and measurement un-certainty (COVe) for cone tip resistance are taken as 0.29and 0.15, based on the reported values of Phoon andKulhawy (1999b). The COV for inherent variability (COVw)is obtained from the analysis of data in the present study.

Figure 4 shows a typical qc profile of the cohesive soil de-posit to >4 m depth. The profile indicates the values of bothcone penetration resistance and undrained shear strength.The soil profile shows that the soil in the upper portion isstiff. The following steps are followed in the analysis.(1) The vertical profile of qc data within the zone of influ-

ence is checked for the stationarity condition usingKendall’s τ test (Daniel 1990).

(2) A quadratic polynomial trend is removed from the datato make the data set stationary.

(3) The empirical autocorrelation function for the residualsoff the trend obtained from step (2) is evaluated. A typi-cal result for location SCPT-1 is shown in Fig. 5. Thedata points represent the experimental autocorrelationfunction.

(4) A theoretical best fit is obtained for the experimentalautocorrelation function by using regression analysisbased on a least-squares error approach. This best fit isshown as a solid line in Fig. 5. The autocorrelation dis-tance is evaluated from the parameters of the best fit.

(5) The scale of fluctuation (δ) for the qc profile is esti-mated from the autocorrelation distance (ro) obtainedfrom step (4). For an exponential autocorrelation func-tion, δ = 2ro (Vanmarcke 1983). With the use of the ap-propriate zone of influence and scale of fluctuation (δ),variance reduction factors are obtained. For the compar-ison of the results, eqs. [2] and [5] are used. It should benoted that the undrained cohesion (cu) is obtained bylinear transformation of the cone tip resistance (qc), andtherefore the correlation structure of cu remains thesame as that obtained for cone tip resistance (qc) data.

Reliability analysis

If it is assumed that resistance (R) and applied pressure(S) are two basic random variables, the performance func-tion or limit state of interest is defined as

[11] Z = g(R, S) = 0

For log-normally distributed R and S, the reliability index(β) is given by

[12] β

µµ

=

++

⎡

⎣⎢⎢

⎤

⎦⎥⎥

+ +

ln( )( )

ln[( )(

R

S

S

R

R

11

1 1

2

2

2

COVCOV

COV COV S2 )]

where µR and µS are the mean and COVR and COVS are theCOVs for R and S, respectively. The mean of R is taken asthe ultimate bearing capacity (qu), obtained from the Prandtlequation (qu = 5.14cu), and COVR is the same as the COVfor cu. Applied pressure on the footing (S) is taken as deter-ministic. The reliability indices for a series of applied pres-sures are evaluated by using eq. [12].

Results

The statistical parameters of the soil profile, such asmean, COV, and scale of fluctuation for cone tip resistancefor the site are evaluated with the use of qc data obtainedfrom SCPTs from four locations and presented in Table 1.The average of the correlation distances for the four loca-tions is approximately equal to the correlation distance forthe qc (SCPT-1) profile, and hence the parameters for this lo-cation are taken to represent the site characteristics. Themean, the COV due to inherent variability, and the scale offluctuation obtained for the qc (SCPT-1) profile are2.43 MPa, 0.36, and 1.08 m, respectively, and these are usedin the analysis. The mean value of the undrained shear

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220 Can. Geotech. J. Vol. 43, 2006

Fig. 4. Variation in cone tip resistance (qc) and undrained cohe-sion (cu) with depth.

strength (cu) and its COVudc (for point estimates), as well as

its COV uac (after spatial averaging), are calculated by usingeqs. [8], [9], and [10]. For an averaging distance of 1.0 m,the mean value of cu is found to be 120.73 kPa. The valuesof COV

udc and COV uac (determined by using eq. [2]) are es-timated to be 0.49 and 0.43, respectively. The COV for spa-tially averaged undrained cohesion (COV uac ), obtained byusing the proposed variance reduction function (eq. [5]), is0.44. These parameters are used in the estimation of mean Rand COVR for the ultimate bearing capacity (qu). The meanvalue is 620.55 kPa, and COVR is 0.43 (if eq. [2] is used) or0.44 (if eq. [5] is used). A reliability index value of 2.85 isobtained for an applied pressure of 150 kPa correspondingto an averaging distance of 1.0 m. In a similar way, if thezone of influence (or averaging distance) is considered to be

2.0 m, the reliability indices for different applied pressureson the footing can be calculated. For the same applied pres-sure of 150 kPa, the reliability index value obtained for anaveraging distance of 2.0 m is 2.83.

Table 2 provides a comparison of allowable bearing ca-pacity values obtained with the deterministic and probabilis-tic approaches. From the deterministic analysis, theallowable pressures on the footing corresponding to factor ofsafety of 3.0 on the ultimate bearing pressure (i.e., qu =5.14 × cu = 620 kPa) are 206 and 172 kPa, corresponding tozones of influence of 1 and 2 m, respectively. The corre-sponding values for a reliability index of 3.0 are 160 and140 kPa, respectively, indicating that the reliability-basedapproach leads to conservative values of allowable bearingcapacity in this particular case. This aspect is examined in

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Sivakumar Babu et al. 221

Fig. 5. Typical exponential fit to empirical autocorrelation function (SCPT-1).

qc (SCPT-1) qc (SCPT-2) qc (SCPT-3) qc (SCPT-4)

Equation of trend linea y = –0.030x2 + 1.600x y = –0.040x2 + 1.000x y = –0.013x2 + 0.598x y = 0.020x2 + 0.448xKendall’s τ

Raw data 0.66 0.44 0.55 0.67Detrended data –0.17 0.09 –0.10 –0.15

Exponential autocorrelation function ρu(∆z) = e–1.88|∆z| ρu(∆z) = e–1.29|∆z| ρu(∆z) = e–2.47|∆z| ρu(∆z) = e–2.29|∆z|

R2 = 0.94 R2 = 0.80 R2 = 0.84 R2 = 0.94Correlation distance (m) 0.54 0.78 0.41 0.44Scale of fluctuation (m) 1.08 1.56 0.82 0.88

ax, depth (m); y = qc (MN/m2).

Table 1. Trend line, autocorrelation function, and autocorrelation distance for four qc profiles.

Depth of zoneof influence (m)

Mean ofundrained shearstrength, mcu

COV for undrained shearstrength, COV

uac (%)

Allowable bearingcapacity correspondingto β = 3.0 (kPa)

Allowable bearing capacitycorresponding to factor ofsafety of 3.0 (kPa)

1 120.73 36 160 2062 100.59 40 140 172

Table 2. Allowable bearing capacity from both deterministic and probabilistic approaches for scale of fluctuation of 1.08 m.

detail, and a parametric study is conducted to evaluate theeffects of scale of fluctuation, averaging distance, and COVon the foundation performance in terms of the reliability in-dex. In the study, the scale of fluctuation of undrained cohe-sion (δcu

) is given various values (0.5, 1.0, 2.0, 3.0, and5.0 m), and similarly the averaging length (L) is taken as 1Band 2B from the base of the footing. Figure 6 shows thevariation in the reliability index with the COV for cu for dif-ferent applied pressures on the footing. The values of theCOV for cu indicated in the figure correspond to scales offluctuation varying from 0.5 to 5.0 m. From Fig. 6, it is ob-served that for a particular value of COV, the reliability in-dex decreases with increasing applied pressure. The samefigure also shows a decreasing trend with increasing COVfor cu. Figures 7 and 8 show the effect of the scale of fluctu-ation on the reliability index for various applied pressuresand averaging distances. It should be noted that for a partic-ular applied pressure on the footing, with an increase in thescale of fluctuation to 5.0 m from 0.5 m, the decrease in reli-

ability indices is not significant (<15%). Hence, the impor-tant factor in the spatial variability analysis is the ratio of theaveraging distance (L) to the scale of fluctuation (δ), insteadof the scale of fluctuation alone. Figures 9 and 10 presentthe reliability index values obtained for different appliedpressures when point variance is used and after spatial aver-aging. The figures also show the values of reliability indicesobtained when eqs. [2] and [5] are used as variance reduc-tion functions. It is clear from the figures that point varianceproduces lower reliability indices and hence underestimatesthe system performance. Figure 11 shows the variation in re-liability index with applied pressure for various averagingdistances when the variance reduction function given byeq. [5] is used. It is observed that for a particular appliedpressure, as the averaging distance is increased to 2 m from1 m, the corresponding reliability index decreases. This is

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222 Can. Geotech. J. Vol. 43, 2006

Fig. 6. Variation in reliability index with applied pressure fordifferent COVs for undrained cohesion.

Fig. 8. Effect of scale of fluctuation on reliability index at differ-ent applied pressures (averaging distance, 2.0 m).

Fig. 7. Effect of scale of fluctuation on reliability index at differ-ent applied pressures (averaging distance, 1.0 m).

Fig. 9. Comparison of reliability index for point variance andvariance of spatially averaged cone tip resistance (averaging dis-tance, 1.0 m).

attributed to the reduction in the mean value of undrainedcohesion (cu) corresponding to an averaging distance of 2 m,compared with that for an averaging distance of 1 m, whichis reflected in the SCPT results presented in Fig. 4.

Concluding remarks

For the case considered, the statistical parameters fromthe experimental cone tip resistance data (qc) obtained fromSCPTs are estimated and used in the assessment of reliabil-ity of a footing resting on the stiff clay deposit. It is ob-served that the smaller the scale of fluctuation and thegreater the depth of the zone of influence of cone tip resis-tance, the higher the reduction in its standard deviation willbe and hence the lower the probability of failure correspond-ing to any value of allowable pressure will be. A simple

variance reduction function is proposed and used in the pres-ent study. The reliability indices obtained from the proposedvariance reduction function compare well with that proposedby Vanmarcke (1983). It is observed that the mean and re-duced standard deviations of a spatially averaged soil prop-erty and the ratio of the averaging distance to the scale offluctuation provide rational estimates of the reliability of theallowable bearing capacity of a shallow foundation restingon the clay considered.

Acknowledgments

The authors thank Prof. S.R. Gandhi of IIT Madras, India,for providing us with the valuable data from the static conepenetration profiles, which were used in the analysis. Theauthors also thank the anonymous reviewers for their criticalcomments and suggestions.

References

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Cherubini, C. 2000. Reliability evaluation of shallow foundationbearing capacity on c′, φ′ soils. Canadian Geotechnical Journal,37: 264–269.

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Duncan, J.M. 2000. Factors of safety and reliability in geotechnicalengineering. Journal of Geotechnical and Geoenvironmental En-gineering, ASCE, 126(4): 307–316.

Fenton, G.A., and Griffiths, D.V. 2003. Bearing capacity predictionof spatially random c–φ soils. Canadian Geotechnical Journal,40(1): 54–65.

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Phoon, K.K., and Kulhawy, F.H. 1999a. Characterization ofgeotechnical variability. Canadian Geotechnical Journal, 36:612–624.

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Sivakumar Babu et al. 223

Fig. 10. Comparison of reliability index for point variance andvariance of spatially averaged cone tip resistance (averaging dis-tance, 2.0 m).

Fig. 11. Effect of averaging distance on reliability index at dif-ferent applied pressures.