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ReliabilityanalysisofallowablepressureonshallowfoundationusingresponsesurfacemethodARTICLEinCOMPUTERSANDGEOTECHNICSMAY2007ImpactFactor:1.63DOI:10.1016/j.compgeo.2006.11.002

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G.L.SivakumarBabuIndianInstituteofScience159PUBLICATIONS571CITATIONS

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Dr.AmitSrivastavaInstituteofTechnologyandManagement61PUBLICATIONS124CITATIONS

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Availablefrom:Dr.AmitSrivastavaRetrievedon:21October2015

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rateand settlement of a shallow foundation resting on a cohesive frictional soil for a range of expected variation of input soil parameters. Theresponse surface models are developed using available conventional equations and numerical analysis. Considering the variations in the

calculated based on shear failure criterion (ultimate limit simple and straightforward, it does not consider dierent

For reliability analysis, a functional relationship (eitherimplicit or explicit) is required between input and outputrandom variables. Explicit relationships for reliability anal-ysis can be obtained from conventional bearing capacity

* Corresponding author. Tel.: +91 80 22933124; fax: +91 80 23600404.E-mail addresses: gls@civil.iisc.ernet.in (G.L. Sivakumar Babu),

amisri@civil.iisc.ernet.in (A. Srivastava).

Computers and Geotechnicsstate, ULS) and settlement criterion (serviceability limitstate, SLS). The allowable bearing capacity is obtainedby dividing the ultimate bearing capacity of the foundationsoil with a factor of safety. At the same time, it is alsoensured that the magnitude of settlement of footing shouldnot exceed a specied permissible limit from serviceabilityrequirements [1]. Design values of input soil parameterssuch as cohesion (c), angle of internal friction (/), unitweight (c), modulus of elasticity (E), for estimating theallowable pressure on foundation soil, are obtained eitherfrom eld tests or from laboratory tests. As soil is inher-ently variable, variations in measured values of those soil

sources of uncertainty in geotechnical design in a rationalmanner [35]. In order to incorporate these variations, reli-ability analysis is performed. In this approach, input soilparameters are treated as random variables and the inu-ence of these input random variables on the output randomvariable is studied. Reliability analysis approaches can beused in conjunction with conventional approaches to havebetter insight into the choice of allowable value of bearingpressure and helps in decision-making process. SivakumarBabu et al. [6] demonstrated this aspect with reference tothe estimation of allowable bearing pressure of a shallowfoundation resting on a typical sti clay.input soil parameters, reliability analysis is performed using these response surface models to obtain an acceptable value of the allowablebearing pressure. The results of the reliability analysis are compared with the results of Monte Carlo simulation and it is demonstratedthat application of response surface method in the probabilistic analysis can considerably reduce the computational eorts and memoryrequirements. It is also concluded that conventional analysis using available equations and numerical analysis when used in conjunctionwith reliability analysis enable a rational choice of allowable pressure and help in decision-making process. 2006 Elsevier Ltd. All rights reserved.

Keywords: Conventional analysis; Reliability analysis; Response surface method; Monte Carlo simulation; Soil parameters; Correlation

1. Introduction

In the conventional design of shallow foundations rest-ing on a cohesive frictional soil, the allowable pressure is

parameters is inevitable. Phoon and Kulhawy [2] indicatedthat inherent variability of soil deposits, testing errors andmodel transformation uncertainties contribute to variabil-ity in the test data. Although factor of safety approach isTechn

Reliability analysis of allowableusing response

G.L. Sivakumar Ba

Department of Civil Engineering, Indian Institut

Received 30 June 2006; receivedAvailable onlin

Abstract

The concept of response surface method (RSM) is used to gene0266-352X/$ - see front matter 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.compgeo.2006.11.002l note

ressure on shallow foundationurface method

*, Amit Srivastava

Science, Bangalore, Karnataka 560 012, India

evised form 10 November 2006January 2007

approximate polynomial functions for ultimate bearing capacity

www.elsevier.com/locate/compgeo

34 (2007) 187194

because calculation of derivatives for variances by rst or

mpuand settlement equations. These relationships are implicitin numerical analysis and tools such as random eld niteelement analysis (RFEM), response surface methodology(RSM) are used to analyze the response variability. Vari-ous researchers [7,8] used random eld nite element mod-elling (RFEM) to study the bearing capacity of shallowfoundations. Wong [9] performed reliability analysis of soilslopes using response surface method (RSM). Humphreysand Armstrong [10] analyzed a slope stability problemusing results of nite dierence method and regressionanalysis. Tandjiria et al. [11] used response surface methodfor reliability analysis of laterally loaded piles.

A question that often arises in practice is to know howsafe is safe? or to what extent factors of safety that are rou-tinely used in conventional solutions address the question ofsafety adequately. It is possible to examine a degree ofsafety associated with analytical formulations by compar-ing reliability indices from conventional solutions withthose from analysis of response surfaces constructed fromconventional solution as well as from the results of numer-ical analysis. As expected, a comparative study of the resultsof the analysis from conventional solution and numericalanalysis in terms of reliability indices demonstrates theeectiveness of conventional, numerical and probabilisticapproaches in the assessment of allowable pressure on foun-dation soil and this is the main focus of the present study.

2. Conventional approach

In a conventional approach, allowable pressure on ashallow footing is taken as lesser of the two values, i.e.(i) allowable bearing capacity based on shear failure crite-rion and (ii) allowable bearing pressure based on settlementcriterion.

(i) Shear failure criteriaFor a shallow strip foundation of width B resting ona horizontal ground at a depth Df and loaded with aconcentric vertical loading the ultimate bearingcapacity (qu) can be calculated using Meyerhofequation

qu cNcscdc cDfNqsqdq 0:5cBN cscdc 1The values of bearing capacity factors Nc, Nq and Ncas well as shape factors (sc, sq, sc) and depth factors(dc,dq,dc) are obtained from available equations pro-vided in literature [1]. Allowable bearing capacity (qa)is obtained after applying a factor of safety (FS) tothe ultimate bearing capacity (qu) and usually a valuebetween 2.5 and 3 is considered appropriate in thecase of a shallow foundation.

(ii) Settlement criteriaFor an applied pressure (q0), the magnitude of settle-ment of a footing can be computed using Eq. (2) andthis is compared with the allowable settlement speci-

188 G.L. Sivakumar Babu, A. Srivastava / Coed in design guidelines/codes to satisfy serviceabilityrequirementssecond order methods is cumbersome.Using Monte Carlo simulation technique, probability of

failure can be computed for both explicit and implicit limitstate functions. This method involves the generation of Nrandom numbers of input soil parameters with given prob-abilistic characteristics. These N sample points for outputresponse are used to obtain required sample statistics,which is incorporated in probabilistic calculations. Theminimum value of number N depends on percentage (%)acceptable error (e) in the estimation of sample mean andvariance as well as condence level dened in terms ofparameter a [13]. As the performance function is denedS q0B1 m2

EIf 2

where B is the width of footing, q0 is an applied pres-sure on the footing, If is an inuence factor, m is poi-sons ratio and E is modulus of elasticity of soil.

As per Eurocode 7 [12], total settlements up to 50 mm areoften acceptable for normal structures with isolated foun-dations. As per Indian codes of practice, 40 mm is consid-ered acceptable and hence in the present study, theallowable settlement is taken as 40 mm.

3. Reliability analysis

Reliability is dened as probability of safety of a systemin a given environment and loading conditions and isassessed in terms of reliability index (b) values. Normally,a reliability index value in the range of 3.04.0 is acceptedfor good performance of the system [13,14]. For the estima-tion of reliability index, methods such as rst order reliabil-ity method (FORM), second order reliability method(SORM), point estimate method (PEM), Monte Carlo sim-ulation (MCS) are available in literature. For a linear per-formance function dened as g() = C D, where C is thecapacity (either ultimate bearing capacity or allowablesettlement) and D is the demand (either applied pressureor calculated settlement), g() > 0 denotes safety whileg() < 0 represents unsafe condition. For uncorrelated nor-mally distributed C and D, reliability index (b), represent-ing the shortest distance of the performance functionfrom the origin of reduced coordinate system of variables,can be calculated using Eq. (3) [13].

b C Dr2C r2D

p 3

where rC and rD are standard deviations of capacity (C)and demand (D).

For non-linear performance functions, iterative proce-dure is used to evaluate reliability index.

Reliability analysis using conventional equations forultimate bearing capacity and settlement is complicated

ters and Geotechnics 34 (2007) 187194as g() = C D, capacity (C) is deducted from demand(D) for each realization of output response. If Nf is the

the response surface model is presented in terms of natural

mpuvariables.To examine the adequacy of the tted model and to

ensure that it provides a good approximation of the truesystem, a normal probability plot should be approximatelyalong a straight line. In addition, computed values of coef-cients of multiple determinations (R2) and adjusted R2

also give information on the adequacy of the tted model.The non-dimensional quantity R2 is calculated as (Eq. (7)).

2

Pni1 y

^i y

2number of simulation cycles when g() is less than zero andN is the total number of simulation cycles, probability offailure can be obtained as pf NfN which is approximatelyrelated to reliability index as b = /1(1 pf).

On the other hand, response surface method (RSM)facilitates the formulation of explicit linear or non-linearrelationships between input random variables and requiredoutput response through regression analysis and this aspectis advantageously used in the present study for reliabilityanalysis.

4. Response surface methodology (RSM)

Response surface method involves generation of polyno-mial equation using regression analysis and an approxi-mate linear or non-linear functional relationship betweendependent output y and input variables (x1,x2,x3, . . .) isestablished (Eq. (4)).

y f x1; x2; x3; . . . e 4The following Eq. (5) shows multiple linear regressionmodel having two input variables without interaction termsand Eq. (6) shows second order regression model withinteraction terms containing two input variables x1, x2.

y a0 a1x1 a2x2 e 5y a0 a1x1 a2x2 a11x21 a22x22 a12x1x2 e 6where a0,a1,a2, . . . are regression coecients and e repre-sents error involved in neglecting other sources ofuncertainties.

In RSM analysis, single replicate 2n factorial design isused to t rst order linear regression model. For this, totalnumber of sample points is required as 2n, where n is totalnumber of input variables. For n = 2 and n = 3, number ofsuch sample points are 4 and 8, respectively.

In the analysis, natural variables (x1,x2,x3, . . . ,xk) areconverted into coded variables ni using a relationshipni ximaxximinxi=2maxximinxi=2 . The maximum and minimum valuesof xi are taken as xmax = x + hr and xmin = x hr, whereh is an integer and ris are the standard deviations of inputvariables (xis). These limits xmax and xmin cover the extentof variation of input parameters in a given situation withsome level of condence interval discussed later. Finally,

G.L. Sivakumar Babu, A. Srivastava / CoR Pni1 yi y

2 7where y, yi and y

^are estimated mean value, actual and pre-

dicted values of output response (y) respectively. The valueof R2 lies between 0 and 1 and a value close to 1 indicatesthat most of the variability in y is explained by regressionmodel. It should be noted that it is always possible to in-crease the value of R2 by adding more regressor variables.Therefore, adjusted R2 value is calculated using followingEq. (8):

R2adj 1k 1k p 1 R

2 8

where k is total number of observations and p is number ofregression coecients. For a good model, values of R2 andadjusted R2 should be close to each other and also theyshould be close to 1. Myers and Montgomery [15] pre-sented more details on the use of response surfacemethodology.

5. Methodology

In the present case, the output response (i.e. the ultimatebearing capacity from shear failure consideration) isobtained using conventional Meyerhof equation corre-sponding to eight sample points for three input variables(c, / and c). A linear regression model is tted between out-put response (qu) and input variables (c, / and c) and thisapproximate linear relationship is used to obtain mean andvariance of qu for reliability analysis.

The same procedure is adopted to obtain response sur-face models for settlement of the footing with input ran-dom variables E and m and for dierent appliedpressures on the footing and reliability index values arecalculated.

The above results are also compared with the results ofreliability analysis from response surface models obtainedfrom numerical analysis. Using commercial nite dierencecode, fast Lagrangian analysis continua (FLAC) [16],response surface models corresponding to the above eightsample points are developed. For the analysis, physicaldomain of the problem is divided into nite dierence meshof quadrilateral elements. The unbalanced force of eachnode is normalized by gravitational force acting on thatnode. A simulation is considered to have converged whenthe normalized unbalanced force of every node in the meshis less than 103. In the analysis, elastic perfectly plasticmodel based on MohrCoulomb failure criteria with asso-ciated ow rule is used for the foundation soil.

After obtaining response surface models for ultimatebearing capacity and settlement from conventionalapproach and numerical analysis, reliability analysis isperformed.

The probability of failure is also obtained using MonteCarlo simulations. For an acceptable error of 5.0% anda = 0.05 (95% condence level), minimum sample size is1536 following the procedure explained in the literature

ters and Geotechnics 34 (2007) 187194 189[13,17]. In the present analysis, 20,000 random numbersare generated for simulation.

6. Results and discussion

A shallow strip foundation of width (B) 1.5 m at adepth (Df) of 1.0 m, resting on a cohesive frictional soilis considered. The values of input soil properties forconventional and reliability analysis are presented inTable 1. Using conventional approach, for mean valuesof input soil properties, the mean value of ultimate

respectively. Hence, the response surface modeladequately represents the response variable (ultimatebearing capacity) for the range of variation of input soilparameters assumed in the analysis. The mean (lC) andstandard deviation (rC) values of capacity (i.e. qu) ob-tained are 438.54 kPa and 76.43 kPa, respectively. For dif-ferent applied pressures, reliability index (b) values arecalculated.

ion (r) xmax xmin CoV (%) Distribution

6 4 12 Normal27.5 22.5 6 Normal18.7 15.3 6 Normal

Table 2Single replicate 2n design showing eight combinations of input variablesand corresponding output (qu) obtained from Meyerhof equation

Sr. no. Combination Natural variables qu

c / c

1 + + + 6 27.5 18.7 628.522 + + 6 27.5 15.3 547.183 + + 6 22.5 18.7 361.82

44.5

55.5

622

.5 23.7

5 25

26.2

5 27.5

0

100

200

300

400

500

600

qu

c

500-600

400-500

300-400

Fig. 1a. Actual response surface for the ultimate bearing capacity of thefoundation using Meyerhof equation.

190 G.L. Sivakumar Babu, A. Srivastava / Computers and Geotechnics 34 (2007) 187194bearing capacity from shear failure criteria is obtainedas 419 kPa. Using a factor of safety of 3.0, the allowablebearing capacity of foundation soil is 139.67 140 kPa.From serviceability requirements, for an appliedpressure of 140 kPa, the settlement of foundation is18.78 mm, which is less than the allowable settlementof 40 mm.

Becker [18] and Orr [19] suggest that selection of char-acteristic values of geotechnical parameters and corre-sponding condence intervals should be incorporated inreliability based designs. Response surfaces are generatedcorresponding to 95% condence intervals, i.e. the lowerlimit and upper limit values related to mean (l) and stan-dard deviation (r) with the relationships xu = l + 1.65rand xl = l 1.65r are used in the analysis. The upperlimit (xu or xmax) and lower limit (xl or xmin) values indi-cated in Table 1 are based on the assumption that inputsoil parameters follow normal distribution and upperand lower limit values have probabilities of 5% and 95%being exceeded.

Table 2 shows a single replicate 2n factorial design forconstructing response surfaces for ultimate bearing capac-ity using RSM. Using the procedure described earlier, theresponse surface equation for ultimate bearing capacity(qu = C, capacity) is obtained as

C 1191:04 25:53c 47:64/ 18:29c 9Fig. 1a shows plots of actual response surface for ultimatebearing capacity using Meyerhof equation and Fig. 1bshows contours of ultimate bearing capacity calculatedfrom conventional (Meyerhof) equation and approxi-mated linear response surface equation. It can be seenthat the results of response surface model are quite com-parable with those obtained from Meyerhof solution andalso the linear approximation of ultimate bearing capacityof foundation soil for the range variation of input param-eters is valid. Fig. 2 shows normal probability plot for theregressed model, which is approximately a straight line.The values of R2 and adjusted R2 are 0.99 and 0.98,

Table 1Input soil properties used in the analysis

Input soil properties (x) Mean (l) Standard deviat

Cohesion (c), kPa 5 0.61 kPaFriction angle (/) 25 1.52Unit weight (c), kN/m3 17 1.03 kN/m3Modulus of elasticity (E), kPa 10,000 1212 kPaPoisons ratio m 0.325 0.0454 + 6 22.5 15.3 318.805 + + 4 27.5 18.7 568.136 + 4 27.5 15.3 486.807 + 4 22.5 18.7 320.108 4 22.5 15.3 277.1012,000 8000 12 Normal0.40 0.25 14 Normal

G.L. Sivakumar Babu, A. Srivastava / CompuFollowing the same procedure, response surface equa-tion for settlement (S) of the footing for an applied pres-sure of 200 kPa is

S 62:15 0:0028E 20:31m 10Fig. 3 shows contour plot of settlement of footing obtainedfrom approximated linear response surface model and con-ventional equation. The values of R2 and adjusted R2 are0.99 each; and are close to 1. These results indicate the ade-quacy of the tted model for settlement prediction. Simi-larly, response surface models are developed for dierent

Fig. 1b. Contour plot for ultimate bearing capacity obtained from linearresponse surface model and Meyerhof equation.applied pressures on the footing as indicated in Table 3.The table also shows mean and standard deviation of

Normal Probability Plot

0

100

200

300

400

500

600

700

0 20 40 60 80 100Sample Percentile

q u

Fig. 2. Normal probability plot for the ultimate bearing capacity (qu).demand (D, i.e. calculated settlement) and correspondingreliability index values obtained for the allowable settle-ment (capacity C) of 40 mm.

6.1. Results from numerical analysis

Numerical simulations are performed using FLAC andthe response surface models for ultimate bearing capacityand settlement are obtained. Fig. 4 shows load displace-ment curves for mean values as well as for eight combina-tions (sample points) of input soil parameters. These curvesare used to obtain response surface models for ultimatebearing capacity and settlement predictions.

For the mean value of input soil parameters, usingnumerical analysis, the mean ultimate bearing capacity offoundation soil is obtained as 408 kPa. This value is quiteclose to the value 419 kPa obtained from the conventionalapproach. This shows the adequacy of the numerical modelused in the analysis. Using regression analysis, responsesurface model for the ultimate bearing capacity of founda-tion soil is obtained as

qu 975 12:5c 37:0/ 19:18c 11The approximately straight normal probability plot (notshown) and the calculated values of R2 and R2adj 0.992and 0.986, respectively conrm the adequacy of the ttedmodel. Using Eq. (11), the values of mean and standarddeviation of qu are calculated as 338 kPa and 59.64 kPaand for dierent applied pressures reliability index (b) val-ues are calculated using Eq. (3). The same procedure isadopted for reliability index calculation from settlementconsiderations. The results of the reliability analysis usingresponse surface method, from conventional approachand numerical analysis are compared with the correspond-ing reliability index values obtained from Monte Carlosimulation. The results of the analysis are summarized inFig. 5.

It can be seen from Fig. 5 that the allowable pressure onthe footing varies from 140 kPa to 160 kPa in order toachieve reliability index (b) value in the range of 34(shown in the shaded region). The shaded region alsoshows that all four dierent approaches provide reliabilityindex values in the range of 3.04.0, if applied pressureranges between 140 kPa and 160 kPa. It was found thatthe calculated values of reliability indices from settlementconsideration and from all four approaches (RSM andMCS using conventional solution and numerical analysis),are well above the minimum acceptable value of b (=3.0)for the range of applied pressure 140160 kPa. The resultsalso indicate that the conventional procedure or analyticalsolution for ultimate bearing capacity, with a factor ofsafety of 3.0 is adequate in the present case as it is associ-ated with required reliability index value.

It can also be noted that the results obtained from RSMare comparable with the values obtained from Monte

ters and Geotechnics 34 (2007) 187194 191Carlo simulations. The advantage of RSM is that itrequires fewer number of simulations (2n and for n = 3,

0100

200

300

400

500

600

0 100 200 300 400 500 600

Applied pressure (kPa)

settl

emen

t (mm)

For 8 combination runsmean soil properties

Fig. 4. Load displacement curve for mean values and eight combinationsof input soil parameters obtained through numerical analysis of the soilfoundation system.

Table 3Response surface models obtained from conventional settlement equationfor dierent applied pressure on the footing and corresponding reliabilityindices (b) from settlement criterion

Appliedpressure(kPa)

Response surfaceequation

Mean(D)

Std.dev.(lD)

b

140 43.500.0019E14.22m 19.88 2.37 Allow.settlement(40 mm)

8.50160 49.720.0022E16.25m 22.44 2.76 6.35180 55.930.0025E18.28m 24.99 3.14 4.78200 62.150.0025E20.31m 27.55 3.51 2.87

Fig. 3. Contour plot for settlement of the footing obta

192 G.L. Sivakumar Babu, A. Srivastava / Computers and Geotechnics 34 (2007) 187194the number of simulations is 8) as compared to 20,000Monte Carlo simulations. Hence, a distinct advantage ofRSM is in terms of computational time and memoryrequirements, notwithstanding the availability of high-endcomputers and simulation procedures.

6.2. Eect of correlation among input variables

In order to study the eect of correlation among inputsoil parameters on the reliability index values, negative

ined from RSM model and conventional equation.

Table 4Correlation coecient between input soil parameters

Correlation coecient (q) c/ cc /c

Case 1 0.25 +0.25 +0.25Case 2 0.50 +0.50 +0.50Case 3 0.75 +0.75 +0.75

0

1

2

3

4

5

100 120 140 160 180 200Applied pressure (kPa)

RSM - Meyerhof EquationRSM - Numerical analysis

MCS - Meyerhof Equation

MCS - Numerical analysis

Fig. 5. Results of the reliability analysis using RSM and MCS from shearfailure criterion.

mpucorrelation between cohesion (c) and friction angle (/)varying from (0.25 to 0.75) and positive correlationbetween bulk density (c) and shear strength parameters,viz, cohesion (c) and friction angle (/) varying from 0.25to 0.75 are considered in accordance with the reported val-ues in literature [20]. The values of correlation coecientsused in this study are presented in Table 4.

From shear failure consideration, the response surfacemodels developed from Meyerhof equation (Eq. (10))and numerical analysis (Eq. (12)) are used to calculatereliability index values for dierent applied pressures.The results of the analysis are summarized in Table 5.It can be noted that the consideration of correlationbetween input soil parameters marginally aects the reli-ability index values.

6.3. Consideration of non-normal distributions in response

surface methodology

In a further study, it is assumed that input soil parame-ters are uncorrelated log-normally distributed. For thedevelopment of response surface model (RSM) for uncor-related log-normally distributed input soil parameters andcalculation of reliability index values, the following stepsare used:

Table 5Reliability index values from shear failure consideration (using responsesurface model developed from Meyerhof equation)

Appliedpressure(kPa)

Reliability index (b)

Uncorrelated normallydistributed inputsoil parameters

Correlated normally distributed inputsoil parameters

Case 1 Case 2 Case 3

120 3.94 4.12 4.08 4.03140 3.70 3.86 3.82 3.78160 3.45 3.57 3.57 3.53180 3.20 3.31 3.31 3.27200 2.95 3.05 3.05 3.02

G.L. Sivakumar Babu, A. Srivastava / Co(i) Log-normally distributed input soil parameters aretransformed into equivalent normal variable usingRackwitz and Fiessler procedure [21] described in [17].

(ii) Using the parameters of equivalent normal transfor-mation (leq, req) of input soil parameters, samplepoints are selected (leq kreq). (In the present casek = 1.65.)

(iii) Response surface model is generated using regressionanalysis and least square error approach and the ade-quacy of the tted model is checked as describedearlier.

(iv) The response surface model for ultimate bearingcapacity of the foundation soil is developed for eightcombination of sample points and correspondingoutput (i.e. ultimate bearing capacity) as

C 1164:81 25:01c 46:63/ 17:43c 12Reliability index values for dierent applied pressures arecalculated using Eq. (3) and also using FORM.

Table 6 shows the reliability index values calculatedusing Eq. (3). It can be noted that there is large dierencebetween the two reliability index values computed, basedon the assumption of normal distribution and log-normaldistribution of input random variables for lower appliedpressures but this dierence reduces for higher loads. Itcan also be noted that the allowable pressure on the footingwould be higher if the input soil parameters are log-nor-mally distributed provided it also satises the serviceabilityrequirements. For comparison, reliability index is also cal-culated using iterative procedure described in literature[17]. As the performance function is linear, only two itera-tions were required for convergence.

7. Concluding remarks

The work presented in this paper demonstrates the appli-cability of response surface methodology in the assessmentof allowable bearing pressure on shallow foundation restingon cohesive frictional soil. The results show that reliabilityanalysis results from conventional approaches and numeri-cal analysis help in deciding the range of allowable pressureson the footing. Although the reliability based approach isnot a substitute for the conventional approach, it can benoted from the present study that the choice of allowablepressure on the footing can be made rationally. The use of

Table 6Reliability index values from shear failure consideration using RSMapproach

Applied pressure (kPa) (1) (2) (3)

120 3.64 6.94 6.38140 3.30 6.16 5.64160 2.97 5.38 4.90180 2.64 4.60 4.19200 2.31 3.81 3.38

(1) From MV FOSM, variables follow normal distribution.(2) From MV FOSM, variables follow log-normal distribution.(3) Iterative procedure, variables follow log- normal distribution.

ters and Geotechnics 34 (2007) 187194 193response surface method in establishing an approximatefunctional relationship between input soil properties andoutput response, viz. ultimate bearing capacity or settlementof the foundation, reduces computational eorts to a greatextent for the probabilistic analysis of the foundation.

Acknowledgments

The authors thank the reviewers for their critical com-ments which have been very useful in improving the workpresented in this paper.

References

[1] Bowles JE. Foundation analysis and design. 5th ed. McGraw-HillBook Company; 1996.

[2] Phoon KK, Kulhawy FH. Characterization of geotechnical variabil-ity. Canad Geotech J 1999;36:61224.

[3] Whitman RV. Organizing and evaluating uncertainty in geotechnicalengineering. Geotech Geol Eng 2000;126(7):58393.

[4] Duncan JM. Factors of safety and reliability in geotechnicalengineering. J Geotech Geoenviron Eng 2000;126(4):30716.

[5] Schweiger HF, Thurner R, Pottler R. Reliability analysis in geotech-nics with deterministic nite elements. Int J Geomech ASCE2001;1(4):389413.

[6] Sivakumar Babu, Amit Srivatsava, Murthy DSN. Reliability analysisof bearing capacity of a shallow foundation resting on cohesive soil.Canad Geotech J 2006;43:17.

[7] Griths DV, Fenton GA. Seepage beneath water retaining struc-tures founded on spatially random soil. Geotechnique 1993;43(4):57787.

[8] Paice GM, Griths DV, Fenton GA. Inuence of spatially randomsoil stiness on foundation settlements. In: Yeung AT, Felio GY,editors. ASCE specialty conference Settlement94. Texas A&MUniversity; 1994. p. 62839.

[9] Wong FS. Slope stability and response surface method. J GeotechGeoenviron Eng ASCE 1985;111(1):3253.

[10] Humphreys MP, Armstrong LW. Assessing the sensitivity of numer-ical models using response surface methodology. In: Proc of the confof probabilistic methods in geotechnical engineering, Canberra,Australia, 1993. p. 13544.

[11] Tandjiria VI, Teh CI, Low BK. Reliability analysis of laterally loadedpiles using response surface methods. Struct Safety 2000;22:33555.

[12] Eurocode 7 Geotechnical Design Part 1: General rules, EN 1997-1:2004(E).

[13] Baecher GB, Christian JT. Reliability and statistics in geotechnicalengineering. Chichester, NJ: John Wiley Publications; 2003.

[14] USACE. Risk-based analysis in Geotechnical Engineering for Supportof Planning Studies, Engineering and Design. US Army Corps ofEngineers,Department ofArmy,Washington,DC, 1997. p. 20314-100.

[15] Myers RH, Montgomery D. Response surface methodology. JohnWiley & Sons; 2002.

[16] FLAC 4.0. Reference Manual. Itasca Consulting Group Inc.,Minneapolis, 2000.

[17] Haldar A, Mahadevan S. Probability, reliability and statisticalmethods in engineering design. NY: John Wiley & Sons; 2000.

[18] Becker DE. Limit state design for foundations: Part I. An overview ofthe foundation design process. Eighteenth Canadian geotechnicalcolloquium. Canad Geotech J 1996;33:95683.

[19] Orr TLL. Selection of characteristic values and partial factors ingeotechnical designs to Eurocode 7. Comput Geotech 2000;26:26379.

[20] Chowdhury RN, Xu DW. Reliability index for slope stability assess-menttwomethods compared. Reliab Eng Syst Safety 1992;37:99108.

[21] Rackwitz R, Fiessler B. Note on discrete safety checking when usingnon-normal stochastic models for basic variables. Load projectworking session. Cambridge, MA: MIT; 1976. June.

194 G.L. Sivakumar Babu, A. Srivastava / Computers and Geotechnics 34 (2007) 187194