Probabilistic-based assessment of the bearing capacity of shallow foundations

ORIGINAL PAPER

Probabilistic-based assessment of the bearing capacityof shallow foundations

Samer Barakat & Radhi Alzubaidi & Maher Omar

Received: 1 February 2014 /Accepted: 14 August 2014# Saudi Society for Geosciences 2014

Abstract In this paper, a probabilistic distribution for thebearing capacity and safety factor of shallow foundations isproposed to account for the variability and randomness of thesoil strength properties and applied loads. A probabilistic-based model is developed to assess the bearing capacity ofshallow foundations. A Monte Carlo simulation is performedto infer probabilistic descriptions of the bearing capacity ofshallow foundations. The effects of the variation in strengthand load random variables on the variation of the bearingcapacity and the safety factor are studied. The generalizedextreme value reduced to type II extreme distribution wasproved to be best suited in describing the variability in boththe bearing capacity of shallow foundations and the safetyfactor. The reliability index and the deterministic safety factorare compared. A risk-based safety factor for the ultimatebearing capacity of shallow foundations is proposed andassessed.

Keywords Bearing capacity . Probability distribution .

Probability of failure . Risk-based . Safety factor . Shallowfoundations

Introduction

The variability of bearing capacity of shallow foundations ismainly due to the heterogeneity of the soil media. Such

variability is not explicitly accounted for in the design. In-stead, large safety factors are usually assigned to account forthe variability of the soil strength properties. However, thisprocedure that is based on deterministic safety factors mayresult in uneconomical or even unreliabile designs. Due toinherent variability of the soil strength properties, geometricdimensions, and loads, these variables shall be modeled asrandom variables, and probabilistic methods shall be used.The work of Shahin and Cheung (2011), Massih et al. (2008),Cherubini (2000), Foye et al. (2006a, b), Fenton and Griffiths(2003), Griffiths et al. (2009), Basma (1994), Cherubini(1990, 1993), Williams (1989), and Harr (1987) amongothers, recognized that the problem of bearing capacity is besttreated in a probabilistic context. Therefore, this paper isattempting to derive a probabilistic destribution that representsthe variability and randomness of the bearing capacity ofshallow foundations as a function of the variability of soilstrength and load properties. The objectives of this study are:(1) to develop a probabilistic-based model to assess the ulti-mate bearing capacity of shallow foundations and the safetyfactor that are associated with it; (2) to demonstrate how thevariabilities of the bearing capacity and the safety factor areaffected by the variability of the soil strength and load prop-erties; and (3) to propose a modified and improved equationfor the safety factor, which is calibrated by probabilisticanalysis.

Bearing capacity of shallow foundations

When the embedded depth of the foundation is less or equal tothe least dimension of that foundation, it can be considered asa shallow foundation. This is the interpretation of the Terzaghitheory that was presented six decades ago (Terzaghi 1943).Thus, the word “shallow” does not necessary mean that thefoundation is close to the ground surface. Foundations with

S. Barakat (*) :R. Alzubaidi :M. OmarCivil and Environmental Engineering Dept., University of Sharjah,P.O. Box 27272, Sharjah, United Arab Emiratese-mail: [email protected]

R. Alzubaidie-mail: [email protected]

M. Omare-mail: [email protected]

Arab J GeosciDOI 10.1007/s12517-014-1581-x

embedded depth that is equal three to four times the width ofthe foundation may be defined as shallow foundation. Basedon the above definition, most of the foundation types exclud-ing pile foundation (such as isolated, strip, and most of matfoundations) are considered shallow foundations.

The ultimate bearing capacity of shallow foundation givesan expression of the max load that the soil in position cancarry from the engineering structures without a shear failure.The ultimate bearing capacity can be obtained using one of thelimit equilbrium methods developed by Terzaghi (1943), Tay-lor (1948) or Meyerhof (1951), and Vesic (1973). The as-sumed shape of the failure surface and the distribution of thenormal stresses along this surface are the main differencebetween these methods.

Physical properties of the footings and mechanicalproperties of the underlying soil layers are limiting theultimate bearing capacity. Physical properties includesthe width, length, and the depth of the foundation, while themechanical properties are the soil properties parameters(soil cohesion (c), angle of internal friction (ϕ), and soilunit weight (γ)).

Prediction of the allowable bearing capacity from theultimate one can be established using the deterministic ap-proach by the mean of using suitable factor of safety (toaccount for the variablity in the underlying soil and loadparameters) or by the probabilistic approach using one ofthe different methods such as the point estimation method(PEM) (Harr 1987; Rosenblueth 1981), Tayler′s series ex-pansion method (Cornell 1969; Hasofer and Lind 1974;Madsen et al. 1986), the exact methods (Hahn and Shapiro1967), and finally, the Monte Carlo simulation (MCS)methods (Rubinstein 1981). At different levels of accuracyand cost, any of these probabilistic approaches may be usedin evaluating the uncertainty involved in calculating theultimate bearing capacity. For this study, theMCS techniqueis selected to achieve this goal with an acceptable degree ofsimplisity, accuracy, and cost.

The main philosophy of MCS technique involves generat-ing a huge data of independent random variables withinlimiting ranges (mean and standard deviation) to establishthe distribution of another dependent variable with respect tothe behavior of all random variables used to calculate thisvariable in the deterministic approach. Thus, the MCS tech-nique deals with the shape of the distribution and its geometriccharacteristics and the number of random variables does notcause any difficulty as in the other probabilistic methods.

In this paper, all variables that have an impact on theuncertainty in the determination of the ultimate bearingcapacity will be treated as random variables and largesets of these variables will be randomly generated toestablish the distribution of the ultimate bearing capacityand the combined effect of these variables on the shapeof the distribution.

Problem formulation

The model of general ultimate bearing capacity for shallowfoundation proposed byMeyerhof (1963) can be expressed bythe following equation;

qult ¼ cN cFcsFcdFciFcw þ qNqFqsFqdFqiFqw

þ 1

2γBNγFγsFγdFγiFγw ð1Þ

Where c and γ are the soil cohesion and soil unit weight,respectively, q is the effective stress at the level of the bottomof the foundation, B is the width of a footing (=diameter for acircular foundation) embedded at a depth Df. Fcs, Fqs, Fγs=shape factors, Fcd, Fqd, Fγd=depth factors, Fci, Fqi, Fγi=loadinclination factors. Nc, Nq, and Nγ represent the bearing ca-pacity factors, and they are almost dependent on the angle ofshearing resistance of the soil. Nc and Nq are defined asfollows:

Nq ¼ tan2 45þ ϕ2

� �eπtanϕ ð2Þ

N c ¼ N q−1� �

cotϕ ð3Þ

Nγ ¼ 2 Nq þ 1� �

tanϕ Vesic ð1973Þ ð4aÞ

Nγ ¼ Nq−1� �

tanϕ Meyerhof ð1963Þ ð4bÞ

Nγ ¼ 1:5 Nq−1� �

tan 1:4ϕð Þ Hansen ð1970Þ ð4cÞ

Nγ ¼ 1:1 Nq−1� �

tan 1:3ϕð Þ Spangler and Handy ð1982Þð4dÞ

Nγ ¼ Nq þ 1� �

tanϕ Euro Code ð1992Þ ð4eÞ

Because there are still some controversies involvingthe variation of Nγ with the soil friction angle ϕ, thevalue of Nγ is presented in Eq. 4 in different relationships.Equation 1 is valid for different footing shapes subjected toconcentric load embedded at a depthDf. The shape factors aregiven by:

Fcs ¼ 1þ B

L

� �N q

N c

� �

Fqs ¼ 1þ B

L

� �tanϕ

Fγs ¼ 1−0:4B

L

� � ð5Þ

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Where L is the length of a footing (=diameter for a circularfoundation). For the commonly used depth, and inclinationfactors see DAS (2011). Equation 1 is based on the fact thatthe water table is too deep from the base of the foundation;however, if the water table rose up to the ground surface,effective unit weight should be used in the second and thirdterms of these equations. If the water table is partially raised,within the limit ofB from the base of the foundation, then onlythe submerged part should be treated as the effective unitweight. Fcw=1, and Fqw=Fγw is a random number generatedin the range of 0.5 to 1.0 and represents the reduction in thebearing capacity due to rapid variation of the water tableduring the service life of the structure.

Methodology for MCS

MCS is effective and practical for problems with prescribedprobability distributions of the random variables. During thegeneration procedure, the distribution of each random variableshould be known. Usually, a normal distribution is assumedfor the soil parameter γ and lognormal distribution for c and ϕ(Baecher and Christian 2003; Foye et al. 2006a, b; Massihet al. 2008; Fenton et al. 2008, 2011; Griffiths et al. 2009;Shahin and Cheung 2011).

The use of the lognormal distribution is one of the manypossible choices, as opposed to the more familiar normaldistribution, or even some other more complex distribution,that is based on the following arguments: first, it offers theadvantage of simplicity and familiarity of its two-parameterdescription. Second, and perhaps most important from aphysical point of view, is that the lognormal distributionsguarantee that the random variable is always positive. Third,it has been used by several investigators since there is someevidence from the field to support the lognormal distributionas a reasonable model for the physical soil properties in theview of lack of exhaustive field data that would be necessaryto support other kinds of distributions over the lognormal.However, Harr (1987) discussed the unbounded nature of theupper end of the lognormal distribution.

The uncertainty in the measured data is expressed in termsof unbiased estimates of statistical moments, i.e., samplemean(μ) and standard deviation (σ). The coefficient of variation(CV=σ/μ), and the bias factor (=mean/nominal) are com-monly used in expressing the variability of the soil propertiesbecause of the advantages of being dimensionless as well asproviding a meaningful measure of relative dispersion of dataaround the sample mean.

The probability distribution functions for γ, c, φ, B, L, deadload (DL), and live load (LL) are assumed to follow certaindistributions. The assumed statistical values of these distribu-tions are within the practical ranges as has been used and citedin the literature of several geotechnical engineering

applications (Phoon et al. 1995; Lacasse and Nadim 1996;Baecher and Ladd 1997; Duncan 2000; Massih et al. 2008;Lee et al. 1983; Phoon and Kulhawy 1999a, b; Lee et al. 1983;Lumb 1970; Wolff 1985; Yuceman et al. 1973; Cherubini2000).

LL and DL variabilities have been examined thoroughly bymany researchers (Andrzej and Nowak 1994; Ellingwood andTekie 1999; Scott et al. 2003). Since the loads used in thefoundation design are the total load (DL+LL), the appropriateprobability distribution will be used in the generation of theseapplied loads. In this study, the parameters considered asrandom variables in the bearing capacity model are summa-rized in Table 1. It is also reasonably assumed that theseparameters are statistically independent. If both the probabilitydistribution of the random variables and the deterministicmodel are selected properly, a set of data from the MCS issimilar to a set of experimental data. This simulation conductsa repeating process using a set of random variables generatedin accordance with the assigned probability distribution.

One of the most significant problems during the generationprocedure is to determine the suitable number of simulationcycles which reflects the simulation accuracy. An iterativestudy was performed to determine a suitable number of sim-ulation cycles, that is, the number of samples required in thestatistical analysis of the general bearing capacity. The numberof generations is increased gradually in order to minimize theCVs of the four statistics (min, max, mean, and mode) as lessas possible and to attain the convergence. If the modificationin the four statistics cited earlier, which describes the distribu-tion of the presumed variable (bearing capacity or safetyfactor) is 1.0 % or less, then the iterations are halted and theconvergence is said to have occurred. In this study, the resultsare stabilized for 10,000 or more runs. Therefore, for thisstudy, 10,000 simulation runs were used for each case.

The earlier cited statistical data were used to bring aboutsample values of c, φ, γ, B, L, and Df, and the complementingdeterministic bearing capacity was calculated using Eqs 1, 2, 3and 4a of Vesic′s (1973) model. Model uncertainty and theuncertainty connected with the natural variability of soil werenot regarded here. This was carried out again several timesuntil a convergence criterion was attained. The predictedrandom variables obtained from the many simulations carriedout were used to plot the normalized frequency diagrams(NFDs). It should be noted that the probabilistic simulationsdescribed in this aspect of the study were carried out with thehelp of the Matlab software (2010).

Probabilistic interpretation of MCS results

A parametric study was performed to evaluate the effects oftypical random variables on the simulated bearing capacityvalues and to derive the corresponding probability

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distribution. The variables considered herein with their rangesare shown in Table 1.

The required probability distribution has to be determinedempirically based entirely on the generated data. The requireddistribution model may be determined by visually comparinga density function with the frequency diagram for a set ofgenerated data. Alternatively, the data may be plotted onprobability papers prepared for specific distributions (Angand Tang 1975). If the data points plot approximately on astraight line on one of these papers, the distribution corre-sponding to this paper may be an appropriate distributionmodel. Then, some tests can be used to examine the relativedegree of validity of the different distributions. Two such teststhat are commonly used for these purposes include the Chi-square (χ2) and the Kolmogorov-Smirnov (K-S) tests. Thechoice of the probability distribution may also be dictated bymathematical convenience. Therefore, the normal (or lognor-mal) distribution is frequently used to model nondeterministicproblems.

To study the effect of Nγ factor as stated in Eqs. 4a–4e onthe soil bearing capacity (Eq. 1), the NFDs and the fittedprobability distributions (normal, lognormal, gamma, andgeneralized extreme value (GEV)) are shown in Fig. 1. Therandom variables in Table 1 along with Eq. 1 and one ofEqs. 4a–4e (Nγ factor) were used to generate the bearingcapacity values for the same combination of c, φ, γ, B, L,andDf. The results using Eq. 4a to 4e for Nγ factor in Eq. 1 areplotted, respectively, in Fig. 1a–e. The results in Fig. 1a–e arealmost identical. The generated samples (NFDs) obtainedfrom the MCS were fitted by normal, lognormal, gammaand GEV distributions. The visual inspection of these figuresshows that they are almost identical and the effect of changingNγ factor on the shape of the bearing capacity distribution isnegligible.

Also, the visual inspection of these figures shows that it isdifficult to state which distribution fits the generated databetter. All distributions appear to fit well into the data.

Therefore, the probability papers for normal, lognormal, gam-ma, and GEV distributions are plotted to fit the generated datapoints. The data points on the GEV paper follow almost thesame function with exceptions of few points indicating thatthe GEV distributionmay be appropriate. The validity of GEVdistribution was also confirmed by goodness-of-fit tests. Inthis study, the K-S test was used for this. Using the Matlab®,the K-S tests of the data set versus the four GEV, normal,lognormal, and gamma distributions at the α% significancelevel were performed. K-S test compares the values in thegenerated data set to the hypothesized continuous distribution.The null hypothesis is that the data set has a hypothesizeddistribution. The alternative hypothesis is that the data set doesnot have that distribution. The resultingH is 1 if the test rejectsthe null hypothesis at the α% significance level; if theH resultis 0, then the test accepts the null hypothesis at the α%significance level. Also, the test returns the p value, the teststatistic, and the cutoff value Dn

α for determining if the teststatistic is significant and the maximum difference betweenthe empirical CDF (generated data set) and the hypotheticalCDF (Dn). If H=0, the distribution with the smallest maxi-mum difference (Dn) can be considered as the best fit.

The K-S test results in Table 2 show that H=1 for normal,lognormal, and gamma distributions and H=0 for GEV dis-tribution. Distributions withH=1 are rejected. One would notreject the hypothesis of GEV distribution fit to the bearingcapacity data because H=0. Also, the critical value at α=0.01significance level is Dn

α=0.0051. The maximum difference(Dn) for GEV distribution is 0.0033 and is less than Dn

α=0.0363. The distribution with H=0 and the smallest Dn is theGEV and can be considered as the best fit. The assumeddistributions fitted the generated data at different levels. Ingeneral, the GEV distribution showed the best fitting and evenif the focus of distribution fitting was on the tails, GEV stillhas the least ill-fitted points in the tails. Also, the validity ofthe fitting distribution was confirmed by K-S goodness-of-fittests. Only GEV distribution passed the test.

Table 1 Probability models ofrandom variables Random

variablePDF Bias factor CV range Reference

γ Normal 1.0 0.02–0.15 Phoon et al. (1995), Lacasse and Nadim (1996),Baecher and Ladd (1997), and Duncan (2000)

φ Lognormal 1.03 0.05–0.15 Massih et al. (2008)

0.12–0.56 Lee et al. (1983) and Phoon and Kulhawy (1999a, b)

c Lognormal 1.07 0.10–0.70 Cherubini (2000)

0.30 Lee et al. (1983)

B, L Normal 1.05 0.045 Foye et al.(2006a, b)

DL Normal 1.05 0.15 Foye et al. (2006a, b)

LL Lognormal 1.15 0.25 Choi 1990

Gamma 0.16–0.24 0.60–0. 90 Choi 1990

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Fig. 1 NFD and the fitted(normal, lognormal, gamma, andGEV) distributions used for Nγ: aEq. 4a (Vesic 1973), b Eq. 4b(Hansen 1970), c Eq. 4c(Meyerhof 1963), d Eq. 4d(Spanler and Handy 1982), and eEq. 4e (Euro Code 1995)

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The above procedure is repeated using different combina-tions of c, φ, γ, B, L, and Df covering all practical ranges for csoils, φ soils, c–φ soils, and continuous and rectangular foun-dations. Examples of the results for φ soils are shown in Fig. 2

and Table 3; the results for c soils are shown in Fig. 3 andTable 4; and the results for c–φ soils are shown in Fig. 4 andTable 5. In addition, the effect of the correlation between ϕand c, ρϕ, c on the results was studied for the case of thec-φ-soils (Fig. 4). Figure 5 and Table 6 show the analysisresults for, ρϕ, c which is −0.6. A comparison of Figs. 4 and 5reveals that the correlation, ρϕ, c has no effect on the shape ofthe distribution but on the values of the soil bearing capacity.

To conclude this section, it is clear from the samplesgenerated above (NFDs), which was obtained from the MCSand the fitted normal, lognormal, gamma, and GEV distribu-tions tested by the corresponding probability papers and val-idated by the goodness-of-fit (K-S) test using the Matlab®,that the GEV distribution appears to best fit the soil bearingcapacity and considered acceptable for all c, φ, γ, DL, and LLcombinations considered herein.

The CDF for the GEV distribution with location parameterμ, scale parameter σ and shape parameter k≠0 are

F x;μ;σ; kð Þ ¼ Exp − 1þ kx−μσ

� �h i−1=k� ð6Þ

f x;μ;σ; kð Þ ¼ 1

σ1þ k

x−μσ

� �h i −1.

k

� �−1Exp − 1þ k

x−μσ

� �h i−1.k

8<:

9=;

ð7Þfor 1+k(x−μ)/σ>0.The shape parameter k governs the tail behavior of the

distribution. The subfamilies defined by k=0, k>0, and k<0correspond, respectively, to the Gumbel, Fréchet, and Weibullfamilies, whose cumulative distribution functions aredisplayed below.

& Gumbel or type I extreme value distribution (k=0)& F x;μ;σ; 0ð Þ ¼ e−e

− x−μð Þ=σ ð8Þ

& Fréchet or type II extreme value distribution, if k>0&

F x;μ;σ; kð Þ ¼0 x≤μ

e− x−μð Þ

.σ

� �−1=k

x > μ

8<: ð9Þ

Table 2 K-S test for the results in Fig. 1e, CVall=0.10

PDF H Dnα Dn Test results Best distribution

Normal 1 0.0995 0.0051 Reject GEVLognormal 1 0.0315 Reject

Gamma 1 0.0538 Reject

GEV 0 0.0033 Accept

Fig. 2 NFDs, the fitted (normal, lognormal, gamma, and GEV) distribu-tions and the corresponding probability plots for ϕ soil, μc=0 kN/m2,μϕ=30°, μγ=18 kN/m

3, CVall=0.10, and B=L=2.0 m

Table 3 K-S test results for ϕ soils, μc=0 kN/m2, μϕ=30°, μγ=18 kN/m

3,CVall=0.10, and B=L=2.0 m




GEV 0 0.0040 Accept

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& Reversed Weibull or type III extreme value distribution, ifk<0

&F x;μ;σ; kð Þ ¼ e− − x−μð Þ=σð Þ−1=k x < μ

1 x≥μ

�ð10Þ

Where, σ=0.

Fig. 3 NFDs, the fitted (normal, lognormal, gamma, and EV) distribu-tions and the corresponding probability plots for c soils, μc=100 kN/m2,μϕ=0°, μγ=18 kN/m

3, CVall=0.20, B=1.8 m, and L=2.0 m

Table 4 K-S test results for c soils,μc=100 kN/m2,μϕ=0°,μγ=18 kN/m

3,CVall=0.10, B=1.8 m, and L=2.0 m


Normal 1 0.0388 0.0051 Reject Lognormal GEVLognormal 0 0.0022 Accept


GEV 0 0.0049 Accept

Fig. 4 NFDs, the fitted (normal, lognormal, gamma, and GEV)distributions and the corresponding probability plots for c–ϕ soils,μc=100 kN/m2, μϕ=25°, μγ=18 kN/m3, CVϕ=0.1, CVγ=0.10,CVc=0.20, B=2.0 m, and L=2.0 m

Table 5 K-S test results for c-ϕ soils, μc=100 kN/m2, μϕ=25°,μγ=18 kN/m3, CVϕ=0.1, CVγ=0.10, CVc=0.20, B=2.0 m, and L=2.0 m




GEV 0 0.0026 Accept

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Reliability analysis

Foundation design consists of selecting and proportioningfoundations in such a way that limit states are prevented. Limitstates are associated with outcomes involving safety versusfailure or satisfactory versus unsatisfactory performance. Re-liability analysis is a design philosophy that aims at keepingthe probability of reaching limit states lower than some limit-ing value and thus, a direct assessment of the risk is possible.The limit state g(x) for the possibility that the foundation willexperience bearing capacity failure is defined as:

g c; γ;ϕ;B; L;D f ;DL;LLð Þ ¼ qult−Sd ð11Þ

Where qult is the strength part as defined by Eq. 1 and Sd isthe total load divided by the foundation area, Sd=(DL+LL)/A,where A=B×L (πB2/4 for a circular foundation). DL and LLvariability has significant impact on the final uncertainty in thelimit state of the bearing capacity. By analogy to deterministiccase, a probabilistic factor of safety, FOS is defined as:

FOS ¼ qultSd

ð12Þ

Reliability index, β versus FOS

In this aspect of the study, the probabilistic-based analysis forthe bearing capacity of the foundation was carried out takinginto consideration parameter uncertainty that is related withthe input variables in Eq. 11. Among the random variables inEq. 11, c, ϕ, and LL are likely to include significant parameteruncertainty, and of lesser degree, γ and DL, therefore they areassumed to be random variables. Additionally, the footingdimensions (B and L) andDf are likely to bring about marginalparameter uncertainty and are therefore assumed to be deter-ministic for practical purposes. The failure will occur if g(x)<0, thus the probability of failure is defined as:

p f ¼ p g < 0ð Þ ¼ p FOS−1 < 0ð Þ ð13Þ

Pf can be computed following a run ofN realizations for theMCS as (count g<0)/N.

The complement of failure is the safety or the reliabilitywhich can be expressed using the concept of reliability index(β). The reliability index can be expressed as;

β ¼ ϕ−1 1−P fð Þ ð14Þ

Where ϕ−1=inverse of the normal probability distributionfunction.

The probability distribution for the FOS defined in Eq. 12is obtained using MCS and the same procedure already ex-plained in “Probabilistic interpretation of MCS results.” Theresults are presented in Figs. 6 and 7 and Table 7. Again the

Fig. 5 NFDs, the fitted (normal, lognormal, gamma, and GEV) distribu-tions and the corresponding probability plots for c–ϕ soils, μc=100 kN/m

2,μϕ=25°, μγ=18 kN/m3, CVϕ=0.1, CVγ=0.10, CVc=0.20, B=2.0 m,L=2.0 m, and ρϕ, c=−0.6

Table 6 K-S test results for c-ϕ soils, μc=100 kN/m2, μϕ=25°, μγ=18 kN/m3, CVϕ=0.1, CVγ=0.10, CVc=0.20,B=2.0 m, L=2.0 m, and ρϕ,c=−0.6




GEV 0 0.0040 Accept

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GEV distribution is the best fit to the FOS data. These resultswere used to develop probabilistic design solutions corre-sponding to certain reliability levels.

In order to illustrate the above procedure, the followingcase study was investigated. A square footing of B=L=2.0 m,and Df=1.5 m below the ground surface, and the soil proper-ties are μγ=16.5 kN/m3, μc=20 kN/m2, and μϕ=25°. Thetotal load is kept constant throughout the analysis, Sd=μDL+μLL=2,747 kN. This Sd value is selected such that the deter-ministic FOSd=1.0. The CVof the strength and loading RVsvariedwithin the practical ranges and the PDFs are assumed tofollow certain distribution, as has been found in several geo-technical engineering applications, and as summarized inTable 1. While Sd is kept constant, the LL/DL ratio and thedeterministic FOSd vary.

Figures 8, 9, and 10 show, respectively, the effect of vary-ing CVϕ, CVc, and CVLoad on both FOSd and β. The resultsshows that, for the case study cited earlier, the safety factor of3 which is normally used in the deterministic analysis does notalways give a reliable design. For example, for the same

FOSd=3.0, β is decreased from 5.8 to 0.75 when CVϕ isincreased from 0.1 to 0.5, while there is also a decrease from5.8 to 3.0 when CVc is increased from 0.1 to 0.5, and finally, adecrease also occur from 5.8 to 4.0 when CVLoad is increasedfrom 0.1 to 0.4. The effect of ρc, ϕ on β for the same FOSd=2.0is presented in Fig. 11. It is clear that increasing |ρc, ϕ| willincrease β, and this is more pronounced in the presence ofhigher variability measured by the CVs. Conservatively ρc, ϕcan be assumed negligible. This demonstrates that the uncer-tainty associated with the RVs (c, ϕ, DL, and LL) can consid-erably affect the bearing capacity of the footing and thusshould not be neglected by using only deterministic FOS.

Probabilistic model versus deterministic model

The main objective of this section is to develop a probabilisticFOS to account for the variation of Qu and the required safetyinvolved. In this case, the risk or the safety involved can be

Fig. 6 FOS NFDs, the fitted(normal, lognormal, gamma, andGEV) distributions and thecorresponding probabilityplots for c–ϕ soils, μϕ=25°,μγ=18 kN/m

3, CVϕ=CVγ=0.10,CVc=0.20, ρϕ, c=0.0,CVDL=CVLL=0.10, B=2.0 m,and L=2.0 m; a μc=25 kN/m

2

and b μc=70 kN/m2

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defined in terms of the probability of failure Pf or the reliabil-ity index, β. In the context of this work, the probability ofusing FOS that is lower than the one required in the design isdefined as,

P f ¼ P FOS < FSdð Þ ð15Þ

To evaluate the probability of failure, knowledge of theprobability distribution of FOS is required. It was proven thatthe generalized extreme value distribution GEV distributionwould seem to be the best choice. Also, the MCS for FOS

results showed that for all cases considered herein, the value ofshape parameter k>0.

Therefore, the PDF and CDF of FOS defined inEqs. 7 and 9 and written here in the form of standardextreme value distribution functions (Fréchet or type II ex-treme value distribution):

F yð Þ ¼0 y≤0

exp −νy

� �k !

y > 0

8><>: ð16Þ

Fig. 7 FOS NFDs, the fitted(normal, lognormal, gamma, andGEV) distributions and thecorresponding probabilityplots for c–ϕ soils, μϕ=25°,μγ=18 kN/m

3, CVϕ=CVγ=0.10,CVc=0.20, ρϕ, c=0.0, B=2.0 m,and L=2.0 m. a CVDL=CVLL=0.10 and LL/DL=4. b CVDL=CVLL=0, loads are deterministic

Table 7 FOS K-S test results forc-ϕ soils, μϕ=25°, γ=19 kN/m

3,CVϕ=CVγ=0.10, CVc=0.20,ρϕ, c=0.0, CVDL=CVLL=0.10,and B=L=2.0 m

Soil PDF H Dnα Dn Test results Best distribution

μc=25 kN/m2 Normal 1 0.0745 0.0051 Reject GEVLognormal 1 0.0175 Reject


GEV 0 0.0033 Accept

μc=70 kN/m2 Normal 1 0.0704 0.0051 Reject GEVLognormal 1 0.0142 Reject


GEV 0 0.0032 Accept

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f yð Þ ¼0 y≤0νy

� �kþ1

exp −νy

� �k !

y > 0

8><>: ð17Þ

Where

E yð Þ ¼ ν Γ 1−1.k

� �k > 1 ð18Þ

CVy

� �2 ¼ Γ 1−2.k

� �Γ 2 1−1

.k

� �−1 k > 2 ð19Þ

ν is the most probable value of y and Γ() is a gammafunction in Eqs. 16 and 17.

The probability of failure (Eq. 15) is clearly equal toF(FOS). Defining the risk-safety factor FSR as;

FSR ¼ E FOSð ÞFOS

ð20Þ

Fig. 8 The effect of varying CVϕ on both FOSd and β

Fig. 9 The effect of varying CVc and on both FOSd and β

Fig. 10 The effect of varying CVLoad on both FOSd and β

Fig. 11 The effect of ρc,ϕ on β

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Substituting this along with the equation for ν into Eq. 16yields;

F FOSð Þ ¼ exp −1

FOS� E FOSð Þ

Γ 1−1.k

� �0@

1A

k0@

1A FOS > 0

P f ¼ exp −FSR

Γ 1−1.k

� �0@

1A

k0@

1A

ð21Þ

FSR ¼ −lnP fð Þ1kΓ 1−1

k

� �ð22Þ

It is important to point out that FSR increases with anincrease in the variation of FOS and a decrease in Pf. Equa-tions 19, 20, and 22 state that due to the variation of thestrength and loading variables, the FOSwill vary, consequent-ly the mean value of FOS (obtained by substituting the meanvalues of the RVs in Eq. 12) must be increased to FOS=FSR×

Fig. 12 Risk-safety factor (FSR)

Fig. 13 Coefficient of variationof FOS, CVFOS, for ϕ soils

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E(FOS). The amount of increase depends on the total variationof FOS measured by the coefficient of variation (Eq. 19) andthe required probability of failure, Pf. Figure 12 shows agraphical presentation of FSR. Theoretically by Eq. 22, FSRvaries between 1 and ∞ for k>2 (limitation in Eq. 19) and anyvalue of Pf. However, for the values of Pf considered here,FSR∈ [1, 6.59].

The minimum value of FSR=1, it theoretically exists (ifCVFOS=0) but practically is not possible. The results alsoindicate that in the presence of randomness, for thesafety factor of 3 that is usually used in the determin-istic analysis, it is not enough to achieve a certainreliable level. For example, FSR should be larger than3 if the CVFOS is more than 0.8 and the required Pf isless than 10−3. The value of FSR might reach about 5 oreven 6 for higher variation in the FOS (soil and loadparameters) and the reliability level. Figure 13 shows thevariation of CVFOS, along with the variation in ϕ, in termsof μϕ and CVϕ for different values of CVγ and μc=CVc=0.This figure clearly demonstrates that CVFOS is very sensitiveto changes in ϕ.

Mathematical modeling of CVFOS based on MRA

The calculation of CVFOS is not straightforward and affectedby several parameters. Therefore, a more simplified modelthat predicts its value with reasonable accuracy is sought. Thetheoretical results based on the simulation analysis shown inFigs. 1, 2, 3, 4, 5, 6, and 7 are re-arranged to fit the multipleregression analysis, MRA. The output dependent variable(CVFOS) based on ten input independent variables (CVc,CVϕ, CVγ, CVDL, CVLL, μc, μϕ, μγ, μDL, and μLL) areincluded in the modeling phase. Then, a model on the keyresponse parameter was established by incorporating largedata sets using MRA. Several trials were made using SPSSstatistics 19.0 (SPSS 2010) computer package to select thebest-fit formulae. To gain an overall view of the problem, thepairs of data of the variables of interest are examined in thematrix scatter diagram shown in Fig. 14. The curvature in therelationships is evident. Different linear and nonlinear regres-sion models among the dependent variable (CVFOS) and theten independent variables (CVc, CVϕ, CVγ, CVDL, CVLL, μc,μϕ, μγ, μDL, and μLL) are examined using set 1 data consisting

CV CVc CV CV CVDL CVLL

CV

FOS

Mean

Fig. 14 Matrix scatter plotamong the output-dependent var-iable (CVFOS) and CVs and μs ofthe input independent variables(c, ϕ, γ, DL, and LL)

Table 8 Multiple linear regression output

Model output Independent variables output

Variable, xi CVc CVϕ CVγ CVLL μc μϕ μγ

Model for CVFOS SEE=0.220

R2 0.867 Coefficient of xi −1.53 7.24 3.95 −1.82 −0.002 1.02 −0.028Model F value 182.8 t value −4.29 18.21 3.83 −4.02 −3.52 6.03 −3.91Model F sig. 0.00 t sig. 0.00 0.00 0.00 0.00 0.00 0.00 0.00

Model for LN (CVFOS) SEE=0.284

R2 0.969 Coefficient of xi −2.07 16.62 4.23 2.21 −0.002 2.92 −0.26Model F value 933.6 t value −4.50 32.46 3.19 3.76 −3.52 13.44 −28.92Model F sig. 0.00 t sig. 0.00 0.00 0.00 0.00 0.00 0.00 0.00

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of 242 points. The results of the best model are summarized inTable 8. Despite the apparent best fitting (high values of R2,low values of standard error of estimate (SEE), and passingthe F and t tests), the linear model is not physically sound forpredicting CVFOS. This is attributed to the negative signs on atleast one of the independent variables coefficients. For

example, this implies that increasing the CVc or CVLL

decreases the CVFOS, which is counter intuitive. Conse-quently, nonlinear regression model is adapted herein.The best-fit nonlinear models that could predict CVFOS andhas high value of R2=0.969, low value of SEE=0.284, passthe F and t tests, and logically meaningful are listed in Table 8and given by:

CVFOS ¼ 0 if CVall ¼ 0¼ expð−2:07CVc þ 16:62CVf þ 4:23CVγ

þ 2:21CVLL−0:002μc þ 2:92μ f−0:26μγÞ other

ð23Þ

The accuracy of the regression model was checkedthrough comparison to the MCS results. The inputvalues from set 1 of data are presented to the MRAmodel to perform the necessary calculations and pro-duce the corresponding outputs. Comparison of theoret-ical and recalled values of CVFOS by MRA is presentedin Fig. 15a. Furthermore, the prediction accuracy of themodel adopted in this work was also checked. Oneadditional set of data (set 2) consisting of 32 pointswas used to perform the prediction test using the MRAmodels. It should be stressed that all of the data in thislatter set was initially withheld from the MRA. Theresults of this test are shown in Fig. 15b. The closeness ofthe points to the equality line serves only to indicate thevalidity of the MRA models.

Fig. 15 Recalled and predicted s values by multiple nonlinear regressionversus MCS

Table 9 Calculated values of FOS for various CVs and failure probabilities

CVc CVϕ CVγ CVDL CVLL μc μϕ μγ μDL μLL Pf CVFOS E(FOS) FSR FOS

Deterministic 20 25 16.5 915.7 915.7 – – 3.0 1.0 3.00

0.1 0.1 0.1 0.1 0.1 20 25 16.5 915.7 915.7 0.01 0.35 3.0 1.40 4.20

0.1 0.1 0.1 0.1 0.1 20 25 16.5 915.7 915.7 0.001 0.35 3.0 1.70 5.10

0.1 0.1 0.1 0.1 0.1 20 25 16.5 915.7 915.7 0.0001 0.35 3.0 1.85 5.55

0.1 0.2 0.1 0.1 0.1 20 25 16.5 915.7 915.7 0.01 1.85 3.0 2.45 7.35

0.1 0.2 0.1 0.1 0.1 20 25 16.5 915.7 915.7 0.001 1.85 3.0 3.25 9.75

0.1 0.2 0.1 0.1 0.1 20 25 16.5 915.7 915.7 0.0001 1.85 3.0 4.00 12.00

Table 10 Calculated values of B for various CVϕ and failureprobabilities

CVϕ Pf CVFOS FSR B (m) Increase in B (%)

Deterministic 1.00 1.36 –

0.10 0.01 0.32 1.35 1.58 16

0.001 1.60 1.72 26

0.0001 1.85 1.85 36

0.20 0.01 1.67 2.27 2.04 50

0.001 3.25 2.41 77

0.0001 3.85 2.61 92

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Practical applications

Example 1: consider a square 2 m×2 m footing withDf =1.5 m resting on a soil having the followingproperties:μϕ=25

ο, μc=20 kN/m2, μγ=16.5 kN/m3, andμDL=μLL=915.67 kN. The parameters values are selectedsuch that the deterministic FOS=3.0.

Determine the required Factor of Safety if: CVall≠0

i. From Eq. 1, the E(qult)=1,373.5 kN/m2

ii. From Eq. 12, the E(FOS)=(1,373.2×2×2)/(2×915.67)=3.0iii. From Eq. 23, obtain the value of CVFOS

iv. With CVFOS, enter Fig. 12 to determine FSR for any Pf.

v. FOS=FSR×E(FOS)

Table 9 lists the calculated values of FOS for variousCVs and failure probabilities. This table indicates that asthe selected reliability level increased (Pf value de-creased), the risk-safety factor FSR increases. This in-crease is more for higher values of CVϕ at the samereliability level.

Example 2: consider a square foundation (B×B) with Df=1.22 m resting on a cohesionless soil having the followingproperties:μϕ=34°, μγ=16.5 kN/m3, μγsat=18.55 kN/m3,CVγ=0.1, μDL=μLL=333.6 kN with FOS=3, the water tableat depth of 0.61 m (CVc=CVϕ and CVDL=CVLL=μc=0.0).Determine the required dimension B:

Table 10 lists the calculated values ofB for various CVs andfailure probabilities. This table indicates that as the selectedreliability level increased (Pf value decreased), the foundationsize increases. This increase is more for higher values of CVϕ

at the same reliability level.

Recommended target reliability levels and FO

The acceptable safety levels are expressed in terms of targetreliability indexes. The selection of the target reliability in-dexes should be based on the indexes for current codes,evaluation of performance of existing structures, economicanalysis and engineering judgment. A reliability index wouldbe assigned to an entire structure and/or to different modes offailure. Therefore, the only practical approach is to associate aprescribed target reliability index with each safety check foreach failure mode. The target reliability index may vary withthe type of loading (dead load or live load), type of failuremode (ductile or brittle), and type of material. Tables 11 and12 give the target reliability indices, the corresponding failureprobabilities, and reference period for different limit states(Sarja 2002; Skjong 2002). Once the acceptable safety levels(target reliabilities) are selected by the codes of practice andthe soil variability is determined by field and lab testing, theproposed formulation can be easily implemented. In general,to design for a target reliability index above 3.0 (Pf>10

−3) andCVFOS above 0.5, the required FSR should be above 2.5 (seeFig. 12).

Summary and conclusions

A probabilistic model was developed to estimate the bearingcapacity of shallow foundations and the associated factor ofsafety using a Monte Carlo simulation. The results obtainedfrom the probabilistic analysis indicate that the probabilisticdistributions of the bearing capacity and the factor of safetyfollow type II extreme value.

Table 11 Annual target reliability index β (and associated target failure probability Pf) (Sarja 2002)

Class of failure Consequence of failure

Less serious Serious

Redundant structure β=3.09 (Pf ≈10−3) β=3.71 (Pf ≈10−34)Significant warning before the occurrence of failure in a nonredundant structure β=3.71 (Pf ≈10−34) β=4.26 (Pf ≈10−35)No warning before the occurrence of failure in a nonredundant structure β=4.26 (Pf ≈10−35) β=4.75 (Pf ≈10−36)

Table 12 Target Reliability In-dex β and associated target failureprobability Pf) for existing struc-tures (Skjong 2002)

Limit state Reliability index β Reference period

Serviceability Reversible β=0.0 (Pf≈0.5×10−1) Intended remaining service lifeIrreversible β=1.5 (0.65×10−1)

Fatigue Inspectable β=2.3 (Pf ≈10−2)Not inspectable β=3.1 (Pf ≈10−3)

Consequences of failure Very low β=2.3 (Pf ≈10−2) Minimum period for safety(e.g., 50 years)Low β=3.1 (Pf ≈10−3)

Medium β=3.8 (Pf ≈10−4)High β=4.3 (Pf ≈10−5)

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Based on this result, a risk-safety factor is derived toaccount for the inherent randomness and variability of soilstrength and load parameters as well as the reliability level.Through a multiple linear regression, an equation for thecoefficient of variation of the factor of safety FOS or theultimate bearing capacity qμ was developed as a function ofthe soil and the load statistics (CVand μ). The findings of thiswork warrant the following conclusions:

1. The variation of FOS is greatly affected by the variation ininternal friction angle ϕ while the cohesion, soil unitweight, and load on the other hand, had a minor effect.

2. The risk-safety factor FSR was found to be directly relatedto the coefficient of variation of FOS and the selectedreliability level (probability of failure). It increases with anincrease in CVFOS and the reliability level. For Pf=10

−3,FSR was found to range between 1.0 and 5.0.

3. The results showed that commonly used safety factorsmay not suffice primarily when the soil parameters exhibithigh variations. The angle of internal friction was foundthe main contributor to the soil variability and therefore tothe safety factor. In these cases, however, the introductionof probability may provide a safer and a more economi-cally viable design solution. The proposed approach canbe used to calibrate design safety factors to reflect theuncertainties in the loads and the soil bearing capacity ofshallow foundations.

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Probabilistic-based assessment of the bearing capacity of shallow foundations