Effect of spatial variability of cross-correlated soil properties on bearing capacity of strip footing

8/7/2019 Effect of spatial variability of cross-correlated soil properties on bearing capacity of strip footing

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INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS Int. J. Numer. Anal. Meth. Geomech. 2010; 34 :126Published online 7 May 2009 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.791

Effect of spatial variability of cross-correlated soil propertieson bearing capacity of strip footing

Sung Eun Cho 1,, , and Hyung Choon Park 2,

1Korea Institute of Water and Environment , Korea Water Resources Corporation , 462-1 , Jeonmin-Dong , Yusung-Gu , Daejon 305-730 , South Korea

2 Department of Civil Engineering , Chungnam National University , 220 Gung-Dong , Yusung-Gu , Daejon 305-764 , South Korea

SUMMARY

Geotechnical engineering problems are characterized by many sources of uncertainty. Some of thesesources are connected to the uncertainties of soil properties involved in the analysis. In this paper, anumerical procedure for a probabilistic analysis that considers the spatial variability of cross-correlatedsoil properties is presented and applied to study the bearing capacity of spatially random soil withdifferent autocorrelation distances in the vertical and horizontal directions. The approach integrates acommercial nite difference method and random eld theory into the framework of a probabilistic analysis.Two-dimensional cross-correlated non-Gaussian random elds are generated based on a KarhunenLo eveexpansion in a manner consistent with a specied marginal distribution function, an autocorrelationfunction, and cross-correlation coefcients. A Monte Carlo simulation is then used to determine thestatistical response based on the random elds. A series of analyses was performed to study the effectsof uncertainty due to the spatial heterogeneity on the bearing capacity of a rough strip footing. Thesimulations provide insight into the application of uncertainty treatment to geotechnical problems and show

the importance of the spatial variability of soil properties with regard to the outcome of a probabilisticassessment. Copyright q 2009 John Wiley & Sons, Ltd.

Received 16 May 2008; Revised 21 December 2008; Accepted 22 February 2009

KEY WORDS : bearing capacity; probabilistic analysis; spatial variability; Monte Carlo simulation

1. INTRODUCTION

Soil properties vary spatially even within homogeneous layers as a result of depositional andpost-depositional processes [1]. Nevertheless, most geotechnical analyses adopt a deterministic

Correspondence to: Sung Eun Cho, Korea Institute of Water and Environment, Korea Water Resources Corporation,462-1, Jeonmin-Dong, Yusung-Gu, Daejon 305-730, South Korea.

E-mail: [email protected] Senior Researcher. Assistant Professor.

Copyright q 2009 John Wiley & Sons, Ltd.


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2 S. E. CHO AND H. C. PARK

approach based on single soil parameters applied to each distinct layer. In response, numerousstudies have been undertaken in recent years to develop a probabilistic analysis method that dealswith the uncertainties of soil properties in a systematic manner [25 ]. Detailed reviews of thesestudies can be found in Mostyn and Li [6], Elkateb et al. [7], and Baecher and Christian [8].

Although probabilistic analysis methods do not consider all of the components of design where judgment needs to be utilized and do not suggest the level of reliability that should be targeted [9],working within a probabilistic framework does imply that the reliability of the system can beconsidered in a logical manner. Thus, probabilistic models can facilitate the development of new perspectives concerning risk and reliability, which are outside the scope of conventionaldeterministic models.

In particular, the effect of inherent random variations of soil properties on the response of geotechnical structures has received considerable attention in recent years. Grifths and Fenton[10, 11 ], Fenton and Grifths [12 ], and Popescu et al. [13] examined the response of shallowfoundations; Haldar and Babu [14 ] analyzed the response of a deep foundation under vertical load;Paice et al. [15 ] studied settlements of foundations on elastic soil; Grifths and Fenton [16 ] studiedslope stability; Popescu et al. [17, 18 ] and Koutsourelakis et al. [19 ] studied seismically induced

soil liquefaction; and Kim et al. [20 ] reported on emergent phenomena related to variability in soilproperties.In this study, a numerical procedure for a probabilistic analysis that considers the spatial vari-

ability of soil properties is presented. The approach integrates a commercial nite differencemethod and random eld theory into a probabilistic analysis. Soils with spatially varying shearstrength are modeled as anisotropic random elds with different autocorrelation distances in thevertical and horizontal directions, and an elasto-plastic nite difference analysis is subsequentlyperformed to evaluate the effects of spatial variability of cross-correlated strength parameters onthe bearing capacity of a footing.

The framework presented by Vo rechovsk y [21] is adopted to generate non-Gaussian cross-correlated random elds with a specied marginal distribution function, an autocorrelation function,and cross-correlation coefcients. The approach is combined into the well-known KarhunenLo eve

(KL) expansion method for simulation of a Gaussian random eld. Then the simulated Gaussianrandom eld is transformed to a non-Gaussian random eld.This study focuses on inherent soil variability, where probabilistic analyses can be employed

to assess the effect of this type of variability on a geotechnical structure. The importance of theeffects of the cross-correlation coefcient and the horizontal and vertical autocorrelation distancesof soil properties on the bearing capacity of a rough strip footing is highlighted.

2. BEARING CAPACITY OF A SHALLOW STRIP FOOTING

Terzaghi [22 ] suggested the following well-known form of the bearing capacity formula for acentrally and vertically loaded, shallow strip footing:

qu = cN c + DN q + B2

N (1a)

where c is the cohesion of the soil, is the unit weight of the soil, D and B are the depth and thewidth, respectively, of the footing, and N c , N q , and N are bearing capacity factors.

Copyright q 2009 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2010; 34 :126DOI: 10.1002/nag


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EFFECT OF SPATIAL VARIABILITY OF SOIL PROPERTIES 3

While the exact values for N remain unknown, the bearing capacity factors N c and N q havebeen solved analytically for a weightless soil using the method of characteristics [23] under theassumption that the soil satises an associated ow rule:

N q = tan 2 4 + 2 e tan (1b)

N c = ( N q 1) cot (1c)

Under the assumption of a weightless soil, the bearing capacity equation simplies to the rst termof Equation (1a) ( qu = cN c ).

Probabilistic studies on the bearing capacity of a footing have been reported previously forcohesive soil by considering randomly distributed undrained strength [10, 11 ]. Fenton and Grifths[12 ] also investigated the inuence of cross-correlation between the cohesion and friction angleon the bearing capacity for c soil. They assumed the random eld to be statistically isotropic(the same autocorrelation distance in any direction through the soil). Popescu et al. [13] studied the

bearing capacity problem by modeling soil properties as homogeneous non-Gaussian random elds.They performed parametric studies to assess the inuence of various probabilistic characteristicsof soil properties on the bearing capacity of a strip foundation placed at ground level on anoverconsolidated clay layer under undrained conditions. Soubra et al. [24] studied the effect of thespatial variability of soil properties on the ultimate bearing capacity of a vertically loaded shallowstrip footing. They modeled the cohesion and friction angle as non-normal anisotropic randomelds based on the spectral representation method.

This paper addresses the bearing capacity of a strip footing located on the surface of a c soilunder the action of a vertical, central load. The analyses were performed by applying a controlleddownward velocity (displacement per calculation step) to the surface nodes on the base of thefooting. Although the spatial variation of soil properties might cause rotation of the footing, whichcannot be predicted in homogeneous soil [13], this aspect is not considered in the present study

for simplicity.The contact stress is calculated by dividing the sum of the vertical footing nodal forces by the

width of the footing extended to the center of the rst element outside the footing, following theanalysis presented in the FLAC manual [25].

3. RANDOM FIELD MODEL

3.1. The spatial variability of soil

One of the main sources of heterogeneity is inherent spatial soil variability, i.e. the variation of soilproperties from one point to another in space due to different depositional conditions and different

loading histories [7].Spatial variation is not a random process; rather it is controlled by location in space. Statistical

parameters such as the mean and variance are one-point statistical parameters and cannot capturethe features of the spatial structure of the soil [26]. Spatial variations of soil properties canbe effectively described by their correlation structure (i.e. autocorrelation function) within theframework of random elds [27 ].



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Two indices of correlation, namely, scale of uctuation [27] and autocorrelation distance [28 ],have been used to describe the spatial extent within which soil properties show a strong correlation.The autocorrelation distance is dened as the distance to which the autocorrelation function decaysto 1 / e, where e is the base of natural logarithms.

The scale of uctuation is dened as

=

( ) d (2)

where ( ) is the autocorrelation function and is the separation of two points. For the exponentialautocorrelation function, the scale of uctuation function is equal to two times the autocorrelationdistance [29 ].

A large autocorrelation distance value implies that the soil property is highly correlated over alarge spatial extent, resulting in a smooth variation within the soil prole. On the other hand, asmall value indicates that the uctuation of the soil property is large.

Although an isotropic correlation structure is often assumed in works reported in the literature,correlations in the vertical direction tend to have much shorter distances than those in the horizontal

direction due to the geological soil formation process for most natural soil deposits. A ratio of about 1 to 10 for these autocorrelation distances is common [8].

A Gaussian random eld is completely dened by its mean ( x) , variance 2( x) , and autocor-relation function ( x, x ) . Autocorrelation functions commonly used in geotechnical engineeringhave been presented by Li and Lumb [4] and Rackwitz [30 ]. In this study, an exponential auto-correlation function is used and different autocorrelation distances in the vertical and horizontaldirections are used as follows:

( x, y) = exp | x x |

lh

| y y |lv

(3)

where lh and lv are autocorrelation distances in the horizontal and vertical directions, respectively.

3.2. Discretization of random elds

The spatial uctuations of a parameter cannot be accounted for if the parameter is modeled byonly a single random variable. Therefore, it is reasonable to use random elds for a more accuraterepresentation of the variations when spatial uncertainty effects are directly included in the analysis.

Because of the discrete nature of numerical methods such as nite element or nite differenceformulation, a continuous-parameter random eld must also be discretized into random variables.This process is commonly known as discretization of a random eld.

Several methods have been developed to carry out this task, such as the spatial average method,the midpoint method, and the shape function method. These early methods are relatively inefcient,in the sense that a large number of random variables are required to achieve a good approximationof the eld. More efcient approaches for discretization of random elds using series expansion

methods such as the KL expansion, the orthogonal series expansion, and the expansion optimallinear estimation method have been introduced [31 ].

A comprehensive review and comparison of these discretization methods have been presentedby Sudret and Der Kiureghian [32 ] and Matthies et al. [33].

All series expansion methods result in a Gaussian eld, which is exactly represented as aseries involving random variables and deterministic spatial functions depending on the correlation



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properties in the problem of bearing capacity. In a probabilistic concept, in principle all thesequantities can be modeled by random elds.

The present study deals with cases where all elds simulated over a region share an identicalautocorrelation function over , and the cross-correlation structure between each pair of simulatedelds is simply dened by a cross-correlation coefcient. This is reasonable since the spatialcorrelation structure is caused by changes in the constitutive nature of the soil over space [12 ].Therefore, the modal decomposition of the given autocorrelation function is done only once.The cross-correlated elds are then expanded using the same spectrum of eigenfunctions andeigenvalues, but the sets of random variables used for the expansion of each eld are cross-correlated [21 ].

In this study, each eld of cohesion and friction angle is expanded using a set of independentrandom variables, and these sets are then correlated with respect to the cross-correlation matrixbetween two expanded random elds according to the framework presented by Vo rechovsk y [21 ].

Let block sample matrix v D , which consists of two blocks, be a jointly normally distributedrandom vector. Each block v Di (i = c, ) represents a Gaussian random vector with M standardGaussian independent random variables, while the vectors v Dc , v

D are cross-correlated with the

cross-correlation coefcient between the cohesion and the friction angle.Each approximate Gaussian random eld H i is then expanded using each block v

Di of the

random vector v D as follows:

H i ( x, ) = i + M

j= 1i j j ( x)

Di , j ( ) ( for i = c, ) (7)

To generate cross-correlated non-Gaussian random elds by the KL expansion, input randomvariables must be transformed into standardized Gaussian random variables to assess the cross-correlation and autocorrelation characteristics for the standardized variables. If the autocorrelationfunction in the original (non-Gaussian) space is given, corrections must be made for each eldover the whole range of autocorrelation coefcients of each pair of non-Gaussian variables totransform the original correlations into the Gaussian space, since the spectral decomposition of the autocorrelation structure is carried out in the Gaussian space. If the target cross-correlationcoefcient in the original (non-Gaussian) space is given, a corrected cross-correlation coefcientmust also be found prior to simulation of the cross-correlated random vector v D . In this study, theNataf model [37 ] was used to transform a non-Gaussian multivariate distribution into a standard-ized Gaussian distribution.

3.5. Sampling strategies of random variables

To generate a random eld, it is necessary to simulate the random vector. In this study, theLatin hypercube sampling technique is used to generate the block sample matrix v D . This

technique can be viewed as a stratied sampling scheme designed to ensure that the upperor lower ends of the distributions are well represented. Latin hypercube sampling is generallyrecommended over simple random sampling when the model is complex or when time is anissue.

In this study, the method proposed by Stein [38 ] for inducing correlation among the variablesbased on the rank of a target multivariate distribution is implemented in Matlab.



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3.6. Transformation to non-Gaussian random elds

Although a Gaussian random eld is often used to model uncertainties with spatial variability forreasons of convenience and lack of available data [13], the Gaussian model is not applicable inmany situations where the random variable is always positive.

For convenience, we nd an underlying Gaussian random eld H that can be easily transformedinto the target eld H . If the random variables are considered to be lognormally distributed, thenappropriate lognormal random elds can be obtained by exponentiating the approximate Gaussianeld from Equation (7) as follows:

H i ( x, ) = exp i + M

j= 1i j j ( x)

Di , j ( ) (for i = c, ) (8)

In this study, random variables are assumed to be characterized statistically by a lognormal distri-bution.

The procedure for the stochastic analysis of bearing capacity with random elds is presented inFigure 1.

Statistical inputmarginal distributionautocorrelationcross-correlation

SimulateNsim realizations

Statistical responseprobability densitycumulative probabilityprobability of failure

Cross-correlated random field

FDM analysis

Latin hypercubeSampling,

D

Solve eigenvalueproblem Eq. (4)

Curve fitting

Figure 1. The procedure for the stochastic analysis of bearing capacity with random elds.



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4. PROBABILISTIC ANALYSIS

4.1. Limit state function and computational method

The problem of the probabilistic analysis is formulated by a vector, X =[ X 1 , X 2 , X 3 , . . . , X k ],

representing a set of random variables. From the uncertain variables, a limit state function g(X )is formulated to describe the limit state in the space of X . In the n-dimensional hyperspace of the basic variables, g(X ) = 0 is the boundary between the region in which the allowable bearingcapacity is not exceeded and the region in which it is exceeded. The probability of failure of thefooting is then given by the following integral:

P f = P [g(X 0)]= g(X ) 0 f X (X ) dX (9)where f X (X ) represents the joint probability density function and the integral is carried out overthe failure domain.

The limit state function concerned with the maximum load that can be placed on the footing just prior to a bearing capacity failure is typically dened by the difference between the capacityC and demand D [39 ]:

g(X ) = C D (10)

where D is the allowable bearing capacity evaluated deterministically with the mean values of c and and using the factor of safety. An Finite Difference Method (FDM) analysis is used todescribe the above limit state function by calculating the bearing capacity, C .

The probability of failure is then calculated from Equation (9) as

P f = P [C D]= P [C qu / FS ] (11)

For practical problems, direct evaluation of the k -fold integral in Equation (9) is virtually impossible.The difculty lies in the fact that complete probabilistic information on the soil properties isnot available and the domain of integration is a complicated function. Therefore, approximatetechniques have been developed to evaluate this integral.

Although various stochastic methods have been proposed in the literature, the only currentlyavailable universal method for accurate solution of geotechnical problems is the Monte Carlotechnique, mainly due to the large variability and strong non-linearity of soil properties [13]. In aMonte Carlo simulation, a series of random elds are generated in a manner consistent with theirprobability distribution and correlation structure, and the response is calculated for each generated

set. The process is repeated many times to evaluate the probability of failure by determiningwhether the limit state functions are exceeded. However, the Monte Carlo Simulation method isnot limited to the calculation of the probability of failure. Various statistical properties evaluatedafter the process of simulation, such as mean, variance, coefcient of skewness, probability densityfunctions, and cumulative probability distribution functions, can provide a broader perspective anda more comprehensive description of a given system.



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5. EXAMPLE ANALYSIS

This section describes plane strain nite difference analyses carried out to calculate the bearingcapacity of a rigid strip footing founded on a weightless soil with shear strength parametersc and represented by spatially varying and cross-correlated random elds. The analysis is two-dimensional, corresponding to a strip footing with innite autocorrelation distance in the out of plane direction and assuming elasticperfectly plastic behavior of the soil material with MohrCoulomb yield criterion.

5.1. Deterministic analysis

To assess the ability of the numerical analysis to predict the bearing capacity, a deterministicanalysis was performed using the mean values of shear strength for homogeneous soil. A stripfooting with a width of B = 2m is located on a c soil having properties as given in Table I. Thenite difference grid consists of 1050 zones and is 14 m wide by 6 m deep, as shown in Figure 2.Horizontal movements on the vertical boundaries of the grid were restrained, while the base of the grid was not allowed to move in either horizontal or vertical direction. A rough strip footingwas simulated by setting the horizontal velocity of the nodes representing the footing to zero.

The results of the deterministic analysis are shown in Figure 3. The bearing capacity wasestimated to be 1.01 MPa from the bearing pressuresettlement curve presented in Figure 6(a). Thisshows relatively good agreement with the value (1.04 MPa) obtained from Equation (1), despitethat the equation is only approximate due to the disagreement between the angle of dilation andthe friction angle.

Table I. Statistical properties of soil parameters.

Parameter X COV Correlation coefcient

Cohesion c (kPa) 50 0.3 0.7 r 0.5Friction angle (deg.) 25 0.2Shear modulus G (MPa) 100 Bulk modulus K (MPa) 200 Dilation angle (deg.) 0

(5 lh 30 and 1 lv 10).

2 m

14 m

6 m

Figure 2. Grid used in the analysis of bearing capacity.



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Figure 3. Results of deterministic analysis for mean values: (a) velocity vector and(b) maximum shear strain rate.

Vectors of velocity and contours of maximum shear strain rate at steady plastic ow for theanalysis are shown in Figure 3. These show a well-dened wedge-shaped zone remaining elasticimmediately below the center of the footing.

More accurate solutions can be obtained by using a more rened mesh around the edge of thefooting. However, these solutions involve larger computational times owing to the singularity atthe edge of the footing. Hence, considering accuracy and efciency, the mesh shown in Figure 2has been retained for the work described in this study.

5.2. Stochastic simulations

In this section, application of the presented procedure is illustrated through a series of simulations.To obtain accurate statistical responses such as the mean, standard deviation, and probabilitydensity function, 5000 sets of random elds were generated for each case based on the statisticalinformation. A series of analyses was then performed based on the generated random elds.A FISH (the built-in programming language of FLAC) function was written to generate randomelds from Equation (8) based on the solutions of the eigenvalue problem and the sampled

vector vD

.As the strength of the soil is spatially distributed, random variable soil strength parameters related

to the bearing capacity of the footing, including the friction angle and cohesion, are considered asrandom elds. The bulk modulus K and shear modulus G were assumed to be deterministic sincethe bearing capacity is not sensitive to these variables. Table I summarizes the statistical propertiesof the soil parameters.



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Random variables, the cohesion and the friction angle, are assumed to be characterized statisti-cally by a lognormal distribution dened by a mean X and a standard deviation X . The lognormaldistribution ranges between zero and innity, skewed to the low range, and is therefore particularlysuited for parameters that cannot take on negative values. Once the mean and standard deviationare expressed in terms of the dimensionless coefcient of variation (COV), dened as V X = X /

X ,

the mean and standard deviation of the underlying normal distribution of ln X are then given by

ln X = ln{1 + V 2 X } (12)

ln X = ln X 0.52ln X (13)

In order to incorporate the dependence between the strength parameters, the cross-correlationcoefcient r (c, ) is needed. Wolff [40 ] reported the correlation between c and for CU testsas r = 0.25 and for CD tests as r = 0.47. Yucemen et al. [41] reported values in a range of 0.49 r 0.24, while Lumb [42 ] noted values of 0.7 r 0.37. A negative correlationimplies that low values of cohesion are associated with high values of friction angle and vice versa.In other words, a negative correlation between the cohesion and the friction angle means that theuncertainty in the calculated shear strength is smaller than the combined uncertainty in the twoparameter values used to model the shear strength. This observation arises from the fact that thevariance of the shear strength is reduced if there is a negative correlation between the cohesionand the friction angle [12]. In this study, a value of 0.5 is considered as a base set, and the rangeof 0.7 r 0.5 around the base value is considered.

According to the results of a literature review by El-Ramly et al. [29 ], the autocorrelationdistance is within a range of 1040 m in the horizontal direction, while in the vertical direction itranges from 1 to 3 m. These values are consistent with those noted by Phoon and Kulhawy [43].

Based on the knowledge of the above statistical input, the effects of varying autocorrelationdistance and cross-correlation were investigated.

In this study the autocorrelation distance l in the Gaussian eld is considered to maximally

exploit the analytical solution of Equation (4). If the autocorrelation distance in the original non-Gaussian eld is used, then the original autocorrelation structure should be corrected accordingto the Nataf model. Figure 4 shows the relationship between the corrected correlation andthe original correlation with a common lognormal distribution for various COVs, where therelationship is represented by the Nataf model as follows [44 ]:

i , i =ln(1 + i , i V

2 X i )

ln(1 + V 2 X i ) ln(1 + V 2

X i )(14)

The gure indicates that the correction factor is only slightly greater than 1.0 over the range of possible autocorrelations, and thus the difference between the correlation structures of Gaussianand lognormal random elds is very small.

Analyses were carried out with the same grid used for the deterministic analysis. As explained inSection 3.3, a continuous random eld can be obtained for an exponential autocorrelation functionon a rectangular domain based on the analytical solution of the eigenvalue problem by the KLexpansion method. Therefore, the random eld discretization is independent of the shape of themesh. The accuracy of the random eld generated by the KL expansion method depends on thenumber of terms used in the series expansion, not on the mesh size. The mesh size only controls



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0

0.0

0.2

0.4

0.6

0.8

1.0

COV=0.45COV=0.30

COV=0.10=

0.2 0.4 0.6 0.8 1

Figure 4. Relationships between original autocorrelation and corrected autocorrelationfor a common lognormal distribution.

0Number of eigenmode

0

10

20

30

40

E i g e n v a

l u e

l v=1ml v=2m

20 40 60 80 100

Figure 5. Eigenvalues of the autocorrelation function.

the accuracy of numerical analysis; therefore, if an acceptable accuracy of numerical analysis canbe obtained for a mesh, then the mesh can be used for the expansion of the random eld bythe KL expansion method. As the spatial mesh is regulated by the stress gradients of the response,the mesh presented in Figure 2 was introduced. Then the spatial discretization is able to model the

variability of the random eld effectively through the expansion of the random eld by the KLexpansion method.

The mean values of cohesion and friction angle were xed, while the COV, autocorrelationdistance, and cross-correlation coefcient were varied.

The number of terms in the truncated series should be carefully chosen so as to accuratelyreect the spatial variability of the random eld being expanded. The number of eigenmodes to



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be retained while discretizing a random eld depends on the magnitudes of the correspondingeigenvalues. Figure 5 represents the decaying trends of the eigenvalues obtained by solving the KLintegral eigenvalue problem. The gure shows that a larger number of terms in the KL expansionare necessary to accurately represent the random eld for smaller autocorrelation distances. In thisstudy 100 terms of eigenmode were used to represent the random elds of cohesion and frictionangle.

Figure 6(a) shows typical bearing pressuresettlement curves for the rst 100 realizations aspart of the result for the case where r (c, ) = 0.5, lh = 10m, and lv = 1m. Figure 7(a) and (b)shows the convergence of the estimated mean and standard deviation of the bearing capacity.

Random properties can be calculated at any point in the domain of analysis such as centroid orintegration point in the elements by the KL expansion method since the method offers a randomeld to be represented in terms of a continuous function. In this study, random properties arecalculated at the centroid of grid zone (element) for the nite difference analysis. Hence, therandom eld used in the nite difference analyses is not the originally generated continuousrandom eld, but a discontinuous random eld with element-wise constant properties. It mayseem that there is no difference to the simple and straightforward midpoint method approach;

however, the use of the KL expansion method has comparative advantages over the midpointmethod.The midpoint method is very straightforward to implement; however, it results in a very large

number of random variables equal to the number of elements used for spatial discretization. Themethod puts limitation on the mesh size compared with the autocorrelation distance and requiresalmost regular discretization. For very short autocorrelation distance, the random eld discretizationhas to be very ne, which increases the numerical effort dramatically. The KL expansion methodautomatically captures the variability using a small number of random variables. In this study,the KL expansion method requires 100 random variables, but the midpoint method requires 1050random variables corresponding to the number of elements. Although the more dense elementsare adopted, the KL expansion method requires the same 100 random variables without loss of accuracy of random eld discretization since the random eld discretization procedure does not

change with the choice of element formulations and boundary conditions. This makes it possible touse the same mesh for different autocorrelation distances, which is advantageous for the parametricstudies.

Figure 8 shows two typical realizations of random elds. In the gure, the lighter regionsdenote a larger strength parameter value (stronger soil) and darker regions indicate a smallerstrength parameter value (weaker soil). It can be observed that the cohesion and the friction angleshow a negative correlation. Figure 8(a) shows that the failure region developed to the left of thefooting through the weak strength path. A non-symmetric failure mechanism, caused by the spatialheterogeneity, is not manifested in the deterministic analysis or the probabilistic analysis witha single random variable due to the representation of a homogeneous soil medium. Figure 8(b)shows another realization of random elds of shear strength parameters and the correspondingfailure mechanism, which indicates a shallower slip path due to the weaker region of soil around

the surface.The histograms of the bearing capacity were determined and tted to a lognormal distribution

for the simulation results obtained from 5000 realizations. The histogram shown in Figure 6(b)captures the major trends and shows that the t appears reasonable.

The sensitivity of the statistical response to the autocorrelation distance was examined. Theeffects of the autocorrelation distance are summarized in Figures 9 and 10 for anisotropic random



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0Bearing capacity (kPa)

0

0.5

1

1.5

2

2.5

P d f ( 1 0 - 3 )

1600

1200

800

400

0

0 0.01 0.02 0.03 0.04 0.05Settlement (m)

Deterministic curve

B e a r i n g p r e s s u r e

( k P a )

500 1000 1500 2000 2500

(a)

(b)

Figure 6. Typical results of the Monte Carlo simulation ( r (c, ) = 0.5, lh = 10m, and lv = 1m):(a) bearing pressuresettlement curves for the rst 100 realizations and (b) probability density function.

elds. As indicated in the gures, the mean values show slight increases but the values of standarddeviation show greater increases with an increase in the autocorrelation distance (see Figure9(a) and (b) for the inuence of horizontal correlation distance and Figure 10(a) and (b) forthe inuence of vertical correlation distance). An innite value of the autocorrelation distance



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0Number of simulation

800

900

1000

1100

1200

M e a n

b e a r i n g c a p a c

i t y

( k P a

)

Number of simulation

100

140

180

220

260

300

S t a n

d a r d

d e v

i a t i o n

b e a r

i n g c a p a c

i t y

( k P a

)

1000 2000 3000 4000 5000

0 1000 2000 3000 4000 5000

(a)

(b)

Figure 7. Convergence of the estimated mean and standard deviation ( r (c, ) = 0.5, lh = 10m, andlv = 1m): (a) mean vs number of trials and (b) standard deviation vs number of trials.

implies a perfectly correlated random eld or a single random variable and provides the maximumvalue of the mean and standard deviation. A smaller autocorrelation distance results in weak correlation between soil parameters over the possible failure surface, which induces signicantuctuation of the soil properties. Therefore, the variability of bearing capacity decreases, since theuctuations are averaged to a mean value along the possible failure surface. On the contrary, thevariability of bearing capacity increases as the autocorrelation distance is increased (see Figures 9(c)and 10(c)). This could be anticipated since a higher autocorrelation distance value indicates thatthe random variables are more strongly correlated, thereby reducing the averaging effects. For therange considered in this study, the horizontal autocorrelation distance does not signicantly affectthe statistical response, as can be seen in Figure 9(d) and (e). On the contrary, Figure 10 showsthat the vertical autocorrelation distance has greater inuence on the statistical response. As shown

in Figure 10(d) and (e), the estimated responses converge to those of a single random variable asthe vertical autocorrelation distance approaches innity.

Figure 11 shows the effect of cross-correlation in the original (lognormal) space on the estimatedstatistical response. In this case the mean value of the bearing capacity shows a slight increase withan increase of the negative value of the cross-correlation coefcient, but the standard deviationdecreases considerably (see (a) and (b)). Since the increase of one parameter value decreases the



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(a) (b)

Figure 8. Typical realization of random eld and corresponding analysis results(r (c, ) = 0.5, lh = 10m, and lv = 1m).



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982

984

986

988

990

992

994

M e a n

b e a r i n g c a p

a c

i t y

( k P a

)

0l h(m)

10 20 30

(a)

170

180

190

200

210

220

S t a n

d a r d

d e v

i a t i o n

b e a r i n g

c a p a c

i t y

( k P a

)

l h(m)0 10 20 30

(b)

0.15

0.20

0.25

C o e

f f i c i e n

t o

f v a r i a

t i o n

l h(m)0 10 20 30

(c)

Figure 9. Inuence of horizontal autocorrelation distance on the estimated statistical response obtainedfrom simulation ( r (c, ) = 0.5, COV c = 0.3, COV = 0.2, and xed lv = 1m): (a) mean bearing capacity;(b) standard deviation of the bearing capacity; (c) COV of the bearing capacity; (d) probability density

function; and (e) probability distribution of the bearing capacity.



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10 000 2000 3000Bearing capacity (kPa)

0

0.5

1

1.5

2

2.5l h= 5 m

l h= 10m

l h= 20m

l h= 30m

l h=

0 1000 2000 3000

Bearing capacity (kPa)

0

0.2

0.4

0.6

0.8

1

C u m u l a

t i v e p r o

b a

b i l i t y

l h= 5 m

l h=10m

l h=20m

l h=30m

l h=FS=1.0

FS=1.5

FS=3.0

(d)

(e)

P d f ( 1 0 - 3 )

Figure 9. Continued .

other value, the variation of the total shear strength is reduced, and consequently the variation of the bearing capacity also decreases (see (c)). An opposite effect for the case of a positive valueof the correlation coefcient is observed, since the increase of one parameter induces an increaseof the other, which results in an increase of variation of the total shear strength.



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980

1000

1020

1040

1060

M e a n


i t y

( k P a

)

160

200

240

280

320

S t a n

d a r d

d e v

i a t i o n


i t y

( k P a

)

0.15

0.20

0.25

0.30

0.35

C o e

f f i c i e n

t o

f v a r i a

t i o n

0

l v (m)

l v (m)

l v (m)

2 4 6 8 10

0 2 4 6 8 10

0 2 4 6 8 10

(a)

(b)

(c)

Figure 10. Inuence of vertical autocorrelation distance on the estimated statistical response obtained fromsimulation ( r (c, ) = 0.5, COV c = 0.3, COV = 0.2, and xed lh = 10m): (a) mean bearing capacity;(b) standard deviation of the bearing capacity; (c) COV of the bearing capacity; (d) probability density

function; and (e) probability distribution of the bearing capacity.



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0

0.5

1

1.5

2

2.5l v= 1m

l v= 2m

l v= 5m

l v=10m

l v=


0

0.2

0.4

0.6

0.8

1

C u m u

l a t i v e p r o

b a

b i l i t y

l v= 1m

l v= 2m

l v= 5m

l v=10m

l v=

FS=1.0

FS=1.5

FS=3.0

1000 2000 3000

0 1000 2000 3000

(d)

(e)

P d f ( 1 0 - 3 )


Figure 11(d) shows that the probability density functions of the bearing capacity have a positivecoefcient of skewness. This implies that the probability density functions have a longer tail tothe right than to the left. The coefcient of skewness decreases with an increase of the negative



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950

960

970

980

990

1000

M e a n


i t y

( k P a

)

120

160

200

240

280

320

S t a n

d a r d

d e v

i a t i o n

b e a r i n g c a p a c i

t y ( k P a

)

-0.8

Cross-correlation coefficient r (c,)

0.12

0.16

0.20

0.24

0.28

0.32

0.36

C o e

f f i c i e n

t o

f v a r i a

t i o n

-0.4 0 0.4 0.8

-0.8

Cross-correlation coefficient r (c,)-0.4 0 0.4 0.8

-0.8

Cross-correlation coefficient r (c,)-0.4 0 0.4 0.8

(a)

(b)

(c)

Figure 11. Inuence of the cross-correlation coefcient on the estimated statistical response obtained fromsimulation (COV c = 0.3, COV = 0.2, lh = 10m, and lv = 1m): (a) mean bearing capacity; (b) standarddeviation of the bearing capacity; (c) COV of the bearing capacity; (d) probability density function; and

(e) probability distribution of the bearing capacity.



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0

1

2

3

0.5

1.5

2.5

r (c, )= 0.7

r (c, )= 0.5

r (c, )= 0

r (c, )= 0.25

r (c, )= 0.5


0

0.2

0.4

0.6

0.8

1

C u m u l a

t i v e p r o

b a

b i l i t y

r (c, )= 0.7

r (c, )= 0.5

r (c, )= 0

r (c, )= 0.25

r (c, )= 0.5FS=1.0

FS=1.5

FS=3.0

1000 2000 3000

0 1000 2000 3000

(d)

(e)

P d f ( 1 0 - 3 )


value of the cross-correlation coefcient. Namely, the shape of the probability density functionbecomes narrower and the uncertainty in the bearing capacity decreases.

Although the effect of the variation of soil properties is not presented in this paper, increasingthe COV of cohesion and friction angle resulted in a reduction of the mean value and an increaseof the standard deviation of the bearing capacity [10, 12 ].



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-0.8

Cross-correlation coefficient r (c,)

1.0x10 -81.0x10 -71.0x10 -61.0x10 -51.0x10 -41.0x10 -31.0x10 -21.0x10 -11.0x10 0

1.0x10 -4

1.0x10 -3

1.0x10 -2

1.0x10 -1

1.0x10 0

P f

P f

FS=1.0FS=1.5FS=2.0

0l v (m)

FS=1.0FS=1.5FS=2.0

2 4 6 8 10

-0.4 0 0.4 0.8(a)

(b)

Figure 12. Estimated probability of failure for bearing capacity: (a) inuence of cross-correlation

and (b) inuence of vertical correlation distance.

Figure 12 presents a summary of the probability of failure for the bearing capacity. This indicatesthat the simulation based on random elds gives a probability of exceeding the deterministic bearingcapacity (FS = 1) greater than 50%. This in turn means that the median bearing capacity givenby the simulation is always smaller than the deterministic bearing capacity, which has also beenobserved by other investigators [10, 12, 13 ]. Additionally, the probability of failure was negligiblein the case of FS = 3 for all cases studied in this paper.

Figure 12(a) shows the estimated probability of failure against the cross-correlation. The prob-ability of failure decreases with an increase of the negative correlation coefcient. Therefore,the assumption of independence between cohesion and friction angle gives conservative results

if the actual correlation is negative, but slightly unconservative results are obtained if the actualcorrelation is positive.

The effects of the autocorrelation distance on the probability of failure are summarized inFigure 12(b). As indicated in the gure, with an increase in the vertical autocorrelation distance,the probability of failure decreases slightly if FS = 1.0, but increases if a factor of safetyof 1.5 or 2.0 is used.



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6. CONCLUSIONS

The effect of spatial variability of cross-correlated shear strength parameters on the bearing capacityof a strip footing was studied using random eld theory integrated into a commercial nite differencepackage. An exponential autocorrelation function that considers different autocorrelation distancesin the vertical and horizontal directions is used to describe the spatial variability of the soil.Two-dimensional cross-correlated non-Gaussian random elds are generated by a KL expansion,which is the most efcient method requiring the smallest number of random variables to representthe eld within a given level of accuracy. A series of Monte Carlo simulations is then conductedto determine the statistical response of the bearing capacity of a shallow footing. The simulationobserved various non-symmetric failure mechanisms caused by the spatial heterogeneity.

In the problem studied in this paper the cross-correlation between cohesion and friction angleand the autocorrelation distance in the vertical direction were found to be signicant factors inthe stochastic behavior of bearing capacity. In particular, when a negative cross-correlation isconsidered, the effect on the probability of failure is important. The probability of failure decreaseswith an increase of the negative correlation coefcient. Therefore, the assumption of independence

between cohesion and friction angle gives conservative results if the actual correlation is negative,but slightly unconservative results are obtained if the actual correlation is positive.With a decrease in the vertical autocorrelation distance, the probability density function of the

bearing capacity becomes narrower and the uncertainty in the bearing capacity decreases. On thecontrary, for the range considered in this study, the horizontal autocorrelation distance does notsignicantly affect the statistical response.

The obtained results provide insight regarding the stochastic analysis in the eld of geotechnicalengineering and show the importance of the spatial variability of soil properties in the outcomesof a probabilistic assessment.

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Effect of spatial variability of cross-correlated soil properties on bearing capacity of strip footing