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Effect of soil spatial variability on the response of laterallyloaded pile in undrained clay

Sumanta Haldar 1, G.L. Sivakumar Babu *

Department of Civil Engineering, Geotechnical Engineering Division, Indian Institute of Science, Bangalore 560 012, India

Received 3 April 2007; received in revised form 11 October 2007; accepted 11 October 2007Available online 26 November 2007

Abstract

A comprehensive study is performed on the allowable capacity of laterally loaded pile embedded in undrained clay having spatialvariation of strength properties. Undrained shear strength is considered as a random variable and the analysis is conducted using randomfield theory. The soil medium is modeled as two-dimensional non-Gaussian homogeneous random field using Cholesky decomposition

technique. Monte Carlo simulation approach is combined with finite difference analysis. Statistics of lateral load capacity and maximumbending moment developed in the pile for a specified allowable lateral displacement as influenced by variance and spatial correlationlength of soil’s undrained shear strength are investigated. The observations made from this study help to explain the requirement of allowable lateral capacity calculations in probabilistic framework. 2007 Elsevier Ltd. All rights reserved.

Keywords: Clay; Lateral loads; Piles; Probabilistic analysis; Spatial variability

1. Introduction

Assessment of lateral load capacity of piles embedded inundrained clay medium, exhibiting spatial variability is of considerable importance in geotechnical engineering.Hence, stochastic treatment for analysis of soil spatial var-iability and probabilistic models for assessment of lateralallowable load are necessary. Due to inherent variability,property variations in the in-situ soil normally exhibit atrend with distance and scatter and that are represented

by the mean value, coefficient of variation and correlationdistance. The property values are correlated to each otherat adjacent points, and the distance up to which this signif-icant correlation exists is termed as correlation distance.The effect of inherent random variations of soil propertieson the response of foundation structures received consider-

able attention in the recent years. Griffiths and Fenton [1],Fenton and Griffiths [2] examined the response of shallowfoundations; Fenton and Griffiths [3], Haldar and Babu [4]analyzed response of deep foundations under vertical load.The present study focuses on the response of laterallyloaded pile in a spatially varied soil media. There are twoaspects in designing laterally loaded pile foundations: (i)maximum lateral displacement at pile head and (ii) maxi-mum bending moment in the pile. If these two aspectsare satisfied, pile is considered to be safe and the load cor-

responding to allowable lateral displacement can be consid-ered as lateral load capacity of the pile. Several methodsare described in literature to determine the lateral capacityand failure mechanism of pile in a homogeneous soil. Fewof them are summarized as follows: Broms [5] assumed alimiting resistance of 9su to determine the ultimate lateralload. Based on upper bound analysis, Randolph and Hou-lsby [6] indicated that the value 9su is largely empirical andsuggested values in the range of 9.14su –11.94su. Phoon andKulhawy [7] indicated that interpretation based on speci-fied displacement limit, rotational limit or moment limit,

0266-352X/$ - see front matter 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compgeo.2007.10.004

* Corresponding author. Tel.: +91 80 22933124; fax: +91 80 23600404.E-mail addresses: [email protected] (S. Haldar), gls@civil.

iisc.ernet.in (G.L. Sivakumar Babu).1 Tel.: +91 80 22932815; fax: +91 80 23600404.

www.elsevier.com/locate/compgeo

Available online at www.sciencedirect.com

Computers and Geotechnics 35 (2008) 537–547

mailto:[email protected]

mailto:gls@civil.

mailto:gls@civil.

mailto:[email protected]

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and hyperbolic capacity do not consider the actual soil-shaft behaviour and hence, moment limit is a better choiceto compute lateral capacity. Fan and Long [8] showed thatthe existing methods for predicting soil resistance like APImethod, Hansen’s method and Broms method give differ-ent values of prediction of soil resistance for the same soil.

Poulos and Davis [9], Hsiung and Chen [10] indicated thatin designing pile foundations under lateral loads, the max-imum lateral displacement controls the design rather thanthe ultimate resistance. Zhang and Ng [11] indicated thatmany geotechnical structures such as building foundationsare more often governed by allowable displacementrequirements rather than by ultimate limit requirements.Hence, this paper considers the evaluation of lateral loadcapacity based on specified allowable/serviceable lateraldisplacement.

The objective of this paper is to investigate the statisticsof allowable lateral capacity of pile for a specified allow-able lateral displacement and maximum bending moment,

as influenced by spatial variation and correlation structureof soil’s undrained shear strength. The lateral capacity isdefined as lateral load for a specified lateral allowable dis-placement and this lateral capacity is termed as allowable

load throughout this paper. The present study is conductedusing random field theory combined with finite differencecode, Fast Lagrangian Analysis of Continua, FLAC [12].Two-dimensional non-Gaussian homogeneous randomfield is generated by Cholesky decomposition technique.Monte Carlo simulation is conducted to determine the sta-tistics of the pile response. The allowable load is computedbased on generated lateral load–displacement curves. The

maximum bending moment corresponding to allowableload is also determined. Propagation of failure and forma-tion of failure mechanism close to the pile foundation areexamined considering soil stiffness and shear strain levelin soil near pile. The following sections present the detailsof analysis, typical results obtained and the conclusionsfrom the study.

2. Method of analysis

2.1. Overview of finite difference model

The finite difference program uses the 4-noded quadri-lateral grids. The total soil medium is divided into finite dif-ference grids for analysis. Appropriate boundaryconditions are applied in the soil zone. At the bottom planeof the grid, all movements are restrained. The lateral sidesof the mesh are free to move in downward direction (veY -axis) but not in the X -direction. In order to investigatethe pile response a vertical pile is considered which isembedded in soil media. The soil is modeled using elasto-plastic Mohr–Coulomb constitutive model and the pile ismodeled using linearly elastic beam elements with interfaceproperties (termed as pile element). Each element has three-degrees-of freedom (two displacements and one rotation)

at each node. The pile elements interact with the finite dif-

ference grids via shear and normal coupling springs whichare represented by appropriate stiffness values. The cou-pling springs are similar to the load/displacement relationsprovided by ‘ p – y’ curves. However, ‘ p – y’ curves areintended to capture the interaction of the pile with thewhole soil mass, while in the present analysis nonlinear

springs represent the local interaction of the soil and pileelements [12]. The interaction of pile with grid is repre-sented by four parameters: (i) k n = normal stiffness, (ii)k s = shear stiffness, (iii) cohesive strength of shear springthat prescribes the limiting shear force at pile–soil interface,(iv) cohesive strength normal spring which prescribes limit-ing normal force. The values of the interface parameterscan be derived from the undrained cohesion/shear modulusof soil. The shear and normal stiffness (expressed in stress-per-distance units) are obtained as [12]

k n or k s ¼ 10 max K þ 4

3G

D z min

ð1Þ

where K and G are the bulk and shear modulus of soil zone,respectively; and Dzmin is the smallest dimension of anadjoining zone in the normal direction. The cohesivestrength of shear coupling spring (expressed in force-per-distance units) can be taken as the pile perimeter timesthe undrained cohesion of the soil (e.g., for a circular pile,2p times the radius). Cohesive strength of the normal cou-pling spring (expressed in force-per-distance units) can beconsidered as limiting lateral resistance and it can be com-puted based on Broms solution [5] as 9 · su · Dp, where Dp

is the pile diameter. Lateral load is applied incrementally atthe top of the pile head.

In a numerical analysis, Donovan et al. [13] suggest thatthe linear scaling of material properties is the convenientway of distributing the discrete effect of elements over aregularly spaced distance between the elements. A three-dimensional pile with regularly spaced interval can bereduced to two-dimensional problem considering averagingover the distance between the elements. The relationbetween actual properties and scaled properties can bedescribed by considering the strength properties for regu-larly spaced piles. For a pile element, the following proper-ties are to be scaled: (i) elastic modulus, (ii) stiffness of theinterface springs and (iii) pile perimeter. The input param-eters are given as the actual values, divided by the spacingof the piles. The actual pile responses (forces and moments)are determined by multiplying the spacing value.

2.2. Validation of numerical scheme

The validation of the finite difference code is conductedwith reference to a pile test result (lateral load verses lateralground line displacement) reported in Ref. [9]. The pile testdata is presented in Table 1. A 10 m · 10 m size field is con-sidered and the entire field is divided into 900 elements (30numbers row-wise and 30 numbers column-wise). As indi-cated earlier, the soil is modeled using elasto-plastic Mohr–

Coulomb constitutive model and the pile is modeled as lin-

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early elastic pile element and is discretized into 20 equalsegments. The values of bulk and shear modulus, bulk den-sity of soil, pile properties namely diameter, length andflexural rigidity are given in Table 1. The shear and normalcoupling springs properties are obtained as follows:

The values of normal (k n) and shear stiffness (k s) areobtained as 10 times of K þ 4

3G

=D z min. The value of

Dzmin = 0.33 m (10 m/30 = 0.33 m). Hence value of k nand k s are set to 1.68 · 105 kN/m/m. The cohesive strength

of shear coupling spring is the pile perimeter timesundrained shear strength; hence, value is p · 0.038 ·14.4 = 1.72 kN/m. According to Broms [5], the value of cohesive strength of the normal coupling spring can be esti-mated as 9 · su · Dp = 9 · 14.4 · 0.038 = 4.928 kN/m. Theinput parameters are linearly scaled as the model is reducedto 2D problem. The lateral load is applied incrementally.The unbalanced force of each node is normalized by grav-itational force acting on that node. A simulation is consid-ered to have converged when the normalized unbalancedforce of every node in the mesh is less than 103 and theresults are obtained. Fig. 1 shows the experimental and pre-

dicted load–displacement curves. The results show thatpredicted values are in good agreement with the experimen-tal values. This validation lends credence to the use of the

approach in obtaining the response of laterally loaded pilein soft clay.

2.3. Deterministic analysis

The focus in this study is to understand the response of a

free head concrete bored pile of 1.0 m diameter (Dp) andlength of 10 m (Lp). Hence, the analysis uses a differentfield size (20 m · 20 m) in the following sections to avoidany boundary effect. Total soil medium is discretized into900 numbers of finite difference elements in 30 rows and30 columns of equal size with each side dimension of 0.67 m. A uniform value of su = 20 kPa is employed to findout the deterministic allowable load for homogeneous soil.The same value of undrained shear strength is consideredas mean undrained shear strength for the probabilisticanalysis which is explained in the later section. Pile is eccen-tric (e) over the ground by 1.0 m. The pile element isdivided into 21 equal segments. Pile–soil interface proper-

ties are scaled to represent plane strain condition. Fig. 2illustrates the schematic diagram of the model. The actualvalues (not scaled) of the pile stiffness parameters are givenin Table 2. Pile load–displacement curve for pile head isobtained and allowable load for homogeneous soil is com-puted corresponding to the allowable displacement of 0.0508 m, which is the upper limit for lateral displacementallowed by AASHTO [14]. The maximum bending momentis also obtained corresponding to the allowable load. Therandom field is generated in terms of undrained shearstrength values assigned for each grid location. The detailsof random field model and the approach used in merging

the random soil properties in grid locations in the finite dif-ference code are described in the following section.

Table 1Pile and soil data [9]

Pile data Value Soil data Value

Pile diameter, Dp (m) 0.038 Undrained shear strength(kN/m2)

14.40

Pile length, Lp (m) 5.25 Bulk density (kN/m3) 19Bending rigidity (EI)

(kN m2)

31,600 Shear modulus, G (kN/m2) 925.71

Bulk modulus, K (kN/m2) 4320

0

1

2

3

4

0 0.005 0.01 0.015

Ground line lateral displacement (m)

L a t e r a l l o

a d ( k N )

Measured (Poulos and Davis, 1980)

Predicted

Fig. 1. Comparison of measured and calculated load–displacement

curves.

YX

BXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y BXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

BXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX

Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y B

XXXXXXXXXXXXXXXXXXXXXXXXXXXXX

0m

20 m

20 m

0m

e=1.0 m

DP=1.0 m

L P = 1 0 m

X

Y

Fig. 2. Schematic diagram of the finite difference model.

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2.4. Random field model

The property variations of the in-situ soil represented bythe mean value, coefficient of variation and correlation dis-tance influence the likely parameters for design. In the pres-ent study, soil undrained shear strength su is considered asrandom variable and assumed to be a Log-Normally dis-tributed value represented by parameters mean l su

, stan-dard deviation r su

and spatial correlation distance dz. Useof Log-Normal distribution is appropriate as the soil prop-erties are non-negative and the distribution also has a sim-ple relationship with normal distribution. A Log-Normallydistributed random field is given by

suð~ xiÞ ¼ expflln suð~ xÞ þ rln su ð~ xÞ G ið~ xÞg ð2Þ

where ~ x is the spatial position at which su is desired. G ð~ xÞ isa normally distributed random field with zero mean withunit variance. The values of lln su

and rln su are determined

using Log-Normal distribution transformations given by

r2ln su

¼ ln 1 þr2 su

l2 su

!¼ ln 1 þ COV2

su

ð3Þ

lln su ¼ ln l su

1

2r2

ln su ð4Þ

The correlation function is considered as exponentiallydecaying correlation function as given by

q su ðsÞ ¼ exp 2sd z

ð5Þ

where s ¼ j~ x1 ~ x2j is the absolute distance between the twopoints and dz is the spatial correlation distance. The corre-lation matrix is decomposed into the product of a lower tri-angular and its transpose by Cholesky decomposition,

L LT ¼ q su ð6Þ

Given the matrix L, correlated standard normal randomfield is obtained as follows (e.g. [15,16]):

G i ¼ Xi

j¼1

LijZ j; i ¼ 1; 2; . . . ; n ð7Þ

where Z j is the sequence of independent standard normalrandom variables. Typical values of COV su

for undrainedshear strength lie in the range of 10–50% [17]. In the pres-ent study, results are presented assuming that the soil hasisotropic correlation structure; therefore the correlationdistance is the same in both horizontal and vertical

directions.

2.5. Implementation of random field

The correlation matrix is generated considering Eq. (5).The value of the lag distance (s) is considered to be the cen-ter to center distance of the consecutive grids. Fig. 3explains the evaluation of correlation matrix consideringthe discretization of finite difference grid. For an example,if the center to center distance between grids 1 and 2 is dx,the correlation between these two grids can be calculatedby putting the value s = dx in Eq. (5). Similarly correlationbetween grid 1 with 3, 4, 5 can be established by placing

s = 2 · dx, 3 · dx, and 4 · dx and between grid 1 with

31, 32, 33 are d y, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

d x2 þ d y 2p

and

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2d xÞ2 þ d y 2

q , so on.

Therefore values in the 1st row of the correlation matrixare the correlation coefficients between grid 1 and othergrids, and leads to 900 values in a row (as number gridsis 30 · 30 in the present study). Hence considering all thegrids, the size of the correlation matrix is 900 · 900. Oncethe correlation matrix is established, it is decomposed intolower and upper triangular matrices using Cholesky decom-

position technique. The correlated standard normal ran-dom field is obtained by generating a sequence of

independent standard normal random variables (with zeromean and unit standard deviation) and decomposed corre-lation matrix by Eq. (7). The correlation distance (dz) isutilized to prepare correlation matrix, whereas COV su

is

Table 2Pile and soil data for the deterministic analysis

Pile data Value

Pile diameter, Dp (m) 1.0Pile length, Lp (m) 10.0Bending rigidity (EI) (kN m2) 1.1 · 106

Section modulus, Z (m3) 0.0981

Yield strength of pile, F y (kN/m2) 1.11 · 104

Yield moment of pile section, M y (kN m) 1088Interface spring normal stiffness, k n (kN/m/m) 1.15 · 105

Interface spring shear stiffness, k s (kN/m/m) 1.15 · 105

Cohesive strength of shear coupling spring (kN/m) 62.83Cohesive strength of normal coupling spring (kN/m) 180

Soil data

Undrained shear strength (kN/m2) 20Shear modulus, G (kN/m2) 1285.71Bulk modulus, K (kN/m2) 6000

1 2 3 4 5 6

31 32 33 34 35 36

61 62 63 64 65 66

91 92 93 94 95 96

121 122 123 124 125 126

151 152 153 154 155 156

30

60

90

120

150

180

dx

dy

Fig. 3. Discretization of finite difference grid.


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utilized to determine the standard deviation of theundrained shear strength (Eq. (3)). Realizations of Log-Normally distributed undrained shear strengths at eachgrid location are obtained by transformation presented inEq. (2) for a specified mean, standard deviation of su.The total computation process is conducted by developinga subroutine in ‘FISH’ code in FLAC. Monte Carlo simu-lation approach is used in the generation of sample func-tions of 2D Log-Normal random field. In the presentstudy, values of COV su

in the range of 10–50% and corre-

lation distance 1.5–50 m are used (Table 3). Meanundrained shear strength (l su

) is taken as 20 kPa. For eachset of statistical properties given in Table 3, Monte Carlosimulation is performed and totally 100 realizations of shear strength random field are generated for the analysis.The code is run hundred times and in each run the ‘FISH’program (subroutine) which assigns different realizations of random field in finite difference grids is executed. The valid-ity of number of realizations in Monte Carlo simulation isexamined. The statistical fluctuation of the expected valuesand standard deviations of allowable load and maximum

bending moment are evaluated by means of Monte Carlosimulations are shown in Fig. 4 as a function of samplesize. Observing the figure, 100 realizations are consideredbecause of the following reasons: (i) the fluctuation fallswithin a tolerable range (between 5% and 10%) after 100samples, (ii) the calculation represents the worst case i.e.

at highest COV su ¼ 50% and dz = 50 m, where possibilityof fluctuation in expected values and standard deviationsare maximum, (iii) with the sample size of 100, estimatedexpected value and standard deviation indicates satisfac-tory stability. Popescu et al. [18,19] also performed proba-bilistic analysis using 100 simulations.

Each realization produces a different lateral load–dis-placement curve. Values of Qi

alst i.e. allowable lateral loadcorresponding to lateral displacement of 0.0508 m fromi th load–displacement curve and, M imax i.e. maximum bend-ing moment in pile corresponding to i th allowable lateralload Qi

alst are obtained (i = 1,2, . . . ,100). Statistics of theresponses are obtained by ensemble averaging. Results

are examined in terms of dimensionless spatial correlationdistance given by dz/Lp, where Lp is the pile length. A flow-chart presented in Appendix I gives the scheme used forstatistical numerical analysis.

3. Stochastic verses deterministic analysis results

COV su and dz have physical significance as they reflect the

nature (erratic or homogeneous) of a random field. Theeffect of COV su

and dz can be observed in Fig. 5a–c, whichshows the gray scale representation of possible realizations

Table 3Assumed ranges of probabilistic descriptors for soil undrained shearstrength

Probabilistic descriptor Range

Coefficient of variation, COV su (%) 10, 30, 50

Correlation distance, dz (m) 1.5, 5.0, 15.0, 50.0Normalized vertical correlation distance, dz/Lp 0.15, 0.5, 1.5, 5.0

Probability distribution function, pdf Log-Normal

0 50 100 150 20040

50

60

70

80

90

Sample size Sample size

Sample size Sample size

Q a

( k N )

0 50 100 150 2000

10

20

30

40

50

Q a

( k N )

0 50 100 150 200120

140

160

180

200

M m a x

( k N m )

0 50 100 150 2000

20

40

60

80

M m a x

( k N m )

a

b

c

d

maxM : Mean of maximum

bending moment.aQ: Mean of lateral

allowable load.

aQ: Standard deviation of

lateral allowable load.

maxM : Standard deviation of

maximum bendingmoment.

Fig. 4. (a) Mean of lateral allowable load as a function of Monte Carlo (MC) sample size. (b) Standard deviation of lateral allowable load as a function of MC sample size. (c) Mean of maximum bending moment developed in pile with respect to MC sample size. (d) Standard deviation of maximum bending

moment developed in pile with respect to MC sample size.


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of the undrained shear strength. Darker shades denoteweaker zones having lower undrained shear strength. Real-izations of random field generated for COV su ¼ 10%, 30%and dz = 1.5 m are presented in Fig. 5a and b. The resultsindicate that increase in COV su

contributes to the erraticnature of soil and hence more number of weak zones arepresent. In the deterministic analysis, a uniform soil is gen-erally considered and a value of su = 20 kPa (present study)is realized in the soil medium, whereas in the real field due tospatial variability effect, shear strength varies within a range.A higher range of su is generally observed for a field of highCOV su

(say 30%) compared to a field of low COV su values

(say 10%) and this aspect is clearly evident from Fig. 5aand b. A range of su = 16–26 kPa is observed for a realiza-tion of random field having COV su ¼ 10%, where range of su = 10–40 kPa is observed for COV su ¼ 30%. Realizationsof random field generated for dz = 1.5 m, 15.0 m andCOV su ¼ 30% presented in Fig. 5b and c show the effect of correlation distance. At low values of dz (say 1.5 m) domainis similar to an erratic field and as scale of fluctuationincreases (say 15.0 m), it can be noted that the cohesion fieldbecomes more homogeneous.

For various combinations of COV su and dz/Lp as given in

Table 3, the Monte Carlo simulation are conducted and the

pile lateral load–displacement curves are obtained. Typicalrealizations of load–displacement curves for a combinationof COV su ¼ 30% and dz = 1.5 m are presented in Fig. 6.Ensemble average of load–displacement curves is taken asthe mean load–displacement curves for spatially varied soil.The load–displacement curve based on uniform shearstrength all over the soil zone (the deterministic load–dis-placement curve) is also shown in the same figure. It isobserved that the resulting mean allowable load corre-sponding to allowable lateral deflection of 0.0508 m of spa-tially varying soil, lQalat is significantly lower than thecorresponding value for the homogeneous soil (Qdet

a lat). The‘‘goodness of fit’’ test is conducted using well known Kol-mogorov–Smirnov (K–S) test for COV su ¼ 30% anddz = 1.5 m and the results are presented in Fig. 7. The K–Stest compares the observed cumulative frequency and thecumulative distribution function (CDF) of allowable loadwith an assumed theoretical distribution. It is apparent thatthe Log-Normal distribution represents reasonably well.Fig. 8a and b illustrates typical lateral displacement con-tours for a certain level of lateral loading (not up to ultimatestate) in homogeneous medium and spatially varied soil cor-responding to COV su ¼ 30%, dz = 1.5 m (dz/Lp = 0.15). It is

observed that lateral displacement has a regular pattern for

s u ( k P a )

Weaker zoneStronger

zone

s u ( k P a )

Stronger zone

Weaker zone

s u ( k P a )

Stronger

zone

Weaker zone

a

c

b

Fig. 5. One realization of random field, (a) COV su ¼ 10%, dz/Lp = 0.15; (b) COV su ¼ 30%, dz/Lp = 0.15; (c) COV su ¼ 30%, dz/Lp = 1.5.


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the homogeneous soil whereas, irregular pattern is observedfor spatially varied soil due to presence of weaker zones.

4. Influence of coefficient of variation and correlation

distance

4.1. Influence on pile responses

The results presented in Figs. 9 and 10 indicate the two

important probabilistic characteristics of the soil variabil-

ity, coefficient of variation and correlation distance of undrained shear strength. They have significant effect onthe lateral allowable load and maximum bending moment.Both the characteristics control the amount of weak zonesin the soil mass. Fig. 9 shows the influence of dz/Lp andCOV su

on the estimated mean allowable load, lQa lat. Thedeterministic solution, for homogeneous soil (su = 20 kPain all zones), the allowable capacity is also presented inthe same figure which is termed as deterministic solution.The load corresponding to allowable lateral displacementof 0.0508 m is obtained as 83 kN which is termed asdeterministic allowable capacity ðQdet

a latÞ. Similarly the maxi-mum bending moment is also observed at the same loadlevel as 180 kN m ð M det

maxÞ which is presented in Fig. 10.Fig. 9 shows that there is significant reduction of meanallowable load for spatially varied soil compared to deter-ministic allowable load. At low values of COV su

, meanallowable load is greater than the value obtained for highervalue of COV su

. For an example, a mean value of lQalat =

65.1 kN is observed for COV su ¼ 30% and dz = 1.5 m

0

20

40

60

80

100

120

0 0.01 0.02 0.03 0.04 0.05 0.06

Pile lateral head displacement (m)

L a t e r a l l o a d ( k N )

Deterministic curve (homogenous soil)

Mean curve (spatially varying soil)

100 realizations of MC analysis

A l l o w a b l e d i s p l a c e m e n t = 0 . 0 5 0 8 m

Q a latdet

=83 kN

Q a lat = 65 kN

COVsu = 30%, z /Lp = 0.15

Fig. 6. Load–displacement curves for homogeneous soil and spatiallyvaried soil for COV su ¼ 30% and dz/Lp = 0.15.

40 60 80 1000

0.2

0.4

0.6

0.8

1

Allowable load (kN)

C u m u l a t i v e p r

o b a b i l i t y

Actual distribution

Log-Normal fit

Mean load=65 kN

Deterministic load = 83 kN

Fig. 7. K–S test on allowable load for Log-Normal distribution forCOV su ¼ 30%; dz/Lp = 0.15.

X-displacement contours

-1.10E-02

-8.00E-03

-5.00E-03

-2.00E-03

1.00E-03

4.00E-03

7.00E-03

Contour interval= 1.00E-03

Original position of pile

Deflected position of

Deflected position of

pile

X-displacement contours

-1.90E-02

-1.50E-02

-1.10E-02

-7.00E-03

-3.00E-03

1.00E-03

5.00E-03

9.00E-03


a

b

X-displacement (m)

X-displacement (m)

Fig. 8. (a) Lateral displacement contours for homogeneous soil and(b) lateral displacement contours for spatially varied soil (COV su ¼ 30%,dz/Lp = 0.15).


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(dz/Lp = 0.15), whereas for the same correlation distance,

the lQalat value becomes 60.2 kN, if the COV su ¼ 50%.The influence of dz (or, dz/Lp) is also pronounced withthe value of lQalat. It is observed that the value of lQa lat

increases gradually as the value of dz (or, dz/Lp) reducesto zero. The reason behind this phenomenon is that as dz

becomes vanishingly small, weakest path becomes moreerratic which means that the formation of failure path iscorrespondingly longer and as a result, the weakest pathstarts to find shorter routes cutting through higher strengthmaterial as described by Griffiths and Fenton [1], Griffithset al. [20]. The value of lQa lat marginally decreases up todz = 5.0 m (dz/Lp = 0.5) and then gradually increases. It

represents an important phenomenon with respect to the

design of laterally loaded pile based on spatially varied soil.At dz = 5.0 m, there is maximum reduction of allowableload, hence it can be represented as worst case. This behav-iour is also noted by Fenton and Griffiths [21] for shallowfoundation and Niandou and Breysse [22] for piled raft. Itis noted that generally the worst correlation distance is

problem specific and the value varies within the size of structure. When dz is of intermediate value (i.e. withinstructure size), the structure is sensitive to fluctuation inthe soil properties [22]. For the present case worst correla-

tion distance is the half of the length of pile ðdworst z ¼ 0:5 LpÞ.

The mean allowable load increases with the increase in dz

value. Similar observation is made for maximum bendingmoment. Fig. 10 shows similar trends for the maximumbending moment corresponds to allowable load for homo-geneous and spatially varied soil. It shows mean maximumbending moment increases at low values of COV su

.

4.2. Influence on formation of failure

This section describes the effect of spatial variability onthe maximum shear strain level induced in the soil near pileand the stiffness. The formation of failure surface can bedescribed in terms of accumulation of shear strains in thesoil near pile. It is due to the applied lateral load in pile headand it can be noted that it relates to the number of weakerzones present in the soil. However in the present analysis pileis not loaded up to ultimate state, but the development of maximum shear strains controls the likely failure mecha-nism. To determine the effect of spatial variability on themaximum shear strain in soil, an analysis is conducted con-

sidering different values of COV su and dz and for uniformsoil. Fig. 11a and b shows the maximum shear strain con-tours using deterministic and probabilistic analysis.

100

120

140

160

180

200

220

240

0 1 2 3 4 5

z /Lδ p

M e a n m a x i m u

m

m o m e n t ,

M

m a x ( k N - m )

Deterministic solution

COVsu=10%

COVsu=30%

COVsu=50%

Fig. 10. Mean maximum bending moment with variation in COV su and

dz/Lp.

50

55

60

65

70

75

80

85

0 1 2 3 4 5

z /Lδ P

M e a n u l t i m a t e l a t e r a l l o a d ,

Q a l a t ( k N )

Deterministic solution

COVsu=10%

COVsu=30%

COVsu=50%

w o

r s t c a s e

M e a n a l l o w a b l e l a t e r a l l o

a d

Fig. 9. Mean allowable lateral load with variation in COV su and dz/Lp.

M a x . s t r a i n 7

1 0 - 3

Max. shear strain increment


M a x . s t

r a i n 1 . 4

1 0 - 2

Max. shear strain increment


a b

Fig. 11. (a) Incremental strain contours for homogeneous soil; (b)

incremental strain contours for COV su ¼ 30%, dz/Lp = 0.15.


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Fig. 12a and b represents the maximum shear strain con-tours for COV su ¼ 50%, dz = 1.5 m (dz/Lp = 0.15) andCOV su ¼ 50%, dz = 50 m (dz/Lp = 5.0) respectively. Thepractical significance of the results can be noted as follows:

(1) An erratic formation surface is observed, passing

mainly through weaker soil zones. As correlation dis-tance increases, the soil becomes more homogeneousand this aspect is evident from Fig. 11a and b.

(2) It is also important to observe that maximum strainlevel where pile undergoes maximum lateral move-ment, also changes due to the effect of soil variability.The maximum shear strain corresponding to ultimateload in the homogeneous soil is about 0.007(Fig. 11a) whereas in the spatially varied soil, it isobserved that the maximum shear strain value is inthe range of 0.014 for COV su

¼ 30% (Fig. 11b).(3) The effect of correlation distance can be discerned

from Fig. 12a and b, which shows sample realizations

of spatially varied soil and maximum shear strain con-tours for COV su ¼ 50%, dz = 1.5 m (dz/Lp = 0.15) andCOV su ¼ 50%, dz = 50 m (dz/Lp = 5.0) respectively. Itcan be noted that the effect of correlation distance issignificant. Higher value of maximum strain level(0.02) is observed for low correlation distance (saydz = 1.5 m) and a lower value (0.008) is observed inhigher correlation distance (say dz = 50 m).

The observations from above results are also reflected insoil stiffness values and the corresponding mean load– deflection curves. The stiffness values can be obtained by

analyzing load–displacement curves. The stiffness values(K ) are computed as incremental applied load divided by

incremental lateral displacement. The mean stiffness valuesare obtained from Monte Carlo simulated load–displace-ment curves. Fig. 13a shows that the mean load–displace-ment curves become flat as COV su

increases, whichindicate that soil becomes less stiff as variability increases.Fig. 13b shows a relationship of stiffness values with corre-lation distance and COV su

. Further increase in soil stiffnessis observed as correlation distance increases. This is due tothe fact that strain level decreases as correlation distanceincreases. Worst correlation distance is also pronouncedat dz = 5 m as from Fig. 13b. At this level soil encountersleast stiffness values.

5. Probabilistic interpretation of failure

From the design considerations, a laterally loaded pile isunserviceable if applied load (Q) is more or, equal to theallowable load (Qalat) and failure of it can be interpretedas maximum bending moment (M max) is higher or equalto the yield moment carrying capacity (M y) of the pile sec-tion. That means the laterally loaded pile can be consideredto be in a serviceable state of failure if,

Qalat 6 Q and M y 6 M max ð8Þ

where M y is the yield moment of the pile section. The

allowable capacity Qalat and M max are considered as ran-

Max. shear strain incrementContour interval= 2.00E-03

M a x . s t r a i

n 2 . 0

1 0 - 2

M a x . s

t r a i n 1 . 0

1 0 - 2

Max. shear strain incrementContour interval= 1.00E-03

a b

M a x . s t r a i n 8 1 0 - 3

Fig. 12. (a) Incremental strain contours for COV su ¼ 50%, dz/Lp = 0.15;

(b) incremental strain contours for COV su ¼ 50%, dz/Lp = 5.0.

0

10

20

30

40

0 0.01 0.02

Pile lateral head displacement (m)

L a t e r a

l l o a d ( k N )

K K K

COVsu = 10%

COVsu = 30%

COVsu = 50%

K : Soil Stiffness

2000

3000

4000

5000

6000

7000

0 1 2 3 4 5

z /Lδ p

S o i l s t i f f n e s

s , K ( k N / m )

COVsu = 10%

COVsu = 30%

COVsu = 50%

a

b

z/Lp=0.15

Fig. 13. (a) Soil stiffness values considering variability of soil parameterform mean load–displacement curve; (b) effect of soil spatial variability of soil on soil stiffness.


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dom variables. Considering Qalat as Log-Normal variable,the probability that the computed allowable capacity is lessthan the deterministic applied load (Q) can be stated as

P ðQalat 6 QÞ ¼ Uln Q llnQa lat

rlnQa lat

ð9Þ

where U(Æ) is the cumulative normal function. To show the

influence of COV su and dz/Lp on P (Qalat 6 Q), a determin-

istic load, Q = 60 kN is considered and P (Qa lat 6 Q) is cal-culated for different values of COV su

and dz/Lp. Fig. 14shows that the probability i.e. P (Qalat 6 Q) increases asCOV su

increases. For an example, when dz/Lp = 0.15 andCOV su ¼ 30%, P (Qalat 6 Q) is 0.37, indicating a 37% prob-ability that the pile may subject to unserviceable conditiondue to applied lateral load of 60 kN. The value of P (Qa lat 6 Q) becomes 0.3, when dz/Lp = 5.0 andCOV su

¼ 30% which represents probability of unservice-able condition decreases as correlation distance increases.This behaviour is also indicated by Phoon et al. [23]. Thehighest probability of unserviceable condition is observedat dz/Lp = 0.5 and further increase in dz/Lp decrease theprobability and this aspect is evident in the same figure.

The probability of failure is also computed for the caseof maximum bending moment M max greater than the yieldmoment of the pile section (M y) for an applied lateral loadQ and is given by

P ð M y 6 M maxÞ ¼ Uln M y lln M max

rln M max

ð10Þ

Fig. 15 presents comparison between failure probabilitiesdue to maximum moment exceeding moment carrying

capacity and COV su for different values of dz/Lp. Similarto Fig. 14, failure probability increases as COV su

increases.The value of P (M y 6 M max) for an applied load Q of 60 kNis 7.5 · 109 if dz/Lp = 0.15 and COV su ¼ 30%. The failureprobability is 4.8 · 1010 for dz/Lp = 5.0 and COV su ¼

30%. The results indicate that probability due toP (Qalat 6 Q) is critical for design as the other failure modegives low probability of failure. An important observationfrom Figs. 14 and 15 is that the correlation distance is rel-evant to the probabilistic interpretation at higher values of COV su

. High value of correlation distance is beneficial as itgives lower probability of failure.

6. Concluding remarks

The following conclusions from the present study can be

made:

(1) The major contribution of the present study is the rel-evance of spatial variability of soil undrained shearstrength in laterally loaded pile design. It is observedthat there is a significant change in allowable load,maximum bending moment due to the effect of COV su

and dz. Marginal increase in allowable loadis observed at low correlation distance. At high valueof correlation distance, the soil becomes almosthomogeneous and allowable load increases.

(2) The propagation of failure in soil near pile isdescribed in terms of accumulated shear strain in soil.It is observed that the correlation distance and coef-ficient of variation significantly influence the develop-ment of maximum shear strain values in soil near pile.At higher values of COV su

and lower value of corre-lation distance, the strain level is likely to be high andhence soil will have lesser allowable load. With fur-ther increase in the correlation distance, strain leveldecreases and hence allowable load increases.

(3) Idealisation of the number of weak zones in each real-ization of random field is useful to understand theallowable load. Monte Carlo simulation techniquecombined with numerical analysis is a very useful

approach in this regard as demonstrated in this study.

z /LP=0.15

z /LP=0.50

z /LP=1.50

z /LP=5.00

0

0.1

0.2

0.3

0.4

0.5

10% 20% 30% 40% 50%

COVsu

δ

δ

δ

δ

.

.

.

.

P ( Q a l a t

Q )

Fig. 14. Relationship between P (Qalat 6 Q) and COV su for different dz/Lp.

z /LP=0.15

z /LP=0.50

z /LP=1.50

z /L

P=5.00

0.00000

0.00002

0.00004

0.00006

0.00008

0.00010

0.00012

10% 20% 30% 40% 50%

COVsu

. δ

δ

δ

δ

.

.

.

P ( M y

M m a x )

Fig. 15. Relationship between P (M y 6 M max) and COV su for differentdz/Lp.


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Acknowledgements

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

Appendix I. Flowchart for statistical numerical analysis byfinite difference technique

Input Data File

(1) Generate grids and dimension of the field

(2) Assign material properties in each grid

from randomly generated values.(3) Generate Pile element(4) Assign boundary conditions

(5) Establish initial equilibrium

Generate correlation matrix for a correlation

distance

Decompose correlation matrix by Cholskey

decomposition

Generate normally distributed set of randomnumbers

Generation of lognormal random field i.e.

material properties by transformation

(1) Run analysis

(2) Save output file

Obtain lateral load-displacement curve for piletop and calculate allowable load and maximum

bending moment corresponding to lateraldeflection of 0.0508m

Last

Simulation

End of each input file

No

Yes

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