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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
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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
Contour interval= 1.00E-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
Contour interval= 1.00E-03
M a x . s t
r a i n 1 . 4
1 0 - 2
Max. shear strain increment
Contour interval= 2.00E-03
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
References
[1] Griffiths DV, Fenton GA. Bearing capacity of spatially random soil:the undrained clay Prandtl problem revisited. Geotechnique2001;51(4):351–9.
[2] Fenton GA, Griffiths DV. Three-dimensional probabilistic founda-tion settlement. J Geotech Geoenviron Eng—ASCE 2005;131(2):
232–9.[3] Fenton GA, Griffiths DV. Reliability based deep foundation design.In: Proceedings of GeoDenver2007: new peaks in geotechnics— ASCE, Reston. Paper no. GSP 170; 2007.
[4] Haldar S, Babu GLS. Ultimate capacity of pile foundation onspatially random cohesive soil. In: Proceedings of 10th internationalconference on applications of statistics and probability in civilengineering, Tokyo, Japan; 2007.
[5] Broms BB. Lateral resistance of piles in cohesionless soils. J SoilMech Found Div—ASCE 1964;90(3):123–56.
[6] Randolph MF, Houlsby GT. The limiting pressure on a circular pileloaded laterally in cohesive soil. Geotechnique 1984;34(4):613–23.
[7] Phoon KK, Kulhawy FH. Characterisation of model uncertainties forlaterally loaded rigid drilled shafts. Geotechnique 2005;55(1):45–54.
[8] Fan CC, Long JH. Assessment of existing methods for predicting soil
response of laterally loaded piles in sand. Comput Geotech2005;32:274–89.
[9] Poulos HG, Davis EH. Pile foundation analysis and design. Wiley;1980.
[10] Hsiung Y, Chen Y. Simplified method for analyzing laterally loadedsingle piles in clays. J Geotech Geoenviron Eng—ASCE 1997;123(11):1018–29.
[11] Zhang LM, Ng AMY. Probabilistic limiting tolerable displacementsfor serviceability limit state design of foundations. Geotechnique2005;55(2):151–61.
[12] Itasca Consulting Group Inc. FLAC, Fast Lagrangian analysis of continua. User’s manual, version 5.0. Minneapolis, USA; 2006.
[13] Donovan K, Pariseau WG, Ceepak M. Finite element approach tocable bolting in steeply dipping VCR slopes. In: Geomechanicsapplication in underground hard rock mining. New York: Society of
Mining Engineers; 1984. p. 65–90.[14] Eloseily KH, Ayyub BM, Patev R. Reliability assessment of pile
groups in sands. J Struct Eng—ASCE 2002;128(10):1346–53.[15] Shinozuka M, Yamazaki F. Stochastic finite element analysis: an
introduction. Stochastic structural dynamics, Progress in theory andapplications. London and New York: Elsevier Applied Science; 1988.
[16] Fenton GA. Probabilistic methods in geotechnical engineering. In:Workshop presented at ASCE GeoLogan’97 conference, Logan,Utah; 1997.
[17] Phoon KK, Kulhawy FH. Evaluation of geotechnical propertyvariability. Can Geotech J 1999;36:625–39.
[18] Popescu R, Deodatis G, Nobahar A. Effects of random heterogeneityof soil properties on bearing capacity. Probabilist Eng Mech2005;20:324–41.
[19] Popescu R, Prevost JH, Deodetis G. Effects of spatial variability on
soil liquefaction: some design recommendations. Geotechnique1997;47(5):1019–36.
[20] Griffiths DV, Fenton GA, Manoharan N. Bearing capacity of roughrigid strip footing on cohesive soil: probabilistic study. J GeotechGeoenviron Eng—ASCE 2002;128(9):743–55.
[21] Fenton GA, Griffiths DV. Bearing capacity prediction of spatiallyrandom c– / soils. Can Geotech J 2003;40(1):54–65.
[22] Niandou H, Breysse D. Reliability analysis of a piled raft accountingfor soil horizontal variability. Comput Geotech 2007;34(2):71–80.
[23] Phoon KK, Quek ST, Chow YK, Lee SL. Reliability analysis of pilesettlement. J Geotech Geoenviron Eng—ASCE 1990;116(11):1717–35.
S. Haldar, G.L. Sivakumar Babu / Computers and Geotechnics 35 (2008) 537–547 547