calculo capacidad carga axial pba cpt

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Prediction of ultimate axial load-carrying capacity of piles using

a support vector machine based on CPT data

A. Kordjazi a, F. Pooya Nejad a, M.B. Jaksa b,

a Dept. of Civil Engineering, Ferdowsi University of Mashhad, Mashhad, Iranb School of Civil, Environmental and Mining Engineering, University of Adelaide, Australia

a r t i c l e i n f o

Article history:

Received 18 July 2012

Received in revised form 2 August 2013

Accepted 2 August 2013

Available online 6 September 2013

Keywords:

Support vector machine (SVM)

Static pile load test

Cone penetration test (CPT)

Ultimate bearing capacity

a b s t r a c t

The support vector machine (SVM) is a relatively new artificial intelligence technique which is increas-

ingly being applied to geotechnical problems and is yielding encouraging results. In this paper SVM mod-els are developed for predicting the ultimate axial load-carrying capacity of piles based on conepenetration test (CPT) data. A data setof 108samples is used to develop theSVM models. Thesedata were

obtained from the literature containing pile load tests and each sample contains information regardingpile geometry, full-scale static pile load tests and CPT results. Moreover, a sensitivity analysis is carried

out to examine the relative significance of each input variable with respect to ultimate strength predic-tion. Finally, a statistical analysis is conducted to make comparisons between predictions obtained from

the SVM models and three traditional CPT-based methods for determining pile capacity. The comparisonconfirms that the SVM models developed in this paper outperform the traditional methods.

2013 Elsevier Ltd. All rights reserved.

1. Introduction

Ultimate strength prediction is a fundamental aspect of pilefoundation design and there exists numerous methods for its

determination. Many of the available experimental or theoreticalmethods for predicting pile capacity incorporate assumptionsassociated with the parameters that control the ultimate strengthor bearing capacity and consequently simplify the problem [1].

Similarly, because of complex behavior of piles in soil, almost noneof the existing methods provides uniformly consistent and accu-rate predictions of pile capacity. Consequently, alternative solu-tions are required to overcome these limitations.

Recently, machine learning and data mining methods, such asartificial neural networks (ANNs), have been applied to many geo-technical engineering problems and have demonstrated a consid-erable degree of success [15]. The support vector machine

(SVM) is a relatively new artificial intelligence technique whichis increasingly being applied to geotechnical problems and hasyielded encouraging results [612]. An important feature of theSVM is that it endeavors to discover the rules (or functions) that

govern a phenomenon using only a set of data (a set of measuredinputs and their corresponding outputs). Hence, there is no needto incorporate any assumptions to simplify the problem as, is oftenthe case with many traditional methods. In the case of pile capacity

prediction, Pal and Deswal [13]employed the SVM to model the

static axial capacity of high strength concrete spun pipe piles using

stress-wave data (results of dynamic pile load tests). Anotherexample is the model based on pile-driving records in cohesion-less soils that uses the general pile geometry and the piles re-

sponse to the driving equipment as input patterns [11]. The studiescarried out by Samui [14] and Liu et al. [15] are two additionalexamples of the application of the SVM to the prediction of axialpile capacity. The former incorporates penetration depth ratio,

mean normal stress and the number of blows as input variables,while in the latter greater emphasis was given to the soil proper-ties in the form of shear strength parameters (cohesion and frictionangle), the standard penetration test results along the pile length

and tip resistance of pile-end soil.In addition, prediction of the friction capacity of axially loaded

piles[16,17]and the ultimate capacity of laterally loaded piles inclay [11,18]are other aspects of pile capacity modeled with the

SVM.In this paper, the SVM is used to predict the ultimate capacity of

axially loaded piles, based on cone penetration test (CPT) data. TheCPT is a popular in situ test in geotechnical practice which, like

other in situ tests, eliminates the uncertainty associated with thelaboratory tests, such as sample disturbance, sample preparationand reapplying the in situ conditions in the laboratory, as well asthe possible errors due to performing laboratory tests. The strength

parameters of soil, which are measured through the CPT, namelycone tip resistance, qc, and sleeve friction, fs, are very similar tothe properties which influence pile capacity (i.e. base and shaftcapacity or resistance) [19]. Moreover, this test provides nearly

0266-352X/$ - see front matter 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compgeo.2013.08.001

Corresponding author.

E-mail addresses:[email protected](A. Kordjazi), [email protected](F.

Pooya Nejad),[email protected](M.B. Jaksa).

Computers and Geotechnics 55 (2014) 91102

Contents lists available at ScienceDirect

Computers and Geotechnics

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continuous results of these measurements along the soil profilethat offers more detailed information about the changes throughthe soil than discrete sampling and testing. Furthermore, it has

been shown that the CPT is the most accurate in situ test method,with a random measurement error of 3% or less[20]. It, therefore,is expected that using CPT data when modeling pile capacity can

lead to more accurate estimations. In order to develop a generic

model, this study also employs a database including a wide rangeof pile and soil types. The main aims of the paper are to: (i) developreliable and robust SVM models for predicting the ultimate capac-

ity of single piles under axial loading; (ii) examine the sensitivity ofthe model predictions with respect to the input variables; (iii) ex-plore the influence of three kernel functions and their parameterson the performance of the SVM models; and (iv) compare the re-

sults of the SVM model with three traditional CPT-based methods.

2. Support vector machine (SVM)

The support vector machine was introduced in the 1990s basedon Vapniks statistical learning theory [21,22]. This method em-

ploys the structural risk minimization (SRM) principle to minimizeerrors in the model, whereas other methods, such as ANNs, apply

empirical risk minimization (ERM)[3,23]. The primary purpose ofSRM is to simultaneously minimize the empirical risk (i.e. the er-rors associated with the training set) and maximize the generaliza-

tion ability of the model. As a result, SRM has been found to besuperior to the ERM principle[3].

Briefly, SVM estimates the regression by using a set of linear

functions, although, it can be developed using non-linear regres-

sion [3,7]. The solution of a regression problem within the SVM ap-plied to a dataset (x1,y1), . . ., (xl,yl),x e R

m,y e Ris a linear functionf(x) as given below:

fx wxb 1

where lis the number of samples;xis the input vector;yis the out-put value;w is the weight vector; b is the bias and hwxishows theinner product of the two vectors,w andx.

A loss function with an e-insensitive zone is also defined as[3,8]:

Ley jyfxje 0 jyfxj e

jyfxj e otherwise

2

whereLe(y) is the loss function ande > 0 is a constant value.

Nomenclature

As pile-soil surface areaAtip cross sectional area of the pile tipD pile width or diameterb biasb0 optimum bias

C penalty parameterCyjdj covariance between the model output (yj) and the ob-

served output (dj)d degree of polynomial kerneldj measured (observed) outputd mean of the observed output (dj)f unit shaft resistance (de Ruiter and Beringen[70]meth-

od)fs cone sleeve friction in CPTfs average cone sleeve frictionfs1,fs2,fs3 average cone sleeve friction along first, second and third

segment of the embedded length of the pilef(x) regression functionK(,) kernel functionkc penetrometer capacity factor (Bustamante and Gianes-

elli[71]method)Le loss functionLembed embedded length of pileL Lagrange functionl number of samplesm dimension ofx-vectorNc bearing capacity factorNk cone factor (de Ruiter and Beringen[70]method)n number of samples (to calculate the coefficient of corre-

lation)O perimeter of pilePu ultimate bearing capacityPum measured (observed) ultimate capacityPum average measured (observed) ultimate capacityPup predicted ultimate capacityP50 50% cumulative probabilityq unit tip resistance of pile (de Ruiter and Beringen[70]

method)qc cone tip resistance in the CPTqcav average cone tip resistance

qca the equivalent cone tip resistance at the level of the piletip (Bustamante and Gianeselli [71]method)

q1, q2 minimum average cone resistance in Schmertmann[69]method

qc1,qc2,qc3 average cone tip resistance along first, second and

third segment of the embedded length of the pileqctip average cone tip resistance below the pile tipr number of data in sensitivity analysisR coefficient of correlationRI rank indexRs ultimate shaft resistanceRt ultimate tip resistancesu undrained shear strengthw weight vectorw0 optimum weight vectorx input vectorxr, xs support vectorsy outputyj model outputy mean of the model output

a,a Lagrange multipliera0 adhesion factor (de Ruiter and Beringen [70]method)ab a coefficient for calculating the unit tip resistance in the

Bustamante and Gianeselli[71]methodac ratio of pile shaft resistance to the sleeve friction from

the CPT for clay (Schmertmann[69]method)as ratio of pile shaft resistance to the sleeve friction from

the CPT for sand (Schmertmann[69]method)r width of the RBF kernelr0 width of Pearson VII kernel functionrdj standard deviation of the measured outputryj standard deviation of the modeled outputx tailing factor of the peak in Pearson VII kernel functionh,i dot product operator||w||2 Euclidian norm of weight vectore allowable error in the loss functionni;n

i slack variables

92 A. Kordjazi et al. / Computers and Geotechnics 55 (2014) 91102
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The goal is to identify a function that has the greatest e devia-tion from the target actually obtained for all the training data

and at the same time as flat as possible [22]. It has been shownthat, in order to achieve this objective, the following function mustbe minimized[3,23]:

Uw; f; f kwk

2

2

C Rf Rf 3

Subject to

yi wx b 6 e fiwx b yi 6 e f

i

fi; fi P 0

8>: i 1; 2; . . . ; l 3a

where ||w||2 is the Euclidian norm of weight vector, w; Cis a con-stant known as the penalty parameter which is >0; and fi and f

i

are slack variables. Fig. 1illustrates thee-insensitive loss functionand the slack variables. A flat function is one that seeks a smallweight vector (w) and to achieve such a small w, the Euclidian normof the weight vector must be minimized. Then, minimizing the firstterm in Eq.(3) ensures the flatness of the function [22]. The conceptof the e-insensitive loss function also, allows the regression func-

tion to estimate the training samples output with some errors

and the slack variables represent the amount of these errors. There-fore, minimizing the second term in Eq.(3)controls the model per-formance on the training set. Finally, the constantC> 0 maintains a

trade-off between the flatness of the regression function and thethreshold to which deviations larger thane are tolerated[22].

The function in Eq. (3)can be rewritten in the form of a La-grange function:

La;a eXli1

ai ai

Xli1

yi ai ai

1

2

Xli1

Xlj1

ai ai

aj aj

xixj 4

where a anda are Lagrange multiplier and L(a, a) indicates the La-

grange function. The function above must be maximized subject tothe following constraints:

Rai Rai

0 6 ai 6 C

0 6 ai 6 C

8>: 5to determine the coefficients a and a. The regression functionwithin the SVM is then calculated using[3]:

w0 X

Support vectors

ai aixi 6

b0 1

2w0 xrxs 7

fx X

Support vectors

ai aixix b0 8

wherew0andb0are the optimum values of the weight vector andbias, respectively, andxrandxsare the support vectors.

Samples which have a non-zero Lagrange multiplier are knownas support vectors [3,23]. These samples have prediction errors lar-ger than e and, hence in this way, the value ofe determines the

number of support vectors[7].It should be emphasized that, since the latest optimization

problem can be solved utilizing quadratic programming (QP) tech-niques, achieving a global minimum (or maximum) is ensured ne-

gates any concern regarding local minima[23].In order to develop a SVM model the constants e in the loss

function and the penalty parameter, C, must be defined by the user.The constantC, the penalty function, maintains a balance between

empirical risk minimization (or good performance on the trainingset) and maximizing the generalization ability of the model. A largevalue ofCassigns higher penalties to errors so that the regressionis trained to minimize errors with lower generalization, whereas, a

small Cassigns fewer penalties to errors, which leads to highergeneralization ability[8].

As mentioned previously, the value assigned to e can alter thesupport vectors characteristics and in this way can play a signifi-

cant role in the models efficiency. Although, assigning a large va-lue to e causes a desired reduction in the number of supportvectors, achieving this goal by widening the eband does not neces-sarily lead to the best results. In contrast, when this constant is

very small, the number of support vectors grows and the risk ofoverfitting also increases[7].

Finally, non-linear regression within a SVM can be achieved bymapping the training patterns into a higher dimensional feature

space (where the linear regression is feasible) and then applyingthe linear regression algorithm in that feature space. This processposes computational difficulties due to high dimensionality ofthe feature space [22]. To avoid this problem the SVM employs ker-

nel functions. With a kernel function, data will be mapped implic-itly into a feature space, thus the dimension of the feature spacewill have no effect on the computational process [3,22,23]. To date,many kernel functions have been introduced [23], although poly-

nomial and radial basis functions, specifically, have been employedsuccessfully in geotechnical engineering problems [68]. As a re-sult, Eqs.(6) and (7)can be rewritten as[3]:

w0xX

Support vectors

ai ai

Kxi;x 9

b0 1

2

XSupport vectors

ai aiK xr;xi Kxs;xi 10

whereK(xi,x) is a kernel function used in non-linear regression.

Fig. 1. e-Insensitive loss function and slack variables[22].

A. Kordjazi et al. / Computers and Geotechnics 55 (2014) 91102 93
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3. Development of the support vector machine model

In this study a set of 108 samples is used to develop SVM mod-els for predicting the ultimate capacity of single, axially-loadedpiles. Each sample contains information regarding pile geometry,

full-scale static pile load test and cone penetration test (CPT) re-sults. The data were obtained from the literature containing pileload tests and the references used to compile the database are gi-ven inTable 1. Some of these data were used previously by Eslami

[24] (87 cases) and Pooya Nejad [25] (18 cases). The databaseincludes a wide range of pile and soil types: the pile materials in-clude steel, concrete and composite; the pile shape square,

round, octagonal, triangle, pipe and H section; pile tip condition closed and open; and the type of installation driven and bored.The soil types include sands, clays and silts, in individual and mul-tiple layers, with the soil properties of each varying spatially and

naturally as measured by the CPTs.All piles were subjected to compression static load tests and the

ultimate capacity was calculated based on the results of these loadtests. The pile fails when rapid pile movement occurs with respect

to a constant value of applied load or small increase in it, and theload corresponding to the failure load is considered the ultimatecapacity[26]. In cases where this failure point was not easily rec-ognizable on the loadsettlement curve, the 80% criterion has been

used [27]. In other words, the load that yielded four times themovement of the pile head, the ultimate pile capacity was taken

as 80% of that load. This criterion agrees well with the type of fail-ure considered in this study[26].

The process of developing the SVM models has been dividedinto a series of steps, as outlined in the following sections.

4. Model inputs and outputs

In order to obtain accurate prediction of pile behavior (includ-

ing settlement and capacity), an understanding of the factorsaffecting pile behavior is required [63]. Pile geometry, pile materialcharacteristics, mechanical properties of soil and applied load arewell-established, key parameters in the calculation of ultimate pileload capacity in many of the published methods. In a similar way,the method of pile installation, pile tip condition (open or closed)

and the type of pile load test are other factors which can affect pilecapacity[63].

In this work, the results of CPTs are considered as quantifyingthe soil properties. The CPT results consist of cone tip resistance

(qc) and sleeve friction (fs) along the piles embedded length. In or-der to account more accurately for the variability of soil propertiesalong the pile length, the embedded length of the pile is sub-di-vided to three segments of equal thickness [5]and the average of

qcandfs are calculated along each segment. Therefore, the inputvariables used in the development of the SVM models are the:(1) type of static pile load test (maintained or constant rate of pen-etration); (2) pile material (steel, concrete and composite); (3)

method of pile installation (driven or bored); (4) pile tip (closedor open); (5) embedded length of pile (Lembed); (6) perimeter ofthe pile in contact with the soil (O); (7) cross sectional area ofthe pile tip (Atip); (8) average cone tip resistance along the embed-

ded length of the pile (qc1, qc2andqc3); (9) average sleeve frictionalong the embedded length of the pile (fs1, fs2 and fs3); and (10)average cone tip resistance beneath the pile tip (qctip) (over a depth

of three times of the pile width or diameter below the tip of thepile). The ultimate load-carrying capacity of the pile (Pu) is the sin-gle output variable. It is important to note that this study does not

consider the separate components of ultimate capacity, that is, piletip and shaft resistance. Rather it aims to predict the total ultimatebearing capacity of the pile as the sole output of the developedmodels.

5. Data preparation

In order to develop an SVM model, the data set must be dividedinto two subsets: training and testing [3,7]. The training set is usedto train the model and the performance of the model is evaluated

using the testing set. In this paper, 76.8% of the data (83 cases) areused for training and the remainder (23.2% or 25 cases), are usedfor validation (i.e. testing).

This paper considers SVM as an interpolation technique withinthe range of the training data. Consequently, all patterns that arecontained in the training data need to be included in the testingset. In addition, the ANN literature suggests that the data subsets

which are used for developing a model, need to exhibit similar sta-tistical properties (e.g. mean, standard deviation and range) [64].Numerous combinations of training and testing sets were exam-ined and, while some inconsistencies are apparent, the statistical

parameters for each subset, as shown in Table 2, are reasonablyconsistent.

After dividing the available data into their subsets, the variablesare pre-processed by scaling them to a suitable form. Scaling elim-

inates the variables dimension [63] and ensures that all inputsroughly correspond to the same range of values[6]. In this study

all variables (input and output) are scaled between 0.0 and 1.0by normalizing each variable against their maximum values.

Table 1

Database references.

Reference Location of test(s) No. of p ile l oad t ests

Abu-Farsakh et al.[28] Louisiana, USA 1

Albiero et al.[29] Sao Paulo, Brazil 2

Altaee et al.[30] Baghdad, Iraq 2

Avasarala et al.[31] Florida, USA 3

Bakewell Bridge

(unpublished)

Adelaide, Australia 1

Ballouz et al.[32] Texas A&M University 2Briaud and Tucker[33] USA 36

Brown et al.[34] Grimsby, UK 2

Campanella et al.[35] Vancouver, Canada 4

CH2M Hill[36] Port of Los Angeles, USA 1

Fellenius et al.[37] ITD-Idaho 1

Fellenius et al.[38] ICS2 site, Portugal 1

Finno[39] Evanston, IL, USA 2

Florida Dept. of Transport [40] George Island Bridge 1

Gambini[41] Milan, Italy 1

Harris and Mayne[42] Georgia, USA 1Haustorfer and Plesiotis[43] Australia 2

Horvitz et al.[44] Seattle, USA 1

Laier[45] Alabama, USA 1Matsumoto et al.[46] Noto Island, Japan 2

Mayne[47] Georgia, USA 1

McCabe and Lehane[48] Belfast, Ireland 1Neveles and Snethen[49] Oklahoma, USA 2

Nottingham[50] West Palm Beach, USA 1

Nottingham[50] Jefferson County, USA 1

Nottingham[50] Blount Island, USA 5

ONeill[51] Houston, USA 2

ONeill[52] SF, California, USA 1

Omer et al.[53] Brussels, Belgium 4Paik and Salgado[54] Indiana, USA 2

Peixoto et al.[55] Brazil 1

Reese et al.[56] Texas, USA 2

Tucker and Briaud[57] Lock and Dam 26, SL, USA 3

Tumay and Fakhroo[58] Louisiana, USA 7

Urkkada Ltd.[59] Puerto Rico, USA 2

US Dept. of Transport[60] R oute 351 Bridge 3

Viergever[61] Almere, Netherlands 1

Yen et al.[62] Taiwan 2



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6. Training and model validation

As mentioned previously, in order to develop a non-linear SVM

model, kernel functions are employed. In this paper, three modelswith three kernel functions are developed. The radial basis func-tion (RBF), the Pearson VII function and the polynomial functionkernels are used in this study, which are shown in Eqs. (11)(13),

respectively[3,8,65].

Kx;xi expjxxij2=2r2 11

Kx;xi 1 1 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffikxxik

2

q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2

1=x 1p

=r0 2" #x,

12

Kx;xi hx;xii 1d

13

where constants r, x, r0 and d are kernel parameters which must bedefined by the user; r is the width of the RBF, r 0 andx are thewidth and tailing factor of the peak in the Pearson VII function,respectively, andd is the degree of the polynomial kernel. In this

work, the SVM toolbox[66]in MATLAB has been used to developthe SVM models.

Once model training has been accomplished successfully, theperformance of the trained model is then evaluated against data

that have not been used in the learning process (i.e. testing set).Statistical criteria, namely the coefficient of correlation and root

mean squared error (RMSE), are used to evaluate the predictionperformance of SVM models. The coefficient of correlation (R) is

a measure that is often used to determine the relative correlationbetween the predicted and measured data and can be calculatedas follows[5]:

RCyj djryjrdj

14

Cyj dj 1

n 1

Xnj1

yj ydjd

1

n 1

Xnj1

yjdj

Pnj1yj

Pnj1dj

n

! 15

ryj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiXnj1

yj y2=n 1

r 16

rdj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiXnj1

djd

2=n 1

r 17

y

Pnj1yj

n 18

d

Pnj1dj

n 19

whereyjis the model (predicted) output;djis the observed output;Cyjdj is the covariance between the model output (yj) and observedoutput (dj);ryj is the standard deviation of the model output (yj);rdj is the standard deviation of the observed output (dj); y is themean of the model outputs (yj); dis the mean of the observed out-puts (dj); andn is the number of data. Based on Smith [67], when|R|P 0.8, a strong correlation exists between the two sets of

variables.

The stages involved in the process of developing a SVM model,described in the preceding sections, are briefly summarized in aflow chart inFig. 2.

7. Results

In this study three SVM models have been developed. The first

model (rbf_svm) incorporates a radial basis function as a kernelfunction. The pvii_svm andpoly_svm models employ Pearson VIIand polynomial kernel functions, respectively, to develop non-lin-ear models. To achieve the optimum models, the design parame-

ters of the SVM model, namely the constants C, e, and the kernelparameters have been identified using a trial and error process.

The results of SVM models are summarized inTable 3with dif-

ferent values ofe while the remaining parameters are held con-stant. The table contains the number of support vectors in thetraining set (Nsv) and the values of R (coefficient of correlation)and RMSE which have been assessed against the testing set. In

the table these values for the optimal models appear in boldface.It can beseen thatas e increases, the number of support vectors de-creases, but, as previously mentioned, achieving the lowest predic-tion error and the highest correlation with measured capacity, is

not necessarily obtained by increasinge. Similar toTable 3, the re-sults of models developed with different penalty parameter values,are shown inTable 4. This table also provides information aboutsensitivity of the models with respect to C. As an example, the

rbf_svm model demonstrates almost no change in performancewhen CP 10. On the other hand, the efficiency of the poly_svm

model when predicting pile capacity drastically decreases whenCP 10.

Table 2

SVM input and output statistics.

Model variables

and data sets

Statistical parameters

Mean Std.

Dev. aMinimum Maximum Range

Atip (m2)

Training set 0.1827 0.1783 0.0080 0.7854 0.7774Testing set 0.1519 0.1176 0.0080 0.5030 0.4950

O(mm)

Training set 1625.12 866.92 585.0 7341.3 6756.3Testing set 1425.68 416.67 858.0 2510.0 1652.0

Lembed(m)

Training set 15.70 10.36 5.50 67.00 61.50

Testing set 16.17 8.47 6.50 36.50 30.00

qc1(MPa)

Training set 4.13 3.17 0.02 15.07 15.05

Testing set 3.76 2.85 0.24 11.72 10.74

fs1(kPa)

Training set 74.17 59.03 0.73 283.94 283.21

Testing set 69.49 43.67 10.76 176.75 159.7

qc2(MPa)

Training set 6.21 5.86 0.32 30.71 30.39

Testing set 5.73 4.09 1.00 18.42 17.43

fs2(kPa)Training set 103.52 96.91 2.12 618.66 616.54

Testing set 100.65 63.69 15.00 303.08 288.00

qc3(MPa)

Training set 7.36 6.07 0.27 32.59 32.32

Testing set 6.63 4.76 0.70 23.05 22.36

fs3(kPa)

Training set 126.57 95.58 7.99 396.57 388.58

Testing set 139.18 100.87 25.00 388.00 363.00

qctip (MPa)

Training set 8.75 6.28 0.25 27.11 26.86

Testing set 9.03 6.03 1.15 22.30 21.15

Pu(kN)

Training set 1988.20 1811.60 60.00 10910.00 10850.00

Testing set 1891.68 1 302.78 5 20.00 5850.00 5330.00

a Std. Dev. indicates Standard Deviation.

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It can be seen that among therbf_svmmodels, the model with

C= 1.1, e= 0.02 and r= 0.9 exhibits the lowest RMSE(=318.85 kN) and consequently is the optimal model. Similarly,with respect to thepvii_svmseries, the most efficient model designparameters are C= 0.95,e= 0.02,x= 7 andr0 = 2 and evaluation ofthe pvii_svm model against the testing set yielded R= 0.972 and

RMSE = 315.86 kN. Finally, the optimalpoly_svm model, which pro-videsR = 0.936 and RMSE = 469.62 kN on testing set, is developedwithC= 0.3,e = 0.005 andd= 1.95.

The results of the optimal model corresponding to each type of

kernel function are shown inTables 5 and 6.Table 5contains thedesign parameters and the number of support vectors for eachmodel, while the reliability of the models, against both trainingand testing sets, are summarized inTable 6. The plots of the mea-

sured versus predicted capacity, for these three models, are alsoillustrated inFigs. 35. The results indicate that the pvii_svm andrbf_svmmodels show negligible difference, but thepvii_svmmodelperforms slightly better than the rbf_svmmodel.

In order to identify the relative significance of each input vari-able with respect to ultimate pile load capacity prediction, a simplemethod proposed by Liong et al. [68] has been utilized. In thismethod the change in the model output is measured by varying

each of the input variables at a constant rate, and the sensitivityof the model is calculated as follows:

S% 100

r

Xrj1

%change in output

%change in input

j

20

whererindicates the number of data.In this paper, a constant rate of 20%[7,9]is considered as the

change in input for each variable and a sensitivity analysis with re-spect to cross sectional area of pile tip, perimeter of the pile in con-

tact with soil, embedded length of pile and CPT results (i.e. qc, fsandqctip), for the rbf_svm model has been carried out. Finally, theimpact (influence) factor of each input variable with respect tothe other variables affecting pile capacity is calculated and the re-

sults are summarized inTable 7.It can be seen that the soil properties in the form of CPT mea-

surements, with a combined impact factor of 56.2% (which issum of three last rows inTable 7), has the most significant effect

on the bearing capacity. Pile embedded length, the cross sectionalarea of pile tip and pile perimeter are, respectively, the other most

significant factors that influence pile capacity. These results agreewell with understood soil and pile behavior.

No

Yes

Data Preparation

(i.e. model inputs and output;

scaling; training and testing sets)

Selecting kernel function

Selecting user defined variables

(C, , and kernel function's

parameter)

Undergoing the SVM algorithm

(Model training)

Model performance assessment

using statistical criteria

Model accepted

Statistical analysis

demonstrates

reasonable accuracy

Input vector x

Support vectors x1, x2, , xn

Kernel function (K(x,xi))

Training process

Weight vectors (wi)

Output

Fig. 2. The summary of steps involved in the process of model development using the SVM (the training process by the SVM algorithm has been extracted from[22]).



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8. Comparison between the results of SVM and methods based

on CPT

This section presents the results of a comparison betweenpredictions obtained from the SVM models and three traditional,

CPT-based methods for determining pile capacity. The methods

proposed by Schmertmann[69], de Ruiter and Beringen [70]andBustamante and Gianeselli[71]are employed in this study. Thesemethods are briefly outlined below. For more details, readers arereferred to the main reference associated with each method.

Table 3

Number of support vectors, coefficient of correlation (R) and RMSE for SVM models, correspond to their optimum Cand kernel values and different e.

Radial basis function (RBF)C= 1.1 andr= 0 .9 Pearson VII functionC= 0.95;x= 7 andr0 = 2 Polynomial functionC= 0.3 andd = 1.95

e Nsv R RMSE e Nsv R RMSE e Nsv R RMSE

0.001 81 0.965 391.03 0.001 81 0.967 375.84 0.001 81 0.935 482.77

0.002 80 0.965 387.33 0.002 78 0.966 375.89 0.002 80 0.936 476.39

0.003 80 0.965 384.92 0.003 76 0.966 374.34 0.003 80 0.937 471.350.004 79 0.964 382.79 0.004 76 0.966 370.00 0.004 78 0.937 469.53

0.005 76 0.965 378.92 0.005 75 0.966 365.69 0.005 78 0.936 469.620.006 76 0.965 374.34 0.006 75 0.967 361.89 0.006 76 0.935 472.83

0.007 76 0.966 370.13 0.007 75 0.967 357.45 0.007 72 0.935 473.75

0.008 76 0.966 365.82 0.008 72 0.968 350.54 0.008 69 0.933 480.06

0.009 72 0.967 359.11 0.009 71 0.969 344.05 0.009 68 0.930 489.99

0.01 71 0.968 354.57 0.01 69 0.969 338.41 0.01 67 0.928 496.32

0.02 57 0.972 318.85 0.02 59 0.972 315.86 0.02 63 0.919 520.25

0.03 50 0.968 327.42 0.03 45 0.966 339.12 0.03 51 0.911 546.32

0.04 40 0.955 380.85 0.04 40 0.951 400.09 0.04 39 0.910 545.420.05 34 0.933 460.69 0.05 34 0.927 483.11 0.05 34 0.903 571.49

0.06 28 0.912 526.66 0.06 28 0.909 536.60 0.06 29 0.903 573.55

0.07 23 0.894 576.29 0.07 27 0.888 591.88 0.07 26 0.886 619.20

0.08 21 0.880 610.53 0.08 20 0.870 639.53 0.08 25 0.885 620.15

0.09 21 0.863 649.59 0.09 19 0.857 677.98 0.09 21 0.874 639.22

0.1 18 0.853 678.72 0.1 17 0.853 711.21 0.1 19 0.867 646.10

0.2 6 0.643 1284.00 0.2 6 0.651 1264.42 0.2 5 0.515 1180.68

0.3 4 0.544 1151.28 0.3 4 0.557 1112.62 0.3 3 0.419 1178.09

0.4 3 0.522 1120.11 0.4 3 0.540 1124.74 0.4 3 0.385 1224.950.5 3 0.495 1338.41 0.5 3 0.519 1354.54 0.5 3 0.323 1436.570.6 3 0.405 1701.68 0.6 3 0.441 1709.17 0.6 1 0.269 1679.10

0.7 2 0.032 2054.16 0.7 2 0.017 2052.96 0.7 1 0.269 1815.00

0.8 1 0.008 2137.48 0.8 1 0.008 2132.14 0.8 1 0.269 1962.300.9 1 0.008 2208.88 0.9 1 0.008 2206.24 0.9 1 0.269 2118.60

1.0 1 0.008 2282.06 1 1 0.008 2282.06 1 0 0.270 2282.05

Note: Nsv= number of support vectors;R= coefficient of correlation; RMSE = root mean squared error (kN).

Table 4

Number of support vectors, coefficient of correlation (R) and RMSE for SVM models, correspond to their optimum e and kernel values and different C.

Radial basis function (RBF)e= 0.02 andr= 0 .9 Pearson VII functione= 0.02;x= 7 andr0 = 2 Polynomial Functione= 0.005 andd= 1.95

C Nsv R RMSE C Nsv R RMSE C Nsv R RMSE0.001 82 0.036 2013.05 0.001 82 0.038 2028.43 0.001 81 0.492 1369.64

0.01 65 0.186 1538.08 0.01 65 0.167 1547.76 0.01 75 0.864 893.93

0.05 61 0.813 1141.74 0.05 61 0.816 1139.87 0.05 75 0.894 693.090.1 57 0.884 962.90 0.1 57 0.873 966.66 0.1 74 0.904 613.24

0.15 60 0.887 867.18 0.15 57 0.885 860.80 0.15 75 0.918 545.15

0.2 60 0.898 804.71 0.2 59 0.898 792.99 0.2 75 0.930 497.69

0.25 60 0.911 750.89 0.25 58 0.918 743.22 0.25 77 0.934 476.45

0.3 60 0.924 709.95 0.3 57 0.932 695.47 0.3 78 0.936 469.62

0.35 57 0.934 668.82 0.35 56 0.944 638.08 0.35 77 0.935 484.05

0.4 56 0.942 618.24 0.4 54 0.953 579.52 0.4 78 0.931 520.36

0.5 57 0.957 523.57 0.5 55 0.961 477.43 0.5 77 0.928 553.73

0.6 55 0.963 457.51 0.6 54 0.970 415.52 0.6 74 0.925 575.07

0.7 56 0.967 416.42 0.7 58 0.973 370.05 0.7 75 0.922 595.430.8 56 0.970 378.25 0.8 58 0.975 336.96 0.8 75 0.920 613.55

0.9 56 0.972 348.48 0.9 59 0.974 318.06 0.9 76 0.916 641.44

0.95 55 0.973 337.36 0.95 59 0.972 315.86 0.95 78 0.914 656.95

1 56 0.973 328.26 1 58 0.970 318.07 1 77 0.912 670.841.1 57 0.972 318.85 1.1 59 0.966 335.88 1.1 75 0.908 699.22

1.2 58 0.969 325.36 1.2 58 0.960 367.78 1.2 76 0.903 731.38

1.5 59 0.959 392.18 1.5 57 0.950 443.56 1.5 76 0.894 787.90

2 58 0.955 435.77 2 55 0.948 463.47 2 75 0.889 816.18

5 58 0.949 489.87 5 52 0.946 484.26 5 72 0.896 746.28

10 58 0.947 493.10 10 55 0.947 478.61 10 76 0.860 851.56

100 57 0.946 491.63 100 55 0.947 477.62 100 81 0.796 1206.70

500 57 0.946 491.63 500 55 0.947 477.62 500 83 0.430 3735.31

Note: Nsv= number of support vectors;R= coefficient of correlation; RMSE = root mean squared error (kN).

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8.1. Schmertmann [69] method

Ultimate shaft resistance (Rs) in cohesionless soils:

Rs as1

2fsAs0 t o 8D fsAs8D t o Lembed

21

Ultimate shaft resistance in cohesive soils:

Rs acfsAs 22

Ultimate tip resistance (Rt):

Rt q1q2

2

Atip 23

whereD,AsandAtipare the pile width or diameter, pile-soil surfacearea and pile tip area, respectively; as and acare the ratio of pileshaft resistance to the sleeve friction from the CPT for sand and clay,respectively; as is a function of the ratio of the embedment length ofpile to the pile width or diameter and varies from 0.4 to 2.4; ac,which varies between 0.2 and about 1.2, depends on the values of

the sleeve friction from CPT and also the method limits the productofacandfs to 120 kPa;fs is average sleeve friction; q1is the mini-mum average cone resistance over a depth of 0.7 to 3.75D belowthe pile tip and q2 is the minimum average cone resistance over a

length of 8Dabove the pile tip. The average ofq1andq2is also re-stricted to an upper bound of 15 MPa.

Table 5Design parameters and number of support vectors of optimum models.

Type of kernel

function (model)

Kernel

parameter

Optimum

value ofC

Optimum

value ofeNumber of

supportvectors

Radial basis

(rbf_svm)

r= 0.9 1.1 0.02 57

Pearson VII

(pvii_svm)

r0 = 2;x= 7

0.95 0.02 59

Polynomial

(poly_svm)

d= 1.95 0.3 0.005 78

Table 6

Results of the optimum models with respect to training and testing sets.

Type of kernel function (model) Training set Testing set

R RMSE (kN) R RMSE (kN)

Radial basis (rbf_svm) 0.979 394.38 0.972 318.85

Pearson VII (pvii_svm) 0.982 372.25 0.972 315.86

Polynomial (poly_svm) 0.966 471.41 0.936 469.62

Note: R and RMSE indicate the coefficient of correlation and root mean squared

error, respectively.

Fig. 3. Measured versus predicted capacity for SVM model with radial basis

function: (a) training set and (b) testing set.

Fig. 4. Measured versus predicted capacity for SVM model with Pearson VII kernel:

(a) training set and (b) testing set.

Table 7

Sensitivity analysis of the relative importance of the SVM input variables.

Input parameter Sensitivity respect to other input variables (%)

Atip 14.77

O 12.20

Lembed 16.79

qcav 23.41

fsav 22.31

qctip 10.52



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8.2. de Ruiter and Beringen [70] method

Unit shaft resistance (f) in sand:

f min

qc=300

fslocal sleeve friction

120 kPa

8>: 24

Unit shaft resistance in clay:

f a0su 25

Unit tip resistance (q) in clay:

q Ncsu 26

whereqcandfsare cone resistance and sleeve friction, respectively;a0 is an empirical adhesion factor (in NC clays is 1 and in OC clays is0.5);Ncis bearing capacity factor foru = 0, thenNc= 9 andsuis theundrained shear strength and can be estimated by the following

formula:

su qcNk

27

whereNkis the cone factor which varies between 15 and 20.The calculation of the ultimate pile tip capacity in this method

is similar to the Schmertmann[69]method.

8.3. Bustamante and Gianeselli [71] method

Ultimate unit shaft resistance:

f qcab 28

where the coefficientabis a function of pile type (pile material andthe method of installation) and qcis the CPT cone resistance along

the pile embedded length.Ultimate unit tip resistance:

q kcqca 29

where kc is the penetrometer capacity factor and depends on the

soli material and the method of pile installation; qca is the equiva-lent cone resistance at the level of the pile tip. This equivalent valueis the average of the cone resistance over the length of 1.5D over

and under the pile tip, after modifying the cone resistance profileover this length. Limiting values on both unit shaft and tip resis-tances have been proposed, which must be considered in the pilecapacity estimation with this method.

The same testing data set was applied to these models as were

presented to the SVM models, although three samples were not in-cluded because of a lack of relevant information. In this work, anapproach used by Briaud and Tucker [33] is used to statisticallyanalyze the performance of the methods and to rank them. How-

ever, an attempt to evaluate the performance of the CPT methodsbased only on statistical analysis can yield misleading conclusionsand one must also consider the comparison plot of predicted ver-sus measured ultimate capacities [19,33]. Statistical analysis hasbeen carried out based on four criteria:

(1) The equation of the best fit line of estimated (Pup) versusmeasured pile capacity (Pum) with the corresponding coeffi-cient of determination (R2)[19].

(2) The arithmetic mean and standard deviation ofPup/Pum [33].

(3) The 50% cumulative probability (P50) ofPup/Pum [19,72].(4) The coefficient of efficiency (E)[9].

The corresponding method is more efficient when the tangent

of the best fit line and R2 are closer to approach unity. Similarly,the model performs better when the mean ofPup/Pum approachesunity and the standard deviation of Pup/Pum approaches zero[19,33]. The concept of the third criterion is to arrange Pup/Pumval-

ues for each method in an ascending order, and then estimate thecumulative probability from the following equation [19]:

P i=r 1 30

wherei is the order number given for the considered ratio; and risthe number of data. An optimal model exhibits a value ofPup/Pumwhich is close to unity[19].

Finally, the coefficient of efficiency, E, is calculated as[9]:

E E1E2=E1 31

E1Xrt1

PumPum2

31a

E2X

r

t1

PupPum2

31b

where Pum, Pum and Pup are the measured, average and predictedultimate capacity, respectively. The Evalue compares the modeledand measured values of the variable and evaluates how well the

model is able to explain the total variance of the data[9].In order to quantify the overall performance of the developed

models and each traditional, CPT-based method, an overall rank in-dex (RI) is used. The rank index is defined as the sum of the ranksfrom the different criteria (i.e.RI= R1+ R2+ R3+ R4). The lower therank index, the better the performance of the method[19]. The re-

sults of the statistical analysis are summarized inTable 8.The best fit line ofPupversusPum and the corresponding coeffi-

cient of determination have been obtained by an ordinary leastsquares regression analysis and the results are illustrated in

Fig. 5. Measured versus predicted capacity for SVM model with polynomial

function: (a) training set and (b) testing set.

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Fig. 6. It can be seen that both the rbf_svm andpvii_svm modelswith Pfit= 0.94Pum and R

2 = 0.94 yield the best performance andhence they rank number one based on this criterion (i.e. R1= 1).

SincePfit= 0.94Pum, these models tend to underestimate the mea-sured ultimate capacity, on average, by 6%. In a similar way, theother methods have been investigated and the results are summa-rized in the column corresponding to R1 inTable 8.

As shown inTable 8, thepvii_svmmodel, with arithmetic meanand standard deviation values ofPup/Pumof 1 and 0.22 respectively,ranks first with respect to the second criterion. On the other hand,

the Bustamante and Gianeselli[71]method ranks last.The plots of Pup/Pum versus cumulative probability, for the

SVM models and three traditional, CPT-based methods are alsoillustrated inFig. 7. Again, both the pvii_svm andrbf_svm model

Table 8

Results of statistical evaluation.

Method Best fit Calculation Arithmetic calculation of Pup/Pum Cumulative probability Coefficient of efficiency Overall rank

PfitPum

R2 R1 Mean Std. Dev. R2 Pup/Pum at P50 R3 E R4 RI Final rank

pvii_svm 0.94 0.94 1 1.00 0.22 1 0.97 1 0.94 1 4 1

rbf_svm 0.94 0.94 1 0.99 0.23 2 0.97 1 0.94 1 5 2

poly_svm 0.95 0.88 3 0.96 0.29 3 0.91 3 0.86 3 12 3

Schmertmann[69] 1.19 0.82 5 1.12 0.39 4 1.04 2 0.52 5 16 5

deRuiter and Beringen[70] 1.08 0.91 2 1.14 0.37 5 1.11 4 0.87 2 14 4

Bustamante and Gianeselli[71] 0.86 0.84 4 0.85 0.34 5 0.77 5 0.79 4 18 6

Note:Std. Dev. = Standard deviation;P50= Cumulative probability at 50%.

Fig. 6. Measured versus predicted capacity for different SVM models and CPT methods (solid line and dashed line indicate best fit and perfect fit lines, respectively):(a)

rbf_svm; (b) pvii_svm; (c) poly_svm; (d) de Ruiter and Beringen [70]; (e) Bustamante and Gianeselli [71]; and (f) Schmertmann[69].



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withP50= 0.97 and the highest coefficient of efficiency (E= 0.94),

show the best performance, according to these criteria.Finally, based on the final two columns ofTable 8, it can be con-

cluded that the SVM models outperform the three traditional, CPT-based methods. Among the SVM models, the pvii_svm performs

best, followed by the rbf_svm and the poly_svm models, as ob-served previously.

9. Conclusions

The support vector machine (SVM) technique was used to

predict the ultimate load carrying capacity of single, axiallyloaded piles. Three SVM models with three kernel functions,

namely radial basis (RBF), Pearson VII and polynomial functionshave been developed and among these models the ones with

Pearson VII and RBF kernel demonstrate superior performance.The results indicate that the model with Pearson VII kernelhas the ability to predict the ultimate bearing capacity with a

high degree of accuracy (R= 0.972 and RMSE = 315.86 kN), forcapacities up to about 11,000 kN. Results of the sensitivityanalysis on the SVM model developed using the RBF kernel,indicate that the CPT results, which are representative of soil

properties, have the most significant effect on pile capacity.Finally, the results of a statistical analysis indicate that theSVM models provide more accurate pile capacity predictionwhen compared to three CPT-based methods, namely the

Schmertmann[69], de Ruiter and Beringen [70], and Bustamanteand Gianeselli [71] methods.

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Fig. 7. Cumulative probability plot of (Pup/Pum) for different methods.

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