leo

a b

Box

is p

at

quat

cce

ved

nt s

eal t

les

ner

sportaData Cs throupile seo tranyze thbingand t

resses

stresses (total and effective) around the pile with time. For

nship[21],

s soilg the

concrete pile are not very sensitive to thin soil layers, i.e., wave

Contents lists available at ScienceDirect

.els

Soil Dynamics and Ear

Soil Dynamics and Earthquake Engineering 31 (2011) 12851295interest, which itself is an averaging process, i.e., summing overRodrigo.herrera@dot.state..us (R. Herrera), Peter.Lai@dot.state..us (P. Lai).lengths of propagating waves, which move the pile are large(pile length). Second, materials near the pile/soil interface aresmeared and remolded due to pile installation, and an averageproperty may be warranted. Finally, the total skin friction is of

0267-7261/$ - see front matter & 2011 Elsevier Ltd. All rights reserved.

doi:10.1016/j.soildyn.2011.05.007

n Corresponding author. Tel.: 1 352 278 3594.E-mail addresses: ttk@u.edu (K.T. Tran), mcm@ce.u.edu (M. McVay),equation and boundary conditions. Knowledge of skin friction isextremely useful in assessing pile freeze as a result of changes in

length of the pile. This assumption was made for a number ofreasons. First, the signals that propagate in a homogeneousseparate paper, this paper focuses on determining skin friction.Unlike the current practice [1719,25] of using instrumentationonly at the top of pile with required expertise [10] in separatingskin friction from tip resistance, the proposed technique allowsdirect assessment of skin friction as result of the analytical

because it can be used in cases where the model-data relatiois highly non-linear and produces multimodal mist functionsand it is typically faster than simulated annealing [22].

In the spirit of real time solution, a simple homogeneouprole is assumed with average properties for the soil alondetecting damage, setting pile lengths, pile freeze, etc.To achieve the goal of real time analyses, research is ongoing

in developing extremely fast techniques for assessing pile skinfriction and tip resistance independently of one another. Whilethe assessments of the tip resistance will be presented in a

time. Also, unlike local inversion techniques, global inversiontechniques, e.g. simulated annealing [24,30] and genetic algorithm[5,6,8,14,20,22,23], search over a larger parameter space due totheir stochastic nature when nding the global minimum of themist function. In this study, a genetic algorithm is employed1. Introduction

The Florida Department of Tranprocess of implementing Embeddedfor driven prestressed concrete pilesystem involves the use of internalbottom of a pile), a wireless radireceiver and laptop software to analrequires no external wires (i.e., climrecords information at both the topwhich is used real time to assess sttion (FDOT) is in theollector (EDC) systemsghout Florida [7]. Thensors (at both top andsmitter (Bluetooth), ae data. The EDC systemleads is not required),he bottom of the pile,and static capacity for

instance, Bullock et al. [2] or Axelson [1] have reported skinfriction increases of 20100% (per log cycle) for multiple soiltypes with little, if any, change in pile tip resistance.

Real time, every blow assessment of skin friction requires arobust and extremely fast solution strategy. To eliminate the needof any prior information on soil/rock proles, a global inversiontechnique was selected because of the non-linear nature of theinversion problem, as well as inherent noise in the measured data.The use of any local inversion techniques (e.g., gradient methods)was ruled out because of their heavy dependence on an initialmodel or prior information, which are not always available realA new method for estimating driven piinstrumentation at the top and bottom

Khiem T. Tran a,n, Michael McVay a, Rodrigo Herrera University of Florida, Department of Civil and Coastal Engineering, 365 Weil Hall, POb Florida Department of Transportation, 605 Suwannee St., Tallahassee, FL, USA

a r t i c l e i n f o

Article history:

Received 29 September 2010

Received in revised form

16 April 2011

Accepted 4 May 2011Available online 17 May 2011

Keywords:

Greens function

Ultimate skin friction

Genetic algorithm

a b s t r a c t

A numerical technique

instrumentation installed

solution of the 1D wave e

for solution. Specically, a

used to develop an obser

which is a function of seca

the algorithm provides a r

applied to four driven pi

predicted skin frictions ge

journal homepage: wwwstatic skin friction withf the pile

, Peter Lai b

116580, Gainesville, FL 32611, USA

resented to estimate ultimate skin friction of a driven pile using

the top and bottom of a pile. The scheme is based on an analytical

ion with static skin friction and damping along with a genetic algorithm

leration and strains measured at both the top and bottom of the pile are

Greens function, which is matched to an analytical Greens function,

tiffness and viscous damping. Requiring 13 s of analysis time per blow,

ime assessment of average skin friction along the pile. The technique was

having ultimate skin frictions varying from 700 to 2000 kN, with the

ally consistent with measured static load test results.

& 2011 Elsevier Ltd. All rights reserved.

evier.com/locate/soildyn

thquake Engineering

the complete side area of a pile. However, the approach to bedescribed may be applied to multiple segments of the pile, butthis must be traded off with real time solution, as well asconsistency of solution.

The proposed solution was applied to four full-scale piles attwo different sites with ultimate skin frictions varying from 700to 2000 kN. The analysis considers multiple blows near End ofDrive (EOD) as well as restrike blows that occur up to a week afterEOD. Increase of skin friction on the order of twenty-ve to sixty-ve percent was observed. The estimated restrike skin frictionwas subsequently compared to the measured resistance fromstatic load tests for verication purposes. Finally, it should benoted that the proposed technique may be applied to any deepfoundation (e.g., drilled shafts) subject to dynamic loading (e.g.,Statnamic testing).

2. Model description

For any driven pile, soil static skin friction and damping forcesdevelop on a segment of length dx, as shown in Fig. 1. The skinfriction, FS (force), is characterized as unit skin friction, fs (stress)times the surface area it acts on. The unit skin friction (fs) isusually characterized as a function of the pile displacement, u(x, t).Using secant soil stiffness, K, dened as the unit skin friction perunit of displacement (Fig. 1), the skin friction acting on segment

=

K.T. Tran et al. / Soil Dynamics and Earthquake Engineering 31 (2011) 128512951286

+=

+

dx is found as

FS fsAsurf Kux,tPdx 1Next, assuming generalized damping, the damping force, Fd, isobtained as

Fd CrPdxrs@ux,t

@t2

Summing forces on the segment, dx, results in:

kX

Fv 0 FBFTFIFSFd,

s @s@x

dx

AsArAdx @

2u

@t2KPdxux,tCrPdxrs

@u

@t 0 3

==

=

Fig. 1. Forces acting on pile.Next, canceling plus and minus terms, and then dividing by dxand A, results in:

@s@x

r @2u

@t2KP

Aux,tCrPrs

A

@u

@t 0 4

Introducing a linear pile stress to pile strain relationship andthen differentiating obtain particle displacement

s Ee E @u@x

, then@s@x

E @2u

@x2

Substituting @s/@x and P/A4/B (typical square pile) into Eq. (4)and dividing by r, results in:

E

r@2u

@x2 @

2u

@t24KrB

ux,t4CrrsBr

@u

@t 0 5

Let

a2 Er

, b 4KrB

, c 4CrrsBr

Then, the nal 1D partial differential equation of wave propa-gation with skin friction, b and damping, c is

a2@2u

@x2 @

2u

@t2c @u

@tbux,t 6

In the above equations, Asurf is surface area where forces actover, P is pile perimeter, B is pile width, dx is segment length, Cr isviscous damping coefcient, r and rs are pile and soil densities,respectively, E is Youngs modulus of pile and x, t are spatial andtime variables, respectively.

Numerical approaches such as Newmark/NewtonRaphsonalgorithms [4,15,28] and pseudo-forces/implicit Greens functionbased iterations [26,27] can be used to solve Eq. (6) for thegeneral case, e.g., layered soil proles with linear or non-linearsoilpile interaction [9,11,13]. However, all of these methodsrequire signicant computer time for solution, and may not beuseful for real time global inversion. Therefore, a simple model ofhomogeneous soil and a linear soilpile interaction (constant b) isemployed to achieve an extremely fast analytical solution. Thesupport of the model is given by comparison of the predicted tomeasured results in the case studies.

To solve Eq. (6) for the case of a pile with a nite length, initialand boundary conditions are required. The initial conditionsare at rest, e.g., particle displacement, velocity and acceleration(u, qu/qt and q2u/qt2) are zero when t equals zero (i.e., prior tohammer impact). For the boundary conditions, strains at the top(x0) and bottom (x lpile length) of the pile are prescribed as@u

@x g1t at x 0

@u

@x g2t at x l 7

where g1(t) and g2(t) are the measured strain data (EDC) at the topand bottom of the pile as a function of time. The solution of Eq. (6)with the initial conditions at rest and boundary conditions ofEq. (7) is as follows [16]:

ux,t a2Z t0g1tGx,0,ttdta2

Z t0g2tGx,l,ttdt

a2 g1Gx,0,tg2Gx,l,t 8

Gx,x,t exp 12ct

sin t 9p9q l9p9

q 2l

X1n 1

cosmnxcosmnx

264

sin t

a2m2np

p a2m2np

p35, mn pnl 9

K.T. Tran et al. / Soil Dynamics and Earthquake Engineering 31 (2011) 12851295 1287sin t

a2m2np

p a2m2np

p35exp 1

2ct

24 cos t9p9

q l

2l

X1n 1

cosmnxcosmnxcos ta2m2np

q 35 11An examination of Eqs. (10) and (11) reveal that the only

unknowns are damping, c, and soil stiffness, b. The particlevelocity, v(x, t), is known at both the top, v(x0, t) and bottom,v(x l, t) of the pile by integration of embedded pile accelerationgages with time. Similarly, the strain at the top, g1(x0, t) andbottom, g2(x l, t) of the pile is measured directly with embeddedgages as a function of time. The unknowns, b and c weresubsequently determined through an inversion scheme to bediscussed.

3. Solution methodology

The goal of the inversion method is to estimate two unknownparameters, damping related parameter (c) and stiffness relatedparameter (b). From b, the static skin friction (Fs) can be deter-mined as

Fs fsAsurf KMax ux,t Asurf brB4

Max ux,t 4Bl brB2lMax ux,t 12

where Maxux,t is the mean of maximum measured displace-ments at the top and bottom of the pile.

The simplest way of assessing b and c is from an inversionprocess to match the measured data with estimated data. Forinstance, using particle velocity data, the estimated velocity canbe calculated by assuming values of b and c, computing the timederivative of Greens function, G0 from Eq. (11), and then perform-ing the convolution, Eq. (10), with the measured strains (g1, g2).However, the analysis must be performed hundreds of thousandtimes to minimize the error between measured and predictedvelocity as a function of time. Unfortunately, this approach canrequire signicant computer time for the global inversion tech-nique because of the expensive operation of the convolution inwhere p b1=4c2, n denotes the convolution operator andGx,x,t is Greens function to measure the response at position xcaused by a unit load at position x.

Eq. (8) gives particle displacements, which may be invertedwith the measured displacement to estimate the pile skin friction.However, the measured displacement is usually non-zero,smooth, with few inection points, whereas particle velocity hasmultiple inection points, as well as crosses zero multiple times.Consequently, it was found with velocity, that convergence wasmuch faster because the signals carried only one or two dominantmaxima (pulses) and along with zeros, the velocity had muchgreater sensitivity in the inversion.

Taking the derivatives of Eqs. (8) and (9) with respect to timeand using the symmetry property of the convolution operator

f tgt0 f t0gt f tgt0

the particle velocity may be derived as

vx,t u0x,t a2g1G0x,0,tg2G0x,l,t 10where

G0x,x,t 12cexp 1

2ct

24 sin t9p9

q l9p9

q 2l

X1n 1

cosmnxcosmnxcalculation of the estimated velocity data (forward modeling).To reduce computer time, it is proposed to match the observedand predicted Greens functions through inversion directly. Bydoing so, the estimated Greens function is immediately obtainedfrom Eq. (11). A discussion of the measured Greens function andits derivation follows.

3.1. Observed Greens function

The observed Greens function is obtained from a deconvolu-tion [3] of the measured data. This requires the use of theconvolution theorem [3]

fftfg fftf fftgwhere fft(f) denotes a Fourier transform of f.

First, the Fourier transform is applied to Eq. (10), and thenwith the use of the convolution theorem, the following equationsare derived:

fftv0,t a2fftg1fftG00,0,tfftg2fftG00,l,tfftvl,t a2 fftg1fftG0l,0,tfftg2fftG0l,l,t

13where v(0, t) and v(l, t) are measured velocities, and g1 and g2 aremeasured strains at the top and bottom of the pile. Fft( )represents the Fourier transform of each function.

Denoting G00,0,t G0l,l,t G1 and G0l,0,t G00,l,t G2,then Eq. (13) may be expressed in the frequency domain as

v0,o a2g1oG1og2oG2ovl,o a2g1oG2og2oG1o 14

Next, Eq. (14) is solved for G1(o) and G2(o)or

G1oG2o

" # 1

a2g1o g2og2o g1o

" #1v0,ovl,o

" #15

where, f(o) is the Fourier transform of f(t) at a particularfrequency o.

Greens functions G1, and G2 (Eq. (15)) are calculated for alldesired frequencies, and then an inverse Fourier transform isperformed in order to generate the observed Greens functions inthe time domain. G1 and G2 in the time domain usually have verysimilar shapes, thus only one of them is used for the inversion(solution of b and c), which will be described in detail as follows.

3.2. Inversion method

Inversion involves minimizing a least-squared error, E(m),which measures the difference between observed data andestimated data associated with model m (a pair of assumedvalues of b and c), or

Em 1N

XNk 1

dkgkm2 16

where dk and gk are the kth observed and estimated Greensfunction values, respectively, and N is the number of observationpoints. A least-squared error equal of 0 is obtained when a perfectmatch between the observed and estimated data is found.

To overcome the need for reasonable initial model, i.e. priorinformation, a genetic algorithm was applied to Eq. (16) to obtainthe global minimum. Genetic algorithms have recently been appliedin evaluation of various dynamic data sets [5,14,20,22,23]. Generaldiscussions of genetic algorithms are described by Goldberg [6].

For this application, the algorithm requires a binary code(Fig. 2a), e.g., 8 bits, of 0 or 1, to represent each model parameter,i.e., b and c. For a code of nb bits: {anb, anb1, anb2, y, a1}

representing the parameter mij, the resolution of the parameter is

Psm PA1=Em

19

K.T. Tran et al. / Soil Dynamics and Earthquake Engineering 31 (2011) 128512951288m = min 0 0 0 0 0000

m = min + 1 0 0 0 0 1000

m = min + 2 0 0 0 1 0000

m = min + 3 0 0 0 1 1000

m = max 1 1 1 1 1111

m

m

m

. . .

. . .

m = i model parameter for the j eventmin = minimum value of the i model parameter for the j event

m = resolution of the i model parameter for the j event

* * * * ****

BINARY MODEL PARAMETER CODEdetermined as

Dmij maxijminij

2nb117

and the parameter may be determined by

mij minijDmijXnbn 1

anU2n1 18

Generally, the number of bits, nb, selected should be based onthe expected range of the parameter and its desired resolution.

The genetic algorithm begins with a suite of random (the rstgeneration with a population number of Np) model pairs (e.g.,b and c), Fig. 3(top left). Each parameter in a pair (b or c) in therst generation is found by randomly selecting a code of bits(0 and 1) and then calculating the parameter value from Eq. (18).After that, the least-squared error of each model pair of the rstgeneration is determined from Eq. (16).

The algorithm then generates offspring from the currentparents by reproduction, which essentially consists of three

4)

4 u

ha

tanwhNp

thethetiogen

arethe

mupro

4.

* * 0 1 1***

* * * * **1*

MUTATION

* * * * **0*

mij

mij

mij

mij

*

* * 1 0 0***

CROSS OVER

Fig. 2. Genetic algorithm: (a) parameter coding and (b) crossover and mutation.Recently, the Florida Department of Transportation (FDOT) withthe support of the Federal Highway Administration (FHWA) paidfor the monitoring (top and bottom) of two 0.61 m square piles onmufava station probability (0.01), a moderate value of crossoverbability (0.6) and a high value of update probability (0.9).

Applications

For validation, acceptance and possible implementation, itst be demonstrated that the results of the algorithm comparesorably with measured response from static load tests.guiving the lowest least-squared error.The selection of a reasonable population number Np is impor-t. Selecting a large value leads to unnecessary computations,ereas using a small value leads to a local solution. In this study,values of 20, 50, 100 and 200 pairs were evaluated, with100 pair population recommended. With a population of 100,model parameters usually begin to localize after 10 genera-

ns and converge after 50 generations. For piles studied, 50erations was sufcient to obtain reproducible b and c values.The probabilities of crossover px, mutation pm, and update puthe other important parameters in the global optimization ingenetic algorithm. This work strictly follows the suggesteddelines by Sen and Stoffa [22], which use a low value ofinvprobability pu.Repeat steps 1, 2 and 3 until a new generation is found with Npmodels. All tness of models in the new generation are storedand used for generating of the next generation.

Generations will be generated by repeating steps 1, 2, 3 andntil a specied number of generations are completed. Then, theersion result is taken as the model of the nal generationwhere A denotes all models in the current generation. Again,two different pairs (b and c) are selected as parents.

2) Conduct the processes of crossover and mutation for theselected 2 pair sets in step 1. Only one parameter is randomlyselected for the crossover and mutation, Fig. 2b between eachparent (i.e., b parent 1 to b parent 2). The coded parameterselected is subjected to the possibility of bit crossover withparents with a specied probability px. If crossover is to occur,randomly pick a cross position and exchange all the bits to theright of the position (Fig. 2b). A mutation follows the cross-over, and it is simply the alteration of a random selected bit inthe parameter code based on a specied probability pm(Fig. 2b). After the processes of crossover and mutation,least-squared errors, Eq. (16) is performed on the conceivedchildren.

3) The two new pairs (i.e., model) generated in step 2 are copiedto the new generation. Then, each new models error iscompared to error of a model in the current generationselected under a uniform random selection and used onlyonce. If the new models error is smaller, the new model iskept in the new generation. If it is more, the randomly selectedmodel replaces the new model in the new generation with aoperations: selection, crossover and mutation, and are updatedas follows:

1) Select a pair of models from the current generation forreproduction. The probability of parent selection is based onthe ratio of each models inverse error to the sum of all inverseerrors

1=Emite in South Florida, which were subsequently statically top

2K.T. Tran et al. / Soil Dynamics and Earthquake Engineering 31 (2011) 12851295 12890 1000 2000 3000 40000

50

100

150

200

0 10000

50

100

150

200

150

200

150

200

Dam

ping

par

amet

er c

(1/s

)

1 10down loaded. In addition, the Louisiana Department of Transpor-tation monitored (top and bottom) two 0.76 m square piles driveninto silty sands, restruck (pile freeze) after one week and thenstatically load tested. The side friction for all four piles at End ofDrive (EOD), and Beginning of Restrike (BOR) were computedwith the proposed approach and compared the measured unitskin friction from the load tests.

4.1. Site 1

The site is on SR 810, Dixie Highway at Hillsboro Canal inBroward, Florida. The site consists of upper layers of approxi-mately 15 m of medium dense sand with cemented sand zonesunderlain by limestone (bearing layer). The rst pile analyzed(pile 1) was a 0.61 m square by 15.2 m long prestressed concretepile, driven to a depth 14 m below the ground surface by a singleacting diesel hammer. One week after installation, re-strikes were

0 0.05 0.1 0.15-4

-2

0

2

4

6

8

10

12

14x 10-6

tim

Obs

erve

d G

reen

's fu

nctio

ns, s

/m

Fig. 4. Dixie Highway Pile 1: the

0 2000 40000

50

100

0 200

50

100

Stiffness para

30 40

Fig. 3. Dixie Highway Pile 1: distribution of 100 models000 3000 4000 0 1000 2000 3000 40000

50

100

150

200

150

200

20conducted to investigate whether the skin friction had changed.Then the pile was load tested to failure in accordance to ASTMD1143 (quick test) three days after the re-strike. The compressionloads were applied using two 5000 kN hydraulic jacks.

The wave guide solution was applied to 12 of the End of Drive(EOD) blows and 8 beginning re-strike blows (BOR). The specicresults of one re-strike blow are presented here in detail fordiscussion.

Prior to running the inversion, the observed Greens functionfrom the measured data must be found. The following 3 stepswere completed to obtain the observed Greens functions (Fig. 4).First, the measured strains and velocities (integrated from mea-sured accelerations) were ltered (low-pass) to remove all signalswith frequencies of 100 Hz and above (remove the high frequencynoise), and a Fourier transform was performed to obtain thefrequency components. Second, the transformed strain data(g1, g2) was also ltered (inverse ltering) to remove very low

0.2 0.25 0.3 0.35e, s

G1=G'(0,0,t)G2=G'(0,l,t)

observed Greens functions.

00 4000 0 2000 40000

50

100

meter b (1/s/s)

50

at the end of generations: 1, 10, 20, 30, 40 and 50.

tim

the

K.T. Tran et al. / Soil Dynamics and Earthquake Engineering 31 (2011) 128512951290magnitudes which would result in signicant magnication ofGreens functions, i.e., Eq. (15). The inverse ltering was bound tofrequency response 1/g(o) at the prescribed threshold g asfollows:

1

go 1

go , if1

9go9ogg9go9, otherwise

8