Ge 490605

Back analysis of offshore pile driving with an improved soil model

B. R. DANZIGER, A. M. COSTA,{ F. R. LOPES{ and M. P. PACHECO}

Extensive back analyses of eld records of off-shore piles driven in calcareous sands wereperformed using an improved soil model devel-oped by Simons. The records were obtainedduring the installation of closed-end pipe pilesfor offshore platforms at the Northeast Pole siteof the Campos Basin in Brazil. This model isespecially adequate for hard driving conditionswhen piles are driven almost to refusal. In thesecases, inertial (radiation) damping prevails,whereas viscous damping may be neglected.Back analyses with this model allowed the rela-tive displacements between pile and soil to becalculated at various points along the pile in aone-dimensional nite-element method solution.The soil stiffness ks and the damping coefcientcs were back-calculated for different pile depths.Values of the ratio of the shear modulus to theshaft resistance (G=) were also back-calculatedand suggestions are given concerning tentativevalues of this ratio to be selected in drivabilitystudies in calcareous sands.

KEYWORDS: calcareous soils; model tests; numericalmodelling and analysis; piles.

Nous avons pratique des retro analyses pousseessur les comportements sur le terrain des pilotisoffshore fonces dans des sables calcaires enutilisant un modele de sol ameliore, developpepar Simons. Les dossiers ont ete obtenus pen-dant l'installation des pilotis a tuyaux fermespour les plate-forme offshore du site NortheastPole dans le bassin de Campos, au Bresil. Cemodele est particulierement adequat pour desconditions de battage difciles, quand les pilessont foncees presque jusqu'au refus. Dans cescas, un amortissement du a l'inertie (amortisse-ment par rayonnement) prevaut alors quel'amortissement visqueux est negligeable. Lesretro analyses utilisant ce modele ont permis decalculer le deplacement relatif entre le pilotis etle sol en divers points le long du pilotis grace ala methode d'elements nis en une dimension.La rigidite du sol ks et le coefcient d'amortisse-ment cs ont ete retro calcules pour differentesprofondeurs de pilotis. Nous avons egalementretro calcule les valeurs du rapport entre lemodule de cisaillement et la resistance de l'ar-bre (G/) et nous offrons quelques suggestionssur les valeurs supposees du rapport a choisirdans les etudes des caracteristiques de battagedans les sables calcaires.

INTRODUCTION

When back-analysing eld records of pile driving,one of the main problems is related to the adequacyof the model adopted to represent the soilpileinteraction. The most commonly used model incomputer programs (for back analysis) was devel-oped by Smith (1960). Limitations of this modelhave been pointed out by many authors. Forehand &Reese (1964), for instance, observed that manycombinations of quake values and damping coef-cients in Smith's model could be selected to t eldrecords reasonably well. Simons (1985) described a

number of theoretical and practical limitations ofSmith's (1960) model. Simons (1985) also empha-sized that for an applied load giving rise to purelyelastic displacements, the soil would provide twodistinct rate-dependent components of soil reactionto the motion of an embedded pile: one due tomaterial viscosity, and the other due to its inertia,known as radiation damping. Simons (1985) pointedout that inertial (radiation) damping is more impor-tant when nearly elastic displacement occurs. On theother hand, for plastic displacements, the materialviscosity becomes the more important effect. There-fore, when piles are driven almost to refusal, therst condition prevails and radiation damping ap-pears to be the most important form of energy loss.

For the closed-end pipe piles used in the Cam-pos Basin, offshore from Brazil, hard driving con-ditions occurred at the end of pile installation andtherefore the radiation damping of Simons' (1985)model can be considered the sole rate-dependentcomponent of soil reaction.

Danziger, B. R., Costa, A. M., Lopes, F. R. & Pacheco, M. P. (1999). Geotechnique 49, No. 6, 777799

777

Manuscript received 8 Oct. 1997; revised manuscriptaccepted 24 June 1999.Discussion on this paper closes 30 June 2000; for furtherdetails see p. ii. Fluminense Federal University, Brazil.{ PETROBRAS, Rio de Janeiro, Brazil.{ Federal University of Rio de Janeiro.} Rio de Janeiro State University.

The geotechnical conditions in the NortheastPole site of the Campos Basin consist mainly ofcalcareous sand through which very low shaftresistance is usually developed. A conical steelpoint was added to each pile in order to increaseits toe resistance. Pile driving was monitored bythe usual method of recordings of accelerometersand force transducers at the pile head.

Danziger et al. (1992) presented some backanalyses of the same piles using Smith's (1960)model. In the present paper, the records examinedpreviously are reinterpreted, using Simons's (1985)model. Comparisons of the results presented in thispaper with those reported before show interestingaspects concerning the damping effects derivedfrom the two models.

GEOTECHNICAL CONDITIONS AND PILE

CHARACTERISTICS

A total of seven platforms were installed at theCampos Northeast Pole site (Fig. 1), Pargo 1A

being the largest and consequently the most exten-sively monitored. Results from four borings withsoil sampling and one cone penetration test indi-cated a granular stratum throughout the whole area,with variable cementation. The soil prole consistsof a supercial layer of silty sand (ne to medium,with quartz grains, to a depth of about 13 m) andlayers of calcareous sand (with localized shell andcoral fragments in addition to some clay lenses) toa depth of 120 m (McClelland, 1985). The geotech-nical characteristics are summarized in Fig. 2. InFig. 2, the results of cone penetration tests (CPTs)as presented by McClelland (1985) are presented asthin lines, whereas the simplied prole given byGeomecanica (1986) is indicated by the thick line.

Laboratory direct shear tests were performedwith soil samples prepared at both their minimumand maximum densities. The friction angle variedfrom 268 to 348 for the minimum density, and from338 to 408 for the maximum density. The greatervalues were found for the shallower depths, downto 13 m, where the sand layer had a high quartz

Brazil

Paci

fic O

cean

Atlan

ticOc

ean

Campos

50m

100m

200m

228

22830

2384084083041841830

1000m

1000

m

2000

m

VermelhoFieldNortheast

PoleCarapeba

Field PargoField

NorthPole

SouthPole

Fig. 1. Location of construction site

778 DANZIGER, COSTA, LOPES AND PACHECO

content, and the lower friction angles were foundfor the greater depths, where the sand layer had ahigh content of calcium carbonate.

Figures 3 and 4 show the pile toe characteristics.The choice of closed-end pipe piles resulted in aconsiderable reduction in the foundation cost(Costa et al., 1988; Mello et al., 1989).

The main piles A1, A2, B1 and B2 were drivenwith an MRBS 4600 and an MRBS 5000 hammer.The nal penetration of the piles varied from 42 mto 45 m below the sea-bed. Part of the recordnumber in this paper indicates the pile penetrationbelow the sea-bed (e.g. record A24425 refers topile A2 with pile toe 4425 m below sea-bed). Thenal blow counts were close to 250 blows=25 cm(regarded as refusal).

SIMONS'S MODEL

In Simons's (1985) model (also described inRandolph & Simons, 1986) the theory of dynamicelasticity is used to determine the soil stiffness anddamping coefcients. In addition, a yielding me-

chanism consistent with the physical process in-volved is employed.

The dynamic shear stress at the pilesoilinterface resulting from a harmonic motion u(t) u exp(it) was given by Novak (1977) as

G2r0

[Sw1(a0) iSw2(a0)]u (1)

where a0 r0=vs is the dimensionless frequencyratio, G is the soil shear modulus, r0 is the pileradius, vs is the soil shear wave velocity, Sw1 andSw2 are functions of the frequency ratio a0 and is the frequency.

In fact, during installation the pile is submittednot to a harmonic motion but to a transient one,resulting in the propagation of stress and strainwaves along its length. Clough & Penzien (1975)obtained a solution for the wave propagation froman analysis of modal superposition, where a nitenumber of harmonic responses can be superim-posed to obtain a reasonable approximation of thewave propagation. In this approach, however, thehigher frequencies of the wave spectrum are not

Depth:m

0.00

12.50

33.00

72.75

97.50

116.00

2.25

Soil description Mean point resistanceqc: MN/m2

0 10 20 30 40 50

Interrupted test

.70

.50

.60

.60

.60

.60

.50

Interrupted test

Fine to medium loose sand with quartzand calcareous grains

Fine to medium silty sand, dense to verydense with clay lenses

Fine to medium silty sand, dense to verydense with calcareous grains and claylenses

Fine to medium calcareous silty sand,very dense

Fine to medium silty sand, dense to verydense with quartz and calcareous grains

Fine to medium silty sand, dense tovery dense

Fig. 2. Geotechnical characteristics at Pargo 1A site (McClelland, 1985;Geomecanica, 1986)

BACK ANALYSIS OF OFFSHORE PILE DRIVING 779

incorporated into the analysis and a reasonablematch between measured and back-calculated re-sults is generally difcult. In addition, in order toincorporate the pile slip, the pile response must beobtained using a direct time integration scheme,instead of a frequency domain approach. A fre-

quency-independent form of the solution (equation(1)) is then required. For that purpose, Simons(1985) proposed that the function Sw1 and the ratioSw2=a0 could be approximated by frequency-inde-pendent constants such as Sw1 2:9 and Sw2=a0 2 (Fig. 5). Equation (1) then becomes

G2r0

(2:9 i2a0)u (2)Since

u(t) u exp(it) (3)and

@u

@ t iu(t) (4)

equation (2) becomes

G2r0

2:9u 2r0vs

@u

@ t

(5)

Equation (5) represents the equation of motion of aspring and dashpot system with spring stiffness ksand damping coefcient cs, where

0.038

0.075

0. 40

0.15

Dia. 5 1.67

0.35

50.

355

0.35

50.

352

0.35

20.

384

0.38

4

1.21

0.29

1.17

458

Fig. 3. Characteristics of the conical steel point (Costaet al., 1988) (dimensions in metres)

Fig. 4. View of the conical steel point

Sw1

S w2

a0

0 0.5 1.0 1.5 2.0 2.5 3.0

S w1,

S w

2

20

15

10

5

0

Fig. 5. Dynamic stiffness coefcients for soils (Simons,1985)


ks 2:9G

2r0(6)

and

cs Gvs (Gr)0:5 (7)

Therefore, Simons (1985) assumed that the soilresistance could be modelled by a series of suchspring and dashpot systems (Fig. 6).

When the shear stress at a point at the pilesoilinterface (equation (5)) reaches the soil yieldingstress s, the frictional bond between the pileand the soil is broken. Then the spring and dashpotsystem is disconnected from the pile and the soiland pile displacements are calculated separately.The soil continues to resist with its yielding stressuntil the stress level reduces below the yieldinglimit and the bond is restored. When the springdashpot system is disconnected, the pile displace-ment is calculated in the usual manner assumingthat the soil continues to resist at its limitingstrength. The soil displacement, on the other hand,may be updated independently by solving the equa-tion of motion of the springdashpot system withthe yielding stress at each time step (Simons,1985). From equation (5) it follows that

2r0 ls Ksu Cs @u@ t

(8)

where Ks and Cs represent the values of ks and csmultiplied by 2r0 l (l being the length of the pileelement).

Equation (8) is a partial differential equationwhich allows the determination of the change insoil displacement us under the stress s over thetime interval t. The solution of equation (8) asobtained by Simons (1985) is

us 2r0 l sKs uts

1 exp Ks t

Cs

(9)

uts being the soil displacement at the beginning ofthe interval.

As yielding proceeds, the stress which the soiladjacent to the pile would sustain if the pile andsoil were to rejoin is continuously calculated ac-cording to Simons (1985) as

12r0 l

Ksus Cs @up@ t

(10)

where us is the soil displacement and @up=@ t isthe pile velocity. When the stress given by equation(10) falls below the yielding stress s, the pile andsoil are assumed to rejoin and the subsequentincrements in soil displacements are equal to theincrements in pile displacement.

The soil reaction to the pile motion at the toedue to radiation damping can be modelled by thewell-known Lysmer (1965) analogue. Similarly tothe case of shaft reaction, an analytical solution isrst determined and then adjusted to an equivalentfrequency-independent spring and dashpot system.In the present analysis, however, the Lysmer ana-logue was not used, owing to the more complexrepresentation of the conical point. Instead, thissharp variation of pile impedance was modelled bymasses consistently concentrated at closely spacednodes at the pile toe. Similarly, the toe resistancewas replaced by equivalent shaft resistances con-centrated at the same nodes (by application ofequations (5)(10)). Danziger et al. (1992) showedthat this procedure allowed a good match betweenmeasured and back-calculated velocities in theregion around the pile toe using Smith's (1960)model.

Appendix 1 presents the equations used for thecalculation of the element area and the consistentmass added to each pile node, for both cylindricaland truncated conical elements. Cylindrical ele-ments were adopted for the pile shaft and truncatedconical elements for the toe elements. The deriva-tion of the equations was presented by Danziger(1991) and was based on Mello (1990).

It is worth mentioning that before analysing thedata presented in this paper some analyses wereperformed in order to check the assumptions maderegarding the validity of the adopted length of theelements and of the mass distribution near the piletoe. Well-known boundary conditions with re-sponses calculated by the analytical formulation ofthe differential wave equation were compared withthe results of the one-dimensional nite-elementmethod (1-D FEM) approach. The responses ob-tained from the FEM agreed with those obtainedfrom the analytical formulation, adding condenceto the assumptions made.

Simons's (1985) model has also been analysedby other authors, e.g. Chow et al. (1988a,b), Leeet al. (1988), Nguyen et al. (1988), Matsumoto &Takei (1991) and Randolph & Deeks (1992). Leeet al. (1988) included the effect of viscous damp-

l

Pilemp

kp

ks Cs

Soil

s

Fig. 6. Pile and soil displacement at the interface(Simons, 1985)


ing, which had not been considered explicitly inthe original model. As stated before, viscous damp-ing is not considered in this paper as radiationdamping is the most signicant portion of energyloss for hard driving conditions.

BACK ANALYSES

Several back analyses of eld records were per-formed using a 1-D FEM program originally devel-oped by Costa (1988), with the soilpile interactionfollowing Simons's (1985) model as implementedby Danziger (1991). The application of Simons's(1985) model in a 1-D FEM solution to match eldrecords was carried out in a way similar to theCAPWAP analysis (Goble et al., 1980). This imple-mentation had the following main objectives:

(a) to implement a more realistic soil model,applicable to hard driving conditions, whereradiation damping is believed to prevail

(b) to investigate the relative displacements be-tween the pile shaft and the soil in the 1-DFEM solution

(c) to back-calculate the soil stiffness ks and thedamping coefcient cs at different pile depths

(d ) to compare the mobilized soil resistance andthe resistance distribution along the pile shaftobtained from Smith's and Simons's models forthe same eld records

(e) to compare the penetration for one blow andthe relative displacement between pile and soilat the pile toe (rst node).

The soil parameters obtained from the backanalyses are presented in Table 1. A typical plotshowing the measured and simulated velocities atthe pile head versus time is indicated in Fig. 7 forpile A2, with 4425 m of penetration below thesea-bed. In Fig. 7 ll refers to the part of pile lengthabove the sea-bed whereas lent refers to the em-bedded portion of pile length. C is the stress wavevelocity. The occurrence of successive disconnec-tions and reconnections between the pile and soilis clearly depicted in Figs 8 and 9.

Figure 8(a) shows measured and simulated pilehead displacements for pile A2, while Figs 8(b)and (c) indicate simulated velocities and displace-ments, respectively, at the pile toe. It can be seenthat the maximum toe displacement corresponds tonull velocity, when soil unloading begins. For thiseld record, the pile and soil disconnected at theinstant of maximum velocity. Up to this point(before yielding) the pile and soil displacementsfollowed the same curve. When slip began, thesoil displacements became smaller than the piledisplacements, the difference being the plastic dis-placements at the given depth. The plastic displa-cements also increased up to the point of nullvelocity. It can also be seen that when the pile and

soil rejoined, the subsequent increments of soil andpile displacements were equal and both curvesfollowed a parallel pattern.

The plots of static and total soil reaction wereinterpreted at some specic pile nodes, includingthose simulating shaft and toe resistance (Fig. 9).At the nodes simulating shaft resistance, such asnode 25 in Fig. 9(a), a large contribution of radia-tion damping can be inferred from the back analy-sis. In this case, the dominant effect of radiationdamping at the earlier stages of shaft loadingduring the short time interval between points Aand B in Fig. 9(a)causes the limiting soil resis-tance s to be reached when the static resistance isstill lower than s. This means that the disconnec-tion between the pile and soil was initiated at pointA, remaining effective between points A and B,until the spring component of increasing static soilreaction reached s at point B. At this point bothparticle velocity and radiation damping are zero,and therefore reconnection resumes. Beyond pointB the velocity changes its sign and therefore thestatic and dynamic components have oppositesigns. It may happen in some nodes that reconnec-tion starts before the particle velocity becomeszero, i.e. before the limiting resistance s related tothose nodes is fully mobilized. This indicates thatthe assigned values of s are probably too high andshould be revised.

At the conical steel point, the sudden sharpeningof the pile section causes the modelled radiationdamping to decrease linearly to zero at the toe.Therefore, the total and static responses are coin-cident at node 1 in Fig. 9(b), with no contributionfrom radiation damping. At node 2 (r0 0:22 m),in spite of the very small contribution from radia-tion damping the total and static responses are nolonger coincident, although the difference is small(Fig. 9(c)).

PENETRATION FOR ONE BLOW

As mentioned before, for the present case ofclosed-end pipe piles driven in the Campos Basin,the toe resistance was modelled by means of shaftresistances concentrated at closely spaced nodes atthe pile toe (Figs 3 and 4). Since the soil behaviouris assumed to be linear elastic in Simons's model,the permanent displacement under hard drivingconditions can be consistently determined from thedifference between the pile displacement and thesoil displacement.

Actually, the penetration for any particular blowat the nal stage of pile driving does not necessarilycorrespond to predominantly plastic displacements;it also incorporates a component of elastic displace-ment that is not fully recoverable, associated withresidual stresses. In fact, the ultimate soil displace-ment inferred from Fig. 8(c) indicates that the static


Table 1. Back analyses of eld records from piles A1, A2 and B2 by Simons's (1985) model

Pile depth: m Record Bearing Toe Damping coefcient cs: kN s=m3 Soil stiffness ks: MN=m

3 Maximum pile(Record) quality capacity: resistance: displacements: mm

kN %Top Toe (soil) Shaft

(elastic)

A1 Good 23 500 54 Toe: 1000 (1013) Toe: 667 185 81 1044300 Shaft: Shaft: (52)

(A14300) From 0 to 34 m: 10 (83) From 0 to 34 m: 23From 34 to 415 m: 70 (211) From 34 to 415 m: 145

A2 Excellent 25 160 75 Toe: 750 (1235) Toe: 991 210 67 1434325 Shaft: Shaft: (52)


A2 Excellent 28 000 75 Toe: 750 (1348) Toe: 1182 212 62 1504425 Shaft: Shaft: (47)


B2 Good 38 800 70 Toe: 800 (1370) Toe: 1220 318 104 2144200 Shaft: Shaft: (61)

(B24200) From 0 to 33 m: 290 (88) From 0 to 33 m: 25From 33 to 405 m 350 (188) From 33 to 405 m 114

BA

CK

AN

ALY

SIS

OF

OF

FS

HO

RE

PIL

ED

RIV

ING

78

3

component of soil resistance was not totally un-loaded, indicating the presence of residual stresses.

The values of penetration for the blows back-analysed are presented in Table 2. The penetrationfor one blow in the eld can be estimated from theinverse of the number of blows for a penetration of025 m. Although such a criterion is somewhatarbitrary and represents only a rough approxima-tion, the authors' experience with many instrumen-ted high-capacity offshore piles in the CamposBasin indicates that this simple criterion may beuseful to check for consistency of the back-calcu-lated parameters.

The back analysed penetration in Table 2 wasdetermined by subtracting the maximum soil dis-placement from the maximum pile displacement atthe pile toe, both listed in Table 1. Except for pileA2 at 4425 m penetration, Table 2 shows that thepenetration obtained from the eld records is great-er than the back-analysed value. The penetrationobtained from the eld records is greater becauseit incorporates the elastic component of soil dis-placement that is not fully recoverable, associatedwith residual stresses.

The unrecoverable soil displacement is also de-picted by the position of the dashed line at the endof the soil displacement plot in Fig. 8(c).

RESIDUAL STRESSES

It can be inferred from the plot of axial stressesversus time for the lowest element (Fig. 10) thathard driving gives rise to important residual stres-

ses. In fact, a nal axial stress different from zeroat the end of the plot, after the dynamic resistanceis reducted to zero, indicates the presence ofresidual stresses.

In Simons's model, the presence of residualstresses can also be visualized by plotting the soildisplacement at the pile toe. A nal stabilized soildisplacement different from zero indicates the oc-currence of residual stresses. For pile A2 (4425 mof embedment) a plot of axial stress is presented inFig. 10, which indicates a nal stress at the piletoe of about 10% of the maximum stress.

A common consequence of hard driving is theoccurrence of residual stresses superimposed fromprevious blows. As a result, the mobilized resis-tance near the pile toe during the blow may belower than the ultimate soil resistance. In thosecases, according to Holloway et al. (1978), theresistance distribution along the pile shaft inferredfrom a back analysis may be different from thereal distribution.

Goble & Hery (1984) presented a new versionof the WEAP program, called CUWEAP, thatfollows the procedure suggested by Holloway et al.(1978). In this program successive blows are ana-lysed without zeroing displacements and stresses inthe meantime. This procedure is repeated until aconvergence criterion is accomplished.

In the work described in the present papersuccessive force records were not availableand, therefore, it was not possible to perform amultiple-blow analysis similar to that suggested byHolloway et al. (1978). In spite of this, an impor-

0.7

20.3

21.3

22.3

Velo

city

: m/s

A24425

0 20 40 60 80 100 120 140Time: ms

Measured velocitySimulated velocity

2l /c

2ll /c 2lent /c

Sea floor

2kN/m2

200 400 600 800 1000

Mobilized resistance:(kN/m2)

14kN/m2

3kN/m2

20kN/m2

246kN/m2 4689kN/m2

Fig. 7. Typical measured and simulated velocities at the pile head


tant effect of residual stress in hard driving condi-tions was observed and was evidenced by Simons's(1985) model, as mentioned previously.

SOIL STIFFNESS

The values of the back-calculated soil stiffnessks related to discrete pile segments, selected ap-proximately according to the soil stratication, are

shown in Table 1. From these results, the corre-sponding values of the soil shear modulus G weredetermined from equation (6) as

G ks2r02:9

(11)

Table 3 and Fig. 11 show simplied proles ofback-calculated values of the shear modulus of thesoil and the corresponding mobilized shaft resis-

Fig. 8. (a) Displacement at the pile head; (b) velocity at the pile toe; (c)pile and soil displacement at the pile toe

20.03

20.023

Dis

plac

emen

t: m

0 50 100 150Time: ms

(a)

Simulated displacementMeasured displacement

0.00

22.00

Velo

city

: m/s

0 50 100 150

Simulated velocity

Time: ms

(b)

0.002

20.003

20.008

Dis

plac

emen

t: m

DisconnectionReconnection

Soil displacementPile displacement

0 50 100 150Time: ms

(c)


tance s. The values of s representing the toeresistance were assumed to be the point ultimateresistance divided by the surface area of the con-ical point.

Down to 34 m, the back-calculated values ofshaft friction were very low, different from theCPT results. This was attributed to the reducedresistance offered by the soil during continuousdriving. Similar behaviour is obtained in CAPWAP

analyses of piles in sandy soils, where the skinfriction mobilized during continuous driving can bevery low. Visser et al. (1985), for instance, ob-tained a very low soil resistance during continuouspile driving in a dense to very dense sand layer inthe North Sea. In this case, redriving of an instru-mented pile after a 24 h interval indicated a set-upfactor of 26 for the frictional resistance.

It must also be remembered that even in the

Fig. 9. Static and total resistances for pile A1 at 43 m of penetration: (a)at node 25; (b) at node 1; (c) at node 2

20

2180

Forc

e: k

N

0 50 100 150Time: ms

(a)

A B (Reconnection)

Radiation damping Static resistance (node 25)Total resistance (node 25)

s

(Disconnection)

21800

2800

Forc

e: k

N

0 50 100 150

(b)Time: ms

Static resistance (node 1)Total resistance (node 1)

A B (Reconnection)(Disconnection)

22800

2800

Forc

e: k

N

(c)Time: ms

0 50 100 150

(Disconnection) (Reconnection)A B

Static resistance (node 2)Total resistance (node 2)


CPT there is a considerable decrease in the lateralresistance when it is measured along the wholelength of the rod. This was the reason whyBegemann (1963) proposed measurement of thelateral friction close to the cone with the so-calledcone sleeve.

It can be concluded from Table 3 and Figs 2and 11 that s increases signicantly as the soilresistance increases, whereas the ratio G=s showsa relatively small variation, with a mean value of166 along the shaft and 209 at the toe. Therefore,for the purpose of preliminary drivability studies incalcareous granular soils using Simons's (1985)model, the shear modulus can be assumed as

G 175s (12)Equations (11) and (12) can be very useful to

estimate preliminary values of ks in the rst trialsof a back analysis or for drivability studies.

RADIATION DAMPING

It should be emphasized that in Simons's (1985)model both soil stiffness and damping coefcientdepend on the shear modulus in such a way thatthey are dependent variables. However, in order tomake the computed velocity at the pile head matchthe measured velocity properly, it was necessary tovary independently the soil stiffness and the damp-ing coefcient in the back analyses. Although thesetwo parameters were treated as independent vari-

ables, similar trends could be veried in theirbehaviour.

The values of damping coefcient presented inTable 1 are back-calculated ones. The values inparentheses are related to cs calculated from equa-tion (7) on the basis of the G values shown inTable 3, adopting a soil density of 1:7 Mg=m3.

Apart from pile A1 at 43 m penetration, thethree remaining records showed similar trend. Thecs values calculated as (rG)0

:5, on the basis of Gvalues from Table 3, can now be compared to theback-calculated damping coefcients shown inTable 1. The two values differ typically by a factorof 030060 for the pile shaft. The differences inbehaviour between shaft and toe can be explainedby the residual stress located at the pile toe,equilibrated by shear stresses acting downwardsalong the pile shaft. The mobilized toe resistanceduring a blow is underestimated and the mobilizedshaft resistance is overestimated in cases whereresidual stresses are present at the pile toe. In fact,underestimated s values at the pile toe may have

Table 2. Comparison of penetration for a blow in-ferred from back analyses with that obtained fromeld records

Record Penetration (back-analysed): mm

Penetration (eldrecords): mm

A14300 29 35A24325 15 23A24425 15 08B24200 43 12

Record A24425

Simulated axial stress

280

2180

Axia

l stre

ss: M

N/m

2

0 50 100 150Time: ms

Fig. 10. Axial stress at the pile toe

Table 3. Soil shear modulus G and mobilized shaftresistance s obtained from back analyses

Record Depth from seaoor: m

G:MN=m2

s:kN=m2

G=s

A14300 00 to 340 41 24 171340 to 415 262 149 176415 to 430 6035 2833 213

A24325 00 to 340 18 10 175340 to 417 228 124 184417 to 432 8966 4213 213

A24425 00 to 350 24 13 185350 to 427 221 126 175427 to 442 10690 4689 228

B24200 00 to 330 46 32 144330 to 405 207 179 116405 to 420 11035 6064 182

Depth corresponding to the pile toe.


Fig. 11. Simplied proles of shaft friction and shearmodulus: (a) pile A1 at 430 m penetration; (b) pile A2 at432 m penetration; (c) pile A2 at 442 m penetration; (d) pileB2 at 420 m penetration

Mobilized shaft resistance s: kN/m2

Soil shear modulus G: MN/m20 10 20 30 40 50 60

(a)

200 400 600 8001000

43.

0 m

41.

5 m

34.

0 m

s G



(b)

200 400 600 8001000

43.

2 m

41.

7 m

34.

0 m

s G


Fig. 11. (Cont.)



(c)

200 400 600 8001000

44.

25 m

42.

7 m

35.

0 m

s G



(d)

200 400 600 8001000

42.

0 m

40.

5 m

33.

0 m

s G


been responsible for the higher G=s ratio at thepile toe, and overestimated s values at the pileshaft may have caused the lower G=s ratios nearthe pile shaft. As a consequence, the dampingcoefcient cs, calculated as (rG)0

:5, resulted inhigh values at the pile toe when compared to theback-calculated parameters shown in Table 1. Theopposite has been observed near the pile shaft.

Randolph & Deeks (1992) presented a eld casein which a pipe pile behaved as unplugged duringdriving. In that case, residual stresses are not sup-posed to occur. Furthermore, Randolph & Deeks(1992) discuss a redriving record in which the soilresponse is usually distinct from that in continuousdriving. Although a direct comparison between theRandolph & Deeks (1992) results and those in thepresent paper cannot be made, mainly because theydeal with different situations, Randolph & Deeks(1992) found it necessary to halve (at the pile shaft)the dashpot values obtained from (rG)0:5 in order toachieve a good match between the measured andcomputed velocity curves.

Randolph & Deeks (1992) mentioned that thetrend found is consistent with the comment byMitwally & Novak (1988) that a zone of soft soilaround a driven pile would produce a much lowerdamping than a homogeneous soil. In fact there arenot enough data to enable a comprehensive evalua-tion of the radiation damping during continuous

driving. More data are needed from different sitesbefore conclusions on the overall behaviour ofradiation damping can be arrived at.

In order to examine the inuence of the choiceof the damping coefcient and soil stiffness on thesignal matches, the following analyses were carriedout (a) a sensitivity analysis for the blow on pileA1 at 43 m and (b) a comparison of the simulatedvelocities (i) using the parameters in Table 1 and(ii) using the best consistent set of stiffness anddamping coefcient, based on equations (6) and(7).

The sensitivity analysisA sensitivity analysis was performed for the

blow on pile A1 at 43 m penetration. This blowwas chosen in an effort to clarify its behaviour,which was distinct from that of the other records.The results are summarized in Table 4 and Figs1217. The gure numbers listed in Table 4 arethose of the gues showing the matches based onthe radiation damping and soil stiffness valuesshown in the same row of the table. Fig. 12illustrates the original match, with the parametersalso shown in Table 1.

The match in Fig. 13 indicates a lower toedamping than that indicated in Table 1 (Fig. 12).On the other hand, the bearing capacity and the

Table 4. Sensitivity analysis of A14300 record (pile A1 at 43 m penetration)

Figurenumber

Bearing capacity:kN

Toe resistance:%

Damping coefcient cs: kN s=m3 Soil stiffness ks: MN=m

3

12 23 500 54 Toe: 1000 (1013) Toe: 667(original t as Shaft: Shaft

in Table 1) From 0 to 34 m: 10 (83) From 0 to 34 m: 23From 34 to 415 m: 70 (211) From 34 to 415 m: 145

13 25 000 60 Toe: 950 (893) Toe: 518Shaft: Shaft:From 0 to 34 m: 100 (75) From 0 to 34 m: 18From 34 to 415 m: 150 (188) From 34 to 415 m: 116



16 25 000 60 Toe: 950 (850) Toe 470Shaft: Shaft:From 0 to 34 m: 50 (68) From 0 to 34 m 15From 34 to 415 m: 70 (168) From 34 to 415 m 93

17 25 000 60 Toe: 950 (850) Toe: 470Shaft: Shaft:From 0 to 34 m: 30 (68) From 0 to 34 m 15From 34 to 415 m: 60 (168) From 34 to 415 m: 93


percentage of toe resistance are somewhat higher.The match is quite good up to 2l=c; after this timethe simulated velocity needs to be reduced in orderto t the measured velocity. Fig. 14 shows thesame blow with a reduced damping coefcientalong the pile shaft. A better agreement was foundafter 2l=c but the need to reduce the dampingcoefcient is still apparent, to a lesser extent. Fig.15 indicates a fairly good t after wave reectionat the pile toe but the simulated velocity nowseems to be higher than the measured velocity nearthe pile toe. A reduction in stiffnes can have aneffect in this direction, as shown in Fig. 16. In factthis last measure helped to t the record in thevicinity of 2l=c but also produced an effect afterthat time. An additional reduction in the damping

coefcient along the pile shaft produced a better tas shown in Fig. 17.

The data shown in Table 4 suggest that the tpresented in Table 1 at 43 m penetration could bereplaced by the one related to Fig. 17. In this case,the soil response would not be so different fromthat of the remaining records.

The comparative analysisA comparison between the simulated velocities

(a) using the parameters in Table 1 and (b) usingthe best consistent set of stiffness and damping,based on equations (6) and (7), was made. Thecomparison was performed for piles A2 at 4325 mpenetration and B2 at 42 m penetration.

1.0

0.5

0.0

20.5

21.0

21.5

22.0

Velo

city

: m/s

0 10 20 30 40 50 60 70 80 90 100 110 120Time: ms


Fig. 12. Measured and simulated velocities at the pile head for pile A1 at43 m penetration. Match corresponding to data in Table 1, also shown inTable 5

22.0

21.5

21.0

20.5

0.0

0.5

1.0

Velo

city

: m/s

0 10 20 30 40 50 60 70 80 90 100 110 120

Time: ms


Fig. 13. Measured and simulated velocities at the pile head for pile A1 at43 m penetration. Match corresponding to data shown in Table 5


The consistent set of stiffness and damping co-efcients was based on equations (6) and (7),adopting a soil shear modulus G taken as anaverage of those indicated in Table 3, and theactual soil density of 1:7 Mg=m3. The resultingparameters are indicated in Table 5.

Figures 18 and 19 show the measured andsimulated velocities at the pile head for pile A2 at4325 m penetration and for pile B2 at 42 m pene-tration, respectively. Figs 18(a) and 19(a) indicatethe simulated velocities related to ks and cs fromTable 1, whereas Figs 18(b) and 19(b) indicate thesimulated velocities related to the consistent set ofstiffness and damping coefcients based on the soilshear modulus G and equations (6) and (7).

From Figs 18 and 19 it can be concluded thatalthough a better t between the measured andsimulated velocities is obtained when both thestiffness and the damping coefcient are variedindependently, as indicated in Figs 18(a) and 19(a),a reasonable agreement can also be found whenadopting a consistent set of stiffness and dampingcoefcients, as indicated in Figs 18(b) and 19(b).Figs 20 and 21 compare on a single plot for eachpile the simulated velocity based on ks and cs fromTable 1 and the simulated velocity based on theconsistent set of stiffness and damping coefcients.Figs 20 and 21 indicate that the differences are notsignicant and are mainly concentrated near thepile toe, where signicant residual stresses are

1.0

0.5

0.0

20.5

21.0

21.5

22.0

Velo

city

: m/s

0 10 20 30 40 50 60 70 80 90 100 110 120Time: ms



1.0

0.5

0.0

20.5

21.0

21.5

22.0

Velo

city

: m/s

0 10 20 30 40 50 60 70 80 90 100 110 120Time: ms




supposed to occur. It should also be emphasizedthat both analyses were performed using the sameresistance distribution along the pile shaft and toe,as indicated in Table 1. Small variations in resis-tance distribution were found to be much moresignicant in signal matching than the differences

resulting from using the consistent set of stiffnessand damping coefcients or the stiffness anddamping coefcients from Table 1.

Danziger et al. (1996) presented a discussion onthe uniqueness of the CAPWAP-type analysisbased on Smith's model. The authors suggested

1.0

0.5

0.0

20.5

21.0

21.5

22.0

Velo

city

: m/s

0 10 20 30 40 50 60 70 80 90 100 110 120Time: ms



1.0

0.5

0.0

20.5

21.0

21.5

22.0

Velo

city

: m/s

0 10 20 30 40 50 60 70 80 90 100 110 120Time: ms



Table 5. Consistent set of stiffness and damping coefcients based on soil shearmodulus G

Depth from seaoor

Soil shear modulus:MN=m2

Soil stiffnessks: MN=m

3Damping coefcient

cs: kN s=m3

00 to 340 32 18 738340 to 415 230 128 1977

Pile toe 9182 10149 12494


some useful procedures to arrive at a unique solu-tion. The authors' suggestions were based on aconsiderable number of back analyses usingSmith's model. Unfortunately, Simons's (1985)model has not been extensively applied so far anda similar set of procedures could not be proposedfor using this model. Extensive analyses need to beperformed before any conclusions can be drawnconcerning the quality of the match obtained usingSimons' model.

The need to treat the soil stiffness and dampingcoefcient as independent variables in order toproduce better signal matches can only be clariedwhen other documented cases of large closed-ended piles become available. Gathering more datafrom different sites will enable verication ofwhether the trend found in the present paper is a

particular feature of closed-end pipe piles at theend of continuous driving in calcareous sands orcan be found in other situations.

BEARING CAPACITY

Table 6 was prepared from the results of soilresistance distribution obtained from back analyses.This table presents, for each eld record, both thetoe and the shaft resistance during a single blow;the mean, maximum and minimum shaft resistancemobilized along the pile length; the mean shaftresistance mobilized in both layers of calcareoussand; and the mobilized toe resistance. The mobi-lized toe resistance was estimated as the toe ulti-mate resistance divided by the lateral area of theconical point. In the same column, in parentheses,

Fig. 18. Measured and simulated velocities at the pile head for pile A2 at4325 m penetration: (a) simulated velocity related to ks and cs from Table1; (b) simulated velocity related to consistent set of ks and cs

1.0

0.5

0.0

20.5

21.0

21.5

22.0

Velo

city

: m/s

0 10 20 30 40 50 60 70 80 90 100 110 120Time: ms


(a)

1.0

0.5

0.0

20.5

21.0

21.5

22.0

Velo

city

: m/s

0 10 20 30 40 50 60 70 80 90 100 110 120Time: ms


(b)


the unit toe resistance, calculated as the toe ulti-mate resistance divided by the cross-sectional areaat the top of the cone, is also indicated.

From this table the following observations canbe made.

(a) The mean shaft resistance is around 40 kN=m2.(b) The larger values of shaft resistance are con-

centrated near the pile toe, as expected. In theremainder of the shaft and, mainly, close to thesea oor, the shaft resistance shows low values.

(c) For the rst layer of calcareous sand, from 13to 33 m depth, the mean shaft friction mobil-ized according to Simons's model is about21 kN=m2.

(d ) For the second layer of calcareous sand, from33 m to the pile toe, the mean shaft friction

mobilized according to Simons's model isabout 127 kN=m2.

(e) The end-bearing stress values, qp, are very lowin relation to the cone resistance. This may bedue to the following reasons: (i) scale effect, asignicant factor for a 167 m dia. pile; (ii) thelack of accounting for residual stresses; (iii)the fact that the end-bearing stress qp actuallymobilized during pile monitoring may be lowerthan the end-bearing capacity, mainly in harddriving conditions when small penetrationsoccur. In fact, redriving some piles with aheavier hammer indicated a higher toe resis-tance than that back-analysed at the end ofdriving (Danziger et al., 1992).

A comparison between the results in Table 6

Fig. 19. Measured and simulated velocities at the pile head for pile B2 at42 m penetration: (a) simulation velocity related to ks and cs from Table 1;(b) simulated velocity related to consistent set of ks and cs

1.0

21.5

22.0

22.5

23.0

Velo

city

: m/s

0 10 20 30 40 50 60 70 80 90 100 110 120Time: ms


0.0

0.5

20.5

21.0

(a)

1.0

21.5

22.0

22.5

23.0

Velo

city

: m/s

0 10 20 30 40 50 60 70 80 90 100 110 120Time: ms


0.0

0.5

20.5

21.0

(b)


1.0

0.5

0.0

20.5

21.0

21.5

22.0

Velo

city

: m/s

0 10 20 30 40 50 60 70 80 90 100 110 120Time: ms

Parameters from Table 1Consistent set of parameters

Fig. 20. Simulated velocities at the pile head for pile A2 at 4325 mpenetration

1.0

21.5

22.0

22.5

23.0

Velo

city

: m/s

0 10 20 30 40 50 60 70 80 90 100 110 120Time: ms

Parameters from Table 1Consistent set of parameters

0.0

0.5

20.5

21.0

Fig. 21. Simulated velocities at the pile head for pile B2 at 42 mpenetration

Table 6. Mobilized resistances during continuous driving

Pile Depth:m

Toeresistance:

kN

Skinresistance:

kN

s mean:kN=m2

s max:kN=m2

s min:kN=m2

s13 to 33 m:

kN=m2

s33 m to piletoe: kN=m3

qp:kN=m2

A1 4300 12 690 10 810 47 175 2 23 141 2 833(5 795)

A2 4325 18 870 6 290 28 225 2 10 113 4 213(8 616)

A2 4425 21 000 7 000 30 246 2 13 105 4 689(9 589)

B2 4200 27 000 11 800 53 352 20 38 150 6 064(12 402)

Unit toe resistance estimated as the toe ultimate resistance divided by the lateral area of the conical point (values inparentheses: unit toe resistance estimated as the toe ultimate resistance divided by the cross-sectional area at the top ofthe cone).


and similar results obtained with Smith's model(described by Danziger et al., 1992) leads to thefollowing conclusions.

(a) The ultimate resistance back-calculated withSimons's model exceeded that obtained withSmith's model by some 15%.

(b) The mean shaft friction mobilized with Si-mons's model was nearly the same as thatobtained with Smith's model.

(c) As far as the toe resistance is concerned, thedifferences between the two models reached35%, Simons's ultimate resistance exceedingSmith's.

A reason why Simons's model gives differentresults from Smith's near the pile toe may be theeffect of viscous damping, which is not consideredexplicitly by Simons (1985). Although the mostsignicant portion of the energy loss for harddriving conditions is due to radiation damping, theviscous damping still plays a minor role. Its effect,which is usually greater at the pile toe, may havebeen incorporated into the static resistance, lead-ing to a Simons's ultimate resistance exceedingSmith's.

CONCLUSIONS

The paper presents results of back analyses ofeld records of offshore closed-end pipe pilesdriven in calcareous sands, performed with an im-proved soil model proposed by Simons (1985). Thereasons for implementing this model were theconsistency in its formulation, its adequacy forhard driving conditions and the possibility of fol-lowing the relative displacement between the pileand the soil at the pile shaft. At the same time themodel preserves the simplicity of Smith's (1960)approach.

The main conclusions drawn in the paper arethe following.

(a) The process of matching the predicted beha-viour with eld records presents no additionaldifculties when compared with the use ofSmith's model.

(b) Although in Simons's model the soil stiffnessand the damping coefcient depend on theshear modulus in such a way that they aredependent variables, a better agreement be-tween simulated and measured velocities wasfound in the present analysis when bothparameters varied independently. However,when adopting the best consistent set ofstiffness and damping coefcients as depen-dent variables, reasonable agreement betweensimulated and measured velocities was alsoobtained.

(c) Back calculated values of the ratio of the shear

modulus to the shaft resistance (G=s) indi-cated similar values for depths along the shaftand near the pile toe, around (G=s) 175.

(d ) For drivability studies involving calcareoussands, in the absence of a soil shear modulusprole, the authors suggest the followingprocedure: rst estimate s for each foundationdepth; the value of the shear modulus is thenobtained by taking G 175s. On the basis ofthe shear modulus, the value of the soilstiffness and a rst indication of the dampingcoefcient can be determined.

(e) A comparison of the pile capacities obtainedfrom Smith's (1960) model and Simons's(1985) model showed that the ultimate resis-tance back-calculated with Simons's modelexceeded that obtained with Smith's modelby some 15%; the mean mobilized shaftfriction was nearly the same and the differencein toe resistance was about 35%, Simons'svalue exceeding Smith's.

( f ) For hard driving conditions, where elasticdisplacements are believed to prevail, radiationdamping appears to be the most appropriateform of energy loss to consider. In these cases,Simons's model should be preferred. On theother hand, for easy driving, where plasticdisplacements are dominant, viscous dampingis perhaps the most appropriate form ofdynamic soil reaction.

(g) More data need to be gathered from differentsites to verify whether the trend found in thepresent paper is a particular feature of closed-end pipe piles at the end of continuous drivingin calcareous sands or can be found in othersituations.

ACKNOWLEDGEMENTS

The authors are grateful to PETROBRAS (theBrazilian State Oil Company) for allowing thepublication of the data, and to CNPq (BrazilianResearch Council) for nancial support to thesenior author.

APPENDIX 1. CYLINDRICAL AND TRUNCATED

CONICAL ELEMENTS

Cylindrical elementThe element area A is

A e( e) (13)where e and are dened in Fig. 22.

The consistent mass matrix is

M r AL3

11

21

21

264375 (14)

The consistent mass added to the nodes is


m r AL3

(15)

Truncated cone elementThe element area A is

A e(2 1)ln[(2 e)=(1 e)] (16)

where 1 and 2 are dened in Fig. 23.The consistent mass matrix is

M rel12

4(1 e) (2 1) 2(1 e) (2 1)2(1 e) (2 1) 4(1 e) 3(2 1)

(17)

The consistent mass added to the node corresponding tothe diamter 1 is

m1 rel12

[4(1 e) (2 1)] (18)The consistent mass added to the node corresponding to

the diameter 2 is

m2 rel12

[4(1 e) 3(2 1)] (19)

REFERENCESBegemann, H. K. S. (1963). The friction jacket cone as

an aid in determining the soil prole. Proc. 6th Int.Conf. Soil Mech. Found. Engng, Montreal 1, 1720.

Chow, Y. K., Karunaratne, G. P., Wong, K. Y. & Lee,S. L. (1988a). Prediction of load-carrying capacity ofdriven piles. Can. Geotech. J. 25, 1323.

Chow, Y. K., Wong, K. Y., Karunaratne, G. P. & Lee,S. L. (1988b). Wave equation analysis of pilesarational theoretical approach. Proc. 3rd Int. Conf.Applicat. Stress Wave Theory to Piles, Ottawa,208218.

Clough, R. W. & Penzien, J. (1975). Dynamics of struc-tures. McGraw-Hill, New York.

Costa, A. M. (1988). DINEXP-1D Program. Rio deJaneiro: Cenpes (Research Centre), PETROBAS.

Costa, A. M., Moreira, L. F. R., Ebecken, N. F. F.,Coutinho, A. L. G. A., Landau, L. & Alves, J. L. D.(1988). Recent application of computer methods fordrivability analysis of offshore piles in Brazil. Pro-ceedings of the International Conference on ComputerModelling in Ocean Engineering, Venice, pp.665672.

Danziger, B. R. (1991). Dynamic analysis of pile driving.DSc thesis, COPPE/Federal University of Rio deJaneiro (in Portuguese).

Danziger, B. R., Pacheco, M. P., Costa, A. M. & Lopes,F. R. (1992). Back-analyses of closed-end pipe pilesfor an offshore platform. Proc. 4th Int. Conf. Appli-cat. Stress Wave Theory to Piles, The Hague,557562.

Danziger, B. R., Lopes, F. R., Costa, A. M. & Pacheco,M. P. (1996). A discussion on the uniqueness ofCAPWAP analyses. Proc. 5th Int. Conf. Applicat.Stress Wave Theory to Piles, Orlando, 394408:

Forehand, P. W. & Reese, J. L. (1964). Prediction of pilecapacity by the wave-equation. J. Soil Mech. Found.Div., ASCE 90, 125.

Geomecanica (1986). Evaluation of soil properties at thePargo SiteCampos Basin, Report RJ-3274/007. Geo-mecanica (in Portuguese). Rio de Janeiro, Brazil.

Goble, G. G. & Hery, P. (1984). Inuence of residual forceson pile driveability. Proc. 1st Int. Conf. Applicat. StressWave Theory on Piles, Stockholm, 131161.

Goble, G. G., Raushe, F. & Likins, G. E. (1980). Theanalysis of pile drivinga state-of-the-art report.Proc. 2nd Int. Conf. Applicat. Stress Wave Theory onPiles, Stockholm, 154161.

Holloway, D. M., Clough, G. M. & Vesic, A. S. (1978).The effects of residual driving stresses on pile per-formance under axial loads. Proceedings of the Off-shore Technology Conference, OTC 3306, Houston,22252236.

L

2

1

x

Fig. 23. Denition of 1 and 2

e

Fig. 22. Denition of and e


Lee, S. L., Chow, Y. R. & Karunaratne, G. P. (1988).Rational wave equation model for pile driving analy-sis. J. Geotech. Engng Div., ASCE 114, No. 3,306325.

Lysmer, J. (1965). Vertical motion of rigid footings,Report 3-115. Vicksburg, MS: US Army Corps ofEngineers, Waterways Experiment Station.

Matsumoto, T. & Takei, M. (1991). Effects of soil plugon behaviour of driven piles. Soils Found. 31, No. 2,1434.

McClelland, B. (1985). Geotechnical site investigation,preliminary eld report, Pargo Site, Campos Basin,Job No. 0184-1237.

Mello, J. R. C., Amaral, C., Costa, A. M., Rosas, M. M..Coelho, P. S. D. & Porto, E. C. (1989). Closed-endedpipe piles: testing and piling in calcareous sand.Proceedings of the Offshore Technology Conference,OTC 6000, Houston, 95106.

Mello, L. F. N. (1990). Internal report on the conical pilebase model. Rio de Janeiro: Cenpes (Research Cen-tre), PETROBRAS (in Portuguese).

Mitwally, H. & Novak, M. (1988). Pile driving analysisusing shaft models and FEM. Proc. 3rd Int. Conf.Applicat. Stress Wave Theory to Piles, Ottawa, 455466.

Nguyen, T. T., Berggren, B. & Hansbo, S. (1988). A newsoil model for pile driving and driveability analysis.Proc. 3rd Int. Conf. Applicat. Stress Wave Theory toPiles, Ottawa, 353367.

Novak, M. (1977). Vertical vibration of oating piles.J. Engng Mech. Div., ASCE EM1, 153168.

Randolph, M. F. & Deeks, A. J. (1992). Keynote lecture:dynamic and static soil models for axial pile response.Proc. 4th Conf. Applicat. Stress Wave Theory to Piles,The Hague, 314.

Randolph, M. F. & Simons, H. A. (1986). An improvedsoil model for one dimensional pile driving analysis.Proc. 3rd Int. Conf. Numer. Meth. Offshore Piling,Nantes, 117.

Simons, H. A. (1985). A theoretical study of pile driving.PhD thesis, University of Cambridge.

Smith, E. A. L. (1960). Pile driving analysis by the wave-equation. J. Soil Mech. Found. Div., ASCE 86, No. 4,3561.

Visser, M., Van der Zwaag, G. L. & Pluimgraaff, D. J. H.M. (1985). Application of monitoring systems duringpile driving in North Sea sands. Proc. 4th Int. Conf.Behaviour of Offshore Structures, Amsterdam,623631.


INTRODUCTIONGEOTECHNICAL CONDITIONS AND PILE CHARACTERISTICSSIMONS'S MODELBACK ANALYSESPENETRATION FOR ONE BLOW RESIDUAL STRESSESSOIL STIFFNESSRADIATION DAMPINGThe sensitivity analysisThe comparative analysisBEARING CAPACITYCONCLUSIONSACKNOWLEDGEMENTSAPPENDIX 1. CYLINDRICAL AND TRUNCATED CONICAL ELEMENTSCylindrical elementTruncated cone elementREFERENCES

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