1-Michelides-Gazetas (1997)-Approximate Non-linear Dynamic Axial of Piles

7/27/2019 1-Michelides-Gazetas (1997)-Approximate Non-linear Dynamic Axial of Piles

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Approximate non-linear dynamic axial response of piles

O . M I C H A E L ID E S , G . G A Z E T A S , G. BOUCKOVALAS a n d E . C H R Y S I K O U {

The axial dynamic response of a single pile inclay is examined within the framework of aWinkler-type approach. Experimental data forthe variation of soil shear modulus and hystere-tic damping with the amplitude of shear strainand the soil `plasticity' index are curve-ttedand utilized in simulating in a realistic way thenon-linearity of the surrounding soil arisingfrom the stresses induced by the pile. Anequivalent radially-inhomogeneous but linearly-hysteretic continuum is established, and the`springs' and `dashpots' of the dynamic Winklermodel are obtained at each depth by solvinganalytically the pertinent plane-strain (in z) andaxisymmetric (in r) elastodynamic problem.Slippage at the pilesoil interface is also takeninto consideration in a simple approximate way.Analytical results are presented to demonstratethe practical signicance of soil non-linearity,soil `plasticity' index and interface slippage inthe dynamic stiffness and damping of the pile.

KEYWORDS: dynamics; numerical modelling andanalysis; piles; soilastructure interaction; stiffness;vibration.

Les auteurs examinent la reaction dynamiqueaxiale d'un pieu dans l'argile dans le contexted'une approche du type Winkler. Apres l'ajuste-ment de courbe, les donnees experimentalesobtenues pour la variation du module de cis-aillement et de l'amortissement hysteretique enfonction de la deformation de cisaillement et del'indice de plasticite du sol sont utilisees poursimuler de facon realiste la non-linearite du solenvironnant sous l'effet des tensions engendreespar le pieu. Un continuum equivalent, nonhomogene radialement, mais hysteretique line-airement, est etabli, et les ressorts et amor-tisseurs du modele dynamique de Winkler sontobtenus a chaque profondeur par resolutionanalytique du probleme elastodynamique de ladeformation plane (en z) et axisymetrique (enr). On tient egalement compte, de facon ap-proximative et simple, du glissement a l'inter-face pieu-sol. Les resultats de l'analyse sontpresentes pour demontrer les effets pratiques dela non-linearite du sol, de l'indice de plasticite

du sol et du glissement interfacial sur la rigidite

dynamique et l'amortissement du pieu.

INTRODUCTION

A widely used method for evaluating the dynamic

axial response of piles replaces the soil sur-rounding the pile with a series of independentsprings and dashpots (Winkler approach). This im-plies that shear waves, emitted from the pile peri-phery, propagate only horizontally and plane-strainconditions prevail. Radial soil displacements areneglected. Soil, therefore, deforms solely in pureshearan assumption also used with success forstatically-loaded piles (Randolph & Wroth, 1978;Baguelin & Frank, 1979), and for the dynamicinteraction of piles in a group (Dobry & Gazetas,1988).

The frequency-dependent moduli of the (con-tinuously distributed along the pile length) Winklersprings and dashpots are obtained by solving the

elastodynamic problem of a unit-thickness soillayer of innite lateral extent, containing an oscil-

lating rigid inclusion (the pile slice). Most solu-tions are based on the additional simplifyingassumption that the soil is radially homogeneous.This, however, may not be realistic even withhorizontally uniform soils, because the effective(secant) modulus of the soil in the vicinity of thepile will be reduced to lower than the free-eldvalues, owing to the comparatively large ampli-tudes of the induced shear strains and the ensuingnon-linear soil response.

Novak & Sheta (1980) were the rst to proposethe use of a massless, narrow, annular boundaryzone around the pile, having a shear modulus Ginsmaller than the modulus Gs of the outer surround-ing zone (i.e. of the free eld), and larger material

damping. The purpose of such a `soft' zone was toaccount in an approximate way for both soil non-linearity in the region of highest stresses andslippage (and other `deciencies') at the pilesoilinterface (Novak, 1991; Pender, 1993). Neglectingthe mass of the boundary zone was necessary inorder to prevent wave reections from the ctitious

Michaelides, O., Gazetas, G., Bouckovalas, G. & Chrysikou, E. (1997). Geotechnique 48, No. 1, 3353

33

Manuscript received 17 November 1995; revised manu-script accepted 11 September 1996.Discussion on this paper closes 1 May 1998; for furtherdetails see p. ii. National Technical University, Athens.{ Massachusetts Institute of Technology.


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discontinuity at the interface between the cylindri-cal zone and the outer region. On the other hand,the solution, by Veletsos & Dotson (1986), in-cluded the inertia of the boundary zone; hence theresulting stiffness and damping exhibited undula-tions with frequency. More appropriate continuous(monotonically increasing) variations of moduluswith radial distance, G

G(r), assumed by

Gazetas & Dobry (1984a) and Veletsos & Dotson(1988), eliminated the problem of the spuriouswave reections at the interface between `boundaryzone' and the surrounding soil.

The previous contributions address the problemof lateral soil heterogeneity with only qualitativereference to the non-linear soil response, since thevariations of soil properties employed are merelyhypothetical. To aid practical applications, thispaper utilizes experimental data (e.g. Vucetic &Dobry, 1991) on the dependence of the secantshear modulus and hysteretic damping of soil onthe shear strain amplitude and the nature of thesoil (the latter represented by the plasticity indexI

P). The variation of modulus and damping is then

related to the magnitude of the applied loadthrough the amplitude of the induced strains. Thus,the radial inhomogeneity studied models the non-

linearity of soil in shear realistically (althoughapproximately). In addition, slippage at the pilesoil interface is also modelled in a simple, realisticway.

PROBLEM DEFINITION AND OUTLINE OF METHOD

The problem studied is that of a oating cylind-rical pile embedded in a layered soil depositand subjected to harmonic axial load at the top(Fig. 1). Initially, before applying any static ordynamic load, all soil layers are assumed to belaterally homogeneous (Fig. 1(a)), implying thatany installation effects have `dissipated'. Applyingthe static load, shear stresses 0 develop on theelements of each layer. To a good approximation,0 is inversely proportional to the radial distancefrom the pile (Randolph & Wroth, 1978). A dy-namic load Pc cos t generates additional stressesc cos(t ) on the soil elements (Fig. 2),where c is the amplitude of the stresses and isthe phase difference with respect to the applied

Fig. 1. Steps of the methodology developed in this paper and denition of the impedances kz and Kv: (a) initial,radially uniform distribution of shear modulus G Gs for each horizontal layer; (b) and (c) attenuation withincreasing radial distance of the shear stress and shear strain amplitudes induced by load Pc cost; (d) strain-compatible (`effective') shear modulus G G(r; Pc, ); (e) generalized Winkler-type reaction of each soil layer asrepresented with a `spring' ( kz) and a `dashpot' (cz), both functions of Pc and ; (f) complete pilesoil systemrepresented with both Kv and Cv dependent on load-amplitude and frequency

Notice, however, the slightly different notation in thispaper.

34 MICHAELIDES, GAZETAS, BOUCKOVALAS AND CHRYSIKOU


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load. Figs 1(b) and 1(c) schematically illustrate theradial distribution of the amplitudes of cyclic shearstress c and cyclic shear strain c. They bothdepend, for a given soil and pile, on the amplitudeof the applied load (Pc) and the frequency ofexcitation ().

The radial variation of shear strain greatly af-fects the value of the secant (èffective') shearmodulus and the generated hysteretic damping ofthe surrounding soilwith the modulus decreasingand the damping increasing in the vicinity of thepile. The soil now becomes, èffectively', radiallyinhomogeneous (Fig. 1(d)). Experimental data for

clays are utilized in estimating the degree of thisèffective' inhomogeneity, and the elastodynamicproblem of the axisymmetric radially-inhomoge-neous unbounded soil layer containing a vibratinginclusion (the pile slice) is solved analytically. Thereaction of each soil layer is then represented witha (frequency-dependent) `spring' and `dashpot', kz

and cz, as in Fig. 1(e). Finally, the total pile headstiffness and damping Kv and Cv (Fig. 1(f)) areobtained from the solution of the differential equa-tion of motion of the (axially deformable) pilecontinuously supported by the Winkler-type axial`springs' and `dashpots' of Fig. 1(e). The effect ofslippage between the soil and the pile is alsoexamined in the paper, in an approximate way(however, this is not shown in the gure).

DYNAMIC SHEAR STRESS DISTRIBUTION

In principle, the cyclic stressstrain response of

the soil depends on both the static (initial) and thecyclic (induced) shear stresses. For low, moderateand moderately high intensities of cyclic loading,however, one may overlook the effect of staticshear stresses and focus solely on the effect ofcyclic shear stresses or strains. This argument issuggested by the well known Masing (1926) criter-

(a)

Layer i

0 c

Pc

P0

Time

Shear stress

(b)

0c

0

0c

0

0

c

c

G

1

G

1

c

c

First

loading

0

c c c c0

(c)

Fig. 2. (a) The fundamental approximation: shear waves from the shaft periphery propagate horizontally underplane strain conditions; (b) variation of shear stress with time in cycling loading; (c) stressstrain behaviour of asoil element under initial shear stress 0 and a sinusoidal imposed cyclic shear stress of amplitude c; thehysteresis loops developing after completion of the rst loading are essentially identical for two different initialshear stresses 0 and 90

APPROXIMATE NON-LINEAR RESPONSE OF PILES 35


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ion for unloading reloading of soils, illustrated inFig. 2. Only for the rst loading from 0 to 0 cdoes the resulting strain depend on 0; thereafterthe loops depend solely on the size of c (orc).

This is also substantiated by experimental evi-dence. For instance, Fig. 3 shows typical resultsfrom cyclic direct simple shear tests on a clay(Andersen, 1992), which clearly demonstrate thatthe cyclic shear strain amplitude c (and therebythe corresponding secant shear modulus Gcac) at different stages of cyclic loading remainpractically constant for the entire possible range ofstatic shear stresses 0 and shear strains 0, at leastfor the numbers of loading cycles of interest inearthquake engineering.

Evidently, the radial distributions of c, c andG, sketched in Figs 1(b), 1(c) and 1(d), are inter-dependent. For example, the distributions c c(r) and c c(r) cannot be computed until thesoil modulus G G(r) is already known; G(r),however, is obtained from soil data only after thedistribution of strains c(r) is known.

Strictly speaking, the problem of determining c(or c) and G can be solved only with an iterative

procedure. Fortunately, however, the radial distribu-tion of shear stresses c(r) turns out to be quiteinsensitive to variations in the radial distribution ofshear modulus G(r). This insensitivity is demon-strated in Appendix 1, where the elastic stressdistributions c(r) for a (radially) homogeneousand two radially inhomogeneous soil layers, com-puted analytically (and rigorously), are shown to bequite similar for most frequencies of practicalsignicance.

This observation simplies considerably the im-plementation of the method of Fig. 1. The inducedshear stresses are obtained, a priori, for a homo-

geneous soil (i.e. without knowing the exact varia-tion of G(r)). As shown in Appendix 1,

c(r) c0

sJ21 a0

r

R

Y21 a0

r

R

J21(a0) Y21(a0)

Hfd

Ige

(1a)

in which R is the pile radius, c0 c(R) is theamplitude of the imposed cyclic shear stress at thepilesoil interface,

a0 RaVs0 (1b)Vs0 Vs(R) is the S-wave velocity at r R, J1

and Y1 are the rst-order Bessel functions of therst and second kind respectively, and is thecircular frequency of the applied force.

Equation (1) can be simplied to

c(r) c0 Rr

F(ar) (2a)

where

F(ar) % 1 if ar,1F(ar) % ar0X57 if ar.1 (2b)

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0/Su

c

/Su

0.25%

0.1%

0.5%

1%

0.25%

0.

5% 1%

c15%

0

15%

1 cycle

0 0.2 0.4 0.6 0.8 1.0

0/Su

0

0.2

0.4

0.6

0.8

1.0

c

/Su

0.1%

0.25%

0.5%

1%

1%

0.5%

0.25%

c15%

0

15%

10 cycles

Fig. 3. Static and cyclic shear stresses and resultingstrains in simple shear test on Drammen clay withOCR 1 (from Andersen, 1992); the solid curves arenearly horizontal, implying insensitivity of cyclicresponse to initial shear stress 0

This nding is reminiscent of the (then) astonishing

nding about 25 years ago by Gibson (1967, 1968, 1974)that the stresses in a half-space with G proportional todepth and 1

2are identical to those in a homogeneous

half-space. This result was surprising because, in thewords of Gibson, `it paid no regard to the well-knowndoctrine that more rigid material attracts stress'. (See alsoGibson & Sills (1972)).



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in which

ar raVs(r) (2c)(Morse & Ingard, 1968; Bouckovalas et al., 1992).

Notice that for approaching 0 (static case),equation (2) reduces to the aforementioned `cylind-rical' solution of Randolph & Wroth (1978) andBaguelin & Frank (1979).

RADIAL VARIATION OF MODULUS AND DAMPING:

EXPERIMENTAL DATA AND MODELLING

The variation with shear strain amplitude of theshear modulus G G() and the hysteretic damp-ing () has been studied experimentally bynumerous investigators in both cyclic (Seed &Idriss, 1970; Richart & Wylie, 1977; Dobry &Vucetic, 1987) and monotonic (Jardine et al.,1984; Burland, 1989) experiments. Vucetic &Dobry (1991) synthesized a variety of experimentaldata and proposed the dashed curves plotted inFig. 4, where Gs is the shear modulus at low strainlevels (,105) and IP is the plasticity index ofthe soil (in %).

The effect of the so-called `plasticity' index IPseems to be substantial. As IP increases, the ratioG()aGs increases and decreases. This indicatesthat the soil behaviour remains essentially elasticfor increasingly larger values of shear strain as IPincreases.

The following equation was tted to the data ofFig. 4 (Chrysikou, 1993):

G()

Gs 1 2700c G

Gs

0X7210(IPa)

@ A(3)

or, in terms of the shear stress amplitude c(r),

G()Gs

1 2700 c0Gs

c(r)c0

0X72

10(IPa)@ A

(4)

in which is a function of the `plasticity' index:

% 0X002(IP)2 0X25(IP) 60 (5)The experimental curves of the variation of hys-teretic damping with shear strain are described bythe following set of expressions:

2 [18 0X08(IP 15)](1 GaGs)if 0 < IP , 100

2 11X2(1 GaGs) if IP > 100 (6)The good agreement between the proposed ex-pressions and the experimental data is evident inFig. 4.

Inserting equation (2) for the radial distributionof the dynamic shear stress amplitude into equation(4), a radial variation of the shear modulus isobtained:

G(r)

Gs 1 2700c0

Gs 101X4(IPa) !&

3R

rF(ar)

!'0X72(7)

or, alternatively,

G(r)

Gs 1 R

rF(ar)

& '0X72(8)

where , hereafter called the `loading intensityfactor', is

2700 c0Gs

10(1X4IPa) 2700 c0

fs

101X4(IPa)

GsaSu

(9)

where fs Su is the frictional capacity of thesoil pile interface, with Su being the undrainedshear strength of the soil and the well-knownreduction factor, deduced empirically from pileload tests, in function of the Sua9v0 ratio and theslenderness LaR of the pile (Poulos & Davis,

0.0001 0.001 0.01 0.1 1 10

0.0

0.5

1.0

G/G

s

20050

30

Ip 0

Proposed expressions

Experimental curves Ip 0

30

50

200

0.0001 0.001 0.01 0.1 1 10

Cyclic shear strain c: %

0

5

10

15

20

25

Dampingratio:%

Fig. 4. Comparison of proposed expressions for shear-strain dependence of secant shear modulus andhysteretic damping ratio (equations (3) and (6)) withexperimental design curves proposed by Vucetic &Dobry (1991)

The quotation marks around the word `plasticity' are toremind one that increasing Ip increases the elasticity (notthe plasticity) of a clay.



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1980; Fleming et al., 1985; Tomlinson, 1986;Poulos, 1988). Typical values of for usual pilelengths are

1 for soft clays1..0X4 for stiff clays % 0X4 for hard clays.It is presently well established that the shear

modulus of a clay can be expressed as a multipleof the undrained shear strength, in the formGs Su where typically the coefcient rangesaround 1000. Recent experimental data show that is also affected by the `plasticity' index. For in-stance, Larsson & Mulabdic (1991) give the fol-lowing expression for clays with medium to highvalues of the `plasticity' index:

Gs

Su% 20 000

IP 250 (10)

Equations (9) and (10) reveal that the loadingintensity factor is primarily a function of theratio of the induced cyclic shear stress amplitudeto the `frictional' capacity of the interface, while italso encompasses the effect of the soil `plasticity'index.

Figure 5 shows typical diagrams, obtained fromequation (8), for the radial variation of shearmodulus and hysteretic damping ratio, computedfor different values of the loading intensity factor and different frequency factors a0 RaVs0. Itis observed that the shear modulus and the hystere-tic damping ratio of the soil change most rapidlyin the immediate vicinity of the pile, while atlarger radial distances they tend asymptotically tothe corresponding free-eld values.

DYNAMIC SOIL REACTION (`SPRING' AND`DASHPOT' FOR A PILE SLICE)

No slippage at the pile soil interfaceThe governing differential equation of motion

for the vertically excited inhomogeneous layer ofunit thickness is derived from the dynamic equili-brium of shear and inertial forces in an elementalsoil ring (Dotson & Veletsos, 1990; Gazetas &Dobry, 1984a; Novak, 1974; Novak et al., 1978;Veletsos & Dotson, 1988):

Gd2 w

dr2 dG

dr G

r

dw

dr r d

2 w

dt2(11)

A closed-form analytical solution to this equationcannot be obtained if G(r) is described with equa-tion (8). Moreover, efforts to simplify equation (8)so that equation (11) could be solved analyticallywere not successful (Michaelides & Gazetas,1995). By contrast, solutions have been presentedfor the static problem (Kraft et al., 1981; Kuwa-bara, 1991). To overcome this difculty, the soilwas divided into four inhomogeneous{ ring zonesand the `exact' radial variation of shear modulus

(as given in equation (8)) was numerically curve-tted with the following exponential expressions:

G(r) G0 rR

m0for r,R1

G(r) G1 rR1

m1for R1 , r,R2

G(r) G2 rR2

m2for R2 , r,R3

G(r) G3 for r.R3 (12)in which

Gj Gj(1 i2j), j 0, 1, 2, 3

Fig. 5. Radial distribution of effective shear modulusG G(r), reecting the fact that shear strain ampli-tudes decrease with radial distance from the pile

10 20 3000

0.5

1.0

G/G

s

0

10

20

Dampingratio:%

0.10.5

1.0

1.0

0.5

a0 0.1

0.5

0 10 20 30

r/R

G/GsDamping ratio

G/G

s

0

0.5

1.0

0

10

20

Dampingratio:%

0.1

0.5

1.0

1.0

0.5

0.1

a0 0.5

As this ar ticle was being nalized, a paper was

published by Viggiani & Atkinson (1995) containing awealth of experimental data, relating shear modulus atsmall strain levels to the mean effective stress, in theform Gs A(9v) n. For the experimental parameter A, afunction of the plasticity index, we curve-tted theexpression A % 25000aIp to their data. The similarityto equation (10) is apparent.

{ Although a discretization of the soil medium into anumber of homogeneous ring elements could in principlealso be used, the large gradient of G(r) in the vicinity ofthe pile would lead to spurious wave reections due tothe unavoidable sharp discontinuity in G across theinterface of two zones.



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and Gj are the moduli at the boundaries of eachzone, given by

G1 G0 R1R

m0

G2 G1 R2R1

m1

G3 G2 R3R2

m2

where i p(1); G0 is the shear modulus at thepilesoil interface; 0, 1, 2 and 3 are thematerial hysteretic damping at the beginning ofeach zone (obtained using equations (6) and (8));and R1, R2 and R3 (radii of the zones) and m0, m1and m2 are functions of the dimensionless fre-quency as RaVs (Vs being the far-eld S-wavevelocity) and the loading intensity factor (seeequation (9)). These parameters (Rj, mj) were ob-tained by numerically curve-tting the data ofFig. 5.

By using the power variation of G(r) given inequation (12), the differential equation of motionfor harmonic excitation

w(r, t) w(r) ei t (13)becomes

2d2 w

d2 (mj 1) dw

dj22mjw 0,

j 0, 1, 2 (14)where

j aja(1 2i), aj RVs(r)

, j 0, 1, 2

a0

R

Vs(R)

, mj

m0, for r,R1

aj aj1 RjR

(mjmj1)a2,

for Rj, r,Rj1 raR, for all zones.(15)

Equation (14) is a generalized Bessel differentialequation, the solution of which is

w ma2[AiH(1)k1(k01ak) BiH(2)k1(k01ak)](16)

where

k 2

2 mj

and H( )k

is the k-order Hankel function of the rstor second kind respectively (Abramowitz & Ste-gun, 1972; Spiegel, 1971).

The boundary conditions are a (known) displa-

cement c ei t is imposed at the pilesoil interface,

displacements vanish as 3 I and displacementsand stresses are continuous at the interfaces of thefour ring zones. Enforcing these boundary condi-tions leads to a system of algebraic equations, fromwhich the eight complex-valued constants Ai andBi are determined. The complex dynamic stiffnessof the pilesoil system is then obtained:

kz 2G0 dwd

1

2G00[A0H(1)k (k0) B0H(2)k (k0)](17)

Equation (17) can be expressed in two alternativeforms

kz kreal ikimag kz icz (18)where kz kreal and cz kimaga are the (fre-quency-dependent) moduli of the `spring' and`dashpot' that model the soil reaction against the

oscillating pile slice; kz reects the stiffness and(distributed) inertia of the surrounding soil, whilecz represents the radiation of wave energy awayfrom the pile plus the energy dissipated in hystere-tic action in the soil (e.g. Gazetas, 1983; Gazetas& Dobry, 1984b).

Results of the analysis are presented in Figs 68. Fig. 6 portrays in dimensionless form the varia-tion with frequency of the spring and dashpotmoduli, for different values of the load intensityfactor . Recall that Gs and Vs are the shearmodulus and shear wave velocity at low strainlevels, that is, in the far eld ( r3 I). To visua-lize better the importance of soil non-linearity, thesame results are replotted in Fig. 7, but normalized

with respect to the corresponding solution for lin-ear soil ( 0). Finally, Fig. 8 uses the results ofFig. 6 for 0 and 05 to illustrate the relativecontributions of the real and imaginary parts to theoverall (complex) stiffness.

The following trends are worthy of note in thesegures.

(a) As soil non-linearity increases with increasing`loading intensity factor' , the stiffness kz of thepilesoil system (the spring modulus), predictably,decreases. The rate of decrease `accelerates' withfrequency , especially for high values. As aresult, kz becomes negative (implying a phasedifference of 1808 between pile force and displace-

ment) when both as and are relatively large.Stated in different words, the frequency factor asat which kz crosses the zero axis decreases whenthe loading factor increases.

To explain this behaviour, recall that kz can bethought of as proportional to the difference of the(overall) shear resistance of the soil minus the



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(overall) soil inertia. An increase in frequencyleads to increased soil inertia, and thereby toreduced kz. An increase in load amplitude leads tolarger soil non-linearity and reduced soil shearstiffness; thus kz would also decrease. The in-creasing (negative) role of soil inertia at highvalues of loading amplitude suggests that, effec-tively, a soil mass next to the pile vibrates almostin phase with the pile.(b) The dashpot modulus cz encompasses both thehysteretic damping in the soil (prevalent at low

frequency factors as,

0X20) and the geometricdamping due to radiation of waves from the pileperiphery to innity (dominant at high frequencyfactors as > 0X20). It is evident in Figs 6 and 7that increasing the amplitude of the load and henceincreasing non-linearity leads to a decrease ingeometric (radiation) damping and an increase in

hysteretic damping. These are hardly surprisingobservations. As the name implies, soil hystereticdamping increases with increasing hysteresis dueto non-linearity. On the other hand, radiation ofwave energy is proportional to the S-wave velocityin the soil, and with increasing non-linearity thisvelocity would decrease in the neighbourhood ofthe pile. In addition, the laterally inhomogeneoussoil wave velocity (e.g. Fig. 1(d)) leads to `con-tinuous reections' of the radially propagatingwaves, thereby further undermining radiation

damping. In fact, for a linear soil with the S-wavevelocity increasing as a power of the radialdistance (Vs(r) Vs0(raR)m, where Vs0 is thevelocity at the interface r R), Gazetas & Dobry(1984a) have shown that, asymptotically, at highfrequencies, the radiation dashpot modulus be-comes equal to 2RrVs0. This implies that the

0 1 2

3

2

1

0

1

2

3

kz

/Gs

as R/Vs

(a)

0.7

0.5

0.3

0.1

0 (linear)

0 1 2

asR/Vs

(b)

0

1

2

3

4

5

0.70.50.30.1

0 (linear)

cz

/2RV

s

Fig. 6. Results for the dynamic impedance of a pile slice in a horizontalsoil layer, showing the effect of dimensionless frequency and level ofloading (in terms of ); (a) stiffness normalized by the initial (free-eld)

soil modulus Gs; (b) dashpot modulus normalized by the product rrVstimes the perimeter of the pile



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emitted very-high-frequency waves (i.e. waves ofvanishingly small wavelength) `see' the soil sur-rounding the pile as a homogeneous mediumhaving velocity equal to Vs0, that is, the velocityin the immediate vicinity of the source! In Figs 6and 7 the reader can easily check that the aboveobservation also holds true here, to an excellentdegree: the high-frequency value of cz is propor-

tional to the interface velocity Vs0the latterbeing a decreasing fraction of the far-eld velocityVs as (and hence soil non-linearity) increases.(c) The imaginary part kimag cz of the com-plex stiffness (representing the soil reaction that isout of phase with the imposed motion) dominatesat all but the very lowest frequency factors.

Slippage at the pilesoil interfaceIn the previous analysis no slippage occurred

between soil and pile. In reality, however, suchslippage will take place whenever the total shearstress (static and dynamic) at the interface tends toexceed the frictional capacity (`skin friction') fs.In that case, the equivalent stiffness of the soil isdrastically reduced since, for a given force, the

displacement of the pile segment becomes larger.The detailed effect of slippage on the non-linearstiffness of the pile segment requires rigorousmodelling of the cyclic response of the interfaceand numerical treatment, which are beyond thescope of the very simple solution that is sought inthis paper. The aim is to gain a realistic insight

0 1 2

0

1

1

0.7

0.5

0.3

0.1

0(linear)

as R/Vs

(a)

kz/

kz,l

inear

0 1 2

0.0

0.5

1.0

1.5

0.0 0.1as

1

1.2 0.7

0.5

0.3

0.1

0 (linear)

0 (linear)

0.7

0.1

0.3

0.5

asR/Vs

(b)

cz

/cz,l

inear

Fig. 7. The results of Fig. 6 for kz and cz, normalized by the linear curveskz,linear and cz,linear



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into the phenomenon and develop an approximatemethod for estimating its effect relative to theeffects of soil non-linearity discussed up to thispoint. The method is illustrated conceptually inFig. 9. A pile slice is subjected to an initial staticdisplacement 0 (typically of the order of

12

to 13

ofthe yield displacement s) followed by dynamicloading with constant displacement amplitude c.A mechanical model of soil reaction against suchpile motion is depicted in Fig. 9(a). The soil isreplaced with a non-linear springdashpot element(as in Fig. 1(e)) connected to the pile, not directlybut through a frictional slider. The non-linearspringdashpot element is described through acomplex-valued impedance kz kz icz, whichis obtained as a function of the amplitude of theinterface shear stress c0 by the method developedin the preceding section and illustrated in the plotsof Figs 68. The slider is essentially a rigidplastic element with a yield force Fs 2Rfs. The`skin friction' fs is taken here for clayey soils asSu (see earlier discussion).

The developed method replaces (at its nalstage) the initial (`real') model of Fig. 9(a) withonly a single springdashpot element, the imped-ance of which,

kzs kzs iczs (19)encompasses both soil material non-linearity andpile interface sliding. To make the analysis simpler,the effect of slippage is decomposed into two com-ponents, studied separately.

Effect of slippage on the reduction of springmodulus and radiation damping. Figures 9(b), 9(c)and 9(d) explain in a simple manner how thecomplex modulus ks ks ics of a springdashpot system is obtained in terms of the possiblesliding. To make the picture clear, the initial (i.e.before sliding) force displacement relationship isreplaced with a segment of straight line of slope kz,that is, the complex equivalent linear modulusbefore sliding.

With reference to Fig. 9, three possible modes

Fig. 8. Comparison of the relative importance of the real (kreal kz) andimaginary (kimag cz) parts of the amplitude of the dynamic impedanceof a pile slice



11/21

of response (in a displacement-controlled vibration)can take place.

Mode I. The maximum displacement 0 cremains lower than the displacement srequired to initiate slippage of the slider:s Fsajkzj, that is, the correspondingpeak external load F0 Fc remainslower than the yield load Fs (Fig.

9(b)). Then, the response of the pilesegment is controlled by the stiffness ofthe surrounding soil, and the equivalentelasto-plastic stiffness ks is equal to theequivalent linear stiffness kz.

Mode II. The maximum displacement 0 cbecomes larger than the displacement

s required to initiate slippage of theslider, but the cyclic displacement am-plitude c remains lower than s (Fig.9(c)). In this case, slippage will occuronly on the rst-time loading, from 0to 0 c (branch abb9 of the loop),while subsequent cycles of unloadingand reloading will be sustained by thespringslider system without further

slippage (branch b9cd of the loop). Thisis achieved by a reduction of the static

force on the pile segment that counter-balances the difference between the peakand yield loads during rst-time loading.Thus, beyond the very rst loading,slippage has essentially no effect on

Pileelement

Friction

slider

fs

SU

Equivalentlinear springand dashpot

kz cz

a

d

c, c

b, b

0

(0 c)

(0 c)

kz kz icz function of c

ks

kscs

cm

kz kz icz function of slippage mode

cm function of slippage, for mode III

kzs ks

(a)

czscscm

kzs kzs iczs

F

Fs

ca,d

b

Fs

1

s

0c0

0cs

kzks

Mode I: 0cskskzcm 0

(b)

F

Fs

a

b b

ds

1 Fskz c

1

ks

0

s

0c

Mode II: 0cs and c skskzcm 0

(c)

0c

0c

s

0c

s

0

F

Fs dbb

a

cc

Fs

b 1

1

kz

ks

Mode II|: cskskzs/ccm 2esks/

(d)

Fig. 9. Load displacement diagrams (in displacement-controlled vibration of a pile segment) for conceptual

illustration of the effect of slippage on pile response: (a) elasto-plastic model of soil reaction against a vibratingpile slice with interface slippage and its equivalent linear approximation; (b), (c) and (d) illustration of the threepossible modes with respect to sliding, for the development of the equivalent linear model (all F `curves' areonly shown to be linear for the sake of clarity of the effect of slippage)



12/21

the response of the pile segment. Thus,again, the equivalent elasto-plastic stiff-ness ks is equal to the equivalent linearstiffness kz.

Mode III. The cyclic displacement amplitude cexceeds the yield displacement s (Fig.9(d)). In this case, even a reduction ofthe static load to zero during the rst-time loading would not be enough toavoid slippage. Unlike modes I and II,slippage now occurs during all subse-quent cycles of loading and unloading,whenever the external load reaches theyield limit (branches b 0b9 and cc9 of theloop). As a result, an equivalent elasto-plastic stiffness ks can be dened as theslope of the diagonal b9c9 of the loop inFig. 9(d). Apparently, ks is smaller thanthe equivalent linear stiffness kz.

On the basis of the geometry of the loaddisplacement loop in Fig. 9(d), the relative effectof slippage on stiffness may be approximated as

ksakz % sac (20)or, in terms of shear stresses at the pilesoil inter-face,

ksakz % fsac0 (21)where fs is the skin friction and c0 Fca2Rdenotes the cyclic shear stress amplitude thatwould have developed at the pilesoil interfacehad slippage not occurred.

Effect of slippage on the increase of hystereticdamping. The above `corrections' for sliding (equa-

tion (21)) imply a reduction in both the spring anddashpot moduli (ks ks ics). The decrease ofdashpot modulus estimated here stems from thereduction in radiation damping, since no additionalwaves are emitted from the pile during slippage. Onthe other hand, slippage dissipates energy. In theconceptual sketch of Fig. 9(d), hysteretic loss ofenergy takes place during cyclic loading. In fact,the area enclosed by the load displacement loopincreases with the cyclic displacement incrementc. An equivalent additional damping ratio (es)can be obtained from the ratio of the area of acomplete loop, Eh, to the equivalent elastic en-ergy, Ees 12Fsc, as follows:

es

Eh4Ees

2

(1 sac) (22)The additional hysteretic dashpot modulus thatmust be added to cs obtained from equation (21) isthen approximately given by

cm % 2es ksa (23)

This is in addition to the dashpot modulus arisingfrom soil non-linearity, which is hidden in cz (andcs). Evidently, when c s this additional equiva-lent hysteretic damping is zero, while ks kz, inaccordance with the second mode.

Thus the resulting `total' dashpot modulus czseither remains equal to cz (for modes I and II) oris given (for mode III) as the sum of the moduli ofthe two (in series) dashpots:

czs cz for modes I and IIczs cs cm czfsac0

2es ksa for mode III (24a)while the `total' spring modulus kzs is given by

kzs kz for modes I and IIkzs ks kzfsac0 for mode III (24b)

The resulting variations of `spring' and `dash-pot' factors are plotted in Fig. 10 as functions ofthe cyclic shear stress ratio c0afs, for a low (nearstatic) and a high (dynamic) value of the dimen-sionless frequency factor as, equal to 01 and 1respectively. The skin friction parameter is takenequal to 1. As seen in this gure, the effect ofslippage on the spring modulus is signicant inboth of the cases as 0X1 and 1. On the otherhand, for the dashpot modulus slippage is impor-tant at high frequencies but practically unimportantat low frequencies. These differences can be attrib-uted to the different relative signicance of theaforementioned two counteracting phenomena con-trolling the dashpot modulus:

(a) the increase in hysteretic dampingin inverseproportion to according to equation (23)

(b) the decrease in radiation dampingnearly

independent of .

Thus, whereas at low frequencies phenomenon(a) is still signicant and its effect may oversha-dow phenomenon (b), at high frequencies phenom-enon (a) becomes insignicant and phenomenon(b) dominates. For a simple proof of this explana-tion, the dashed lines in Fig. 10 represent thedashpot modulus when the additional hystereticdamping (cm) is ignored. Indeed, at high frequen-cies (where hysteretic damping is insignicant) thishas practically no effect on the dashpot modulus,whereas at low frequencies the effect is substantial,increasing with the amplitude of c0afs.

DYNAMIC STIFFNESS OF THE WHOLE PILE

The dynamic spring and dashpot moduli ob-tained for a pile slice in the preceding paragraphsare used in the analysis of a vertically vibratingsingle pile, in the Winkler-type model sketched in



13/21

Fig. 11. For harmonic steady-state oscillation of

the pile,(z, t) c(z) ei t (25)

dynamic equilibrium of a pile element yields

EpApd2c(z)

dz2 (kz icz m2)c(z) 0

(26)

where Ep and Ap are the Young's modulus ofelasticity and cross-sectional area of the pile, andm is the mass of the pile per unit length. Thegeneral solution is

c A1 eDzcos (a2) eiDzsin (a2)

A2 eDzcos (a2) eiDzsin (a2) (27a)where

D (kz m2)2 (cz)2

(EpAp)2

4 51a4

arctan

cz

kz m2

(27b)

(see e.g. Makris & Gazetas, 1993).Equation (27) is an implicit relation, since c is

given as a function of kz and cz, which in turndepend on c(z). Iterations are therefore needed toderive the solution. It is instructive in this respectto study the following two cases separately.

Flexible pilesFor the general case of a exible pile embedded

in layered soil, where shear stresses vary along thepile, an iterative method is followed to account forthe effect of soil non-linearity. For each pile slice,the integration constants A1 and A2 (equation (27))

are computed by enforcing the continuity of stres-ses and displacements along the pile axis. More-over, at the top of the pile

(0, t) c(z 0) ei t (28)while at the pile tip

0 1 2

0

1

2

3

k/G

s

as 0.1

Noslippage:kz

Withslippage:kzs

0 1 2

0

1

2

3

k/G

s

as 1

0 1 2

0

1

2

c/2RVs

c0/fs

Noslippage:cz

Withslippage:

czsc

scmWithslippage

butignoringadditional

hystereticdamping:

czsc

s

2

1

0

0 1 2

c0/fs

c/2RV

s

Fig. 10. Effect of slippage on dynamic impedance: the spring modulus k is signicantly affected for all frequencyvalues, while the dashpot modulus c is reduced only at high frequencies, where radiation damping dominates; forlow frequencies, the presence of the additional hysteretic damping partly counterbalances the reduction ofradiation damping due to slippage



14/21

Pb EpAp d(z, t)dz

zL

(29)

where Pb is the developing harmonic force at thetip (z L) of the pile. Pb is related to the resultingpile tip displacement c(z L) through the dy-namic stiffness Sb of the pile base. Accepting thearguments of Randolph & Wroth (1978) and Scott(1981), it is assumed that Sb is approximatelyequal to the dynamic stiffness of a circular footingon the (underlying) homogeneous half-space:

Sb Pbac(z L) % 4GbR1 b iR

2rb(VLa)b

(30)

in which Gb, (VLa)b, b and rb are the shearmodulus, `Lysmer's analogue' wave velocity, Pois-son's ratio and the mass density of the soil belowthe base of the pile. VLa is related to the S-wave

velocity (Gazetas & Dobry, 1984a,b):

VLa 3X4

(1 ) Vs (31)

In the rst iteration, the shear stresses along thepile shaft are computed using the linear spring and

dashpot expressions (Makris & Gazetas, 1993;Makris & Makris, 1991):

kz,linear 0X60Es(1 12p

as)

cz,linear 1X20a1a4s drs Vs 2kza (32)Then equation (27) gives the complex-valued dis-placement amplitude c(z). The shear stress ampli-tude c0(z) is obtained from

c0(z) 12R

jkz,linear(z)c(z)j (33)where the vertical bars indicate the absolute valueof the complex number. The parameter (z)is then calculated (equation (9)) and a new set of`non-linear' springs is obtained, using the proposedmethod (slippage is taken into account by replacingkz with kzs, as described previously). The processis repeated until a reasonable convergence of shearstresses is achieved.

Rigid pileFor the special case of a rigid cylindrical pile

and uniform soil properties with depth, a forceequilibrium method can be followed. Since the pileis rigid the displacement amplitude is constantalong the pile, that is,

c(z 0) c(z L) c(z) c (34)The applied force at the head of the pile is equalto the sum of the total force Ps at the pile shaft,the force Pb at the pile tip and the pile inertialforce Pin. This force equilibrium can be written as:

Pc Ps Pb Pin (35)where the time factor ei t, common to all force

components, has been omitted, and

Ps L

0

(kz icz)c dz (kz icz)cL (36)

Pb Sbc (37)and

Pin mL2c (38)m being the mass per unit length of the pile.

The dynamic stiffness of the pile is nally givenby the relationship

Kv

Pc e

i t

c ei t

Sb

(kz

icz

m2)L (39)

If the pile is subjected to a known c, equation(39) can be used to obtain directly Pc orKv, oncekz kz(c) and cz cz(c) have been determined.Iterations, however, will be needed in a force-controlled excitation.

cz

kz

cz

kz

dz

z, c

Pceit

mc

dz

(kz

ccz

.c

)dz

Fig. 11. Winkler model for a vertically vibrating pile(from Makris & Gazetas, 1993)



15/21

COMPARISONS AND PARAMETRIC RESULTS

Non-linear static response of a pile in Mexico Cityclay

The method developed here is compared withexperimental results from a quasi-static test con-ducted in Mexico City (Trochanis et al., 1991a,b).In this test a 03 m wide, 15 m long, square con-crete pile was axially loaded in a clay with un-drained shear strength Su % 40 kPa, Poisson's ratio 045 and shear modulus Gs % 6800 kPa, andwith a plasticity index in the region of 200 (typicalof the Mexico City clay). Fig. 12 plots the forcedisplacement relationships from the eld test, theproposed method and a non-linear nite elementanalysis (Zha, 1995). The agreement of the presentmethod with the in situ measurements and themore rigorous analysis is satisfactory.

Distribution of stressaforce with depthcomparison with other solutions

The validity of the proposed method is furtherchecked in Figs 1316 against results from Poulos& Davis (1980). Specically, Fig. 13 shows that,under elastic conditions, the frequency of the ap-plied load has practically no effect on the distribu-tion with depth of the shear stress on the pileshaft, for an extreme range of EpaEs ratios. Theagreement with the Poulos & Davis curves is clearin this gure. On the other hand, frequency doesaffect the shear stress distribution and generallyplays a more signicant role as the applied loadincreases and non-linear conditions are established;a demonstration is given in Fig. 14 for the case ofa medium-intensity load PcaPu 0X25. In Fig. 15the distribution with depth of the axial forceN N(z) is shown for different base conditions.

Measured

Finite element analysis

Ip 150

Ip 200Present method

600

400

200

0 0.020 0.04

Mexico City clay

Gs6800kPa, Su40kPa, Ep2 107 kPa

L15m, d0.34m

w: m

P:kN

Fig. 12. Verication of the proposed method with realdata (full-scale experimental results in Mexico Cityclay: Trochanis, 1991a,b) and with a more rigorousanalysis (nite element analysis results: Zha, 1995)

0 1 2 3 4

0

0.2

0.4

0.6

0.8

1.0

z/L

c0dL/Pc

Pc/Pu 0 (linear)

Ip 30, L/d= 25

Ep/Es 50

5000

as 0.1

as 1

Static (Poulos)

Presentmethod

Fig. 13. Distribution of shear stress amplitude alongthe pile shaft, for two frequencies and two pilesoilstiffness ratios, under elastic conditions

0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4

L/d25, Ep/Es50

Pc

/Pu

0.25, Ip

30

Static

as0.1

as0.5

as1

z/L

c0dL/Pc

Fig. 14. Distribution of shear stress amplitude alongthe pile shaft, for static and dynamic conditions, withmoderately non-linear soil response (PcaPu 0X25)



16/21

The force transmitted to the lower part of the pileincreases with Eb, the soil modulus of elasticity atthe tip level. The good agreement with the curvesgiven by Poulos & Davis (1980) is again evident.On the other hand, Fig. 16 shows that the distribu-

tion of shear stresses along the pile shaft becomesincreasingly uniform with increasing intensity ofthe applied load.

Effect of non-linearity on pile-head stiffness anddamping

To illustrate the effect of the level of the appliedload and the `plasticity' index of soil on the stiff-ness and damping of a pile, we consider a concretepile 064 m in diameter and 16 m long, embeddedin a homogeneous clay with Es 20 MPa. Fig. 17shows, in dimensionless form, the load displace-ment curves for as 0X1 and 05. Both real andimaginary parts of the displacement are plotted to

illustrate the relative contribution of each compo-nent in the resulting displacement. For as 0X1(near-static case) the real part (in-phase compo-nent) of the response dominates, while the imagin-ary part (out-of-phase component) becomes sig-nicant only at high values of frequency (as 0X5)X Notice that the amplitude of the complex

0

0.2

0.4

0.6

0.8

1.0

0.2 0.4 0.6 0.8 1.00

Pc/Pu 0 (linear)

L/d

25, Ep/Es

1000

Eb/Es 0

1

as0.10.5 (present method)

Static (Poulos)

z/L

N(z)/Pc

Fig. 15. Distribution of pile axial force with depth, forthree different end-bearing conditions, ranging fromxed end (EbaEs I) to free end (EbaEs 0)

0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4

L/d 25, Ep/Ez 50

as 0.5, Ip 30

Pc/Pu 0 (linear)

Pc/Pu 0.5

Pc/Pu 1

z/L

codL/Pc

Fig. 16. Distribution of shear stress amplitude alongthe pile shaft, for three different levels of loading, withsoil behaviour ranging from linear to highly non-linear

1.0

0.5

0

1.0

0.5

0

0.003 0.002 0.001 0 0.001 0.002 0.003

0.003 0.002 0.001 0 0.001 0.002 0.003

w/d

w/d

Imaginary part

Imaginary partReal part

Real part

Amplitude

Amplitude

as 0.1

as 0.5

P/P

u

P/P

u

Ip30

L/d25, Ep/Es1000

Fig. 17. Loaddisplacement curves (real part, imagin-ary part and amplitude) at the head of a pile withLad 25, EpaEs 1000 and Ip 30, for the fre-quency factors as 0X1 and as 0X5



17/21

impedance, given by the slope of the `amplitude'curve, increases with frequency. This is the resultof a signicant increase in radiation damping.

Figures 18 and 19 illustrate the effect of themagnitude of the applied load and soil `plasticity'index on the dynamic impedance at the pile head(given in the form Kv Kv iCv). Specically,Fig. 18(a) plots the variation of pile stiffness as afunction of as for different values of the level ofthe applied load; the latter is normalized by Pu,the ultimate static axial load of the pile. Thesubstantial inuence of non-linearity on the pilestiffness is obvious for the whole range of frequen-cies. However, this effect becomes more signicantat high values of the frequency factor asX On theother hand, the damping ratio plotted in Fig. 18(b)is affected by the level of non-linearity to a gen-erally lesser degree. For large frequencies the valueof Cv tends to

Cv 2RLrVs(R) (40)which corresponds to the value of Cv for a pilesurrounded by soil with a constant value of S-wavevelocity equal to the Vs(r R) value. In otherwords, radiation damping decreases in proportionto the decrease in the effective S-wave velocitynext to the pile (at r R). This trend is qualita-tively similar to the trend observed earlier forthe dynamic impedance of the pile slice (Figs 6and 7).

In Fig. 19 the effect of the soil `plasticity' indexis examined for two cases, Ip 30 and Ip 100.It is concluded that with increasing Ip, the soilremains increasingly elastic and hence homoge-neous for large values of the applied load. Itshould be noted, however, that for the same shearstrength Su, the soil modulus Es (or the S-wavevelocity at small strains Vs) will be larger as Ipdecreases, as, for instance, described by equation

(10). As a result, the elastic (linear) stiffness of thesoil, and hence of the pile, will be reduced forincreasing Ip.

CONCLUSIONS

An axially oscillating pile induces shear strainsin the surrounding soil, the amplitude of whichattenuates radially away from the pile. In thevicinity of the pile such strains can be largeenough to cause non-linear cyclic response of thesoil. This paper has developed an equivalent linearmethod to approximate such a non-linear response.To this end, linear elastic theory has providedamplitudes of shear stresses and shear strains as

functions of the radial distance r from the pile.Used in conjunction with published experimentaldata (in the form of secant shear modulus anddamping ratio functions of cyclic strain amplitudeand soil `plasticity' index), the radial distributionof strains was translated into an equivalent shearmodulus G, increasing monotonically and continu-

Kv/

Es

L

1

0

1

3

2

1

00 0.5 1.0

0 0.5 1.0

asR/Vs

asR/Vs

Pc/Pu0 (linear)

Pc/Pu0.5

Pc/Pu1

Cv

/2RLV

s

Fig. 18. Effect of level of non-linearity on the dynamicimpedance (stiffness and damping) at the head of apile with Lad 25, EpaEs 1000 and Ip 30

Linear

PPu1/2250kN, Ip100

PPu2Pu1/2250kN, Ip30

PPu1500kN, Ip100

4

3

2

1

0

0 0.5 1.0

0 0.5 1.0

1

0

1

Kv

/Es

L

Cv

/2RLV

s

asR/Vs

asR/Vs

Fig. 19. Dynamic impedance (stiffness and damping)at the head of a pile, with Lad 25 andEpaEs 1000 for different types of soil and level ofnon-linearity



18/21

ously with r, from a value of G0 at the pilesoilinterface to the maximum (zero-strain) modulus Gsin the free eld, that is, outside the zone of inu-ence of the pile. Thus, a linear but radially inho-mogeneous medium replaced the actual non-linearbut radially homogeneous soil, allowing analyticalderivation of the soil reaction (both the in-phase`spring' and the out-of-phase `dashpot' compo-nents) against the pile periphery at the particulardepth. A simple experimentally motivated approx-imation was developed to account for slippage atthe pile soil interface. The response of the wholepile could, in general, only be obtained iteratively.

Despite several simplifying approximations, themethod seems capable of capturing the key aspectsof the non-linear dynamic response. Parametricresults have shown that increasing the amplitude ofthe applied load reduces the stiffness and radiationdamping of the system while increasing the hys-teretic damping. The signicance of non-linearityincreases with increasing frequency of pile oscilla-tion, while it decreases with increasing soil `plasti-city' index. On the other hand, frequency has arather minor effect on the distribution of the axialforce along the pile.

It is nally noted that the developed methodcan utilize experimental data other than those ofVucetic & Dobry (1991) (shear modulus anddamping against strain curves) that have been usedin this article.

ACKNOWLEDGEMENT

The computer code used for the evaluation ofthe stiffness of the pile was provided by GeorgeMylonakis, a doctoral student at the State Univer-sity of New York, Buffalo. The authors modiedthe code so as to take into account the new

impedances found in this study. We are grateful forhis help.

APPENDIX 1. SHEAR STRESSES INDUCED IN

RADIALLY NON-HOMOGENEOUS ELASTIC MEDIA

It is shown below that the radial distribution of (cyclic)shear stress given in equation (2) is quite insensitive to theexact radial variation of shear modulus G G(r). To thisend, the shear modulus is taken to vary as follows:

G(r) G(r0)f1 i2(R)g rR

m(41)

For harmonic excitation, the solution of the differentialequation of motion (equation (11)) yields

w Bw rR

ma2H(2)

kk0

r

R

1ak2 3(42)

where Bw is a constant of integration that can bedetermined from the boundary conditions, H(2)

kis the

Hankel function of the second kind of order k,

k 22 m

0 R rG(R)[1 i2(R)]

and r is the mass density.The shear stress is then given by

c(r)

G(r)

dw

dr

(43)

and the following cases can be investigated, starting withthe radially homogeneous case and moving to increasinglyinhomogeneous proles:

Radially homogeneous soil ( m 0, 0) with shearmodulus equal to the shear modulus at the pilesoilinterface

The radial distribution of shear strength is given byequation (1) and approximately by equation (2).

Soil with shear modulus increasing as the 23

power of r(m

23,

0)

The resulting differential equation was solved byGazetas (1982) for the (mathematically similar) case ofa vertically inhomogeneous earth dam, and by Gazetas &Dobry (1984a) for a strip footing. The solution is

w Bwr

4

30

r

R

2a3

3 sin3

20

r

R

2a32 3 icos 3

20

r

R

2a32 34 5

(44)

from which

c(r)

c0 r

R 1a3

3 sa20

4

9

r

R

4a3

a20

4

9

H

fffd

I

ggge (45)

Soil with shear modulus proportional to r ( m 1, 0)The ratio c(r)ac0 is given by

c(r)

co r

R

1a2s

[J1(t) a0 rR

1a2J0(t)]

2 [Y1(t) a0 rR

1a2Y0(t)]

2

[J1(t) a0J0(t)]2 [Y1(t) a0 Y0(t)]2

Hfd

Ige

(46)

in which

t 2a0 rR

1a2and t 2a0

Fig. 20 compares the radial variations of shear stress forthe above three cases, m 0, m 2

3and m 1; also

plotted is the approximate (asymptotic) equation (2).The agreement between the two sets of results conrms

the insensitivity of the radial variation of shear stress to



19/21

the radial distribution of shear modulus, especially for themost interesting range of low a0 values (a0 , 0X5).

At high frequencies (a0 . 0X5) the radial distribution ofshear modulus does affect (r). This could be attributed tothe fact that for large values of the applied frequency, theshort-wavelength waves emitted from the pile `see' almostonly the area next to the pile; as a result, the gradient ofG(r) in this area (which varies signicantly for the threeG(r) functions studied) greatly affects the distribution of(r). For this case, a more accurate approach would be toobtain c(r) iteratively. Nevertheless, the proposed simpli-

ed expression is very close to the most realistic of thethree studied distributions, namely the one with m 2

3.

Therefore in view of the several other approximations ofthe present method, such iterations are not considerednecessary.

NOTATION

a0 RaVs0Ap pile cross-sectional area

ar raVs(r)as RaVs

cm es2ksa additional hysteretic dashpotmodulus due to slippage

Cv pile-head dashpot modulus in axial loading (imaginary part of Kva)

cz `dashpot' modulus

d pile diameterEp pile modulus of elasticityF0 2R0(R) static load applied on pile slice

(per unit length)Fc 2Rc0 amplitude of cyclic load applied on

pile slice (per unit length)fs frictional capacity of soilpile interface

Fs 2Rfs yield load of the pilesoil interface(per unit length)

G G(r) shear modulus of soil at radial distancer

G0 shear modulus of soil at r R (i.e. at the pilesoil interface)

Gs shear modulus of soil in the far eld (i.e. atvery low strain levels)

Ip plasticity index of soili

p(1)

kimag cz

imaginary part of kz

kreal kz real part of kzks ks ics elasto-plastic complex dynamic

impedance accounting for the reduction ofspring modulus and radiation damping due toslippage

Kv pile-head dynamic stiffness modulus in axialloading ( real part of Kv)

Kv Kv iCv complex pile-head dynamicimpedance in axial loading

kz `spring' moduluskz kz icz complex dynamic impedance of

the pile slicekzs kzs iczs equivalent elasto-plastic complex

dynamic impedanceL pile lengthm mass of pile per unit length

Pb force at the pile tip

Pc amplitude of the sinusoidal applied axialdynamic load

Pin inertial force of the vibrating pilePs total force at the pile shaft

PU ultimate axial static loadR pile radius

r radial distance from the pile axis

1.0

0.5

00 10 20 30

as1

as0.1

r/R

r/R0 10

1

5

G(r)/G(R)

Pileaxis

m1

m2/3

m0Approximate solution (equation (2))

Exact solution for m0

Exact solution for m1Exact solution for m2/3

c

/c0

Fig. 20. Variation of shear stress amplitude with radial distance r, for different

radial distributions of shear modulusG

G

(r

), ranging from radiallyhomogeneous (m 0) to strongly inhomogeneous with modulus proportionalto r (m 1)



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Sb dynamic impedance (stiffness and damping) atthe pile base

Su undrained shear strength of the soilVLa `Lysmer's analogue' wave velocity

Vs(r) shear wave velocity of soil at radial distance rVs shear wave velocity of soil in the far eld (i.e.

at very low strain levels)Vs0 Vs(R) shear wave velocity of soil at r R

(i.e. at the pilesoil interface)w w(r) amplitude of axial displacement of soilat radial distance r

`skin friction' parameter (fs Su)c c(r) amplitude of cyclic shear strain

corresponding to c(r)(z, t) cyclic displacement of pile segment at depth z

0 initial (static) displacement of pile segmentc c(z) amplitude of the cyclic displacement of

pile segment at depth z. If no slippage occursc w(R)

s Fsakz displacement of pile segment requiredto initiate slippage

loading intensity factor (dened in equation(9))

hysteretic damping factor of soiles additional hysteretic damping ratio due to

slippagerb mass density of soil below the base of the pile9v0 mean effective vertical stress0 0(r) static shear stress on soil element at a

radial distance rc c(r) amplitude of cyclic shear stress on soil

element at a radial distance rc0 c(R) amplitude of the imposed cyclic shear

stress at the pilesoil interface (if c0 . fs thenc0 is the amplitude of shear stress that wouldhave developed if no slippage occurred)

frequency of the applied dynamic load

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1-Michelides-Gazetas (1997)-Approximate Non-linear Dynamic Axial of Piles

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