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7/30/2019 Single Pile Dynamic Stiffness

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19.0 INTRODUCTION

We will commence our consideration of the dynamic behaviour of pilefoundations by reviewing information from field tests on single piles. This will befollowed by consideration of the dynamic stiffness of single piles in this chapter,after which the dynamic response of pile groups will be covered in Chapter 20; asthere are many similarities between the material presented in this chapter and thatin the next there are some forward references to Chapter 20.

19.1 Dynamic tests on prototype scale piles and pile groups.

19.1.1 Blaney and O'NeillBlaney and O'Neill (1986a and b) describe a field investigation of the forced

vibration response of a steel tube pile driven into a deposit of clay. The pile hada 273 mm in outer diameter and 254.5 mm internal diameter and penetrated 13.4m into the clay. The details of the test set up and brief soil profile information isgiven in Figure 19.1. By means of a vibrator attached to the extension of the piletube above the ground surface it was possible to apply a sinusoidal excitation at

various frequencies to the pile. The response of the pile at a range of fixed force

amplitudes and a sweep of frequencies is shown in Figure 19.2. Of significance


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Design of Earthquake Resistant Foundations

330

is the normalised dynamic lateral displacement profile with depth being similar tothe static profile.

Note that the natural frequency decreases slightly after the 600 lb sweep, a

possible explanation is the formation of a gap adjacent to the pile shaft near theground surface.

Figure 19.1 Details of the pile-mass system and soil profile investigated byBlaney and ONeil (after Blaney and ONeil (1985)) (1 foot (1) 305 mm).


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Chapter 19: Single pile dynamic stiffness

331

Figure 19.2 Dynamic and static response of the Blaney and ONeil test pile(after Blaney and ONeil (1985)).

19.1.2 Jennings et al Jennings et al (1985 and 1986) report on dynamic tests on a pair of 450mmdiameter piles driven into saturated silty sands. The spacing between the piles issuch that the interaction between them is assumed to be negligible, hence thiscase history is discussed in this chapter and not the next which deals with pilegroups. Details of the soil profile are reproduced in Figure 19.3 and the details of the piles in Figure 19.4. Self-weight was sufficient to for the pile shells topenetrate to a depth of 5m, subsequent gentle tapping was all that was requiredto get them to the required depth of 6.75m. After cleaning out, reinforcement

was placed in the shells and concrete poured. Dynamic and slow cyclic loads were applied to the piles. Cyclic loads were applied by means of a jack mounted

between the piles 1.35 m above ground level, and the dynamic loads with a shak-


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Figure 19.3 Field data for the Central Laboratories twin pile test (after Jenningset al (1985)).

Figure 19.4 Details of the Central Laboratories test piles (after Jennings et al(1984)).


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Figure 19.5 Response of the Central Laboratories piles: upper: cyclic loaddeformation loops, lower: resonance curve (after Jennings et al (1984)).

ing machine mounted on top of one of the piles. Initial testing involved dynamicshaking of the west pile. This was followed by slow cyclic loading at a rate of about one cycle per hour with the following sequence of maximum loads percycle in each direction: 10kN, 20kN, 40kN (2 cycles), 80kN (2 cycles), 120kN (2cycles), 160kN (2 cycles), 200kN (2 cycles).


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Figure 19.5a has the force deflection results for the slow cyclic loading andFigure 19.5b has the resonance curve for the dynamic test on the single pile. It isclear that, as with the Blaney and O'Neill pile, there is a distinct natural frequency

when the pile is loaded dynamically at low levels of excitation. It was apparent

from ground surface observations during the loading that high pore waterpressures were generated adjacent to the pile shafts. When piles are embedded insaturated sands gaping cannot occur along the pile shaft but there is,nevertheless, a dynamic degradation in the pile performance because of areduction in stiffness of the silty sand as a consequence of the build-up in pore

water pressure.

Figure 19.6 Moment profiles during slow cyclic loading of the CentralLaboratories test piles (after Jennings et al (1984)).


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In attempts to match the observed moment profiles, shown in Figure 19.6, a Winkler model with k increasing uniformly with depth was adopted, i.e. k = n hz. The moment profiles, determined from the strain gauge readings, were the basisfor the inference, using a Winkler spring approach, that towards the bottom of

the piles nh 12 but near the ground surface a value of 6 is more appropriate. This was interpreted as evidence for softening of the upper sand layers due to thedevelopment of positive pore pressures.

19.1.3 Sadon et alField experiments were conducted at a site in Auckland with driven steel tubepiles embedded in Auckland residual clay. Site and soil profile details are givenby Sadon et al (2010) and Sadon (2011). The first batch of tests used aneccentric mass shaking machine to excite the foundations with sinusoidaloscillations at a range of frequencies. Although successful there are limitationsto this approach for the following reasons. First, a given level of excitation

force cannot be obtained until the shaker frequency has been increased fromzero to the frequency required to generate the required force. Second, theresponse of the system is measured under steady state excitation at a fixedfrequency. In this way what is obtained from the use of a shaking machine isnot representative of what happens during earthquake excitation.

An alternative is the use of snap-back testing. This test is simpler than using aneccentric mass shaking machine. It gives the response of the system to oneimpulsive excitation instead of continuous excitation; it is more representativeof what occurs during an earthquake. An added bonus is the static load-deflection curve obtained during pull-back phase of the test. The initial pull-

back can generate a force of comparable magnitude to the maximum force thatcan be produced by the shaking machine used.

Figure 19.7 shows the pile head with the eccentric mass shaker in place set-upfor steady state sinusoidal excitation (top) and for snap-back testing (bottom).

Figure 19.8 gives frequency curves for the piles before and after high levelsinusoidal excitation. These are obtained from the small amount of out of balance in the shaking machines; there is no added mass. Before high levelshaking the natural frequency is largest; after the high level shaking the systemstiffness has deteriorated so the indicated natural frequency is reduced. Since

the piles are embedded in clay the main reason for the reduction in lateralstiffness is likely to be the opening of a gap near the top of the pile shaftbetween the surrounding soil.

Figure 19.9 shows hysteresis loops recorded during high level sinusoidalexcitation. A line defining the lateral stiffness of the pile head obtained using the small strain shear modulus of the soil is plotted. It is seen that hysteresisloops recorded during forced vibration at +- 10 kN indicate that a smaller soilmodulus is operational. Furthermore, when the forcing amplitude is +- 60 kN,then the operational stiffness is about one third to one quarter of that definedby the small strain shear modulus of the soil.


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Figure 19.7 Pile with eccentric mass shaking machine attached andinstrumentation installed. Top: Set-up for sinusoidal excitation; bottom: set-up for snap-back testing (the eccentric mass shaker now provides pile-head mass).

Figures 19.10, 11 and 12 give results of a series of snap-back tests on a givenpile. First, Figure 19.10 gives the load-deformation curves generated during thepull-back phase. Figure 19.11 gives hysteresis loops during the vibrationfollowing the snap release. During pull-back the load applied to the pile ismeasured directly through the load cell shown in the bottom part of Figure19.7. During the free vibration the loads are calculated from the recordedstrains in the strain gauges on the pile shaft above the ground surface.


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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

4 6 8 10 12 14 16Frequency (Hz)

D i s p

l a c e m e n

t A m p

l i t u

d e ( m m

)

Series 1_BeforeSeries 1_After

Series 2_Before

Series 2_After

Figure 19.8 Frequency response curves for low-level excitation generatedwith the natural eccentricity of the eccentric mass shaker.

-80

-60

-40

-20

0

20

40

60

80

-25 -20 -15 -10 -5 0 5 10 15 20 25

Pile Head Displacement (mm)

P i l e H e a

d L o a

d ( k N )

HL

1.0 m

6.5 mz

0 E(z)

E=E s

G s,max = 40 MPa

Experiment:

H L Max - 60 kN

Experiment:H L_Min - 10 kN

Figure 19.9 Hysteresis loops generated steady state sinusoidal excitationwith the eccentric mass shaking machine.


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0

10

20

30

40

50

60

70

0 5 10 15 20 25


P i l e H e a

d L o a

d ( k N )

Experimental Backbone Curve

Davies & Budhu

G s =40 MPa

Davies & BudhuG s =21 MPa

Davies & BudhuG s =19MPa



Figure 19.10 Pull-back load displacement curves

-80

-60

-40

-20

0

20

40

60

80

-25 -20 -15 -10 -5 0 5 10 15 20 25


P i l e H e a

d L o a d

( k N )

Snap Back: 15kNSnap Back: 60kN

G s = 40 MPa

Figure 19.11 Snap-back hysteretic load-lateral displacement loops


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-20

-15

-10

-5

0

5

10

15

20

25

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Time (s)

D i s p

l a c e m e n

t ( m m

)

Snap-Back: 15kN

Snap-Back: 36kN

Snap-Back: 65kN

Figure 19.12 Displacement responses for three snap-back tests

Table 19.1 The natural frequencies and damping ratios for Pile 3

Tests Sequence Natural frequency (Hz) Damping ratio (%)

Free Vibration 1 10.823 2.33

Snap-back 1 15 kN 10.059 3.61

Snap-back 2 30 kN 8.984 4.02

Snap-back 3 45 kN 8.008 4.12

Snap-back 4 60 kN 7.422 6.76

Snap-back 5 45 kN 7.031 4.57

Snap-back 6 30 kN 7.219 6.96

Snap-back 7 15 kN 7.194 11.90

Free Vibration 2 8.338 3.20

Figures 19.10 and 19.11 suggest a method for predicting the operation stiffnessduring earthquake loading. Figure 19.11 has data for release after a pull-back to maximum force a little greater than 60 kN. From this data a secant modulusfor the pile head stiffness at that displacement can be evaluated. Figure 19.12shows the hysteresis loops following the release. It is apparent that theoperational modulus for the first few cycles after release is quite similar to thesecant modulus defined by the load and displacement at the end of the pull-back. This means that the Budhu and Davies relationship for predicting nonlinear pile head response, presented in Chapter 16, can be used to predictthe dynamic pile head stiffness, provided one can predict the maximum pilehead load during the earthquake.


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Table 19.1 gives data on the pile head damping. It is apparent that the amountof damping available is modest, probably because of the effect of the gapping that occurs between the pile shaft and the adjacent soil.

19.1.4 General observations from these studies A number of general observations can be made from the above case studies.First, at loads small in relation to the pile capacity laterally excited piles have adefinite resonance curve. This is apparent for the Blaney and O'Neill test andthat reported by Sadon et al for piles in clay, and the Jennings et al test on pilesin saturated silty sands. Second, there is a trend for the dynamic lateral stiffnessto be less than the value expected from the small strain stiffness of the soil in

which the piles are embedded. For piles in clay this is explained by the formationof gaps along the pile shaft, and for piles in saturated sands it is a consequence of a decrease in stiffness due to development of pore water pressure.

19.2 Dynamic response of single piles

Gazetas and his co-workers have conducted a number of studies using numericalanalysis of the dynamic response of piles. Dynamic pile response is discussedbelow under two headings: piles in soil profiles with a smooth variation of stiffness with depth and piles in layered soil profiles.

If we consider a pile embedded in a soil profile and ask what will happen underearthquake loading two questions arise. First, what effect does the presence of the pile have on the propagation of the earthquake motion up through the soilprofile? Does the pile simply follow the soil deformations or is the pile headmotion different from that of the adjacent soil? A very stiff pile may not be ableto flex and follow the soil deformations, on the other hand a flexible pile mightreadily follow the soil. Second, how does the dynamic stiffness relate to the staticstiffness of the pile. The first question is of concern for earthquake loading only.In the case of wave and wind loading, two other important dynamic loading mechanisms for pile foundations, the excitation comes from the structureattached to the top of the pile rather than from beneath the pile as in theearthquake case. Interaction between the pile and the soil is thus divided into twosteps: kinematic and inertial interaction. Kinematic interaction refers to the firstof the steps outlined above and inertial interaction refers to the second step. Theconcepts are shown diagrammatically in Figure 19.13. In estimating the kinematicinteraction we are seeking the modification to the free field ground surfaceearthquake motion required to get the motion appropriate to the pile head.Having obtained the pile head motion the inertial response of the structureattached to the pile head is estimated.

An important parameter for the dynamic response of a pile is the natural period,for vertical propagation of shear waves, of the soil layer in which the pile isembedded. Three modulus distributions are considered by Gazetas, these are thesame as those shown in Figure 14.4. Gazetas gives the following formulae for thefundamental frequency (the inverse of the natural period) of the three modulusdistributions:


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Figure 19.13 Dynamic soil-pile interaction: (a) geometry, (b) kinematic andinertial interaction (after Gazetas (1984)).

sH

con

sHsqr

sHlin

0.25=f H

0.22=f

H0.19

=f H

V

V

V

19.1

where: V sH is the shear wave velocity at the bottom of the soil layer,and H is the soil layer thickness.

The abbreviations con, sqr, and lin represent the Young's modulus distributionfor the three soil profiles. Equation 19.1 gives the fundamental frequency, thesecond mode frequency for each of the profiles is obtained from:

INERTIAL INTERACTIONKINEMATIC INTERACTION

uk(t)

masslesssuperstructure [M][u k(t)]

ug(t)

D

L

uo(t)

ug(t)

up(t)uo(t)

Ep

E s , s

M

[M][u k(t)

h =K HM/K HH

K MM

K HHRepresentation of pile head

stiffness for inertial interaction

Soil layer

a

b


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2

1 con

2

1 sqr

2

1 lin

f = 3.00f

f = 2.66f

f = 2.33f

19.2

where: f 1 is the fundamental frequency of the soil layer.

19.2.1 Inertial interaction

In this section we deal with dynamic loading applied to a pile head. The

techniques are useful not only for earthquake loading but also for wind, wave,and any other type of cyclic loading of pile foundations.

In sections of chapters 14 to 15 various equations for evaluating the staticdisplacements and stiffness of pile foundations were presented. In chapter 14flexibility coefficients are given so that displacements can be evaluated directly.

The pile head static stiffness matrix can be obtained by the inversion of thematrix of flexibility coefficients as given in equation 14.5 (Gazetas (1991) gives aset of formulae for estimating pile head stiffness components directly.) Whenconsidering dynamic behaviour we have to ask if the dynamic stiffness isdifferent from the static stiffness and how much damping is present. As both

stiffness and damping contribute to resisting dynamic loads it is convenient tocombine the stiffness and damping terms into impedance (the dynamicequivalent of the static stiffness) defined by:

P(t) = ( ) u(t) with :

( ) = K( ) + i C( )

I

I

19.3

where: I is the impedance,

K is the dynamic pile stiffness,C is the damping parameter,

is the frequency in radians/second,i is -1,P(t) is the dynamic excitation force, andu(t) is the dynamic displacement response.

Recall that the notion of impedance was introduced in section 7 of Chapter 11.Note that the impedance, dynamic stiffness and damping coefficients, arefunctions of frequency and that the impedance is a complex quantity. Also notethere is an impedance corresponding to each component of the pile head staticstiffness matrix.

The damping parameter, C, in the above equation has units of MT -1, the morefamiliar dimensionless value can be calculated from:


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f CC=K 2K

19.4

where: K is the component of the stiffness matrix associated with damping component, the notation refers to the various damping components HH, HM and MM.

and f is the frequency in cycles per second.

Expressions for are given below.

An alternative expression for the impedance in terms of the coefficient is:

= ( ) + i ( )K k I 19.5 where: K is a component of the static pile head stiffness matrix,

k is a coefficient relating the real part of the impedance I to K .

Equations 19.3 and 19.5 provide alternative expressions for the impedance, thereare others also. The units associated with the damping term need to be kept inmind. In equation 19.3 the units with the damping term, i C, are MT-2 which arethose for stiffness as is required for dimensional homogeneity of the equation, sothe units of C are MT -1 as was mentioned earlier. In equation 19.5 the coefficient

is dimensionless but a function of frequency. Further commonly used formsof the impedance are:

oK ia CI 19.5a

K iCI 19.5b

In a given reference the units of the damping term need to be checked. ForExample the ordinates of the damping parts of Figures 20.1 to 20.6 havedamping terms for pile group foundations with stiffness units, that is thesediagrams refer to C .

The impedance terms are functions of the frequency of the dynamic loading.

This causes no problem if the dynamic loading occurs at a fixed frequency, suchas the vibrations of a machine foundation, but is not so convenient forearthquake loading which contains many frequencies. A response spectrumapproach to the estimation of pile group foundation displacements is illustratedin examples 20.3 and 20.4.

19.2.2 Lateral pile head impedenceGazetas and his co-workers performed dynamic finite element analyses toinvestigate the dynamic active pile length and impedance.

Dimensional analysis reveals that the following parameters are of significance:


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po

s

= F K, , , , ,aK 19.6

where: p/ s is the ratio of density of the pile material to the density of thesoil,ao is the ratio D/V sH ( is the excitation frequency in

radians/second), is the hysteretic damping value for the soil

is the length to diameter ratio of the pile.

The finite element model used a pile length of 40D, considerably greater than thedynamic effective length. For the dynamic finite element analyses was set to0.40, to 5%, and p/ s to 1.60. The results are not sensitive to and arethought to be valid for the range 0.30 0.48. Similarly the results are notsensitive to the density ratio, they apply to the range 1.40 to 2.50.

Gazetas (1991) gives dynamic active pile lengths, estimated from finite elementanalyses, for the three modulus distributions:

0.25ad con

0.22ad sqr

0.20ad lin

K = 2 DLK = 2 DLK = 2 DL

19.7

where: K is the ratio of the pile Young's modulus to that of the soil at depthD,

and Lad is the dynamic effective pile shaft length.

Reference to section 14.2.2 reveals that the dynamic active length is greater thanthe static active length.

Values of k and were evaluated for the three soil profiles for a range of modulus ratios and excitation frequencies. These results are presented in Figures19.14 to 19.16, the left hand part of the figures giving k for the real part of theimpedance and the right hand column giving , the imaginary part. The figuresshow that k is usually close to unity, thus we conclude that the dynamic

stiffness of a laterally loaded pile is very similar to the static stiffness.

The damping values in Figure 19.14 to 19.16 are for radiation damping, that isdamping that originates because energy is carried away from the pile by elastic

wave propagation. Radiation damping, as can be seen from the figures, is afunction of frequency. Dobry and Gazetas (1985) explain that the radiationdamping is zero for frequencies less that the natural frequency of the layer.

In addition, material damping (the mechanism of energy loss for which ishysteresis and consequently independent of frequency) occurs adjacent to thepile. At low frequencies material damping is the only mechanism available for

dissipation of seismic energy.


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Figure 19.14 Dynamic stiffness coefficients and damping ratios for flexiblepiles in a constant modulus soil profile (after Gazetas (1984)).


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Figure 19.15 Dynamic stiffness coefficients and damping ratios for flexiblepiles in a soil profile having a square root variation of soil modulus with depth(after Gazetas (1984)).


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Figure 19.16 Dynamic stiffness coefficients and damping ratios for flexible pilesin a soil profile having a linear variation of soil modulus with depth (afterGazetas (1984)).

Gazetas (1991) gives the following formulae for the pile head damping, including both radiation and material damping, for a soil profile having a constant Young'smodulus:


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1

0.17sHH

0.18sHM

0.20sMM

1

HH

HM

MM

when f > f = 0.80 + 1.10 f D / V K = 0.80 + 0.85 f D / V K = 0.35 + 0.35 f D / V K

when f f

= 0.80= 0.50= 0.25

19.8

where: is the material damping parameter,f is the frequency of the dynamic loading in cycles per second,

and K is defined in equation 14.19.

Gazetas (1991) gives the following formulae for the pile head damping values ina soil profile with a square root distribution of modulus with depth:

1

0.08sHH

0.05sHM

0.10sMM

1

HH

HM

MM

when f > f = 0.70 + 1.20 f D / V K = 0.60 + 0.70 f D / V K = 0.22 + 0.35 f D / V K

when f f = 0.70= 0.35= 0.22

19.9

where: K is defined in equation 14.33.

Gazetas (1991) gives the following formulae for the pile head damping values ina soil profile with a linear distribution of modulus with depth:

1

sHH

sHM

sMM

1

HH

HM

MM

when f >f = 0.60 + 1.80 f D / V

= 0.30 + 1.00 f D / V = 0.20 + 0.40 f D / V

when f f

= 0.60= 0.30= 0.20

19.10


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Examination of Figures 19.14 to 16 leads to the following comments:

(a) Apart from k HH at small values of Ep/E s the pile stiffnesses are notparticularly sensitive to frequency, so the dynamic stiffnesses are not much

different from the static values.(b) At the natural frequency of the soil layer there is a dip in the elastic

stiffness values, at higher modal frequencies no further dip is apparent.

(c) The damping associated with rocking motions of the pile head, MM, issmaller than that for the other two modes of deformation. (Similarbehaviour is familiar for the rocking behaviour of a footing, Chapter 10.)

(d) The general features of the stiffness behaviour do not change for the threemodulus profiles investigated. However there are differences in the

damping behaviour and in particular the linear modulus profile has anopposite trend with E p/E s as f increases to the constant and square rootprofiles.

19.2.3 Vertical pile head impedanceGazetas (1991) gives the following expressions for the coefficient which modifiesthe static vertical stiffness to get the real part of the vertical pile head impedance:

con

o

sqr

o

lin

= 1.0 for < 15k

= 1 + for 50a = 1.0 for < 20k

1= 1 + for 50a

3 k = 1.0

19.11

In equation 19.11 interpolation gives k for a o values in the range between the

limits specified.

Gazetas (1991) reports that the finite element studies show the plot of k versusexcitation frequency has a narrow valley at the frequency:

sHr

V =f 4 H

19.12

where:

sH V is the average shear wave velocity over the depth of the soil layer.

The value of k (the minimum) found for this frequency is 0.8.


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The imaginary part of the vertical pile head impedance, the radiation damping term, for the constant modulus soil profile is:

0 2

2

3

2.

rz s d

r

d-K

= a D L r for f > 1.5C V f

= 0 for f f

where r = 1 - e

o 19.13

As with for soil layer natural frequency equation, the shear wave velocity and soilmodulus, in the above equations, are for the bottom of the soil layer (depth H).

For the square root modulus profile:3

4-0.25

rz o sH d

r

d-2

p sH-1.5 ( / )E E

= D L r for f > 1.5C a V f

= 0 for f f where r = 1 - e

19.14

where: V sH is the shear wave velocity at the bottom of the layer.

For the linear modulus profile:

0 332

3.

rz sH d

r

d

-2p sH-2 ( / )E E

= a D L r for f > 1.5C V f

= 0 for f f

where r = 1 - e

o

19.15

19.2.4 Equivalent single degree of freedom structureFor lateral motion a pile head has two degrees of freedom, it can rotate andtranslate (note that these two degrees of freedom are coupled as indicated by equations 14.14 and 14.15. In addition the structure which is supported by thepile foundation has further degrees of freedom. If the structure itself is modelled

with a single degree of freedom, the structure-foundation system then has threedegrees of freedom. Wolf (1985) gives a equivalent SDOF model for a shallow foundation, illustrated in Figures 1.1 and 11.1. A similar model applies to a pilesupporting a SDOF structure as shown in Figure 19.17. There are three stiffnesscomponents, the structure, and the horizontal and lateral stiffness of the pilefoundation, from these an equivalent natural frequency is obtained from:

22 s

2s h s

=1 + / + /k k k h k

19.16

where: is the natural frequency of the equivalent SDOF model,s is the natural frequency of the structure supported by the

foundation,


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uh he

heu s

Me

u =u s +u h +h e

K s

K h

K

embedded pile

coupling

Figure 19.17 Pile-supported SDOF structure; left: pile head respresented withtwo coupled springs; right: The SDOF structure supported by a single pileembedded in soil.

k s is the stiffness of the structure,k h is the unrestrained horizontal stiffness of the pile head,k is the unrestrained rotational stiffness of the pile head.

Note that in evaluating k h and k the coupling between these two degrees of freedom needs to be considered; equations 14. 46 and 14.47 ensure this.

There are a number of component damping values, these are combined into asingle equivalent value by:

2 2 2 2

s h2 2 2 2h h

= 1 - - + +

19.17

where: .. is the damping value for the equivalent SDOF model,s is the damping for the structure,h is the damping for horizontal motion of the foundation,

and is the damping associated with the rotational stiffness of thefoundation.

Note that the damping value for the equivalent SDOF structure is frequency dependent.


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Example 19.1 For the Central Laboratories Test Piles estimate the naturalperiod of the piles. Some details of the piles and the test results are given hereinin Figures 19.3 to 19.6. The concrete modulus is 26GPa, the section modulus,

evaluated using transformed sections, is 0.00529m4

for the first two metresbeneath the ground surface and 0.00473m 4 at greater depths. Using theproperties near the ground surface this gives: E pIp = 137.54 MNm 2. This data istaken from the report by Jennings et al (1985).

Since we have a tubular pile section equation 14.17 is used to calculate andequivalent pile modulus:

Ep = 137.54/( x0.4504/64) = 68.3 GPa.

We will assume a linear increase in k with depth. Initially take nh = 0.75 MPa/m

(about the value at the lower end of the range of the values suggested by Jenningset al as describing the distribution of bending moment in the pile with depthduring the slow cyclic loading, Figure 19.6).

We need Young's modulus at a depth of one diameter, at which depth k = 2x(0.75x0.45) = 0.675 MPa. (The value of k is doubled to account for the soilacting on both sides of the pile shaft.) From equation 14.39, assuming v = 0.3,

we have:

Es = 2.62k = 1.77MPa. This gives E p/E s = 38615.

m = 1.77/0.45 = 3.93 MPa/m.

For dynamic loading the equivalent length from equation 19.7 is:

ld = 2.0x0.45x(38615)0.20 = 7.44 m

(For a comparison the static active length equation 14.27 the equivalent lengthfor static loading is:

ls = 1.3x0.45x(38615)0.22 = 4.95 m)

From equation 13.28 we obtain the pile head flexibility coefficients:

f uH = 3.2x38615-0.333/3.93x0.452 = 0.119

f uM = 5.0x38615-0.556/3.93x0.453 = 0.039

f M = 13.6x38615-0.778/3.93x0.454 = 0.023

To evaluate the natural frequency of the system we need the effective horizontalstiffness at the shaker attached to the top of the pile. This is 1.25m above theground surface. The 2x2 stiffness matrix with the above components can be

inverted to give a flexibility matrix. If we consider a horizontal force H applied a


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distance g (1.25m) above the pile head, the horizontal displacement and rotationat the pile head are:

uH uM

uM M

u f H + f Hg

f H + f Hg

=

=

At the distance g above the ground surface there will be additional displacementsfrom cantilever bending, these are:

3g Hg u

3 E Icantilever = +

We can now express the horizontal displacement at the level of the shaking machine in terms of a flexibility coefficient:

32

uH uM Mg u

g = f + 2f g + f +

H 3 E I

Making the appropriate substitutions we find that the equivalent horizontalstiffness at the shaking machine is:

k h = 3.96 MN/m

The mass of the 1.5m pile extension and the shaking machine is 969kg. Thenatural frequency of the system is then:

9 46

n1 3.96 x 10 . Hzf

2 968.0= =

The observed natural frequency was 8.1 Hz, the response curve is reproduced inFigure 19.5. There are two possible explanations as to why the predictedfrequency is greater than that measured. Firstly the cyclic loading might havecaused further softening of the soil adjacent to the pile shaft. The report on thetest mentions the occurrence of liquefaction near the surface. Secondly the work from which Gazetas developed his equations assumes long piles so that the lowerend is fixed. The length required for dynamic fixity is calculated above to beabout 7.5 metres whereas the embedded length of the pile is 6.75 metres. Thus itis not surprising that the actual dynamic lateral stiffness of the pile is less than thecalculated value.

In the above calculations, the stiffnesses calculated via equations 14.28 have notbeen modified to account for dynamic effects. This could be done using thecurves in Figure 19.16, but as the effect is small, typically a 10% reduction instiffness at the natural frequency, no change has been made.

The question of embedment depth warrants comment. There are the two valuesgiven above, 4.95 m for static behaviour of the pile and 7.44 m for dynamicbehaviour. These come from the analyses done by Gazetas and his coworkers.In addition there is the equivalent cantilever length of 4.1m. It so happens thatthe natural frequency for a 6.5m length of pile of the same properties as the testpile is about 8Hz. If the shaker and the pile above the ground surface are


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represented by a 2.5m length of pile (this give the same mass with the centre of gravity the correct distance above the ground surface) then we have a 4m sectionembedded in the ground. This suggests that the equivalent cantilever concept,section 14.5, is capable of predicting the stiffness of the system accurately.

However, this is misleading because it gives no idea of the true length of pileshaft required for proper embedment.

Example 19.2 Consider the test pile of Blaney and O'Neill from the data given inFigures 19.1 to 19.2 estimate the natural period of the pile and the responsecurve. Assume that the damping for the steel pipe is 5% and material damping for the soil is 5%.

The data for this test are given by Blaney and O'Neill (1986a & b). The steel

tube pile diameter had an outside diameter of 273mm and inside diameter of 254.5mm, the second moment of area of the pile section is then 6.7x10 -5m4. Forthe pile shaft EI was 15.05MNm 2. The embedded length of the pile shaft was13.4m. The second moment of area for a solid pile shaft 273mm in diameter is2.73x10-4m4, so the pile modulus for use in the Gazetas equations is 55.2GPa.

The centre of gravity of the attached mass was 2.21m above the ground surface. The total mass at this at this level was 11294.6kg.

Some preliminary calculations along the lines of those outlined in example 19.1,assuming the data from Figure 19.1, gave a natural frequency of 2.08 Hz in

comparison with the 2.30 Hz measured during the first dynamic test. Initially it was thought that the soil modulus chosen might have been a little too low anddid not reflect the increase in stiffness with depth evident from the datapresented in Figure 19.1. However, reasonable estimates of this increase did notlead to the required increase in natural frequency. Next considered was the way the shaker and mass were attached to the pile shaft, perhaps these might not havebeen completely unrestrained as was assumed in the calculations. Rather thanattempt to model this in detail, it was decided to represent it by reducing thelength of the pile shaft extension. It was found that a value of 2.02 m gave anatural frequency of 2.30 Hz. Thus in the remainder of this example a pile shaftextension of 2.02 m is assumed.

The pile was installed in a clay profile. Details are given by Blaney and O'Neill(1986a), some of which are reproduced in Figure 19.1. Cross-hole shear wave

velocity measurements gave a velocity of 150 m/sec for the clay near the top of the pile shaft. The small strain Young's modulus of the soil was thus 134 MPaand (Ep/E s ) = 412.

Since the test pile was installed in an overconsolidated clay profile we will assumea constant modulus distribution. The dynamic active length of the pile fromequation 19.7 is:

Lad = 2.0x0.273x(412)0.25

= 2.46 m.


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Examination of the deflected shapes for the pile in Figure 19.2 confirms that thisis a reasonable estimate of the length of pile shaft involved in lateral deformation.

We are not given the depth of the soil layer in which the piles are embedded. It is

at least 13.4 m, the depth of embedment, and, as there is no mention of thebottom of the clay layer, we will assume a depth of 20 m to estimate the naturalfrequency of the layer and use this in the calculation of frequency dependentdamping. From equation 19.1 the natural frequency of a layer 20 m deep is:

f 1 = 0.25x150/20 = 1.9 Hz

For this example we will use the equivalent SDOF model of Figure 19.17 andequations 19.16 and 19.17.

From equations 6.21 we obtain the pile head flexibility coefficients:

f uH = 1.3x412-0.18/134.0x 0.273 = 0.012

f uM = 2.2x412-0.45/134.0x 0.2732 = 0.015

f M = 9.2x412-0.73/134.0x 0.2733 = 0.042

Inverting the matrix of pile head flexibilities gives the stiffness components:

K HH = 145.9 kN/mm, K MH = -51.4 kNm/mm, andK MM = 42.1 kNm/mrad.

Using equation 14.46 we find that the unrestrained horizontal stiffness at the pilehead is:

k h = (145.9x42.1 - 51.42 )/(42.1 - 2.02x(-51.4)) = 24.0 kN/mm.

From equation 14.47 the unrestrained rotational stiffness of the pile head is:

k = (42.1x145.9 - 51.42 )/(145.9 - (-51.4)/2.02) = 20.4 kNm/mrad.

The lateral stiffness of the extension of the pile shaft above the ground surface is:

k s = 3x15.05/2.02 3 = 5.5 kN/mm.

The natural frequency of the mass on the extension of the pile shaft is:

s = (5.5x1000/11.3) = 22.0 rads/sec.

The equivalent natural frequency of the system is:

~ = {22.0/(1 + 5.5/24.0 + 5.5x2.02 2/20.4) 0.5 ) = 14.45 rad/sec.

f n = 14.45/2x3.1412 = 2.30 Hz. The equivalent damping values are now calculated.


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We need the natural frequencies for horizontal displacement and rotation of thepile head, which are obtained from the equivalent stiffnesses k h and k :

h = (24.0x1000/11.3) 0.5 = 46.1 rad/sec.

= (20.4x1000/11.3x2.02 2 )0.5 = 21.1 rad/sec.

Since the frequency at which these calculations are being done, 2.30 Hz, is greaterthan the frequency of the soil layer, 1.9 Hz, both the radiation and materialdamping contributions of equation 19.8 apply.

From equation 19.8 the damping values for the piles are:

HH = 0.80x0.05 + 1.10x2.30x0.273x4120.17/150.0 = 0.053

MM = 0.35x0.05 + 0.35x2.30x0.273x4120.20

/150.0 = 0.022

HM = 0.80x0.05 + 0.85x2.30x 0.273x4120.18/150.0 = 0.051Using these values it was found that the peak amplification was overpredicted.

This suggests that the Gazetas equations are conservative and underpredict thedamping. It was found that increasing HH and MM by 30% gave the requiredpeak amplification. Thus the damping values to be used in subsequentcalculations are:

HH = 0.069, MM = 0.029 and HM = 0.051.

Having the damping values we can now assemble the various components of thepile head impedance matrix (we will assume that k = 1.0):

HH =K HH(1 + 2HHi) = 145.9(1 + 0.138i)

MM =42.1(1 + 0.058i)

HM =-51.4(1 + 0.102i)

At this point the advantage of the complex impedance becomes evident as wecan substitute these terms directly into equations 14.46 and 14.47, which give thestatic unrestrained pile head stiffness, to get the equivalent pile head impedances.Equation 14.46 with the complex impedances inserted becomes:

2HH MM HM

hMM HM

=

- r

I I I I

I I

Evaluating the parts of this equation term by term, we obtain:

HH MMI I = 145.9(1 + 0.138i)x42.1(1 + 0.058i) = 6093.0 + 1204.0i

2

HMI = (51.4(1 + 2x0.10i))2 = 2614.0 + 534.0i


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Thus the top line of the expression for hI is:

6093.0 + 1204.0i - (2614.0 + 534.0i) = 3479.0 + 670.0i

The bottom line is:

42.1(1 + 0.058i) - (-51.4)(1 + 0.102i)x2.02 = 145.9 + 13.0i

Thus hI expressed as a complex fraction is then:

h3479.0 + 670.0 i

=145.9 + 13.0 i

I

Multiplying the top and bottom lines by the complex conjugate of the bottomline gives:

h(3479.0 + 670.0 i) (145.9 - 13.0 i)

=(145.9 + 13.0 i) (145.9 - 13.0 i)

I

Performing the calculations required, this gives:

hI = (24.1 + 2.5i) kN/mm and h = 2.5/2x24.1 = 0.052

Proceeding similarly the damping value = 0.029.

To calculate the equivalent damping value we need the following:

k s/k h = 5.5/24.0 = 0.23

k sh2/k = 5.5x2.022/20.4 = 1.10

From equation 19.17 the equivalent damping value is:

~ = (0.05 + 0.052x0.23 + 0.029x1.10)/(1 + 0.23 + 1.10) = 0.040

The amplification is calculated from the standard equation for a SDOF system in which the damping is expressed as a fraction of the critical damping (2k/ n ):

2 22

1 Amp =

1 - + 2

At f = 2.30 Hz, = ~, so the amplification is: Amp = 1/2x0.040 = 12.5

Measurement of Figure 19.2 gives an amplification at the natural frequency of the pile of 12.5.

The set of values calculated for the various frequencies are given in the table anddiagram below. Values for the damping coefficients are constant for frequencies


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less than the natural frequency of the layer, f 1. Also the table shows how HH, acomponent of the pile head impedance matrix, increases with frequency, inaccordance with Figure 19.14. This trend is less evident for the damping of thefree head pile, h, and even less apparent for the overall damping value for the

pile.

For the test pile much of the response of the system comes from the projectionof the pile shaft above the ground surface, as this is so much less stiff than thepile head. Note also that the overall damping is small, once again being dominated by the pile shaft.

0 2 4 6 8 100

5

10

15

frequency (Hz)

A m p l i f i c a t i o n

Frequency =2.3 Hz

f(Hz) HH h Amplif.

0.46 0.052 0.040 0.038 1.04

0.92 0.052 0.040 0.038 1.19

1.38 0.052 0.040 0.038 1.56

1.84 0.052 0.040 0.038 2.74

2.30 0.069 0.051 0.040 12.50

2.76 0.072 0.054 0.041 2.22

3.22 0.075 0.057 0.043 1.03

3.68 0.079 0.060 0.044 0.64

4.14 0.082 0.063 0.045 0.45

4.60 0.085 0.066 0.046 0.33

An acceptable alternative to the dynamic response curve plotted in the abovediagram could have been obtained by estimating the damping at the naturalfrequency and applying this value to all frequencies. The purpose of the lengthier


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approach employed above was to illustrate the manner in which various damping components vary with frequency.

19.2.5 Inertial interaction for piles in layered soil profiles The ideas presented here follow the paper of Gazetas and Dobry (1984). Since we are considering a layered soil profile the convenient expressions for pile headflexibilities given in section 14.2 are not applicable. However, we observe fromFigure 19.2 that at resonance the normalised deflected shape of the Blaney andO'Neill test pile is the same as that for static deformation. In Figure 19.18 thedeflected shape of a flexible pile in soil with a linear increase in modulus withdepth is plotted from Krishnan et al (1983) for a range of frequencies. Theproperties are such that the natural frequency of the soil layer corresponds to a o = 0.31. It is apparent from the diagram that for frequencies up to f 1 thehorizontal displacements of the pile are of the same form as the static deflected

shape in Figure 19.2. At higher frequencies the pile shaft undergoes smallexcitation at greater depths.

Gazetas and Dobry use this observation as the basis for their method forestimating the dynamic stiffness of flexible piles in layered media. They start

with an estimate of the static deflected shape of the pile. This is obtained by any convenient method, Winkler, elastic continuum etc. From this the static stiffnessof the pile head is obtained. Next the damping values are assigned to the soilsurrounding the pile shaft by approximate methods outlined below. The overalldamping value for the pile shaft is obtained by a weighted averaging process. Inthis way the soil surrounding the top of the pile shaft has the greatest

contribution to the damping, thus the approximation involved in using the staticdeflected shape is justified as the small deflections at depth at high frequencies will have negligible effect. Third, the dynamic modification of the static stiffnessand damping values must be assessed. Diagrams such as Figure 19.14 to 16 willbe helpful here if one can make some approximation of the actual soil profile onthe basis of the three profiles in Figure 14.4.

Gazetas and Dobry propose a simplified method for estimation of the radiationdamping coefficient at various positions along the pile shaft. As is shown inFigure 19.19 it is assumed that energy is radiated away from the pile shaft inorthogonal directions, the velocity of propagation in the direction in which the

pile is loaded is the p-wave velocity and at right angles the propagation velocity isthe shear wave velocity. Based on this concept they arrive at the following expressions for the radiation damping adjacent to the pile shaft:

DD

5/4 3/4r -1/4

0s s

3.4c = 1 + z > 2.5 a 2 (1 - ) 4v

19.18a

DD

3/4r -1/4

0s s

c = 2 z 2.5 a 2 4v

19.18b

where: ao =

D/2v s.


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Figure 19.18 Distribution of dynamic deformations with depth in a flexible pileat a range of frequencies (after Krishnan et al (1983)).


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As explained above at frequencies less than the natural frequency of the systemthere is no radiation damping. Thus Gazetas and Dobry propose that thedistribution of damping coefficients used be as shown in Figure 19.20. They alsoexplain how a strain dependent material damping coefficient might be estimated.

The above relations consider assigning a damping value to various positionsalong the pile shaft. The next step is to obtain the effective damping value at thepile head. This is achieved by weighting the damping values in accordance withthe static deflected shape of the pile shaft, using:

2sr mC ( + ) (z) dzc c Y

L

0

= 19.19

where: Y s is the static deflected shape of the pile normalised with respect tohorizontal displacement at the pile head. An example of the calculation processtaken from Gazetas and Dobry (1984) is reproduced in Figure 19.21.

Of interest in this example is the fact that despite a pile is of unusually largediameter it is still flexible, i. e. the effective length is somewhat less than theactual length of 34m. In Figure 19.22 a comparison is made between thedamping values obtained for this pile by the method given by Gazetas and Dobry in comparison with a three dimensional dynamic finite element analysis; theagreement is very good.

Figure 19.19 Pile shaft radiation damping model (after Gazetas and Dobry(1984)).


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Figure 19.20 Pile shaft damping near resonance (after Gazetas and Dobry(1984)).

Figure 19.21 Layered soil profile for pile lateral stiffness example (after Gazetasand Dobry (1984)).


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Fig 19.22 Comparison of finite elem en t results with Ga zeta s a nd Do b ry pile st iffness and da m ping pred ict ions (after Gazetas and Dobry (1984)) .

Figure 19.23 Development of a gap adjacent to the shaft of a pile subject to

cyclic lateral loading in clay (after Swane and Poulos (1984)).


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19.2.6 GappingFor the two dynamic tests analysed above the applied forces are small in relationto the lateral capacity of the piles. At larger force levels the soil adjacent to thetop of the pile shaft will be stressed beyond the elastic range. For sands,

particularly, saturated sands, this will cause a decrease in the available resistance.For clay soils it may lead to the opening up of a gap adjacent to the pile shaft. This is will reduce the stiffness at the pile head and consequently reduce thenatural frequency of the system.

Swane and Poulos (1984) used the Winkler model to represent this phenomenon. The pile shaft is envisaged as having Winkler springs on each side. At the start of the lateral loading these springs are pre-compressed with forces representing thein situ horizontal stresses in the soil adjacent to the pile shaft. Consider a pair of these springs close to the top of the pile shaft. Prior to the commencement of lateral loading the springs will carry the same force and the displacement of each

is zero. During lateral loading the spring on one side is compressed further andthat on the opposite side unloads. At some lateral load the force in the spring onthe unloading side reaches zero and a gap begins to open. Further lateral loading

will increase the width of the gap and it will move deeper effecting other springs. When the load is reversed the situation is reversed as well with the spring which was previously being compressed now unloading. At some stage the pile shaft will cross the gap and commence to re-engage the previously unloaded spring(s).

This cyclic process will continue and, at loads small in relation to the lateralcapacity of the pile, a stable gap length may be generated at which stage there willbe no further reduction in pile stiffness of growth in gap length. This is known

as shake-down. At larger cyclic loads shakedown may take a much larger numberof cycles to occur, or at larger loads still there may be a tendency for continuousgrowth of the gap length. The results of one of the Swane and Pouloscalculations are presented in Figure 19.23. This was for a 610mm diameter steeltube pile in stiff clay. The numbers of cycles in Figure 19.23 refer to cyclicloading of a foundation for an offshore platform. In the case of earthquakeloading of a bridge foundation one would expect a few tens of cycles at most. Itis apparent from Figure 19.23 that with large loads gaps of length several pilediameters could form with a consequent reduction in the pile stiffness andincrease in the maximum moment in the pile shaft.

19.2.7 Kinematic interactionKinematic interaction, refer to Figure 19.13, is expressed in terms of thefollowing kinematic amplification factors:

pu

g

p

g

u= A u

D= A u2

19.20

where: up is the amplitude of the horizontal displacement of the pile headrelative to the input at the base of the pile u g , the total

displacement of the pile head is u g (t) + u p(t),and p is the rotation of the pile head.


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Perhaps a clearer way of indicating the effect of the pile shaft is to consider therelation between the motion of the pile head and that of the ground surfacesufficiently distant that there is no effect from the pile head movement, the so-called free field motion. This is achieved with the following kinematic interaction

factors:p

uo

p

o

u=I u D

=Iu2

19.21

where: uo is the amplitude of the free field motion of the ground surfaceadjacent to the pile head.

The amplitudes u p and uo are calculated for steady state sinusoidal excitation, ug ,at the base of the pile.

Gazetas, having obtained a large number of individual results for the various soilprofiles, found that they depend on the ratios E p/E s and the length to diameterratio of the pile,( ), as well as the Poisson's ratio for the soil, the soil damping factor, and the ratio of the density of the pile material to that of the soil. The firsttwo of these are the most important, in the remainder of this section a Poisson'sratio of 0.4 and a density ratio of 1.6 will be assumed. In the majority of cases thesoil damping ratio will be taken as 5%. The kinematic amplification factors peak,as expected, at the natural frequencies of the layer, and the displacementinteraction is large whilst the rotational interaction factor is small. The kinematicinteraction factor for horizontal displacement, I u, is unity for static displacementand less than unity at frequencies much greater than the natural frequency of thelayer. The shape of the curves is very dependent on the distribution of soilmodulus with depth. For the linear profile I u is less than unity when f > f 1. Onthe other hand for the constant modulus profile there is a range of frequenciesfor which I u > 1 before it decreases below unity. The rotational kinematicinteraction factors, I , are sufficiently small that they can be neglected. Gazetasfound that with suitable dimensionless frequency parameters it is possible toexpress the kinematic interaction factors for each soil profile as a unique curve,these are plotted in Figure 19.24.

The frequency parameters. F A, FB and FC, in Figure 19.24 are defined by thefollowing equations:

0.10 -0.40 A

1

0.16 -0.35B

1

0.30 -0.50C

1

f F = K

f

f F = K

f

f F = K

f

19.22

where: is the length to diameter ratio of the pile shaft.


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Figure 19.24 Kinematic interaction factors (I u) for free head piles in terms ofdimensionless frequency parameters (equation 19.22) (after Gazetas (1984)).


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Figure 19.24 shows that at low frequencies the kinematic interaction is negligibleand that at high frequencies the lateral response of the pile head is about half thesurrounding elastic soil. When the response of a soil layer is calculated to anearthquake time history it is found that the kinematic effects are small other than

at low periods. Thus kinematic interaction for piles in uniform soils is not alarge effect.

19.3 Effect of abrupt changes in layer properties

The above discussion envisages a soil profile with a smooth variation in soilstiffness with depth. Another factor in determining kinematic interaction is thepresence of sudden changes in soil properties in a layered soil profile. It isexpected that at such changes might lead to localised damage of the pile shaftthat cannot be predicted with the above methods. This topic is discussed by Oweiss (1981) and Margason and Holloway (1977). Also Makris and Gazetas

(1992) point out that kinematic interaction effects in layered soil profiles may bedifferent from those presented above dealing with soil profiles having a smooth

variation in modulus with depth.

19.4 Key ideas

References

Blaney, G. W. and O'Neill, M. W. (1986a) "Measured lateral response of mass onsingle pile in clay", Jnl. Geotech. Eng., Proc. ASCE, Vol. 112 No. 4, pp.443-457.

Blaney, G. W. and O'Neill, M. W. (1986b) "Analysis of dynamic laterally loadedpile in clay", Jnl. Geotech. Eng., Proc. ASCE, Vol. 112 No. 9, pp. 827-840.

Dobry, R. and Gazetas, G. (1985) "Dynamic stiffness and damping of foundations by simple methods", Proc. ASCE symposium on: VibrationProblems in Geotechnical Engineering, edited by G. Gazetas and E. T.Selig, pp. 75-107.

Dobry, R. and Gazetas, G. (1988) "Simple method for dynamic stiffness anddamping of floating pile groups", Geotechnique, Vol. 38 No. 4, pp. 557-574.

Gazetas, G. (1991) "Foundation vibrations", in Foundation Engineering Handbook,2nd. edition, H-Y Fang editor, Van Nostrand Reinhold, pp. 553-593.

Gazetas, G. and Dobry, R. (1984) "Horizontal response of piles in layered soils", Jnl. Geotech. Eng., Proc. ASCE, Vol. 110 No. 1, pp. 20-40.


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Gazetas, G. (1984) "Seismic response of single end-bearing piles", Soil Dynamicsand earthquake engineering, Vol. 3 No. 2, pp. 82-93.

Jennings, D. J., Thurston, S. J., Edmonds, F. D. and Millar, P. J. (1985) "Thebehaviour of two 450mm diameter piles subject to static and dynamic

cyclic lateral loading", Ministry of Works and Development, CentralLaboratories Report 5-85/17, Vol. I & II. Jennings, D. N., Thurston, S. J. and Edmonds, F. D. (1986) "Static and dynamic

lateral loading of two piles", Proc. N Z National Roads Board RoadResearch Unit, Bridge Design and Research Seminar, Auckland, RRUBulletin No. 73, pp. 29-38.

Krishnan, R., Gazetas, G. and Velez, A. (1983) "Static and dynamic lateraldeflexion of piles in non-homogeneous soil stratum", Geotechnique,

Vol. 33 No. 3, pp. 307-325.Makris, N. and Gazetas, G. (1992) "Dynamic pile-soil-pile interaction. Part II:

lateral and seismic response", Earthquake Engineering and Structural

Dynamics, Vol. 21, pp. 145-162.Margasen, E. and Holloway, D. M. (1977) Pile bending during earthquakes, Proc.6th World Conference on Earthquake Engineering, Vol. II, p. 1690 -1696.

Oweis, I. S. (1981) "Evaluating pile performance during earthquakes", Proc. ASCE, Jnl. Geotech. Eng., Vol. 107 GT5, pp. 678-683.

Sadon, N M; Pender, M J; Orense, R P and Abdul Karin, A R (2010) Full-scale pile head lateral vibration tests in Soil-Foundation-Structure-Interaction, R Orense, N Chouw, and M Pender (eds), p. 33-39. CRCPress/Balkema, The Netherlands.

Sadon, N. M. (2011) Full scale static and dynamic lateral loading of a single

pile. PhD thesis, University of Auckland.Swane, I. C. and Poulos, H. G. (1984) "Shakedown analysis of laterally loadedpile tested in stiff clay", Proc. 4th. Australia - New Zealand Conferenceon Geomechanics, Perth, Vol. I, pp. 165-169.

Wolf, J. P. (1985) "Dynamic soil-structure interaction" , Wiley.