Soil Dynamics and Earthquake Engineering 46 (2013) 52–63

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Soil Dynamics and Earthquake Engineering

0267-72

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/soildyn

Piles-induced filtering effect on the Foundation Input Motion

Raffaele Di Laora a,n, Luca de Sanctis b

a Department of Civil Engineering, Second University of Napoli, Aversa (CE), Italyb Department of Technologies, University of Napoli Parthenope, Napoli, Italy

a r t i c l e i n f o

Article history:

Received 13 September 2012

Received in revised form

8 December 2012

Accepted 10 December 2012Available online 22 January 2013

61/$ - see front matter & 2012 Published by

x.doi.org/10.1016/j.soildyn.2012.12.007

esponding author. Tel.: þ00390815010385.

ail address: raffaele.dilaora@unina2.it (R. Di L

a b s t r a c t

The inertial interaction analysis of a structure founded on piles is conventionally performed by

imposing that the Foundation Input Motion is merely that of the free field, thus neglecting the

kinematic interaction between piles and soil generated by the passage of seismic waves. This would

lead to unnecessary overconservatism in the design, as there is evidence that the free-field motion may

be thoroughly filtered out by piles (generally reduced), especially in the case of soft soils, where piles

are recurrently required to carry out the total load transmitted by the superstructure and/or to reduce

foundation settlements. Results provided from analytical and numerical tools elucidate the crucial

aspects controlling the mechanism of filtering effect. Reduced design spectra are also suggested to

account for the beneficial effect coming from the piles when the inertial interaction analysis of the

superstructure is being performed.

& 2012 Published by Elsevier Ltd.

1. Introduction

The response of a structure to an earthquake is usuallypredicted assuming that the support motion at the foundationlevel is merely that of the free-field. However, the superstructureinteracts with its foundation and the surrounding soil, creatingadditional soil deformations, that add to those generated from thepassage of seismic waves, so as the motion in the vicinity of thefoundation can differ substantially from that of the free-field.Assuming a linear soil-foundation-superstructure response, theanalysis of the complete system can be performed according tothree consecutive steps: (i) predict the motion of the foundationin the absence of the superstructure, i.e. the so-called FoundationInput Motion (FIM); (ii) determine the dynamic impedancefunctions associated to swaying, vertical, rocking and crossswaying-rocking oscillation of the foundation; (iii) evaluate theresponse of the superstructure supported on the springs anddashpots and subjected to the motion of the foundation deter-mined at the first step. This procedure is commonly referred to inliterature as kinematic-inertial decomposition or substructuremethod [1–3]. Once the response of the structure has beencomputed, the pile–soil interaction effects can be readily calcu-lated by superposition of kinematic and inertial effects. Thismethod has been extensively adopted to study the response ofstructures and foundations under seismic excitation and is trulyattractive as alternative to fully 3D analyses involving the

Elsevier Ltd.

aora).

complete pile–soil-superstructure interaction, that are very com-plex and rarely performed in engineering practice [4].

Dynamic impedance functions at the foundation level are afundamental ingredient for inertial interaction analysis of both thesuperstructure and the piled foundations. Impedance functions ofsoil foundation systems show frequency dependent characteris-tics. However, the analysis of structures is currently orientedtowards performance based criteria, for which non-linearity ofstructural members is mandatory. In this case, the structuralanalyses cannot be but performed in time domain, and thefrequency dependency of impedance functions makes problematicthe numerical computations. To overcome this difficulty, a betterchoice is represented by the so-called lumped-parameter models,LPMs [5], capable of accounting for frequency dependency ofimpedance functions. LPMs can be easily incorporated into anon-linear analysis of the structure. An early application of thistype of model was presented by Ciampoli and Pinto [6] to analyzethe response of bridge piers. A similar work has been recentlypublished by Carbonari et al. [7]. They performed a non-linearinertial analysis in time domain of a wall-frame structures withconcentrated plasticity by using LPMs to model pile group foun-dations embedded in a two-layer soil, focusing on the comparativebehaviour of compliant vs fixed base models. It is argued fromthese studies that compliance-base models behave quite differ-ently from fixed base models because SSI has a remarkable effecton the dissipative behaviour of the structure. One out of thelimitation of LPMs is the inability to model impedance functionswhose components are neither quadratic nor linear with fre-quency. A second option that has received great attention is thenon-linear macro-element approach, where the foundation and

List of symbols

Latin symbols

a0 dimensionless frequencyap, as pile top acceleration, free-field surface soil

accelerationc Winkler dashpot coefficientEs, E1 soil Young’s modulus, Young’s modulus in layer 1Ep pile Young’s modulusf excitation frequencyf1 fundamental natural frequency of soil (site frequency)fm, fp mean and predominant frequency of earthquake

recordFIM foundation input motionh1, h2 thickness of the first (surface) soil layer, thickness of

the second soil layerH thickness of the soil depositIp pile cross-sectional moment of inertiaIu translational kinematic response factork Winkler spring modulusL pile lengthML local magnitude of the earthquake eventq soil wavenumberT structural periodT1 site period

Tmin, Tcrit characteristic structural periodsVs, Vs1, Vs2 soil shear wave velocity, soil shear wave velocity in

layer 1 and 2

Greek symbols

bs soil hysteretic damping coefficientG dimensionless response coefficientGt1, Gt2 dimensionless transient response parametersd Winkler stiffness coefficient (¼k/Es)l Winkler wavenumberlp pile characteristic wavelengthls Wavelength in the soil mediumn1, n2 soil Poisson’s ratio in layer 1 and 2np pile Poisson’s ratiors soil mass densityx0, xmin, xcrit characteristic spectral ratios for transient motionsx0,av, xmin,av characteristic average spectral ratiosrp pile mass densityo cyclic excitation frequency (2pf)os characteristic cyclic excitation frequency of earth-

quake recordos, av average characteristic cyclic excitation frequency of

earthquake records employed

R. Di Laora, L. de Sanctis / Soil Dynamics and Earthquake Engineering 46 (2013) 52–63 53

the soil are represented by a single macro-element, NLME, whichis capable to predict non-linear permanent deformations of thefoundation in terms of settlement, sliding and rotation. As aconsequence of dissipative effects at the soil foundation level,the seismic demand of the superstructure may also be reducedsignificantly. The early formulation of a macro-element waspublished by Nova and Montrasio [8]; since then a number ofNLEMs have been proposed, like those suggested in Refs. [9,10],capable to account for permanent deformation related dissipativeeffects. The use of NLEMs for predictive analyses has been limitedto shallow foundations and homogenous soil conditions, as in thiscase the availability of well documented 1 g and centrifuge testsallowed for model validation.

Despite the modern trend towards complex SSI models, thefiltering effect on the support motion at the foundation levelgenerated from the piles has not been adequately addressed. Themost attractive application of the substructure method is toassume that the support motion equals the free-field seismicmotion. By contrast, the free-field motion is filtered out by thepiles, especially in the case of soft soils, where piles are recur-rently required to increase the bearing capacity of the foundationand/or to reduce settlements [11,12], but this potential has notbeen yet exploited in engineering practice. The goal of this paperis threefold: (i) to outline the importance of the filtering effect foranalyzing efficiently the dynamic response of the superstructure;(ii) to offer insight into the filtering action exerted by the piles;(iii) to propose a correction to design spectra that can be ofassistance for seismic risk reduction strategies.

2. Literature review

Piles–soil kinematic interaction has been addressed by manyresearchers in the last decade. Emphasis has been placed onkinematic bending effects, evaluated on the basis of analyticalstudies [13–16] and numerically-based parametric analysis[17–21]. As a result of this research effort, a number of ready-

to-use methods are available to predict kinematic bending effectsat both the interfaces between layers and the pile head. Theproblem of the filtering effect exerted by the piles has alsoreceived some attention. The existence of filtering effect hasconfirmed through published works referring to theoretical[22–29] and experimental evidence [2,30–32], even if this obser-vation has not been so far taken into consideration in engineeringpractice.

Filtering effect was examined in the seminal work by Flores-Berrones and Whitman [22], who expressed the ratio of pile andsoil accelerations as a function of excitation frequency in a closedform solution, for the case of an infinitely-long pile in homo-geneous halfspace. As an outcome, piles were found to reduceseismic motion with increasing excitation frequency and pilediameter, whereas an increase in soil stiffness leads to a decreasein the filtering effect.

In general the support motion at the foundation level isdifferent from that of the free-field because of the scattered wavefield generated from the difference between pile and soil rigid-ities. For motions that are rich in high frequency components,even practically flexible piles may not be able to follow the wavymovements of the free-field. On the other hand, if low frequencycomponents of the input motions are predominant, the scatteredfield is weak, and the support motion can be expected to beapproximately equal to that of the free-field [23,25–27].

Fan et al. [25] extended the investigation on filtering effect topile groups under steady-state conditions. They found that groupeffects, although clearly depending on pile spacing, are notrelevant for lateral vibrations. By contrast, they may stronglyaffect the rotational component of motion, which, however, liesbeyond the scope of this paper.

All theoretical studies mentioned above highlight the promi-nent role exerted by excitation frequency. Such a result isconfirmed by the experimental evidence about the filtering effectavailable in the literature.

Kawamura et al. [30] have reported the case history of a 7-storeyresidential building in Japan, as shown in Fig. 1. Acceleration

Plan view

8,36 8,36 8,36 5,63 8,36 8,36 8,3655,79

1,84

5,63

4,72

1,40

13,5

9

Acceleration BUILDING LINEAcceleration SOIL LINEVelocity

RF T = 0.33 s (RF-1F)T = 0.20 s (Underground)T = 5 s

1F

4

12

24

GL

S04

S24

S12

Recorder

4 3

15

0 10 20304050600 10 20304050N-ValueN-Value

5

10

15

20

5

10

15

20

Sand

Silty

Sand

Sand

Sea Side Land Side

Sandy

FineSand

Silt

Sand

Sand

Clay

Sand

Fig. 1. Case history: plan view and elevation of the building (modified from Kawamura et al. [30]).

R. Di Laora, L. de Sanctis / Soil Dynamics and Earthquake Engineering 46 (2013) 52–6354

recordings were available since 1971 at two vertical alignments, the‘building line’ and the ‘soil line’. By comparing the records of the twoalignments during 20 earthquakes, they found that the maximumamplification at the ground surface was 1.5 times higher than theone recorded at the base slab of the building. They also plotted theFourier spectral ratio between the building line and the soil line atthe level of the base slab, concluding that for structural periodssmaller than 0.3 s the ratio of the two accelerations is 0.5 as anaverage. With increasing structural periods the Fourier spectralratios were found to be about unity. It is argued from this casehistory that the high frequency components of the free-field motionare filtered out by the pile–soil-superstructure interaction.

Tajimi [31] described an example of (large diameter) boredpiles for which acceleration recordings were available at both thepile-cap and at the ground surface on a far-distant axis represent-ing free-field conditions. By comparing the acceleration spectra ofthe two motions he noticed that the spectrum of the far distantmotion included frequency contents higher than those composingthe pile cap motion, in agreement with the conclusion byKawamura et al. [30]. The same example has been later reportedby Otha et al. [32] and Gazetas [2]. This last work is particularlysignificant for the scope of this study. The example underexamination consists of an 11-storey building supported oncast-in-place piles and founded on an alluvial deposit of alternat-ing layers of sand and silt, as shown in Fig. 2. The foundationssystem response was monitored during seven earthquakesthrough 27 accelerometers that were placed on the pile buildingaxis and on a far-distant axis representing free-field soil condi-tions. Fig. 2 first published by Gazetas [2] shows the completefield recordings supplied by the 27 accelerometers in the form ofratios between Fourier amplitude spectrum of the foundationmotion over that of the free-field. It is worthy of note thatfrequencies in the range between the fundamental frequency ofthe soil deposit, f1, and the fundamental frequency of the super-structure, fst, are amplified due to pile–soil-structure interaction.

In addition, high frequency components of the seismic motion(f41.5f1) are filtered out by the pile–soil-structure interaction,while low frequency components (f o f1) are not affected by boththe piles and the structure. However, at frequencies larger than1.5f1, there is no evidence however of the amount of the filteringeffect generated from the piles, as the recorded accelerations arethe result of the pile–soil-structure interaction (i.e., the sum ofkinematic and inertial interaction). Nevertheless, this case historyclearly demonstrates that piles have the potential of filtering outthe high frequency components of the input motion.

Makris et al. [33] have reported the case history of PainterStreet Bridge, in northern California, shaken on 25 April 1992 bythe Petrolia earthquake (with Magnitude ML¼7.1 and 18 km fromthe fault). Painter Street Bridge is a continuous, two span, 16 mwide, cast in place prestressed bridge. One span is about 45 mand the other 36 m (Fig. 3). It is about 16 m wide. The bent issupported by two pile-groups, each consisting of 20 (4�5) drivenconcrete piles. The bridge was instrumented in 1977. Motionswere recorded in all accelerographs, including the one in the free-field, that on the footing of one pier, that above the pier at theunderside of the bridge girder and that on the deck near the westabutment. The shear wave velocity has an average value ofVs¼225 m/s, and the pile diameter was only 0.36 m; the dimen-sionless frequency, a0¼2pfd/Vs, is of the order of only 0.1. Fromstudies on vertical propagating shear waves in homogeneous soildeposits [25], the authors concluded that the support motion ispractically equal to that of the free field. Considering the free fieldinput motion, they found a very satisfactory agreement betweenthe computed and the recorded response of the pile cap and thebridge deck. It is worthy of note that the soil consists ofmoderately stiff/dense soil layers, with the SPT blowcountsranging between 8 near the surface and 34 at 10 m depth andthe underlying layer being a very dense gravelly sand. This is thecase of a flexible pile embedded in a moderately stiff/dense soildeposit. In addition, the free-filed motion has a low-frequency

R. Di Laora, L. de Sanctis / Soil Dynamics and Earthquake Engineering 46 (2013) 52–63 55

content. Under this circumstances there is evidence that thefiltering effect is negligible.

The limited amount of experimental evidence demonstratesthat the frequency content of the input signal might exert a

f1 fst

f1 = fundamental frequency of the soil stratum

fst = fundamental frequency of the superstructure

Recorded Fourieramplitude ratio

3

2

1

0 1 2 3 4 5

10

0

20

30

30 5

frequency [Hz]

78 m

8.35 m Plan view

dept

h [m

]pi

le-s

oil a

ccel

erat

ion

ratio

AccelerometerPressure gaugePore waterpressure gauge

100

100 200 300 400Vs [m/s]

Fig. 2. Case history: (a) plan and section of the building; (b) recorded ratio of

Fourier amplitude spectra atop the pile and at the free-field ground motion

(modified from Gazetas [2]).

Fig. 3. Cross section of the Painter Street Bridge (modified from Makris et al. [3]).

Fig. 4. Acceleration ratio as function of excitation frequ

remarkable influence on the support motion. The parametricstudy reported in this work confirmed the trend coming fromthese field observations.

3. The mechanism of filtering effect

Following the contributions by Flores-Berrones and Whitman[22], Makris and Gazetas [26] and Nikolaou et al. [17], it may beshown that in a homogenous halfspace the ratio between theacceleration atop a fixed-head infinitely long pile, ap, and that atthe soil surface, as, is given by:

Iu ¼ap

as¼G ð1Þ

where G is a complex-valued interaction factor which may beapproximately expressed as [34]:

GC4l4

4l4þq4

ð2Þ

in which q (¼o/Vs) is the wavenumber of the harmonic SH wavetravelling in the soil, o the excitation circular frequency, and l isthe well-known Winkler parameter, which may be faithfullyapproximated through the following expression, as the termassociated to inertia contribution of the pile mass is neglected:

lCkþ ioc

4EpIp

� �14

ð3Þ

Ep and Ip being pile Young’s modulus and cross-sectional momentof inertia. The Winkler spring reaction modulus k may be taken asproportional to soil stiffness Es by a coefficient d, which typicallyassumes values close to unity [35], whereas the dashpot coeffi-cient is expressed as [23]:

c¼ 6o�14rsV

54sd

34þ2bsk=o ð4Þ

in which d is the pile diameter, rs and bs the density and thehysteretic damping ratio of the soil. Eq. (2) is plotted in Fig. 4aagainst excitation frequency for d¼1.2 and different values ofstiffness ratio Ep/Es. Pile-to-soil acceleration ratio is alwayssmaller than unity (i.e. piles play always a beneficial role inreducing the seismic motion that excites the superstructure) anddecreases with frequency. This effect is physically due to theresistance that the pile offers in adapting to the short wave-lengths in the soil, as the dimensionless frequency a0 may beinterpreted as the ratio of pile diameter d and soil wavelength ls

(¼Vs/o). Moreover, at a given frequency pile–soil accelerationratio is progressively smaller as pile–soil stiffness ratio increases.The physical interpretation of this phenomenon deserves specialdiscussion as reported below.

The ability of the pile to follow soil displacements is physicallyconnected to both pile diameter and pile–soil stiffness ratio. A

ency for the case of an infinitely-long pile (Iu¼G).

R. Di Laora, L. de Sanctis / Soil Dynamics and Earthquake Engineering 46 (2013) 52–6356

characteristic pile wavelength may be therefore defined as:

lp ¼ dEp

Es

� �14

ð5Þ

It is straightforward to verify that the real part of thewavenumber l in Eq. (3) is proportional to the reciprocal of theabove defined wavelength lp. Therefore, assuming d¼1 andneglecting the term ioc in Eq. (3), and substituting Eq. (5) intoEq. (2) the amplitude of G can be expressed as:

GC 1þ1

20

olp

Vs

� �4" #�1

ð6Þ

The new dimensionless parameter olp/Vs can be viewed as theratio of pile characteristic wavelength, lp, and soil wavelength, l.Eq. (6) is plotted in Fig. 4b against olp/Vs. The curve expressed byEq. (2) for d¼2 is also added for comparison. It can be seen thatthe filtering effect, represented by the kinematic interactionfactor G, is governed by the ratio of the two wavelengths. Thelarger is the characteristic wavelength of the pile compared to thesoil wavelength, the larger is the amount of the filtering effect.

4. Parametric analysis

To investigate quantitatively the filtering effect exerted on theFoundation Input Motion by a piled foundation, a comprehensiveset of Finite Element analyses has been performed. Owing to thesecond-order influence of group effects for lateral vibration [25], afixed-head single pile embedded in a two-layer soil has beenconsidered, as shown in Fig. 5a. Harmonic S-waves applied at thebedrock level and propagating up and down (after reflection)constitute the seismic excitation.

According to the Buckingham theorem [36] pile–soil interac-tion is governed by 7 dimensionless ratios (H/d, L/d, Ep/E1, Vs2/Vs1,h1/d, od/Vs1, bs). Despite the fact that only the complete set ofsuch parameters would suffice to control kinematic response,some simplifications are possible and were thereby employed inthis study. Specifically: (a) considering long piles, whose length isgreater than the ‘active’ one, as shown in [37], the first twodimensionless parameters do not affect pile–soil interaction;(b) damping ratio does affect both pile and soil response, yetnot their ratio as proven in the ensuing, so that damping levelmay be set at a constant value; (c) in light of the analyticalderivations reported above, dimensionless frequency od/Vs1 andpile–soil stiffness ratio Ep/E1 may be combined to give the uniqueparameter olp/Vs1. Kinematic harmonic response of long piles in

L

H

h1

h2 Vs2

Vs1

1 2

Bedrock

SH SH

d

Fig. 5. (a) Problem considered; (b) finite ele

two-layer soils may be therefore conveniently expressed asfunction of the 3 parameters h1/d, Vs2/Vs1 and olp/Vs1. Conse-quently, interface depth h1/d assumes values of 5, 10, 15 and 19whereas layer impedance contrast is considered equal to 2 and 4.The shear wave velocity in the first layer is taken equal to 50 and100 m/s, corresponding to pile–soil stiffness ratios of about 600and 2500, respectively.

To investigate the response to real earthquakes a geometricalquantity (e.g., total height of the soil deposit) and a materialstiffness (e.g. pile Young’s modulus) have to be defined. Accord-ingly, the total height H is set to 30 m, whereas the pilehas diameter d¼1 m, length L¼20 m and Young’s modulusEp¼30 GPa. Soil and pile density are set to 1.75 and 2.5 Mg/m3,respectively, whereas the value of soil Poisson’s ratio is 0.4.

The parametric analysis has been performed assuming a vis-coelastic soil model, which is intrinsically unable to predict realsoil behaviour. Nevertheless, the results coming from this assump-tion have a general validity. Martinelli [4] has performed a study ofthe kinematic response of piles embedded in a two layer soilmodelled through the advanced consititutive model by Dafaliasand Manzari [38]. Results of this study, undertaken by finiteelement analysis, show that kinematic interaction effects can bepredicted within reasonable accuracy even assuming that the soilbehaves like a viscoelastic material, provided that the soil stiffnessprofile is preliminary evaluated through an equivalent linear visco-elastic analysis, or ELE analysis. This conclusion, however, onlyapplies by taking a soil permeability larger than 10�4 m/s orassuming drained soil conditions. Under circumstances where soilpermeability is lower than 10�4 m/s, it is necessary to perform acoupled analysis to achieve a successful prediction of earthquake-induced kinematic interaction effects. It is worthy of note, how-ever, that even for medium to dense sand, the excess porepressures lead to a reduction of the soil stiffness pertaining tothe upper layer. In this respect, the filtered motion predicted by thedrained analysis represents an upper bound of that predicted bythe coupled analysis and can be safely employed in the evaluationof the inertial response of the superstructure.

4.1. Numerical analyses

Although the problem is 3D, the geometry is axisymmetricwhereas the load is anti-symmetric. To simplify the analysis,stresses and displacements are expanded into a Fourier series inthe circumferential direction, according to the technique intro-duced by Wilson [39]. For the example at hand, only the first-order term of the series is needed. Owing to this procedure, the

4 Vs/ Vs1

ment mesh employed in the analyses.

R. Di Laora, L. de Sanctis / Soil Dynamics and Earthquake Engineering 46 (2013) 52–63 57

original three-dimensional problem is conveniently reduced to a2D one [40].

Numerical analyses were conducted using the commercial FEcode ANSYS [41]. Four-noded axisymmetric 2D elements are usedto mesh soil and pile. The assumption of a line of anti-symmetryimplicitly implies the use of ‘periodic’ or ‘tied’ boundaries. As aconsequence, the model under examination does not allow for thetransmission of outgoing waves generated at the pile–soil inter-face, which remains trapped inside the model. A common proce-dure to check the amount of inaccuracy coming from this choiceis the one originally suggested by Zienckiewicz et al. [42] for bothlinear and non linear models. It consists of comparing the resultsof the complete model with those predicted by the free-fieldconditions at the location of the boundary. For the problem underexamination, Di Laora et al. [43] have shown that results comingfrom the assumption of tied boundaries are sufficiently accurate,provided that the width of the entire model is large enough.To this aim, the lateral boundary of the model was set at 400 d

from the pile axis. Vertical displacements are restrained along thelateral boundary of the mesh, while nodes at the base of themodel are restrained to both horizontal and vertical directions torepresent a rigid bedrock. The mesh employed in the analyses isshown in Fig. 5b. The vertical size of the elements is set to0.1 diameters for a width of 3 diameters close to layers interfaceand 0.5 diamaters elsewhere. On the other hand, the horizontal

Fig. 6. Acceleration ratio for different damping levels and types. Comparison between FE re

Fig. 7. Acceleration ratio in two-layer soil for different interface depth and

size is 1/8 pile diameters at pile–soil interface, thereby increasingwith radial distance up to 1.5 diameters at the free-field. Theanalyses performed in the parametric study were carried out inthe frequency domain, after extracting vibrational modes withfrequency up to 25 Hz, each having a pre-specified level of viscousdamping. In this way the drawbacks stemming from use ofcommon energy loss formulations such as Rayleigh dampingwere avoided. An FFT algorithm was employed to transferresponses from the frequency to the time domain and vice versa.

5. Harmonic response

Eq. (6) is compared with results from FE analysis in Fig. 6a.Despite its simplicity, graph formula (6) provides accurate valuesof pile–soil acceleration ratios. The advantage of the aboveexpression is twofold: (a) it offers an insight into the physicalinterpretation of the interaction phenomenon; (b) it condensatesinto a unique dimensionless parameter a number of physicalquantities such as pile diameter and Young’s modulus, soilstiffness and density as well as excitation frequency.

The ability of such a parameter to describe the filtering effectmechanism is confirmed in Fig. 6b, where results from Fan et al.[25] are compared to pile–soil acceleration ratios obtained by FEanalyses for different types and levels of damping. It is noted that

sults from present work and those by Fan et al. (1991). In all cases, L/d¼20, H/L¼1.5.

soil stiffness. (a) h1/d¼5, (b) h1/d¼10, (c) h1/d¼15 and (d) h1/d¼19.

R. Di Laora, L. de Sanctis / Soil Dynamics and Earthquake Engineering 46 (2013) 52–6358

although both soil and pile responses are known to be dependentof damping (primarily the level and secondarily the type), theacceleration ratio is not.

The above considerations are valid under homogeneous soilcondition. From a conceptual point of view a hypothetical stifferlayer would act as a partial fixity for the embedded portion of pile.It is hence expected that in two-layer soils pile–soil accelerationratio may deviate from Eq. (6), yet the entity of such perturbationshould vanish for smooth stiffness contrasts between soil layersand deep layers interfaces. Fig. 7 depicts FE results in terms ofpile–soil acceleration ratio for a two layer soil, by varying inter-face depth, pile–soil stiffness ratio and layer stiffness contrast. Itis noted that for deep interfaces (h1/dZ10) Eq. (6) may be stilladopted for two-layer soils, at least for practical applications.

6. Transient response

Transient response is investigated by using real accelerationtime-histories, selected from an Italian database [44] and theEuropean database of Ambraseys et al. [45]. Time histories andFourier spectra of the signals are reported in Fig. 8. Also shown inthis graph are frequency content indices like the mean frequencyfm as defined by Rathje et al. [46] and the predominant frequencyfp, defined as the structural natural frequency corresponding tothe maximum spectral acceleration.

Some results of the parametric study are illustrated in Fig. 9,where the spectral acceleration normalized by the peak rockacceleration is plotted as a function of the structural period fortwo different earthquake events, specifically ULCINJ (Montenegro,1979), characterized by low-frequency components, and NOCERAUMBRA (Umbria-Marche, 1997), rich in high frequencies. Thecomparison between the results coming from these two earth-quakes allowed to investigate the role of the frequency content.The filtering effect is remarkable for both these events when theshear wave velocity of the upper layer, Vs1, is equal to 50 m/s. Bycontrast, for Vs1¼100 m/s, the filtering effect is negligible forULCINJ, and is less pronounced (but still significant) for NOCERAUMBRA. This is due to the fact that high-frequency componentsare filtered out by the piles. The plots in Fig. 9 show that thegoverning parameters of the filtering-effect are the soil stiffnessand the frequency content of the input signal. Note that this is inagreement with the harmonic response shown in Figs. 4, 6 and 7.An interesting result is that illustrated in Fig. 10, where the ratio xof the spectral acceleration of the filtered motion, Sa,p, over that ofthe free-field Sa,s is plotted against the structural period. For agiven subsoil profile and any particular earthquake event it ispossible to recognize two critical points in (T:x) plane, T being thestructural period: (i) that corresponding to the minimum value ofthe spectral acceleration ratio; (ii) that pertaining to the struc-tural period after which the filtering effect becomes negligible.The first point has coordinates (Tmin, xmin), as reported in Fig. 10d,while the second point has coordinates (Tcrit, xcrit) and can beidentified with the point of maximum curvature of the spectralratio function. The ordinate of the point at which the structuralperiod equals the critical value is nearly coincident with unity, soas the coordinates of the second point can be convenientlyassumed to be (Tcrit, 1). The most interesting result is that forany subsoil the structural periods Tcrit and Tmin are practicallyunaffected by the earthquake event, as it will be discussed later.On the other hand xmin strongly depends on the frequencycontent of the input signal. The comparison between Fig. 10aand b also shows that the amount of the filtering effect isremarkably affected by the shear wave velocity of the upperlayer. By contrast, the layer impedance contrast, i.e. the hetero-geneity of the soil profile, has only a little effect on the ratio of the

spectral acceleration of the foundation over that of the free-field(Fig. 10c and d).

The spectral acceleration ratio at T¼0, x0, is a purely kinematicinteraction factor and is strongly dependent of the frequencycontent of the input signal, as for xmin. For any particular earth-quake event, it is convenient to define an average circularfrequency os as:

os ¼2pf mþ2pf p

2ð7Þ

The kinematic interaction factor, x0, is plotted in Fig. 11aagainst the parameter:

Gt1 ¼ 1þ0:15oslp

Vs1

� ��1

ð8Þ

Fig. 11b shows the relationship between the minimum accel-eration ratio, xmin, and the quantity

Gt2 ¼ 2:5Gt1�1:5 ð9Þ

Note the similarity of the parameter Gt1 in Eq. (8) with the Gfactor in Eq. (6). Gt1 may be therefore interpreted as a ‘transient’kinematic interaction factor, whose coefficients naturally arisefrom the choice in the definition of the frequency index os.

The results plotted in Fig. 11 indicate a clear trend for both x0

and xmin of increasing with the inverse of the frequency index os,as expected. Based on linear regressions, the parameters x0 andxmin can be calculated through the following expressions:

x0 ¼ 1:71Gt1�0:64 ð10Þ

xmin ¼ 0:91Gt2 ð11Þ

Despite the amount of scattering visible in Fig. 11, the accuracyof both the correlations is worthy of note, taking into account thatthe frequency content of any earthquake event has been synthe-sized by a unique parameter, the circular frequency os defined byEq. (7).

7. Reduction of design spectra

In the previous section it has been shown that transient resultsdo not indicate a dependence of Tmin and Tcrit on the frequencycontent of the input signal. This result could be actually antici-pated by simple dimensional considerations. Indeed, consideringthat such periods (which in dimensional terms represent a time)clearly depend on soil shear wave velocity, which already con-tains the dimensions of time, excitation frequency cannot affectTmin and Tcrit. It may be therefore convenient to condensate theresults of the finite element analyses in terms of mean spectralratios, as in Fig. 12. For both Vs1¼50 m/s and Vs1¼100 m/s themean spectral ratios have an upper bound, corresponding to thehomogeneous subsoil (Vs2/Vs1¼1). This can be physically moti-vated by the kinematical restraint exerted by the stiffer lowerlayer. The larger is the degree of restraint exerted by the lowerlayer, the greater is the resistance of the pile to follow thedeformations of the upper layer. On the other hand, the layerstiffness contrast has only a weak effect on the structural periodsTmin and Tcrit. Thus, Fig. 12 can be further synthesized, by referringonly to the upper bound represented by the homogeneoussubsoil. The results pertaining to the homogenous soil conditionare illustrated in Fig. 13, where the mean spectral ratios obtainedfor Vs¼75 m/s and 125 m/s have been also added for comparison.The mean spectral curve of the homogeneous subsoil is well-represented by two parabolic functions, intersecting at T¼Tcrit,and a horizontal plateau. The vertex of the first parabola hasabscissa T¼0, while the vertex of the second parabola hascoordinates (Tcrit, 1).

Fig. 8. Acceleration time-histories considered in the parametric analysis.

R. Di Laora, L. de Sanctis / Soil Dynamics and Earthquake Engineering 46 (2013) 52–63 59

Interestingly, it is easy to verify that transient parameters Gt1

and Gt2 work very well even by employing a mean frequencycontent. Specifically, x0,av and xmin,av satisfy the followingequations:

x0,av ¼Gt1

xmin,av ¼Gt2

)ð13Þ

with

os ¼os,av ¼ 10 rad=s ð14Þ

The characteristic periods Tmin and Tcrit may be taken asequal to:

Tmin ¼ 12d

Vsð15Þ

Fig. 9. Spectral acceleration normalized by the peak rock acceleration. In all cases, h1/d¼10 and Vs2/Vs1¼4.

Fig. 10. Pile-head spectral acceleration over that of the free-field. (a) h1/d¼10, Vs1¼50 m/s, Vs2/Vs1¼4, (b) h1/d¼10, Vs1¼100 m/s, Vs2/Vs1¼4, (c) h1/d¼15, Vs1¼50 m/s, Vs2/

Vs1¼2 and (d) h1/d¼15, Vs1¼50 m/s, Vs2/Vs1¼4.

Fig. 11. Linear regressions for x0 and xmin.

R. Di Laora, L. de Sanctis / Soil Dynamics and Earthquake Engineering 46 (2013) 52–6360

Fig. 12. Mean spectral ratios for the subsoils considered in the parametric study. (a) Vs1¼50 m/s and (b) Vs1¼100 m/s.

a

c

b

d

Fig. 13. Mean spectral ratios for homogeneous soil conditions. (a) Vs¼50 m/s, (b) Vs¼75 m/s, (c) Vs¼100 m/s and (d) Vs¼125 m/s.

R. Di Laora, L. de Sanctis / Soil Dynamics and Earthquake Engineering 46 (2013) 52–63 61

Tcrit ¼ 3:5Tmin ð16Þ

All the ingredients needed to evaluate the reduced designspectra are now available. It is suggested to adopt the reductionfactor for acceleration design spectra defined by the followingequations:

x Tð Þ ¼ x0�x0�xminð Þ

T2min

T2¼ x0� x0�xminð Þ T

Tmin

� �2; TrTmin

x Tð Þ ¼ 1� 1�xminð ÞTcrit�T

Tcrit�Tmin

� �2; TminrTrTcrit

xðTÞ ¼ 1 TZTcrit

8>>>><>>>>:

ð17a;b; cÞ

8. Conclusion

Inertial interaction analysis of the superstructure is usuallyperformed by imposing that the foundation input motion ismerely that of the free field. By contrast, the free-field signal isfiltered out by the piles, yet this potential in reducing seismicdemand is generally not exploited in engineering practice.

In the paper, emphasis has been first placed on the ideal caseof an infinitely-long pile embedded in a homogenous soil, to

elucidate the key parameters controlling the mechanism offiltering effect. A comprehensive parametric study has been thenperformed via Finite Element analyses for a subsoil consisting of atwo layer medium underlain by a rigid bedrock. Harmonicresponse has been investigated emphasizing the role of interfacedepth and layer impedance contrast. Transient response has beenanalyzed with reference to 9 real acceleration time-histories,selected in order to explore the effect of frequency content.

The results of this study are fully consistent with the experi-mental evidence about the filtering effect and can be summarizedas follows:

a.

The mechanism of filtering effect is primarily governed by soilstiffness layer, pile diameter and excitation frequency; a uniqueparameter has been found to govern the phenomenon in harmo-nic oscillations, and is represented by the ratio of a characteristicpile wavelength lp over the soil wavelength Vs/o;

b.

Compared to the homogeneous case, in two layer soils filteringeffect is more pronounced due to the kinematic restraintexerted by the stiffer layer. Nevertheless, should the interfacebe not shallow (h1/d45), the deviation from the homogeneoussoil case is negligible even for large impedance contrasts;

c.

With regard to transient response, the ratio of the spectralaccelerations has a ‘square root’ shape and is characterized by

R. Di Laora, L. de Sanctis / Soil Dynamics and Earthquake Engineering 46 (2013) 52–6362

a critical point, at which the filtering effect becomesnegligible;

d.

A reduction factor for acceleration design spectra hasbeen suggested to be adopted in the presence of piledfoundations.

The result of the present study can be accounted for in theperformance prediction analysis of new structures. In this case, itis suggested to apply the substructure method, including thecalculation step finalized to the evaluation of the FIM. The resultspresented herein can be of assistance to evaluate the opportunityof carrying out the computational cost associated to the time-domain pile–soil kinematic interaction analysis. When the per-formance of the superstructure is being predicted by modalresponse spectrum analysis, it is suggested to use the reduceddesigned spectrum, according to Eqs. (17a,b,c). An equivalentlinear elastic analysis is strongly recommended to assess the soilstiffness profile.

The reduced design spectrum equations could be also helpfulwithin strategies for seismic risk reduction at regional scale. Inthis respect, it is suggested to incorporate the filtered effectexerted by the piles within the seismic demand capacity, aPGA,conventionally adopted for the so called ‘first-level’ evaluation ofthe seismic vulnerability of the structure.

It is believed that further investigations are needed, forsubsoil conditions other than those considered in the parametricanalysis, to confirm the validity of the main conclusions ofthis work.

Finally, it is fair to mention that the result of this study cannotbe applied to liquefiable soils. In this case, indeed, the accelera-tion transmitted by the piles may be larger than that associated tothe free-field conditions.

Acknowledgements

This research has been developed under the auspices of theresearch Project ReLUIS 2009–2013, funded by the NationalEmergency Management Agency.

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