1-s2.0-S0267726111000297-main

Soil Dynamics and Earthquake Engineering 31 (2011) 891–905

Contents lists available at ScienceDirect

Soil Dynamics and Earthquake Engineering

0267-72

doi:10.1

n Corr

E-m

mylo@u

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

Transient kinematic pile bending in two-layer soil

Stefania Sica a, George Mylonakis b, Armando Lucio Simonelli a,n

a Dipartimento di Ingegneria, Universit �a degli Studi del Sannio, Piazza Roma 21, 82100 Benevento, Italyb Department of Civil Engineering, University of Patras, University Campus, Rio 26500, Greece

a r t i c l e i n f o

Article history:

Received 16 September 2010

Received in revised form

25 January 2011

Accepted 2 February 2011Available online 31 March 2011

Keywords:

Piles

Soil–structure interaction (SSI)

Kinematic interaction

Numerical modelling

61/$ - see front matter & 2011 Elsevier Ltd. A

016/j.soildyn.2011.02.001

esponding author.

ail addresses: [email protected] (S. Sica),

patras.gr (G. Mylonakis), alsimone@unisanni

a b s t r a c t

The dynamic response of piles to seismic loading is explored by means of an extensive parametric study

based on a properly calibrated Beam-on-Dynamic-Winkler-Foundation (BDWF) model. The investi-

gated problem consists of a single vertical cylindrical pile, modelled as an Euler–Bernoulli beam,

embedded in a subsoil consisting of two homogeneous viscoelastic layers of sharply different stiffness

resting on a rigid stratum. The system is subjected to vertically propagating seismic S waves, in the

form of a transient motion imposed on rock outcrop. Several accelerograms recorded in Italy are

employed as input motions in the numerical analyses. The paper highlights the severity of kinematic

pile bending in the vicinity of the interface separating the two soil layers. In addition to factors already

investigated such as layer stiffness contrast, relative soil–pile stiffness, interface depth and intensity of

ground excitation, the paper focuses on additional important factors, notably soil material damping,

stiffness of Winkler springs and frequency content of earthquake excitation. Existing predictive

equations for assessing kinematic pile bending at soil layer interfaces are revisited and new regression

analyses are performed. A synthesis of findings in terms of a set of simple equations is provided. The

use of these equations is discussed through examples.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

During the passage of seismic waves through soft deposits,embedded piles tend to deform in a different manner with respectto the surrounding soil. The difference between soil and pile motiondepends on several factors, notably soil layering, pile–soil stiffnesscontrast, excitation frequency and kinematic constraints at the pilehead and tip [1–4]. As curvatures are imposed to the pile body by thevibrating soil, bending and shearing will develop even in the absenceof a superstructure. The associated pile bending moments are,thereby, referred to as ‘‘kinematic’’, to be distinguished from thosegenerated by loads acting at the pile head due to the dynamicresponse of the superstructure (so-called ‘‘inertial’’ moments).Kinematic and inertial bending moments constitute complementaryaspects of a unique phenomenon known as soil–pile–structureinteraction (SPSI). Reviews of the subject have been published, amongothers, by Novak [5], Pender [6] and Gazetas and Mylonakis [7].

Evidence from case histories – as documented by Mizuno [8]and other Japanese researchers [9–12] – or from recent experi-mental investigations on physical pile models in centrifuge and 1g

earthquake simulators [13–18] have elucidated the important

ll rights reserved.

o.it (A.L. Simonelli).

role of kinematic interaction in seismic response of pile founda-tions. Kinematic bending is significant (as compared to its inertialcounterpart) particularly in correspondence to stiff pile caps andsoil layer interfaces. The latter may explain the concentration ofseismic demand at depths where inertial effects are negligible.

The accumulated evidence has generated significant interest inexploring theoretical and analytical aspects of the phenomenonand developing seismic regulations to incorporate it into designprocedures [19–21].

Following the early work by Margason and Halloway [22], theo-retical investigations of the problem began in the 1980s [1,23–27]and continued into the 1990s and beyond [2–4,28–31].

In 2005, a systematic research effort was initiated in Italyunder the auspicious of the ReLUIS project (University Network ofSeismic Engineering Laboratories), which has lead to a number ofpublications [32–43]. The main goal of the project was to produceengineering provisions to be incorporated into the new nationalseismic code [21], which is compulsory in Italy since July 2009.

1.1. Unresolved issues

At present, it appears that many aspects of kinematic pilebending are well understood, whilst others require furtherresearch and remain unresolved. First, most of the publishedresults concentrate on flexible piles (i.e., piles whose lengths aregreater than the so-called ‘‘active pile length’’ [44], embedded in

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Fixed head

h1

Vs1, ρ1 LEpI, ρpL

D1, ν1

VVs2, ρ2D , ν

S. Sica et al. / Soil Dynamics and Earthquake Engineering 31 (2011) 891–905892

two-layer soils with the interface placed deep beneath the sur-face [2–4,31]. Less research has been carried out on short pilesand/or interfaces located close to the pile head. For such systemsthe interplay between pile head and interface moments, as well asbetween inertial and kinematic effects, becomes significant and,thereby, a proper summation rule is essential for combining thecontributions of the individual response components.

Second, whereas a number of simplified procedures forestimating kinematic pile bending moments at the interface oftwo soil layers are available [24,4,3], there is a lack of simpleformulations for assessing corresponding moments at the pilehead regardless of layer thickness and pile length. Likewise,kinematic pile head moments have not been addressed in casesdeviating significantly from that of a homogeneous soil.

Third, the effect of the transient nature of input motion on thedevelopment of kinematic bending moments along the pilerequires further investigation. Available predictive equationsare either pure frequency-domain approaches [3], or mixedfrequency–time domain formulations based on a limited numberof accelerograms [4,35]. Key parameters such as number ofearthquake cycles and frequency content are currently accountedfor in an approximate way, or even neglected altogether.

Fourth, the effect of certain modelling assumptions involvingsoil damping, stiffness of Winkler springs and wave propagationon kinematic bending moments needs to be better quantified.

Fifth, the effect of material nonlinearity in the soil (and the pileitself) needs to be explored in a more systematic way, despiterecent progress [33,40,45–50].

Last but not least, the effect of material plasticity for both soiland pile occurring under stronger earthquakes needs to beclarified.

The work at hand focuses only on some of the aforementionedissues. Specifically it aims at reporting the results of an extensiveparametric investigation carried out on single piles in layered,viscoelastic soil accounting for different material properties,geometric factors and earthquake excitations. The analyses areperformed using an extension of the numerical tool developed byMylonakis et al. [29] and Mylonakis [62], which implements aBeam-on-Dynamic-Winkler-Foundation (BDWF) formulationencompassing finite-element-based springs and dashpots distrib-uted along the pile axis. Although simplified, this type of model-ling is known to provide satisfactory accuracy as compared torigorous numerical schemes, often allowing closed-form solutionsto be obtained [1,3,4,24,43]. With reference to kinematic pilemoments at soil layer interfaces, the paper presents: (i) a valida-tion of the BDWF procedure against other available solutions; (ii)a sensitivity study of pile bending as function of spring stiffnessadopted in Winkler models; (iii) a sensitivity analysis of theresults as function of soil damping employed in the analysis offree-field response; (iv) a comprehensive parametric study on atwo-layer soil profile as function of stiffness contrast between theupper and the lower layer, the depth of the upper layer and theearthquake waveform. On the basis of these results, new regres-sion analyses are carried out to correlate peak pile bendingmoments with the above parameters, with emphasis on fre-quency content of ground motion. Finally, a set of conclusionsand recommendations are produced for implementing the find-ings in routine engineering calculations.

2 2

h2

d

Fig. 1. System considered.

2. Analysis method and validation

The response to vertical S-wave excitation of a single verticalcylindrical solid pile embedded in layered soil is investigatedthrough a Beam-on-Dynamic-Winkler-foundation (BDWF) formula-tion, solved using the hybrid analytical–numerical algorithm of

Mylonakis et al. [29] and Mylonakis [62]. As the fundamentalaspects of the method are well-known, only a brief description isgiven here. Soil–pile interaction is modelled through a set ofcontinuously distributed springs and dashpots, the parameters ofwhich, k¼k(o) and c¼c(o), have been calibrated against the resultsof finite-element and boundary-element analyses. Such springs anddashpots connect the pile to the free-field soil; the wave-inducedmotion of the latter serves as the support excitation for the pile–soilsystem. Mylonakis et al. [29] and Mylonakis [62] reformulated andextended earlier frequency-domain solutions [2] to the time domainusing a Discrete Fourier Transform (DFT) approach as described byVeletsos and Ventura [51]. This can accommodate precisely thefrequency dependence of the spring and dashpot moduli, contrary tomethods that require frequency-independent parameters to obtainthe response directly in the time domain. The approach has beenvalidated in Mylonakis et al. [29], Mylonakis [3,62] and Nikolaouet al. [4,71] by comparing pile bending moments to those obtainedby a variety of continuum approaches solved by FEM and BEMprocedures.

2.1. Validation

Recently, the research groups working as part of the ItalianReLUIS project conducted a comparative study of predictionsprovided by different numerical tools on conceptual prototypesconsisting of single piles embedded in two-layer profiles underthe assumption of vertically propagating S waves (Fig. 1). Theparameter analyses were carried out by varying the dimension-less interface depth (h1/d) while keeping the overall soil depthconstant. The configuration adopted for the pile and the soil isshown in Fig. 1. The analyses were performed by adopting a suiteof Italian accelerograms (Table 1) selected from the Italiandatabase SISMA [52]; the recordings were scaled at a peakacceleration of 0.35 g, which is consistent with a zone of highseismicity according to the Italian seismic zonation of 2003 [53].For the purposes of this validation the accelerograms weredirectly applied at the base of the soil column, as done by theother ReLUIS research groups, without de-convolution or con-sideration for rock outcrop effects. In Table 1, the frequencycontent of the accelerograms adopted as input motion is quanti-fied through the predominant frequency, fp, corresponding to the

Table 1Ground motions employed in the analysis (all records are scaled to PGA¼0.35 g).

# Record label Station name Earthquake Date (d/m/yr) Magnitude(Mw)

Epicentraldistance (km)

PGA (g) fp (Hz) b fm (Hz) c Soil type

1 A-TMZ270 Tolmezzo-Diga Ambiesta Friuli 06/05/76 6.5 23 0.32 1.6 2.0 A

2 A-TMZ000 Tolmezzo-Diga Ambiesta Friuli 06/05/76 6.5 23 0.36 3.8 2.5 A

3 A-STU270 Sturno Campano Lucano 23/11/80 6.9 32 0.32 5.0 1.2 A

4 A-STU000 Sturno Campano Lucano 23/11/80 6.9 32 0.23 2.6 1.5 A

5 A-AAL018 Assisi-Stallone Umbria Marche 26/09/97 6 21 0.19 3.1 3.0 A

6 E-NCB090 Nocera Umbra-Biscontini Umbria Marche a 06/10/97 5.5 10 0.38 8.3 5.8 A

7 E.NCB000 Nocera Umbra-Biscontini Umbria Marche a 06/10/97 5.5 10 0.26 7.1 6.1 A

8 R-NCB090 Nocera Umbra-Biscontini Umbria Marche a 03/04/98 5.1 11 0.31 5.6 5.6 A

9 J-BCT000 Borgo-Cerreto Torre Umbria Marche a 14/10/97 5.6 12 0.34 10.0 6.0 A

10 J-BCT090 Borgo-Cerreto Torre Umbria Marche a 14/10/97 5.6 12 0.33 6.3 4.8 A

11 E-AAL108 Assisi-Stallone Umbria Marche a 06/10/97 5.5 20 0.19 4.5 4.1 A

12 B-BCT000 Borgo-Cerreto Torre Umbria Marche 26/09/97 5.7 23 0.18 12.5 6.5 A

13 B-BCT090 Borgo-Cerreto Torre Umbria Marche 26/09/97 5.7 23 0.19 8.3 5.1 A

14 TRT000 Tarcento Friuli a 11/09/76 5.3 8 0.21 10.0 4.7 A

15 C-NCB000 Nocera Umbra-Biscontini Umbria Marche a 03/10/97 5.3 8 0.19 25.0 7.8 A

16 C-NCB090 Nocera Umbra-Biscontini Umbria Marche a 03/10/97 5.3 8 0.27 8.3 6.5 A

17 R-NC2090 Nocera Umbra 2 Umbria Marche a 03/04/98 5.1 10 0.31 5.6 5.4 A

18 R-NC2000 Nocera Umbra 2 Umbria Marche a 03/04/98 5.1 10 0.38 6.3 6.6 A

a Aftershock.b Predominant frequency from response spectra.c Average frequency according to Rathje et al. [54].

2.0

1.5

1.0

ε p x

10-3

ε p x

10-3

0.5

0.02.0

1.5

1.0

0.5

0.00 5 10 15 20 25 30 35

h1/d

BEM (Cairo & Dente, [32])FEM (Di Laora, [56])BDWF (Cairo et al., [33])BDWF (Mylonakis et al., [29])BDWF (Dezi et al., [42])

Fig. 2. Peak kinematic pile bending strain at interface level as function of depth

for A-TMZ270 (top) and A-TMZ000 (bottom) Tolmezzo recordings; L¼20 m,

d¼0.60 m, n1¼n2¼0.4, r1¼r2¼1.9 Mg/m3, D1¼D2¼10%, Vs1¼100 m/s,

Vs2¼400 m/s, Ep¼2.5�107 kPa and rp¼2.5 Mg/m3.

S. Sica et al. / Soil Dynamics and Earthquake Engineering 31 (2011) 891–905 893

maximum value of the acceleration response spectrum for 5%damping, and through the mean frequency, fm, as defined in [54]on the basis of frequency content.

In Fig. 2 the results obtained by the BDWF method employedin this study are represented in terms of maximum pile bendingstrain:

ep ¼Mmax

EpI

d

2ð1Þ

Mmax being the maximum kinematic bending moment at theinterface; EpI the bending stiffness of the pile and d/2 the distancefrom the pile centerline to the outer fibre of the pile cross section.ep is plotted as function of the dimensionless embedment factorh1/d. The use of bending strain over other normalisation schemesis desirable in such problems as [3]: (i) it is dimensionless; (ii) it isdirectly measurable; (iii) it can be used to quantify damage, asyield amplitudes do not vary significantly among common struc-tural materials (being of the order of 10�3); (iv) it can generatebi-dimensionless transfer functions, when normalised by perti-nent strain parameters such as soil shear strain or site amplifica-tion functions.

The results obtained with the selected approach are compared tothe kinematic bending moments provided by two different BDWFformulations and two continuum solutions employed in the ReLUISproject. The first BDWF approach has been developed by Deziet al. [42] and was validated by detailed 3D finite-element ana-lyses [43] using the ABAQUS platform [70]; other two approaches –one continuum and another BDWF – were developed by Cairo andDente [32] and Cairo et al. [33]. The continuum approach [32] isimplemented through a frequency-domain BEM technique thatmakes use of the soil stiffness matrices derived by Kausel andRoesset [55] to simulate the response of a horizontally layereddeposit. The last approach [56] is implemented by means of thecommercial finite-element code ANSYS [57] using very fine dis-cretization. From Fig. 2 a satisfactory comparison is noted among theresults provided by the aforementioned methods. Particularly satis-factory is the comparison between the BDWF approaches ofMylonakis et al. [29], Cairo et al. [33], Dezi et al. [42] and the twocontinuum approaches (FEM or BEM). Some discrepancies appearingin Fig. 2a in correspondence to the ratios h1/d¼25 and 32 may beattributed to the selected values of Winkler springs, differentdiscretizations of the pile, different damping schemes (e.g., Rayleighversus linearly hysteretic), as well as different boundary conditionsat the pile tip. Some of these factors are discussed in the ensuing.

2.2. Sensitivity of BDWF analyses on selection of Winkler springs

In the BDWF approach the stiffness k of the springs connectingthe pile to the soil is generally defined as

k¼ d Es ð2Þ


where Es is Young’s modulus of the soil and d is a dimensionlesscoefficient. Following the early work of Novak [58], differentformulations have been proposed over the years for evaluatingthe d factor.

For instance, using the results from the finite-element formu-lation of Blaney et al. [59], Roesset [60] recommended the value:

d¼ 1:2 ð3Þ

regardless of other problem parameters. This simple proposal waslater adopted by several investigators, including the authors,

εp x 10-3

0 0.5 1 1.5 20

60

2.78 (Mylonakis, [ ])35 3

10

15

z/d

20

interface25

30

35

δ = 1.20 (Roesset, [60])2.30 (Kavvadas & Gazetas, [2])2.78 (Mylonakis, [3])

Fig. 3. Effect of Winkler spring factor d on kinematic bending moments computed

along the pile for the system of Fig. 1, subjected to A-TMZ270 input motion

(case S1-6 in Table 2).

1.4

Eq. 6 1.2

M (δ

) / M

(1.2

)

1

0.8

Roe

sset

, [60

]

0.8 1.2 1.6 2

(Ep/Es1 = 500)

Fig. 4. Effect of Winkler spring factor d on kinematic bending moments at a soil layer in

[42,61,62]. Improvements over the original formula have beenpresented, among others, by Dobry et al. [63] and Syngros [64].

With reference to kinematic pile bending in a two-layerdeposit, Kavvadas and Gazetas [2] related d to pile and soilproperties according to the equation:

d¼3

1�n2

Ep

Es1

� ��1=8 L

d

� �1=8 h1

h2

� �1=12 G2

G1

� ��1=30

ð4Þ

where n is the Poisson ratio of the soil material (common in bothlayers), L the pile length, h1 and h2 the thickness of the upper andlower layers, respectively, Ep the pile Young’s modulus, Es1 theYoung modulus of the upper soil layers and G1, G2 the shearmoduli of the upper and lower layers, respectively. Note that for ahollow cylindrical pile, Young’s modulus should be taken asEp[1�(1�2t/d)4], t being the tube thickness.

Mylonakis [3] simplified the above equation for the case oflong piles in two-layer soil as follows:

dffi6Ep

Es1

� ��1=8

ð5Þ

The effect of Winkler parameter d on kinematic pile bendingmoments was investigated by means of an extensive parametricanalysis. Fig. 3 presents results referring to accelerogramA-TMZ270 recorded during the 1976 Friuli earthquake. It can beobserved that d has a negligible effect on kinematic bendingeverywhere, except for the vicinity of the layer interface. In thatregion an increase in d from 1.2 to 2.78 increases kinematicbending by 20% or so.

A wider set of results referring to the same pile–soil config-uration is presented in Fig. 4. There, kinematic bending momentscomputed for different d’s are normalised with the resultsobtained for d¼1.2, which was selected as a reference case.Naturally kinematic bending moment increases with increasingd, the increase being of the aforementioned order for dapproaching 2.8.

Based on these results the following approximate relation wasderived:

MðdÞMð1:2Þ

¼ ð0:97þ0:17dÞEp

Es1

� ��1=40

ð6Þ

This equation is in agreement with the solution of Dobry andO’Rourke [24] as to the dependence of pile bending on pile–soilstiffness contrast. Note that for soft piles (Ep/Es1r500) the effect

A-AAL018A-STU000A-STU270ATMZ000ATMZ270ATMZ270BBCT000B-BCT090C-NCB000C-NCB090E-AAL108ENCB000ENCB090J-BCT000J-BCT000J-BCT090R-NC2000R-NC2090M

ylon

akis

, [3]

R-NCB090TRT000K

avva

das

& G

azet

as, [

2]

TRT000

2.4 2.8 3.2δ

terface for pile–soil configuration S1-6 subjected to the suite of records in Table 1.


of d becomes more significant. This is evident in Fig. 5, whereresults are presented for the geometry studied in the previousfigures, subjected to A-STU000 and A-TMZ270 records.

1.4

Eq. 6δ from Mylonakis, [3]

1.2

Eq. 6

δ from Kavvadas & Gazetas, [2]

M (δ

) /M

(1.2

)

1

δ from Roesset, [60]

A-TMZ270

A-STU000

0.80 100 200 300 400 500 600 700 800 900 1000 1100 1200

Ep/Es1

Fig. 5. Effect of Winkler stiffness factor d and Ep/Es1 ratio on interface kinematic

bending moments, for pile–soil configuration S1-6 subjected to A-TMZ270 and

A-STU000 input motions.

Table 2Problem parameters considered. In all cases n1¼n2¼0.4, L/d¼33, Vrock¼1000 m/s.

Scheme h1 (m) h2 (m) Vs1 (m/s) Vs2/Vs1 ID parameter case

S1 15 15

50

2 S1-1

3 S1-2

4 S1-3

100

2 S1-4

3 S1-5

4 S1-6

150

2 S1-7

2.7 S1-8

4 S1-9

S2 15 30

100

2 S2-10

3 S2-11

4 S2-12

150

2 S2-13

2.7 S2-14

4 S2-15

S3 15 6

100

2 S3-16

3 S3-17

4 S3-18

150

2 S3-19

2.7 S3-20

4 S3-21

S4 5 25 100

2 S4-22

3 S4-23

4 S4-24

S5 10 20 100

2 S5-25

3 S5-26

4 S5-27

S6 19 11 100

2 S6-28

3 S6-29

4 S6-30

3. Parametric investigation

An extensive parameter study has been performed by the afore-mentioned BDWF approach for both free- and fixed-head piles in atwo-layer subsoil subjected to vertically propagating seismic waves.The soil profile consists of a soft surface soil layer of thickness h1 andshear wave velocity Vs1, followed by a stiffer stratum of thickness h2

and shear wave velocity Vs2 (Fig. 1). Elastic bedrock conditions havebeen assumed at the bottom of the lower layer. All cases consideredare listed in Table 2. For each geometry, shear wave velocities of theupper and lower layers were selected in such a way that subsoilprofiles correspond to class C or D of the EC8 classification on thebasis of the equivalent shear wave velocity Vs,30 in the upper 30 m ofthe subsoil. Three different values of soil damping were employed:2%, 10% and 20%, which cover several cases of practical interest.

Eighteen runs were carried out for each parametric case basedon the input motions of Table 1. The accelerograms were chosen insuch a way that their original peak ground acceleration is as closeas possible to the reference maximum peak acceleration on soiltype A of a seismic zone according to the Italian seismic zonation of2003 [53]. For comparison purposes, the selected accelerogramshave been scaled in amplitude to peak acceleration of 0.35 g,linearly de-convoluted to bedrock level and then propagatedupward in the soil to provide the excitation motion of theembedded pile. In Table 2 the ratio f1/fp between the fundamentalfrequency of the subsoil f1 and the predominant frequency of inputmotion fp is provided. In all analyses the following parameters

Vs, 30 (m/s) Soil type (EC8) Ep/Es1 D f1 (Hz) f1/fp, range

66 D

1880

0.7 0.03–0.42

75 D 0.8 0.03–0.49

80 D 0.8 0.03–0.50

133 D

470

1.3 0.05–0.86

150 D 1.5 0.06–0.99

160 D 1.6 0.06–1.03

200 C

209

2.0 0.08–1.28

218 C 2.2 0.09–1.42

240 C 2.4 0.10–1.54

133 D

470

1.1 0.04–0.67

150 D 1.3 0.05–0.86

160 D 1.5 0.06–0.97

200 C

209

2% 1.6 0.06–1.00

218 C 10% 1.9 0.08–1.22

240 C 20% 2.3 0.09–1.47

160 D

470

1.5 0.06–0.99

169 D 1.6 0.07–1.05

174 D 1.7 0.07–1.08

235 C

209

2.3 0.09–1.47

245 C 2.4 0.10–1.53

255 C 2.5 0.10–1.58

171 D

470

1.6 0.06–1.03

225 C 2.5 0.10–1.57

267 C 3.2 0.13–2.05

150 D 1.6 0.06–1.02

180 C/D 2.0 0.08–1.28

200 C 2.3 0.09–1.45

122 D 1.2 0.05–0.74

132 D 1.3 0.05–0.80

138 D 1.3 0.05–0.84

0

M (kN*m)

0

VS1/ VS2 = 1/2

5

10

z (m

)

8 ΦΦ

16

(0.

6 %

A)

12 Φ

24

(1.

9 %

A)

12 Φ

30

(3.

0 %

A)

15

20 ε p =

0.8

%o

ε p =

1.1

%o

ε p =

0.5

%o

0

VS1/VS2 = 1/3

5

10

z (m

)

15

200

VS1/VS2 = 1/4

5

10

z (m

)

15

20

200 400 600 800 1000

A-AAL018 A-STU000

A-STU270 ATMZ000

ATMZ270 B-BCT000

B-BCT090 C-NCB000

C-NCB090 E-AAL108

E-NCB000 E-NCB090

J-BCT000 J-BCT090

R-NC2000 R-NC2090

R-NCB090 TRT000

Fig. 6. Effect of shear wave velocity contrast on kinematic bending moments for pile–soil configuration S1. From top to bottom: parameter cases S1-4, S1-5 and S1-6

(subsoil D) of Table 2. Grey bands define yield resistance of pile cross sections for different amounts of longitudinal reinforcement.


were kept constant: pile length (L¼20 m), pile diameter(d¼0.60 m), soil Poisson’s ratio (n1¼n2¼0.4), soil mass density(rs¼1.9 Mg/m3), pile Young’s modulus Ep¼2.5�107 kPa and piledensity rp¼2.5 Mg/m3. The shear wave velocity of the elasticbedrock, Vrock, was taken equal to 1000 m/s.

3.1. Effect of geometry and soil impedance contrast

Typical results from the parameter study are shown in Figs. 6–8,where envelopes of maximum kinematic bending moments alongthe pile are plotted for each of the 18 signals. In all figures, three

zones shown in grey colour are indicated, corresponding to therange of yield moments for typical reinforcements of the concretepile cross section (8F16, 24F12 and 12F30) and variable normalloads at the pile head. M–N interaction diagrams were computedassuming concrete class C20/25 with Rck¼25 N/mm2 and fck¼20 N/mm2 and steel rebars with fyk¼375 N/mm2 and ftk¼450 N/mm2,corresponding to the Italian steel class FeB38K. For each reinforce-ment configuration, the lower limit of the grey zone represents thecross sectional yielding moment corresponding to zero normal loadwhile the higher one to a typical normal load N¼1200 kN [normal-ised axial load nd¼N/(fckA) between 0 and 0.32].

0

M (kN*m)

0

VS1/ VS2 = 1/2

5

10

z (m

)

8 ΦΦ

16

(0.

6 %

A)

12 Φ

24

(1.

9 %

A)

12 Φ

30

(3.

0 %

A)

15

20 ε p =

0.8

%o

ε p =

1.1

%o

ε p =

0.5

%o

0

VS1/VS2 = 1/3

5

10

z (m

)

15

200

VS1/VS2 = 1/4

5

10

z (m

)

15

20

200 400 600 800 1000

A-AAL018 A-STU000

A-STU270 ATMZ000

ATMZ270 B-BCT000

B-BCT090 C-NCB000

C-NCB090 E-AAL108

E-NCB000 E-NCB090

J-BCT000 J-BCT090

R-NC2000 R-NC2090

R-NCB090 TRT000

Fig. 7. Effect of shear wave velocity contrast on kinematic bending moments for pile–soil configuration S1. From top to bottom: parameter cases S1-7, S1-8 and S1-9

(subsoil C) of Table 2. Grey bands define yield resistance of pile cross sections for different amounts of longitudinal reinforcement.


The following observations may be drawn from these figures:

�
Interface bending moments increase dramatically when theshear wave velocity contrast between the upper and lowerlayers Vs1/Vs2 decreases from 1/2 to 1/4. This occurs for bothsubsoil types D (Fig. 6) and C (Fig. 7), in agreement with earlierresearch. In the latter case, however, peak moments at inter-face and pile head are much smaller than those computed forsubsoil type D. � Maximum bending moment develops close to the interface
when the active pile length la is smaller than the thickness h1

of the first layer. This length can be determined from thefollowing empirical formula as

‘a ¼ wdEp

Es1

� �0:25

ð7Þ

where w is a dimensionless constant varying between 1.75 and2.5 [2–4,29,44,62,64]. Conversely, when the active pile length‘a is equal or smaller than h1, the kinematic bending momentat the pile head may exceed that of the interface (Fig. 8).
� For subsoil profiles corresponding to class D (Fig. 6) the
computed kinematic bending moments may be well above

0

M (kN*m)

0

VS1/ VS2 = 1/2

5

10

z (m

)

8 ΦΦ

16

(0.

6 %

A)

12 Φ

24

(1.

9 %

A)

12 Φ

30

(3.

0 %

A)

15

20 ε p =

0.8

%o

ε p =

1.1

%o

ε p =

0.5

%o

0

VS1/VS2 = 1/3

5

10

z (m

)

15

200

VS1/VS2 = 1/4

5

10

z (m

)

15

20

200 400 600 800 1000

A-AAL018 A-STU000

A-STU270 ATMZ000

ATMZ270 B-BCT000

B-BCT090 C-NCB000

C-NCB090 E-AAL108

E-NCB000 E-NCB090

J-BCT000 J-BCT090

R-NC2000 R-NC2090

R-NCB090 TRT000

Fig. 8. Effect of shear wave velocity contrast on kinematic bending moments for pile–soil configuration S4. From top to bottom: parameter cases S4-22 (subsoil D), S4-23

and S4-24 (subsoil C) in Table 2. Grey bands define yield resistance of pile cross sections for different amounts of longitudinal reinforcement.


the yielding moments for typical concrete reinforcements andnormal loads acting along the pile. As the analyses have beenperformed assuming linear elastic behaviour for the materials,the computed moments can be assumed valid until the yieldlimit is reached.
� For subsoil profiles corresponding to soil class C and shear wave
velocity contrasts Vs2/Vs142, kinematic bending moments mayalso be considerable, especially for low levels of pile reinforce-ment (Fig. 7).

3.2. Effect of frequency content of input motion

For a better interpretation of the analytical results, the ratiof1/finput between the fundamental natural frequency of the subsoil,f1, and the predominant frequency of input motion, finput, has beenestablished for all cases. finput has been determined on the basis ofboth response spectrum predominant frequency, fp, and averagefrequency, fm, as described in [54] (Table 1). In Fig. 9 resultspertaining to case S1-6 (Table 2) are shown. It is evident that the

2

finput = fm

1.5 finput = fp

1f 1

/ f in

put

possible resonance

0.5no resonance

0A

TMZ0

00

ATM

Z270

TRT0

000

A-A

AL0

18

A-S

TU00

0

A-S

TU27

0

B-B

CTT

000

B-B

CT0

90

C-N

CB

000

C-N

CB

090

E-A

AL1

08

E-N

CB

000

E-N

CB

090

J-B

CT0

00

J-B

CT0

90

R-N

C20

00

R-N

C20

90

R-N

CB

090

Input motion

Fig. 9. Ratio of natural frequency of soil profile, f1, to the predominant frequency of input motion, finput, for case S1-6 (Table 2) and the signals of Table 1.

2

1.5 Eq. 8

1

M (D

) / M

(10%

)

0.5

00 5 10 15 20

D (%)

Fig. 10. Effect of soil material damping on kinematic pile bending moments.

Results refer to case S1-6 and the input motions of Table 1.


accelerograms that induce the highest kinematic moments arecharacterised by values of f1/fp or f1/fm close to unity, a trendobviously related to the development of resonant phenomena inthe soil. This is in accordance with the findings in [2,4] byfrequency–time domain analyses: maximum effects of kinematicbending in piles occur when the frequency of ground motion isnear the fundamental period of the subsoil (f1/finputE1).Conversely, all the accelerograms that induce low kinematicmoments in the pile have f1/finput below 0.5.

These results suggest that a threshold value of ratios f1/finput

can be defined beyond which kinematic effects are important(Fig. 9). This finding may provide a practical way for selectingrecords from a database of seismic events for a particular region:if acceleration time-histories (in addition to peak ground accel-erations) are available as part of the design process, the potentialof developing significant kinematic pile bending can be estab-lished on the basis of the following criterion:

-

1400
if ratio f1/finput is lower than the threshold value, kinematicbending will typically be minor and only inertial interactioncan be accounted for in design;
1200 Mresonance
-
1000

800

600Mstatic

M (k

Nm

)

400

200f ff1 fc0

0 1 2 3 4 5 6 7finput (Hz)

Fig. 11. Relation between transient and steady-state interface kinematic bending

moments for configuration S1-6 and the input motions of Table 1 (finput¼ fm). Note

the critical frequency fc beyond which maximum moment is smaller than static.

if ratio f1/finput is higher than the threshold value, the potentialfor kinematic bending will be high and kinematic interactionshould be considered, in addition to inertial interaction.

3.3. Effect of soil damping

The effect of soil damping on the response is examined inFig. 10, which provides a comparison among the kinematicbending moments computed at layer interface for different levelsof material damping in the soil (2%, 10% and 20%). The reportedresults pertain to parameter case S1-6 (Table 2). It is observedthat soil damping can significantly affect the magnitude ofkinematic bending moments, as it controls free-field responseand it should be carefully assigned when linear elastic analysesare carried out. Based on these results, the following regressionformula was derived:

MðDÞ

Mð10%Þ¼

5

9D�1=4 ð8Þ

which is reminiscent of corresponding expressions in seismicregulations for structural problems [19,20] and can be used toquantify the effect of damping in preliminary design calculations.Naturally, kinematic response tends to drop with increasing

M (t)max

M (t)max

η =Mresonance

Mresonance

maximum possible moment

Φ1 = Mstatic

maximum transient moment for a given input motion

M(t)max

moment for ω = 0

Mstatic

10 10 1 finput0 10

f1static resonance

Fig. 12. Definition of response factors Z and F1 for a given pile–soil configuration.

1.0

0.8

0.6

η

0.4mean + 1σmean0.2mean-1σ

mean + 1σmeanmean-1σ

0.0

3.0resonance range

finput−1.3

Φ1 = 1.942.5f1

finput−1.5

η = 0.68f1

2.0

1.5Φ1

1.0

0.5

0.00 1 2 3 4 5 6 7 8 9 10 11 12

finput / f1

Fig. 13. Factors Z (top graph) and F1 (bottom graph) versus frequency ratio (finput/f1)

for different pile–soil configurations and input motions. Only cases corresponding to

long piles are shown.


material damping, the decrease being of the order of 20% for anincrease from 5% to 10%.

3.4. Frequency versus time domain bending

In Fig. 11 interface kinematic bending moments determined inthe frequency domain are compared to corresponding momentscomputed in the time domain for each selected accelerogram,referring to parameter case S1-6 (Table 2). Time-domainmoments naturally lie below the value associated with resonancecondition (Mresonance). Interestingly, for most signals time-domainmoments are lower than the static value Mstatic obtained for zero-frequency excitation. This occurs because the input accelerogr-ams are characterised by much higher dominant frequencies(4–8 Hz—Table 1), than the fundamental frequency of the subsoilat hand (f1¼1.6 Hz). For this frequency range, de-amplification isobserved and kinematic effects consequently diminish. Anenlarged version of Fig. 11 is schematically represented inFig. 12 to define the following factors correlating frequency-and time-domain results:

Z¼ MðtÞmax

Mresonanceð9Þ

F1 ¼MðtÞmax

Mstaticð10Þ

which were first employed in Nikolaou et al. [4,71] and Mylona-kis [3], respectively. In these equations, parameter Z can beinterpreted as a ‘‘de-amplification’’ factor, bringing the resonancebending moment, Mresonance, to match the peak time-domainmoment, M(t)max. [As an example, spectral amplification of2.5 in conventional design spectra versus a peak resonant ampli-fication of 1/(2D)¼10 for 5% damping, is equivalent to an Z factorof 2.5/10¼0.25 for a simple oscillator.] On the other hand,parameter F1 may be interpreted either as an amplificationor as a de-amplification factor to bring the bending momentcomputed for static conditions, Mstatic, to match the peak time-domain moment, M(t)max. In the ensuing and except if specificallyotherwise indicated, factors Z and F1 in Eqs. (9) and (10) aredetermined from a numerical BDWF analysis—not from approx-imate expressions such as those provided in the originalpublications.

In Fig. 13, the mean values of a numerical regression analysisperformed for Z and F1 are displayed with corresponding upper

(mean+s) and lower (mean�s) estimates, s being the standarddeviation.

With reference to the mean value, the following simpleregression relations were derived:

Z¼ 0:68finput

f1

� ��1:5

ð11Þ

F1 ¼ 1:94finput

f1

� ��1:3

ð12Þ

which are valid for long piles and finput/f1 ratios greater thanapproximately 1.5.

Eqs. (11) and (12) can be used to quantify the effect oftransient nature of input motion on kinematic response of piles.Naturally, both parameters tend to drop with increasing finput/f1

that is for conditions far from resonance. For finput/f1E1 theresults exhibit significant dispersion. In that frequency range,response is known to depend on number of excitation cycles [4],an effect which is not captured by Eqs. (11) and (12). For thisreason, dashed lines have been employed in Fig. 13 for 0.5ofinput/f1o1.5.

4. Simple formulas for transient pile bending at a layerinterface

Mylonakis [3] developed a simple formulation for predictingkinematic bending moments at a layer interface under low-frequency excitation (o-0). The kinematic bending momentmay be derived from a strain transmissibility parameter, ep/g1,which is simply the ratio between peak pile bending strain, ep, andfree-field soil shear strain at the interface, g1. The corresponding

3.0

2.5

2.0

1.5Φ2

Φ2

1.0

0.5

0.03.0

2.5Vs1/Vs2 = 1/2

Vs1/Vs2 = 1/4 ÷ 1/3

2.0

1.5

1.0

0.5

0.00 1 2 3 4 5 6 7 8 9 10 11 12

finput / f1

Fig. 14. Factor F2 versus frequency ratio (finput/f1) for pile–soil configurations S1

and S5. Only cases corresponding to long pile are shown; Vs1/Vs2¼1/4–1/3 (top

graph) and Vs1/Vs2¼1/2 (bottom graph).


analytical solution is

ep

g1

� �o ¼ 0

¼1

2c4ðc2�cþ1Þ

h1

d

� ��1

3k1

Ep

� �1=4 h1

d

� ��1

" #cðc�1Þ�1

( )

ð13Þ

where c¼(G2/G1)¼ is the layer stiffness contrast, (h1/d) the embed-ment ratio, k1 the Winkler spring modulus associated with the upperlayer. As Eq. (13) has been obtained for static conditions, it does notaccount for the transient, dynamic nature of the phenomenon. Toaddress this limitation, Mylonakis [3] introduced a correction functionF of the form:

F¼ep

g1

� �dyn

�ep

g1

� �o ¼ 0

ð14Þ

which accounts for the effect of frequency on kinematic pile bending.It should be noticed that, although conceptually related, F in Eq. (14)is not identical to F1 in Eq. (10). Indeed, whereas Eq. (10) relatesstatic to transient response accounting for all relevant dynamicphenomena, Eq. (14) refers exclusively to SSI effects. For relativelysoft piles and low-frequency input (oinput d/Vs1o0.1), reference [3]recommends that parameter F lies in the range 1–1.25.

To investigate frequency-to-time domain response relations,Maiorano et al. [35] adopted the kinematic bending momentM(t)max computed in time domain by the FEM code VERSAT-P3D [65] and the shear strain at the interface (g1)dyn provided by aone-dimensional EERA analysis [66]. For this scope theyre-arranged Mylonakis [3] solution in the form:

F2 ¼MðtÞmax

2EpI

d

ep

g1

� �o ¼ 0

ðg1Þdyn

ð15Þ

for which symbol F2 is employed in this article to avoid confusionwith the definitions provided earlier. Contrary to factor F1 inEq. (10) F2 is associated with: (i) the peak dynamic strain (g1)dyn,as computed from a 1-D wave propagation analysis [67]; (ii) theapproximate solution for (ep/g1) given by Eq. (13). Accordingly, F2

relates the exact transient moment with an approximate momentthat is neither purely static nor purely dynamic, referring partiallyto dynamic and SSI effects.

By linear regression analysis, Maiorano et al. [35] found thatparameter F2 is approximately 1.30 for conditions far fromresonance and 1.40 near resonance.

Using nonlinear regression analysis based on the numericaldata obtained in this study, the parameter F2 was determined tobe approximately 1.20 for layer stiffness contrast Vs1/Vs2 in therange 1/4–1/3 (Fig. 14). This value is close to the genericrecommendation by Mylonakis [3]. Higher values of 1.3 onaverage were obtained for lower stiffness contrast Vs1/Vs2¼1/2(Fig. 14). In this case, however, the kinematic bending moment atthe layer interface is not expected to be important. It is note-worthy that F2 is practically independent of (finput/f1). This isunderstood given that both M(t)max and (g1)dyn, in Eq. (15),encompass the frequency dependence of the free-field response,which, thereby, cancels out from the ratio.

As an alternative to the mechanistic model by Mylonakis [3],Nikolaou et al. [4,71] introduced the following empirical equationto compute the pile kinematic bending moment at the interface inresonant steady-state conditions:

Mresonance ¼ 0:042tcd3 L

d

� �0:30 Ep

E1

� �0:65 Vs2

Vs1

� �0:50

ð16Þ

where tc is a characteristic shear stress at the interface:

tc ¼ asr1H1 ð17Þ

in which as is the free field acceleration at soil surface. Theauthors introduced a parameter Z (Eq. (9)), to correlate the

transient kinematic bending moment with the value obtainedfor resonant conditions, from Eq. (16). A theoretical weakness ofthe above formula is that the predicted moment tends to increasewithout bound for very slender piles, large pile–soil stiffnesscontrasts and large velocity contrasts between the two soil layers,a trend that has not been observed.

In the same spirit as before, Maiorano et al. [35] re-arrangedthe Nikolaou et al. [4,71] equation in the form:

b¼MðtÞmax

G1b ðg1Þdyn

ð18Þ

where parameter b is given by

b¼ d3 L

d

� �0:3 Ep

Es1

� �0:65 Vs2

Vs1

� �0:5

ð19Þ

In this approach, b can be interpreted as an ‘‘average dynamiccoefficient’’ that links the maximum transient bending moment tothe transient peak soil shear strain at the interface, (g1)dyn. Maioranoet al. [35] report b to be about 0.07 on average. Recall thatparameter b is not equivalent to parameter Z in Eq. (9), for thedenominator in Eq. (18) is not an exact peak steady-state moment.

In the present work the procedure followed by Maioranoet al. [35] was employed to estimate the parameters F2 and bby adopting the M(t)max provided by the BDWF approach ofMylonakis et al. [29]. Results are shown in Fig. 15. It was foundthat F2 is equal to approximately 1.15 whereas b is around 0.053.Both these estimates are lower than those reported by Maioranoet al. [35]. The reasons for this discrepancy are worth investigating,yet lie beyond the scope of this paper.

2.5

2.0 mean+1σmean

1.5mean-1σ

1.0(Mm

ax/a

) x 1

02(M

max

/bG

1) x

103

0.5

0.0

1.2

1.0

0.8

0.6

0.4

0.2

0.00 0.005 0.01 0.015 0.02 0.025

(γ1) dyn

Mmax/b = 0.053 (γ1)dyn ± 2.8×10-5

Mmax / a = 1.15 (γ1)dyn ±5.1×10-4

Fig. 15. Graphical representation of peak kinematic bending moments using two

different normalisation schemes (modified from [35]).


5. Synthesis: new proposed approaches

As previously illustrated, different simple approaches may befollowed for determining the transient kinematic pile bending at theinterface of two soil layers. Three such approaches are suggested inthis paper. One is purely static and, accordingly, does not requiredynamic analysis; the other two are dynamic and, thereby, are moredemanding from a computational viewpoint.

1)
Static approach based on F1
This is the simplest procedure and can be implemented with-out carrying out a dynamic analysis. To this end, the procedureby Mylonakis [3] is reformulated as

MðtÞmax ¼2EpI

d

ep

g1

� �o ¼ 0

ðg1Þo ¼ 0F1 ð20Þ

where (ep/g1)o¼0 is provided by Eq. (13) and F1 is given byEq. (12) and Fig. 13; (g1)o¼0 is the static shear strain at theinterface, which can be easily established by the followingequation:

ðg1Þo ¼ 0 ¼a h1

V2s1

ð21Þ

where a¼as¼ar is a uniform pseudostatic seismic acceleration inthe profile. Note that Eq. (21) is equivalent to the familiar Seed andIdriss [68] procedure with the depth factor rd taken equal to 1.This approach is attractive for engineering applications as it doesnot require a free-field site response analysis. A disadvantage isthat parameter F1 is sensitive to variations of (finput/f1) ratio, whichis often not known with sufficient accuracy in practice. Adjust-ments due to damping and spring stiffness can be made with thehelp of Eqs. (6) and (8).

2)
Dynamic approach based on F2
The procedure by Mylonakis [3] may be written in thealternative form as

MðtÞmax ¼2EpI

d

ep

g1

� �o ¼ 0

ðg1ÞdynF2 ð22Þ

where the static shear strain (g1)o¼0 has been replaced by itsdynamic counterpart (g1)dyn; (ep/g1)o¼0 is given by Eq. (13)and dynamic modifier F2 is equal to about 1.20–1.25 (Fig. 14).This approach is also attractive as dynamic effects due tofrequency content of the input motion are incorporated intoparameter (g1)dyn, whereas F2 is a constant. A disadvantage isthat a free-field site response analysis is required to establishthe value of (g1)dyn.

3)
Dynamic approach based on ZThe original formulation by Nikolaou et al. [4,71] can be re-written in the form:
MðtÞmax ¼ Z Mresonance ð23Þ

where Z can be determined from Eq. (11) and Fig. 13; Mresonance

may be obtained from Eqs. (16) and (17). To this end, knowl-edge of surface acceleration as at resonance is required. Thelatter can be determined by a harmonic site response analysisor by a fully analytical approach assuming harmonic wavepropagation in a two-layer subsoil. Cairo et al. [69] suggested asimple method to obtain the peak steady-state acceleration atground surface as, based on the peak acceleration ar on soiltype A and the fundamental period T1 of the subsoil. In thisway, no numerical site response analysis is necessary and theapproach can be fully analytical. As in the case with parameterF2, Z is sensitive to variations of frequency ratio (finput/f1).Numerical examples are provided in Appendix I.

6. Conclusions

Results from an extensive parametric analysis were reported,carried out on single vertical elastic solid piles in layered soil,accounting for different material properties, geometric factorsand earthquake excitations. The analyses were performed using anumerical tool developed by Mylonakis et al. [29] and Mylona-kis [62] based on a properly calibrated Beam-on-Dynamic-Wink-ler-Foundation (BDWF) model.

The main conclusions of the study are:

1.
A sensitivity analysis of pile bending moments as function ofspring stiffness k adopted in Winkler models and dampingratios D employed in free field response analyses was carriedout; Eqs (6) and (8) were developed to account for theseeffects in a simple manner.
2.
A comprehensive parametric study on a two-layer profile asfunction of: (i) stiffness contrast between upper and lowerlayer, (ii) depth of upper layer, (iii) waveform of input motionand (iv) soil damping adopted in the free-field site responseanalysis, was reported. It was found that:a) Severe kinematic interaction develops at the interface of
two soil layers having sharply different stiffness and per-taining to subsoil of class C and D according to EC8classification. In these cases the computed kinematic bend-ing moments at the interface may be well above theyielding moments of the pile cross section computed fortypical reinforced concrete piles and axial loads.

b) A threshold value of ratio (f1/finput) can be determinedbeyond which kinematic effects are considerable. This limitprovides a possible criterion for selecting significant


records from a ground motion database. Therefore, if theacceleration time-histories are provided for a given seismiczone (in addition to conventional peak ground accelera-tion), the potential of the site with the associated wave-forms to induce significant kinematic bending in piles maybe established.

3.
On the basis of the above results, new regression analyseswere carried out for computing the transient pile bendingmoments at the soil layer interface. Three alternative proce-dures were outlined to solve the problem in the realm ofroutine engineering calculations. Specifically:a) The approach based on F1 is perhaps the most attractive for
engineering purposes, as it does not require a free-field siteresponse analysis. A disadvantage is that parameter F1 issensitive to variations of (finput/f1) ratio, which is often notknown with sufficient accuracy in practice.

b) The approach based on F2 is also attractive as dynamiceffects due to frequency content of the input motion areincorporated into parameter (g1)dyn, whereas F2 is aconstant. A potential disadvantage is that a free-field siteresponse analysis is required to establish the value of (g1)dyn.

c) The approach based on Z requires knowledge of Mresonance,which can be realized by means of Eq. (16). As in the case ofparameter F1, Z is sensitive to variations of frequency ratio(finput/f1).

4.
Large differences often appear in the parameters representingthe frequency content of input motion (fp or fm, Table 1). Forthe sake of conservatism, the authors suggest to apply thesimplified formulas choosing between fp and fm (or anyother rational estimate) the value closer to the fundamentalfrequency f1 of the soil profile.
Acknowledgements

The work herein described is part of the ReLUIS researchProject ‘‘Innovative methods for the design of geotechnicalsystems’’, promoted and funded by DPC (Civil Protection Depart-ment) of the Italian Government and coordinated by the AGI(Italian Geotechnical Association). The authors wish to thankReLUIS and AGI research coordinators.

Appendix I. Numerical examples

The vertical pile shown in Fig. 1 is embedded in soil profileS1-4, subjected to Italian records A-TMZ000 and A-AAL018(Tables 1 and 2), scaled at a peak ground acceleration of 0.35 g.In the ensuing the following parameters are considered: L¼20 m,d¼0.60 m, n1¼n2¼0.4, rs¼1.9 Mg/m3, Ep¼2.5�107 kPa,rp¼2.5 Mg/m3, D¼10%. The shear wave velocity of the elasticbedrock, Vrock, was taken at 1000 m/s. At the layer interface aBDWF analysis according to reference [29] provides maximumkinematic bending moments in the time domain equal to 194 and248 kN m for the A-TMZ000 and A-AAL018 records, respectively.These values were computed assuming a spring stiffness d equalto 1.2. The three simplified approaches proposed in the paper willbe hereafter applied to the above case.

Record A-TMZ000


The strain transmissibility is computed from Eq.(13) as

ep

g1

� �o ¼ 0

¼ 0:115 ðA:1Þ

From Eq. (21) the static shear strain is likewise obtained:

ðg1Þo ¼ 0 ¼0:35� 9:81� 15

1002¼ 0:00515 ðA:2Þ

To account for the transient nature of the phenomenon,frequency parameter F1 needs to be determined by means ofEq. (12). For the case in hand, the ratio (finput/f1) is equal to 2.4.Applying Eq. (12), parameter F1 is estimated at 0.622.

Finally, the transient kinematic bending moment is calculatedfrom Eq. (20):

MðtÞmax ¼2� ð2:5� 107

Þ � 0:00636

0:6� 0:115� 0:00515� 0:622¼ 195kNm

ðA:3Þ

which is almost identical to the ‘‘exact’’ value of 194 kN m.


In this approach a free-field site response analysis should firstbe carried out to obtain (g1)dyn. For the problem at hand an EERAanalysis provided the value (g1)dyn¼0.0029, which, remarkably, islower than the static estimate in Eq. (A.2).

Assigning F2 the value 1.20, as suggested in this paper, Eq. (22)provides

MðtÞmax ¼2� ð2:5� 107

Þ � 0:00636

0:6� 0:115� 0:0029� 1:20¼ 220kNm

ðA:4Þ

which is about 10% higher than the exact value.

Dynamic approach based on ZTo apply this approach, knowledge of surface acceleration as at

resonance is required. This may be obtained from a free-field siteresponse analysis, or by a fully analytical approach [69]. For thecase in hand, as is found to be equal to 1.5 g.

From Eq. (17), we have

tc ¼ asr1H1 ¼ 1:5� 9:81� 1:9� 15¼ 419kPa ðA:5Þ

From Eq. (16) the estimate of the resonant moment is

Mresonanceffi0:042� 419

� 0:63 20

0:6

� �0:30 2:5� 107

53200

!0:65400

100

� �0:50

¼ 1187kNm ðA:6Þ

For (finput/f1)¼2.4, Eq. (11) yields Z¼0.18. Finally, applyingEq. (23):

MðtÞmax ¼ ZMresonance ¼ 0:18� 1187¼ 216kNm ðA:7Þ

Following the Maiorano et al. [35] approach, with the para-meter b¼0.053 obtained in this study:

MðtÞmax ¼ bbG1ðg1Þdyn ¼ 0:053� 67:5� 19000� 0:0029¼ 197kNm

ðA:8Þ

Both these results are close to the exact value of 194 kN m.Additional adjustments to the above estimates can be made withreference to d and D values (Eqs. (6), (8)), if required.

Record A-AAL018


Repeating the above analysis yields the same strain transmis-sibility (ep/g1)o¼0 and static shear strain (g1)o¼0 as before. For aratio (finput/f1)¼1.9, F1¼0.842 (Eq. (12)).


The transient kinematic bending moment is obtained fromEq. (20) as

MðtÞmax ¼2� ð2:5� 107

Þ � 0:00636

0:6� 0:115� 0:00515� 0:842¼ 264kNm

ðA:9Þ


A free-field site response analysis using A-AAL018 record asinput provides (g1)dyn¼0.0036.

Assigning, as before, F2 to be equal to 1.20, Eq. (22) yields

MðtÞmax ¼2� ð2:5� 107

Þ � 0:00636

0:6� 0:115� 0:0036� 1:20¼ 263kNm

ðA:10Þ

Dynamic approach based on ZThe surface acceleration as at resonance, tc and Mresonance are

the same as before, due to the hypothesis of linearly viscoelasticsoil behaviour. For (finput/f1)¼1.9, Eq. (11) yields Z¼0.26.

Finally, applying Eq. (23):

MðtÞmax ¼ ZMresonance ¼ 0:26� 1187¼ 308kNm ðA:11Þ

Following the Maiorano et al. [35] approach, with the para-meter b¼0.053 obtained in this study yields the estimate:

MðtÞmax ¼ 0:053� 67:5� 19000� 0:0036¼ 244kNm ðA:12Þ

References

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