pile-head kinematic bending in layered soil

Pile-head kinematic bending in layered soil

Raffaele Di Laora1, George Mylonakis2 and Alessandro Mandolini1,*,†

1Department of Civil Engineering, Second University of Napoli, 81031 Aversa (CE), Italy2Department of Civil Engineering, University of Patras, 26500 Rio, Greece

SUMMARY

Kinematic effects at the head of a flexible vertical pile embedded in a two-layer soil deposit are investigatedby means of rigorous three-dimensional elastodynamic finite-element analyses. Both pile and soil areidealized as linearly viscoelastic materials, modelled by solid elements, without the restrictions associatedwith the use of strength-of-materials approximations. The system is analyzed by a time-Fourier approachin conjunction with a modal expansion in space. Constant viscous damping is considered for each naturalmode, and an FFT algorithm is employed to switch from frequency to time domain and vice versa in naturalor generalized coordinates. The scope of the paper is to: (a) elucidate the role of a number of key phenomenacontrolling the amplitude of kinematic bending moments at the pile head; (b) propose a simplified semi-analytical formula for evaluating such moments; and (c) provide some remarks about the role of kinematicbending in the seismic design of pile foundations. The results of the study provide a new interpretation of theinterplay between interface kinematic moments and corresponding head moments, as a function of layerthickness, pile-to-soil stiffness ratio, and stiffness contrast between the soil layers. In addition, the role ofdiameter in designing against kinematic action, with or without the presence of an inertial counterpart, isdiscussed. Copyright © 2012 John Wiley & Sons, Ltd.

Received 30 October 2011; Revised 4 April 2012; Accepted 14 April 2012

KEY WORDS: numerical modeling; pile; soil–structure interaction; kinematic interaction; inertialinteraction; pile diameter

1. INTRODUCTION

Piles in earthquake-prone areas are typically designed to withstand lateral forces arising from structuraloscillations. Such an approach neglects that piles are forced to deform even in absence of asuperstructure, because of strains in the ground imposed by the incident seismic waves. This actiongenerates internal forces along the whole pile length (kinematic forces), in addition to the onestransmitted onto the pile head through the cap, which are significant only in the upper portion of thepile (inertial forces). Following the early work by Margason [1] and the seminal studies by Blaneyet al. [2] and Flores-Berrones and Whitman [3], kinematic response has been the subject ofsystematic research for almost 30 years [4–12]. Post-earthquake investigations have demonstratedthe importance of the phenomenon, by revealing damage at depths where inertial forces arenegligible, in deposits that have not suffered a loss of strength such as that induced by soilliquefaction [7, 13–15].

The accumulated evidence has been recognized by modern seismic codes including Eurocode 8 andthe Italian national provisions [16, 17]. These regulations encourage designers to consider kinematiceffects, yet only under specific conditions. For example Eurocode 8 states that: ‘piles shall be

*Correspondence to: Alessandro Mandolini, Department of Civil Engineering, Second University of Napoli, via Roma29, 81031 Aversa (CE), Italy.

†E-mail: [email protected]

Copyright © 2012 John Wiley & Sons, Ltd.

EARTHQUAKE ENGINEERING & STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2013; 42:319–337Published online 11 May 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/eqe.2201

designed to resist the following two types of action effects: (i) inertia forces from the superstructure. . . ;(ii) kinematic forces arising from the deformation of the surrounding soil due to the passage of seismicwaves’, and: ‘bending moments developing due to kinematic interaction shall be computed only whenall of the following conditions occur simultaneously: (i) the ground profile is of type D, S1 or S2, andcontains consecutive layers of sharply differing stiffness; (ii) the zone is of moderate or highseismicity, i.e. the product (agS) exceeds 0.10g; (iii) the supported structure is of class III or IV’. TheItalian Code provides similar indications. Furthermore, Eurocode 8 encourages engineers to designpiles by accounting for both kinematic and inertial effects, thereby decomposing soil–foundation–structure interaction in two complementary problems in the realm of the substructure method. By thisapproach, a system is analyzed via the following consecutive steps [18, 19]:

(1) A kinematic interaction analysis based on the assumption of zero structural mass. This step aimsat providing: (i) a component of internal forces along the piles and (ii) the motion of the pilehead (known as foundation input motion or FIM), that may be different from the free-field soilmotion and may include a rotational component [7, 20, 21];

(2) An inertial interaction analysis in which the oscillating structure, shaken by FIM, induces additionalforces and displacements at the foundation, transmitted through the pile cap. For pile foundations,the problem has been reviewed [19–22] for various structural configurations and subsoil conditions;

(3) A structural analysis of internal forces induced on the piles by inertial interaction [21].

A simplification in the above methodology is possible by considering that if seismic motion has low-frequency content, FIM will be essentially identical to the free-field motion [20]. Di Laora et al. [23, 24]provides practical rules to identify cases where such conditions are met.

A number of research efforts has focused on kinematic effects for piles embedded in homogeneoussoil [3, 20, 25]. Other contributions have treated bending moments in correspondence to an interfaceseparating soil layers of sharply different stiffness [4, 6–8]. Yet, only a few studies have explicitlydealt with bending at the pile head in the presence of a restraining cap [5, 10, 11, 26]. Limitedknowledge also exists as to the effect of soil layering on the latter type of moments. The problemappears to be important, given that the pile head is stressed by both kinematic and inertial forces,which are often of comparable magnitude [27]. Hence, neglecting kinematic effects may lead tosevere underestimation of the actual internal forces in the pile.

2. KINEMATIC RESPONSE IN HOMOGENEOUS SOIL

Kinematic bending moments along a pile may be interpreted as the combined effect of two distinct, yetsimultaneous actions: (i) the displacement profile soil tries to impose on the pile (free-field response);(ii) the resistance of the pile to the induced displacements because of stiffness mismatch and inertia(soil–structure interaction). Because ground response can be determined by standard soil dynamicsprocedures [28], the key to understanding the physics of kinematic response lies in evaluatingpile–soil interaction.

Modeling the pile as a beam on dynamic Winkler foundation and considering vertically-propagatingS waves, it is straightforward to show that for an infinitely-long fixed-head pile embedded in ahomogeneous halfspace, the ratio of pile and soil curvature is [3, 7, 8]

1=Rð Þp1=Rð Þs

¼ Γ (1)

where Γ is a dimensionless kinematic response parameter given by

Γ ¼ 4l4 þ mo2=EpIp4l4 þ q4

(2)

in which Ep and Ip are the pile Young’s modulus and cross-sectional moment of inertia, respectively, mthe pile mass per unit length and o the cyclic excitation frequency. Equation (1) can be interpreted as a

320 R. DI LAORA, G. MYLONAKIS AND A. MANDOLINI

Copyright © 2012 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2013; 42:319–337DOI: 10.1002/eqe

curvature transmissibility function between soil and pile. As will be shown in the ensuing, the abovedefinition is advantageous over alternative indices such as strain transmissibility [8, 11, 12, 23, 26]at the specific location.

In the above equation,

q ¼ o=V�s (3a)

and

l ¼ k þ ioc� mo2

4EpIp

� �14

(3b)

are, respectively, the soil and pile wavenumbers, Vs* =Vs √1 + i2bs being the complex propagationvelocity of damped shear waves in the soil, k the modulus of the Winkler springs (modulus ofsubgrade reaction), c is the corresponding dashpot coefficient and bs a frequency-independentdamping coefficient for the soil material. In the above equation k is traditionally taken as proportionalto soil Young’s modulus, Es. For kinematic loading the value of the proportionality constant, d = k/Es,generally lies in the range 1 to 3 depending on pile slenderness, pile–soil stiffness contrast and therotational constraint at the pile head [6, 8]. These values are higher than corresponding values forinertial loading [29–32]. With reference to curvature ratio at the pile head (Equation (1)), theamplitude of parameter Γ is less than 1 and decreases monotonically with increasing frequency, asevident in Figure 1(a). A physical explanation for this behavior lies in the inability of the pile tofollow short wavelengths in the free-field. Di Laora [23] showed that even at large pile–soil stiffnessratios (Ep/Es up to 10,000), Γ does not strongly deviate from unity in the frequency range relevant toearthquake engineering and, thereby, curvature ratio may be taken equal to 1 for most cases ofpractical interest. Note that the alternative representation of kinematic bending in terms of straintransmissibility (ep/g), ep being peak pile bending strain and g soil shear strain at the same elevation[8], is not applicable at the pile head because g at soil surface is never finite (i.e., can be either zero orinfinite depending on the rate of variation of stiffness with depth [33]).

Mylonakis [34] and Nikolaou et al. [7] derived a closed-form expression for the curvature ratio for afixed-head end-bearing pile of diameter d in a homogeneous layer of thickness h over rigid rock. Thissolution reads

1=Rð Þp1=Rð Þs

¼ Γ 1þ q

l

� �2 cosqh sinhlh coslhþ coshlh sinlhð Þ þ q=lð Þ sinlh sinhlh sinqhsinh2lhþ sin2lh

� �(4)

and is depicted in Figure 1(b) as a function of dimensionless excitation frequency (od/Vs). It is notedthat contrary to the case of a homogeneous half-space (Figure 1(a)), curvature ratio may exceed unityat low frequencies. The jump observed at the fundamental natural period of the layer corresponds to a

0.0 0.1 0.2 0.3 0.4 0.50

0.5

1

1.5

curv

atur

e ra

tio, (

1/R

) p /

(1/R

) s

Margason (1975)

10000

1000

Ep/Es = 100

(a)

0.0 0.1 0.2 0.3 0.4 0.5

fundamental frequency of soil layer

20

10

h/d = 5

(b)

dimensionless frequency, d/Vs

1

Figure 1. Curvature ratio for a pile in: (a) homogeneous halfspace and (b) homogeneous layer over rigidrock, Ep/Es = 1000. In all cases, rs/rp = 0.7, ns = 0.4, bs = 0.05 (modified from [7]).

PILE-HEAD KINEMATIC BENDING IN LAYERED SOIL 321


cut-off frequency associated with the sudden appearance of waves travelling horizontally in theunbounded soil medium and the associated radiation damping.

On the basis of finite element (FE) analyses, de Sanctis et al. [11] concluded that curvature at thepile head can be taken as proportional to corresponding soil curvature through a regressioncoefficient, which they estimated at around 1.02 on average. The same authors suggested a smallervalue (0.94) for signals having predominant frequencies higher than 1.5 times the fundamentalfrequency of the ground. In a contemporary paper, Dezi et al. [10] reported a parametric study bymeans of a beam on the dynamic Winkler foundation model and proposed a simplified expressionfor kinematic bending moments at the pile head. Details are given in the original articles.

Despite the research and the publications, several aspects of kinematic bending at the pile head havenot been adequately addressed and are still poorly understood. A notable omission relates to the role ofsoil layering, which is unclear even for idealized conditions such as static loading (i.e., a constant bodyforce at zero frequency). In the ensuing, results from a suite of rigorous elastodynamic FE analyses willbe reported, focusing on pile-head bending under static, harmonic and transient dynamic conditions.Static response will be discussed first, to clarify the physical mechanisms controlling head bendingand the interplay with interface bending as a function of depth, pile–soil stiffness contrast and layerstiffness contrast. At a second step, the effect of frequency will be highlighted through pertinentharmonic and transient analyses.

3. NUMERICAL ANALYSES FOR TWO-LAYER SOIL PROFILES

The problem under investigation consists of a flexible cylindrical fixed-head pile embedded in atwo-layer subsoil over a rigid bedrock (Figure 2). Seismic excitation is represented by verticallypropagating S waves specified as a horizontal motion at the bedrock. To explore the role of pile–soilinteraction on kinematic bending, numerical analyses were conducted by the authors using thecommercial FE code ANSYS [35]. The dimensions of the model were selected to ensure thatsoil response close to the boundaries is not affected by outward-spreading waves generated at the

h1

L

d

H

layer 1E1, 1,

1, 1

layer 2E2, 2,

2, 2

pileEp, p, p

S waves

Figure 2. Problem considered.



pile–soil interface. To this end, boundaries were placed 80 and 55 diameters away from the pile axis, inthe direction parallel and perpendicular to that of shaking.

Vertical displacements were restrained along the lateral boundaries of the mesh to simulateone-dimensional conditions for S-waves at large distances from the pile. Nodes at the base of themodel were fully restrained to represent the boundary conditions associated with the presence of therigid bedrock. In addition, restraints were imposed on vertical planes of symmetry and antisymmetryto reduce the model to one quarter of its original size. The mesh was refined in the vicinity of theinterface separating the two soil layers to capture the strong curvatures the pile undergoes in thatlocation (Figure 3). Sensitivity analyses were carried out by varying the geometry and type of finiteelements close to the pile head and the layer interface. It was found that computed bendingmoments tend to increase with decreasing element size and increasing accuracy of polynomialinterpolation within the elements. Vertical element size of 0.1 pile diameters was found to providesatisfactory accuracy for both 8-noded and 20-noded elements. Accordingly, solid 8-noded linearisoparametric elements were selected to represent the soil and the pile, because they are moreeconomical over the 20-noded option, without significant loss of accuracy. In total, 10,620 nodesand 8642 elements were used. Details on the selection of type and size of elements andcorresponding sensitivity analyses are available in [23] and [36]. Linear viscoelastic behavior wasassumed for the pile and soil materials, and perfect bonding was considered at the interfaces. Theanalysis was carried out in the frequency domain, first by extracting 2000 undamped vibrationalmodes (having natural frequencies up to 12–15 times the fundamental natural frequency of the soillayer), and then superimposing their contributions by employing a prespecified level of viscousdamping set at 10%. In this way, the drawbacks stemming from use of common energy-lossformulations such as Rayleigh damping were avoided. An FFT algorithm [37] was employed toswitch from frequency to time domain and vice versa. The numerical model was systematicallyverified against published results, which are not discussed here because of space limitations [8, 12].Note that for the common values L/d = 20, Ep/Es = 1000 and d= 1.2, FE results and the analyticalsolution in Equation (4) at a dimensionless frequency a0 = 0.25 show a discrepancy of less than 5%.

4. STATIC BEHAVIOR

In principle, pile-head bending in two-layer soil is different from bending in a homogeneous medium,because pile–soil interaction is affected by the stiffness mismatch above and below the interface. Thedifferences are naturally more pronounced for shallow interfaces, large pile–soil stiffness contrasts andstrong stiffness discontinuities between the soil layers. Figure 4 depicts the distribution of curvature

Figure 3. Finite-element mesh near the soil layer interface.



along a pile excited statically (i.e., by a constant body force, equal to the product of material density androck acceleration) for different interface depths and layer stiffness contrasts, obtained from the FEanalyses. Pile curvature is evaluated as the ratio of pile axial strain at the outer fiber of the cross-section(provided directly from the FEM solution) and pile radius.

It can be observed that curvature at the pile head always possesses the same sign which, according tothe notation adopted in this study, is negative. For homogeneous soil, pile curvature is essentiallyconstant with depth, except for points located close to the (moment free) pile toe. Curvature at theinterface deviates from the homogeneous case by a positive amount that depends primarilyon stiffness contrast between the layers. As evident by observing the cases for shallow interfaces(h1/d= 2 and 4), head curvature decreases both with increasing stiffness contrast and decreasinginterface depth. The opposite trend is observed for deeper interfaces (h1/d = 10), for which theincrease in interface bending with increasing G2/G1 leads to an increase in head bending. The aboveobservations may appear contradictory, yet have a simple explanation in light of the physicalmechanism described below.

The basis of the interpretation is that the increased stiffness of the second layer acts as a partialfixity, which tends to reduce pile rotation at the interface, thus generating a positive contribution tobending moment in that location. A certain percentage of this moment is transmitted to the pile top,as a function of thickness of the surface layer and stiffness mismatch between pile and soil. Thisinteraction may be investigated by means of a simple Winkler beam, which is loaded at one end bya bending moment (representing the action of the interface) and is fixed against rotation at the otherend (representing the boundary condition at the pile head). The length of the beam stands for thethickness of the surface layer. Figure 5 depicts the moment reaction at the pile head, as a function ofthe applied interface moment and the dimensionless beam length L/lp, where

lp ¼ffiffiffiffiffiffiffiffiffiffiffi4EpIpk

4

r(5)

is a characteristic wavelength— the reciprocal of the (static value of) Winkler parameter l in Equation(2) [31, 34]. Evidently, the pattern is analogous to the distribution of bending moment along a flexibleWinkler beam loaded by a moment at one end, which can be achieved by replacing L/lp by x/lp on thehorizontal axis, x being the distance of the cross-section at hand from the applied moment. It can benoticed that for zero beam length, 100% of the interface moment is, naturally, transmitted to the pilecap. Up to L/lp = 3p/4 the bending moment transmitted to pile cap, R, has the same sign as the

-1 0 120

15

10

5

0

z / d

1

2

10

20

(h1/d = 2)

dimensionless pile curvature, (1/R)p Vs12 / as

-1 0 1 -1 0 1

interface

(h1/d = 4)interface

(h1/d = 10)interface

G2/G1

Figure 4. Pile curvature with depth under static loading (o= 0), for different interface depths and layerstiffness contrasts. Vs1 and as stand for shear wave propagation velocity in the first soil layer, and uniform

soil acceleration, respectively.



externally-applied moment, M; for longer beams it changes sign, attaining a local minimum at L/lp =pand then vanishing asymptotically with increasing beam length.

In light of these observations, the interpretation of the results in Figure 4 becomes straightforward:for small interface depths (e.g., h1/d = 2), the positive bending contribution generated at the interface istransmitted as a positive fraction to the pile head, thus reducing the absolute value of the bendingmoment. On the contrary, for h1/d = 10 the positive additional moment at the interface is transmittedto the pile head as a negative contribution, thus increasing the overall bending moment. It should beemphasized that pile–soil stiffness ratio Ep/E1 influences lp in the same manner as does layerthickness h1/d. Accordingly, it may be concluded that while layer stiffness contrast G2/G1 controlsthe amount of additional bending generated at the interface, h1/d and Ep/E1 control the fraction(including sign) of interface moment that is transmitted to the pile head. It follows that for shallowinterfaces (h1/d< 6) the positive additional bending moment reduces the absolute value of bendingboth at the pile head and the interface [23].

In Figure 6 pile–soil curvature ratio at the pile head is depicted as a function of the governingproblem parameters. The bending mechanisms described above are confirmed by these results.Specifically, Figure 6(a) depicts that up to a certain interface depth (whose value depends on pile–soil stiffness ratio) an increase in stiffness contrast between the soil layers leads to a decrease incurvature ratio, whereas for larger interface depths curvature ratio increases with G2/G1. Naturally,the deeper the interface, the smaller its influence on the head moment. This phenomenon isrecognizable in Figure 6(b), which shows that beyond a critical interface depth an increase in h1/ddecreases curvature ratio with increasing layer stiffness contrast. Figures 6(c) and (d) confirm thatpile–soil stiffness ratio has essentially the same effect (through parameter lp) on head moment asdoes interface depth. The similarity between the upper and lower graphs in the figure is indeed striking.

5. DYNAMIC BEHAVIOR (A): FREQUENCY-DOMAIN RESPONSE

Response to harmonic earthquake excitation of the pile in Figure 2 is governed by 16 independentvariables (i.e., h1, H, L, d, Ep, E1, E2, rp, r1, r2, b1, b2, np, n1, n2, o) which describe thegeometric, material and excitation properties of the problem. In the context of dimensionalanalysis, Buckingham’s theorem [38] specifies that 13 independent dimensionless ratios sufficeto fully describe the response. Note that because of linearity, rock acceleration is not anordinary independent variable, as it simply scales pile bending in proportion to its amplitudeand, thereby, can be used to normalize the response. Of these ratios, six (i.e., Ep/E1, E2/E1, h1/d, L/d, H/d, od/Vs1) are known to have first-order influence on the solution and are investigated inthe ensuing. The appearance of shear wave velocity Vs1 in the last ratio should not come as asurprise, because it is merely a function of E1, n1 and r1. In the same spirit, the stiffness contrastbetween the two layers, E2/E1, can be expressed in the alternative forms G2/G1 and Vs2/Vs1, which

MR

h

pilehead

interface

0 2 4 6 8dimensionless beam length, h / p

-0.5

0

0.5

1

dim

ensi

onle

ss m

omen

t re

actio

n, R

/ M

3 /4

Figure 5. End reaction, R, of a Winkler beam fixed against rotation at one end and loaded at the other end bya concentrated bending moment, M.



are employed below for static and dynamic conditions, respectively. The remaining dimensionlessquantities (i.e., rp/r1, r2/r1, b1, b2, np, n1, n2) generally have second-order influence on pile–soilinteraction (although b1 and b2 may have first-order influence on site response) and, thereby, werekept constant in the analyses.

Pile-to-soil curvature ratio is depicted in Figures 7(a) to (c) against dimensionless excitationfrequency od/Vs1. Two trends are worthy of note: First, curvature ratio decreases monotonicallywith frequency. Second, the rate of decrease depends on the three aforementioned governingparameters (i.e., h1/d, Ep/E1, G2/G1). As in the homogeneous case, bending decreaseswith frequency because of the increasing difficulty of the pile to conform to short wavelengths inthe free-field. In this context, Figure 7(a) shows that the rate of decrease is higher with increasingG2/G1, because this ratio controls the degree of restraint provided by the second layer. Indeed, thehigher the degree of restraint, the smaller the ‘allowable’ length of deflection of the pile abovethe interface.

With increasing frequency, curvature ratio becomes gradually independent of stiffness contrastG2/G1, a trend that might be explained considering the vanishing contribution provided by theprogressively smaller interface moment. On the other hand, for a given stiffness contrast(Figure 7(b)), the higher the static curvature ratio, the higher its rate of decrease withfrequency. Figure 7(c) is in agreement with the earlier observation that interface depth andstiffness contrast have an analogous effect on head bending. Figures 7(d), (e) and (f) depictpile–soil acceleration ratio at the surface against excitation frequency. Evidently, the ratioassumes values that are always smaller than 1. This confirms that curvature ratios higher than1 are generated by interface action — not by an amplification of motion close to the pile head.

Figure 6. Static curvature ratio at pile head. (a, b) Ep/E1 = 300; (c, d) h1/d= 6.



6. DYNAMIC BEHAVIOR (B): TIME DOMAIN RESPONSE

Because kinematic pile–soil interaction is inherently frequency dependent, it is expected that themaximum bending moment at the pile head generated during transient earthquake excitation isaffected by the frequency content of the ground motion. To explore the possibility of developing asimple criterion for estimating transient moments, parametric analyses were carried out by varyingthe governing parameters.

Whereas in a harmonic analysis fixing the values of the 13 aforementioned dimensionless ratios issufficient to produce perfectly similar systems, in the realm of transient analysis additional

0.0

0.5

1.0

1.5

2.0

G2/G1 = 2.5

4.5

10

20

1000

0.0

0.5

1.0

1.5

2.0

curv

atur

e ra

tio, (

1/R

) p /

(1/R

) s

h1/d = 2

46

8

10

15

(a) (d)

acce

lera

tion

ratio

,

0.0

0.5

1.0

1.5

2.0

Ep/E1 = 100

300

1000

10000

0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0

)e()b(

(c) (f)

dimensionless excitation frequency, d/Vs1

Figure 7. Curvature ratio versus dimensionless excitation frequency. (a, d) Ep/E1 = 1000, h1/d= 10; (b, e) Ep/E1 = 1000, G2/G1 = 10; (c, f) h1/d= 10, G2/G1 = 10. ap and as are pile and soil acceleration at pile-head level;

in all cases b1 = b2 = 10%.



considerations are needed, because a real earthquake motion contains a variety of dimensional Fouriercomponents. To address this issue, two parametric studies were conducted, as shown in Table I.

The first study utilized the fixed physical parameters H= 30 m, d= 1 m, L= 24 m and E1= 50MPa.A total of 48 combinations involving dimensionless ratios Ep/E1, Vs2/Vs1 and h1/d ranging,respectively, from 300 to 10,000, 1.5 to 6, and 2 to 16 were employed.

The second study considered H= 30m, d= 0.5m, Ep= 30GPa, with the dimensionless ratios Ep/E1,Vs2/Vs1, h1/d and L/d varying from 150 to 1500, 1.5 to 3, 4 to 16, and 24 to 40, respectively— a total of54 cases. In both studies r1= 1.6Mg/m3, b1= b2= 0.1, np= 0.2, n1= n2= 0.3. Density contrast r2/r1was set equal to 1.125 when Vs2/Vs1 = 1.5 or 2 and 1.25 otherwise.

Six recorded earthquake accelerograms, selected from the Italian Database [39], scaled to 1m/s2,were adopted as input motions. Their main features are reported in Table II, while time histories andresponse spectra in terms of acceleration and pseudovelocity are provided in Figures 8 and 9.Overall, a set of 6 � (48 + 54) = 612 cases were analyzed.

It is important to note that the first parametric study employs constant E1 and variable Ep values,whereas the second constant Ep and variable E1. Accordingly, the latter study allows variation ofsoil profiles as a function of E1 (in addition to H, h1 and E2). For the parameter combinations ofTable I, the extra variable E1 produces a wider set of soil profiles (27 versus 16) and, thereby, agreater set of natural frequencies and associated ground responses. On the other hand, the firstparametric study allows studying the response of a wider set of pile stiffnesses (3 versus 1) for eachsoil profile. Evidently, controlling Ep/E1 (in addition to the other 12 ratios) does not suffice tonormalize dynamic response for a given earthquake record. To the best of the authors’ knowledge,this aspect of the problem, which is intrinsic in dynamic analyses of flexible foundations, has notbeen investigated in the past.

6.1. Results

As evident in the foregoing, the curvature ratio atop the pile is influenced by interface action. It hasbeen shown that in the static case a fraction of interface bending moment is transmitted to the pilehead in the form of a positive or negative increment. In this context, there should be two criticalinterface depths, in accordance with Figure 5, which separate regions characterised by a differentbehavior regarding the sign of this transmission.

For instance, considering the switch from positive to negative moment transmission in Figure 5 (i.e.,at h/lp = 3p/4), one obtains the critical interface depth

hc1 ¼ 1:25dEp

E1

� �14

(6a)

which is derived from Equation (5) assuming d =2.5 [6, 8]. Negative transmission appears to besignificant up to a depth of 2� (3p/4) lp beyond which interface action becomes negligible. Accordingly,

Table I. Parametric investigation of transient kinematic bending at pile head using the motions of Table II. Inall cases, b1= b2= 0.1, np= 0.2, n1= n2= 0.3.

Study H/d L/d h1/d Ep/E1 Vs2/Vs1

1 30 24 2 300 1.510004 210000 38

16 62 30 24 4 150 1.5

8 667 24016 1500 3



Table

II.Earthquakerecordsem

ployed

asbedrockinputmotions.

Statio

nname

Event

Mw

Distance

Com

p.PGA

PGV

Arias

intensity

Site

class

Vs,30

[km]

[g]

[cm/s]

[m/s]

[m/s]

Tolmezzo

–DigaAmbiesta

Friuli06/05/1976

6.4

20NS

0.357

22.84

0.787

A1092

Sturno

Irpinia1st23/11/1980

6.5

30WE

0.321

71.96

1.393

A1134

Borgo

–Cerreto

Torre

Umbria-M

arche(A

S)

12/10/1997

5.1

10WE

0.162

4.78

0.065

A1000

San

Rocco

Friuli(A

S)11/09/1976

5.8

24NS

0.090

3.47

0.032

B600

Tarcento

Friuli(A

S)11/09/1976

5.5

8NS

0.211

8.12

0.108

A843

NoceraUmbra–B

iscontini

Umbria-M

arche(A

S)

03/10/1997

5.0

7NS

0.186

3.89

0.158

B442

Mw

=mom

entmagnitude,PGA=peak

ground

acceleratio

n,PGV=peak

ground

velocity



hc2 ¼ 2:5dEp

E1

� �14

(6b)

In light of the above formulae, it is expected that the first region (i.e., h1< hc1) is characterized bycurvature ratios, which are systematically smaller than unity, whereas the second region(hc1< h1< hc2) encompasses values that can be larger than unity. Finally, for deep interfaces(h1> hc2) curvature ratio is expected to be less than 1.

Figure 10 depicts results in terms of pile-to-soil curvature for the three aforementioned regions,computed for the ensemble of ground motions in Table II. Evidently for h1> hc2 (Figure 10(a)) thecurvature ratio for kinematic loading can be confidently taken equal to 1. This simple behaviour

0

1

2

3

4

5

spec

tral

acce

lera

tion

[m/s

2 ] Borgo CerretoSturnoNocera UmbraTolmezzoSan RoccoTarcento

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

period [s]

0

5

10

15

20

25

30

spec

tral

pse

udo-

-vel

ocity

[cm

/s]

Figure 9. Response spectra of time histories in Table II.

-1.0

-0.5

0.0

0.5

1.0

Borgo CerretoSturno

-1.0

-0.5

0.0

0.5

1.0

acce

lera

tion

[m/s

2 ]

Nocera UmbraTolmezzo

151050time [s]

-1.0

-0.5

0.0

0.5

1.0

San RoccoTarcento

Figure 8. Acceleration time histories employed in the analyses.



could be useful in design as: (1) no pile–soil interaction analyses need to be performed to estimatekinematic bending moment at the pile head; (2) a ground response analysis may not be requiredeither, because soil curvature in homogeneous soil is related to ground surface acceleration as through

1R

� �s

¼ dgdz

¼ asV 2

s1

(7)

dg/dz being the derivative of shear strain with depth. If surface acceleration is known (e.g., on the basisof a design code), kinematic pile head moments can be established from the simple expression

0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7di

men

sion

less

pile

cur

vatu

re, (

1/R

) p x

d [

x 10

4 ]

0 1 2 3 4 5 6 7

dimensionless soil curvature, (1/R)s x d [x 104]

0

1

2

3

4

5

6

7

(a): h1 > hc2

(b): hc1 < h1 < hc2

(c): h1 < hc1

y=

1.1x

Figure 10. Pile versus soil curvature: (a) h1> hc2; (b) hc1< h1< hc2 ; (c) h1< hc1.



Mhead ¼ EpIp1R

� �p

¼ EpIpasV 2

s1

(8)

without use of numerical tools. The above solution coincides with that obtained by a Winkler model forhomogeneous soil at low frequency excitation [27] and is analogous to the one established numericallyby de Sanctis et al. [11]. However, the additional regression factors employed in that study are not partof Equation (8). Discussing the origin of these factors lies beyond the scope of this work. In theintermediate region hc1< h1< hc2 (Figure 10(b)), bending transmission is additive and, thereby,curvature ratio may increase above 1. On a conservative basis and as a first approximation, thisincrease can be taken at about 10%. The decrease observed in certain cases, leading to curvature ratiosof less than 1 is due to dynamic effects. Finally, for a shallow interface (h1< hc1 – Figure 10(c))bending transmission is subtractive and, thereby, curvature ratio is systematically below unity.Frequency effects tend to further decrease this ratio. As a simple conservative approximation, curvatureratio in this case can be taken equal to one.

7. EFFECT OF PILE SIZE

A number of seismic codes [16, 17] require evaluation of kinematic pile bending under conditionsinvolving layered soil. For homogeneous soil it is implicitly assumed that piles do not experiencesignificant kinematic stresses (given the absence of layer interfaces) and thus can be designedexclusively for inertial and gravity loads. Contrary to this perception, the results obtained in this studystrongly suggest that even in homogeneous soil kinematic moments may be important. The role ofselecting a pile diameter in designing piles against such moments is inadequately understood [8, 36, 40, 41],not covered by seismic codes.

For the purposes of this discussion and under the reasonable hypothesis that axial pile bearingcapacity is proportional to the gravitational axial load carried by each pile under working conditions,Wp, it is instructive to identify the following extreme cases:

• Pile bearing capacity due exclusively to tip action (perfectly end-bearing pile)• Pile bearing capacity due exclusively to shaft resistance (perfectly floating pile)

Evidently, in the first caseWp is proportional to d2 (tip area) whereas in the second case to d (side area).

It follows that since inertial moment at the pile-head is proportional toWp� d (i.e., overall proportional tod2 or d3) and kinematic moment is proportional to d4 (i.e., proportional to Ip - see Equation (8)), the lattermoment will tend to dominate seismic demand with increasing pile diameter.

Moreover, recalling that the moment capacity of a cylindrical steel pile is proportional to d3, theremust always be a maximum diameter beyond which the pile will not be able to withstand kinematicaction. The yield moment can be computed from the strength-of-materials expression

My ¼ EpIpey2d

1� Wp

fyA

� �(9)

ey and fy being the uniaxial yield strain and corresponding stress of the material and A the cross-sectional area.

For the aforementioned case of a perfectly floating pile in cohesive soil, Wp can be expressed as

Wp ¼ p aSF

SuL d (10)

where Su is the undrained shear strength of the soil material, a the pile–soil adhesion coefficient(ranging from 0 to 1), and SF a safety factor against axial bearing capacity failure.



7.1. Kinematic Loading

For a flexible pile in homogeneous soil (or a two-layer soil with h1> hc2), setting the kinematicdemand moment in Equation (8) equal to the yield moment in Equation (9) and using Equation (10)yields the second-order dimensionless algebraic equation

12ey

asL

V2s

d

L

� �2

� d

L

� �þ 4 aqA SF

Sufy

¼ 0 (11)

which admits the solutions

d kð Þcrit ¼ 2ey

V2s

as

12�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi14� 1ey

2 aqA SF

V2s1

asL

� ��1 Sufy

� �s24

35 (12)

The larger root of Equation (12) corresponds to the critical (maximum) pile diameter to withstandkinematic action. In the above expressions, qA = 1� (1� 2t/d)2 is a dimensionless factor accountingfor wall thickness, t, of a hollow pile. It should be noticed that in the realm of the above derivation,Wp was assumed not to generate lateral inertial loads on the pile (kinematic problem).

In the extreme case of a pile carrying zero axial load (Wp = 0) the term in brackets in the right side ofEquation (12) tends to unity; the solution reduces to the simple expression

d kð Þcrit ¼ 2ey

V2s

as(13)

which can be obtained directly from Equations (8) and (9).

7.2. Inertial Loading

Considering exclusively inertial action and assuming, as a first approximation, that the lateralearthquake load imposed on the pile is proportional to the axial gravitational load Wp, it isstraightforward to show that the maximum moment at the pile head can be expressed as

M ið Þhead ¼ Fh

lp2¼ 1

4p qId

� �14 as

g

� �Ep

Es

� �14

Sa Wp d (14)

Fh being the inertial force acting at the pile head, lp the characteristic wavelength in Equation (5), d theWinkler stiffness parameter (which is of the order of 1 to 2 for inertial loading [29, 31]),qI = 1� (1� 2t/d)4 a dimensionless factor accounting for wall thickness t of a hollow pile, Sa adimensionless spectral amplification parameter, and g the acceleration of gravity.

Setting equal the right sides of Equations (9) and (14), and employing Equation (10) yields the solution

d ið Þcrit ¼

8aSF

LSaey

pd

� �14 as

g

� �qIEp

Es

� ��34 Su

Es

� �þ 12qA

Sufy

� �" #(15)

which defines a critical (minimum) pile diameter to withstand inertial action. Note that in theabove derivation surface acceleration as has been assumed not to generate stress waves in thesoil (inertial problem).

In the limit case of zero ground acceleration (as = 0), Equation (15) reduces to

dcrit ¼ 4aSF qA

Sufy

� �L (16)

which corresponds to the minimum diameter to resist the gravitational loadWp. The same result can beobtained by setting as = 0 in Equation (11).



7.3. Combined kinematic and inertial loading

For the more realistic case of combined kinematic and inertial loading, Equations (8) and (15) can becombined for the overall flexural earthquake demand at the pile head through the simplified expression

M tð Þhead ¼ M kð Þ

head þ ekiM

ið Þhead (17)

where superscript (t) stands for ‘total’ and eki is a correlation coefficient.Setting the above moment equal to the yield moment in Equation (9), one obtains the second-order

dimensionless equation

12asL

V2s

d

L

� �2

� eyd

L

� �þ 4aqASF

SuEp

� �1þ 2eki

qAqI

pqId

� �14 as

g

� �Ep

Es

� �14

Sa

" #¼ 0 (18)

which can be solved analytically for the pair of pile diameters

d1;2 ¼ ey V2s

as1�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 8a

qA e2y SFasL

V2s

� �SuEp

� �1þ 2eki

qAqI

p qId

� �14 as

g

� �Ep

Es

� �14

Sa

" #vuut8<:

9=; (19)

corresponding to a minimum (d1) and a maximum (d2) value obtained for the upper (�) and lower sign(+), respectively. Values between these two extremes define the range of admissible diameters for theconditions specified by the values of dimensionless ratios in the right side of Equation (19).

The range of admissible pile diameters is plotted in Figure 11 as a function of soil shear wavevelocity Vs, for different values of surface seismic acceleration (as/g) and pile lengths, for hollowsteel piles. For simplicity eki= +1 has been considered in all graphs. Evidently, critical values areincreasing functions of Vs leading progressively to a wide range of admissible diameters. The curvesfor purely kinematic and inertial action (Equations (12) and (15)) bound the admissible range fromabove and below, respectively, which suggests that kinematic and inertial moments interactdetrimentally for pile safety. The interaction is more pronounced with increasing pile length andseismic acceleration. For piles in very soft soil such as peat (Vs less than 50m/s), maximum pilediameter may be of the order of 1m, as evident in the graphs.

In addition, it is worth mentioning that there is always a minimum soil shear wave velocity forwhich the admissible range collapses to a single point corresponding to a unique admissible pilediameter (i.e., d1 = d2). This diameter can be obtained by setting the term in square root in Equation(19) equal to zero to give

d1 ¼ d2 ¼ ey V2s

as(20)

Note that this value is exactly one half the value obtained for kinematic action under zero axial force(Equation (13)), and thereby is independent of the pile’s Young’s modulus and wall thickness.

As a final remark, the relative importance of pile head moments and interface moments withreference to pile size is worth discussing. On the basis of available closed-form solutions (e.g.,Mylonakis [8]), the latter moments can be determined from the low-frequency approximation

epg1

� �st

¼ 12c�4 c2 � cþ 1

3

k1Ep

� �1=4 h1d

� �� 1

" #c c� 1ð Þ � 1

( )h1d

� ��1

(21)

which is valid for h1> hc1. The above expression clearly indicates that bending strain at deep interfacesis essentially independent of pile diameter, that is,



epg1

� �st

¼ 32c�3 c� 1ð Þ c2 � cþ 1

k1Ep

� �1=4

(22)

In light of Equation (22), the magnitude of kinematic moments in that location is proportional to d3, thesame dependence as that of yield resistance My in Equation (9). Accordingly, a variation in pilediameter does not improve or reduce safety and, thus, interface moments at deep elevations cannotbe used to define pile size in design.

8. CONCLUSIONS

A number of unresolved issues related to kinematic bending of pile foundations during earthquakeshaking have been discussed in light of a set of rigorous three-dimensoinal finite-element analysesfor flexible single piles embedded in two-layer soil. The main conclusions of the study can besummarized as follows:

(1) Static analyses and a simple mechanistic model (Figure 5) demonstrate that pile–soilcurvature ratio at the pile head may assume values different from unity (the baseline valuefor homogeneous soil). This is explained given the transmission of interface moment to thepile head as a positive or negative contribution depending on the dimensionless parametersh1/d, Ep/E1 and G2/G1. Of these ratios, the first two control the sign and the fraction of thetransmission, whereas the last the amount of interface moment to be transmitted.

Figure 11. Admissible pile diameters against soil shear wave velocity (Es/Su = 500, fy = 275MPa,Ep = 210GPa, ns = 0.5, Sa = 2.5, FS = 3, t/d= 0.015, a= 0.7, d= 1.2).



(2) Frequency-domain analyses have shown that in two-layer soil, pile-to-soil curvature ratiodecreases with frequency. The rate of the decrease depends on the governing parameters h1/d,Ep/E1 and G2/G1.

(3) In accordance with static behavior, time-domain results have shown that three response regionsare distinguishable, separated by two characteristic lengths (hc1 and hc2, Equations (6a) and (6b).For deep interfaces (h1> hc2) the assumption of pile–soil curvature ratio at the pile head equal to1 provides a satisfactory estimate of maximum bending. For hc1< h1< hc2 the interface effectprovides an additional moment at the pile head and, thereby, curvature ratio may increase aboveunity with an upper bound of approximately 1.1. For shallow interfaces a unit curvature ratiorepresents a conservative choice. Knowledge of peak ground acceleration in the free-fieldmay, therefore, be sufficient for assessing kinematic bending at the pile head without use ofnumerical tools.

(4) Kinematic moments at the pile head were found to increase in proportion to the fourth power of pilediameter d, whereas corresponding inertial moments increase in proportion to the second orthird power. This suggests that kinematic moments will tend to dominate seismic demand withincreasing pile diameter. Moreover, because the moment capacity of a pile is proportional to d3,there always exists a maximum diameter beyond which a pile cannot sustain kinematic action atthe head (Equation (12)). Conversely, inertial loading imposes a minimum required diameter(Equation (15)).

(5) Contrary to kinematic moments at the pile head, kinematic moments at a layer interface increasein proportion to d3, the same as the dependence on pile diameter of yield resistance My (Equa-tion (9)). Accordingly, a variation in diameter does not improve or reduce safety in that elevationand, thus, the value of interface moment cannot control pile size in design.

(6) The combined action of kinematic and inertial loading provides a range of admissible pile dia-meters mainly depending on soil shear wave velocity, surface earthquake acceleration and axialload (Equation (19)). The curves for purely kinematic and inertial action (Equations (12) and(15)) provide upper and lower bounds of the admissible range, which suggests that kinematicand inertial moments interact detrimentally for the safety of the pile. The interaction is more pro-nounced with decreasing pile length and increasing seismic acceleration. For piles in very soft soilsuch as peat (Vs less than approximately 50m/s), maximum pile diameter may be of the order of1m. These aspects of seismic response appear to be neglected by seismic codes.

It is fair to mention that the results presented in this study are valid under the hypothesis of isotropiclinearly viscoelastic behavior for pile and soil materials and perfect bonding at the interfaces.Therefore, some of the findings may require revision in the presence of strong nonlinearities inducedby high-amplitude earthquake shaking.

ACKNOWLEDGEMENTS

This research has been developed under the auspices of the research project ReLUIS ‘Methods for risk eval-uation and management of existing buildings’, funded by the Italian National Emergency ManagementAgency. The authors are grateful for this support.

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