Kinematic Winkler modulus for laterally-loaded piles

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Kinematic Winkler modulus for laterally-loaded piles

George Anoyatis

Senior Lecturer, Department of Geography & Environmental Management, University of the West of England

UWE, Bristol, UK, email: [email protected]

Anne Lemnitzer

(Corresponding Author)

Assistant Professor, Department of Civil and Environmental Engineering, University of California Irvine,

Irvine, U.S., email: [email protected]

ABSTRACT

Beam-on-Dynamic-Winkler-Foundation models are widely used to study kinematic soil-pile

interaction. Winkler models consider the pile as a flexural beam and simulate the restraining and

dissipative action of soil through independent springs and dashpots along its axis. Their performance

is related to the proper selection of the spring stiffness and dashpot coefficient which depends on

parameters such as pile geometry, pile-soil stiffness ratio, and boundary conditions. Expressions for

static and dynamic Winkler moduli from literature were implemented in a Winkler model to assess its

ability to predict the curvature ratio and kinematic response factors for various pile boundary

conditions. Based on an existing static expression a frequency-dependent, logarithmic-based Winkler

modulus is proposed. This modulus offers an attractive and versatile alternative to existing

mathematically complex formulations as it is capable of capturing resonant effects and can be used for

both inertial and kinematic analyses, while all other frequency-independent expressions from

literature are limited by their unique application to the kinematic problem. A comprehensive graphical

comparison between results from the Winkler model using existing and proposed moduli and the

more accurate FE solution is offered to guide the user in selecting the most appropriate modulus for

the problem to be analyzed.

Keywords: soil-pile interaction, kinematic, Winkler modulus, lateral, harmonic, oscillations

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1. Introduction

Kinematic soil-pile interaction is generated by soil motion capable of exciting the pile across its entire

length, even when no superstructure is present. Such loading mechanism can develop due to natural

hazard events like earthquake motions, blast loading and vibrations generated through adjacently

operating, high frequency machinery. Among the various computational tools available, kinematic

interaction is most often studied using rigorous finite element (FE) (e.g., (Fan, et al., 1991)) and

boundary element (BE) (e.g., (Kaynia & Kausel, 1991)) solutions and various beam-on-dynamic-

Winkler-foundation models (BWFM) (e.g., (Pender, 1993), (Mylonakis, 2001a), (Anoyatis, et al.,

2013), (Kampitsis, et al., 2013), (Chidichimo, et al., 2014), (Di Laora & Rovithis, 2015)).

The BWFM has remained the most widely employed methodology, but its performance and accuracy

strongly depends on the input Winkler moduli, which in turn have been the focus of diverse studies

over the past decades. In this study a comprehensive review of available Winkler moduli from

inertially ( (Francis, 1964), (Baranov, 1967) – (Novak, 1974) – (Novak, et al., 1978), (Roesset, 1980),

(Dobry, et al., 1982), (Gazetas & Dobry, 1984b), (Mylonakis, 2001b), (Syngros, 2004), (Anoyatis &

Lemnitzer, 2017)) and kinematically ( (Dobry & O'Rourke, 1983), (Kavvadas & Gazetas, 1993),

(Mylonakis, 2001a)) stressed piles is presented and their ability to accurately capture pile behavior

under kinematic conditions is evaluated. Results show that most of the moduli investigated herein

performed well when predicting curvature ratios at the pile head and tip, and capture the kinematic

response factors with sufficient accuracy. This observation is contrary to the analysis of inertially

loaded piles where frequency-independent moduli (or moduli that cannot capture resonances) yield

erroneous predictions for dynamic pile head stiffness as shown in (Anoyatis & Lemnitzer, 2017).

Currently there is no study available in literature (at least none known to the Authors) that has

performed a comparative investigation of available Winkler moduli for computing kinematic response

factors in translation and rotation, and curvature ratio using a kinematic Winkler model. Therefore,

the scope of the current study is twofold: (1) to conduct a comprehensive literature review on

available frequency-independent (“static”) and frequency-dependent (dynamic) Winkler moduli, and

(2) to propose a new, simple expression for a frequency-dependent, resonance-dependent complex

valued Winkler modulus (spring and dashpot). This expression is advantageous due to: (i) its ability to

accurately calculate kinematic response parameters and (ii) its mathematical simplicity (no Bessel

functions are included and thus calculations can be carried out even by using a simple calculator). The

Authors consider this fact particularly important for engineering practice, where calculations should

be made fast, at least for preliminary analyses.

A general performance assessment of the kinematic Winkler model using available and the proposed

moduli was conducted. The following parameters were employed to assess the suitability of the

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existing and the proposed Winkler moduli for various head and tip pile boundary conditions: (i)

curvature ratios 𝐶𝑅0 and 𝐶𝑅𝐿 which relate the curvature of the pile at the head and tip, respectively, to

the curvature of the soil at surface in the free-field, and (ii) kinematic response factors in translation 𝐼𝑢

and rotation 𝐼𝜑, which relate the motion and rotation of the pile head to the motion of the soil surface

at free-field. The aforementioned kinematic parameters have been evaluated using closed form

expressions developed in (Anoyatis, et al., 2013) where kinematic soil-pile interaction was

investigated by means of a BDWF model and results were compared against FE results extracted from

the same study.

2. Kinematic problem statement

The kinematic soil-pile interaction problem considered is depicted in Fig. 1: a single vertical pile of

length 𝐿 and solid cylindrical cross section of diameter 𝑑 is embedded in a homogeneous soil stratum

of thickness 𝐻(= 𝐿) overlying rigid rock. Following the Bernoulli assumption the pile is treated as a

beam of Young’s modulus 𝐸𝑝 and mass density 𝜌𝑝. The soil is modeled as a linear elastic material

with Poisson’s ratio 𝜈𝑠, mass density 𝜌𝑠 and hysteretic type material damping 𝛽𝑠, expressed through a

complex-valued shear modulus 𝐺𝑠∗ = 𝐺𝑠(1 + 2𝑖𝛽𝑠). Lateral harmonic pile motion of the form

𝑤(𝑧, 𝜔) 𝑒𝑖𝜔𝑡 is induced due to the passage of vertically propagating harmonic shear waves (S-waves)

in the soil medium. This excitation is expressed in the form of a harmonic horizontal displacement

𝑢𝑔(𝜔) 𝑒𝑖𝜔𝑡 applied at the rock level (Fig. 1), where 𝜔 is the cyclic excitation frequency and 𝑡 is the

time variable. Utilizing simple 1-D wave propagation analysis the output motion observed at the free

field surface can be described as 𝑢𝑓𝑓0(𝜔) 𝑒𝑖𝜔𝑡. In the presence of a pile foundation the resulting

motion at the pile head differs from the free field and can be expressed as 𝑤0(𝜔) 𝑒𝑖𝜔𝑡.

Kinematic soil-pile interaction is commonly evaluated in terms of curvature ratios 𝐶𝑅0 and 𝐶𝑅𝐿 and

kinematic response factors in translation and rotation, 𝐼𝑢 and 𝐼𝜑, respectively:

𝐶𝑅0 =(1 𝑅⁄ )𝑝|𝑧=0

(1 𝑅⁄ )𝑠|𝑧=0 (1)

and

𝐶𝑅𝐿 =(1 𝑅⁄ )𝑝|𝑧=𝐿

(1 𝑅⁄ )𝑠|𝑧=0 (2)

where (1 𝑅⁄ )𝑝|𝑧=0 and (1 𝑅⁄ )𝑠|𝑧=0 express the curvature of the pile and the curvature of the soil at

the level of the pile head and soil surface (𝑧 = 0), respectively. In the same manner (1 𝑅⁄ )𝑝|𝑧=𝐿 is the

pile curvature at the level of the pile tip (𝑧 = 𝐿).

The kinematic response factors are defined as follows:

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𝐼𝑢 =𝑤0(𝜔)

𝑢𝑓𝑓0(𝜔) (3)

and

𝐼𝜑 =𝑤0′(𝜔) 𝑑

𝑢𝑓𝑓0(𝜔) (4)

where (′) denotes the first derivative with respect to depth 𝑧.

Expressions for the kinematic parameters shown in Eqs. (1) to (4) are provided in (Anoyatis, et al.,

2013). The characteristic Winkler parameter 𝜆 embedded in those equations is expressed as:

1/4* 2

4

p

p p

k m

E I

(5)

where 𝐼𝑝 is the moment of inertia of the pile cross section, �̃�𝑝 = 𝜌𝑝 𝐴𝑝 is the pile mass density (𝐴𝑝

being the pile cross sectional area) and 𝑘∗ = 𝑘(1 + 2𝑖𝛽𝑠) or, equivalently, 𝑘∗ = 𝑘 + 𝑖𝜔𝑐 (𝑐 =

2𝛽𝑘 𝜔⁄ being the dashpot coefficient) is the complex valued Winkler modulus. Note that 𝑘 =

𝑅𝑒𝑎𝑙(𝑘∗) represents the dynamic springs and 𝛽 = 𝐼𝑚𝑎𝑔𝑖𝑛𝑎𝑟𝑦(𝑘∗) 2𝑅𝑒𝑎𝑙(𝑘∗)⁄ is the corresponding

damping ratio associated with the dashpots (Fig. 1). Following the recommendation by (Anoyatis, et

al., 2013) the term related to the pile inertia (𝜔2 �̃�𝑝) in Eq. (5) may be neglected without introducing

significant error in the results for the range of frequencies relevant to earthquake engineering. Thus,

Eq. (5) can be rewritten as

1/4*

4 p p

k

E I

(6)

A further detailed explanation on the parameter 𝜆 used in this study will be given in Section 3.

3. Review of available Winkler moduli

Tables 1 and 2 offer a comprehensive review of static (frequency-independent) and dynamic

(frequency-dependent springs and dashpots) Winkler moduli available in literature. The chronological

presentation of static Winkler moduli (Table 1) starts with a formulation proposed by (Francis, 1964).

This formulation doubles Vesic’s spring modulus (Vesic, 1961) derived for the case of an infinite

beam subject to a point load resting on an infinite elastic foundation and therefore accounts for the

fact that, unlike the beam, the pile is surrounded by soil (Yoshida & Yoshinaka, 1972). In the studies

of (Roesset, 1980) and (Syngros, 2004) an optimum value for the Winkler spring was obtained by

matching the static pile head stiffness obtained from a Winkler model with a finite element analysis.

(Roesset, 1980) suggested a single value for the normalized Winkler modulus 𝑘/𝐸𝑠 equal to 1.2,

while (Syngros, 2004) proposed separate expressions for free- and fixed-head piles as a function of

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pile-soil stiffness ratio with the objective to compute static pile head stiffness in swaying for long

piles. (Dobry & O'Rourke, 1983) treated the pile as a beam on elastic foundation and proposed simple

expressions for Winkler moduli in the upper and lower soil layer (i.e., 𝑘1 = 3𝐺𝑠1 and 𝑘2 = 3𝐺𝑠2) to

compute the bending moment at the soil layer interface of a kinematically stressed pile (where 𝐺𝑠1

and 𝐺𝑠2 is the shear modulus for the upper and lower stratum, respectively). Thus for a homogeneous

soil the Winkler stiffness can be reasonably estimated as 𝑘 = 3𝐺𝑠. The spring formulation by

(Mylonakis, 2001a) is a modified expression of the originally proposed springs by (Kavvadas &

Gazetas, 1993) (Eq. 17a). The latter were derived by matching kinematic bending moments of a pile

embedded in a two-layer soil using a Winkler model with those obtained from a rigorous finite

element analysis. Mylonakis’ simplification is based on the assumption of relatively long piles

(𝐿 𝑑⁄ ≈ 40) and soil layers of equal thickness. The proposed simplified springs were developed to

compute the strain transmissibility (peak pile bending strain and soil shear strain at the soil-layer

interface). Expressions proposed by (Mylonakis, 2001b) are based on a dynamic analytical solution in

which a sinusoidal or, alternatively, an exponential shape function was employed to account for the

profile of lateral pile displacements induced by dynamic loading at its head. From the dynamic

expression shown in Eq. (18a, b) a simplified static expression is obtained (Eqs. 11a, b) using series

expansion. The exponential based solution (Eqs. 11a, b and A.1) additionally accounts for the pile-soil

stiffness ratio, which is not considered in the sinusoidal solution (Eq. 11a, b, c), while the effect of

pile slenderness is taken into account in both.

Table 2 provides an overview of dynamic Winkler moduli i.e., springs and dashpots. Among the

presented Winkler springs two groups of expressions emerge: (i) frequency-independent ( (Dobry, et

al., 1982), (Gazetas & Dobry, 1984b), (Makris & Gazetas, 1992), (Kavvadas & Gazetas, 1993)), and

(ii) frequency-dependent ( (Baranov, 1967) – (Novak, 1974) – (Novak, et al., 1978), (Mylonakis,

2001b), (Anoyatis & Lemnitzer, 2017)) springs. While the first group of frequency-independent

spring formulations are accompanied by simplified expressions for radiation damping 𝛽𝑟, the second

group offers complex-valued springs which inherently account for both, dynamic stiffness and

damping (the real part being the dynamic stiffness and the imaginary part being associated with

energy loss). The damping ratio 𝛽 in Table 2 represents both material and radiation damping (i.e.,

𝛽𝑠 + 𝛽𝑟); 𝛽𝑠 is associated with the hysteretic type of energy dissipation in the soil medium and 𝛽𝑟

with the emergence of horizontally propagating waves emanating from the soil-pile interface. Note

that in the original studies of (Dobry, et al., 1982), (Gazetas & Dobry, 1984b), (Makris & Gazetas,

1992), and (Kavvadas & Gazetas, 1993) dashpots are represented using the damping coefficient 𝑐 (see

Appendix B). However, in Table 2, for consistency, all damping expressions are rewritten using the

equivalent parameter = 𝜔𝑐 2𝑘⁄ .

(Dobry, et al., 1982) investigated the behavior of a single pile in a linear homogenous soil resting on a

rigid base subject to a harmonic lateral load at its head. The authors proposed simple closed-form

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expressions for frequency-independent Winkler springs 𝑘 and frequency-dependent dashpots by

matching the pile head stiffness in swaying and the corresponding damping for the case of a “long”

pile (equivalent to an infinitely long pile in a halfspace) obtained from finite element analyses with

those computed using a Winkler formulation. (Gazetas & Dobry, 1984b) assumed that a laterally

oscillating pile would generate shear waves (S-waves) in the direction of loading and compression-

extension waves (not P-waves) which propagate with velocity 𝑉𝐿𝑎 (Lysmer’s analog wave velocity) in

the direction perpendicular to the loading. These waves emanate from the entire pile perimeter which

is mathematically simplified into four quarters. Based on the analogy proposed by (Berger, et al.,

1977), which assumes that a dashpot at the end of a cylinder fully absorbs the energy of a wave

travelling along its body, (Gazetas & Dobry, 1984b) derived a frequency-dependent expression for

radiation damping (Eqs. 15c, d). Based on earlier studies the authors reported a range of values for

Winkler moduli 𝑘 depending on the conditions at the head (Eqs. 15a, b). (Makris & Gazetas, 1992)

used a Winkler formulation with static springs and frequency-dependent dashpots to predict the

response of piles under inertial and kinematic loading. Using expressions for 𝑘 and 𝛽 from literature

(i.e., (Roesset & Angelides, 1980) and (Gazetas & Dobry, 1984b) ), simple approximations as shown

in Eqs. (16) were introduced. (Kavvadas & Gazetas, 1993) studied the kinematic response of single

free-head piles embedded in a two-layer soil subject to soil motion induced by vertically propagating

S-waves in the soil medium. Results were generated using a finite element formulation developed by

(Blaney, et al., 1976), along with a beam-on-dynamic-Winkler-foundation model. The proposed

Winkler springs were calibrated for maximum kinematic bending moments but work well when

estimating pile deflections. Expressions for damping were proposed based on the work of (Roesset &

Angelides, 1980), (Krishnan, et al., 1983), (Gazetas & Dobry, 1984a) and (Gazetas & Dobry, 1984b).

The frequency-dependent complex valued modulus 𝑘∗ of Baranov-Novak ( (Baranov, 1967), (Novak,

1974), (Novak, et al., 1978)) is derived considering only an incompressible horizontal soil slice of the

soil medium and neglects the thickness of the soil layer. Thus, 𝑘∗ does not account for resonance

effects. This modulus is mathematically expressed in terms of the modified second kind Bessel

functions of first and zero order, 𝐾1( ) and 𝐾0( ), respectively, a frequency-dependent parameter 𝑠 and

a compressibility parameter 𝜂𝑠. The latter parameter expresses the ratio of the P-waves to the S-waves

in the soil medium (𝜂𝑠 = 𝑉𝑝 𝑉𝑠⁄ ) and is a function of the Poisson’s ratio only. (Mylonakis, 2001b)

proposed an expression for the complex-valued Winkler modulus 𝑘∗ which seems identical to the

Baranov-Novak equation. However, contrary to the dynamic plane strain model, Eq. (18a) was

derived under the consideration that the horizontal soil slice is compressible (normal stresses are

included), thus accounting for the thickness of the soil layer. By assuming a sinusoidal and an

exponential shape function and integrating the governing equations over the thickness of the soil

profile, Mylonakis accounted for the resonant effects by introducing a new parameter 𝑠 as shown in

Eq. (18b). An alternative simpler expression for a modulus 𝑘∗ to estimate dynamic pile impedances

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(stiffness and damping) was proposed by (Anoyatis & Lemnitzer, 2017). By taking the limit of

Poisson’s ratio to 0.5 (i.e., incompressible soil) the classic dynamic plane strain expression (Eq. 13a)

was reduced to the one shown in Eq. (19a). The new expression replaces the parameter 𝑠 with the one

obtained from an analytical continuum solution (Anoyatis, et al., 2016) using the first mode only (Eq.

19b). This new parameter s differs from Mylonakis’ in the sense that it incorporates an empirical

parameter 𝜒 to capture the effect of the Poisson’s ratio and a different compressibility parameter 𝜂𝑠

(detailed discussion on the parameters 𝜂𝑠 and 𝜒 is presented in (Anoyatis, et al., 2016)). Both

(Mylonakis, 2001b) and (Anoyatis & Lemnitzer, 2017) can be considered advantageous over the

plane strain model as they can account for static spring stiffness, while the plane strain 𝑘 collapses at

𝜔 = 0. In addition, the plane strain parameter 𝑠 can be viewed as a special case of the parameters in

Eqs. (18b) and (19b) when setting the cutoff frequency equal to zero, hereby representing an infinitely

long pile embedded in a halfspace.

All comparisons presented in Section 5 use the following approach: for all expressions that separate

springs and dashpots (i.e., (Dobry, et al., 1982), (Gazetas & Dobry, 1984b), (Makris & Gazetas,

1992), (Kavvadas & Gazetas, 1993)) the terms associated with damping were omitted (𝑐 = 0 or

𝛽 = 0), and a static λ was used instead:

1/4

4 p p

k

E I

(20)

This is in agreement with (Anoyatis, et al., 2013) where predictions from the Winkler model using

only Winkler springs (no dashpots) lead to a better agreement with FE results for frequencies below

cutoff. For all remaining expressions in Table 2 ( (Baranov, 1967), (Mylonakis, 2001b), (Anoyatis &

Lemnitzer, 2017)) as well as the proposed expression shown in the ensuing (Eq. 21), Equation (6) was

used when evaluating dynamic curvature ratios. For the computation of kinematic response factors the

𝜆 shown in Equation (20) was implemented. In this case 𝜆 is frequency-dependent [i.e., 𝑘 =

𝑅𝑒𝑎𝑙(𝑘∗)]. Since damping is already incorporated into the complex-valued moduli 𝑘∗, this study

separates the real and the imaginary parts and employ only the real in the analysis.

4. Proposed Winkler modulus 𝑘∗

The proposed Winkler modulus is based on a static simplified expression originally proposed by

(Mylonakis, 2001b) as shown in Eq. (11a) of Table 1. Since that expression is not applicable to

dynamic conditions the following modifications are employed: the shear modulus 𝐺𝑠 is replaced with

𝐺𝑠∗, the complex valued modulus, and the parameter s as originally shown in Eq. (11b) is substituted

with a frequency-dependent, and resonant-dependent parameter s. By implementing those

modifications, the following expressions are obtained:

8

2 **

2

4

2ln 1 ln

s s

s s

Gk

s

(21)

where 𝛾 ≈ 0.577 is the Euler’s gamma, 𝜂𝑠 = √2(1 − 𝜈𝑠) 1 − 2𝜈𝑠⁄ as in the original Eq. (11a) and 𝑠

is given by the following expression

2

2 0a1

a2 1 2

cutoff

s

si

(22)

The new parameter 𝑠 as shown in Eq. (22) differs from the plane strain expression previously

presented in Eq. (13b) and builds upon the formulation shown in (Anoyatis, et al., 2016) for the first

mode (𝑚 = 1), where the reaction of a soil layer to a horizontally vibrating pile is investigated. “𝑠” is

independent of the pile boundary conditions and depends on the thickness of the soil profile H, the

pile diameter 𝑑, the propagation velocity of shear waves in the soil Vs and the excitation frequency ω.

In the specific case examined the pile length 𝐿 is equal to the thickness of the soil layer 𝐻 (= 𝐿). In

the ensuing it will be shown that upon implementing 𝑠 in the Winkler model, kinematic parameters

(e.g., 𝐼𝑢, 𝐶𝑅) for various pile boundary conditions can be predicted with adequate accuracy.

Through the above-described modifications to the parameter 𝑠, material and radiation damping are

accounted for in the solution. Material damping 𝛽𝑠 is included in Eq. (22) through the complex-valued

propagation velocity of S-waves in the soil medium 𝑉𝑠∗ = 𝑉𝑠√1 + 2𝑖𝛽𝑠. This becomes evident when

the term a02 1 + 2𝑖𝛽𝑠⁄ in Eq. (22) is rewritten in its alternative form: (𝜔𝑑 𝑉𝑠

∗⁄ )2. The frequency

dependent 𝑠 allows the Winkler modulus 𝑘 to be real-valued for frequencies below first resonance

(a0 < a𝑐𝑢𝑡𝑜𝑓𝑓), and turns 𝑘 into complex-valued (𝑘∗) beyond the resonance threshold (a0 > a𝑐𝑢𝑡𝑜𝑓𝑓),

even in the case of an undamped medium (𝛽𝑠 = 0). For a0 > a𝑐𝑢𝑡𝑜𝑓𝑓, damping 𝛽 = 𝐼𝑚(𝑘∗)/

2 𝑅𝑒𝑎𝑙(𝑘∗) includes the loss of energy due to radiation (i.e., radiation damping 𝛽𝑟) and can be written

as 𝛽 = 𝛽𝑠 + 𝛽𝑟. This loss of energy is associated with the emergence of travelling waves at resonance

which develop at the pile-soil interface and propagate horizontally in the soil medium. Damping 𝛽 is

practically equal to soil material damping until resonance is reached (𝛽 ≈ 𝛽𝑠).

The frequency dependent parameter 𝑠 plays a major role in inertial soil-pile interaction (Anoyatis &

Lemnitzer, 2017) and governs the behavior of the dynamic springs and dashpots in the following

specific manner: Eqs. (18a, 19a, 21) take into account the effects of the first resonance (occurring at

a0 = a𝑐𝑢𝑡𝑜𝑓𝑓 or 𝜔 = ω1) of the soil layer which become apparent as the “drop” in springs values 𝑘

and “jump” in damping values 𝛽, while Eq. (13a) neglects these phenomena. The proposed expression

for 𝑘∗ allows the dynamic springs 𝑘 to attain a minimum value (“drop” in stiffness) and the damping

ratio 𝛽 to exhibit an increase (“jump” in damping) at a0 = a𝑐𝑢𝑡𝑜𝑓𝑓 to include the radiation damping.

This capability is advantageous over many simple existing expressions (e.g. Roesset), which provide

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frequency-independent (the term a0 is missing) or resonant-independent (the term a𝑐𝑢𝑡𝑜𝑓𝑓 is missing)

𝑘-values. The behavior of the frequency-dependent Winkler springs 𝑘 and dashpots expressed through

𝛽 is investigated in detail in the ensuing.

While the real part of Eq. (21) works well across the entire frequency range, the imaginary part

requires an empirical modification after resonance (𝜔 > 𝜔1) identical to that suggested in (Anoyatis

& Lemnitzer, 2017) by adding the term 𝑖 2.2 (a02 − a𝑐𝑢𝑡𝑜𝑓𝑓

2 )1/2

2

2

*1/2

2 2

0 0

s

4 (1 2 )ImaginaryImaginary

2ln 1 ln

( )2.2 a a , a a

G

s s

s s

cutoff cutoff

i

s

ki

(23)

In order to accurately capture the static (𝜔 = 0,𝛽𝑠 = 0) curvature ratio 𝐶𝑅𝐿 the proposed modulus in

Eq. (21) has been further modified by adding the soil shear modulus 𝐺𝑠. The resultant expression can

be written as follows

2

2

41

2ln 1 ln

ss

s s

k G

s

(24)

In Eq. (24) the value obtained through the fraction in the parenthesis can be viewed as a

dimensionless soil-pile interaction parameter. This is a physically motivated, mathematically iterated

modification as 𝑘 needs to be higher than 𝑘 from Eq. (21) to accurately compute 𝐶𝑅𝐿. The amount of

this increase is expressed by the second term in the parentheses, which is equal to 𝑘 𝐺𝑠⁄ when 𝑘 is

obtained from Eq. (21). The verification of this newly proposed representation can be achieved

through the following thought experiment: when replacing the pile with a soil column (i.e., soil

column has the same properties as the surrounding soils) the Winkler spring 𝑘 becomes equal to 𝐺𝑠,

which is an accurate description of “soil column – soil interaction”.

5. Numerical results

Numerical results of the Winkler expressions presented in Table 1 are shown in Fig. 2 for a wide

range of pile-soil stiffness ratios and selected values of pile slenderness 𝐿/𝑑. For clarity results are

presented in linear – logarithmic scale. Expressions by (Roesset, 1980), (Dobry & O'Rourke, 1983)

and (Mylonakis, 2001b) (Eqs. 11a, b, c) are independent of pile-soil stiffness ratio and are represented

through straight horizontal lines. As Mylonakis’ sinusoidal based expression depends on the pile

slenderness ratio, different horizontal lines are plotted for 𝐿 𝑑⁄ = 5, 10, 15 and 20. In turn expressions

which are a function of 𝐸𝑝/𝐸𝑠. exhibit a general trend: the Winkler modulus 𝑘 decreases with

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increasing 𝐸𝑝/𝐸𝑠. Among all expressions listed in Table 1, this decrease is most evident in

formulations by (Kavvadas & Gazetas, 1993), (Mylonakis, 2001a), (Mylonakis, 2001b) (Eqs. 11a, b

and A.1) and (Syngros, 2004) (Eq. 12b). Fig. 2 indicates an extreme large bandwidth (up to 300 %) of

Winkler moduli for stiff soils (𝐸𝑝 𝐸𝑠⁄ = 100), while the range gradually decreases as pile-soil

stiffness ratio increases.

A selected suite of Winkler moduli from Table 2 is plotted against frequency for different pile

slenderness ratios and presented in Fig. 3. Hereby only frequency-dependent springs and

corresponding dashpots (equivalent damping ratio β) are considered. The behavior is investigated in

the low (0 ≤ 𝜔 ≤ 𝜔1) and high (𝜔 > 𝜔1) frequency range and different normalizations as introduced

by (Anoyatis, et al., 2016) and (Anoyatis & Lemnitzer, 2017) are applied for each range. Below

resonance (𝜔 < 𝜔1) dynamic Winkler springs are normalized by their static value and frequencies are

normalized by the corresponding first resonant frequency. Beyond resonance (𝜔 > 𝜔1) the Winkler

springs are normalized by the soil shear modulus 𝐺𝑠 and the frequencies are presented using an

incremental dimensionless frequency term which is a function of the well-known excitation frequency

a0 = 𝜔𝑑/𝑉𝑠 and the cutoff frequency a𝑐𝑢𝑡𝑜𝑓𝑓 = ω1𝑑 𝑉𝑠⁄ = π 𝑑 2𝐿⁄ . The damping ratio 𝛽 is kept

constant across the entire frequency range. The formulations from (Mylonakis, 2001b) and (Anoyatis

& Lemnitzer, 2017) are compared with the proposed expression. For completeness the performance of

the plane strain Winkler modulus is evaluated in the high frequency range as it cannot capture

resonance (see (Anoyatis, et al., 2016)). A general trend can be observed from Fig. 3: below

resonance all formulations decrease with increasing frequency and attain a minimum value at 𝜔 ≅

𝜔1. Beyond resonance all curves experience a minimal drop in stiffness associated with material

damping before steadily increasing with increasing frequency and practically converging into a single

curve.

The damping shows a relatively uniform behavior across both frequency ranges. Below resonance

(𝜔 < 𝜔1) damping is practically unaffected by frequency and depends solely on the soil material

damping (since only “weak” travelling waves develop in the medium (Anoyatis, et al., 2016)). An

increase in damping due to energy radiation is observed when the excitation frequency approaches

resonance as horizontally travelling waves emerge in the soil medium. Beyond cutoff frequency

damping increases with increasing frequency. The proposed expression shows slightly higher

damping in the high frequency range compared to the other expressions. However, this difference

(≲ 10 %) is negligible for practical purposes as typical kinematic excitation frequencies are rarely

found in this range. Stiffness 𝑘 and damping 𝛽 become gradually independent of pile slenderness

ratios and all 𝐿/𝑑 curves converge into one single curve at high frequencies. Hence only four curves

are shown at 0.4 ≲ (a02 − a𝑐𝑢𝑡𝑜𝑓𝑓

2 )1/2

≲ 1, each representing one of the four expressions investigated.

From Fig. 3 it becomes evident that the proposed expression and (Anoyatis & Lemnitzer, 2017) are

11

practically identical in the low frequency range and very similar in the high frequency range. This

behavior implies that the proposed expression is very versatile and can also be used for predicting pile

head stiffnesses due to inertial loading for which the (Anoyatis & Lemnitzer, 2017) formulation was

originally developed.

To complete the discussion on the variation of damping ratios with frequency, Fig. 4 offers a

comprehensive comparison of the damping ratios 𝛽 listed in Table 2. Compared to the dynamic plane

strain expression the majority of the available expressions over-predict the damping ratio.

The pile-soil curvature ratios for various fixity conditions at the pile head and pile tip are presented in

Figs. 5 – 16. Expressions listed in Tables 1 and 2 are implemented in a Winkler model and the

predictions are compared against finite element results, which are used as reference curves. The FE

results are extracted from (Anoyatis, et al., 2013) in which analyses were performed by means of the

commercial computer platform ANSYS. A detailed description of the FE model is provided in

(Anoyatis, et al., 2013).

Figure 5 focus on the static curvature ratio 𝐶𝑅0 for a fixed head pile with free conditions at the tip.

For pile slenderness ratio 𝐿 𝑑⁄ = 5 Fig. 5(a) shows the largest divergence among all results (160 %).

While the expressions proposed by (Kavvadas & Gazetas, 1993) and Mylonakis (2001a, b)

approximate the FE solution reasonably well, all other expressions converge into one curve across the

entire range of pile slenderness ratios examined. For 𝐿 𝑑⁄ > 10 all expressions offer similar results.

For 𝐸𝑝 𝐸𝑠 = 10000⁄ (Fig. 5b) all curves except for (Mylonakis, 2001a) under-predict the curvature

ratio.

In the case of a fixed tip condition (Fig. 6a) the general shape of the curves for all expressions is

similar and follows the trend of the rigorous FE solution. However, none of the expressions is capable

of capturing the FE results over the entire range of 𝐿 𝑑⁄ ’s. Much better agreement is noticeable for

𝐸𝑝 𝐸𝑠 = 10000⁄ (Fig. 6b) where general agreement between the expressions and the FE solution can

be observed up to 𝐿 𝑑⁄ = 12.

Figures 7 and 8 investigate the curvatures ratios at the pile tip (𝐶𝑅𝐿) for a fixed- and free-head pile,

respectively. A comparison of Figures 7 and 8 indicates that the curvature ratio is governed by the

fixity condition at the base and results are similar regardless of the boundary condition at the pile

head. The closest agreement for fixed head piles with 𝐸𝑝 𝐸𝑠 = 1000⁄ (Figs. 7a) was reached by

Mylonakis (2001b) (Eqs. 11a, b and A.1). For free head piles with the same pile-soil stiffness ratio

(Figs. 8a) excellent agreement is observed for (Syngros, 2004), closely followed by Mylonakis

(2001b) (Eqs. 11a,b and A.1). For 𝐸𝑝 𝐸𝑠 = 10000⁄ (Figs. 7b and 8b) all expressions yield similar

results and can be confidently used to estimate the 𝐶𝑅𝐿. The FE solution appears to be the average of

all plotted expressions.

12

Figures 9 to 12 show static curvature ratios for a selected set of frequency-dependent Winkler moduli

from Table 2. In order to obtain static results, frequency ω was set equal to zero. This evaluation

omits the use of the Baranov-Novak plane strain formulation (as it collapses at 𝜔 = 0), as well as the

Mylonakis’ (2001b) formulation (as it reduces to the static results previously presented in Figs. 5 – 8).

The proposed expression (Eq. 21) along with the one previously introduced by (Anoyatis &

Lemnitzer, 2017) is compared to the FEM solution (Anoyatis, et al., 2013) as done for Figs. 5 – 8. The

Authors chose a separate set of figures for this comparison to enable a better performance evaluation

and visualization of the proposed expression.

In the case of a fixed head pile with a free tip (Figs. 9a, 9b) the static curvature ratio is well captured

by the proposed expression and performs slightly better than the (Anoyatis & Lemnitzer, 2017) across

the entire range of 𝐿 𝑑⁄ ’s. For piles with double fixity (head and tip, Figs. 10a, b) results using the

proposed expression are in alignment with the curves shown in Figs. 6(a) and 6(b). The modified

proposed expression for bottom fixity (Eq. 24) shows an excellent agreement for all pile-soil stiffness

ratios and all pile head boundary conditions examined (Figs. 11, 12). This indicates a powerful

improvement over the existing expressions as shown by the bandwidth in Figs. 7 and 8.

Figures 13 and 14 present the amplitude of the curvature ratio 𝐶𝑅0 for a short (𝐿 𝑑⁄ = 5) and a long

(𝐿 𝑑⁄ = 10) pile using soil material damping of 𝛽𝑠 = 0.10. For 𝐿 𝑑⁄ = 5 the results can be found in

slightly better agreement for the case of the low pile-soil stiffness ratio (𝐸𝑝 𝐸𝑠⁄ = 1000). Only

Mylonakis (2001a) is aligned with the reference FE solution. Results from (Mylonakis, 2001b) (Eqs.

18), (Anoyatis & Lemnitzer, 2017) (Eqs. 19) and the proposed expression (Eq. 21) (all of them being

complex valued expressions) can capture the small drop associated with the first resonance of the soil-

pile system, while the implementation of the plane strain modulus leads to a poor performance of the

model for both pile-soil stiffness ratios examined. A much wider discrepancy is observed for the high

pile-soil stiffness ratio shown in Fig. 13(b). While the complex valued expressions can capture the

resonant effect, neither those nor other expressions from literature come close to the reference

solution. Much better results are obtained for the case of a longer pile (Fig. 14). For the same soil

material and pile boundary conditions, results for low pile-soil stiffness ratios fall within a small

bandwidth, including those from the plane strain modulus, and show good agreement with the FE

solution (Fig. 14a). Large dispersion is observed for the high pile-soil stiffness ratio at small

frequencies (Fig. 14b). With increasing frequency, the bandwidth becomes smaller and all solutions

tend to converge in high frequencies. Similar to the previous figures the complex valued expressions

capture resonance even though the drop in 𝐶𝑅0 may be overestimated.

It is evident that the results using (Mylonakis, 2001b) and the proposed expression are in very close

agreement. This is anticipated since the new modulus is developed by applying pertinent

modifications on the basis of the “static” Mylonakis’ expression as shown in Table 1. The main

13

advantage of the proposed expression lies in its mathematical simplicity (no Bessel functions are

included) which in turn allows numerical results to be obtained with the use of a simple calculator by

making use of the alternative simple algebraic expressions provided in the Appendix C. These

equivalent expressions allow for separately computing the stiffness (real part of 𝑘∗) and the damping

(imaginary part of 𝑘∗) and will be of greater use when evaluating the kinematic response factors

where only the dynamic spring is implemented (a “static” 𝜆, Eq. 20).

It is worth mentioning that the performance of the dynamic plane strain modulus (Eqs. 13) in

predicting the dynamic 𝐶𝑅0 fluctuates. According to the Authors the suitability of the plane strain

modulus is not only associated with the pile geometry (short or long pile) but with the cutoff

frequency and the pile-soil stiffness ratio: for the case of a soil layer characterized by a small cutoff

frequency and low pile-soil stiffness contrast (as in Fig. 14) good performance is expected. In

particular, the pile geometry (𝐿 𝑑⁄ ) in conjunction with 𝐸𝑝 𝐸𝑠⁄ may be a better combination to

evaluate whether the pile behaves as “long” or “short” (see mechanical slenderness, (Anoyatis, et al.,

2013)). Note that in all cases the dynamic plane strain modulus fails to evaluate static 𝐶𝑅0 as the

solution collapses at 𝜔 = 0.

An alternative representation of the dynamic curvature ratio is offered in Figs. 15 and 16 where 𝐶𝑅0 is

normalized with its static value. This approach allows all expressions to be closer aligned with the FE

solution. In Figs. 15 (a) and (b) all expressions for frequency independent springs collapse into a

single curve and under-predict the dynamic reduction of 𝐶𝑅0 – the largest deviation being in the

vicinity of resonance. The complex valued expressions ( (Mylonakis, 2001b), (Anoyatis & Lemnitzer,

2017) and the proposed) can capture the drop at resonance but over-predict the magnitude in curvature

reduction ratio (Fig. 15a). For 𝐸𝑝 𝐸𝑠⁄ = 10000 the FE solution indicates a larger drop of 𝐶𝑅0 at a

slightly higher frequency, which is currently not being accurately captured by the 𝑘∗ moduli. In both

Figures Mylonakis’ and the proposed expression yield results in very close agreement. Nevertheless,

these expressions still offer a significant improvement over the existing expressions from the

literature. For the case of a long pile (Fig. 16) all solutions capture the dynamic reductions reasonably

well.

Figures17 and 18 extend the presentation of results in terms of the kinematic response factor 𝐼𝑢 for

fixed- and free-head piles with 𝐿 𝑑⁄ = 20 and 𝛽𝑠 = 0.05. In both graphs (Fan, et al., 1991) and

(Liang, et al., 2013) serve as finite element and boundary element reference solutions, respectively.

Using the parameter λ from Eq. (20) (i.e., neglect the dashpot) results from complex valued

expressions collapse into a single curve and are shown to be in very good agreement with the FE

solution by (Fan, et al., 1991) for all boundary conditions. Their performance is improved for free

head piles (Fig. 18), where the agreement with FE is extended over the entire range of frequencies

examined (except high frequencies and very soft soil, Fig. 18b).

14

A comparison of the kinematic response factor in rotation 𝐼𝜑 is presented in Fig. 19, where similar

performance as previously shown in Fig. 17 and 18 is observed. For stiff soils (𝐸𝑝 𝐸𝑠⁄ = 1000), four

of expressions ( (Dobry & O'Rourke, 1983), (Kavvadas & Gazetas, 1993), (Mylonakis, 2001a),

(Mylonakis 2001b, Eqs. 11a, b and A.1)) over-predict the amplitude 𝐼𝜑 by up to 25 % for frequencies

a0 > 0.4, while the remaining expressions yield results in very close vicinity of the FE solution. For

very soft soils (𝐸𝑝 𝐸𝑠⁄ = 10000) all curves (except for (Kavvadas & Gazetas, 1993) and (Mylonakis,

2001a)) align closely with the FE solution. The BE solution captures the general shape but

experiences an offset in frequency.

6. Conclusions

This study investigates the suitability of static and frequency-dependent (complex valued) Winkler

moduli from the literature to predict the response of kinematically stressed piles using a traditional

Winkler model. Results are evaluated in terms of curvature ratio and kinematic response factors in

translation 𝐼𝑢 and rotation 𝐼𝜑 as introduced in the earlier study of (Anoyatis, et al., 2013). For all

results presented herein the term associated with the pile inertia is neglected following (Anoyatis, et

al., 2013) and satisfactory performance of the model was achieved. Following the recommendation of

the aforementioned study, all Winkler moduli with separate formulations for springs and dashpots

were implemented considering the spring stiffness only (damping was omitted). Complex valued

moduli that integrate stiffness and damping into one single expression were implemented as published

to evaluate the dynamic curvature ratios at the pile head and pile tip. The kinematic response factors

were computed using the real part only (i.e., dynamic spring stiffness) and the imaginary part

associated with the damping was neglected.

The study showed that most available expressions for the Winkler modulus are capable of capturing

the static curvature ratios 𝐶𝑅0 and 𝐶𝑅𝐿, the dynamic modification 𝐶𝑅0 𝐶𝑅0,𝑠𝑡𝑎𝑡𝑖𝑐⁄ as well as the

kinematic factors 𝐼𝑢 and rotation 𝐼𝜑 at low frequencies. As opposed to inertial loading, it was shown

that results are less sensitive to the selection of 𝑘 when piles are kinematically stressed. The extensive

graphical comparisons of the results provided in the manuscript offer the geotechnical engineer the

capability to select the expression most suitable for the respective pile geometry, soil properties and

boundary conditions.

In addition to the evaluation of literature formulations a simple logarithmic based expression for the

Winkler modulus is introduced. The proposed expression is motivated by Mylonakis’ (2001b)

formulation derived for static conditions (Eq. 11a, b, c) and modified with a frequency-dependent, and

resonant-dependent parameter (Eq. 22) to enable the implementation of the expression in dynamic

15

conditions. The modulus was further modified to obtain accurate results for curvature ratio at the pile

tip 𝐶𝑅𝐿 (Eq. 24).

Specific observations pertaining to the newly proposed Winkler modulus can be summarized as

follows:

1. By comparing the predictions of the Winkler model using the proposed 𝑘 against rigorous FE

results it was found that 𝑘 performs well for static and dynamic curvature ratios examined

(𝐶𝑅0, | 𝐶𝑅0| |𝐶𝑅0,𝑠𝑡𝑎𝑡𝑖𝑐|⁄ ). An even better performance is observed for the evaluation of

kinematic response factors (𝐼𝑢, 𝐼𝜑) by implementing the real part of the modulus only (i.e.,

dynamic spring stiffness).

2. By introducing a physically motivated mathematical modification to the proposed expression,

an excellent performance in predicting the curvature ratio at the depth 𝑧 = 𝐿 for piles with

bottom fixity (e.g., rock socketed) is achieved.

3. The proposed modulus 𝑘 is advantageous over existing frequency-independent formulations

as such cannot capture resonant effects (e.g., drop in 𝐶𝑅 at 𝜔 = 𝜔1).

4. The proposed modulus 𝑘 offers a lucrative alternative over available complicated expressions

(e.g., (Mylonakis, 2001b), (Anoyatis & Lemnitzer, 2017)) as no Bessel functions are included

and results can be evaluated via a calculator using equivalent expressions which account

separately for dynamic stiffness and damping (Appendix C). The Authors consider this fact

particularly important for engineering practice, where calculations should be made fast, at

least for preliminary analyses.

5. Besides its kinematic application in this study, the versatility of the new modulus allows it to

be implemented in a Winkler model to predict pile head stiffnesses and damping (inertial

problem). All other frequency-independent expressions from the literature are limited by their

unique application to the kinematic problem, meaning such expressions will yield erroneous

results under inertial loading (Anoyatis & Lemnitzer, 2017).

16

APPENDIX A

The cutoff frequency using an exponential shape function is given by (Mylonakis, 2001b)

a𝑐 = 𝜇 𝑑

√ 𝑒8𝜇𝐿 − 1 − 2𝑒2𝜇𝐿[1 + 4𝜇𝐿 + 𝑒4𝜇𝐿(4𝜇𝐿 − 1)] cos 2𝜇𝐿

+2𝑒2𝜇𝐿(1 + 𝑒4𝜇𝐿) sin 2𝜇𝐿 + 2𝑒4𝜇𝐿(sin 4𝜇𝐿 − 8𝜇𝐿)

3(𝑒8𝜇𝐿 − 1) +

2𝑒2𝜇𝐿 {3(𝑒4𝜇𝐿 − 1) cos2𝜇𝐿 −

−[3 − 4𝜇𝐿 + 𝑒4𝜇𝐿(3 + 4𝜇𝐿) + 6𝑒2𝜇𝐿 cos 2𝜇𝐿] sin2𝜇𝐿}

(𝐴. 1𝑎)

where

𝜇 ≈ (𝐸𝑠

4 𝐸𝑝 𝐼𝑝)

14

(𝐴. 2𝑏)

17

APPENDIX Β

The dashpot 𝑐 can be expressed as the sum of a material dashpot 𝑐𝑚 and a radiation dashpot 𝑐𝑟:

𝑐 = 𝑐𝑚 + 𝑐𝑟. In all the following expressions 𝑐𝑚 = 2𝑘𝛽𝑠 𝜔⁄ .

(Dobry, et al., 1982)

0.124

1.55 1pr

s

s s s

Ecv

d V E

(B.1)

(Gazetas & Dobry, 1984b)

5/43/4

1/4

0

3.44 1 a

4 1

r

s s s

c

d V v

(B.2a)

3/4

1/4

08 a , 2.54

r

s s

cz d

d V

(B.2b)

(Makris & Gazetas, 1992)

1/4

06ar

s s

c

d V

(B.3)

(Kavvadas & Gazetas, 1993)

5/4

1/4

0

3.42 1 a

1

r

s s s

c

d V v

(B.4a)

1/4

04a , 2.5r

s s

cz d

d V

(B.4b)

18

APPENDIX C

For small values of the soil material damping such as we can assume that 𝛽𝑠2 → 0, Eq. (22) can be cast

in the following approximate form

2 2 2

0 0

1

2a a 2 acutoff s

imaginaryreal

s i (C.1)

For 0a acutoff

*2

2 2

2Real 8 s

s

s

R Ik

G R I

(C.2a)

*2

2 2

2Imaginary 8 s

s

s

R Ik

G R I

(C.2b)

where

2ln 1 4ln 2 2s sR x (C.3a)

21 sI y (C.3b)

and

2

2 2

0 2

1

21ln a a ln 1

2 1

scutoffx

(C.4a)

2

1

2

1

sy ArcTan

(C.4b)

For 0a acutoff , set 2 2 2 2

0 0a a a acutoff cutoff and

2

1

2

1

sy ArcTan

(C5b)

All Eqs. (C.2) – (C.4a) remain as shown above.

19

References

Anoyatis, G., Di Laora, R., Mandolini, A. & Mylonakis, G., 2013. Kinematic response of single piles

for different boundary conditions: Analytical solutions and normalization schemes. Soil Dynamics

and Earthquake Engineering, Volume 44, pp. 183-195.

Anoyatis, G. & Lemnitzer, A., 2017. Dynamic pile impedances for laterally–loaded piles using

improved Tajimi and Winkler formulations. Soil Dynamics and Earthquake Engineering, Volume 92,

p. 279–297.

Anoyatis, G., Mylonakis, G. & Lemnitzer, A., 2016. Soil Resistance to Lateral Harmonic Pile Motion.

Soil Dynamics and Earthquake Engineering, Volume 87, p. 164–179.

Baranov, V. A., 1967. On the calculation of excited vibrations of an embedded foundation (in

Russian), No. 14, Polytech. Inst. Riga, pp. 195-209: Voprosy Dynamiki Prochnocti.

Berger, E., Mahi, S. A. & Pyke, R., 1977. Simplified method for evaluating soil-pile-structure

interaction effects. Houston, TX, Offshore Technology Conference.

Blaney, G. W., Kausel, E. & Roesset, J. M., 1976. Dynamic Stiffness of Piles. Blacksburg, Virginia,

Proc. 2nd Int. Conf. Numer. Meth. Geomech., 1001-1012.

Chidichimo, A. et al., 2014. 1-g Experimental investigation of bi-layer soil response and kinematic

pile bending. Soil Dynamics and Earthquake Engineering, Volume 67, p. 219–232.

Di Laora, R. & Rovithis, E., 2015. Kinematic Bending of Fixed-Head Piles in Nonhomogeneous Soil.

Journal of Geotechnical and Geoenvironmental Engineering, 141(4), p. 04014126.

Dobry, R. & O'Rourke, M. J., 1983. Discussion on seismic response of end-bearing piles by Flores-

Berrones and Whitman R V J. Journal of Geotechnical Engineering, 109(5), pp. 778-781.

Dobry, R., Vicente, E., O’Rourke, M. & Roesset, J. M., 1982. Horizontal stiffness and damping of

single piles. Journal of Geotechnical and Geoenvironmental Engineering ASCE, 108(3), pp. 439-459.

Fan, K. et al., 1991. Kinematic seismic response of single piles and pile groups. J. Geotech. Engrg,

117(12), pp. 1860-1879.

Francis, A. J., 1964. Analysis of pile groups with flexural resistance. Journal of the Soil Mechanics

and Foundations Division, 90(3), pp. 10-32.

Gazetas, G. & Dobry, R., 1984a. Simple radiation damping model for piles and footings. J. Eng.

Mech., 110(6), pp. 937-956.

Gazetas, G. & Dobry, R., 1984b. Horizontal response of piles in layered soils. J. of Geotech. Engrg,

110(1), pp. 20-40.

Kampitsis, A. E., Sapountzakis, E. J., Giannakos, S. K. & Gerolymos, N. A., 2013. Seismic soil–pile–

structure kinematic and inertial interaction—A new beam approach. Soil Dynamics and Earthquake

Engineering, Volume 55, p. 211–224.

Kavvadas, M. & Gazetas, G., 1993. Kinematic seismic response and bending of free-head piles in

layered soil. Geotechnique, 43(2), pp. 207-222.

Kaynia, A. M. & Kausel, E., 1991. Dynamics of piles and pile groups in layered soil media. Soil

Dynamics and Earthquake Engineering, 10(8), pp. 386-401.

Krishnan, R., Gazetas, G. & Velez, A., 1983. Static and dynamic lateral deflexion of piles in non-

homogeneous soil stratum. Géotechnique, 33(3), pp. 307-325.

Liang, F., Chen, H. & Guo, W. D., 2013. Simplified Boundary Element Method for Kinematic

Response of Single Piles in Two-Layer Soil. Journal of Applied Mathematics, Volume 2013, pp. 1-12.

20

Makris, N. & Gazetas, G., 1992. Dynamic pile-soil-pile interaction. Part II: Lateral and seismic

response. Earthquake Engineering and Structural Dynamics, 21(2), pp. 145-162.

Mylonakis, G., 2001a. Simplified model for seismic pile bending at soil layer interfaces. Soils and

Foundations, 41(4), p. 47–58.

Mylonakis, G., 2001b. Elastodynamic model for large-diameter end-bearing shafts. Journal of the

Japanese Geotechnical Society : soils and foundation, 41(3), pp. 31-44.

Novak, M., 1974. Dynamic Stiffness and Damping of Piles. Canadian Geotechnical Journal, 11(4),

pp. 574-598.

Novak, M., Aboul-Ella, F. & Nogami, T., 1978. Dynamic soil reactions for plane strain case. Journal

of the Engineering Mechanics Division, 104(4), pp. 953-959.

Pender, M., 1993. Seismic pile foundation design analysis. Bulletin of the New Zealand National

Society for Earthquake Engineering, 26(1), pp. 49-160.

Roesset, J. M., 1980. The use of simple models in soil-structure interaction. Knoxville, TN,

Proceedings of Civil Engineering and Nuclear Power, ASCE Specialty Conference, Volume 2.

Roesset, J. M. & Angelides, D., 1980. Dynamic stiffness of piles. London, England, Numerical

methods in offshore piling, Institution of Civil Engineers,, pp. 75-80.

Syngros, C., 2004. Seismic Response of piles and pile-supported bridge piers evaluated through case

histories, City University of New York: Ph. D. thesis.

Vesic, A., 1961. Bending of beam resting on isotropic elastic solid. J. Engng Mech. Div., 87(2), pp.

35-53.

Yoshida, I. & Yoshinaka, R., 1972. A method to estimate modulus of. Soils and Foundations, 12(3),

pp. 1-17.

21

Table 1: “Static” expressions for Winkler moduli 𝑘 found in literature

Studies spring modulus 𝑘

(Francis, 1964) (inertial interaction)

𝑘 = 1.67 𝐸𝑠

1 − 𝜈𝑠2(𝐸𝑝

𝐸𝑠)−1/12

Eq. (7)

(Roesset, 1980) (inertial interaction)

𝑘 = 1.2 𝐸𝑠 Eq. (8)

(Dobry & O'Rourke, 1983) (kinematic interaction)

𝑘 = 3 𝐺𝑠 Eq. (9)

(Mylonakis, 2001a) (kinematic interaction)

𝑘 = 6 𝐸𝑠 (𝐸𝑝

𝐸𝑠)−1/8

Eq. (10)

(Mylonakis, 2001b) (inertial interaction)

𝑘 = 4 𝜋 𝐺𝑠 𝜂𝑠

2

ln(𝜂𝑠) + (1 + 𝜂𝑠2) [ln (

2𝑠) − 𝛾]

, 𝑠 = a𝑐𝑢𝑡𝑜𝑓𝑓 2⁄ Eqs. (11a, b)

sinusoidal shape function: a𝑐𝑢𝑡𝑜𝑓𝑓 =𝜋

2 (𝐿

𝑑)−1

Eqs (11c)

exponential shape function: a𝑐𝑢𝑡𝑜𝑓𝑓: Eqs. (A.1) and (A.2)

(Syngros, 2004) (inertial interaction)

𝑘 = 2.0 𝐸𝑠 (𝐸𝑝

𝐸𝑠)−0.075

, 𝑓𝑖𝑥𝑒𝑑 ℎ𝑒𝑎𝑑 Eq. (12a)

𝑘 = 3.5 𝐸𝑠 (𝐸𝑝

𝐸𝑠)−0.11

, 𝑓𝑟𝑒𝑒 ℎ𝑒𝑎𝑑 Eq. (12b)

22

Table 2: Frequency-dependent expressions for Winkler moduli (𝑘, 𝛽) found in literature (𝛽 = 𝛽𝑠 + 𝛽𝑟)

Studies spring 𝑘 damping ratio 𝛽

(Baranov, 1967)

(Novak, 1974)

(Novak, et al., 1978)

(dynamic plane strain model)

𝑘∗ = 𝜋 𝐺𝑠∗ 𝑠2

4𝐾1(𝑞)𝐾1(𝑠) + 𝑠𝐾1(𝑞)𝐾0(𝑠) + 𝑞𝐾0(𝑞)𝐾1(s)

𝑞𝐾0(𝑞)𝐾1(𝑠) + 𝑠𝐾1(𝑞)𝐾0(𝑠) + 𝑞𝑠𝐾0(𝑞)𝐾0(𝑠)

𝑠 =𝑖 a0

2√1 + 2𝑖𝛽𝑠, 𝑞 =

𝑠

𝜂𝑠, 𝜂𝑠 = √

2(1 − 𝜈𝑠)

1 − 2𝜈𝑠

Eq. (13a)

Eqs. (13b, c, d)

(Dobry, et al., 1982)

(inertial interaction) 𝑘 = 1.67 𝐸𝑠 (

𝐸𝑝

𝐸𝑠)−0.053

𝛽𝑟 = 1.55 (1 + 𝜈𝑠2

) (𝐸𝑝

𝐸𝑠)0.124

a0 𝐺𝑠 𝑘⁄ Eqs. (14a, b)

(Gazetas & Dobry, 1984b)

(inertial interaction)

fixed head

𝑘 = 1 − 1.2 𝐸𝑠

free head

𝑘 = 1.5 − 2.5 𝐸𝑠

𝛽𝑟 = 2 (𝜋

4)3/4

a03/4 [1 + (

3.4

𝜋(1 − 𝜈𝑠))

54] 𝐺𝑠 𝑘⁄

or for shallow depths (𝑧 ≤ 2.5𝑑)

𝛽𝑟 = 4 (𝜋

4)3/4

a03/4 𝐺𝑠 𝑘⁄

Eqs. (15a, b, left)

Eqs. (15c, d, right)

(Makris & Gazetas, 1992)

(inertial interaction) 𝑘 = 1.2 𝐸𝑠 𝛽𝑟 = 3 a0

3/4 𝐺𝑠 𝑘⁄ Eqs. (16a, b)

(Kavvadas & Gazetas, 1993)

(kinematic interaction) 𝑘 =

3 𝐸𝑠1 − 𝜈𝑠

2 (𝐸𝑝

𝐸𝑠)−1/8

(𝐿

𝑑)1/8

𝛽𝑟 = 2 a03/4

[1 + (3.4

𝜋(1 − 𝜈𝑠))

54]𝐺𝑠 𝑘⁄

or for shallow depths (𝑧 ≤ 2.5𝑑)

𝛽𝑟 = 4 a03/4 𝐺𝑠 𝑘⁄

Eq. (17a, left)

Eqs. (17b, c, right)

(Mylonakis, 2001b)

(inertial interaction)

𝑘∗ = 𝜋 𝐺𝑠∗ 𝑠2

4𝐾1(𝑞)𝐾1(𝑠) + 𝑠𝐾1(𝑞)𝐾0(𝑠) + 𝑞𝐾0(𝑞)𝐾1(s)

𝑞𝐾0(𝑞)𝐾1(𝑠) + 𝑠𝐾1(𝑞)𝐾0(𝑠) + 𝑞𝑠𝐾0(𝑞)𝐾0(𝑠) Eq. (18a)

𝑠 =1

2√a𝑐𝑢𝑡𝑜𝑓𝑓

2 −a02

1 + 2𝑖𝛽𝑠, 𝑞 =

𝑠

𝜂𝑠, 𝜂𝑠 = √

2(1 − 𝜈𝑠)

1 − 2𝜈𝑠 , a𝑐𝑢𝑡𝑜𝑓𝑓 =

𝜋 𝑑

2 𝐿 Eqs. (18b,c,d,e)

(Anoyatis & Lemnitzer, 2017)

(inertial interaction)

𝑘∗ = 𝜋 𝐺𝑠∗ 𝑠 (𝑠 + 4

𝐾1(𝑠)

𝐾0(𝑠))

𝐼𝑚(𝑘∗) = 𝜋 𝐺𝑠∗ 𝑠 (𝑠 + 4

𝐾1(𝑠)

𝐾0(𝑠)) + 𝑖 2.5 (a0

2 − a𝑐𝑢𝑡𝑜𝑓𝑓2 )

1/2, a0 > a𝑐𝑢𝑡𝑜𝑓𝑓

Eq. (19a, b)

𝑠 =1

2 (𝜂𝑠)𝜒√a𝑐

2 −a02

1 + 2𝑖𝛽𝑠, 𝜂𝑠 = √

2 − 𝜈𝑠1 − 𝜈𝑠

Eqs. (19c, d)

Expressions for dashpot coefficients 𝑐 (= 2𝛽𝑘 𝜔⁄ ) for (Dobry, et al., 1982), (Gazetas & Dobry, 1984b) and (Makris &

Gazetas, 1992), and (Kavvadas & Gazetas, 1993) are shown in Appendix B

23

Fig. 1. Problem considered

24

Fig. 2. Variation of “static” Winkler moduli with pile-soil stiffness ratio

25

Fig. 3. Variation of frequency-dependent Winkler moduli with frequency

26

Fig. 4. Variation of damping ratios (Table 2) with frequency; 𝐿/𝑑 = 20, 𝛽𝑠 = 0.05, 𝐸𝑝/𝐸𝑠 = 1000

27

Fig. 5. Effect of pile slenderness on static curvature ratio at the pile head for a fixed head – free tip pile

28

Fig. 6. Effect of pile slenderness on static curvature ratio at the pile head for a fixed head – fixed tip pile

29

Fig. 7. Effect of pile slenderness on static curvature ratio at the pile tip for a fixed head – fixed tip pile

30

Fig. 8. Effect of pile slenderness on static curvature ratio at the pile tip for a free head – fixed tip pile

Fig. 9. Effect of pile slenderness on static curvature ratio at the pile head for a fixed head – free tip pile;

(Anoyatis & Lemnitzer, 2017) and the proposed expression

31

Fig. 10. Effect of pile slenderness on static curvature ratio at the pile head for a fixed head – fixed tip pile;

(Anoyatis & Lemnitzer, 2017) and the proposed expression

Fig. 11. Effect of pile slenderness on static curvature ratio at the pile tip for a fixed head – fixed tip pile;

(Anoyatis & Lemnitzer, 2017) and the proposed expression

32

Fig. 12. Effect of pile slenderness on static curvature ratio at the pile tip for a free head – fixed tip pile;

(Anoyatis & Lemnitzer, 2017) and the proposed expression

33

Fig. 13. Variation of the amplitude of curvature ratio at the pile head with frequency for a free head –

fixed tip pile; 𝐿/𝑑 = 5, 𝛽𝑠 = 0.10

34

Fig. 14. Variation of the amplitude of curvature ratio at the pile head with frequency for a free head –

fixed tip pile; 𝐿/𝑑 = 10, 𝛽𝑠 = 0.10

35

Fig. 15. Variation of the normalized curvature ratio at the pile head with frequency for a free head – fixed

tip pile; 𝐿/𝑑 = 5, 𝛽𝑠 = 0.10

36

Fig. 16. Variation of the normalized curvature ratio at the pile head with frequency for a free head – fixed

tip pile; 𝐿/𝑑 = 10, 𝛽𝑠 = 0.10

37

Fig. 17. Variation of the kinematic response factor in translation with frequency for a fixed head – free tip

pile; 𝐿/𝑑 = 20, 𝛽𝑠 = 0.05

38

Fig. 18. Variation of the kinematic response factor in translation with frequency for a free head – free tip

pile; 𝐿/𝑑 = 20, 𝛽𝑠 = 0.05

39

Fig. 19. Variation of the kinematic response factor in rotation with frequency for a free head – free tip

pile; 𝐿/𝑑 = 20, 𝛽𝑠 = 0.05