mathematics

Article

An Analytical Method for the Longitudinal Vibrationof a Large-Diameter Pipe Pile in RadiallyHeterogeneous Soil Based onRayleigh–Love Rod Model

Zhimeng Liang 1, Chunyi Cui 1,*, Kun Meng 1,2, Yu Xin 1, Huafu Pei 3 and Benlong Wang 1

1 Department of Civil Engineering, Dalian Maritime University, Dalian 116026, China;liangzhimeng@dlmu.edu.cn (Z.L.); mengkuntumu@dlmu.edu.cn (K.M.); sdxinyu@dlmu.edu.cn (Y.X.);w268041177@dlmu.edu.cn (B.W.)

2 Cardiff School of Engineering, Cardiff University, Queen’s Buildings, The Parade, Cardiff,Wales CF24 3AA, UK

3 Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian 116024, China;huafupei@dlut.edu.cn

* Correspondence: cuichunyi@dlmu.edu.cn; Tel.: +86-411-84723186

Received: 9 August 2020; Accepted: 26 August 2020; Published: 28 August 2020�����������������

Abstract: Based on the Rayleigh–Love rod model and Novak’s plane-strain theory, an analyticalmethod for the longitudinal vibration of a large-diameter pipe pile in radially heterogeneous soilis proposed. Firstly, the governing equations of the pile-soil system are established by taking boththe construction disturbance effect and transverse inertia effect into account. Secondly, the analyticalsolution of longitudinal dynamic impedance at the pile top can be achieved by using Laplace transformand complex stiffness transfer techniques. Thirdly, the present analytical solution for dynamicimpedance can also be performed in contrast with the existing solution to examine the correctness ofthe analytical method in this work. Further, the effect of pile Poisson’s ratio, pile diameter ratio aswell as soil disturbed degree on the dynamic impedance are investigated. The results demonstratethat the Rayleigh–Love rod is appropriate for simulating the vibration of a large-diameter pipe pile inheterogeneous soils.

Keywords: pile vibration; analytical method; Rayleigh–Love rod model; dynamic impedance;complex stiffness transfer method; construction disturbance effect

1. Introduction

Pile foundations of the building and civil engineering structures often suffer earthquake loads,dynamic machine loads and traffic loads in a natural environment. The theory of the pile-soilinteraction has played an important role in seismic design and dynamic design of foundation [1–3].Many scholars have established lots of simplified computational models in combination with theknowledge of mathematics and mechanics to explore the longitudinal vibration characteristics of pipepiles in the complex geological conditions [4–14]. Based on the one-dimensional (1D) theoretical model,Randolph et al. [15,16] first proposed a new pile-soil coupling vibration model for the thin-wall pipepiles by taking soil plug into account, and then verified the rationality of the model by the experimentaland numerical methods. Liu et al. [17,18] applied the large diameter thin wall pipe piles for naturalsoft-clay, and expounded the advantages of pipe piles applied to practical engineering. Ding’s researchteam [19,20] has developed some analytical methods for the longitudinal vibration of pipe pile byusing Laplace transform and inverse Fourier transform technique, based on Winkler model and

Mathematics 2020, 8, 1442; doi:10.3390/math8091442 www.mdpi.com/journal/mathematics

Mathematics 2020, 8, 1442 2 of 11

three-dimensional (3D) continuous medium model. Further, Zheng et al. [21,22] took the longitudinaland radial displacement of soil into account, and then deduced novel analytical solutions of the verticalvibration frequency domain of the pipe pile by using the potential function method. Based on the 1Dvibration theory, Ai et al. [23] researched the dynamic respond problems of pipe piles in a multilayeredmedium by means of the matrix method and analytical layer-element method. In order to considerthe longitudinal and radial heterogeneity soil, Li et al. [24] introduced a new analytical approach forthe complex impedance of pipe piles by using variable separation technique, and investigated thevertical dynamic response of the pipe pile. According to the plane-strain theory, Wu et al. [25] andLiu et al. [26] further developed a new pile-soil coupling vibration model by introducing an additionalmass to take the soil plug into account, and then highlighted the importance of the apparent wavevelocity in the results of non-destructive testing of tubular piles.

It is obvious that the above pile foundation models are all based on the classical Bernoulli–Eulertheory, which is only suitable for the slender piles. However, the propagation of the stress wave in thepile shaft can be induced due to the dispersion effect, and there is a certain deviation for the utilization ofthe 1D stress-wave theory in the treatment of large-diameter piles. Based on this, it is more appropriateto adopt Rayleigh–Love rod to simulate the vertical vibration of large-diameter piles in the naturalmedium [27–30]. Hence, based on Novak’s model and the Rayleigh–Love rod model, Li et al. [31]and Wu et al. [32] developed new coupling vibration models for the large-diameter solid piles inradial heterogeneous and homogeneous soil, respectively. Afterwards, based on the 3D continuummodel and Rayleigh–Love rod model, Lu et al. [33,34] and Li et al. [35] made an exploration to thevertical vibration for the large-diameter solid pile in a multilayer medium. Further, Zheng et al. [36]established a new coupled vibration model between a large-diameter tubular pile and core soil, andthen emphasized a necessity of taking lateral inertia effect into account for the longitudinal vibrationof large diameter tubular piles.

However, no work so far has focused on the longitudinal vibration of large-diameter tubular pilesin heterogeneous medium by taking lateral inertia effect into account. In light of this, an analyticalmethod for longitudinal vibration of a large-diameter pipe pile in radially heterogeneous soil isdeveloped based on the Rayleigh–Love rod and Novak’s plane-strain models. Further, correspondinganalytical solutions for the dynamic impedance of a large-diameter pipe pile are deduced by employingLaplace transform and complex stiffness transfer techniques. Finally, the influences of pile Poisson’sratio, diameters ratio as well as the disturbance degree on the dynamic impedance are investigatedby means of parametric analysis method. Besides, the present analytical solution can be reduced toinvestigate slender pile or pipe pile embedded in homogeneous soil with hysteretic-type damping,and also suitable for various vibration problems with respect to the 3D effect of wave propagation forthe large-diameter pipe pile.

2. Mechanical Model and Basic Assumptions

A new simplified mechanical model of the pile-soil coupled vibration can be established as shownin Figure 1. A longitudinal height of pile-soil system is H. The inner disturbed zone is dissected into nlongitudinal ring-shaped homogeneous layer numbered 1, 2, . . . , j, . . . , n. r j is the radius of the jthsublayer. rn+1 is the radius of the inner disturbed zone. b is a radial thickness of inner disturbed zone.r0 and r1 represent the inner and outer radius of large-diameter pipe pile, respectively. p(t) representsthe uniformly distributed excitation load. δp and kp denote spring and damper coefficient of the piletoe, respectively. f (r) is a parabolic function [37,38].

Some basic assumptions are applied in this paper:(1) A Rayleigh–Love rod model is introduced to simulate the pipe pile.(2) A complex stiffness transfer technique can be used to transfer shear stress from soil to pile.(3) There exists a small deformation when the pile is subjected to vertical vibration.

Mathematics 2020, 8, 1442 3 of 11

(4) In internal disturbance region, the interface is completely coupled. Gsj and ηs

j satisfy thefollowing expressions:

Gsj(r) =

Gs

1 r = r1

Gsn+1 × f (r) r1 < r < rn+1

G.sn+1 r ≥ rn+1

(1)

ηsj(r) =

ηs

1 r = r1

ηsn+1 × f (r) r1 < r < rn+1

ηsn+1 r ≥ rn+1

(2)

Mathematics 2020, 8, x FOR PEER REVIEW 3 of 11

1s.

1

11s

1

1s1

s )(

nn

nnj

rrG

rrrrfG

rrG

rG (1)

1s

1

11s

1

1s1

s )(

nn

nnj

rr

rrrrf

rr

r

(2)

Figure 1. The mechanical computational model.

3. Governing Equations

According to the elastodynamic theory of Novak’s plane-strain conditions [39], an equilibrium

equation of the jth disturbed layer is established as follows

trut

trurtr

Gr

trurtr

G jjjjjjjj ,),()(1

),()( s

2

2ss

3sss

2

3s

2

2s

(3)

where ),(s tru j , sj and s

j are the longitudinal displacement, viscous coefficient and density,

respectively, of the surrounding medium.

An equilibrium equation of the core soil is also established:

trut

trurt

trur

Gr

trurt

trur

G ,)],(),([1

),(),( s02

2s0

s0

3s0

s0

s0

s02

3s0

s02

2s0

(4)

where s0u , s

0 and s0 are the longitudinal displacement, viscous coefficient and density,

respectively, of the core soil.

An equilibrium equation of large-diameter pipe pile shaft is given by

0),(2),(2),(

)(),(),( s

00s

1122

p42pp

2

p2pp

2

p2pp

tzfrtzfr

tz

tzuR

t

tzuA

z

tzuAE (5)

where ),(p tzu , p , pE , p , pA and pR are the longitudinal displacement, mass density, elastic

modulus, Poisson’s ratio, cross-section area and radius of inertia of pipe pile, respectively. s0f and

s1f denote shear stress exerted by the core and disturbed soil on large-diameter pipe pile shaft,

respectively. 20

21

p rrA ，2

)( 20

21p rr

R

.

Stress and displacement continuity conditions at the interface between pile and the core soil

are expressed as

Figure 1. The mechanical computational model.

3. Governing Equations

According to the elastodynamic theory of Novak’s plane-strain conditions [39], an equilibriumequation of the jth disturbed layer is established as follows

(Gsj∂2

∂r2 + ηsj∂3

∂t∂r2 )usj(r, t) +

1r(Gs

j∂∂r

+ ηsj∂3

∂t∂r)us

j(r, t) = ρsj∂2

∂t2 usj(r, t) (3)

where usj(r, t), ηs

j and ρsj are the longitudinal displacement, viscous coefficient and density, respectively,

of the surrounding medium.An equilibrium equation of the core soil is also established:

Gs0∂2

∂r2 us0(r, t) + ηs

0∂3

∂t∂r2 us0(r, t) +

1r[Gs

0∂∂r

us0(r, t) + ηs

0∂3

∂t∂rus

0(r, t)] = ρs0∂2

∂t2 us0(r, t) (4)

where us0, ηs

0 and ρs0 are the longitudinal displacement, viscous coefficient and density, respectively, of

the core soil.An equilibrium equation of large-diameter pipe pile shaft is given by

EpAp ∂u2p(z, t)∂z2 − ρpAp

(∂u2p(z, t)∂t2 + (νpRp)2 ∂

4up(z, t)∂z2∂t2

)− 2πr1 f s

1 (z, t) − 2πr0 f s0 (z, t) = 0 (5)

where up(z, t), ρp, Ep, νp, Ap and Rp are the longitudinal displacement, mass density, elastic modulus,Poisson’s ratio, cross-section area and radius of inertia of pipe pile, respectively. f s

0 and f s1 denote

shear stress exerted by the core and disturbed soil on large-diameter pipe pile shaft, respectively.

Ap = πr21 −πr2

0, Rp =

√(r2

1+r20)

2 .

Mathematics 2020, 8, 1442 4 of 11

Stress and displacement continuity conditions at the interface between pile and the core soil areexpressed as

us0(r0, t) = up(r0, t) (6)

f s0 = τs

0(r)∣∣∣r=r0 (7)

In semi-infinite outer zone, the displacement tends to zero, i.e.,

limr→∞

usn+1 = 0 (8)

The relationships of displacement and stress continuity at the interface between pile shaft and thedisturbed medium are given by

us1(r1, t) = up(r1, t) (9)

f s1 = −τs

1(r)∣∣∣r=r1 (10)

The vibration of the pile can be written by[ρpAp(νpRp)2 ∂

3up(z, t)∂z∂t2 + EpAp ∂up(z, t)

∂z

]∣∣∣∣∣∣z=0

= −p(t) (11)

[ρpAp(νpRp)2 ∂

3up(z, t)∂z∂t2 + EpAp ∂up(z, t)

∂z+ (δp ∂

∂t+ kp)up(z, t)

]∣∣∣∣∣∣z=H

= 0 (12)

4. Solution of the Disturbed Soil

Applying Laplace transform to Equation (3), it can be transformed to

(Gsj∂2

∂r2 + ηsjs∂2

∂r2 )Usj(r, s) +

1r(Gs

j∂∂r

+ ηsjs∂∂r

)Usj(r, s) = ρs

js2Us

j(r, s) (13)

where Usj is Laplace transform of us

j , the transformation formula is U(r, s) =∫ +∞

0 u(r, t)e−stdt, s = iω,

i =√−1.

After further arrangement of Equation (13), it can be given by

(∂2

∂r2 +1r∂∂r

)Usj(r, s) = (qs

j)2Us

j(r, s) (14)

where qsj =

√ρs

js2

Gsj+η

sjs

.

Hence, the general solution of Equation (14) is expressed as

Usj(r, s) = As

jK0(qsjr) + Bs

jI0(qsjr) (15)

where I0(qsjr) and K0(qs

jr) are the first and second kind, respectively, modified Bessel functions of orderzero; As

j and Bsj are undetermined coefficients.

Substituting Equation (8) into Equation (15) and applying Laplace transform, it produces

Usn+1(r, iω) = As

n+1K0(qsn+1r) (16)

Therefore, the vertical shear stress at outer zone is given by

τsn+1 = (Gs

n+1 + iωηsn+1)

∂Usn+1(r, iω)

∂r= −qs

n+1Asn+1(G

sn+1 + iωηs

n+1)K1(qsn+1r) (17)

Mathematics 2020, 8, 1442 5 of 11

The longitudinal shear stress of the jth layer is expressed as

τsj = (Gs

j + ηsjs)∂Us

j(r, s)

∂r= −qs

j(Gsj + ηs

js)[AsjK1(qs

jr) − BsjI1(qs

jr)] (18)

Further, the longitudinal complex stiffness at the outer boundary and inner boundary of the jthlayer are given by

TTsj+1 = −

2πr j+1τsj(r j+1)

Usj(r j+1, s)

=2πr j+1qs

j(Gsj + ηs

js)[AsjK1(qs

jr j+1) − BsjI1(qs

jr j+1)]

AsjK0(qs

jr j+1) + BsjI0(qs

jr j+1)(19)

TTsj = −

2πr jτsj(r j)

Usj(r j, s)

=2πr jqs

j(Gsj + ηs

js)[AsjK1(qs

jr j) − BsjI1(qs

jr j)]

AsjK0(qs

jr j) + BsjI0(qs

jr j)(20)

Therefore, Equation (21) can be obtained by the condition of Equation (9), it yields

TTsj =

2πr jqsj(G

sj + ηs

js)(MsjTTs

j+1 + Esj)

Nsj TTs

j+1 + Fsj

(21)

where Esj , Fs

j , Msj and Ns

j can be viewed in Appendix A.

5. Solutions of the Inner Soil

Performing Laplace transform to Equation (4), it can be transformed to

(Gs0∂2

∂r2 + ηs0s∂2

∂r2 )Us0(r, s) +

1r(Gs

0∂∂r

+ ηs0s∂∂r

)Us0(r, s) = ρs

0s2Us0(r, s) (22)

After further arrangement of Equation (22), the expression is as follows

(∂2

∂r2 +1r∂∂r

)Us0(r, s) = (qs

0)2Us

0(r, s) (23)

where qs0 =

√ρs

0s2

Gs0+η

s0s .

The solution of Equation (23) is expressed as

Us0(r, s) = As

0K0(qs0r) + Bs

0I0(qs0r) (24)

Further, substituting Equation (6) into Equation (23), it produces

Us0(r, s) = Bs

0I0(qs0r) (25)

Therefore, combining Equations (7) and (25), the longitudinal shear stiffness at interface of soil-pileis given by

TT0 = −2πr0τs

0(r0)

Up = −2πr0qs

0(Gs0 + ηs

0s)I1(qs0r0)

I0(qs0r0)

(26)

6. Solutions of the Pipe Pile

Combining with TTsj and TT0, Equation (5) can be transformed to

∂U2p(z, s)∂z2 = α2Up(z, s) (27)

Mathematics 2020, 8, 1442 6 of 11

where α2 =ρpAps2+TT1−TT0

Ap[Ep+ρp(νpRps)2].

The solution of Equation (27) is given by

Up(z, s) =Mpe2αz + Np

eαz (28)

where qsj =

√ρs

js2

Gsj+η

sjs

.

Substituting Equation (28) into Equations (11) and (12) and applying Laplace transform, it produces

Mp =ξsP(s)

EpApα(ξs − 1)(29)

Np =P(s)

EpApα(ξs − 1)(30)

where ξs =ρpAp(νpRps)2+αEp

−(kp+δps)αEp+ρpAp(νpRps)2+(kp+δps)

e−2αH.

Hence, the displacement solution of the pile shaft can be obtained as

Up(z, s) =(ξs + 1)e−αz

EpApα(ξs − 1)P(s) (31)

where P(s) is the Laplace transform of p(t).Further, the longitudinal impedance function of the pile top is given by

Kd =P(s)Up =

EpApα(ξs− 1)

(1 + ξs)=

EpAp

HK′d (32)

K′d =α(ξs

− 1)(ξs + 1)

(33)

where α = αH.The dimensionless complex impedance K′d can be further rewritten as

K′d = Kr + iKi (34)

where Kr represents the stiffness factor; Ki denotes the damping factor.

7. Results and Discussions

Comprehensive parametric analyses are given insight into the dynamic impedance for alarge-diameter pipe pile in radially heterogeneous medium based on a solution of Equation (34).Wang et al. [37] and EI Naggar [40] have pointed out that the analytical solutions tend to be stable bysetting n ≥ 20. Therefore, in this paper, the value of n is equal to 20. In addition, the coefficient ofdisturbance degree ζs [37] is defined as

ζs =√

Gsn+1/ρs

n+1) =√ηs

1/ηsn+1 = Vs

1/Vsn+1 (35)

when ζs = 1, the soil is homogeneous; when ζs > 1, the soil is strengthened; when ζs < 1, the soilis weakened. The parameters of pile-soil system are listed: H = 6 m; r1 = b = 0.5 m; r0 = 0.38 m;Vp = 3200 m/s; ρp = 2500 kg/m3; kp = 1 × 103 kN/m3; δp = 1 × 102 kN · s/m2; Vs

m+1 = Vs0 =

50 m/s; ηsn+1 = ηs

0 = 5 kN · s/m2; ρsj = ρs

0 = 2000 kg/m3; vp = 0.25 and ζs = 1.4.

Mathematics 2020, 8, 1442 7 of 11

7.1. Verification of the Solution

To verify the rationality of the analytical solution of a large-diameter pipe pile in radialheterogeneous medium based on Rayleigh–Love rod model, a degradation of present solutionof the dimensionless complex impedance K′d given in Equation (34) is compared with the existingliterature. By setting vp = 0 and ζs = 1, the Rayleigh–Love rod model in this paper was reduced to theEuler–Bernoulli rod model of Ref. [41]. It is apparent from Figure 2 that the degenerated solution ofthis work with different inner radius agreed well with the previous solution.

Mathematics 2020, 8, x FOR PEER REVIEW 7 of 11

7.1. Verification of the Solution

To verify the rationality of the analytical solution of a large-diameter pipe pile in radial

heterogeneous medium based on Rayleigh–Love rod model, a degradation of present solution of

the dimensionless complex impedance dK given in Equation (34) is compared with the existing

literature. By setting 0p v and 1s , the Rayleigh–Love rod model in this paper was reduced to

the Euler–Bernoulli rod model of Ref. [41]. It is apparent from Figure 2 that the degenerated

solution of this work with different inner radius agreed well with the previous solution.

0 200 400 600 800 1000 1200

-8

-4

0

4

8

12

16

f / Hz

Kr

Present solution (r0=0.38 m)

Previous solution of Ref. [41] (r0=0.38 m)

Present solution(r0→0)

Previous solution of Ref. [41] (r0→0)

0 200 400 600 800 1000 1200

0

4

8

12

16

20

24

28

32

f / Hz

Ki

Present solution (r0=0.38 m)

Previous solution of Ref. [41] (r0=0.38 m)

Present solution(r0→0)

Previous solution of Ref. [41](r0→0)

(a) (b)

Figure 2. Contrast of the degradation of the present solution ( 0pv , 1s ) with the solution of Ref.

[41]: (a) true stiffness and (b) equivalent damping.

7.2. Parametric Analyses

In the following analysis, the characteristic of the large-diameter pipe pile is conducted to

study by taking a transverse inertia effect into account. According to the analytical solution of the

method in this work, the Rayleigh–Love rod can be degenerated into the Euler–Bernoulli rod by

setting 0p v . The effect of Poisson’s ratio on the dynamic impedance is shown in Figure 3. It is

clear that the effect of Poisson’s ratio on the dynamic impedance becomes significant mainly in the

high-frequency range, that is, both the resonance amplitude and frequency of the curves decrease as

Poisson’s ratio increases in high-frequency range, and the amplitude is obviously reducing with a

rising frequency. However, the effect of pile Poisson’s ratio is almost negligible in the

low-frequency region. The results showed that the Rayleigh–Love rod model performs better than

Euler–Bernoulli rod model when simulate large-diameter pipe pile embedded in heterogeneous

medium.

0 200 400 600 800 1000 1200-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

f/Hz

Kr

vP=0 v

P=0.15

vP=0.25 v

P=0.35

0 200 400 600 800 1000 12000

2

4

6

8

10

12

14

16

18

20

f/Hz

Ki

vP=0 v

P=0.15

vP=0.25 v

P=0.35

(a) (b)

Figure 2. Contrast of the degradation of the present solution (vp = 0, ζs = 1 ) with the solution ofRef. [41]: (a) true stiffness and (b) equivalent damping.

7.2. Parametric Analyses

In the following analysis, the characteristic of the large-diameter pipe pile is conducted to studyby taking a transverse inertia effect into account. According to the analytical solution of the method inthis work, the Rayleigh–Love rod can be degenerated into the Euler–Bernoulli rod by setting vp = 0.The effect of Poisson’s ratio on the dynamic impedance is shown in Figure 3. It is clear that the effect ofPoisson’s ratio on the dynamic impedance becomes significant mainly in the high-frequency range,that is, both the resonance amplitude and frequency of the curves decrease as Poisson’s ratio increasesin high-frequency range, and the amplitude is obviously reducing with a rising frequency. However,the effect of pile Poisson’s ratio is almost negligible in the low-frequency region. The results showedthat the Rayleigh–Love rod model performs better than Euler–Bernoulli rod model when simulatelarge-diameter pipe pile embedded in heterogeneous medium.

Figure 4 shows the effect of the diameter ratio on the dynamic impedance at the head of thelarge-diameter pipe pile. As shown in Figure 4, the diameter ratio had a remarkable effect on thedynamic impedance curves. The resonance amplitude and frequency on the dynamic impedancecurves increased significantly with a decrease of the diameter ratio. The resonance amplitude was thelargest when the inner radius of the pipe pile was equal to zero, which means that the inner soil showsan obvious vibration reduction effect on the large-diameter pipe pile. Besides, it can be found thatboth the resonance amplitude and frequency at high-frequency reduced obviously on the dynamicimpedance curves when transverse inertia effect was taken into account (i.e., vp = 0.25), but the curveskept invariant at the low-frequency.

Figure 5 depicts the influences of the transverse inertia effect on the dynamic impedance with adifferent value of the disturbance degree coefficient. As shown in Figure 5, the disturbance degreehad a remarkable effect on the dynamic impedance. Compared to the homogeneous soil (i.e., ζs = 1),the resonance amplitude and frequency on the dynamic impedance curves increased obviouslyin the softening soil (i.e., ζs = 0.6). In contrary, the resonance amplitude and frequency on the

Mathematics 2020, 8, 1442 8 of 11

dynamic impedance curves decreased in the hardening soil (i.e., ζs = 1.4). Besides, when vp = 0.25,the resonance amplitude and frequency on the dynamic impedance curves decreased gradually as theexcitation frequency increased, and this trend became more evident at high-frequency.

Mathematics 2020, 8, x FOR PEER REVIEW 7 of 11

7.1. Verification of the Solution

To verify the rationality of the analytical solution of a large-diameter pipe pile in radial

heterogeneous medium based on Rayleigh–Love rod model, a degradation of present solution of

the dimensionless complex impedance dK given in Equation (34) is compared with the existing

literature. By setting 0p v and 1s , the Rayleigh–Love rod model in this paper was reduced to

the Euler–Bernoulli rod model of Ref. [41]. It is apparent from Figure 2 that the degenerated

solution of this work with different inner radius agreed well with the previous solution.

0 200 400 600 800 1000 1200

-8

-4

0

4

8

12

16

f / Hz

Kr

Present solution (r0=0.38 m)

Previous solution of Ref. [41] (r0=0.38 m)

Present solution(r0→0)

Previous solution of Ref. [41] (r0→0)

0 200 400 600 800 1000 1200

0

4

8

12

16

20

24

28

32

f / Hz

Ki

Present solution (r0=0.38 m)

Previous solution of Ref. [41] (r0=0.38 m)

Present solution(r0→0)

Previous solution of Ref. [41](r0→0)

(a) (b)

Figure 2. Contrast of the degradation of the present solution ( 0pv , 1s ) with the solution of Ref.

[41]: (a) true stiffness and (b) equivalent damping.

7.2. Parametric Analyses

In the following analysis, the characteristic of the large-diameter pipe pile is conducted to

study by taking a transverse inertia effect into account. According to the analytical solution of the

method in this work, the Rayleigh–Love rod can be degenerated into the Euler–Bernoulli rod by

setting 0p v . The effect of Poisson’s ratio on the dynamic impedance is shown in Figure 3. It is

clear that the effect of Poisson’s ratio on the dynamic impedance becomes significant mainly in the

high-frequency range, that is, both the resonance amplitude and frequency of the curves decrease as

Poisson’s ratio increases in high-frequency range, and the amplitude is obviously reducing with a

rising frequency. However, the effect of pile Poisson’s ratio is almost negligible in the

low-frequency region. The results showed that the Rayleigh–Love rod model performs better than

Euler–Bernoulli rod model when simulate large-diameter pipe pile embedded in heterogeneous

medium.

0 200 400 600 800 1000 1200-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

f/Hz

Kr

vP=0 v

P=0.15

vP=0.25 v

P=0.35

0 200 400 600 800 1000 12000

2

4

6

8

10

12

14

16

18

20

f/Hz

Ki

vP=0 v

P=0.15

vP=0.25 v

P=0.35

(a) (b)

Figure 3. Variation of dynamic impedance with pile Poisson’s ratio: (a) true stiffness and(b) equivalent damping.

Mathematics 2020, 8, x FOR PEER REVIEW 8 of 11

Figure 3. Variation of dynamic impedance with pile Poisson’s ratio: (a) true stiffness and (b)

equivalent damping.

Figure 4 shows the effect of the diameter ratio on the dynamic impedance at the head of the

large-diameter pipe pile. As shown in Figure 4, the diameter ratio had a remarkable effect on the

dynamic impedance curves. The resonance amplitude and frequency on the dynamic impedance

curves increased significantly with a decrease of the diameter ratio. The resonance amplitude was

the largest when the inner radius of the pipe pile was equal to zero, which means that the inner soil

shows an obvious vibration reduction effect on the large-diameter pipe pile. Besides, it can be

found that both the resonance amplitude and frequency at high-frequency reduced obviously on

the dynamic impedance curves when transverse inertia effect was taken into account (i.e. 25.0p v ),

but the curves kept invariant at the low-frequency.

Figure 5 depicts the influences of the transverse inertia effect on the dynamic impedance with a

different value of the disturbance degree coefficient. As shown in Figure 5, the disturbance degree

had a remarkable effect on the dynamic impedance. Compared to the homogeneous soil (i.e.,

1s ), the resonance amplitude and frequency on the dynamic impedance curves increased

obviously in the softening soil (i.e., 6.0s ). In contrary, the resonance amplitude and frequency

on the dynamic impedance curves decreased in the hardening soil (i.e., 4.1s ). Besides, when

25.0p v , the resonance amplitude and frequency on the dynamic impedance curves decreased

gradually as the excitation frequency increased, and this trend became more evident at

high-frequency.

0 200 400 600 800 1000 1200

-12

-8

-4

0

4

8

12

16

20 r0/r

1=0, v

P=0 r

0/r

1=0, v

P=0.25

r0/r

1=0.2,v

P=0 r

0/r

1=0.2,v

P=0.25

r0/r

1=0.4,v

P=0 r

0/r

1=0.4,v

P=0.25

f/Hz

Kr

0 200 400 600 800 1000 12000

5

10

15

20

25

30

35

40 r

0/r

1=0, v

P=0 r

0/r

1=0, v

P=0.25

r0/r

1=0.2,v

P=0 r

0/r

1=0.2,v

P=0.25

r0/r

1=0.4,v

P=0 r

0/r

1=0.4,v

P=0.25

f/Hz

Ki

(a) (b)

Figure 4. Variation of dynamic impedance with the diameters ratio: (a) true stiffness and (b)

equivalent damping.

0 200 400 600 800 1000 1200

-1.2

-0.6

0.0

0.6

1.2

1.8

2.4

3.0

3.6

4.2

4.8 ζ

S=0.6, v

P=0 ζ

S=0.6, v

P=0.25

ζS=1.0, v

P=0 ζ

S=1.0, v

P=0.25

ζS=1.4, v

P=0 ζ

S=1.4, v

P=0.25

f/Hz

Kr

0 200 400 600 800 1000 12000

2

4

6

8

10

12

14

16

18

20 ζ

S=0.6, v

P=0 ζ

S=0.6, v

P=0.25

ζS=1.0, v

P=0 ζ

S=1.0, v

P=0.25

ζS=1.4, v

P=0 ζ

S=1.4, v

P=0.25

f/Hz

Ki

720 740 760 780 800 820

11.2

11.6

12.0

12.4

12.8

13.2

(a) (b)

Figure 4. Variation of dynamic impedance with the diameters ratio: (a) true stiffness and(b) equivalent damping.

Mathematics 2020, 8, x FOR PEER REVIEW 8 of 11

Figure 3. Variation of dynamic impedance with pile Poisson’s ratio: (a) true stiffness and (b)

equivalent damping.

Figure 4 shows the effect of the diameter ratio on the dynamic impedance at the head of the

large-diameter pipe pile. As shown in Figure 4, the diameter ratio had a remarkable effect on the

dynamic impedance curves. The resonance amplitude and frequency on the dynamic impedance

curves increased significantly with a decrease of the diameter ratio. The resonance amplitude was

the largest when the inner radius of the pipe pile was equal to zero, which means that the inner soil

shows an obvious vibration reduction effect on the large-diameter pipe pile. Besides, it can be

found that both the resonance amplitude and frequency at high-frequency reduced obviously on

the dynamic impedance curves when transverse inertia effect was taken into account (i.e. 25.0p v ),

but the curves kept invariant at the low-frequency.

Figure 5 depicts the influences of the transverse inertia effect on the dynamic impedance with a

different value of the disturbance degree coefficient. As shown in Figure 5, the disturbance degree

had a remarkable effect on the dynamic impedance. Compared to the homogeneous soil (i.e.,

1s ), the resonance amplitude and frequency on the dynamic impedance curves increased

obviously in the softening soil (i.e., 6.0s ). In contrary, the resonance amplitude and frequency

on the dynamic impedance curves decreased in the hardening soil (i.e., 4.1s ). Besides, when

25.0p v , the resonance amplitude and frequency on the dynamic impedance curves decreased

gradually as the excitation frequency increased, and this trend became more evident at

high-frequency.

0 200 400 600 800 1000 1200

-12

-8

-4

0

4

8

12

16

20 r0/r

1=0, v

P=0 r

0/r

1=0, v

P=0.25

r0/r

1=0.2,v

P=0 r

0/r

1=0.2,v

P=0.25

r0/r

1=0.4,v

P=0 r

0/r

1=0.4,v

P=0.25

f/Hz

Kr

0 200 400 600 800 1000 12000

5

10

15

20

25

30

35

40 r

0/r

1=0, v

P=0 r

0/r

1=0, v

P=0.25

r0/r

1=0.2,v

P=0 r

0/r

1=0.2,v

P=0.25

r0/r

1=0.4,v

P=0 r

0/r

1=0.4,v

P=0.25

f/Hz

Ki

(a) (b)

Figure 4. Variation of dynamic impedance with the diameters ratio: (a) true stiffness and (b)

equivalent damping.

0 200 400 600 800 1000 1200

-1.2

-0.6

0.0

0.6

1.2

1.8

2.4

3.0

3.6

4.2

4.8 ζ

S=0.6, v

P=0 ζ

S=0.6, v

P=0.25

ζS=1.0, v

P=0 ζ

S=1.0, v

P=0.25

ζS=1.4, v

P=0 ζ

S=1.4, v

P=0.25

f/Hz

Kr

0 200 400 600 800 1000 12000

2

4

6

8

10

12

14

16

18

20 ζ

S=0.6, v

P=0 ζ

S=0.6, v

P=0.25

ζS=1.0, v

P=0 ζ

S=1.0, v

P=0.25

ζS=1.4, v

P=0 ζ

S=1.4, v

P=0.25

f/Hz

Ki

720 740 760 780 800 820

11.2

11.6

12.0

12.4

12.8

13.2

(a) (b)

Figure 5. Variation of dynamic impedance with the coefficient of the disturbance degree: (a) truestiffness and (b) equivalent damping.

Mathematics 2020, 8, 1442 9 of 11

8. Conclusions

According to the Rayleigh–Love rod model and elastodynamic theory of Novak’s plane-strain,an analytical method for the longitudinal vibration of the large-diameter pipe pile in radiallyinhomogeneous soil with viscous damping can be proposed by means of the Laplace transformand complex stiffness transfer techniques. The effect of Poisson’s ratio, the diameter ratio as well asthe disturbance extent on the dynamic impedance is investigated. Calculation and analysis resultsshow that:

(1) The impact of pile Poisson’s ratio on the pile head’s dynamic impedance was most evidentin the high-frequency region. The resonance amplitude and frequency of the pile head’s dynamicimpedance curves decreased with an increase of pile Poisson’s ratio, and the tendency became moreobvious with a rising frequency. The results demonstrated the rationality and necessity of introducingRayleigh–Love rod model to simulate the large-diameter pipe pile.

(2) The resonance amplitude and frequency on the dynamic impedance curves had a remarkableincrease as a decrease of the diameter ratio. It is shown that the inner soil had an obvious vibrationreduction effect on the large-diameter pipe pile.

(3) In the softening (hardening) soil, the resonance amplitude and frequency on the dynamicimpedance curves were larger (smaller) than those in the homogeneous soil. When the transverseinertia effect was considered, the change became more noticeable at the high frequency.

(4) The degeneration of the present solution corresponded well to the previous solutions, andit indicates that Rayleigh–Love rod model was appropriate for the vertical vibration problem oflarge-diameter pipe pile. The present solution had given a significant insight into the vital relationshipbetween the mechanical parameters of pipe pile and radially heterogeneous medium, and couldovercome some limitations of the previous models based on the 1D classical rod theory. In addition,this method could be extensively applied into the complex geological environment due to theconstruction disturbance.

Author Contributions: Conceptualization, Methodology, Software, Writing-Original draft preparation, Z.L.;Project administration, Resources, Writing-Reviewing and Editing, Supervision, Funding acquisition, Formalanalysis, C.C.; Investigation, Data curation, K.M.; Software, Validation, Y.X.; Supervision, Visualization, H.P.;Software, Validation, B.W. All authors have read and agreed to the published version of the manuscript.

Funding: This work is supported by the National Natural Science Foundation of China (Grant Nos. 51878109,51778107 and 51578100), the Fundamental Research Funds for the Central Universities (Grant No. 3132019601),and Cultivation Project of Innovation Talent for Doctorate Student (BSCXXM022).

Acknowledgments: The corresponding author would like to acknowledge the support from State Key Laboratoryof Coastal and Offshore Engineering, Dalian University of Technology.

Conflicts of Interest: The authors declare no conflict of interest.

Appendix A

The coefficients of Esj , Fs

j , Msj and Ns

j can be expressed by

Esj = 2πr j+1(Gs

j + ηsjs)q

sj [K1(qs

jr j)I1(qsjr j+1) −K1(qs

jr j+1)I1(qsjr j)] (A1)

Fsj = 2πr j+1(Gs

j + ηsjs)q

sj [K0(qs

jr j)I1(qsjr j+1) −K1(qs

jr j+1)I0(qsjr j)] (A2)

Msj = K0(qs

jr j+1)I1(qsjr j) + K1(qs

jr j)I0(qsjr j+1) (A3)

Nsj = −K0(qs

jr j+1)I0(qsjr j) + K0(qs

jr j)I0(qsjr j+1) (A4)

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