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Seismic response of an embedded pile in a transversely isotropic half-spaceunder incident P-wave excitationsM. Shahmohamadi a , A. Khojasteh a , M. Rahimian a , , R.Y.S. Pak ba Department of Civil Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iranb Department of Civil, Environmental and Architectural Engineering, University of Colorado, Boulder, CO 80309-0428, USA

a r t i c l e i n f o

Article history:

Received 22 April 2010Received in revised form13 September 2010Accepted 14 September 2010

a b s t r a c t

A rigorous mathematical formulation is presented for the analysis of a thin cylindrical shell embeddedin a transversely isotropic half-space under vertically incident P-wave excitation. By virtue of a set of ring-loads Greens functions for the shell and a group of dynamic fundamental solutions for the half-space under arbitrary interfacial dynamic loads, the problem is shown to be reducible to a pair of Fredholm integral equations. By utilizing an adaptive-gradient family capable of capturing regular-to-singular solution transitions smoothly, an accurate numerical procedure is developed. To assess theeffect of material anisotropy on the dynamic load-transfer process, a set of comprehensive numericalresults presented for various material and geometrical conditions. The accuracy of the proposednumerical scheme is conrmed by its comparison with a benchmark solution for the correspondingisotropic problem.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Analytical and numerical methods adopted to study soilstructure interactions have expanded greatly during the lastdecades. For a review of such analysis and their development onecan refer to Kausel [1] .

By increasing the use of composite materials and therecognition of anisotropic behavior of rocks and deposited soils,the study of stress-transfer in transversely isotropic solids havegained the focus of researchers from several areas. This kind of solutions are valuable in analysis of crack, inclusion, andinteraction problems which has a great appeal in offshoreengineering, geomechanics, geophysics and material science.Scattering problems in transversely isotropic media is a kind of this investigations and studied broadly. Incident wave scatteringand diffraction problem on cracks studied by Kunda and Bostrom

[2] , Bostrom et al. [3] , Brock and Hanson [4] , Wu [5] , Tan et al. [6]and Gautesen et al. [7] . Study of wave scattering problem aroundthe inclusions and cavities presented by Gatmiri and Eslami [8] .Analysis of waveseabed interaction studied by Jeng and Liu [9]by regarding the effect of seabed anisotropy effect. Hirose et al.[10] used BEM to analyze wave propagation in a water-lledborehole in an anisotropic solid. Scattering problem by an innitetransversely isotropic cylinder in a transversely isotropic medium

studied by Niklasson and Datta [11] . By the method of potentialfunctions, Eskandari and Sattar [12] presented solution for

transient wave propagation in transversely isotropic medium.To the best of authors knowledge, in contrast to the need forrigorous treatments to study of seismic soilpile interaction intransversely isotropic solids, these solutions are not presentedin the literature. However, approximate analytical or numericalsolutions of dynamic soilstructure interaction problems intransversely isotropic solids are studied widely. Liu and Novak[13] and later Zheng [14] studied dynamic-response of a singlepile embedded in transversely isotropic layered media using thenite element method combined with dynamic stiffness matricesof the soil. Barros [15] studied axial soilpile interaction underdynamic excitation, in which pile modeled by means of FEM whilefor soil half-space BEM is adopted. Dynamic interaction of rigidfoundations embedded in transversely isotropic solids studied by

using BEM by Gazetas [16] , Kirkner [17] , Wang and Rajapakse [18]and Barros [19] . Dynamic response of a pile in a transverselyisotropic saturated soil to transient torsional loading can be foundin Chen et al. [20] and for a cylindrical pile placed on a rigidbedrock under time-harmonic torsion in Wang et al. [21] .

Their contributions to a difcult problem notwithstanding, theseformulations are less than rigorous in enforcing the full mechanicalinteraction conditions between the pile and the soil. The lack of serious recognition and treatment of the singular nature of the load-transfer procedure at the end sections of the pile and the elementarytreatment of the pile medium are some examples.

For the case of dynamic soilpile interaction in isotropicmaterials the most rigorous analytical treatments can be found in

Contents lists available at ScienceDirect

journal homepage: w ww.elsevier.com/locate/soildyn

Soil Dynamics and Earthquake Engineering

0267-7261/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.

doi: 10.1016/j.soildyn.2010.09.005

Corresponding author. Tel.: +98 2161112256; fax: +98 2188078263.E-mail addresses: [email protected] (M. Rahimian) , [email protected]

(R.Y.S. Pak) .

Soil Dynamics and Earthquake Engineering 31 (2011) 361371
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Pak and Ji [22] and Ji and Pak [23] . Following their approach, inthis paper an exact mathematical boundary integral equationmethod is presented for a cylindrical pipe pile embedded ina transversely isotropic half-space under vertically incidentP-waves. In the framework of a shell theory for the pile andlinear elastodynamics for the soil, a set of Greens functions forthe representation of the response of the shell and the half-spacein an integral setting is presented. By enforcing the compatibility

conditions on both the inner and outer surfaces of the pile, theaxisymmetric problem under consideration is shown to bereducible to a set of Fredholm integral equations. To captureefciently the singular characteristics at the terminal sectionsexposed in the analysis, the new method of adaptive-gradientelements presented by Pak and Ashlock [24] is employed for thenumerical solution. The material anisotropy effect on differentaspects of the vertically incident P-waves scattering problem ispresented by means of several numerical results. As a substantia-tion of its accuracy, the proposed analytical and numericalmethod is compared with the existing solutions for isotropicmaterials in Ji and Pak [23] .

2. Vibration of a thin-walled cylindrical shell under axisymmetric distributed loads

With reference to Fig. 1, consider a exible cylindrical shellwith a mid-surface radius a, length l and thickness h 5 a whosecentroidal axis is coincident with the z -axis. Except for its exposedupper terminal section, the shell is assumed to be fully immersedin, and continuously bonded through its inner and outer surfaceto a homogeneous, transversely isotropic, linearly elastic half-space where z -axis is the axis of symmetry of the solid. The shell isassumed to be made from an isotropic solid where its shearmodulus ratio and Poissons ratio are denoted by me and ne ,respectively. The vertical and radial displacements of the shelldesignated as w z ( z ,t ) and w r ( z ,t ), and the stress-resultants per unitlength in the vertical, angular, bending, and shear directionsacting on an innitesimal shell element are denoted by N z ( z ,t ),N y z , t , M z ( z ,t ) and Q z ( z ,t ), respectively (see Fig. 1). For theaxisymmetric problem of interest, the free-head pile conditionunder arbitrary time-harmonic plane p-wave incidence can beexpressed as

N z 0 0 1

M z 0 0 2

Q z 0 0 3

For a thin circular cylindrical shell under torsionless axisym-metric time-dependent loads, the governing equations of motioncan be written as

@N z @ z

p z r eh@2 w z @t 2

4

@Q z @ z

1a

N y pr r eh@2 w r @t 2

5

@M z @ z

Q z 0 6

In the above, r e is the mass density of the shell, and p z ( z ,t ) and pr ( z ,t ) are the resultants of the distributed vertical and radialcontact stresses acting on the surfaces of the shell, respectively.

For time-harmonic loading and motion, one may express p z z , t p z z eio t , pr z , t pr z eio t , w z z , t w z z e io t , etc. whosetime factor eio t is suppressed for brevity. The displacement responseof the shell due to the combined external and inertial loads can berepresented as

w r z Z l

0w Rr z ; s pr s r eho

2 w r sds

Z l

0 w Z r z ; s p z s r eho

2w z sds 7

w z z Z l

0w R z z ; s pr s r eho 2 w r sds

Z l

0w Z z z ; s p z s r eho 2 w z sds w z 0 8

provided

Z l

0 p z s r eho

2 w z z ds 0 9

representing the zero axial load condition of (1) at the pile head. In(7) and (8), the kernels w j

i z ; si, j r , z are the axisymmetric static

displacement Greens functions for the shell response in the i-direction due to concentrated ring loads of uniform distributionacting at z s in the j-direction. For the detail of these Greensfunctions one can refer to [22] .

3. Scattered motion in half-space

In the absence of the pile, under a vertically incident planetime-harmonic P-wave, the free-eld response of a half-space canbe given by

u f z z , t u f 0 cos k p z e

io t 10

u f r z , t 0 11

where u0 f represents the modulus of the incident wave andk p o ffiffiffiffiffiffiffiffiffiffiffir s=c 33p is regarded as the wave number of verticallytraveling P-waves [25] . Under the incident excitation, the totaldisplacement of the half-space can be expressed as

u z u f z us z 12

u r u f r usr 13

where { u r s, u z s} are the displacement components generated by thewave-induced interfacial loads between the shell and the soil atf xjr a , 0 r z r lg. With the interfacial contact load distributionsacting in the half-space taken to be equal and opposite to p z and pr acting on the hollow pile, in accordance to the low of action

and reaction, the resulting scattered motion generated in theFig. 1. Cylindrical thin-walled pile embedded in a transversely isotropic half-

space.

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half-space are given by

u s z r , z Z l

0u R z r , z ; s pr sds Z

l

0u Z z r , z ; s p z sds , 0 r z r l 14

u sr r , z Z l

0u Rr a , z ; s pr sds Z

l

0u Z r a , z ; s p z sds , 0 r z r l 15

In (14) and (15), the four kernels u jir , z ; si, j r , z are the

axisymmetric half-space displacement ring-load Greens func-tions. In the next section, these Greens functions will begenerated.

4. Dynamic axisymmetric Greens functions for the half-space

Attempts considering stress transfer analysis in transverselyisotropic material caused by a harmonic excitation can be referredto the one in Rajapakse and Wang [26] , Eskandari-Ghadi [28] ,Rahimian et al. [29] and Khojasteh et al. [27] .

By adopting the analytical method similar to [29] andassuming that the reaction of the thin-walled shell on theembedding medium can be represented by a group of distributedbody forces interior the half-space xjr a , 0 r z r l, a set of Greens functions is determined for the elastic half-space fordynamic axial and radial ring loads acting on the boundary of thesoilpile.

In a cylindrical coordinate system r , y , z , where z -axis is theaxis of symmetry of the solid, the equations of time-harmonicmotion for a homogeneous transversely isotropic elastic solid interms of displacement and in the absence of the body forces canbe expressed as [27]

c 11@2 u r @r 2

1r

@u r @r

u r r 2 c 44 @

2 u r @ z 2

c 13 c 44 @2 u z @r @ z

r so2 u r 0

c 44@2 u z

@r 2

1

r

@u z

@r c 33

@2 u z

@ z 2 c 13 c 44

@2 u r

@r @ z

1

r

@u r

@ z r so 2 u z 0

16

where u r and u z are the displacement components in the r and z directions, respectively; r s the mass density of the solid; c ij theelasticity constants of the solid; o the circular frequency; and thetime factor eio t is omitted for brevity. In order to uncouple (16) acomplete potential function F is used [28] . This potential function,F , is related to displacement components, u r and u z as

u r r , z a 3@2 F r , z

@r @ z

uy r , z 0

u z r , z 1 a 1 @2

@r 2 1r @@r a 2 @2

@ z 2 rso

2

c 66 F r , z 17 where

a 1 c 12 c 66

c 66, a2

c 44c 66

, a3 c 13 c 44

c 6618

By substituting (17) into (16), the governing equation forpotential function F can be written as

r 21 r 22 do 2

@2

@ z 2 F 0 19 where

r 2i @2

@r 2

1

r

@

@r

1

s2i

@2

@ z 2

1

mi

r so 2

c 66, i 1 , 2 20

and

m1 a 2 c 44c 66

, m2 1 a 1 c 11c 66

dr s

1c 44 s22

1c 11 s21

1c 11

1 c 33c 44 " # 21

In the above, s1 and s2 are the roots of the following equation,

which in view of the positive-deniteness of the strain energy arenot zero or pure imaginary numbers [30]

c 33 c 44 s4 c 213 2c 13 c 44 c 11 c 33 s2 c 11 c 44 0 22

By virtue of the zeroth order Hankel transform with respect tothe radial coordinate, the general solution of F can be expressedas [27]

~F 0

x , z Axel 1 z Bxe l 1 z C xel 2 z Dxe l 2 z 23

where l 1 and l 2 can be written as

l 1 ffiffiffiffiffiffiffiffiffiffiffiax2 b 12 ffiffiffiffiffiffiffiffiffic x4 dx2 eq r l 2 ffiffiffiffiffiffiffiffiffiffiffiax2 b 12 ffiffiffiffiffiffiffiffiffic x4 dx2 eq r 24 a

12

s21 s22 , b

12

r so2 1

c 33

1c 44 , c s22 s21 2

d 2r so2 1

c 33

1c 44 s21 s22 2 c 11c 33 1c 11 1c 44

e r 2s o 4 1c 33

1c 44

2

25

and A, . . . , D are constants of integration to be revealed fromboundary conditions.

In the next, a set of fundamental solutions for the dynamic

axial and radial ring loads for the semi-innite transverselyisotropic medium will be generated.

4.1. Axial ring load at z s

A unit intensity axial ring load acting on a circle of radius a at adepth s in the interior of a half-space can be expressed as

t zz r , s t zz r , s dr a 26

t zr r , s t zr r , s 27

Under such load, one can nd the axial component of thedisplacement eld as

u Z z r , z , s ac 44 Z 1

0O2 x , z , sx J 0 xa J 0 xr dx 28

where

O2 z , s , x c 44

2a 2 c 33 l 21 l22

W1l 1

e l 1 j z sjW2l 2

e l 2 j z sj

I xI x

W1l 1

e l 1 z s W2l 2

e l 2 z s 2I x

Z2 n2W1l 2

e l 1 z l 2 s Z1 n1W2l 1

e l 2 z l 1 s Here,

Zi a 3 a 2 l2i x

2 1 a 1 r so 2

c 66, Wi a 3 l

2i Zi

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ni Zi a 3c 13c 33

x2 a 3 l 2i l i , I 7 x Z2 n1 7 Z1 n2 , i 1 , 2 29 The radial displacement under axial ring load can be written as

u Z r r , z , s ac 44 Z

1

0g3 x , z , ; sx J 0 xa J 1 xr dx 30

where

g3 z , s , x a3 c 44 x

2a 2 c 33 l 21 l 22 sgn z sel 1 j z sj

el 2 j z sj

I xI x

e l 1 z s e l 2 z s

2I x

Z2 n2l 1l 2

e l 1 z l 2 s Z1 n1l 2l 1

e l 2 z l 1 s 31 where

sgn z s 1 , z 4 s

1 , z o s(It can be shown that these inuence functions for an isotropicmaterial can be exactly reduced to the one in [23] .

4.2. Radial ring load at z s

A radial ring load of unit intensity acting in a circle of radius aat a depth s can be written as

t zr r , s t zr r , s dr a 32

t zz r , s t zz r , s 33

The radial component of the inuence eld under the dened ringload can be expressed as

u Rr r , z , s ac 44 Z

1

0g1 x , z , sx J 1 xa J 1 xr dx 34

1.0

0.0

0.5

e /c 44 =100

e /c 44 =200

e /c 44 =1000

-1.0 R e [ ( p

z / c

4 4

) / ( u

z f ( 0 ) / a

) ]

I m [

( p z

/ c 4 4

) / ( u

z f ( 0 ) / a

) ]

z/a

e /c = 107

Ji and Pak 1996

Ji and Pak 1996

Ji and Pak 1996

Ji and Pak 1996

e /c 44 =100

e /c 44 =200

e /c 44 =1000

e /c = 107

Ji and Pak 1996

Ji and Pak 1996

Ji and Pak 1996

Ji and Pak 1996

-1.5

0

0.5

0.0

-0.5

-1.0

0

z/a

-0.5

5 10 15 20

5 10 15 20

Fig. 2. Validation of the vertical dynamic contact load distribution in an isotropicmedia: ne 0:2, l/a 20, h /a 0.05, r s=r e 0:25, o a

ffiffiffiffiffiffiffiffiffiffiffiffiffiffir s=c 44

p 0:5: (a) real part;

(b) imaginary part.

1.0

0.0

0.5

Material#1 E/E = 1

-1.5

-1.0 R e

[ ( p z

/ c 4 4

) / ( u

z f ( 0 ) / a

) ]

z/a

Material#2 E/E = 2

Material#3 E/E = 3

Material#4 E/E = 0.5

0

1.0

0.0

-1.0

I m [ ( p

z / c

4 4

) / ( u

z f ( 0 ) / a ) ]

Material#2 E/E = 2

Material#3 E/E = 3


-0.5

5 10 15 20

z/a

0 5 10 15 20

Material#1 E/E = 1

Fig. 3. Vertical dynamic contact load distribution in transversely isotropic mediawith different E u =E : ne 0:2, l/a 20, h/a 0.05, r s=r e 0:25, o a

ffiffiffiffir s=c 44

p 0:5,

me =c 44 100: (a) real part; (b) imaginary part.

Table 1Properties of synthetic materials.

Material E u

E Gu

Gnu

nc 11 c 12 c 13 c 33 c 44

1 (Isotropic) 1 1 1 6.0 2.0 2.0 6.0 2.02 (Transversely isotropic) 2 1 1 5.6 1.6 1.8 10.9 2.03 (Transversely isotropic) 3 1 1 5.5 1.5 1.8 15.9 2.04 (Transversely isotropic) 0.5 1 1 7.0 3.0 2.5 3.75 2.05 (Transversely isotropic) 1 0.5 1 6.0 2.0 2.0 6.0 1.06 (Transversely isotropic) 1 0.2 1 6.0 2.0 2.0 6.0 0.47 (Transversely isotropic) 1 1 0.68 5.6 1.6 1.2 5.4 2.08 (Transversely isotropic) 1 1 0.2 5.4 1.4 0.3 5.0 2.0


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where

g1 z , s , x 1

2Z1 l22 Z2 l

21

l 1 W2 e l 1 j z sj l 2 W1 e l 2 j z sj

I xI x

l 1 W2 e l 1 z s l 2 W1 e l 2 z s

2I x

l 1 W1 Z2 n2 e l 1 z l 2 s l 2 W2 Z1 n1 e l 2 z l 1 s

The axial displacement under radial ring load can be expressed as

u R z r , z , s ac 44 Z

1

0O1 x , z , ; sx J 1 xa J 0 xr dx 35

where

O1 z , s , x W1 W2

2a 3 xZ1 l22 Z2 l

21

sgn z se l 1 j z sj e l 2 j z sj

I xI x

e l 1 z s e l 2 z s

2I x

Z2 n2W1W2

e l 1 z l 2 s Z1 n1W2W1

e l 2 z l 1 s Owing to the presence of the products of Bessel functions as wellas other irrational functions in the integrands, the evaluation of the improper integrals in the inuence functions (28), (30), (35)and (34) is difcult both analytically and numerically. The problem

is aggravated by the singular behavior of some of the inuenceelds. To deal with these issues effectively, these inuence eldscan be determined effectively by numerical contour integration[31] and using adaptive quadrature methods due to the rapidquadratic decay of the integrands [29] . To ensure that thedisplacement and rotations are nite on the cylindrical surface xjr a , 0 r z r l in the half-space, one can therefore easily deducethat end load transfers at the bottom of shell in the form of

concentrated ring loads of N z (l), Q z (l) and M z (l) are inadmissible inthe formulation. These mathematical conclusions are consistentwith the physical observation that a three-dimensional continuumcannot support nite loads on truly sharp edges. As will be furtherillustrated in the next section, an understanding of the basicanalytical features of the inuence elds is essential to a properboundary element treatment of the elastodynamic problem.

5. Soilpile interaction formulation

Regarding the equal and opposite interfacial contact loaddistribution p r and p z acting on the pile, in accordance to the lawof action and reaction, the necessary compatibility conditionsbetween the two media are

w z z limr - a

u z r , z , 0 r z r l 36

w r z limr - a

u r r , z , 0 r z r l 37

2.0Material#1 E/E = 1

0.0

1.0

Material#2 E/E = 2

Material#3 E/E = 3


-1.0 R e

[ ( p z

/ c 4 4

) / ( u

z f ( 0 ) / a

) ]

-2.0

1.0

0.0

I m [

( p z

/ c 4 4

) / ( u

z f ( 0 ) / a

) ]

Material#1 E/E = 1

Material#2 E/E = 2

Material#3 E/E = 3

-1.0

0

z/a


5 10 15 20

0

z/a

5 10 15 20



p 0:5,


1.0Material#1 E/E = 1

0.5

Material#2 E/E = 2

Material#3 E/E = 3


-0.5

0.0

R e

[ ( p r

/ c 4 4

) / ( u

z f ( 0 ) / a

) ]

-1.0

0

z/a

0.2

0.0

-0.2

I m [

( p r /

c 4 4

) / ( u

z f ( 0 ) / a

) ]

Material#1 E/E = 1

Material#2 E/E = 2

Material#3 E/E = 3


-0.4

5 10 15 20

0

z/a

5 10 15 20

Fig. 5. Radial dynamic contact load distribution in transversely isotropic mediawith different E u=E : ne 0:2, l/a 20, h/a 0.05, r s=r e 0:25, o a

ffiffiffir s=c 44

p 0:5,



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Considering that the stress distributions acting on the interface of the two media p z and p r are fully integrable, the above Eqs. (36)and (37) lead to

Z l

0w Rr z ; s pr s r eho 2 w r sds Z

l

0w Z r z ; s p z s r eho 2 w z sds

Z l

0u Rr r , z ; s pr sds Z

l

0u Z r r , z ; s p z sds u

f r z 0 r z r l 38

Z l

0w R z z ; s pr s r eho 2 w r sds Z

l

0w Z z z ; s p z s r eho 2 w z sds

Z l

0u R z a , z ; s pr sds Z

l

0u Z z a , z ; s p z sds u

f z z 0 r z r l 39

which are a pair of weakly singular Fredholm integral equationsfor p r and p z .

6. Numerical procedures

By dividing the contact interval into a nite number of elements and interpolating the unknown contact traction overthe region with suitable shape functions, the two weakly singularFredholm integral equations in (38) and (39) can be solved in termsof their unknown singular functions. Due to the exactness of the

formulation and the availability of the integral kernels in closedform through the adopted denitions of the shell and half-spaceGreens functions, the accuracy of the numerical solution iscontrolled mainly by the choice of the shape functions andquadrature accuracy. Also, an effective numerical scheme must beprepared to handle both singular and regular contact tractiondistributions, the form of which is a function of the relative material,geometric, interfacial and loading congurations. As a new paradigm

for handling the fundamental analytical dilemma, the adaptive-gradient (AG) element family [24] , which can handle both singularand regular variations aptly and smoothly is an attractive solution.For the present integral equation system, the 3-node one-dimen-sional AG element introduced in [24] is found to perform superbly.Using the new elements in edge contact zone with enhancednumerical quadrature in combination with standard quadraticelements for the rest of the domain, the integral equation systemfor the soilpile interaction problem is solved numerically under avariety of material interfacial-loading congurations.

Owing to their analyticity, the unknown functions, pr , p z , wr and w z , can be effectively represented by

pr z

X

n

j 1N j z p jr 40

p z z Xn

j 1N j z p j z 41

1.0

0.0

Material#1 G/G = 1.0

-1.5

-1.0 R e

[ ( p z

/ c 6 6

) / ( u

z f ( 0 ) / a

) ]

z/a



0

0.25

0.00

-0.50

-0.25

I m [

( p z

/ c 6 6

) / ( u

z f ( 0 ) / a ) ]

z/a




0

0.5

-0.5

5 10 15 20

5 10 15 20

Fig. 6. Vertical dynamic contact load distribution in transversely isotropic mediawith different Gu =G: ne 0:2, l/a 20, h/a 0.05, r s=r e 0:25, o a


p 0:5,


1.0


0.5Material#5 G/G = 0.5


-0.5 R e

[ ( p

r / c 6 6

) / ( u

z f ( 0 ) / a

) ]

-1.0

0.2

0.0

-0.2

I m [

( p r /

c 6 6

) / ( u

z f ( 0 ) / a

) ]




-0.4

0

z/a

5 10 15 20

0

z/a

5 10 15 20

0.0

Fig. 7. Radial dynamic contact load distribution in transversely isotropic mediawith different Gu =G: ne 0:2, l/a 20, h/a 0.05, r s=r e 0:25, o a

ffiffiffiffir s=c 66

p 0:5,



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w r z Xn

j 1N j z w jr 42

w z z Xn

j 1N j z w j z 43

where N j z , j 1 , . . . , n are shape functions associated with thenodal locations z j, pr j and p z j are the nodal values of the regular

parts of the wave-induced interfacial load distributions, and wr jand w z j are the nodal displacements of the shell, respectively. Onsubstituting the foregoing representations into the governingequations and requiring equality at all nodes, (38), (39) and (9),one may nd the total system of equations as

F baT 2 0 ! pw z 0 ! f f 0 ! 44

where

F H r eho 2 GH G 45

aT 2 aT 1 1 r eho

2 H 46

f I reho 2 Gu f 47

f 0 r eho 2 a T 1 u f 48

in matrix notation. Here, I is a (2 n 2n) identity matrix;G(2 n 2n) are inuence matrix associated with the shell and

are dened by

Gi, j Z l

0w Rr N jsds 49

Gi, n j Z l

0w Z r N jsds 50

Gn i, j Z l

0 wR z N jsds 51

Gn i, n j Z l

0w Z z N jsds , i 1 , . . . , n , j 1 , . . . , n 52

respectively; H (2 n 2n) is an inuence matrix of the half-spacegiven by

H i, j Z l

0u Rr N jsds 53

H i, n j Z l

0u Z r N jsds 54

H n i, j

Z l

0u R z N jsds 55

H n i, n j Z l

0u Z z N jsds , i 1 , . . . , n , j 1 , . . . , n 56

1.0

0.0

Material#1 / =1.0

Material#7 / =0.68

Material#8 / =0.2

-2.0

-1.0

R e

[ ( p

z / c 4 4

) / ( u

z f ( 0 ) / a

) ]

z/a

Material#1 / =1.0

Material #7 / =0.68

Material#8 / =0.2

0

0.25

0.00

-0.50

-0.25

I m [

( p z

/ c 4 4

) / ( u

z f ( 0 ) / a

) ]

5 10 15 20

z/a

0 5 10 15 20

Fig. 8. Vertical dynamic contact load distribution in transversely isotropic mediawith different nu =n: ne 0:2, l/a 20, h/a 0.05, r s=r e 0:25, o a


p 0:5,


1.0

0.5

-0.5 R e

[ ( p r /

c 4 4

) / ( u

z f ( 0 ) / a

) ]

-1.0

0

z/a

0.2

0.0

-0.2

I m [

( p r /

c 4 4

) / ( u

z f ( 0 ) / a

) ]

Material#1 / =1.0

Material#7 / =0.68

Material#8 / =0.2

Material#1 / =1.0

Material#7 / =0.68

Material#8 / =0.2

-0.4

0.0

5 10 15 20

0

z/a

5 10 15 20

Fig. 9. Radial dynamic contact load distribution in transversely isotropic mediawith different nu =n: ne 0:2, l/a 20, h/a 0.05, r s=r e 0:25, o a

ffiffiffir s=c 44

p 0:5,



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The vector p2n 1 f p1r , . . . , pnr , p1 z , . . . , pn z gT is the array of nodal

values of the resultant contact load distributions; u f 2n 1 fu f 1r , . . . , u

fnr , u

f 1 z , . . . , u

fn z g

T is the nodal displacement vector of thefree-eld motion; w z 0 (2 n 1) is dened by

w 0 z bw z 0 57

where

b f 0 , 0 , . . . , 0 , 1 , 1 , . . . , 1gT 58

and a 1 2n 1 fa 11 , . . . , a2n1 g

T is given by

a j1 0

an j1 Z l

0N jsds , j 1 , . . . , n 59

The result is a numerical method which performs extremelywell in all cases examined. In the next section, some typicalsolutions for the axisymmetric seismic excitation problem of anembedded pile foundation in a semi-innite medium will bepresented.

7. Numerical results

By means of the numerical procedure presented earlier, theseismic soilpile interaction problem under vertically incidentP-waves can be computed readily. In a transversely isotropicmaterial, ve independent elastic constants are needed todescribe its mechanical properties, where E and E u are Youngsmoduli in the plane of isotopy and perpendicular to it; n is

Poissons ratio which characterize the effect of horizontal strainon the complementary vertical strain; nu is Poissons ratio whichcharacterize the effect of vertical strain on the horizontal one; andc 44 stands for the shear modulus for the plane normal to the planeof isotropy. For relation of these material parameters to theelasticity constants c ij, one can refer to Rahimian et al. [29] . Tosurvey the effect of anisotropy of the materials on the seismicinteraction between the two media, several types of isotropic andtransversely isotropic materials are considered to constitute thebasic benchmark soil media. Their elastic properties are listed inTable 1 with E u=E varying between 0.5 and 3 and Gu=G and nu =nranging between 0.2 and 1.0. For all these materials, E , G and n are50 GPa, 20 GPa and 0.25, respectively. To ensure the positivedeniteness of the strain energy, the subsequent restrictions for

material constants c ij have been checked [32]c 11 4 jc 12 j , c 11 c 12 c 33 4 2c 213 , c 44 4 0 60

2.0

3.0

Material#1 E/E=1

Material#2 E/E=2

0.0

1.0

Material#3 E/E=3

Material#4 E/E=0.5

-3.0

-2.0

-1.0

R e

[ ( p z

/ c 4 4

) / ( u

z f ( 0 ) / a

) ]

1.0

-0.5

0.0

-1.5

-1.0 I m

[ ( p

z / c

4 4

) / ( u

z f ( 0 ) / a

) ]

Material#1 E/E=1 Material#2 E/E=2

Material#3 E/E=3 Material#4 E/E=0.5

0

z/a

5 10 15 20

0

z/a

5 10 15 20

0.5



p 1,


3.0

Material#1 E/E=1 Material#2 E/E=22

1.0

2.0 Material#3 E/E=3Material#4 E/E=0.5

-1.0

0.0

R e

[ ( p r /

c 4 4

) / ( u

z f ( 0 ) / a

) ]

-2.0

0

z/a

0.6Material#1 E/E=1 Material#2 E/E=2

0.3

Material#3 E/E=3 Material#4 E/E=0.5

-0.3 I m [ ( p

r / c 4 4

) / ( u

z f ( 0 ) / a

) ]

-0.6

0.0

5 10 15 20

0

z/a

5 10 15 20

Fig. 11. Radial dynamic contact load distribution in transversely isotropic mediawith different E u =E : ne 0:2, l/a 20, h/a 0.05, r s=r e 0:25, o a

ffiffiffir s=c 44

p 1,



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In Fig. 2, the axial contact load distributions for the case of isotropic half-space under vertically incident P-waves at afrequency of o a ffiffiffiffiffiffiffiffiffiffiffiffiffiffir s=c 44p 0:5 for different soilpile modulusratios are used in the comparison with the ones in [23] . Asindicated in the gure, there is an excellent agreement betweenthe two solutions. This shows the reliability of the numericalprocedure to handle singular load transfer at the end of the shellas well as the regular behavior in the mid length.

Figs. 3 and 4 illustrate the dynamic axial contact loaddistribution of the pile embedded in different anisotropic solidswith different E u =E under the vertically incident P-waves. Thesingular nature of the dynamic load transfer at the end of theshell/pile is evident for all materials and is in agreement with [23]for isotropic materials and the physical observation that soilmedium cannot support much load on sharp edges. For highmodulus ratios in Fig. 4, the effect of material anisotropy ondynamic vertical load distribution is more evident, specially forthe imaginary part of the solution. Although, by increasing E u=E the real part of the vertical contact load distribution increases, theimaginary part decreases. For material with E u=E 0 :5 one can seethat real and imaginary part of the solution has more variation incomparison to materials with higher E u .

In order to survey the effect of material with different E u =E onthe radial load distribution under the vertically incident P-waves,Fig. 5 presented here for frequency of o a ffiffiffiffiffiffiffiffiffiffiffiffiffiffir s=c 44p 0:5. From thegure, the singular nature of contact load distribution is apparent

for all materials for both ends of the pile. Also, one can see that forhigher E u =E real part of the response decreases while imaginarypart increases.

To understand the effects of Gu on the load transfer processFigs. 6 and 7 are presented here for frequency of o a ffiffiffir s=c 66p 0:5.Fig. 6 shows axial load distribution under the vertically incidentP-waves. From the gure, one can see that by reducing Gu real andimaginary part of the response reduces which is too signicant for

the imaginary part. For the radial response, from Fig. 7, similarbehavior to the axial one is seen. It is apparent from Figs. 6 and 7that radial response is more affected by Gu . Also, the singular loadtransfer at the ends of the pile shows intrinsic singular nature of load transfer. From the gures one can see that the effect of Gu isinsignicant in comparison to materials with different E u .

For material with different nu=n Figs. 8 and 9 are shown. Fromthe gures it is seen that the effect of nu is not signicant for theaxial traction around the pile. For the radial response one can seethat by reducing nu , real and imaginary part of solution reduces.

In Figs. 10 and 11 , the effect of material anisotropy on thedynamic axial and radial contact load distribution for higherincident frequencies is presented for materials with different E u=E and frequency of o a=

ffiffiffiffiffiffiffiffir s=c 44

p 1. From Fig. 10, one can see that

by increasing E u =E , the real part of the axial contact loaddistribution increases while the imaginary part of the solutiondecreases. For the radial contact load distribution, one can seethat by increasing E u =E real part of the stress distribution

1.0Material#1 E/E=1

Material#2 E/E=2

0.0

0.5 Material#3 E/E=3

Material#4 E/E=0.5

Material#1 E/E=1

Material#2 E/E=2

Material#3 E/E=3

Material#4 E/E=0.5

-0.5

R e

[ ( w

z / u

z f ( 0 ) ]

-1.0

0

z/a

0.3

0.5

0.1

I m [

( w z

/ u z f

( 0 ) ]

-0.3

-0.1

5 10 15 20

0

z/a

5 10 15 20

Fig. 12. Vertical displacement prole of the shell in transversely isotropic mediawith different E u =E : ne 0:2, l/a 20, h/a 0.05, r s=r e 0:25, o a


p 1,


0.01

0.02Material#1 E/E=1

Material#2 E/E=2

Material#3 E/E=3

0.00

R e [

( w r /

u z f

( 0 ) ]

-0.02

-0.01

0

z/a

0.005

0.010Material#1 E/E=1 Material#2 E/E=2

Material #3 E/E=3 Material#4 E /E=0.5

0.000

I m [

( w r /

u z f

( 0 ) ]

-0.010

-0.005

Material#4 E/E=0.5

5 10 15 20

0

z/a

5 10 15 20

Fig. 13. Radial displacement prole of the shell in transversely isotropic mediawith different E u =E : ne 0:2, l/a 20, h/a 0.05, r s=r e 0:25, o a

ffiffiffir s=c 44

p 1,



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decreases. The singular nature of contact load distribution isapparent for higher frequencies. Also, the oscillatory behavior of the axial and radial response generally increases for higherfrequencies.

The effect of material anisotropy on the axial displacement of the pile is presented in Fig. 12 , for materials with different E u =E and frequency of o a= ffiffiffiffiffiffiffiffiffiffiffiffiffiffir s=c 44p 1. From the gure, it is apparentthat by increasing E u=E , the real and imaginary part of the verticaldisplacement increases. It is apparent that oscillatory behavior of the vertical displacement decreases, by increasing E u . It is note-worthy that for material with different E u =E presented innumerical solutions, vertical P-wave transfers with differentspeeds, this speed increases rapidly by increasing E u . Here, axialdisplacement prole is found to closely resemble the wavelengthof the incident longitudinal wave.

Fig. 13 represents the radial displacement of the shell due tothe vertically incident P-wave excitation. From the gure, one cansee that by increasing E u=E , the real part of the radial displacementof the embedment decreases. Near the head of the pile, the radialdisplacement appears not to be much affected by materialanisotropy.

In Figs. 14 and 15 a set of foundation input motion functionpresented which is of particular interest in the case of seismicsoilstructure interaction under consideration. To validate thepresent results, a comparison between the present solution withthe one in [23] is furnished in Fig. 14 . From the gure, one can see

that those solutions are identical for different soilpile modulusratios. Fig. 15 shows the effect of material anisotropy on thevertical foundation input motion functions of an embedded pile.From the gure, one can see that the complex-valued foundationinput-motion function is dependent on the material stiffness andexcitation frequency. Also, the material anisotropy effect becomesmore apparent for higher frequencies.

8. Conclusion

In this paper, a rigorous mathematical formulation is pre-sented for the analysis of the scattering of vertically incidentP-waves by a cylindrical pile embedded in a semi-innite trans-versely isotropic medium. By means of a set of Greens functionsfor the pile in the context of a shell theory under different loadingcondition and a group of fundamental solutions for the elastichalf-space, the proposed problem is reduced to the solution of a pair of Fredholm integral equations. To illustrate the effectof anisotropy of the medium on the seismic load-transfer, acomprehensive set of illustrative results for various responses aregenerated by an effective numerical adaptive-gradient elementprocedure. Because of the completeness of formulation and theconvenience of the numerical method, the proposed approach can

be used for the further analytical developments in transversely

0.50

1.00

0.00 R e [

( w z

( 0 ) / u

z f ( 0 ) ]

e/c

44=100

e /c 44 =1000

e /c 44 =10000

Ji and Pak 1996

-0.500.00

Ji and Pak 1996Ji and Pak 1996

0.60

0.40

e /c 44 =100

e /c 44 =1000

e /c 44 =10000Ji and Pak 1996Ji and Pak 1996

Ji and Pak 1996

0.00

0.20

I m [

( w z

( 0 ) / u

z f ( 0 ) ]

Ji and Pak 1996

-0.20

0.00 0.10 0.20 0.30 0.40 0.50

0.10 0.20 0.30 0.40 0.50

a s /c 44

a s /c 44

Fig. 14. Validation of the vertical foundation input motion functions of anembedded shell: ne 0:2, l/a 20, h/a 0.05, r s=r e 0:25:(a) real part; (b)imaginary

part.

1.0Material#1 E/E=1

Material#2 E/E=2

0.5

Material#3 E/E=3

Material#4 E/E=0.5

-0.5

0.0 R e

[ ( w z (

0 ) / u

z f ( 0 ) ]

0.6

Material#1 E/E=1

Material#2 E/E=2

0.2

0.4 Material#3 E/E=3

Material#4 E/E=0.5

-0.2

0.0

I m [

( w z

( 0 ) / u

z f ( 0 ) ]

a s /c 44

0.0 0.1 0.2 0.3 0.4 0.5

0.0 0.1 0.2 0.3 0.4 0.5

a s /c 44

Fig. 15. Vertical foundation input motion functions of an embedded shell intransversely isotropic media with different E u =E : ne 0:2, l/a 20, h/a 0.05,r s=r e 0:25, me=c 44 1000: (a) real part; (b) imaginary part.


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isotropic solids or as a benchmark for comparison with othernumerical attempts to this class of problems.

References

[1] Kausel E. Early history of soilstructure interaction. Soil Dynamics andEarthquake Engineering 2010;30(9):82232.

[2] Kunda T, Bostrom A. Elastic wave scattering by a circular crack in atransversely isotropic solid. Wave Motion 1992;15(3):285300.

[3] Bostorm A, Grahn T, Niklasson AJ. Scattering of elastic waves by a rectangularcrack in an anisotropic half-space. Wave Motion 2003;38(2):91107.

[4] Brock LM, Hanson MT. An exact transient analysis of plane wave diffractionby a crack in an orthotropic or transversely isotropic solid. International Journal of Solids and Structures 2002;39(2122):5393408.

[5] Wu KC. Diffraction of a plane stress wave by a semi-innite crack in a generalanisotropic elastic material. Wave Motion 2004;40(4):35972.

[6] Tan A, Hirose S, Zheng C, Wang CY. A 2D time-domain BEM for transient wavescattering analysis by a crack in anisotropic solids. Engineering Analysis withBoundary Elements 2005;29(6):61023.

[7] Gautesen A, Zernov V, Fradkin L. Diffraction coefcients of a semi-inniteplanar crack embedded in a transversely isotropic space. Wave Motion2009;46(1):2646.

[8] Gatmiri B, Eslami H. Wave scattering in cross-anisotropic porous mediaaround the cavities and inclusions. Soil Dynamics and Earthquake Engineer-ing 2008;28(12):101427.

[9] Jeng DS, Lin YS. Poroelastic analysis of the waveseabed interaction problem.

Journal of Computers and Geotechnics 2000;26(1):4364.[10] Hirose S, Ushida Y, Wang CY. BEM analysis of wave propagation in a water-lled borehole in an anisotropic solid. Journal of Tsinghua Science andTechnology 2007;12(5):5206.

[11] Niklasson AJ, Datta SK. Scattering by an innite transversely isotropiccylinder in a transversely isotropic medium. Wave Motion 1998;27(2):16985.

[12] Eskandari-Ghadi M, Sattar S. Axisymmetric transient waves in transverselyisotropic half-space. Soil Dynamics and Earthquake Engineering 2009;29:34755.

[13] Liu WM, Novak M. Dynamic-response of single piles embedded intransversely isotropic layered media. Journal of Earthquake Engineeringand Structural Dynamics 1994;23(11):123957.

[14] Tiesheng Z. Dynamic behavior of piles embedded in transversely isotropiclayered media. Acta Mechanica Sinica 1997;13(3):24152.

[15] Barros PLA. Dynamic axial response of single piles embedded in transverselyisotropic soils. In: 16th ASCE engineering mechanics conference, University of Washington, Seattle, 2003.

[16] Gazetas G. Strip foundations on a cross-anisotropic soil layer subject todynamic loading. Geotechnique 1981;31(2):16179.

[17] Kirkner DJ. Vibration of a rigid disc on a transversely isotropic elastic half-space. International Journal for Numerical and Analytical Methods inGeomechanics 1982;6:293306.

[18] Wang Y, Rajapakse N. BE analysis of dynamics of rigid foundations embeddedin transversely isotropic soils. Journal of Chinese Institute of Engineering2000;23(3):27588.

[19] Barros PLA. Impedances of rigid cylindrical foundations embedded intransversely isotropic soils. International Journal of Numerical and AnalyticalMethods in Geomechanics 2003;30:683702.

[20] Chen G, Cai YQ, Liu FY, Sun H. Dynamic response of a pile in a transverselyisotropic saturated soil to transient torsional loading. Journal of Computersand Geotechnics 1997;35(2):16572.

[21] Wang KH, Zhang ZQ, Leo CJ, Xie K. Dynamic torsional response of an endbearing pile in transversely isotropic saturated soil. Journal of Sound andVibration 2009;327(35):44053.

[22] Pak RYS, Ji F. Mathematical boundary integral equation analysis of anembedded shell under dynamic excitations. International Journal forNumerical Methods in Engineering 1994;37:250120.

[23] Ji F, Pak RYS. Scattering of vertically-incident P-waves by an embedded pile.Soil Dynamics and Earthquake Engineering 1996;15:21122.

[24] Pak RYS, Ashlock SM. Method of adaptive-gradient elements for computa-tional mechanics. Journal of Engineering Mechanics 2007;133(1):8797.

[25] Thomsen L. Weak elastic anisotropy. Geophysics 1986;51(10):195466.[26] Rajapakse RKND, Wang Y. Greens functions for transversely isotropic elastic

half space. Journal of Engineering Mechanics 1993;119(9):172446.[27] Khojasteh A, Rahimain M, Eskandari M, Pak RYS. Asymmetric wavepropagation in a transversely isotropic half-space in displacement potentials.International Journal of Engineering Science 2008;46:690710.

[28] Eskandari-Ghadi M. A complete solution of the wave equations for trans-versely isotropic media. Journal of Elasticity 2005;81:119.

[29] Rahimian M, Eskadari-Ghadi M, Pak RYS, Khojasteh A. Elastodynamicpotential method for transversely isotropic solids. Journal of EngineeringMechanics 2007;133(10):113445.

[30] Lekhnitskii SG. Theory of elasticity of an anisotropic elastic body. SanFrancisco: Holden Day; 1963.

[31] Pak RYS. Dynamic response of a partially embedded bar under transverselyexcitation.EERL Report 85-04, California Institute of Technology, Pesadena; 1985.

[32] Payton RG. Elastic wave propagation in transversely isotropic media. TheNetherlands: Martinus, Nijhoff, 1983.


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