Impedances of Rigid Cylindrical Foundations Embedded

7/30/2019 Impedances of Rigid Cylindrical Foundations Embedded

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INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICSInt. J. Numer. Anal. Meth. Geomech., 2006; 30:683702Published online 23 January 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nag.496

Impedances of rigid cylindrical foundations embedded

in transversely isotropic soils

P. L. A. Barrosn,y

Department of Geotechnics and Transportation, FEC, State University of Campinas, Caixa Postal 6021,

CEP 13083-852, Campinas, SP, Brazil

SUMMARY

A complete formulation and implementation for assessment of the response to dynamic loads of cylindricalrigid structures embedded in transversely isotropic elastic half-spaces is presented. The analysis is

performed in the frequency domain and the steady-state structure response is obtained. The method isbased on a non-singular version of the indirect boundary element method which uses influence functions,instead of Greens functions, as fundamental solutions. These influence functions are the response of anelastic half-space to distributed, internally applied loads. The proposed method imposes full bondingcontact between the foundation and the surrounding soil. Numerical results for displacement (vertical andhorizontal) and rotation (twisting and rocking) impedances, showing the influence of the soil anisotropy,are presented. Results for the soilstructure interface tractions and for the displacement field throughoutthe half-space are also shown. Copyright # 2006 John Wiley & Sons, Ltd.

KEY WORDS: dynamic soilstructure interaction; boundary element method; elastodynamics; anisotropy;rigid foundations

1. INTRODUCTION

The assessment of the response to a system of dynamic loads of a rigid foundation which

interacts with a (visco-)elastic half-space is emblematic of the dynamic soilstructure interaction

(DSSI) field and one of the cornerstones in Foundation Engineering. Such study is fundamental

to the design of machine foundations and to the seismic analysis of structures.

Numerical methods based on the boundary integral equation method (BIEM), such as the

boundary element method (BEM) have been successfully employed in the analysis of this kind

of problem for more than 30 years. Indeed, the vertical dynamic response of a rigid disc resting

on the surface of an elastic half-space [1] is regarded as the first BEM application to the soil

Contract/grant sponsor: Fundacao de Amparo a` Pesquisa do Estado de Sao Paulo FAPESP; contract/grant number:

00/10186-4

Contract/grant sponsor: Massachusetts Institute of Technology

Received 1 December 2004Revised 1 August 2005

Accepted 1 November 2005Copyright# 2006 John Wiley & Sons, Ltd.

yE-mail: [email protected]

n

Correspondence to: Pe sio L. A. Barros, FEC/Unicamp, Caixa Postal 6021, CEP 13083-852, Campinas, SP, Brazil.


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dynamics. After that first development, solutions for horizontal and rotation dynamic responses

were also obtained [2], followed by solutions for the cases of embedded rigid foundations [3] and

of layered soils [4].

All those developments, however, assume isotropic behaviour for the elastic medium. The

main reason for the isotropic hypothesis at that moment was the lack of fundamental solutionsfor the anisotropic cases, which are required for the application of the BEM. But there are many

cases in which the soil shows a marked anisotropic behaviour. In particular, the transversely

isotropic behaviour is often observed in soils.

The first solutions for the dynamic soilfoundation interaction problem with transversely

isotropic soils [5,6] were obtained only for a restricted form of elastic transverse isotropy [7].

Only more recently fundamental solutions for general transversely isotropic media have been

synthesized. Those solutions, however, require computationally expensive numerical integra-

tions in the evaluation process [8]. After those fundamental solutions became available,

numerical methods for the dynamic response of foundations interacting with transversely

isotropic media could be implemented.

The BEM implementations used in the evaluation of the dynamic response of rigid

foundations may be divided in two broad groups. One group is formed by general purpose BEMcodes that have been adapted for the analysis. Such implementations use, in general, full-space

fundamental solutions along with the direct version of the BIEM formulation. Those BEM

implementations which uses full-space solutions are more common because the fundamental

solution is more easily obtained than the half-space solution and the implementation can deal

with more general problems. This approach, however, requires the discretization of the soil top

free surface that surrounds the foundation, at least up to some distance of it [9]. The minimum

portion of the top surface which should be covered by the discretization is not known a priori

and its determination is part of the discretization convergence analysis.

The second group is formed by implementations specially developed for the soilstructure

interaction analysis. Those implementations employ, in general, half-space solutions along with

the indirect version of the BIEM. The use of half-space fundamental solutions makes the top

free surface discretization unnecessary, thus avoiding the inconvenience of the convergenceprocedure mentioned above and reducing the total number of elements to be used. On the other

hand, a frequent approach taken by those implementations is the application of point loads

along an auxiliary surface located at some small distance of the real soilstructure interface [10].

This is a limitation, since the location of this surface is not known a priori and must be

tentatively determined in the convergence analysis.

In Reference [11], a method for the analysis of the interaction of cylindrical foundations with

transversely isotropic soils based on the direct version of the BIEM, using half-space

fundamental solutions, was presented. In that paper, numerical results for the foundation

response to vertical and horizontal loads were shown for a few cases of transversely isotropic

materials. The procedure described in Reference [11] avoids both the free surface discretization

and the auxiliary loading surface. But it reportedly requires numerical integration of the Greens

functions at least along some elements. Since the Greens functions for this case themselvesrequire numerical integration, the procedure ends up with double numerical integrations, which

are computationally very expensive.

The method presented herein employs the indirect version of the BIEM and uses influence

functions, instead of Greens functions, as fundamental solutions. The influence functions are the

medium response to distributed loads and were obtained by analytical integration of half-space

Copyright # 2006 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. 2006; 30:683702

P. L. A. BARROS684


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Greens functions. The main advantage of this approach is the ability to apply the loads directly

along the true soilstructure interface, since there are no singularities in the influence functions,

thus avoiding the use of the auxiliary surface at some distance of the interface. In addition, the

top free surface discretization and the double numerical integration are also avoided. A similar

methodology has been successfully used before for plane-strain cases [12,13]. For the presentpaper, influence functions for disk and cylinder loads were synthesized by analytical integration

of half-space Greens functions along the horizontal and vertical directions.

Results for the foundation displacements (vertical and horizontal) and rotations (rocking and

twisting) due to external dynamic loads are presented. The impedances were obtained for different

transversely isotropic elastic media generated by means of variations of three independent

anisotropy indices. In this way, the influence of each of those indices on the foundation response

can be accessed. Results for the soilstructure interface tractions and for the displacement field

throughout the half-space, which show the proposed method capabilities, are also presented.

2. STATEMENT OF THE PROBLEM

The soilstructure system considered in this study is shown in Figure 1, where a rigid massless

cylindrical foundation block with radius a is embedded to a depth h in a surrounding

homogeneous transversely isotropic soil, whose symmetry axis coincides with the axis of the

block. The soil extends to infinity in the x; y; and positive z directions and its elastic constants cijare the parameters in the stressstrain relations:

sxx c11exx c12eyy c13ezz 1

syy c12exx c11eyy c13ezz 2

szz c13exx c13eyy c33ezz 3

syz 2c44eyz 4

sxz 2c44exz 5

sxy 12

c11 c122exy 6

F

h

a

z

z

xy

x

F

Fy

M

M

M

Figure 1. Rigid foundation embedded in a transversely isotropic half-space.


IMPEDANCES OF RIGID CYLINDRICAL FOUNDATIONS 685


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The isotropic case is a special one where c11 c33 l 2m; c12 c13 l and c44 m; wherel and m are the Lame` s constants. The anisotropic character of the material may also be

expressed by three anisotropy indices ni given by

n1 c33=c11 7

n2 c11 c12=2c44 8

n3 c11 2c44=c13 9

The foundation is perfectly bonded to the soil and is subjected to external time-harmonic

forces Fx; Fy; Fz and moments Mx; My; Mz: The structure steady-state response to these loads isgiven by the foundation displacements ux; uy; uz and rotations fx; fy; fz: Due to problemsymmetry, only the vertical and one of the horizontal forces and displacements, as well as the

twisting and one of the rocking moments and rotations should be considered in the analysis. The

relation between the forces/moments and the corresponding displacements/rotations is

expressed by the impedance matrix of the system. This relation can be written as

Fz

Mz=a

Fx

My=a

8>>>>>>>>>:

9>>>>>=>>>>>;

ac44

Kzz 0 0 0

0 Kyy 0 0

0 0 Kxx Kxf

0 0 Kfx Kff

2666664

3777775

uz

afz

ux

afy

8>>>>>>>>>:

9>>>>>=>>>>>;

10

were Kij are non-dimensional impedances.

Since the vertical displacement uz and the twisting rotation fz are uncoupled, they may be

solved independently. On the other hand, the horizontal displacement ux and the rocking

rotation fy are coupled and should be analysed together.

3. GREENS AND INFLUENCE FUNCTIONS

The basic ingredient in any BEM formulation is the so-called fundamental solution, which is a

known solution for an auxiliary elastic state with more general boundary conditions than the

original problem.

The present work proposes the use of half-space solutions of distributedloads called influence

functions, as fundamental solutions. Since the solutions for distributed loads does not present

singularities, the load can be applied on the real soilstructure interface.

The Greens functions for a load applied inside a transversely isotropic half-space have the

general form, in cylindrical coordinates [8]:

uikmr; z Z

1

0%uikmz%pkmz dz 11

sijkmr; z

Z10

%sijkmz%pkmz dz; i;j; k r; y; z 12


P. L. A. BARROS686


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were uikm and sijkm are, respectively, the mth component of Fourier expansions of the

displacements uir; y; z and stress components sijr; y; z in y direction due to a load in kdirection. The displacements are given by

urr; y; z X1m0

urmr; z cos my un

rmr; z sin my 13

uyr; y; z X1m0

uymr; z sin my un

ymr; z cos my 14

uzr; y; z X1m0

uzmr; z cos my un

zmr; z sin my 15

where uim and un

im are the symmetrical and antisymmetric components of the expansion,

respectively [14]. The stress components are given by similar expressions.

The terms%uikm and

%sijkm are functions of the elastic constants, the load depth, the frequency,and of the receiver r; z coordinates. The terms %pkm are the Hankel transform of the mth

component of the Fourier expansion of the load pkr; y applied in the k direction. This Fourierexpansion is given by

prr; y X1m0

prmr cos my pn

rmr sin my 16

pyr; y X1m0

pymr sin my pn

ymr cos my 17

pzr; y X1m0

pzmr cos my pn

zmr sin my 18

For the present application, half-space influence functions for loads distributed along annular

plates and cylindrical shafts were obtained (Figure 2). The annular influence functions were

obtained directly by applying Hankel transform on the load distribution. The influence

l rh

l

r

Figure 2. Annular and cylindrical loading elements.




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functions for cylindrical loads were derived through the analytical integration of the half-space

Greens functions for rings along the vertical surface.

The numerical evaluation of the influence functions deserves special attention, since this is the

most time-consuming part of the whole analysis process. The influence functions are expressed

as improper integrals of oscillating functions which are also singular at some points. Since theseintegrals have no known analytical solution, numerical techniques are required in their

evaluation. The singularities can be avoided by adding a negligible amount of internal dumping

to the material [8]. The oscillations, which are due to products of Bessel functions, are dealt with

an adaptive numerical integration scheme and the infinite upper limit of integration is dealt with

a series extrapolation procedure.

The process described above can be used to evaluate the influence functions at any point

inside the half-space, except for the cylindrically distributed load stress influence functions at

points on the loading path. The numerical evaluation is unstable when applied to the stresses at

those points because they are discontinuous along the loading path. For those cases, a modified

numerical procedure, which uses a three-effect superposition, was developed. The effect of the

top free surface, the effect of an infinitely long cylinder load, and the effect of a long cylinder

load with an unloaded portion are superposed to get the stress value. Details of this modifiedprocedure can be found in Reference [15].

4. PROBLEM SOLUTION

4.1. General solution

The soilfoundation contact is discretized in Nb annular elements along the base and Nfcylindrical elements along the vertical face, so the total number of elements is N Nb Nf (see

Figure 3). Along each element, uniform fictitious loads qi are applied inside the half-space.

The real interface displacements and tractions at the node points, located at the centre of each

element, are obtained by the superposition of the effects of these fictitious loads by

u Uq 19

t Tq 20

were the elements Uij and Tij are, respectively, the displacement and traction influence

functions at the centre of the ith element due to an unitary load applied on the jth element.

N + N = N

(i)

(i)

q , iN

N1 2

fb

b

b

b

Figure 3. Discretization in annular and cylindrical elements.


P. L. A. BARROS688


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The rigid-body condition imposed on the displacements can be expressed as

u Cu0 21

and the overall equilibrium requires that:

f Dt 22

where u0 is the foundation displacement vector, f if the external load vector, and C and D are,

respectively, the compatibility and equilibrium matrices. The above equations can be combined

in the following system of equations:

Uq Cu0 0

DTq f 23

The solution of this system gives the displacement u0 and the fictitious loads q: The realtractions t along the interface can be obtained by means of Equation (20). Also, the

displacements and stresses at any other point in the medium can be obtained by superposition of

the effects of q at that point.

4.2. Twisting

The case of foundation twisting under a Mz moment is the simplest one because an uniform

torsion load py applied inside the half-space causes only uy displacements. The influence

functions for this case are obtained by setting m 0 and taking the antisymmetric components

of the displacements uny0 and stresses sn

ry0; sn

yz0:The U and T matrices elements are taken as

Uijy u

n

y0 24

Tijy

snry0; 14i4Nb

snyz0; Nb5i4N(

25

The rigid-body condition imposed on the nodal displacements can be expressed as

uiy rifz; i 1; . . . ; N 26

and, assuming constant traction distribution over each element, the overall equilibrium requires

that:

Mz XNi1

tiy mi 27

mi 4pli

33r2i l

2i ; 14i4Nb

4pa2li; Nb5i4N

8>: 28

where ri and li are the mean radius and the half-length of the ith element, respectively (see

Figure 2).




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4.3. Vertical loading

The vertical tractions applied to the soil by the foundation under a vertical load cause not only

vertical displacements, but also radial displacements inside the elastic medium. Then, along the

soilstructure interface, vertical and radial tractions will be present.

After the interface discretization, uniformly distributed radial qr and vertical qz loads(see Figure 4), are applied on each element. The influence functions are obtained by setting

m 0 and taking the symmetrical terms of displacements and stresses.

The elements of the U and T matrices are given by

Uij Uijzz U

ijzr

Uijrz Uijrr

" #29

Tij Tijzz T

ijzr

Tijrz Tijrr

" #30

were Uijab

and Tijab

are, respectively, the displacement and traction in a direction at the centre of

the ith element due to a distributed unitary load applied at the jth element in b direction. They

are taken from the influence function as

Uab ua0 31

Tab saz0; 14i4Nb

sra0; Nb5i4N

(32

The rigid-body condition requires that:

uiz uz 33

uir 0; i 1; . . . ; N 34

at the ith node, and, assuming tz constant over each element, the equilibrium condition is

expressed by

Fz XNi1

4prilitiz 35

pr

pz

Figure 4. Radial and vertical loads.


P. L. A. BARROS690


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The solution of the system of equations gives the foundation vertical displacement uzand the fictitious loads qz and qr: Note that the radial traction equation is not used in the finalsystem of equations, so the T

ijrb

are not necessary, unless the real radial tractions tir are

required.

4.4. Rocking and horizontal loading

The influence functions required in the rocking and horizontal loading case are obtained

by setting m 1 and taking the symmetrical components of the loads, displacements and

stresses.

The pr1 and py1 loads may be combined to obtain a horizontal px load. This is done by

imposing pr1 py1 px: The displacement components ur1 and uy1 due to this horizontal loadcan also be combined to get the horizontal displacement ux:

ux ur1 uy1

236

Since, in general, ur1 and uy1 are not equal in modulus and opposed in sign, the horizontaldisplacement will not be constant along the circumferential direction y: There will be a circulardistortion ud expressed by

ud ur1 uy1

237

Along the soilstructure interface such distortion does not occur due to the rigid-body

condition. So, a counteracting distorting load pd must be applied to impose that condition. This

type of loading is obtained by setting pr1 py1 pd:Finally, the pz1 load results in a moment pm applied to the elastic medium and the uz1

displacement results in a rotation um: The three types of loads are shown in Figure 5.For this case, the elements of the U and T matrices are given by

Uij

Uijxx Uijxd U

ijxm

Uijdx U

ijdd U

ijdm

Uijmx Uijmd U

ijmm

26664

37775 38

Tij

Tijxx Tijxd T

ijxm

Tijdx T

ijdd T

ijdm

Tijmx Tijmd T

ijmm

26664

37775

39

px

pd

pm

Figure 5. Horizontal, distorting and moment loads.




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where Uijab and T

ijab have the same meaning as in the previous section. They are given by

Uxb ur1 uy1

240

Udb ur1 uy1

2 41

Umb uz1 42

Txb

srz1 syz1

2; 14i4Nb

srr1 sry1

2; Nb5i4N

8>: 43

Tdb

srz1 syz1

2; 14i4Nb

srr1 sry1

2 ; Nb5

i4N

8>>>: 44

Tmb szz1; 14i4Nb

srz1; Nb5i4N

(45

The rigid-body condition can be expressed as

uix ux fyzi 46

uid 0 47

uim fyri 48

where zi is the z coordinate of the node at the centre of the ith element. It should be noted that

the distorting load pd makes the displacement compatibility condition valid along the whole

circle (04y42p), at each nodal point (ri; zi).The equilibrium requires that:

Fx XNi1

4prilitix 49

My XNi1

mxitix mmit

im 50

wheremxi 4prilizi 51

mmi

2pli

33r2i l

2i ; 14i4Nb

2pr2i li; Nb5i4N

8>: 52


P. L. A. BARROS692


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The solution of the system of equations gives the foundation horizontal displacement ux and

rocking rotation fy; along with the fictitious loads qx; qd and qm: Note that the Tdb influencefunctions are not required for the foundation response evaluation.

5. NUMERICAL RESULTS

5.1. Validation

Numerical results obtained with the method described in the previous sections were compared

with results published by other researchers, both for the isotropic medium case [10] and for the

transversely isotropic medium case [11]. For this comparison, the soilstructure interface was

discretized with unequalelements, the length of which is proportional to the square root of the

distance from the element centre line to the foundation lower edge. This strategy results in finer

discretization near the foundation lower edge, where the tractions are known to be singular. In

this way, better convergence can be achieved with smaller number of elements. In addition,

smoother traction distributions along the soilfoundation interface can be obtained with this

discretization scheme.

Table I shows numerical values of the vertical, horizontal and rocking impedances Kij

kij ia0cij; i ffiffiffiffiffiffiffi

1p

; of a rigid cylindrical foundation, with an embedment ratio h=a 1: Themedium is isotropic with Poison ratio n 0:25: The impedances were calculated for selectedvalues of the non-dimensional frequency of the excitation forces a0 ao

ffiffiffiffiffiffiffiffiffir=G

p; where o is the

circular frequency, r the mass density and G the shear modulus of the medium. The results were

obtained with 12 elements along the foundations lateral face and 12 elements along its base.

Numerical results for the same impedance components, but now for a transversely isotropic

medium, are shown in Table II. The medium is a silty clay with c11=c44 2:11; n1 1:22;n2 0:84 and n3 0:23: Here a0 ao

ffiffiffiffiffiffiffiffiffiffiffir=c44

p:

Table I. Impedance values for a cylindrical block embedded in an isotropic medium (h=a 1; n 0:25).

kzz; czz kxx; cxx kff; cff

a0 This work Reference [10] This work Reference [10] This work Reference [10]

0.5 (8.07,10.52) (8.10,10.72) (9.62,11.21) (9.56,11.37) (12.59,3.81) (12.31,3.40)1.0 (7.48,10.71) (7.57,10.79) (9.31,11.03) (9.30,11.13) (11.31,5.32) (11.31,5.12)2.0 (6.15,11.33) (6.44,11.35) (8.35,11.24) (8.49,11.31) (10.11,6.38) (10.21,6.24)

Table II. Impedance values for a cylindrical block embedded in a transversely isotropic medium (h=a 1;c11=c44 2:11; n1 1:22; n2 0:84 and n3 0:23).

kzz; czz kxx; cxx kff; cff

a0 This work Reference [11] This work Reference [11] This work Reference [11]

0.5 (7.62,10.39) (7.47,10.50) (8.25,10.34) (8.18,9.71) (11.24,3.22) (11.47,3.44)1.0 (6.97,10.37) (6.91,10.38) (8.07,10.17) (8.06,9.71) (10.27,4.98) (10.18,5.20)2.0 (5.94,11.11) (6.53,10.41) (7.36,10.28) (7.59,10.15) (9.44,6.04) (9.14,5.85)




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As can be noted in Tables I and II, the numerical results obtained by the method proposed

here are in close agreement with those published in References [10,11].

In order to better evaluate the capabilities of the proposed method, plots showing the

tractions along the soilfoundation interface and the displacement field inside the half-space

were made. Figures 68 show the traction distribution along the face and base of a cylindricalfoundation with embedment ratio h=a 1: The soil is an isotropic half-space with Poison ration 0:25 and the frequency a0 1: Only the predominant traction components are shown. Forthe twisting case, ty is shown, while tz and tx are shown for the vertical and horizontal loading

cases, respectively.

0.0 0.2 0.4 0.6 0.8 1.0

r/a , z/h

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Face

Base

0.0 0.2 0.4 0.6 0.8 1.0

r/a , z/h

-0.04

-0.03

-0.02

-0.01

0.00

0.01

Face

Base

a3Re(t

)/M

za

3Im(t

)/M

z

Figure 6. ty tractions along the face and base of a cylindrical foundationsubjected to Mz moment (h=a 1; a0 1; n 0:25).

0.0 0.2 0.4 0.6 0.8 1.0

r/a , z/h

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Face

Base

0.0 0.2 0.4 0.6 0.8 1.0

r/a , z/h

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

0.04

Face

Base

a2Re(t

z)/F

za

2Im(t

z)/F

z

Figure 7. tz tractions along the face and base of a cylindrical foundation subjectedto Fz force (h=a 1; a0 1; n 0:25).


P. L. A. BARROS694


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The plots show that the obtained interface traction distributions are smooth, with exception

of tz distribution at the base, which shows a small oscillation near the edge. The observed

traction behaviour indicates that the chosen discretization strategy is very effective when

dealing with this kind of problem. The evaluation of the interface traction distribution at the

base of a rigid foundation subjected to a dynamic loading was analysed in Reference [16], for

surface strip foundations (plane strain). It was shown that the use of higher order interpolation

functions (linear, quadratic, etc.) does not improve the results. Better accuracy on the tractions

can only be achieved with singular traction elements. But singular traction elements are

available only for infinite strip loads applied at the top surface of the half-space. It is believed,

however, that localized inaccuracies on the tractions have small effect on the foundationdisplacements.

Figure 9 shows the displacement field inside the half-space for 04r; z410a; for the case of anhorizontal load Fx applied at the base centre of a cylindrical foundation (h=a 1; a0 1;n 0:25).

The displacement fields in Figure 9 show the wave propagation pattern within the half-space.

5.2. Influence of the degrees of anisotropy on the impedances

Numerical results for the impedances of a rigid cylindrical foundation, for three embedment

ratios h=a 0; 0:5; 1 were obtained. The elastic media used in the present analysis has a ratioc11=c44 3: The other elastic constants used are shown in the Table III, where the soil 1 is an

isotropic material with Poisson ratio n 0:25:The soilstructure interface discretization uses 12 unequal-length elements at the base and 0, 6

and 12 elements at the foundation side surface for h=a 0; 0:5 and 2; respectively.In the twisting case, since only c11; c12 and c44 are involved in the influence functions, only the

n2 anisotropy index has influence on the impedance. The obtained results were plotted as a

function of a0 in Figure 10, were the twisting impedance is given by Kyy kyy ia0cyy:

0.0 0.2 0.4 0.6 0.8 1.0

r/a , z/h

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Face

Base

0.0 0.2 0.4 0.6 0.8 1.0

r/a , z/h

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

Face

Base

a2Re(t

x)/F

xa

2Im(t

x)/F

x

Figure 8. tx tractions along the face and base of a cylindrical foundation subjected to Fx force applied at

the foundations base (h=a 1; a0 1; n 0:25).




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The n2 anisotropy index has a marked effect on the twisting stiffness kyy but a much smaller

effect on cyy: This occurs for the three embedment ratios tested. It should be noted that the n2index is the ratio between the shear modulus inside the horizontal plane, which is the plane of

isotropy, and the shear modulus inside the vertical plane. In this way, these results show the

expected behaviour, with kyy increasing with n2:It is worth noting that, for the surface foundation (h 0), the torsional impedances

for the transversely isotropic cases can be derived from the isotropic case impedances. In order

to get this, one should substitute the shear modulus G for c11 c12=2 in the isotropicformulation and, after evaluating the torsional impedance Kyy for this isotropic soil, multiply it

byffiffiffiffiffi

n2p

:For the vertical loading case, the c12 elastic constant is not present in the involved influence

functions. So, only the n1 and n3 anisotropy indices have influence on the foundation response.

The same discretization used in the twisting case was used here. The numerical results

for the vertical impedance, expressed by Kzz kzz ia0czz; are shown in the plots presented inFigure 11.

0 2 4 6 8 10r/a r/a

0

2

4

6

8

10

z/a

z/a

0 2 4 6 8 10

0

2

4

6

8

10

(a) (b)

Figure 9. In phase (a) and p=2 rad out of phase (b) parts of the displacement field inside the y 0 planedue to a horizontal load Fx applied at the base of the foundation (h=a 1; a0 1; n 0:25).

Table III. Media elastic constants.

Soil c11=c44 n1 n2 n3

1 3.0 1.0 1.0 1.02 3.0 0.5 1.0 1.03 3.0 2.0 1.0 1.04 3.0 1.0 0.5 1.05 3.0 1.0 2.0 1.06 3.0 1.0 1.0 0.57 3.0 1.0 1.0 2.0


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The plots in Figure 11 show that the influence of the n1 anisotropy index is greater than

the n3 index influence on both kzz and czz: This is the expected behaviour, since n1 expresses

the soil relative stiffness in the vertical direction compared to the stiffness in the horizontaldirection.

Numerical results for the horizontal and rocking impedances are shown in the plots in

Figures 1214. For those results, the reference point, where the horizontal loads and rocking

moment are applied, is located at the centre of the foundation base. In this case, all three

anisotropy indices are involved in the influence functions.

0.0 0.5 1.0 1.5 2.0

2

4

6

8

k

1

4

5

0.0 0.5 1.0 1.5 2.00.0

0.5

1.0

1.5

c

k

c

k

c

0.0 0.5 1.0 1.5 2.0

5

10

15

20

25

0.0 0.5 1.0 1.5 2.0

0

1

2

3

4

5

0.0 0.5 1.0 1.5 2.0

a0

a0

a0

a0

a0

a0

10

20

30

40

0.0 0.5 1.0 1.5 2.00

2

4

6

8

10

h/a = 0

h/a = 0.5

h/a = 1

Figure 10. Twisting rotation impedances.




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For the horizontal impedances expressed by kxx and cxx; n2 has the largest and n1 the smallestinfluence. On the other hand, for the rocking impedances expressed by kff and cff; the largestinfluence is exert by n1 and the smaller influence by n2: Also, the anisotropy indices show muchmore influence on kff than on cff:

For the coupling impedances expressed by kxf and cxf; the relative weight of each anisotropyindex depends on the embedment ratio h=a: For h=a 0; the three indices seems to exert

0.0 0.5 1.0 1.5 2.0

a0

a0

a0

a0

a0

a0

2

4

6

8

10

kzz

czz

kzz

czz

kzz

czz

1

2

3

6

7

0.0 0.5 1.0 1.5 2.0

2

3

4

5

6

7

8

0.0 0.5 1.0 1.5 2.0

2

4

6

8

10

12

0.0 0.5 1.0 1.5 2.0

6

7

8

9

10

0.0 0.5 1.0 1.5 2.0

2

4

6

8

10

12

0.0 0.5 1.0 1.5 2.0

8

10

12

14

h/a = 0

h/a = 0.5

h/a = 1

Figure 11. Vertical displacement impedances.


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comparable influence on the coupling impedances, but as the h=a ratio increases, n2 becomesmore important.

6. CONCLUDING REMARKS

A complete formulation and implementation for the evaluation of a rigid massless cylindrical

foundation embedded in a transversely isotropic elastic half-space impedances was presented.

3

4

5

6

kxx

1

2

3

4

5

6

7

2.5

3.0

3.5

1

2

3

4

5

6

k

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.5 1.0 1.5 2.0

a0

a0

0.2

0.4

0.6

0.8

1.0

kx

cxx

c

cx

0.0 0.5 1.0 1.5 2.0

0.0 0.5 1.0 1.5 2.0

a0

a0

0.0 0.5 1.0 1.5 2.0

0.0 0.5 1.0 1.5 2.0

a0

a0

0.0 0.5 1.0 1.5 2.0

-0.2

0.0

0.2

0.4

h/a = 0

Figure 12. Horizontal and rocking impedances for h=a 0:




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With this formulation, the displacement and stresses at any point of the elastic medium can also

be obtained.The method proposed here considers complete bonding between the soil and the structure.

Simplified solutions for partial bonding conditions can also be derived. For example, one can

implement a relaxed interface condition for the vertical loading case by using only the zz

components of the displacements and fictitious loads in the formulation. The system of equation

will then result half the size of the complete bonding case system. The same applies to the

2

4

6

8

10

12

kxx

1

2

3

4

5

6

7

6

7

8

9

10

2

4

6

8

10

k

0

1

2

3

4

0.0 0.5 1.0 1.5 2.0

a0

0.0 0.5 1.0 1.5 2.0

a0

0.0 0.5 1.0 1.5 2.0

a0

0.0 0.5 1.0 1.5 2.0

a0

0.0 0.5 1.0 1.5 2.0

a0

0.0 0.5 1.0 1.5 2.0

a0

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

kx

cxx

c

cx

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

h/a = 0.5

Figure 13. Horizontal and rocking impedance for h=a 0:5:


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horizontal-rocking case. If the components of the displacements and fictitious loads whichinclude the distortion effect are neglected, an incomplete bonding condition along the soil

foundation interface is allowed, and the resulting system of equations is reduced.

Procedures which impose bonding along a limited portion of the soilstructure contact can

also be easily implemented, due to the clear physical meaning of the fictitious loads used in the

formulation.

6

8

10

12

kxx

1

2

3

4

5

6

7

10

12

14

8

10

12

14

16

k

2

4

6

8

0.0 0.5 1.0 1.5 2.0

-4

-3

-2

-1

0

kx

cxx

c

cx

0.0 0.5 1.0 1.5 2.0

a0

a0

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

a0

a0

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0

a0

a0

-5

-4

-3

h/a = 1

Figure 14. Horizontal and rocking impedances for h=a 1:




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ACKNOWLEDGEMENTS

The development of this research was made possible by the support of the Fundacao de Amparo a` Pesquisado Estado de Sao Paulo FAPESP (00/10186-4). The author also acknowledges the support of theMassachusetts Institute of Technology, in the USA, where this research was partially developed.

REFERENCES

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Engineering Division (ASCE) 1972; 98(SM12):12911310.4. Luco JE. Impedance functions for a rigid foundation on a layered medium. Nuclear Engineering Design 1974;

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International Journal of Mechanical Sciences 1981; 23(9):547559.6. Kirkner DJ. Vibration of a rigid disk on a transversely isotropic elastic half space. International Journal for

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8. Rajapakse RKND, Wang Y. Greens functions for transversely isotropic elastic half space. Journal of EngineeringMechanics (ASCE) 1993; 119:17241746.9. Emperador JM, Dominguez J. Dynamic response of axisymmetric embedded foundations. Earthquake Engineering

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13. Barros PLA, Mesquita Neto E. A non-singular version of the indirect boundary element method applied tostationary response of elastic domains. Proceedings of the 13th ASCE Engineering Mechanics Conference, ASCE,1999 (in CD format).

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15. Barros PLA. Accurate stress evaluation of influence functions at the load application points. In The 2005 JointASCE/ASME/SES Conference on Mechanics and Materials Electronic Proceedings, Voyiadjis GZ, Dorgan RJ (eds).ASCE, 2005; 14 (in CD format).

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