Vibration isolation by a row of piles using a 3-D frequency domain BEM.pdf

8/11/2019 Vibration isolation by a row of piles using a 3-D frequency domain BEM.pdf

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trenches, usually in connection with special non-re#ecting boundaries. The BEM is the ideal

method for this class of problems since it requires only a surface discretization and satis"es the

radiation conditions at in"nity automatically. It has been used, e.g. by Emad and Manolis [7],

Beskoset al. [8, 9], Leung et al. [10, 11], Beskos and Vardoulakis [12], Von Estor!and Antes

[13], Ahmad and Al-Hussaini [14], Al-Hussaini and Ahmad [15], Stamos et al. [16] and Mateo

and Alarcon [17] to successfully solve in the frequency or time domain two-dimensional

vibration isolation problems involving open or in"lled trenches in homogeneous or layered soils.

Three-dimensional cases of vibration isolation by open or in"lled trenches have also been solved

by the frequency domain BEM, e.g. by Beskos and Vardoulakis [18], Banerjee et al. [19],

Dasguptaet al. [20], Kleinet al. [21, 22], Ahmad et al. [23] and Al-Hussaini and Ahmad [24].

The review articles of Beskos [25, 26] on the BEM as applied to dynamic problems contain

a complete bibliography on vibration isolation up to the year 1996.

Use of trenches as wave barriers is restricted to cases involving small to medium trench depth

because of soil instability and water table level problems. When the transmitted waves, primarily

Rayleigh surface waves, have long wavelengths, trenches cannot be used as e!ective wave barriers.

Thus, when the required barrier depth is very large, a series of piles is the solution. Woods et al.

[27] and Liao and Sangrey [28] were the "rst to study experimentally the problem of thescreening e!ectiveness of piles and provide some design guidelines. The problem of vibration

isolation by a row of piles is a truly three-dimensional multiple wave di!raction problem di$cult

to solve even by numerical methods. Aviles and Sanchez-Sesma [29, 30] and Baroomand and

Kaynia [31] have presented approximate analytical solutions of the problem, which, however, in

view of the various simplifying assumptions involved and the complexity of the problem, seem

only qualitatively correct.

In this work the problem of vibration isolation by a row of piles is solved for the "rst time by

a highly accurate and e$cient BEM in the frequency domain in a three-dimensional context. The

BEM is used to model both the piles and the soil, which are both assumed to be linearly elastic or

viscoelastic materials. The soil and pile domains are coupled together through equilibrium and

compatibility at their interfaces. The dynamic behaviour of foundation piles has been studied in

the past by using the BEM for the soil and the FEM for the piles treated as one-dimensional beamstructures [32]. However, this pile modelling and the imperfect matching between soil and piles

do not provide accurate results, while the symmetry advantage of the FEM equations is

destroyed when BEM and FEM are coupled together. Use of the BEM for both the soil and the

piles ensures perfect matching of the two kinds of domains and results of high accuracy. The piles

can be tubular or solid and have circular or square cross-sections. The vibration source is

a vertical concentrated force harmonically varying with time, while the row of piles acts as a wave

barrier in a passive way.

The BEM program used in this work employs various types of boundary elements (here

eight-noded quadratic quadrilateral boundary elements are used) continuous or partially discon-

tinuous and treats singular integrations directly [33] in a highly accurate manner. In addition the

program has symmetry, antisymmetry and substructuring capabilities, while it can be interfaced

to commercial pre- and post-processors. The full space fundamental solution is used in theformulation, which requires a free soil surface discretization. However, this discretization is

restricted to a"nite part around the area of interest and is needed anyhow in vibration isolation

analysis where surface displacements are needed everywhere inside the area of interest.

The computer program is "rst tested for accuracy by comparing its results pertaining to two

three-dimensional wave di!raction problems involving spheres and trenches as scatterers against

714 S. E. KATTIS, D. POLYZOS AND D. E. BESKOS

Copyright 1999 John Wiley & Sons, Ltd. Int. J . Numer. Meth. Engng. 46, 713}728 (1999)


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where [;] and [] are in#uence matrices with entries integrals over boundary elements with

integrands involving the fundamentals tensors u*

andt*

, respectively, whiletand u denotethe vectors of the nodal boundary tractions and displacements, respectively. Employment of the

boundary conditions (2) in equation (5) and subsequent rearrangement to separate known from

unknown boundary nodal values results in the equation

[A] x"b (6)

to be solved for the unknown nodal values of displacements and tractions constituting the vector

xin terms of the known quantities constituting the vector b. When a problem involves morethan two elastic bodies in full contact, as it is the present soil}pile system, equation (5) is written

for every body in partitioned form and the resulting equations are coupled together through

equilibrium and compatibility at their interface [32].

Interior displacements and stresses as well as stresses on the boundary can also be determined.

The latter ones are evaluated highly accurately in a direct manner from the expressions for the

interior stresses by employing Huber's [38] algorithm to deal with the hypersingular integrals

involved. Thus, the classical approximate way for evaluating boundary stresses through the use of

boundary tractions and tangential displacement derivatives [32] is not needed any more. Ofcourse, in the present vibration isolation problems there is no need of computing either interior

displacements and stresses or boundary stresses.

An advanced computer code in Fortran 90 has been constructed for the accurate solution of

three-dimensional linear elastic systems in the frequency domain [34, 36] by following the

aforementioned BEM procedure. The main features of this code, characterized by high accuracy

and e$ciency, are the following:

(i) Use of linear or quadratic, triangular or quadrilateral boundary elements. These elements

can be continuous or discontinuous (totally or partially). Partially discontinuous elements

[39] are used to accommodate edges, corners, interfaces and discontinuities in the

boundary conditions.

(ii) Evaluation of singular and hypersingular integrals is done by the direct very accurate ande$cient algorithm of Guiggiani [33] and Huber et al. [38], respectively. Thus, there is no

need to employ enclosing elements which have to be used when the evaluation of singular

integrals is done by using the rigid-body motion concept [26]. The evaluation of regular

integrals is done by Gauss quadrature and variable integration order which depends on

the distancer. In order to evaluate nearly singular integrals with accuracy, a method based

on Bu [40] is used.

(iii) Substructuring as well as symmetry and antisymmetry capabilities, which drastically

reduce the size of a large-order problem.

(iv) E$cient algorithm for solving systems of linear equations involving dense non-symmetric

complex matrices either in-core [41] or out-of-core [42].

(v) Interior and boundary displacements and stresses are accurately determined.

(vi) Interfaces to commercial pre- and post-processors for the convenience of the user.

In some respects the present computer code is similar to the code of Klein et al. [21, 22], apart

from the facts that here: (i) Geometric and boundary data discontinuities are treated in a more

general manner by discontinuous elements [39] (elements with collocation points inside rather

than on the element edges) and not by the employment of the free term c

[37]. Thus, one can




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Figure 2. Amplitude of radial displacements versus the scattering angle for the example of the wave scattering by a sphere

Figure 3. Amplitude of tangential displacements versus the scattering angle for the example of the wave scattering bya sphere

For the solution of this problem by the present BEM code the discretization of

(due to

symmetry) of the spherical surface of the scatterer involving 14 quadratic quadrilateral eight-

noded boundary elements as shown in Figure 1 was used in the analysis. Figures 2 and 3 show theamplitude of the radial and tangential displacement, respectively, versus the di!raction anglefork"/c

"1)3, ""0)25, /"1)0 and for the cases of a cavity, a soft inclusion

(c

/c"0)5) and a hard inclusion (c

/c

"2)0) as obtained for the mesh of Figure 1 by the

present BEM as well as by analytic means [43]. In all cases the agreement between the numerical

and the analytical results is excellent.




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Figure 5. Discretization of one-quarter of the active vibration isolation system

Figure 6. Amplitude reduction factor versus distance across the diagonal OA for the active vibration isolation system

a!ect the vibration results and in order to be compatible with the analysis of Klein et al. [21, 22],

who have solved the same problem and produced highly accurate results.

Figure 6 depicts the amplitude reduction factor AR versus the non-dimensional distance x/

from the source of disturbance along the direction of the diagonal OA of Figure 4 as computed by

the present method as well as by that of Klein et al. [21, 22] and using the same mesh in both

cases. The factor AR is de"ned as the ratio of the vertical displacement component of the soilsurface in the presence of the trench over that displacement in the absence of the trench. The

present results are in excellent agreement with those of Klein et al. [21, 22]. This was expected in

view of the fact that the codes of the two methods are similar. It is obvious that for the data of the

problem, this active vibration isolation system is clearly e!ective, at least along the direction of

the diagonal OA, as AR)0)25 outside the trench barrier [2].




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Both these examples as well as additional ones in [36] serve to illustate the BEM code

previously described and demonstrate its high accuracy.

4. VIBRATION ISOLATION BY A ROW OF PILES

As it was mentioned in the introduction, when the ground transmitted waves have a large

wavelength, which would require unrealistic trench depths (larger than 0)8

) for screening

e!ectiveness, the only practical type of wave barriers is that of a row of closely spaced piles. In this

section the wave screening e!ectiveness of a row of piles is studied numerically by using the BEM

computer code described in Section 2. The piles may be tubular or solid and may have square or

circular cross-sections. Tubular piles are modelled as long cylindrical cavities in soil for reasons of

simplicity, while the solid ones can be of concrete or any other material exhibiting linear elastic or

viscoelastic behaviour.

Piles have some distinct advantages over trenches as wave barriers. These are: their ability to be

driven very deeply into the soil, the well-known technology for foundation piles and the

possibility of any desirable arrangement to create wave barriers. Their screening e!ectiveness,however, as it will be shown in the following, is not as good, in general, as that of trenches. The

modelling of the vibration isolation by piles problem is also more complicated and consequently,

its solution is more di$cult than the one for the corresponding trenches problem. Indeed the

vibration isolation by piles problem is a multiple wave di!raction problem of a truly three-

dimensional character which requires many more substructures and a much larger number of

degrees of freedom than the corresponding trench problem. A typical isolation by a row of eight

concrete piles problem requires six times the computer solution time than the corresponding

equivalent concrete trench problem. This is probably the reason why the problem has not

been solved before, even with the BEM which is ideally suited for this type of problem and in spite

of the fact that it does not present serious theoretical di$culties, at least in principle. Here

use is made of the BEM code described in the previous two sections to solve the problem and

through parametric studies assess the screening e!ectiveness of piles and reach some practicalconclusions.

In the following, the passive vibration isolation problem by a row of eight concrete piles of

circular cross-section is solved in detail for illustration purposes. Results of parametric studies

involving rows of six and eight open or in"lled piles of square or circular cross-section are

subsequently presented which help to understand the behaviour of the system with respect to its

vibration isolation e!ectiveness.

Consider the row of eight concrete piles of circular cross-section shown in Figure 7. Every pile

has a sectional diameter d and length h, while the surface to surface spacing between two

successive piles iss (Figure 7). The distance of the centre of the row of piles from the vibrational

source, i.e. a vertical load harmonically varying with time with magnitude P"1 KN and

operational frequency 50 Hz, is r as depicted in Figure 7. The soil is assumed to be linear

viscoelastic with a shear modulus G"132 MPa, Poisson's ratio "0)25, mass density

"17)5 Kg/m and damping coe$cient

"5 per cent, while the concrete piles are assumed

also to behave linearly viscoelastically with G"34)29G

,

"

"0)25,

"1)37

and

"

5)0. The Rayleigh wavelength for the soil is computed to be

"5 m and the normalized by

geometrical data of the problem read R"r/"1)5, D"d/

"0)2, H"h/

"1)0 and

S"s/"0)1.

VIBRATION ISOLATION BY A ROW OF PILES 721



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Figure 7. Cross-section and top view of the pile barrier vibration isolation system

The solution of the problem is obtained as the superposition of a symmetric and an antisym-

metric problem for each of which only

of the problem requires discretization. This discretization

is shown in Figure 8 and involves"ve substructures (four for the piles and one for the soil) and 330

eight-noded quadratic quadrilateral boundary elements (1285 functional nodes). The size of the

elements corresponds to approximately "ve elements per Rayleigh wave length, at least in the

area close to piles. The free soil surface is extended to a distance of 3

from the centre of the pile

barrier system. The meshes constructed on the basis of these rules were also found to providesatisfactory results by means of convergence studies involving coarse and re"ned meshes not

shown here for lack of space. The"ve substructures are coupled together by enforcing equilibrium

and compatibility at their interfaces in an automatic manner by the code. Figure 9 depicts the

variation of the factor AR on the whole soil surface for the above passive isolation by eight

concrete piles problem. The average surface amplitude reduction factor behind the trench is




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Figure 8. Discretization of one-fourth of the pile barrier vibration isolation system

Figure 9. Contour of the amplitude reduction factor for the pile barrier vibration isolation system (eight concrete circularcross-section piles)

de"ned as AR"(1/A)

AR dA, whereA is the area behind the trench enclosed by the semicircle

with a radiusl/2 (one-half the length of the row of piles) andARis the amplitude reduction factorde"ned previously. In this problem it was found that AR"0)712.

In the following, additional numerical results are presented for other cases of passive vibration

isolation with variations in the material of the piles and their cross-section. Thus, Figure 10 shows

the variation of factor AR on the whole soil surface for the case of eight &open' piles of circular

cross-section. The BEM mesh of this case involved 234 eight-noded quadratic quadrilateral




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Figure 10. Contour of the amplitude reduction factor for the pile barrier vibration isolation system (eight open circularcross-section piles)

Figure 11. Contour of the amplitude reduction factor for the pile barrier vibration isolation system (eight open square

cross-section piles)

boundary elements (865 functional nodes) for

of the system and the AR was found to be 0)812.

Figure 11 portraysAR for the case of eight&open'piles of square cross-section with sideD"0)2,

while Figure 12 depicts corresponding things for the case of eight concrete piles of square cross-

section. The discretizations for these two last cases involved 206 and 274 boundary elements,




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Figure 12. Contour of the amplitude reduction factor for the pile barrier vibration isolation system (eight concrete squarecross-section piles)

Table I. Average surface amplitude reduction factor (AR) for the various pilebarrier vibration isolation systems

Average surfaceNumber of Type of Type of amplitude reductionpiles pile material pile cross-section factor (AR)

6 Open Circular 0)926

6 Concrete Circular 0)7556 Open Square 0)8686 Concrete Square 0)7218 Open Circular 0)8128 Concrete Circular 0)7128 Open Square 0)6988 Concrete Square 0)675

10 Open Circular 0)64810 Concrete Circular 0)62410 Open Square 0)64210 Concrete Square 0)620

respectively, while their AR factors were found to be 0 )698 and 0)675, respectively. Details

concerning cases involving six or ten &open'or concrete piles of circular or square cross-section

can be found in [36]. Table I provides values of AR for all the aforementioned cases of passive

vibration isolation by a row of six, eight or ten piles. It is observed that AR decreases with the

number of piles or equivalently with the decrease in the spacing between piles. It is also observed




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REFERENCES

1. Barkan DD. Dynamics of Bases and Foundations. McGraw-Hill: New York, 1962.2. Richart Jr. FE, Hall JR, Woods RD. ROD>N+90, Kratzig WB et al. (eds). A.A. Balkema: Rotterdam, 1991; 693}700.

13. Von Estor!O, Antes H. Vibration isolation analyses in the case of transient excitations by a coupled BE/FEprocedure. In Computational Mechanics '91, Atluri SN, Beskos DE, Jones R, Yagawa G (eds). ICES Publications:Atlanta, 1991; 1148}1153.

14. Ahmad S, Al-Hussaini TM. Simpli"ed design for vibration screening by open and in-"lled trenches. Journal ofGeotechnical Engineering 1991; 117:67}88.

15. Al-Hussaini TM, Ahmad S. Design of wave barriers for reduction of horizontal ground vibration. Journal ofGeotechnical Engineering 1991; 117:616}636.

16. Stamos AA, Von Estor!O, Antes H, Beskos DE. Vibration isolation in road-tunnel tra$c systems. InternationalJournal of Engineering Analysis and Design1994; 1:109}121.

17. Mateo J, Alarcon E. On the use of trenches and walls on the control of ground transmitted railway vibrations.InBoundary Element Method X


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31. Baroomand B, Kaynia AM. Vibration isolation by an array of piles. InSoil Dynamics and Earthquake Engineering

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Vibration isolation by a row of piles using a 3-D frequency domain BEM.pdf