propagation of vibration in a soil layer over bedrock

Propagation of vibration in a soil layer over bedrock

N. Chouw, R. Le and G. Sclunid

Department o f Civil Engineering, Ruhr-University o f Bochum, D-4630 Bochum 1, Germany

This paper presents results of numerical studies on the propagation of waves caused by a harmonically excited rigid strip foundation in a soil layer over bedrock. The analyses are performed in the frequency domain using the boundary element method. The conclusion is drawn that the depth of the soil layer together with the direction of excitation and the soil properties significantly affect the propagation of waves. Some suggestions for the placement of a machine foundation is made in order to limit the amplitudes of vibration.

Key Words: bedrock, boundary elements, frequency domain, soil layers, strip foundation, vibration amplitude.

1. INTRODUCTION

Many structures, for example a pile being driven into the soil or a foundation of an operating machine, transmit dynamic loads to the surrounding soil. The induced energy produces waves in the soil. These waves propagate from the source and transmit energy to other regions. Wave propagation can cause undesired vibrations of buildings in the neighbourhood.

To isolate buildings from ground-transmitted vibrations wave barriers, like a trench or a solid wall, have sometimes been installed between the source of the waves and the building. Many experimental and numerical tests show that such barriers can successfully reduce the amplitudes of vibrations k2.a. However, in order to find an appropriate protection measure for the isolation of structures, it is necessary to have reliable information about the behavior of the propagation of waves in the soil medium. The propagation of waves in a soil layer over bedrock is studied in this paper. The source of the waves is a rigid surface strip foundation which is excited by a harmonically varying load. It will be assumed that the amplitudes of vibration in the soil are infinitesimal and the soil is defined as an isotropic homogeneous elastic material that behaves in a geometrical and physical linear way.

The boundary element method (BEM) used here 4 has the following essential advantages: i) The spreading of waves to infinite regions is correctly represented, ii) Only the boundaries of the studied regions have to be discretized.

2. FORMULATION OF WAVE PROPAGATION BY THE BEM

2.1. Equations o f motion The equations of motion of a linear isotropic elastic

soil can be expressed in terms of displacements

ui = ui(x,t); i = 1,2,3, as

- c,)ujji + c, ui.~ + -- = ai (1) ,~p p

where the velocity of shear wave c~ = x/(E/(2o(1 + v)) and of compression wave cp = x/(E(1 - v))/(a(l + ;,) ( 1 - 2 v ) ) , Ki is the vector of body force per unit volume, E is the modulus of elasticity, ~, is the Poisson's ratio, p is the mass density of the soil and where ( ),0 and ( ) partial space and time derivatives, respectively and summation is assumed over repeated indices.

It is useful to transform the equations of motion into the frequency domain and obtain

Cs)Uj.jk + W2U k = -- (2) + Cs Uk,jj - - P

where w is the circular frequency and where the displacements u~ and the body forces P~ are now frequency dependent. In the special case of harmonic loading and if only steady-state conditions are considered, equation (2) represents the displacement and force amplitudes of the harmonic motion. In this case zero body forces are assumed.

2.2 The boundary element method The basic idea of the proposed boundary element

method (BEM) is to transform the partial differential equations (2) into boundary integral equations. Using Betti's theorem for a frequency component ~ the boundary integral equation formulation reads

CUj(XCt) "~" IF (tiUffiJCt - ri?~tui) dr (3)

© Computational Mechanics Publications 1 9 9 1 Engineering Analysis with Boundary Elements, 1991, Vol, 8, No. 3 125

where c = I for x ~ inside the domain, c = 0.5 for a smooth boundary point, t~ and ui are the tractions and displacements, respectively, at the boundary P and U~; ~ and T, *~ are the fundamental displacement and traction components, respectively, given by Cruse and Rizzo 5 as

U~(x, x ~) = (~16ij - xr,irj)Pj (4) 2voc]

1 [(d_~ r 1 ) ( Or ) : X 6iJ On + rqni Ti~(x, x ~) ~ r

2( - r X n jr , i - 2r,ir,j -~n

- 2 dx Or (~_ 2"~(d~ dr r,ir,; On + \4 /\Tr

dx 1 ) ] dr r X r,inj (5)

with 6~j being the Kronecker symbol, r = I x - x~l the distance of the considered point from the point load and nj the outward unit normal. For the two- dimensional problems considered here the functions '~, and X are

= - - + - K 1 - - - K1 \ c~ / itor \ c, / cp \ c p / j

and

K2(iwr~ c~ (iwr~ x = - - - c i K 2 - -

\ c , / ci kCp /

where Ko, K~, K2 are the Modified Bessel functions of second kind and order 0, 1 and 2, respectively and i = 4 : - 1.

For a region with arbitrary geometry and boundary conditions it is not possible to integrate its boundary integral equations analytically. However, a discretization of the boundary into a finite number of boundary elements makes it possible to integrate the integral equations numerically. Thus displacements and tractions inside an element are expressed as

uAx) = f~(x)u f (6) ti(x) = fla(x)tf (7)

where x is any point at the boundary P, fla are the shape functions and uf and tf are the discrete displacements and tractions at the nodal point/3. With this approximation, the steady-state wave propagation in the soil, equation (3), can be formulated as the algebraic matrix equation

Tu = Ut (8)

where u and t are respectively the discrete values of displacement and traction of the elements at the boun-

AMPLITUDE [ - ] 0.64.

~--- L-----IBm--- R ~ Approximate

s ' , ' . i

0 , 0 0 I 1 I °" : I ! J I I ! ~ '" !

-240 -120 0 120 240

DISTANCE FROM SOURCE [ m ]

Fig. 1. Influence of the two different approaches for the Modified Bessel Functions on vertical surface displacement

dary of the analysed region, while the matrix elements of U and T result from the integration of the fundamental solution over boundary elements.

In the analysis of wave propagation numerical dif- ficulties occur if the exact Modified Bessel functions given in Ref. 6 are used. To overcome the default, an approximate polynomial expansion of the Modified Bessel functions" is applied. Figure 1 shows, as an example, the influence of the two different approaches for the Modified Bessel functions. The vertical surface vibration of the homogeneous half-space results from the vibration of a rigid foundation, due to a unit excitation with a frequency of 5 Hz. The soil has a mass density p = 1800 kg/m 3, a Poisson's ratio v = 0.33 and a shear modulus G = 53.28. 106 N/m 2. No material damping is assumed. Using the approximate Modified Bessel function (solid line) the analysis in Fig. 1 has been performed with a discretization length of 7 LR on both sides of the foundation. LR is the length of the Rayleigh wave and has the value of 32 m. In the case of the exact Modified Bessel function (dotted line) the discretization length is 5 LR on the left and 7 LR on the right side of the foundation. It can be seen that the vibrations at the foundation are the same in both cases. Using the exact Modified Bessel functions numerical errors occur out- side the foundation. The larger the distance offthe foundation the larger the error. The error is especially more pronounced close to the end of the discretization. A comparison with results obtained using other methods shows that the current boundary element method works reliably and gives accurate results s.

3. NUMERICAL STUDIES

An operating machine produces vertical, horizontal and rocking excitation on the foundation simultaneously. In the following study the excitations will be considered separately. The unit excitations have in all cases a frequency of 5 Hz. The soil considered has the same properties as specified before.

126 Engineering Analysis with Boundary Elements, 1991, Vol. 8, No. 3

L B R

~ Pz(t)-Pz(t)-Pm(t) Foot i ~r - P o • exp ( i ~ t )

E= o ~ 8 B.E/LR

Fig. 2. Discretization and geometry of a surface foundation on a soil layer

Figure 2 shows the soil layer discretized for a length o f L + B + R, where L = R = 7 LR and B = 3 m is the width of the foundation. The surface of the soil layer and the bedrock is discretized with eight boundary elements for each Rayleigh wave length LR, and the massless foundation is represented by six elements.

First for the analysis of the vibrations at the source and the surrounding region a soil layer with a depth of 1,25 LR is chosen. An inspection of Fig. 3 shows that

AMPLITUDE [ - ] 0.8

0.4

0.0

Excitation :

, , ! w I ! i | ! ! g ! i

-20 0 20 DISTANCE FROM SOURCE [ m ]

the vertical excitation causes no horizontal vibrations at the source due to symmetry. However, in the surrounding region the amplitude increases with increasing distance off the source. Similar magnitude and development occur for the amplitude of vertical surface vibrations due to the horizontal excitation, but the source vibrations are not zero. Development of vertical vibration due to the vertical excitation and the amplitude development of horizontal vibration due to the horizontal excitation are similar. The vibration at the source in the case of a horizontal excitation is larger than that due to a vertical excitation. The rocking excitation gives only small horizontal responses. The amplitude at the source and at the surrounding soil have nearly the same size. In contrast to this, the vertical responses are much larger and have their maximum at the ends of the foundations. All of these phenomena can also be observed for the case of a half-space s.

3.1 Vertical excitation Figures 4, 5 and 6 show that the thickness of the layer

has a significant influence on the transmitting behavior

AMPLITUDE 1 - ] o . ~

0.32

_ _ l ~ l - Layer thickness H . . . . . . . . = 0.1 LR

- 0.2 LR

0 O0 l!l-~--_ "" , , . . . . . . . . . . . . .... --.

8o- ' ' ' 1- 6o ' ' ' 0

DISTANCE FROM SOURCE [ m ] 240

Fig. 4. Amplitude o f surface displacement. Layer thickness is smaller than the smallest critical thickness

AMPUTUDE [ - ] 0.8

0.4 i'T "T,~

,'~ ?,

Excitation :

22

H = 1.25 LR

0.0 . . . . . . . . . .

-20 0 20 DISTANCE FROM SOURCE [ m ]

Fig. 3. Influence o f excitation on displacement at the source and surrounding soil surface

AMPUTUDE [ - ]

3.2 I v v Layer thickness H - 1 2 - I - = . . . . . . . . = 0.6 LR

1.6

\ /Heft -space solution ~ 0.0 ~ _ . I ! | ! ! | ' ! ! ! !

0 80 160 240 DISTANCE FROM SOURCE [ m ]

Fig. 5. Amplitude o f surface displacement. Layer thickness is about the smallest critical thickness

Engineering Analysis with Boundary Elements, 1991, Vol. 8, No. 3 127

A M P L I T U D E [ - ]

0.32 -

0.00

Layer th ickness H . . . . . . . . = 2 .0 LR

= 4 .0 LR

Hal f -space solut ion

'~"'"" / ',.. .... . ..... ,

" , . j

! l i l I ! i I I I

0 80 160

D I S T A N C E F R O M S O U R C E [ m ]

240

Fig. 6. Amplitude of surface displacement. Layer thickness is larger than the smallest critical thickness

Ao,s /Ao ,h [ - ] 8

I o Ca lcu la ted va lue

o i o ~ . , , Half, space

0.0 116 '

LAYER T H I C K N E S S H [ LR ]

Fig. 7. Influence of layer thickness on the ratio of average amplitude of surface displacement

~olutlon | I ! i

3.2 4.8

of the soil. The results for the two thicknesses, H = 2.0 LR and H = 4.0 LR are compared with those of the half-space (Fig. 6). With increasing layer thickness H, the vertical source amplitude approaches the half-space solution. For both layers the vertical amplitude development differs from that of the half-space. This can be traced back to the more or less reflected soil waves at the soil-bedrock interface. Nevertheless, the size of the amplitude development oscillates about the half- space results.

If the soil depth becomes smaller, the soil waves may behave in a very different way, because a homogeneous soil layer resting on a rigid bedrock has its own eigenfrequencies. They depend on the layer thickness, the material of the soil and the direction of the vibration. Below the lowest eigenfrequency no geometrical damping exists. The soil behaves like a finite system. Only above this frequency can the soil waves propagate laterally 9']°. The eigenfrequencies for vertical vibration are

fe?, = ~ - . ( 2 n - 1 ) ; n = l , 2, 3 . . . (9) 4H

where c r is the propagation velocity of the compression wave and H is the depth of the soil layer.

For a layer thickness of H = 2.0 LR and H = 4.0 LR the excitation frequency f = 5 Hz is much higher than the lowest eigenfrequency fel,. of the soil layer itself. Therefore, the amplitudes of the surface waves have approximately the same size as in the case of a half- space (Fig. 6). If the excitation frequency is much lower than the lowest eigenfrequency of the soil layer, the soil vibrates always in a locally limited region (Fig. 4). For H = 0.5 LR and H = 0.6 LR the excitation frequency is slightly above and below the lowest eigenfrequency of the soil itself and the response shows a behavior similar to resonance. The development of surface vibration in Fig. 5 shows clearly that the amplitudes at the layer surface are much higher than those for the case of the half- space.

For H = 0.5 LR the excitation frequency is below the lowest eigenfrequency of the soil layer and no propagation of waves occur but only a limited part of the layer vibrates. In contrast to this, the amplitude of the surface vibration increases with the distance from the foundation and decreases at the end of the discretized region. This behavior may be interpreted as discretization error close to the resonant frequency. This effect still needs more investigation. For H = 0.6 LR the amplitude development has the similar behavior as for a half-space.

Figure 7 shows the influence of the layer thickness on the response of the soil layer. Ao.s and Ao,h are the average surface amplitudes along a certain length, for the soil layer and the half-space, respectively. The ratio Ao,s/Ao,h shows clearly the existence of critical thicknesses of a soil layer for which the wave amplitudes increases drastically. Below the first critical thickness no surface waves occur. With increasing layer thickness the amplitude of the surface waves becomes the value of the half-space except in case of resonance. The mth critical thickness can be interpreted as a layer thickness at which the frequency of the mth eigenmode of the layer is a resonant frequency. As the soil is assumed to have no material damping, the imaginary part of the foundation compliance F=, shown in Fig. 8, represents the

F zz -]

0.0 ~'"

-1.2 0.0

Fig. 8.

Calcu la ted va lue o Real part T

Imaginary part

Half-space solut ion

I I I ! ! m 3 1 I ! I

1.6 4.8

LAYER T H I C K N E S S H [ LR ]

Influence of layer thickness on compliance of the surface foundation

128 Engineering Analysis with Boundary Elements, 1991, Vol. 8, No. 3

amount of the radiation damping due to wave propagation. Below the first critical thickness the imaginary part of Fzz is zero, i.e. there is no wave propagation. The deviation from zero at resonance can be traced back to the rounding and discretizing errors. As a result of this, the soil system is still 'numerically damped', even if there is no material damping.

Above the smallest critical thickness, approximately the same amount of energy is carried away as in case of a half-space.

3.2 Horizontal excitation For the case of horizontal excitation the horizontal

vibration of the soil layer will not be influenced by the compression waves, but more by the shear waves. The eigenfrequencies are therefore determined by the propagation velocity of the shear wave and can be written as

C $

f'~.h=-;--7_'(2n--1); n = l , 2 , 3 . . . 41-I

(lO)

where c s is the velocity of the shear wave and H is the thickness of the soil.

Figures 9, 10 and 11 show that the vibration transmitting behavior of a soil layer for a horizontal excitation is similar to that for a vertical excitation. If the lowest eigenfrequency of the soil layer is greater than the excitation frequency, the vibrations cannot propagate (see for example the soil response at H = 0.1 LR and H = 0.2 LR in Fig. 9). The soil vibrates only in a limited region. If the lowest eigenfrequency is less than the excitation frequency, wave spreading occurs. The development of the amplitude of the surface vibrations becomes closer to the half-space solution with increasing soil thickness (Fig. 11). For the case where the lowest eigenfrequency lies about the excitation frequency, a resonant phenomenon occurs in the transmitting behavior of the soil layer. If the eigenfrequency is larger than the excitation frequency, although there is no wave propagation, the soil vibrates in a larger region with greater amplitude compared to the layer with much higher eigenfrequency (e.g. soil response for H = 0.268 LR in

AMPLITUDE [ - ] 0.64 -

0.32

i

0

Layer thickness H . . . . . . . . = 0.1 LR

- 0 . 2 L R

i | a ~ | s i i ! ! s

80 160

D I S T A N C E FROM S O U R C E [ m ]

240

Fig. 9. Amplitude of surface displacement. Layer thickness is smaller than the smallest critical thickness

AMPLITUDE [ - ] 3.2

~---~'~----'*.~* Layer thickness H . . . . . . . 0.268 LR

- 0 .300 LR

1.6 \ N

" , \ /Hal f -space solution

| i | ! s i ! ! i |

0 8() 160 240

DISTANCE FROM SOURCE [ m ]

Fig. 10. Amplitude of surface displacement. Layer thickness is about the smallest critical thickness

Fig. 10 in comparison with those at H = 0.1 LR or H = 0.2 LR in Fig. 9). Amplification phenomena also occur if wave propagation exists (e.g. compare soil response at H = 0 . 3 LR in Fig. 10 with those at H = 2 LR or H = 4 LR in Fig. 11).

For a soil layer with a thickness which is not equal to or close to the critical thickness the following conclusions can be drawn: i) a vertical excitation causes always larger vertical surface vibrations than horizontal vibrations. This means that the transmitting behavior of the soil is mainly determined by the eigenmodes and eigenfrequencies of the vertical soil vibration, ii) the horizontal surface vibrations due to a vertical unit excitation are equal to the vertical surface vibrations due to a horizontal unit excitation, in agreement with Betti's law. iii) although both excitations have the same magnitude, vertical surface vibrations due to vertical excitation are larger than horizontal surface vibrations due to horizontal excitation, iv) in case of a horizontal excitation, the horizontal and vertical surface vibrations have nearly the same size. That means: the transmitting behavior of the soil is influenced not only by the horizontal but also by the vertical eigenmodes.

Figure 12 shows the average surface amplitude of a soil layer with the thickness H, related to the average surface amplitude of a half-space. The amplification increases rapidly close to the resonant frequencies of the layer and similarly for horizontal or vertical modes (see for example the second critical thickness in Fig. 12). Below the first critical thickness, there is no wave propagation (see the compliance of the foundation F~ in Fig. 13. Up to the first critical layer thickness the imaginary part is equal zero).

3.3 Rocking excitation The vibration transmitting behavior of a soil layer

with a thickness below, about and above the smallest critical thickness respectively, are presented in Figs 14, 15 and 16. These results show, in comparison with Figs 4, 5 and 6, that the transmitting behavior of a soil layer are the same for rocking and vertical excitation. The reason is that for rocking excitation the transmitting behavior of the soil is mainly determined by the eigen-


AMPLITUDE [ - ] 0.64.

•4--•Pq3 t t> Layer l h i c k n e s s H

. . . . . . . . . 2.0 LR = 4 .0 LR

0.32 /"'"~ .,",.

\ ,.' \ i ',.. /"', ~ / ".. ~" Hal f -space s o l u u o n

0.00 , , , , , , , , , 0 8() 160 240

DISTANCE F R O M S O U R C E [ m ]


AMPLITUDE [ - ]

0.32

__|.__.~__~ | _ '~ * Layer th ickness H 6 5 . . . . . . . . = 0.1 LR

= 0.2 LR

/ H a l f - s p a c e so lu t ion ~ - ~ _ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

0.00 L i u i i i i u i u i u

0 80 160 240

D I S T A N C E F R O M S O U R C E [ rn ]

Fig. 14. Amplitude of surface displacement. Layer thickness is smaller than the smallest critical thickness

Ao,s/Ao,h [ - ] 64

32

o C a l c u l a t e d va lue

i ~ V i i

0 .0 ! i

1.6

l?=u,:., ::c. . . . . . .

i i i i i

3.2 4.8

L A Y E R T H I C K N E S S H [ LR ]


AMPLITUDE [ - ] 0.8

Layer th ickness H _

. . . . . . . - . . . . . . . = 0 .6 LR

0.4 n

-1 ~ - - _ : : _ _ ~ __:___.c" . . . . "___ " . . . . _" . . . . "_ . . . . "_ . . . . . . . .

o . o / , , / , ' , ' , :- , , , , ,

0 80 160 240

D I S T A N C E F R O M S O U R C E [ m ]

Fig. 15. Amplitude of surface displacement. Layer thickness is about the smallest critical

FXX I ' ] : i

1.2

0.0

-1.2 i

0.0 " 1.6

C a l c u l a t e d va lue o Rea l par t

J , I m a g i n a r y par t

3.'2 4.8

L A Y E R T H I C K N E S S H [ LR ]

Fig. 13. Influence of layer thickness on compliance of the surface foundation

AMPLITUDE [ - ] 0.64-

0.32

" . , e , , , . c k n e . . . . . . . . . . . 2.0 LR

- 4 .0 LR

1f-space so lu t ion

0 8() 160 240

D I S T A N C E F R O M s o U R C E [ m ]


1 3 0 Engineering Analysis with Boundary Elements, 1991, Vol. 8, No. 3

Ao,slAo,h [ - ] 4

o Calculated value

. . . . . . . .

0 C J g I I o I | I I I I I

0.0 1.6 3 9 4.8

LAYER THICKNESS H [ LR ]


Fmm [ - ] 1-

1.6 -J Calculated value o Real part

Imaginary part

0.8

41.8 ~ . . . . . . . . . ,

0.0 1.6 3~ 4.8

LAYER THICKNESS H [ LR ]

Fig. 18. Influence of layer thickness on compliance of the surface foundation

modes and eigenfrequencies of the vertical soil vibration. Only the source vibrations are not the same. They are caused by different ways of load application.

The average surface amplitudes normalized with those of a half-space solution show, in Fig. 17, a similar dependency. Below the first critical thickness propaga- ting surface waves do not exist. (This conclusion can also be drawn from the rotation compliance, shown in Fig. 18). With increasing thickness the average amplitude of the soil layer becomes close to the average amplitude of a half-space. The pronounced amplifica- tions occur only at the first critical thickness, whereas in the case of vertical excitation, pronounced amplifica- tions occur at each critical thicknesses. Therefore, it seems that a rocking excitation produces mainly the first mode of the vertical soil vibration.

4. CONCLUSIONS

The numerical results show that propagation of vibration can be successfully calculated with the boundary element method based on the full-space Green's function.

It should be noted that these results are obtained for the sensitive case of zero material damping. The results give confidence to apply the method to problems which are even less tractable when classical analytical methods are used. From the analysis the following practical conclusions can be drawn:

i) when installing a machine on a soil layer resting on top of a bedrock the critical thickness has to be considered. The critical thickness is given by

C H ~ , i t . = ~ ( 2 n - 1 ) ; n = l , 2, 3 . . .

where f is the operating frequency of the machine and c is the velocity of the compression wave for vertical and rocking excitation, or the velocity of the compression and shear wave for horizontal excitation.

ii) the best location for the foundation of a machine in view of the vibration propagation is that place where the depth of the soil layer is less than the first critical soil thickness. In this case, no wave propagation takes place, and, therefore, no disturbance in neighbouring buildings occurs.

ACKNOWLEDGEMENT

This research has been supported by the Deutsche Forschungsgemeinschaft through the Sonderforschung- sbereich 'Tragwerksdynamik' at Ruhr-University of Bochum.

REFERENCES 1 Beskos, D. E. Vibration isolation of foundations by trenches, In

Proceedings of the Greek German Seminar on Structural Dynamics and Earthquake Engineering, HSTAM, Athens, Greece, June 1989, 146-156

2 Hanpt, W. Surface-waves in non-homogeneous half-space, In Pro- ceedings of Dynamic Methods in Soil and Rock Mechanics, Kadsruhe, F. R. C~rmany, Vol. 1, 1978, 335-366

3 Richan, F. E., jr., Hall, J. R., jr. and Woods, R. D. Vibrations of Soils and Foundations, Prentice Hall, Inc., Englewood Cliffs, New Jersey, USA, 1970

4 Schmid, G., Wilims, G., Huh, Y., Gibhardt, M. SSI 2D/3D Soil structure inl~raction, SFB 151, Bericht Nr. 12, Sonder- forschungstcrcich Tragwerksdynamik, Ruhr-Universit~t Bochum, F.R. Germany, 1988

5 Cruse, T. A. and Rizzo, F. J. A direct formulation and numerical solution of the general transient elastodynamic problem I, Journal of Mathematical Analysis and Applications, 1968, 22 (1), 244- 259

6 Bronstein, I. N., Scmendjajew, K. A. Ta~chenbuch der Mathematik, BSB B.G. Teubner Verlagsgeselischafi, Leipzig, G.D.R., 1979

7 OIver, F. W. J. Bessel functions of integer order, In Handbook of Mathematical Functions, (Eds. Abramowitz, M. and Stegun, I. A.), Dover publications, Inc. N.Y. 1964, 358-433

8 Chouw, N., I.,¢, R., Schmid, G. Vibration transmitting heheviour of the soil, In Proceedings of the 1st European Conference on Structural Dynamics, University of Bochum, F.R. Germany, June 1990

9 Chouw, N., Le, R., Schmid, G. Ausbreitung yon ErschiRtgrungen in homogenem Boden, Bauingenieur, 1990, 399-406 (in Cm'man)

10 Chapel, F. Application de la methode des equations integrales a la dynamique des sols-structures sue pieux, these de docteur- ingenieur, Ecole Centrale des Arts et Manufactures, France, 1981


propagation of vibration in a soil layer over bedrock

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