Material Damping vs. Radiation Damping

Computers and Geotechnics 33 (2006) 86–92 www.elsevier.com/locate/compgeo

Material damping vs. radiation damping in soil–structure interaction analysis Ricardo Daniel Ambrosini

*

National University of Cuyo, CONICET, Los Franceses 1537, 5600 San Rafael, Mendoza, Argentina

Received 18 August 2005; received in revised form 10 March 2006; accepted 16 March 2006 Available online 2 May 2006

Abstract

The main objective objective of this paper paper is to contribut contributee to a quantificati quantification on of the effect of soil damping on the most important important design variables in the seismic response of building structures with prismatic rectangular foundations. A soil–structure interaction model was used for this purpose. The physical model of the structure is based on a general beam formulation. A lumped-parameter model was adopted to represent the soil and the interaction mechanisms. The seismic load was incorporated by ground acceleration records of many earthquakes. Finally, using the Correspondence Principle, the hysteretic and Voigt damping was incorporated into the soil model. Using the implemented models, a numerical study was carried out. The results obtained lead to an indirect assessment of the importance of energy energy dissipation dissipation due to soil material damping compared compared with the dissipation dissipation due to radiation radiation damping. Ó

2006 Elsevier Ltd. All rights reserved.

Soil–structure interaction; Radiation damping; Soil damping; damping; Seismic analysis; Lumped-parameter Lumped-parameter model; Structural dynamics dynamics Keywords: Soil–structure

1. Introduction

Seismi Seismicc analys analysis is of buildin buildings gs and other other engine engineerin eringg structures is often based on the assumption that the foundation corresponds to a rigid semi-space, which is subjected to a horizontal, unidirectional acceleration. Such a model constitutes an adequate representation of the physical situation in case of average size structures founded on sound rock. Under such conditions, it has been verified that the free field motion at the rock surface, i.e., the motion that would occur without the building, is barely influenced by its presence. The hypothesis looses its validity when the structure is founded on soil deposits, since the motion at the soil surface, without the building, may be significantly altered by the presence of the structure. The latter, in turn, has its dynamic characteristics, namely the vibration modes and frequencies modified by the flexibility of the supports. Thus, there is a flux of energy from the soil to the structure,

*

Tel.: +54 2627 420211; fax: +54 261 4380120. E-mail address: [email protected] [email protected]..

0266-352X/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.compgeo.2006.03.001

and then back from the structure into the soil, in a process that is known in seismic engineering as soil–structure interaction (SSI). Procedures to take into account soil–structure interaction in the seismic seismic analysi analysiss of buildin buildings gs were reviewed, reviewed, among others, by Dutta and Roy [1] and Antes and Spyrakos [2] [2].. Many papers following the so-called impedance functi functions ons approa approach ch may be mention mentioned, ed, signific significant ant ones ones being those by Wong and Luco [3] [3],, Wong et al. [4] and Crouse et al. [5] [5].. Numerous contributions found in the literature use the lumped-parameter models: Richart et al. [6] [6],, Veletsos Veletsos and Wei [7] and Wolf and Somaini [8] [8].. Many authors follow the direct method, e.g. Viladkar et al. [9] [9].. On the other hand, hand, the substr substructu ucture re approa approach ch and the BEM are used by Hayashi and Takahashi [10] [10].. Moreover, Yazdchi et al. [11] employed the coupled finite-element– bounda boundaryry-ele elemen mentt techni technique que (FE– (FE–BE). BE). The similar similarityitybased methods were developed by Wolf and Song [12] [12],, a simple and fast evaluation method of SSI effects of a partially tially embedd embedded ed struct structure ure was develop developed ed by Takewa Takewaki ki et al. [13] and the coupled finite–infinite element method was used by Khalili et al. [14] and Yerli et al. [15] [15].. The

87

R.D. Ambrosini / Computers and Geotechnics 33 (2006) 86–92

discrete element technique is used, among others, by Selvadurai and Sepehr [16] for the analysis of ice–structure interaction. In general, most of the papers of SSI are based on the assumption that the soil is a perfectly elastic medium that dissipates energy only by radiation of waves toward infinity. Then, the main objective of this paper is to contribute to a quantification of the effect of the material damping of soil on the most important design variables in seismic problems, such as total base shear, base overturning moment and top displacements. The problem of soil material damping has been analysed, among others, by Wolf and Somaini [8], Meek and Wolf [17], Sienkiewicz [18] and Lutes and Sarkani [19]. In the specialised literature, there is no agreement in relation to the importance of the material damping. For example, Richart et al. [6] say that the geometrical damping is the principal factor in the attenuation of R waves. This criterion is commonly adopted in engineering applications because the work is based on the assumption that the soil is a perfectly elastic medium and the material damping is neglected. On the other hand, Sienkiewicz [18] states that the energy dissipation by material damping may be of the same order of magnitude as the geometrical damping by wave radiation. Moreover, according to Wolf [20], in case of shallow layers of soil, the radiation damping can be drastically reduced and material damping is the primary source of energy dissipation in the medium. de Barros and Luco [21] studied the effects of material damping on the impedance functions and scattering coefficients for a semi-circular foundation on a uniform half-space. Significant effect has been observed in the stiffness coefficients at high frequencies and in the damping coefficients at low frequencies, but the effect of damping on the scattering coefficients is small. In this paper, the results obtained lead to an indirect assessment of the importance of energy dissipation due to soil material damping compared with the dissipation due to radiation damping, and it can be stated that the material damping of soil is an important parameter and must be included in the analysis of soil structure interaction, especially to determine the maximum top displacements. 2. Description of the models

At this point, the structure and soil models used in the analysis, will be briefly described. Basically, these models were presented by Ambrosini [22] and more details can be found in Ambrosini et al. [23]. 2.1. Structure model

The physical model of the structure, presented by Ambrosini et al. [24], is based on a general formulation of beams based on Vlasov’s theory of thin-walled beams, which was modified to include the effects of shear flexibility and rotatory inertia in the stress resultants, as well as var-

iable cross-sectional properties. In addition, a linear viscoelastic constitutive law was incorporated. A seismic loading, introduced by a ground acceleration record, constitutes the external load. The elements mentioned above lead to a system with three, fourth-order partial differential equations with three unknowns. Using the Fourier transform, an equivalent system in the frequency domain, with 12 first-order partial differential equations having 12 unknowns is formed. The scheme described above is known in the literature as ‘state variables approach’. Six geometric and six static unknowns are selected as components of the state vector v viz., the displacements n and g, the bending rotations /x and / y, the normal shear stress resultants Qx and Q y, the bending moments M y and M x, the torsional rotation, h and its spatial derivative, h 0 , the total torsional moment, M T and the bimoment, B

ð Þ ¼ fg; / ; Q ; M ; n; / ; Q ; M ; h; h0; M ; Bg

v z ; x

y

y

x

x

x

y

T

T

ð1Þ

The system is: ov

¼ Av þ q qð z ; xÞ ¼ f0; 0; Àq ; 0; 0; 0; Àq ; 0; 0; 0; Àm ; 0g o z

T

x

y

A

ð2Þ ð3Þ

in which A is the system matrix and q, the external load vector. In order to facilitate the numerical solution, the real and imaginary parts of the functions are separated, obtaining a final system of 24 first-order partial differential equations with 24 unknowns. 2.2. Soil model

On basis of the review of literature, and in view of the main objective of this work, a lumped-parameter model, based indirectly on homogeneous, isotropic and elastic halfspace theory, was adopted to represent the soil and the interaction mechanisms. The model, presented by Wolf and Somaini [8], has been formulated for the rectangular foundations embedded in the halfspace and it can represent the coupling between horizontal and flexural vibration modes. It has been formed by a set of masses, spring and dampers, combined adequately with the purpose of representing the ‘exact’ solution to a greater range of frequencies. The model is illustrated in Fig. 1, for horizontal and rocking or flexural vibration modes. It is important point out that, for an embedded foundation, a non-negligible dynamic-stiffness coefficient which couples the horizontal and rocking degrees of freedom referred to the centre 0 of the basemat (see Fig. 1) arises. To take this effect into account, the discrete model corresponding to the horizontal degree of freedom is connected eccentrically to point 0 and the vertical bar connecting the horizontal spring and dashpot to point 0 is rigid. The vertical and torsional degrees of freedom are uncoupled and consequently are independent. For these modes, the fundamental lumpedparameter model presented by Wolf and Somaini [8] could be used.

88


others, by Ganev et 0 0 0 0 nG ; gG ; / xG and / y G are:

al.

[26].

The

ESMs

(a) For horizontal components n0

G

nG

g0

¼g

G G

(  ¼ sin kd

0 6 kd 6 p=2 kd > p=2

kd

0:63

ð7Þ

in which nG and gG are the free-field ground motions, d is the depth of embedment (Fig. 2) and the coefficient, k is evaluated as k

¼ cx

ð8Þ

s

(b) For rotational components

0

/ xG a nG

Fig. 1. Soil model.

¼

0:4 d a 1

ð4aÞ

M 1

¼ bc K c

ð4bÞ

C 0

¼

C 1

¼

2 s 2

0

2 s

1

b K l0 cs b K l1 cs

ð4cÞ ð4dÞ

in which b is half of the width of the foundation and K represents the static-stiffness coefficient. Applying curve-fitting techniques over a range of frequency, the coefficients c0, c1, l0 and l1 have been determined for a specific component of the motion and it can be found in [8]. The shear wave velocity has been defined as: cs

¼

s ffiffi ffi G s

qs

ð5Þ

cos kd

0:405

0 6 d =a 6 1

Þ

0:405 0:05d =a 1 cos kd 0:4d =a

2

¼ bc K c

gG

8> ð À >> >< ð À ¼> ð À >> >: À

The dimensionless coefficients of masses c0, c1, and of dampers l0, l1, are functions of the physical properties of the soil (shear modulus G s, density qs and Poisson’s ratio ms) and of the foundation dimensions. These coefficients have been introduced as: M 0

0 / yG a

Þ

0 6 kd 6 p=2

ÞÂ

d =a > 1

0 6 d =a 6 1 kd > p=2

0:05d =a

d =a > 1

ð9Þ

in which a is the radius of the foundation. As the soil model, proposed by Wolf and Somaini [8], was developed for rectangular foundations of dimensions 2b · 2l , the equivalent radius proposed by Meek and Wolf [17] for rocking vibration has been used: a

¼ r ¼ eq

r ffiffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 4

8 bl 2 b 3 p

2

ð10Þ

ð þl Þ

Then, the components of load vector (3) are:

¼ Àq F ðn0 þ z /0 Þ q ¼ Àq F ðg0 þ z /0 Þ m ¼ Àq F a ðn0 þ z /0 Þ À a ðg0 þ z /0 Þ q x

T

y

T

A

T

G

xG

G

yxG

h

y

G

xG

x

G

y G

i

There are coefficients for each degree of freedom of the foundation and in Fig. 1, only those corresponding to horizontal and rocking vibrations have been presented. The dimensionless coefficients c0, c1, l0 and l1 are given by Wolf and Somaini [8] to all DOF. The dimensionless frequency, a0 is defined as: a0

¼ xcb s

ð6Þ d

2.3. Input motion

The effective seismic motion (ESM) was obtained, starting with the free-field motion, by an approximate analytical solution developed by Harada et al. [25] and used, among

Soil deposit Fig. 2. Flexible foundation model.

ð11aÞ ð11bÞ ð11cÞ

89


in which q denotes the mass density of the beam material, ax and a y are the coordinates of the shear centre and F T is the cross-sectional area.

Using Eq. (13), the equilibrium equations of the model presented in Fig. 1, in the ZX plane, are: (a) Horizontal vibration

2.4. Solution method

The system (2) may be integrated using standard numerical procedures, such as the fourth-order Runge–Kutta method, the predictor–corrector algorithm or other approaches. In order to solve the two-point boundary value problem encountered, the latter must be transformed into an initial value. To incorporate the interaction model, described in Section 2.2, in the analysis, the boundary conditions, which to a fixed end, were: n

¼ g ¼ / ¼ / ¼ h ¼ h0 ¼ 0 x

¼ K

Q x

hx

hx kx

M y

p À

! G ð1 þ 2il Þ

ð13Þ

s

For the hysteretic damping, G s and ls are frequencyindependent. The damping ratio is related to the angle of mobilised internal friction for the soil, d, by [17]:

¼ 0:5tan d

ð14Þ

Since the maximum upper limit of d is the angle of repose of sand slopes, around 35 °, tan d can never exceed about 0.7, which implies that the upper limit of the damping ratio is around 0.35. Neglecting the terms, l2s in relation to 1, the dimensionless frequency coefficient becomes: l

a0

¼

a0

p ffiffi ffiþffi ffi ffi ffi ffi ffi ffi 1

2 ls i

a0 1

ð À l iÞ s

l2

a0

2 0

ffi a ð1 À 2l iÞ s

ð15Þ

Àl a c

ia0 c0hx

s 0 0hx

þa

0

! !

n

þ 2al

ia0 c0hx F x

ls a0 c0hx F x

  ¼ À þ þ 8>< þ > À À À : 0 þ1 2 ÂB @ À þ CA À þ 64 0 B@ À þ þ À K hx f kx 1

l1ry a20

a20 l0ry

l21ry a20

1

s

/ x

0

ð16Þ

2ls

2ls

a0

a0

1

þ

1

c21ry

c1ry 1

þ

l21ry a20 c21ry

(c) Torsional vibration

8>< ¼ > À :2 þ þ 64 þ

l1t a20

K t 1

M T

ia 0

1

l21t a20 c21t

l1t l1t a20 c1t 1 l21t a20 c21t

Àl a

s 0

l1t l1t a20 c1t 1 l21t a20

þ

þ 2al À 2al s

0

1

þ

c21ry

l21ry a20 c21ry

139>= CA75>;

1

1

1

c21t

ð17Þ

/ x

2

1

l21t a20

l21ry a20

þ

þ

0 B@ À 0 þ B@ À

l1t a20

s

0

c21t

n

l1ry l1ry a20

1

1

l21ry a20

a0

!

c1ry 1

c21ry

l1ry a20

2ls

l1ry l1ry a20

ia0

ls a0 c0ry

l21ry a20

1

ls a0

c21ry

2

c0ry

ia0 c0hx F x

ls a0 c0hx F x

1

In this paper, the second important source of energy dissipation, soil material damping, has been introduced by applying the correspondence principle which states that the damped solution can be obtained from the elastic one by replacing the elastic constants with the corresponding complex ones, multiplying by 1 + 2ils, in which ls is the soil damping ratio and i = 1. The essential concept is the direct mathematical correspondence between the governing equations for the Fourier-transformed linear viscoelastic problem and the original small strain elasticity problem with the same boundary conditions. A detailed treatise on the correspondence principle is given by Christensen [27]. In the following, only the development and results obtained for hysteretic damping are shown, although similar procedure could be used for Voigt or viscous damping. Then, using the correspondence principle, the shear modulus of elasticity of the soil is replaced by:

ls

1

K ry 1

3. Soil damping

s

À

2 ls

 þ  þ

In which F x = f cx/ f kx (b) Rocking vibration

It must be replaced by the motion equations of the soil model, except the condition, h 0 = 0.

G s

a20 l0hx

1

þ K f

ð12Þ

y

À

l21t a20 c21t

1 CA

1

þ

l21t a20 c21t

139>= CA75>;

h

ð18Þ

The equations in the ZY plane are similar. Eqs. (16)– (18) can be replaced by the following equations, which are similar to the undamped equations: Q x

2 l 0 0hx

À ¼ À þ 0 þ B @ 2 ¼ 64 À

l

M y

hx

l K hx f kx

ia0

M T

K t 1

i

a0 c0hx f cxl

1

1

þ

l21ry a20

K hx f kx

ia0 cl0hx f cxl / x

l1ry a20

K ry 1

n

1

l

c0ry

Á

l21ry a20 c21ry

ð19aÞ

2 l 0 0ry

Àa l

ð19bÞ

/ x

c21ry

l1t a20

þ

n

0 0hx

l1ry l1ry a20 c1ry

l

Áþ À þ 2 Á þ 64 À þ 13 þ C A75 0 À þ B @ þ

¼ K 1 À a l þ ia c

l21t a20 c21t

l

a20 l0t

ia0

l1t l1t a20 c1t 1 l21t a20 c21t

13 CA75

l 0t

þc

h

ð19cÞ

90

l

c0h


Comparing Eqs. (16)–(18) and (19) it leads to: 2ls

¼c þ a ¼ l þ c al 0h

l

l0h

¼c

ll0t l

f k

f cl

¼l

2 ls

s

0

¼ al

s

0

l1r a20

1

a0

0r

¼ 2al

0

2 0 13 B@ À CA75 þ 64 À þ þ 2 0 13 B@ À CA75 þ 64 þ þ þ 2 0 13 64 À B@ À CA75 þ þ 2 0 13 64 B@ À CA75 þ þ

0r

l l0r

l

0h s

0h

l c0r

c0t

0

1

l1t c1t 1

l21t a20 c21t

2

1

l21r a20 c21r

l21t a20

1

c21t

l1t a20

c21r

1

1

l21t a20

l21r a20

1

c21r

l1t a20

1

1

l1r l1r a20 c1r 1 l21r a20

ls c a0 0r

1

1

1

l21r a20 c21r

c21t

2

1

1

l21t a20 c21t

¼ f À a c f l ¼ a f a cc þþ 22 f l l k

0 0h c s

0 c 0h

0 0h

k s

s

ð20aÞ ð20bÞ ð20cÞ

tion, Ambrosini [22]) that was developed incorporating the models described above. With the purpose of representing many real situations, a set of three structures, defined in Table 1 with the average plan shown in Fig. 3, and three ground acceleration records, defined in Table 2, will be used. The remaining input data used in the analysis are presented in Table 3. 15.24

ð20dÞ

6.096 6.096

1.524

a) CENTRAL CORE BUILDING

ð20f Þ ð20gÞ ð20hÞ 30.225

Wall width 0.40 m

4.55

8.20

9.40

8.20

4.55

34.90

b) TORRES DEL MIRAMAR - FIRST FLOOR 6.70

6.70

14.70

6.70

6.70

6.00 6.00

14.00

6.00

7.00 0.20

4. Numerical analysis and results

c) CORE AND WALLS BUILDING

The numerical analysis was performed using the program DAYSSI (dynamic analysis of soil-structure interac-

Fig. 3. Plan view of the structures.

Table 2 Ground acceleration records used

Table 1 Structures used a

b 1

Building

Description

H (m)

T (s)

Reference

B1 B2 B3

Central core building Torres del Miramar Core and walls building

57.2 55.9 48.0

0.60 1.03 0.87

[29] [30] [31]

a

0.305

ð20eÞ

In the lumped-parameter models, these modified coefficients are mechanically equivalent to augment each original spring by a dashpot and each original dashpot by a mass attached to it, in parallel, by pulleys [17]. The hysteretic damping presents the important advantage of being independent of the excitation frequency and dependent of the strain magnitude, which is realistic because it matches with the experimental results. However, it presents a theoretical problem that, as Crandall [28] demonstrated, the response begins before the excitation. This so-called non-causal behaviour is obviously impossible and confirms that the augmenting dashpots and pulley masses, inversely proportional to frequency (see Eq. (20)), do not exist in reality. In spite of this, the results are correct because, as Meek and Wolf [17] demonstrated, the spring and damping coefficients of the non-causal linear-hysteretic system turn out to match those of a causal non-linear frictional system, averaged over one cycle of response.

b

4.572

Total height of the building. Fundamental period determined by DAYSSI.

a

Earthquake

Record

Dt

A1 A2

Caucete 1977 Chile 1985

0.04 0.017

10 35

A3

Loma Prieta 1989

San Juan Vinã del Mar S20W Santa Cruz

0.02

20

a

Used in the analysis.

(s)

Durationa (s)

Ground acceleration

91


Table 3 Input data used in the analysis Building

B1 B2 B3

Structure properties

Foundation properties

Soil properties

Cross-sectional area (m2)

Shear centre coordinates (m)

Mass density (kg/m3)

Dimensions (m)

Depth of embedment (m)

Shear modulus (MPa)

Mass density (kg/m3)

Poisson modulus

6.5 24.0 18.12

À5.99, 0.0 0.0, 0.0 0.0, À3.0

2400 2400 2400

15.24 · 15.24 22.96 · 22.96 14.0 · 41.7

3.0 6.0 3.0

35 60 50

1600 1600 1600

0.3 0.3 0.3

Table 4 Results obtained Alternative B1A1 G s = 35 B2A2 G s = 60 B3A3 Gs = 50

Fixed base

SSI ls = 0

SSI ls = 0.25

Qmb (MN)

M mb (MN m)

gmt (cm)

Qmb (MN)

M mb (MN m)

gmt (cm)

Qmb (MN)

M mt (MN m)

gmt (cm)

2.495 35.390 38.209

34.5 1261.9 800.1

2.6 22.8 9.7

1.827 21.120 24.036

33.1 954.5 443.1

4.3 23.2 11.2

1.732 17.406 22.398

24.0 848.8 296.9

2.6 19.9 7.1

In connection with the soil damping ratio, besides the limiting value given in Section 3, there are important differences in the literature. For example, Sienkiewicz [18], uses 0.05 and Meek and Wolf [17], based on laboratory experiments presented by Gazetas [32] use 0.25 at a strain of about 0.01. A value of 0.25 has been used in this paper, which is in agreement with the low frequencies values of the linear-hysteretic frequency-dependent model presented by Assimaki and Kausel [33]. The results obtained are presented in Table 4, summarised in the form of the most important design variables in seismic problems, such as maximum total base shear, Qmb, maximum base overturning moment, M mb and maximum top displacements, gmt. In Fig. 4, the time-history of the top displacements for the alternative B3A3 G s = 50 MN/m2, has been presented for the cases: ls = 0 (only geometrical damping) and ls = 0.25 (material and geometrical damping).

5. Discussion and conclusions

Based on the results obtained above, the following observations and conclusions can be stated: The material damping of soil is an important parameter and must be included in the analysis of soil structure interaction, especially to determine the maximum top displacement, in which the differences are significant. The decrease of the base shear force and base overturning moment due to the flexibility of the foundation corresponds, on an average, 70% to radiation damping and 30% to material damping. However, there are special cases in which the greater reduction is due to hysteretic damping. For example, in the alternative B1A1 (Table 4) for the base overturning moment, the dissipation due to geometrical damping is only about 14%. It is well known that for stiff structures having height/ width ratios greater than one, the rocking motion is the

0.15 us=0.25

us=0

0.1

) m ( . p s i D

0.05 0

-0.05 -0.1 -0.15 0

5

10

15

20

25

Time (sec) Fig. 4. Top displacement. Alternative B3A3.

30

35

40

92


predominant interaction effect. In this case, especially for low frequency rocking, very little energy is dissipated by the radiation of waves, so the relative importance of material damping is much more than in translation. As it can be seen in Table 4, this effect is fulfilled because in the base overturning moment, the reduction is more important due to material damping. As it can be seen in Fig. 4, the importance of the soil material damping in the response of displacements is very significant in bringing down the peak of the displacements as well as in the fast attenuation of the free vibrations after the end of the earthquake (20 s in Fig. 4). The change in natural frequencies due to the effect of the soil material damping is negligible, for which the change in the response, presented in Table 4, is due only to the effect studied in this study. Acknowledgements

The author wishes to thank the collaboration of Profs. Jorge Riera and Rodolfo Danesi and the help received from Ms. Amelia Campos in the English revision. Moreover, the financial support of the CONICET and the Universidad Nacional de Cuyo is gratefully acknowledged. References [1] Dutta SC, Roy R. A critical review on idealization and modeling for interaction among soil–foundation–structure system. Comput Struct 2002;80:1579–94. [2] Antes H, Spyrakos CC. Soil–structure interaction. In: Beskos D, Anagnostopoulos S, editors. Computer analysis and design of earthquake resistant structures. A handbook. Southampton, UK, Boston, USA: Computational Mechanics Publications; 1996. p. 271–332. [3] Wong HL, Luco JE. Tables of impedance functions for square foundations on layered Media. Soil Dynam Earthquake Eng 1985;4:64–81. [4] Wong HL, Trifunac MD, Luco JE. A comparison of soil–structure interaction calculations with results of full-scale forced vibration test. Soil Dynam Earthquake Eng 1988;7(1):22–31. [5] Crouse CB, Hushmand B, Luco JE, Wong HL. Foundation impedance functions: theory versus experiment. J Geotech Eng-ASCE 1990;116(3):432–49. [6] Richart F, Hall J, Woods R. Vibrations of soils and foundations. Englewood Cliffs (NJ): Prentice-Hall; 1970. [7] Veletsos AS, Wei YT. Lateral and rocking vibration of footings. J Soil Mech Found Div-ASCE 1971:1227–48. [8] Wolf JP, Somaini D. Approximate dynamic model of embedded foundation in time domain. Earthquake Eng Struct Dynam 1986;14:683–703. [9] Viladkar MN, Godbole PN, Noorzaei J. Space frame–raft–soil interaction including effect of slab stiffness. ComputStruct 1992;43:93–106. [10] Hayashi Y, Takahashi I. An efficient time-domain soil–structure interaction analysis based on the dynamic stiffness of an unbounded soil. Earthquake Eng Struct Dynam 1992;21:787–98. [11] Yazdchi M, Khalili N, Valliappan S. Dynamic soil–structure interaction analysis via coupled finite-element–boundary-element method. Soil Dynam Earthquake Eng 1999;18:499–517.

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Material Damping vs. Radiation Damping

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