Soil-structure interaction analysis of NPP containments: substructure and frequency domain methods

Nuclear Engineering and Design 174 (1997) 165 – 176

Soil-structure interaction analysis of NPP containments: substructure and frequency domain methods F. Venancio-Filho a,*, F.C.P. de Barros b, M.C.F. Almeida a, W.G. Ferreira a a b

COPPE/UFRJ, Caixa Postal 68506, 21945 -970 Rio de Janeiro, Brazil IME/CNEN, Pc. General Tibu´rcio, 80, 22290 -270 Rio de Janeiro, Brazil Received 13 November 1996; accepted 2 April 1997

Abstract Substructure and frequency domain methods for soil-structure interaction are addressed in this paper. After a brief description of mathematical models for the soil and of excitation, the equations for dynamic soil-structure interaction are developed for a rigid surface foundation and for an embedded foundation. The equations for the frequency domain analysis of MDOF systems are provided. An example of soil-structure interaction analysis with frequency-dependent soil properties is given and examples of identification of foundation impedance functions and soil properties are presented. © 1997 Elsevier Science S.A.

1. Introduction Dynamic soil-structure interaction (SSI) is one of the main issues in the dynamic analysis of NPP containment facilities. The material properties of the containment and its appendages are wellknown and mathematical models for their analysis have been well established. Conversely, there are difficulties in properly establishing dynamic soil properties such as shear modulus, damping, non-linear behavior and frequency dependency as well as in establishing adequate mathematical models for the soil. Broadly speaking there are two classes of methods for SSI analysis: direct and substructure methods. The bulk of the problem in both meth* Corresponding author. Tel.: + 55 21 5608993; fax: + 55 21 2809545.

ods is the correct modelling of the unbounded soil medium. In the first class, this medium is considered through the so-called transmitting boundaries whose properties are normally frequency-independent, in the latter, that medium is treated with dynamic stiffness or boundary impedances that are, basically, frequency-dependent. Therefore, the direct and the substructure classes of methods are, respectively, time and frequency-domain ones, although this categorization is not very strict. One early attempt to tackle SSI problems was due to Reissner (1936) following a previous work of Lamb (1904). Reissner obtained the elastodynamic response of a rigid circular footing supported on a semi-infinite, homogeneous, isotropic, elastic body (elastic half-space) submitted to a single harmonic load. Richart Jr. et al. (1970) elaborated Reissner’s solution in an adequate way

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for engineering applications. Veletsos and Wei (1971) and Luco and Westman (1971) solved exactly the mixed boundary value problem of a rigid circular disk resting on an elastic half-space for vertical, sliding, rocking, and torsional harmonic motions. Subsequently Apsel and Luco (1983) obtained dynamic Green’s functions for a layered half-space. Since 1985 Wolf (1985, 1988, 1994) has published three thorough books on dynamic soilstructure interaction. Very recently, Clough and Penzien (1994) presented a very good treatment of SSI analysis by the substructure method. In this paper a description of the substructure and frequency domain methods of SSI analysis is given along with a brief introduction to mathematical models for the soil and for the sources of excitation. The frequency-domain analysis of multi-degree of freedom (MDOF) systems is presented and an example of SSI analysis is given. Some comparisons of experimental and calculated results of SSI interaction analysis are finally provided.

foundation and an adjacent portion of soil, and the semi-infinite soil medium constitutes the second substructure. The analysis is then performed by considering compatibility of displacements and equilibrium of interaction forces in the interface. The transmitting boundaries of the direct method are essentially frequency-independent whereas the dynamic stiffnesses or boundary impedances of the substructure method are frequency-dependent.

3. Sources of excitation There are two types of loading that can be considered in SSI analysis. The first and simpler one occurs when the source of excitation is outside the mathematical model such as, for example, wind, above ground explosions, aircraft crash. The second and most elaborated one is due to a source of excitation inside the mathematical model such as seismic motions and underground explosions.

2. Mathematical models for the soil The model of the soil in the direct method class is composed by a mesh of finite elements which encompasses an adequate size of the soil medium and a set of transmitting boundaries attached to the boundary nodes, Fig. 1(a). The function of these transmitting boundaries is to permit the development of radiation damping through the semi-infinite medium. It must be pointed out that in direct or time-domain methods it is not possible to rigorously satisfy the radiation condition at infinity. In the substructure method class the soil is modelled in two parts: one adjacent to the foundation constituted by a mesh of finite elements in the boundary nodes of which the dynamic stiffnesses are attached, Fig. 1(b). Although both mathematical models are quite similar there are some differences. In the direct method the structure and the soil are analyzed as a whole system. In the substructure method one substructure is formed by the superstructure, the

Fig. 1. Mathematical models for SSI analysis: (a) direct method; (b) substructure method.

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The input motion corresponding to a seismic excitation is defined through the motion of a control point which is usually located in the outcropping rock. Given the control point motion, the motion of all points in the region of interest, adjacent to the foundation, must be obtained. This motion is the so-called free-field motion. The following factors must be taken into account in the process of obtaining the free-field motion: definition of the control point motion which depends mainly on the seismicity of the considered region; type of wave motion; direction of wave propagation and horizontal soil layers definition. For embedded foundations the free-field motion must be modified in order to take into account wave scattering effects due to the surface cavity. This modification leads to the effective input motion. For surface foundations and vertically propagating waves, the effective input motion corresponds to the free-field. 4. The substructure method (Clough and Penzien, 1994)

4.1. Rigid surface foundation The rigid surface foundation resting on a homogeneous elastic, isotropic half-space is addressed first. Consider the SSI system of Fig. 2. The superstructure consists of a rigid beam with mass and mass moment of inertia m and J, respectively, and two columns with height h and stiffness k. The rigid foundation has mass and mass moment of inertia m0, and J0, respectively, and rests on the elastic half-space. The structure has viscous damping c. The present SSI system is then composed of substructure 1 (superstructure and foundation) and substructure 2 (elastic halfspace). The total response of mass m is defined as n t = ng +n Ig + n+ hu I (1) where ng is the free-field motion, n Ig and u I are, respectively, the sliding and rocking motions due to interaction and n is the relative displacement of the superstructure. The following dynamic equilibrium equations are formulated for substructure 1:

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Fig. 2. Rigid surface foundation.

Equations for substructure 1 (time domain) Equilibrium for mass m mn¨ + 2mjvn; + kn + mhu8 I + mn¨ Ig + mn¨ g = 0 j=

c ; 2mv

v=

'

k m

(2) (3a,b)

Equilibrium for masses m and m0, mn¨ + mhu8 I + (m+ m0)n¨ Ig + (m+ m0)n¨ g = V0(t) (4) Moment equilibrium mhn¨ + (mh 2 + J+J0)u8 I + mhn¨ Ig + mhn¨ g = M0(t) (5) V0(t) and M0(t), in Eqs. (4) and (5), are the interaction forces (shear and moment, respectively). Eqs. 2–5 are now considered in the frequency domain through Fourier transformation: Equations for substructure 1 (frequency domain) (− v ¯ 2m+ 2iv ¯ vjm +k)V− v ¯ 2mhUI − v ¯ 2mV Ig + mV8 g = 0

(6)

−v ¯ 2mV−v ¯ 2mhUI − v ¯ 2(m+m0)V Ig + (m+ m0)V8 g = V( 0

(7)

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−v ¯ 2mhV− v ¯ 2(mh 2 +J +J0)UI −v ¯ 2mhV Ig + mhV8 g =M( 0

Table 1 Frequency-independent impedances (Richart Jr. et al., 1970)

(8)

In these equations v ¯ is a single harmonic excitation frequency, capital letters are used for the corresponding response terms of Eq. (1), when defined in the frequency domain, and V( 0 and M( 0 are the interaction forces defined in this domain. Equations for substructure 2 are mandatorily formulated in the frequency domain. They are as follows: Equations for substructure 2 (frequency domain)



G n Ign I g G u In I

g

G n Iu I g G u Iu I

n  



V Ig −V( 0 = I U −M( 0

(9)

In this equation the G’s are the frequency-dependent impedances obtained originally by Veletsos and Wei (1971) and Luco and Westman (1971). With the notation of the former authors these impedances are expressed formally by the complex term. G(ia0)= G R(a 0) + iG I(a0) [stiffness] [damping] in which a0 =

(10)

v ¯r Vs



(k 11 +ia0c12)kx (k 21 +ia0c21)kxr

Stiffness

Horizontal translation

kx =

Rocking

Damping

32(1−n)Gr (7−8n)

cx =

8Gr 3 3(1−n)

cf =

kf =

0,576kxr Vs 0,3kfr Vs(1−Bf )

Bf =(3(1−n)Ib/8rr 5); Ib-base mass moment of inertia.

from the curves provided by Veletsos and Wei (1971), are displayed in Fig. 3. Eqs. (6)–(9) are now conveniently gathered leading to Equations for the SSI system (frequency domain)

ÆG 11 G 12 G 13ÇÁ V  ÃG 21 G 22 G 23ÃÃV Igà ÈG 31 G 32 G 33ÉÄUIÅ Á −m  = à − (m+ m 0) ÃV8 g; Ä − mh Å

(13)

(11) G11 = − v ¯ 2m+ 2iv ¯ vjm +k;

is the dimensionless frequency, r is the radius of the circular rigid foundation, and Vs is the soil shear wave propagation velocity. The real term in Eq. (10) corresponds to the half-space stiffness and the imaginary one to the radiation or geometric damping. The explicit form of the half-space impedance matrix of Eq. (9) is Sg =

Motion

(k 21 +ia0c21)kxr (k 22 +ia0c22)kf

n

G12 = G21 = − v ¯ 2m;

(14a)

G13 = G31 = − v ¯ 2mh; (14b,c)

(12)

where k11, k21, k22, c11, c21 and c22 are dimensionless stiffness and damping coefficients and kx, cx, kf and cf are the frequency-dependent impedances given in Table 1 (x referring to sliding and f to rocking). In most cases the interaction terms [1, 2] and [2, 1] in Eq. (12) can be disregarded. Curves for k11, k22, c11 and c22, developed

Fig. 3. Frequency-dependent impedance coefficients.

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into account the embedment, the soil layering and, even, non-linearities. Substructure 2 (the far field) consists of the elastic half-space whose behavior is considered through the boundary or far field impedances. a degrees of freedom (DOF), Fig. 4, refer to the superstructure; b DOF’s refer to soil–structure interface but excluding the a’s; d DOF’s refer to the interface of substructures 1 and 2, c DOF’s, to the near-field but excluding the b’s and the d’s. The total response definition of this SSI system is defined by the following equation:

    nt n0 s n. s nd = + + n td n0 sd n. sd n dd

(15)

where the vectors in the first lines contain the a, b, and c DOF’s. The first two vectors in the RHS of Eq. (15) are pseudo-static responses and the third one is the response due to inertial effects in the superstructure. The first vector is defined in the time and in the frequency domain, respectively, as

 n˜ s n˜ sd

Fig. 4. Embedded foundation.

G22 = − v ¯ 2(m + m0) +Gn Ign I ;

(14d)

G23 = G32 = −v ¯ 2mh + Gn I u I

(14e)

¯ 2(mh 2 +J + J0) + Gu Iu I. G33 = − v

(14f)

g

g

Eq. (13) is then solved in order to obtain the frequency-domain responses V, V Ig and UI. These responses are finally inverse Fourier transformed, leading to the time domain corresponding ones n, n Ig, and u I. Therefore, going back to Eq. (1), the total response is obtained.

4.2. Embedded foundation The SSI system with an embedded foundation of Fig. 4 is formed by substructures 1 and 2. Substructure 1 consists of the superstructure, the foundation and a portion of soil adjacent to the foundation (near field) which can be modelled by a mesh of finite elements. This modelling can take

Á − k aa kabnbg n bg à à =à à n cg à à ndg Ä Å −1

[time domain]

(16a)

and

  V0 s V0 sd

Á − k aa kabVbg V bg à à =à à V cg à à Vdg Ä Å −1

[frequency domain]

(16b) subscript g, in these equations, referring to the modified ground motion and k being the system stiffness matrix. The second vector in the RHS of Eq. (15) is defined, in the frequency domain, as

  

n 

V. s k kg = T s V. d ¯ )] k g [k dd + Gdd (iv

where

− P0 s 0

(17)

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This superposition leads to the total response in the frequency-domain as

0 Á  à à 1 −1 8 P0 s = à − 2 [k (1) −k k k ]V bb ba aa ab bgà v ¯ Ã Ã Ä Å 0

(18)

¯ ), Eq. (17), is the matrix formed with the Gdd (iv boundary or far-field impedances. Finally, the third vector in the RHS of Eq. (15) is obtained, in the frequency domain, from the following equation:

   −v ¯2

+

n  nn

m m Tg

k k Tg

mg c cg +iv ¯ m dd c Tg c dd

kg [k dd +Gdd (iv ¯ )]

n 

K(iv ¯ )=



V d(iv ¯) d V d (iv ¯)

(19)

n   n  (1)

m 0

0 0

m + m Tg ×

f0 0

s

mg m dd

f. s ; 0

k k Tg

n

kg [k dd +Gdd (iv ¯ )]

−1

(20)

Á − r f0 s = à − IÃ; Ä 0 Å

(21)

Á Â 0 (1) T f. = Ã[k +k abr]Ã; Ä Å 0

(22)

s

Vt V0 s V. s Vd = + + t s s Vd V0 d V. d V dd

The total response in the time-domain, Eq. (15), is then obtained by the inverse Fourier transformation of Eq. (24).

4.3. Boundary or far-field impedances (Gupta et al., 1982)

SR ¬ hR + izR

(25a)

Sf ¬ hf + izf

(25b)

Su ¬ hu + izu

(25c)

where R, f and u refers, respectively, to the radial, tangential and circumferential directions. The foregoing impedances are frequency dependent but are constant in the circumferential direction and are shown in Fig. 6 (n=1/3). They are discretized through tributary areas and the matrix Gdd(iv ¯ ) from Eq. (17) is therefore obtained.

and 1 r= − k − aa kab

(24)

In order to introduce the boundary or far-field impedances the substructures 1 and 2 interface can be modelled as a hemispherical one, Fig. 5, and the continuous impedances, per unit area, can be, quite approximately, given by

= K( (iv ¯ )V8 bg(iv ¯) in which

       

(23)

m and c in Eqs. (19) and (20) are the system mass and stiffness matrices; the superscript (1) in m and k indicates that these matrices contain only coefficients from the superstructure. Once Eqs. (17) and (19) are solved, the total response, in the frequency domain, is obtained by superposition of the solutions of these equations with the frequency domain vector from Eq. (16b).

Fig. 5. Substructure interface.

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Fig. 7. Example of SSI interaction analysis: (a) shear building; (b) load.

trix. It must be pointed out that k and c can be frequency-dependent.

6. Example Fig. 6. 3D frequency-dependent impedances.

5. Frequency domain analysis of MDOF systems (Venancio-Filho and Claret, 1995) The response of a MDOF system is analyzed, in the frequency-domain, through the following equation: ni (tn)=

1 N

 

N−1

% ei2p(mn/N)

m=0 N−1

×



J

% Hij (v ¯ m)

j=1

n

% pj (tn) e − i2p(mn/N)

(26)

The example is a three-story shear building with a rigid circular foundation supported on an elastic half-space. The system is depicted in Fig. 7(a). It has 5 DOF’s: n1, n2 and n3, corresponding to the superstructure, and n4 and n5 to the foundation. The masses and stiffness of the superstructure and the inertia properties of the foundation are given in Table 2. The excitation is the impulsive load shown in Fig. 7(b). The analysis was performed for three cases: 1. Rock foundation (rigid soil) (NO SSI) 2. SSI with the frequency-independent impedances from Table 1 (SSI-FI)

n=0

in which ni (tn) and pj (tn) are, respectively, the ith DOF response and the nodal load associated to the jth DOF, both in the discrete time tn. N is the number of points in the discrete Fourier transforms and ¯ c+ kH) +k] − 1 H(v ¯ )= [− v ¯ 2m +i(v

(27)

is the complex frequency response matrix at frequency v ¯ . m and k are, respectively, the system mass and stiffness matrices, c is the viscous damping matrix and kH is the hysteretic damping ma-

Table 2 Properties of the SSI system of the example Floor

Mass (t)

Stiffness (kN m)

1 2 3

1800 2700 3600 Mass:

105 000 210 000 31 500 3.6×104 t

Foundation Mass moment of inertia: 2.48×105 t m2

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7. Identification of structural and soil properties from forced vibration tests (Luco and de Barros, 1994) The effects of dynamic soil-structure interaction have been a subject of considerable research in the last 25 years. On the one hand, a number of complex techniques, computer codes, and simple engineering approximations have been developed to solve the problem. There has been, on the other hand, very little experimental effort to evaluate the validities, conservatisms, and sensitivities of the SSI analysis methodologies. The large-scale seismic test research program at Hualien, Taiwan, as well as its predecessor, the Lotung Taiwan soil-structure interaction experiment, are focused

Fig. 8. Time history of DOF 2: (a) G =1500 kPa; (b) G = 15 000 kPa.

3. SSI with the frequency-dependent impedances from Fig. 3 (SSI-FD) Two types of soils were considered: a soft clay with shear modulus G =1500 kPa and a medium compact sand with G = 15 000 kPa. Time histories for n2 and n5 are displayed in Figs. 8 and 9, respectively. These results and the ones for the other DOF’s show that there are some differences between the results of NO SSI and SSI. On the other hand the results for SSI-FI and SSI-FD are almost the same but for the n5 response where there are considerable differences in behavior. This is, quite certain, due to the sharp variation of c22 in the low frequency-range, Fig. 3.

Fig. 9. Time history of DOF 5: (a) G =1500 kPa; (b) G= 15 000 kPa.

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on the validation of mathematical models and computer codes used in the analysis of SSI effects for nuclear power plants (Tang and Fong, 1993). The 1/4-scale Hualien containment model is a cylinder-shaped reinforced concrete shell structure with a flat roof slab and a flat basemat. The structure has a total height of 16.13 m, a basal diameter of 10.82 m. The cylindrical shell has an external diameter of 10.52 m and uniform wall thickness of 0.30 m. The cylindrical roof slab has a diameter of 13.28 m and a thickness of 1.50 m and the basemat is 3.00 m thick. The masses of the foundation, shell, and top slab are estimated as 695×103, 264×103 and 505×103 kg, respectively. The soil at the Hualien consists of sands and gravels that have been the subject of numerous geotechnical and geophysical studies. The participants in the Hualien project were asked to submit blind predictions for the response of the containment model for forced vibration test, for two prescribed soil models (Models A and B). It was considered a third model (Model C) based on estimates of the effects of changes of confining stress on shear-wave velocity. All the soil models are characterized by the same values of Poisson’s ratio n= 0.47, density r =2420 kg m − 3, damping ratios for S-waves jS =0.02 and P-waves jP =0.002. The resulting estimates of shear-wave velocity in the first 7 m immediately beneath the foundation are scattered over the range from Vs =200 m s − 1 to Vs =475 m s − 1 and have a weighted average value of Vs =317 m s − 1 (Model A). The shear-wave velocities of the soil layers beyond the first 7 m beneath the foundation are considered identical for all the three soil models A, B, and C. It means, three additional uniform soil layers with thickness of 30 m (Vs =476 m s − 1), 70 m (Vs =520 m s − 1), and 50 m (Vs = 626 m s − 1), respectively, overlying a uniform viscoelastic half-space (Vs =888 m s − 1). The Hualien containment shell was modeled for horizontal-rocking vibration as a Timoshenko beam while for vertical and torsional vibrations it was modeled as a hollow shaft. The predictions for the response of the containment model during forced vibration tests were obtained using the CLASSI approach. This approach is a linear substructure method for the

173

analysis of dynamic three-dimensional soil-structure problems involving single or multiple structures interacting with and through the soil (Wong and Luco, 1976; Luco, 1980). In this approach, elements characterizing the response of the superstructure, foundations, and layered soil are determined independently and then combined to solve the complete interaction problem. The solution is obtained in the frequency domain and the response in the time domain, if required, is obtained via Fast Fourier Transform synthesis. Comparisons of the calculated amplitudes and phases (blind predictions) of the horizontal displacement response at the top of the containment model for horizontal harmonic excitation of amplitude equal to 1 tf (9806 N) applied at the top with the corresponding experimental results in directions D1 and D2 are shown in Fig. 10. Since the theoretical model was considered axisymmetric the response was only calculated for one direction while the observed responses are significantly different in two orthogonal directions suggesting a lateral variation of soil properties or a marked anisotropy (Luco and de Barros, 1994). The amplitudes A(v) and phases f(v) are such that x(t)= A ei(vt − f) when the force f(t)=FT eivt. The results of the forced vibration tests can be used to determine the dynamic characteristics of the superstructure, foundation and soil. To derive the necessary equations we considered a lumped mass model in which the superstructure is excited by the force FT eivt that the harmonic shaker exerts at the top or base of the structure. The observed response of the forced vibration tests at the top of the structure together with the observed translation and rotation of the base were used to isolate and determine some of the fixed-base modal characteristics of the superstructure including modal frequency, modal damping ratio, modal mass, and participation factors associated with translation and rocking of the base (Luco and de Barros, 1994). To obtain experimental estimates of impedance functions for the foundation on the soil a procedure involving combination of the response observed during vibration tests with forces located at different levels in the structure was used. The resulting impedance functions are used to infer,

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Fig. 10. Comparison of experimental observations and blind predictions of the horizontal response at the top of the model for horizontal excitation at the top.

via an identification approach, the characteristics of the top layers of soil below the foundation (de Barros and Luco, 1995). The theoretical horizontal displacements at the top of the containment model based on the identified structural and soil properties, for hori-

zontal harmonic excitation (FT = 9806 N) applied at the top are compared with the corresponding observed responses (open circles) in Fig. 11. The calculated results with the identified properties closely match the experimental response.

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Fig. 11. Comparison of experimental observations (open circles) and new calculations based on the identified structural and soil properties (solid lines) for horizontal response at the top of the model for horizontal excitation at the top.

8. Conclusion In closing it can be said that, on the one hand, there are well equipped methods for SSI analysis.

On the other hand, the quality of the calculated results depends very much upon the adequate evaluation of the structure and, mainly, of the soil properties.

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Acknowledgements This paper was completed while the first author was on a short visit to the Civil and Environmental Engineering Department, Rutgers, The State University of New Jersey, USA. The support of the Department is gratefully acknowledged.

References Apsel, R.J., Luco, J.E., On the Green’s Functions for a Layered Half-space, Bull. Seism. Soc. America, Parts I and II, 73 (1983). Clough, R.W., Penzien, J., Dynamics of Structures, 2nd edn., McGraw-Hill, New York, 1994, pp. 609–709. de Barros, F.C.P., Luco, J.E., 1995. Identification of foundation impedance functions and soil properties from forced vibration tests of hualien containment model. Soil Dyn. Earthquake Eng. 14, 229–248. Gupta, S., Lin, T.W., Penzien, J., Yeh, S.C., 1982. Three-dimensional hybrid modelling of soil-structure interaction. Int. J. Earthquake Eng. Struct. Dyn. 10 (1), 69 – 87. Lamb, H., 1904. On the propagation of tremors over the surface of an elastic solid. Philos. Trans. R. Soc. 203A, 1 – 42. Luco, J.E., Linear Soil-Structure Interaction, Report UCRL15 272, Lawrence Livermore National Laboratory, Livermore, California, 1980.

.

Luco, J.E., de Barros, F.C.P., Identification of Structural and Soil Properties from Forced Vibration Tests of the Haulien Containment Model Prior to Backfill, Department of Applied Mechanics and Engineering Sciences, University of California, San Diego, La Jolla, 1994, pp. 114. Luco, J.E., Westman, R.A., Dynamic Response of Circular Footings, J. Eng. Mech. Div. ASCE 97, EM5 (1971) 1381 – 1395. Reissner, E., 1936. Stationa¨re, axialsymmetrische durch eine schu¨ttenlde Masse erregte Schwingengen eines elastisclien Halbraumes. Ingenieur Archiv. 7 (6), 381 – 396. Richart Jr., F.E., Hall Jr., J.R., Woods, R.D., Vibrations of Soils and Foundations, Prentice Hall, New Jersey, 1970, pp. 191 – 243. Tang, H.T., Fong, B.E., Summary of the Hualten Large-Scale Seismic Test Program FVT-1 Blind Prediction Analysis, Electric Power Research Institute (EPRI), Palo Alto, California, 1993. Veletsos, A.S., Wei, V.T., Lateral and Rocking Vibrations of Footings, J. Soil Mech. Foundations Div. ASCE 97, SM9 (1971) 1227 – 1248. Venancio-Filho, F., Claret, A.M., 1995. Frequency domain dynamic analysis of MDOF systems: nodal and modal coordinates formulations. Comput. Struct. 56 (1), 181– 189. Wolf, J.R., Dynamic Soil-structure Interaction, Prentice Hall, New Jersey, 1985. Wolf, J.R., Soil-structure Interaction Analysis in Time Domain, Prentice Hall, New Jersey, 1988. Wolf, J.R., Foundation Vibration Analysis Using Simple Physical Models, Prentice Hall, New Jersey, 1994. Wong, H.L., Luco, J.E., 1976. Dynamic response of rigid foundation of arbitrary shgre. Int. J. Earthquake Eng. Struct. Dyn. 4, 576 – 587.