SYSTEM PARAMETERS OF SOIL FOUNDATIONS FOR TIME DOMAIN DYNAMIC ANALYSIS

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 19, 541- 553 (1990) SYSTEM PARAMETERS OF SOIL FOUNDATIONS FOR TIME DOMAIN DYNAMIC ANALYSIS WEN-Y...
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EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS, VOL. 19, 541- 553 (1990)

SYSTEM PARAMETERS OF SOIL FOUNDATIONS FOR TIME DOMAIN DYNAMIC ANALYSIS WEN-YU JEAN* Eastern International Engineers, h e . , Taipei Office. SO1 Tun-Hua South Road, 11th Floor, Taipei, Taiwan 106.54, R.O.C.

TSUNG-WU LINt Departmenr of Civil Engineering, National Taiwan University, Taipei. Taiwan 10764, R.O.C.

AND

JOSEPH PENZIEN' Eastern International Engineers, Inc., Main O f i r e , 800 Solara Drive, Lafayetre. California 94549, U.S.A.

SUMMARY Soil-structure interaction analysis is usually carried out in the frequency domain, because the compliance functions of the half-space are known only in the frequency domain. Since non-linear analysis cannot be carried out in the frequency domain, a system with frequency independent parameters is used to represent the half-space soil medium so that a nonlinear analysis in the time domain becomes possible. The objective of this paper is to propose a system with lumped parameters, which are independent of frequency, to represent the half-space soil medium. The proposed frequency independent system consists of a number of real discrete structure elements; thus the existing dynamic analysis programs may be adoptable with little modification. In this paper, the parameters are found by minimizing the sum of the squares of deviations between the steady-state responses of the theoretical half-space model and those of the lumped parameter system over a specified frequency range. Once the parameters have been found, the lumped parameter system can be used in practical applications for time domain dynamic analysis of either linear or non-linear structures. In comparison with the dynamic response of the theoretical half-space model, the lumped parameter system yields satisfactory results.

INTRODUCTION Vibrating soil-foundation systems may be described by both the theory of half-space and the theory of a lumped parameter system under dynamic load. Within the last two or three decades, the half-space models which assume that the subsoil can be considered as an elastic, isotropic and homogeneous half-space have been frequently used in research to obtain the dynamic compliance or impedance functions. These functions can be used only in frequency domain dynamic analysis, but the dynamic analysis of non-linear structures must be performed in the time domain. Therefore, it is necessary and convenient to express the solutions of the half-space theory in terms of the parameters of the lumped parameter system. These parameters can be used in dynamic analysis for non-linear or linear structures, considering the effects of soil-structure interaction. In examining the response characteristics of the SDOF (Single-Degree-Of-Freedom) model and the halfspace solution, the similarity suggests that an approximate solution for the latter is possible by using the former, if the appropriate discrete parameters such as equivalent mass, damping and spring constants are determined. Indeed, many researchers had developed such SDOF models for special cases. Lysmer and *Staft 'Professor. 'Board Chairman.

0098-8847/90/04054 1-1 3$06.50 g> 1990 by John Wiley & Sons, Ltd.

Received 29 September 1988 Revised 5 December 1989

542

W.-Y. JEAN, T.-W. LIN AND J. PENZIEN

Richart' obtained a SDOF model for a vertically loaded rigid footing. The parameters of a vibrating footing with different assumed stress conditions were obtained by Funston and Hall' and by Whitman and Richart.j Meek and Veletsos4 used a simple oscillator with frequency independent properties for both the horizontal and rocking directions. Christof Ehrler' and Veletsos and Verbic6 obtained the parameters which are frequency dependent. Wolf and Somaini7 proposed a 2-DOF discrete system and obtained some frequency independent parameters. From the aforementioned researches, we can find the following deficiencies: (i) The results are not good enough; (ii) some parameters are frequency dependent; (iii) these discrete models are available only for low frequency; therefore, they cannot describe the dynamic characteristics of the soil-foundation system completely.

A lumped parameter system is proposed to get better results. The purpose of this paper is to find a useful lumped parameter system considering the inertia effects of the half-space and the radiation damping, and also to determine the parameters which must be independent of the frequency of the exciting force. The number of DOFs (Degrees-Of-Freedom) of the lumped parameter system may be 2, 3, or more. In this paper, the compliance functions obtained by Luco and Westmann' and Luco' are used. The selections of compliance functions are for a uniform half-space with a massless disk or a massless embedded hemisphere. Vertical, horizontal, torsional and rocking vibrating modes for the former and the torsional mode only for the latter are discussed. If the lumped parameter system can describe exactly the dynamic characteristics of the half-space, then the compliance functions of the two models will be identical. Theoretically, this requires an infinite number of parameters. However, for practical reasons, the number of discrete parameters must be finite; therefore the compliance functions of the two models are different. What needs to be done is to minimize the difference.

EQUIVALENT LUMPED PARAMETER SYSTEM AND ITS PARAMETERS This paper is specifically concerned with a practical method for carrying out the dynamic analysis of the soil-foundation system in the time domain. This method uses a system of lumped masses, dampers and springs which are approximately equivalent to the actual soil-foundation system (half-space). In such a lumped system, the masses, dampers and springs represent the inertia, damping and flexibility present in the half-space system, respectively. Since these parameters can be used in the time domain, they must be independent of frequency and time. The key steps are selecting a proper lumped parameter system and finding the parameters. Once this has been done, available mathematical solutions of the lumped system can be used to estimate the responses of the actual system. Luco studied soil-foundation problems and obtained the dynamic compliances of a rigid massless circular disk on an elastic half-space. They are horizontal, rocking, vertical and torsional compliances, and also two coupling compliances. The latter are the coupling horizontal and the coupling rocking compliances, and are produced by the rocking and horizontal vibrations, respectively. These frequency domain compliances can be written in a matrix form, the subscripts H, M, V and T representing horizontal, rocking, vertical and torsional vibration modes, respectively.

Therefore, for an axisymmetric system, five independent compliance functions are available, namely C,,, CMM, C,,, C,, and CHM ( = CMH).The coupling term C,, has little effect on the dynamic responses of the

SYSTEM PARAMETERS OF SOIL FOUNDATIONS

543

soil-foundation system. If these coupling vibrations are taken into consideration in dynamic analysis, then the parameters will increase in number. Therefore, in this paper, these two coupling vibrations are neglected to simplify the problem. In doing so, the vibration modes of the footing will be uncoupled; and the dynamic behaviour of the soil-footing system can be simulated by the lumped parameter system. The parameters can be determined for each vibration mode individually. Let us treat the response in only one DOF, for example, vertical vibration. All other components can be treated similarly. The response of a uniform half-space under harmonic vertical excitation is (2)

UV(0) = Cvv(w)Pv(w)

The discrete parameter system From experience," it is difficult to obtain results better than those obtained by other researchers if a SDOF discrete system is applied, and the results are little improved if a 2-DOF discrete system is used. On the other hand, a discrete parameter system of three degrees of freedom is surprisingly good enough to represent the half-space soil medium. The first degree, X , , corresponds to the degree in the footing of the halfspace soil system under consideration, and the other two degrees, X2 and X3, are the added degrees of freedom in order to have more freedoms and parameters for better agreement with the half-space soil medium. This system is represented by the following mass, damping and stiffness matrices:

In the above three symmetric matrices, there are 18 frequency independent parameters. When a unit harmonic exciting force eiwtis acting on the first DOF, X,, in the footing, i.e. a dynamic force { p ( t ) } = { P ) eiW'= ( 1 0 O)T eiotis acting on the discrete parameter system, which represents the compliance of the soil-foundation system, the steady-state response of X , (w)eiozcan be obtained from the following frequency domain solution of the equation of motion: {X(w)} = (

-

+ iw[C] + [K])-'{P)

w'[M]

or s22s33

X,(w) = slls22s33

+ 2s12s23s13

-

-

2 s23

sllszJ

- s22s:3

- s33s?2

(4)

+

where sij = kij - w2mij iwcij. This paper is concerned with X , ( w ) and tries to let X,(w) approximate the compliance function, for example, Cvv(w). For any arbitrarily given set of parameters, the matrices [ M I , [C] and [K] can be obtained and substituted into equation (4). Then, X , ( w i )can be obtained for the frequency wi of the acting force, wi being a selected positive value. If w varies continuously, X, ( w )will be a function like the compliance of the lumped system with the given parameters. Because the parameters are arbitrarily given, the function X,(w) is different from the compliance function of the actual soil-foundation system. The square of the deviation of the two functions can be othained as follows, for any one specified frequency w i :

4 = [Xl(wi) - CvV(wi)I'

(5)

If the sum of equations (5) over a specified frequency range is a minimum, the parameters will be the best choice to simulate the soil-foundation system. The problem then becomes an optimization problem. To determine the parameters it is necessary to minimize the quantity in the following equation: Min. W(v)

=

Min.

1[ X , (mi) - C , , ( O ~ ) ] ~ i

(6)

544

W.-Y. JEAN, T.-W. LIN AND J. PENZIEN

where v is the variable vector: all the parameters are shown in equation(3). This is an unconstrained optimization problem. There are several iteration procedures to solve the equation. In this paper, the Conjugate Direction Method'' is adopted to search for the parameters. Adopting the method mentioned above, equation (6) can be rewritten in the following form: Min. @(a) = Min. W(vk+') = Min. W(vk + ED')

(7)

in which vk represents the parameters at the kth iteration step; scalar a is a positive step-size parameter selected to ensure a decrease in error, W, within each iteration cycle. At the kth iteration step, CI is the only unknown, and Dk is determined by VWk'VWk Dk-l ; k 2 2 D k = - V W k + VWk-1 .VWk-l which is the search direction given by the Conjugate Direction Method. V Wk is the gradient vector of W(vk). It must be recognized that the response quantity X i ( o i )is not a simple expression of v but is obtained by a numerical procedure involving the solution of equation (4). The partial derivatives in the gradient vector V W, therefore, have to be replaced by a finite difference. To start the iteration procedure, one must have an initial estimate parameter vector v'. The initial values of the parameters can be arbitrarily and reasonably given, and the parameters will converge rapidly to a local optimum value. Adopting the iteration procedure, one may simulate the soil-foundation system with a lumped parameter system. Some results of simulating the compliances of the massless soil-foundation system are presented in Figures l(aHd). It is shown that the agreements between the numerical results and the analytical solutions are quite satisfactory. In these figures, the abscissae represent dimensionless-frequency, a, = wr/C,, o represents the circular frequency, r is the radius of the disk and C, is the shear wave velocity of the soil medium. Examining the numerical results of parameters (Table I(a)), it is of interest to note that the quantity m,, is smaller than the other diagonal terms and approaches zero. This means that the mass added to the disk is very small. In practice, this term may be neglected. However, some drawbacks still exist in this way, such as the mass matrix has non-zero values in off-diagonal terms and the parameters are too many in number. In this paper, some relationships among these parameters are introduced to avoid these drawbacks; these relationships are m l l = m,, m22

= Bl

= m i 3 = m23 =

0 m33

= B2

This is a lumped system with 10 characteristic parameters, as shown in Figure 2: this is a 3- D O F system with two DOFs under the footing. These two DOFs do not exist in the actual half-space system. There is a damper between them to ensure that, in the optimization procedure of finding the parameters, the solutions should fall in the feasible area. To reduce the number of parameters, the damping between the two masses is set as d = (d2 + d4)/2. The physical meanings of the parameters are shown in Figure 2. One may note that all the parameters in matrices [MI,[ C ] and [ K ] are dimensionless. In equations (9),Bi, hiand di are dimensionless too, and represent the equivalent mass, damping and stiffness of the soil medium, respectively. Scaling

545

SYSTEM PARAMETERS OF SOIL FOUNDATIONS YI

N v)

N

-Luco,

p

=o

-Luco,

+ proscnt

p

=o

+ Preaent

0

0

2.0

'0.0

4.0

6.0

8.0

10.0

'0.0

DIMENSIONLESS-FREOUENCY

v)

2.0 4.0 6.0 8.0 DIMENSIONLESS-FREQUENCY

10.0

Y)

N

r4

'1

-Luco,

-1

p = I/4

p = 114

-Luco.

t Present

WYI O h

t Present

2 0'

*

$2. 0 O 0

n. N 0

'0.0

,

2.0 4.0 6.0 8.0 OlflENSIONLESS-FREQUENCY

10.0

4.0

. 6 .. 0

'

8 .' 0

'

1 0, . 0

In N

-Luq

p = 113

+ Present

Wv)

(a)

2.0

DIMENSIONLESS-FAEOUENCY

v)

$ 04 . 0

y0.0 -.

.

'

.

. 4.0

.

. . 6.0

'

2.0 8.0 DIMENSIONLESS-FREOUENCY

'

-

10.0

Re

N

0

'0.0

,

2.0 4.0 6.0 8.0 10.0 DIMENSIONLESS-FREQUENCY

Figure l(a). Comparisons of the vertical compliance functions (left: 10 parameters right: 18 parameters)

relations are presented in the next section, so that the parameters found for one footing can be extended to another under the same soil conditions. The results of the proposed lumped system The dimensionless-frequency range, a,, for minimizing W , is from 0 to 10, and the compliance function is divided into 40 discrete points. In most applications, the frequency is up t o 30 Hz or so (o= 32 Hz if r = 5 m, C, = 100 m/sec), and this is almost enough for earthquake analysis. Some local optimum sets of parameters are shown in Tables I(a) and I(b). The comparison of the compliance functions obtained by Luco and by the present model are shown in Figures l(aj(d). The agreement between the results of the half-space theory and those of the present model is outstanding. For rocking excitation, the steady-state responses, which are obtained by using the half-space theory, of the rigid circular foundation with different mass ratios, are shown in Figure 3.'' Also shown in the same figure are the responses obtained by using the present model. In this figure, mass ratio B is a dimensionless measure of moment of inertia, I (or mass, M, in horizontal and vertical excitation) of the foundation plate. The relations between B and I (or M ) of the foundation plate are the same as in equations (1 la-e). In Figure 3, for the case B = 0, the foundation is massless as in Figure l(c), but is represented in another way. Other relations are the responses of a disk, of single degree of freedom, with different mass, to the rocking force developed by constant force excitation (unit harmonic force excitation). In these figures, if the mass of the disk is small, l? is small, and the responses of the

546

W.-Y. JEAN. T.-W. LIN AND J. PENZIEN

-Luco,

p

=o

-Luco.

+ Present -01

4.0

2.0

4.0

6.0

8.0

\

10.0

DIRENSIDNLESS-FREOUENCY

DIRENSIONLESS-FREOUENCY

0

0 N

N

p = 114

-Luco,

-Luco,

t Presed

p =

l/4

t Present

e

b

-

0

%.O

p - 0

t Presen1

-

0

2.0

4.0

6.0

8.0

10.0

B.0

DIRENSIONLESS-FREOUENCY

2.0

4.0

6.0

8.0

10.0

DIMENSIONLESS-FREOUENCY

0

N

-

-Luco, p = 1/3

-Luco, p = 113

+

t Preacni

Present

0 0

0

(b)

cb.0

0 1 H E N S I 0N L E S S - F R E 0U E N C Y

2.0

4.0

6.0

8.0

10.0

DIMENSIONLESS-FREOUENCY

Figure I(b). Comparisons of the horizontal compliance functions (left: 10 parameters right: 18 parameters)

two cases (10 parameters and 18 parameters) are quite similar. When B is greater, say, B = 2, comparing the results of the half-space theory and those of the present model, the discrepancy occurs near the resonant frequency in the 10 parameter system. However, the results are excellent in the 18 parameter system. It seems that the proposed discrete system (10 parameters) will slightly underestimate the responses of the superstructure, if this model is applied to practical analysis. In Table I(b), the parameter d , is negative in the cases of torsional and rocking vibration. This may be due to the existence of some mechanisms which apply energy to the vibrating system, but it is difficult to explain reasonably the physical meaning. Fortunately, in practice, with standard analysis codes, this system is stable and the responses are acceptable. Thus, the present model can be adopted to simulate the dynamic behaviour of the soil-foundation system, and also can be used in practical structural dynamic analysis.

APPLICATION O F THE PRESENT MODEL AND ITS PARAMETERS FOR TIME DOMAIN ANALYSIS The present model can be adopted in dynamic analysis just by putting together the structure and the lumped system. Adopting the parameters obtained herein, one can obtain the equation of motion of the structure

I

m

1

09

--

2.

1

4 c

a

-

5-

09

0

4.00

0

4.00

0.40

0.80

COMPLIANCE

1.20

COMPLIANCE 0.40 0.80 1.20

0

4.00

0

4.00

COMPLIANCE 0.00 1.20

0.40

COMPLIANCE 0.40 0.80 1.20

5

1.

09

w

2c

7

c

0

0

0

--

COMPLIANCE

COMPLIANCE

0

&

.

.

.

.

.

COMPLIANCE 0.75 6 0 . 2 5 0.25

0

COMPLIANCE 6 0 . 2 5 0.25 0.75

I

1.25

1.25

0

.

*

f.

.

4

O

.

-P -

50

m?.

c

0

rn

a0

I O I

v)

L o v ) .

z

0

2 0

.l.n ,

2 ?,

I

0

.

.

.

.

L

COMPLIANCE d 0 . 2 5 0.25 0.75

0

COMPLIANCE 6 0 . 2 5 0.25 0.75

d

1.25

1.25

1

548

W.-Y. JEAN, T.-W. LIN AND J. PENZIEN

The proposed lumped parameter system with 10 parameters System

Horizontal or vertical mode

Torsional or rocking mode

B

equivalent mass

equivalent moment of inertia

d h

equivalent damping ratio

equivalent rotational damping ratio

equivalent spring constant

equivalent rotational spring constant

Examole: for hemisoherical foundation (radius = r. torsional mode)

Figure 2. The proposed lumped system and its normalized parameters

system for the time domain analysis according to the general dynamic analysis procedures

CMI{Wf + LmW + CKI{u(t)) = (P(t,>

(10)

where the matrices [ M I , [ C ] and [ K ] include the characteristic parameters corresponding to those of the soil medium so that the soil-structure interaction effects can be taken into consideration. The term { u ( t ) > also includes the degrees of freedom corresponding to those of the soil medium. These are different from those of half-space theory analysis procedure. In half-space theory, the soil-structure system must be partitioned into an isolated structure and a soil-foundation system. The force acting on the structure system must be separated into two parts: one is the external force, ( p ( t ) } ,the other is the interaction force, { J ( t ) } .In such a case, the effects of the soil medium, which can be expressed only in the frequency domain, cannot be included in the matrices [ M I , [ C ] and [K], which are expressed in the time domain. Therefore, these effects can be expressed only by the unknown vector { f ( t ) } but can not be solved in the time domain, which must be transformed into the frequency domain to include the impedances for the interaction effects. This paper uses the dimensionless compliance to find the equivalent parameters B, d and h which represent the soil-structure interaction effects. Therefore, all the parameters shown in Tables I(a) and (b) are dimensionless numerical results. In practice, transformations are necessary and are described as follows. O 1. Vertical vibration mode:

equivalent mass equivalent damping ratio equivalent spring constant

Mv = -p r 3 B 1-P 4r2 Cv =-----&Zd 1-P 4Gr Kv =-h 1-P

549

SYSTEM PARAMETERS OF SOIL FOUNDATIONS

I

0.00

0.40

0.80

1-20

1.60

I 2.00

0.00 0 . 4 0

0.80

1.20

1-60

2.00

DlMENSION~ESS~FREOUENCY

DIMENSIONLESS-FREOUENCY

Figure 3. Comparisons of the steady-state responses in rocking of foundations with different mass ratio B(left: 10 parameters right: 18 parameters)

2. Horizontal vibration mode: 8

equivalent mass

M,

equivalent damping ratio

c, = - 8r2 f i d

equivalent spring constant

8 Gr K , =h 2-P

=---

2-P

pr3B

2-P

3. Rocking vibration mode:

equivalent moment of inertia equivalent rotational damping ratio

c,

equivalent rotational spring constant

K M=

=

m d

8r4 3(1 - P)

8Gr3 3 u - PI

h

4. Torsional vibration mode: (1) rigid massless circular disk:

equivalent moment of inertia

IT

16 3

= -p5 B

equivalent rotational damping ratio

16r4 C, = P J p 3

equivalent rotational spring constant

16Gr3 K , =h 3

Cd

(2) embedded rigid hemispherical foundation: equivalent moment of inertia

I, = 47rpr5B

equivalent rotational damping ratio

C,

equivalent rotational spring constant

K , = 47tGr3h

=4

nr4md

El le)

I I

I

I

I

I

I l l

I

I

I l l

I

I

I

I l l

I

I

I

I I

I

I

I

I I

I

I

I

l

I

I

I

.. m

3

.. d

i

0

I

a

l

P

Mode 114

0.871738 0.623006 0.670237 1.584255 0.349571 1.232231 1.408243 1.085654 1.427814 0467202 1.788034

0

1.071277 0.719621 0.718087 1.556075 0.342105 1.252303 1.404189 1.175808 1.295966 0.465099 1.753634

Vertical

0.883083 0.570012 0.653354 1502514 0.349261 1.170155 1.336334 1.110375 1.416417 0.457885 1.808406

113

114 0.363157 0.679534 0.557397 1.843417 0.148821 1.266378 1.554897 0.644986 1.847095 0.714975 1.962675

0 0.728042 0.532113 0.512511 1.707244 0.226781 1.292031 1.499638 1.019967 1.551950 0.470109 1.906862

Horizontal

Circular footings

0.1 10640 0.624675 0.6455 58 1.772940 0.097622 1.218775 1.495858 0.682 180 1.961640 0.659597 2.075410

113

1/4

1.073235 1.132836 0.347393 0.379 175 - 0.239731 - 0.249124 0.970217 0.949508 0.62 1842 0.648071 006025 1 0.06 1200 0.505354 0.515234 0445928 0.454560 1.658939 1.700381 1.419149 1.396666 1.121020 1.126073

0

Rocking

1.066106 0.340540 - 0'285459 1.095161 0.679554 0053799 0.574480 0.435 209 1.689507 1.516529 1.087072

113

1.090098 0.361 103 - 0'273191 1.123990 0.601864 0.126948 0.625469 0.408978 1.618974 1.379489 1.200869

Embedded

Torsional

0.95 25 87 0.311200 - 0.275540 1.021537 0.559968 00654 10 0.543473 0.409080 1.747190 1.363940 1.227680

Table I(b). The optimum sets of parameters of the lumped system obtained by solving equation (7) (10 parameters)

552

W.-Y. JEAN. T.-W. LIN AND J. PENZIEN

where r represents the radius of a disk or a hemisphere. The mass density, Poisson's ratio and shear modulus of a half-space medium are denoted by p, p and G, respectively. EXAMPLE An example is presented to examine the simulating effect. It is, idealizing, a SDOF system if the soil-structure interaction effects are neglected. Considering the soil-structure interaction, the foundation will produce horizontal and rocking vibrations due to the horizontal excitation of the container. In such a case, there are three DOFs in the whole system, but there are seven DOFs (two added horizontal DOFs and two added rocking DOFs) by adopting the present model. According to the half-space theory, the time domain responses of the container can be obtained in the frequency domain by inverse FFT (Fast Fourier Transform) technique^.'^ Adopting the present model, the time histories can also be obtained by solving equation (10) in the time domain. Here, the Wilson 0 methodL4is used. In order to show the effects of the foundation, we choose a quite soft site, the shear wave velocity being about 67 m/sec (G = 8000 kN/m2). The relative flexibility (the ratio of AF, AH, AR, AT) is shown in Figure 4, where AT is the total displacement of DOFl, AH and AR are displacements due to rigid body motion (horizontal and rocking displacement of the

Ap : A H : Aa : AT = 0.051 : 0.063 : 0.886 : 1

--H.lf-space theory

---pnwt (10 pu-tem)

?I 0

'0.00

I

,

2.00

,

,

4.00

,

,

6.00

-

8.00

,

s*c.

10.00

T I ME- T

Figure 4. Comparison of time domain response on the top, DOFl of structure in example

SYSTEM PARAMETERS O F SOIL FOUNDATIONS

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foundation) and AF is the displacement due to structure deformation, for a static force applied at DOF1. Shown in Figure 4 are the responses of DOFl to foundation excitation (earthquake inputs). The lumped system yields satisfactory results, but there are some underestimations from 4 to 6 sec. The authors believe that this comes from rocking vibration, as in the discussion for Figure 3. If this structure is put on a stiff site (smaller (AH + AR)/AT and larger AF/AT) the responses of both the present model and the half-space theory will be quite similar (these are not shown here). CONCLUSION AND DISCUSSION A simple and useful model is proposed which can express sufficiently the dynamic behaviour of the soil-foundation system. The numerical results are excellent. The frequency domain dynamic analysis, which takes into account the soil-structure interaction, can be transformed into the time domain. The soil-structure interaction can then be taken into consideration for non-linear structural dynamic analysis. Although there are some parameters with negative values, the present model can still be adopted in practical dynamic analysis. The proposed model is a lumped parameter system, in which the parameters have physical meanings corresponding to the actual structural elements; therefore the existing dynamic analysis programs can be adopted with little modification, such as adding a simple damping element only. This paper does not consider the coupling motions. If these coupling motions must be taken into consideration, the parameters obtained without considering the coupling motions can be regarded as the known quantities, and other parameters must be introduced to take into account the coupling motions. The number of parameters in the optimization procedure can then be reduced, and the additional parameters can be determined by adopting the same procedure mentioned in this paper. The results shown in Figures 2(aHd) are only for a massless, resting circular footing and an embedded rigid hemispherical foundation (torsional mode). For other shapes of foundations, or for a layer soil medium, one may use the known compliance functions (or impedance functions) and use the same procedures mentioned to find the parameters of the 3-DOF discrete system. REFERENCES 1. I. Lysmer and F. E. Richart, ‘Dynamic response of footings to vertical loading’, J. soil mech.Jound. dio. A S C E 92, No. SMl, 65-91 ( 1966). 2. N. E. Funston and W. J. Hall, ‘Footing vibration with nonlinear subgrade support’, J . soil mech. found. dio. ASCE 93, No. SM5, 191-21 1 (1967). 3. R. V. Whitman and F. E. Richart, ‘Design procedures for dynamically loaded foundations’, J. soil mech. found dio. ASCE No. SM6, 93, 169-193 (1967). 4. J. W. Meek and A. S. Veletsos, ‘Simple models for foundations in lateral and rocking motion’, Proc. 5th world conJ earthquake eng. Rome, Italy 2, 261G2613 (1974). 5. 0. M. Christof Ehrler, ‘Nonlinear parameters of vibrating foundations’, J . soil mech. found. diu. ASCE 94, 1023-1025; 1199-1214 (1968). 6. A. S. Veletsos and B. Verbic, ‘Vibration of viscoelastic foundations’, Earthquake eng. struct. dyn. 2, 87-102 (1973). 7. John P. Wolf and Dario R. Somaini, ‘Approximate dynamic model of embedded foundation in time domain’, Earthquake eng. struct. dyn. 14, 683-703 (1986). 8. J. E. Luco and R. A. Westmann, ‘Dynamic response of circular footings’, J. eng. mech. diu. A S C E 97, 1381-1395 (1973). 9. J. E. Luco, ‘Torsional response of structures for SH waves: The case of hemispherical foundations’, Bull. seism. soc. Am. 66, 109-123 (1976). 10. W. Y. Jean and T. W. Lin ‘System parameters of soil-foundations for time domain dynamic analysis’, (in Chinese), Master Dissertation, National Taiwan University, 1986. 11. R. Fletcher and C. M. Reeves, ‘Function minimization by conjugate gradients’, Comput. j., 7, 149-154 (1964). 12. F. E. Richart, J. R. Hall and R. D. Woods, Vibrations of Soils and Foundations, Prentice-Hall, Englewood Cliffs, N.J., 1970. 13. S. Gupta, J. Penzien, T. W. Lin and C. S. Yeh, ‘Three-dimensional hybrid modelling of soil-structure interaction’, Earthquake eng. struct. dyn. 10, 69-87 (1982). 14. K. J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1982, Chapter 9.

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