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On the influence of foundation flexibility on the seismic response of structures
Computers and Geotechnics 27 (2000) 179±197 www.elsevier.com/locate/compgeo
On the in¯uence of foundation ¯exibility on the seismic response of structures Ricardo D. Ambrosini a,*, Jorge D. Riera b, Rodolfo F. Danesi c a
Structures Laboratory, National University of TucumaÂn, Lola Mora 380. 4000 San Miguel de TucumaÂn, TucumaÂn, Argentina b CPGEC, Federal University of Rio Grande do Sul, Porto Alegre, Av. Nilo PecËanha 550, Ap 302. 90470-000, Porto Alegre, Brazil c Structures Laboratory, National University of TucumaÂn, MunÄecas 586 1 A. 4000 San Miguel de TucumaÂn, TucumaÂn, Argentina Received 31 March 2000; received in revised form 17 April 2000; accepted 26 April 2000
Abstract The main objective of this paper is to contribute to a quanti®cation of the eect of foundation ¯exibility on the most important design variables in the seismic response of building structures with prismatic rectangular foundations. A soil±structure interaction model was used for this purpose. A general continuum formulation was adopted to represent the physical model of the structure and a lumped parameter model was adopted to represent the soil and the interaction mechanisms. Using the implemented models and in view of the objective of the work, a parametric study was made. For this purpose, a group of three structures, three seismic accelerations and seven sets of soil properties will be considered. The results obtained, in view of the large number of uncertainties involved, are presented in a probabilistic format and suggest that the soil±structure interaction can be neglected in the analysis of this class of structures. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Soil±structure interaction; Seismic design; Earthquake resistant structures; Foundations; Parametric study
1. Introduction The seismic analysis of buildings and other engineering structures is often based on the assumption that the foundation corresponds to a rigid block, which is subjected * Corresponding author. Tel.: +54-381-436-4087; fax: +54-381-436-4087. E-mail address: [email protected] (R.D. Ambrosini). 0266-352X/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0266-352X(00)00010-0
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to a horizontal, unidirectional acceleration. Such 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 veri®ed that the free ®eld motion at the rock surface, i.e. the motion that would occur without the building, is barely in¯uenced by its presence. The hypothesis loses validity when the structure is founded on soil deposits, since the motion at the soil surface, without the building, may be signi®cantly altered by the presence of the structure. The latter, in turn, has its dynamic characteristics Ð vibration modes and frequencies Ð modi®ed by the ¯exibility of the supports. Thus, there is a ¯ux of energy from the soil to the structure, and then back from the structure into the soil, in a process that is loosely known in seismic engineering as soil±structure interaction (SSI). Note that in the rigid foundation model, the energy received by the structure from the base during an earthquake, can only be dissipated through internal damping mechanisms, such as viscous damping, plastic deformations, fracture work, etc. In the case of ¯exible soils, some energy is fed back to the base and radiated away, giving rise to the so-called geometric damping. Procedures to take into account SSI in the seismic analysis of buildings were reviewed, among others, by Wolf [1] and Iguchi and Akino [2]. By way of introduction to the selection of the model used in the studies reported in Sections 3 and 4, a proposal of classi®cation is presented in Fig. 1. From limitations of space, only the principal references will be mentioned. Many papers following the so-called impedance functions approach may be mentioned: Wong and Luco [3], Crouse et al. [4], Wolf [5] and Mita et al. [6]. Numerous contributions found in the literature use the lumped-parameter models: Richart et al. [7], Clough and Penzien [8], Wolf [9], Veletsos and Wei [10] and Wolf and
Fig. 1. State of art of SSI.
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Somaini [11]. Within the frame of continuum mechanics, other procedures may be developed. For example, the cone models proposed by Wolf and Meek [12] and the beam±column analogy presented by Horvath [13]. Many authors follow the direct method, Isenberg and Adham [14], Hayashi et al. [15] and Godbole et al. [16]. On the other hand, the substructure approach is used by Damisa [17], Wolf and Darbre [18], Hayashi and Takahashi [19] using the BEM. More recently, the similarity based methods were developed by Wolf and Song [20]. In spite of its importance, there are no accepted guidelines to establish when the in¯uence of the foundation ¯exibility may be expected to be signi®cant. There is also little or no information in the technical literature on the error that may result from either neglecting the consideration of this factor or using incorrect values for the soil or foundation parameters. The main objective of this paper is, then, to contribute to a quanti®cation of the eect of foundation ¯exibility on the most important design variables in seismic problems, such as total base shear and overturning moment. The results obtained should be useful on several counts: (a) to assist the structural engineer in deciding whether to include in his model the surrounding soil, (b) in the development of simpli®ed criteria for codi®ed design and (c) in the quanti®cation of model uncertainty for the reliability assessment of structures under seismic excitation. The study is con®ned to prismatic building structures founded on similarly rectangular bases, located at an arbitrary plane under the ground level. The behaviour of both soil and structure is assumed linearly elastic. The results are ®nally presented in a probabilistic format, which is the only rational treatment of the problem in view of the large number of uncertainties involved. 2. Description of the model Consider the situation shown schematically in Fig. 2. The structure is founded at depth d on a halfspace of ¯exible soil (Fig. 2a). In general the region of the soil aected by the structure, also shown in Fig. 2a, is small. Now, if a horizontal acceleration is introduced and the assumption of a vertical wave propagation in the soil is adopted, the ``free-®eld'' motions at the soil surface and at a shallow depth underneath (points A0 and A, respectively) are, in general, dierent from the foundation-soil interface motion (point B0 ) and the motion at the boundary of the soil region around the foundation (point B). In this model, the input motion is speci®ed in the form of eective motion, obtained from the free®eld motion. When the rigid foundation assumption is resorted to (Fig. 2b), the problem disappears because all points experience the same acceleration, i.e. u B u B0 u A u A0 . The structural response calculated on the basis of the rigid foundation hypothesis will be herein compared with that obtained taking into account the ¯exibility of the foundation. For this purpose, a group of three structures, three seismic accelerations and seven sets of soil properties will be considered.
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Fig. 2. Flexible and rigid foundation models.
Let Qmb denote the maximum base shear for one combination of excitation and structure, assuming a rigid foundation, and Qimb the same quantity for ``soft'' soil conditions. The soil factor Q is de®ned as: i Q 1 Q Gs mb Qmb The soil factor is herein regarded as a random variable, and hence it should be characterized by its expected value and standard deviation. Indirect evidence suggests that it may be assumed to be normally distributed, i.e. Gaussian. Obviously, as the stiness of the soil increases, the mean value Q ! 1, while the standard deviation, Q ! 0. Since all variables Qimb are functions of the shear soil modulus Gs, the soil factor depends on the shear modulus too. 3. Representation of the structure and the foundation and method of solution The models used to represent the soil and the structure, as well as the solution method, are brie¯y described in the following. 3.1. Soil foundation model For the reasons described in Section 1, a lumped-parameter model was selected for the physical representation of the soil foundation and the interaction mechanism. Among the lumped-parameter models mentioned in Section 1, are the Clough and Penzien [8] (CP) and the Wolf and Somaini [11] (WS) models are considered the best. Comparing both models, it can be noted that the CP model is simpler and of easier utilisation; however, the WS model can adjust the ``exact solution'' to a larger range of frequencies and allows taking into account the foundation embedment. For the reasons mentioned above, the WS [11] model was selected to represent the soil and interaction mechanism. This model corresponds to a rigid prismatic embedded foundation with the soil Poisson ratio equal to s =1/3.
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A set of masses, springs and dashpots constitutes the model, which is shown in Fig. 3. The elements of the ¯exural vibration mode are connected to node 0 of the foundation: M0r, Kr and C0r. Moreover, the mass associated with the horizontal vibration mode (M0h) is connected eccentrically by a rigid rod, the spring Kh and the dashpot C0h. The eccentricities of these elements are denoted by fk and fc. In addition, there is also a free node 1 associated with the ¯exural vibration mode, with mass M1r, connected to the node 0 by a dashpot C1r. It must be pointed out that the masses, springs and dashpots coecients are functions of the soil properties (shear modulus Gs, mass density and Poisson ratio vs) and of the foundation dimensions [11]. 3.2. Structure model The physical model of the structure is based on Vlasov's [21] theory of thin-walled beams, which is modi®ed to include the eects of shear ¯exibility and rotatory inertia in the stress resultants, as well as variable cross-sectional properties [22]. In addition, a linear viscoelastic constitutive law was incorporated. The following three fourth-order partial dierential equations in the generalised displacements ; , and are obtained:
Fig. 3. Soil foundation model.
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4 3 4 @ @3 mx @ @2 mx dJy z @ @3 mx E Jy z ÿ ÿ ÿ ÿ Jy z 2 @z3 @z2 dz @z4 @z3 @z2 @t2 @z@t2 2a 2 dJy z @3 @2 mx @ @2 ÿ FT z 2 ay 2 qy ÿ @z@t2 @t @t @t2 dz 4 3 4 @3 my @ @3 my @ @2 my dJx z @ ÿ ÿ ÿ ÿ Jx z 2 E Jx z @z4 @z3 dz @z2 @t2 @z@t2 @z3 @z2 2b 3 @2 my dJx z @ @2 @2 ÿ a x 2 qx FT z ÿ @t2 dz @z@t2 @t2 @t @4 @3 dJ' z @4 dJ' z @3 ÿ J' z 2 2 ÿ E J' z 4 2 3 @z @z dz @z @t dz @z@t2 @2 @2 @2 @2 dJd z @ mA FT z ay 2 ÿ ax 2 r2 2 ÿ GJd z 2 ÿ G @t @t @t @z dz @z
2c
in which: r2 a2x a2y
Jx Jy FT
3
In these equations, Jx and Jy are the second moments of area of the cross-section in relation to the centroidal principal axes, J' the sectorial second moment of area, Jd the torsion modulus, ax and ay the coordinates of the shear centre. denotes the mass density of the beam material, while qx, qy and mA represent the externally applied loads per unit length. E and G are the Young's and the shear modulus, respectively. Finally, mx and my represent the mean values of shear strains over a cross-section z=constant. Using the Fourier transform, an equivalent system with 12 ®rst order partial differential equations with 12 unknowns, in the frequency domain, is obtained. The scheme described above is known in the literature as `state variables approach'. Six geometric and six static unknown quantities are selected as components of the state vector v: The displacements and , the bending rotations x and y , the normal shear stress resultants Qx and Qy, the bending moments Mx and My, the torsional rotation and its spatial derivative 0 , the total torsional moment MT and the bimoment B. v z; ! f; y ; Qy ; Mx ; ; x ; Qx ; My ; ; 0 ; MT ; BgT
4
in which, v=state vector. The system is: @v Av q @z
5
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in which A is the system matrix and q the external load vector. q z; ! f0; 0; ÿqx ; 0; 0; 0; ÿqy ; 0; 0; 0; ÿmA ; 0gT
6
In the frequency domain, the state variables depend on the frequency ! and the longitudinal coordinate z. For simplicity, the same notation will be used for the state variables and theirs transforms, since the domain can be adequately identi®ed by the indication of the function arguments. For example, z; t and z; ! refer to the ydisplacement in the time domain and to its Fourier transform, respectively. Obviously, the functions in the frequency domain are complex variables. For the sake of numerical convenience the real and imaginary parts of the functions are separated resulting in a system of 24 ®rst order partial dierential equations with 24 unknowns. vR 7 v ...... vI
. ........
Proceeding in the same manner with matrix A: A11 A...... 12 .......... A A21 A22
8
Submatrix A11 is associated with the real part vR, submatrix A22 with the imaginary part vI, while submatrixes A12 and A21 de®ne the coupling between both. It is easy to verify that: A11 A22
A12 ÿA21
9
A11 and A12 are given in the Appendix. It is important to note that the present formulation constitutes a general theory of beams applicable to solid as well as thin-walled beams. 3.3. Determination of the input motion The eective seismic motion (ESM) was obtained starting from the free-®eld motion by an approximate analytical solution developed by Harada et al. [23] and 0 ; 0G ; 0xG and 0yG are: used, among others, by Ganev et al. [24]. The ESMs G . For horizontal components 8 sin kd 0 04kd4=2 G 0G < kd G G : 0:63 kd > =2
10
in which G and G are the free-®eld ground motions, d is the depth of embedment (Fig. 2) and the coecient k is evaluated as
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k
! Vs
11
where Vs is the shear wave velocity of the soil. . For rotational components 8 d > > 0:4 1 ÿ cos kd > > a > > > > > > 0 0xG a yG a < 0:405 ÿ 0:05d=a 1 ÿ cos kd > G G > > 0:4d=a > > > > > > > : 0:405 ÿ 0:05d=a
04d=a41 04kd4=2 d=a > 1
12
04d=a41 kd > =2 d=a > 1
in which a is the radius of the foundation. Because the soil model proposed by Wolf and Somaini [11] was developed to rectangular foundations of dimensions 2b2l, the equivalent radius proposed by Meek and Wolf [25] for rocking vibration is used: r 4 8 bl b2 l 2 a req 13 3 Then, the components of load vector (6) for uniform beams are: 0 z 0xG qx ÿFT G
14a
qy ÿFT 0G z 0yxG
14b
0 z 0xG ÿ ax 0G z 0yG mA ÿFT ay G
14c
3.4. Solution method The system (5) may be easily 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 value problem encountered, the latter must be transformed into an initial value problem as shown, for example, by Ebner and Billington [26]. In order to incorporate the interaction model, described in Section 3.1, of the analysis, two topics must be taken into consideration: a. The generalized functions of forces and deformations showed in Fig. 3 corresponding with the following state variables: u0x ! 0; !
f0x ! x 0; !
15a
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Px ! Qx 0; ! u0y ! 0; ! Py ! Qy 0; ! 0 ! 0; !
187
Ry ! My 0; !
15b
f0y ! y 0; !
15c
Rx ! Mx 0; !
15d
MT ! MT 0; !
15e
b. The boundary conditions, which in case of ®xed support are: x y 0 0
16
must be replaced by motion equations of the soil model, given by Wolf and Somaini [11], except for the condition 0 =0. 4. Numerical analysis and results The in¯uence of the ¯exibility of the soil in the global dynamic response was studied by means of parametric studies on the eect of the shear modulus Gs, or alternatively the shear wave velocity Vs. In agreement with values quoted in the literature [7,27,28], the shear modulus for very soft to medium soil sites, was assumed to vary between 35 and 100 MN/m2. With the purpose of determining the coecient Q de®ned by Eq. (1), a representative number of samples is necessary. For this reason, a set of three structures was considered, as indicated in Table 1 with the average plan shown in Fig. 4, and three ground acceleration records, indicated in Table 2 with response spectra shown in Fig. 5. So, nine simulated values of Q for each value of Gs, were generated. The numerical analysis was performed using the program DAYSSI [29]. It should be noted that for the Torres del Miramar building free vibration testing conducted after the earthquake of Chile in 1985 revealed a fundamental period of approximately 1.06 s [33]. On the other hand, Wallace et al. [31] using the program SAP80 determined a period of 0.92 for a ®xed base. The agreement with the period determined by DAYSSI is good for both cases.
Table 1 Structures used Building
Description
H a (m)
T1b (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
Liaw [30] Wallace et al. [31] Coull [32]
a b
Total height of the building. Fundamental period determined by DAYSSI (®xed base).
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Fig. 4. Plan view of the structures.
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Table 2 Ground acceleration records used Ground acceleration
Earthquake
Record
ta (s)
Durationa (s)
A1 A2 A3
Caucete (Argentina), 1977 Chile, 1985 Loma Prieta (USA), 1989
San Juan VinÄa del Mar S20W Santa Cruz
0.04 0.017 0.02
10 35 20
a
Discretization time used in the analysis.
Fig. 5. Response spectra of the ground motions.
4.1. Results The maximum base shear, shown in Figs. 6±8, in which the straight horizontal line represents the ®xed base response, was selected as the output variable of interest. The corresponding soil factors Q are given in Table 3. The dierences between the results obtained taking into account the in¯uence of the eective input motion with those obtained with the free-®eld motion are presented in Table 4. 4.2. Statistical analysis As discussed before, the soil factor Q can be considered as a random variable, and characterized by its mean value and standard deviation. In addition, if a normal
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Fig. 6. Maximum base shear: central core building.
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Fig. 7. Maximum base shear: Torres de Miramar building.
191
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Fig. 8. Maximum base shear: core and walls building.
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distribution is assumed, con®dence limits for the expected value of the soil factor E(Q), can be established: Q95% Q CI
17
in which Q is the mean value of Q and CI is the radius of the con®dence interval de®ned by: 18 CI za=2 pQ n in which za/2 is the normalised variable equal to 1.96 for a con®dence interval of 95%, mQ is the standard deviation of Q and n is the sample size, which in this case is equal to 9. Table 3 Soil factors Q B1 2
B2
B3
Gs (MN/m )
A1
A2
A3
A1
A2
A3
A1
A2
A3
35 50 60 75 85 100
0.717 0.856 0.887 0.893 0.922 0.972
0.794 0.988 1.081 1.094 1.082 1.037
1.004 1.098 1.111 1.111 1.106 1.096
0.543 0.728 0.748 0.812 0.857 0.921
0.547 0.550 0.591 0.727 0.786 0.848
0.744 0.803 0.793 0.765 0.754 0.793
0.591 0.641 0.742 0.629 0.617 0.548
0.531 0.722 0.759 0.735 0.822 0.915
0.672 0.615 0.816 0.940 0.905 0.820
Table 4 Dierences between ESM and free-®eld motion (%) B1
B2
B3
d=3 m
d=6 m
d=3 m
A1 1.9
A2 0.65
A3 1.5
A1 4.0
A2 1.1
A3 8.5
A1 1.8
A2 0.5
A3 1.95
Table 5 Mean value, standard deviation, COV and CI of Q Gs (MN/m2)
Q
m
COV (%)
CI
35 50 60 75 85 100
0.683 0.778 0.836 0.856 0.872 0.883
0.154 0.179 0.167 0.167 0.155 0.160
22.6 23.0 20.0 19.5 17.7 18.1
0.101 0.117 0.109 0.109 0.101 0.105
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Fig. 9. Variation of the coecient Q.
Finally, the coecient of variation (COV) is normally de®ned by: COV %
Q 100 Q
19
So, using the results of Table 3 and Eqs (14)±(16), Q ; Q , COV and CI are shown in Table 5. In order to visualize better the variation of Q, the ®nal results are shown in Fig. 9. 5. Conclusions The geotechnical properties of the site down to a few hundred meters will in¯uence the seismic input motion usually de®ned in applications as design response spectra. This paper examines the eect of the ¯exibility of the upper soil layers, upon which the building under consideration is founded. The shear modulus Gs is therefore considered representative of the soil between the levels of the foundation plane and a depth not exceeding the width of the building. On the basis of the results obtained, applicable to typical reinforced concrete buildings less than 60 m tall, the following conclusions may be drawn: 1. The peak base shear decreases in the mean as Gs decreases, down to about 70% of the peak base shear corresponding to the assumption of a rigid foundation, as should be expected from the observation that the fundamental period of all building considered in the study were close to the peak or on the descending branches of the response spectra employed in the analysis.
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2. Taking also into consideration that the radius of the con®dence interval of the peak base shear is roughly 10%, it may be concluded that the soil±structure interaction eect may be disregarded, as long as the fundamental period of the structure (on a ®xed base) does not lie on the rising branch of the acceleration response spectrum. 3. Soil±structure interaction should contribute to model uncertainty in the reliability analysis of building structures subjected to seismic excitation with a coecient of variation for peak response quantities of about 20%. 4. The mean dierence between the results obtained taking into account the in¯uence of eective seismic motion (ESM) and those corresponding to the free-®eld motion was 1.36, 4.53 and 1.43% for buildings B1, B2 and B3, respectively. These results lend support to the conclusion of Harada et al. [23], concerning the marginal in¯uence of ESM, which may be replaced, in most practical cases, by the free-®eld motion. The validity of the conclusions above should be further examined in case of ¯exible, slender structures subjected to very strong ground motion, in which any increase of the ¯exibility of the foundation will re¯ect in the P-delta eect, not considered in the present formulation. Acknowledgements The ®nancial support of the CONICET (Argentina) and CNPq (Brazil) is gratefully acknowledged. Appendix The submatrices of system (3) are: 2
0
0
0
0
0
0
0
...........................................
0
...........................................
A11
1 0 1 0 0 0 0 0 0 0 0 0 6 k0y FG1 6 6 1 6 0 0 0 ÿ 0 0 0 0 0 0 0 0 6 EJx 1 6 6 6 ÿF!2 0 0 0 0 0 0 0 F!2 ay 0 0 0 6 6 6 2 0 Jx ! 1 0 0 0 0 0 0 0 0 0 6 6 .................................................................... 6 6 6 1 6 0 0 0 0 0 0 0 0 0 0 1 6 k0x FG1 6 6 1 6 0 0 0 0 0 0 0 0 0 0 0 6 6 EJy 1 6 6 0 0 0 ÿF!2 ay 0 0 0 0 0 0 0 ÿF!2 6 6 6 6 ÿ1 0 0 0 0 0 0 0 0 0 0 ÿJy !2 6 6 .................................................................... 6 6 6 6 0 0 0 0 0 0 0 0 0 1 0 0 6 6 1 6 0 0 0 0 0 0 0 0 0 0 0 ÿ 6 6 EJ' 1 6 6 0 0 0 ÿF!2 ay 0 0 0 ÿF!2 r2 0 0 0 6 F!2 ax 4 0
B0
1
0
3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
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0
0
0 0
0
0
............................ .........
A12
............................ .........
2
0
GJd
0
(A1)
0
in which F denotes the cross-sectional area, ! the frequency, the coecient of structural damping and: B0 J' !2 ÿ GJd
A2a
1 1 2
A2b
References [1] Wolf JP. Survey and classi®cation of computational approaches in soil±structure interaction: comparison of time and frequency domain analyses. In: GuÈlkan P, Clough, RW, editors. Developments in dynamic soil-structure interaction. Kluwer Academic Publishers, 1993, p. 1±23. [2] Iguchi M, Akino K. Experimental and analytical studies for soil-structure interaction (state-of-theart report). In: 12th International Conference on Structural Mechanics in Reactor Technology (SMiRT 12), Division K: Seismic Response Analysis and Design, Principal Division Lecture, 1993. [3] Wong HL, Luco JE. Tables of impedance functions and input motions for rectangular foundations. Report no. CE 78-15, Dept. of Civil Engrg, University of Southern California, Los Angeles, CA, 1978. [4] Crouse CB, Hushmand B, Luco JE, Wong HL. Foundation impedance functions: theory versus experiment. Journal of Geotechnical Engineering. ASCE 1990;116(3):432±49. [5] Wolf JP. Dynamic soil±structure interaction. Englewood Clis, NJ: Prentice-Hall Inc., 1985. [6] Mita A, Yoshida K, Kumagai Sand Shioya K. Soil±structure interaction experiment using impulse response. Earthquake Engineering and Structural Dynamics 1989;18(5):727±44. [7] Richart F, Hall J, Woods R. Vibrations of soils and foundations. Englewood Clis, NJ: Prentice Hall, 1970. [8] Clough R, Penzien J. Dynamics of structures. McGraw-Hill Kogakusha, Ltd., 1975. [9] Wolf JP. Soil±structure-interaction analysis in time domain. Englewood Clis, NJ: Prentice Hall Inc, 1988.
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[10] Veletsos AS, Wei YT. Lateral and rocking vibration of footings. Journal of the Soil Mechanics and Foundation Division 1971;ASCE:1227±48. [11] Wolf JP, Somaini D. Approximate dynamic model of embededd foundation in time domain. Earthquake Engineering and Structural Dynamic 1986;14:683±703. [12] Wolf JP, Meek JW. Cone models for a soil layer on a ¯exible rock half-space. Earthquake Engineering and Structural Dynamic 1993;22:185±93. [13] Horvath JS. Beam±column analogy model for soil±structure interaction analysis, Journal of Geotechnical Engineering. ASCE 1993;119(2):358±64. [14] Isenberg J, Adham SA. Interaction of soil and power plants in earthquakes. Journal of the Power Division, ASCE 1972;98(PO2):273±91. [15] Hayashi Y, Akino K, Esashi Y. Experimental project of soil±structure interaction using the foundation of NUPEC large vibration table Ð vibration of foundation and ground, dynamic earth pressure and wave propagation. In: Eighth World Conference on Earthquake Engineering, vol. III, San Francisco, CA, 1984, p. 1089±96. [16] Godbole PN, Viladkar MN, Noorzaei J. Nonlinear soil±structure interaction analysis using coupled ®nite±in®nite elements. Computers and Structures 1990;36(6):1089±96. [17] Damisa O. Vibration of a rigid footing Ð a ®nite model for the elastic half space. International Journal for Numerical and Analytical Methods in Geomechanics 1986;10:73±89. [18] Wolf JP, Darbre OR. Non-linear soil±structure interaction analysis based on the boundary-element method in time domain with application to embedded foundation. Earthquake Engineering and Structural Dynamic 1986;14:83±101. [19] Hayashi Y, Takahashi I. An ecient time-domain soil±structure interaction analysis based on the dynamic stiness of an unbounded soil. Earthquake Engineering and Structural Dynamics 1992;21:787±98. [20] Wolf JP, Song Ch. Finite-element modelling of unbounded media. John Wiley and Sons, 1996. [21] Vlasov V. Thin-walled elastic beams. 2nd ed. Jerusalem: Israel Program for Scienti®c Translations, 1961. [22] Ambrosini RD, Riera JD, Danesi RF. Dynamic analysis of thin-walled and variable open section beams with shear ¯exibility. International Journal for Numerical Methods in Engineering 1995;38(17):2867±85. [23] Harada T, Kubo K, Takayama T. Dynamic soil±structure interaction analysis by continuum formulation method. Report of the Institute of Industrial Science 29, University of Tokyo, 1981. [24] Ganev T, Yamazaki F, Katayama T. Observation and numerical analysis of soil±structure interaction of a reinforced concrete tower. Earthquake Engineering and Structural Dynamics 1995;24:491± 503. [25] Meek JW, Wolf JP. Approximate Green's function for surface foundations. Journal of Geotechnical Engineering, ASCE 1993;119(10):1499±514. [26] Ebner A, Billington D. Steady state vibrations of damped Timoshenko beams. Journal of the Structural Division, 1968;94(ST3);ASCE:737-60. [27] Haupt WA, Kumar S. Laboratory measurement of the dynamic shear modulus of soils. In: Proc. of the European Conference on Structural Dynamics, EURODYN'90, Bochum, Germany. Rotterdam/ Brook®eld: A. Balkema, 1991, p. 201±8. [28] Romberg W. Guide values for dynamic soil properties and simpli®ed estimation of dynamic soil spring constants. In: Proceedings of the European Conference on Structural Dynamics, EURODYN'90, Bochum, Germany. Rotterdam/Brook®eld: A.A. Balkema, 1991, p. 173±8. [29] Ambrosini RD. ConsideracioÂn de la InteraccioÂn Suelo-Estructura en el AnaÂlisis DinaÂmico de Estructuras. Doctorate thesis, University of Nac. de TucumaÂn, Argentina, 1994. [30] Liaw T. Torsion of multi-storey spatial core walls. Proc ICE 1978;2(65):601±9. [31] Wallace J, Moehie J. Evaluation of ATC requirements for soil±structure interaction using data from the 3 March 1985 Chile earthquake. Earthquake Spectra. EERI 1990;6(3):595±611. [32] Coull A. Interactions between coupled shear walls and cantilever cores in three-dimensional regular symmetrical cross-wall structures. Proc. ICE 1973;2(55):827±40. [33] Bongiovanni G, Celebi M, Safak E. Seismic rocking response of a triangular building founded on sand. Earthquake Spectra. EERI 1987;3(4):793±809.
On the in¯uence of foundation ¯exibility on the seismic response of structures Ricardo D. Ambrosini a,*, Jorge D. Riera b, Rodolfo F. Danesi c a
Structures Laboratory, National University of TucumaÂn, Lola Mora 380. 4000 San Miguel de TucumaÂn, TucumaÂn, Argentina b CPGEC, Federal University of Rio Grande do Sul, Porto Alegre, Av. Nilo PecËanha 550, Ap 302. 90470-000, Porto Alegre, Brazil c Structures Laboratory, National University of TucumaÂn, MunÄecas 586 1 A. 4000 San Miguel de TucumaÂn, TucumaÂn, Argentina Received 31 March 2000; received in revised form 17 April 2000; accepted 26 April 2000
Abstract The main objective of this paper is to contribute to a quanti®cation of the eect of foundation ¯exibility on the most important design variables in the seismic response of building structures with prismatic rectangular foundations. A soil±structure interaction model was used for this purpose. A general continuum formulation was adopted to represent the physical model of the structure and a lumped parameter model was adopted to represent the soil and the interaction mechanisms. Using the implemented models and in view of the objective of the work, a parametric study was made. For this purpose, a group of three structures, three seismic accelerations and seven sets of soil properties will be considered. The results obtained, in view of the large number of uncertainties involved, are presented in a probabilistic format and suggest that the soil±structure interaction can be neglected in the analysis of this class of structures. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Soil±structure interaction; Seismic design; Earthquake resistant structures; Foundations; Parametric study
1. Introduction The seismic analysis of buildings and other engineering structures is often based on the assumption that the foundation corresponds to a rigid block, which is subjected * Corresponding author. Tel.: +54-381-436-4087; fax: +54-381-436-4087. E-mail address: [email protected] (R.D. Ambrosini). 0266-352X/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0266-352X(00)00010-0
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to a horizontal, unidirectional acceleration. Such 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 veri®ed that the free ®eld motion at the rock surface, i.e. the motion that would occur without the building, is barely in¯uenced by its presence. The hypothesis loses validity when the structure is founded on soil deposits, since the motion at the soil surface, without the building, may be signi®cantly altered by the presence of the structure. The latter, in turn, has its dynamic characteristics Ð vibration modes and frequencies Ð modi®ed by the ¯exibility of the supports. Thus, there is a ¯ux of energy from the soil to the structure, and then back from the structure into the soil, in a process that is loosely known in seismic engineering as soil±structure interaction (SSI). Note that in the rigid foundation model, the energy received by the structure from the base during an earthquake, can only be dissipated through internal damping mechanisms, such as viscous damping, plastic deformations, fracture work, etc. In the case of ¯exible soils, some energy is fed back to the base and radiated away, giving rise to the so-called geometric damping. Procedures to take into account SSI in the seismic analysis of buildings were reviewed, among others, by Wolf [1] and Iguchi and Akino [2]. By way of introduction to the selection of the model used in the studies reported in Sections 3 and 4, a proposal of classi®cation is presented in Fig. 1. From limitations of space, only the principal references will be mentioned. Many papers following the so-called impedance functions approach may be mentioned: Wong and Luco [3], Crouse et al. [4], Wolf [5] and Mita et al. [6]. Numerous contributions found in the literature use the lumped-parameter models: Richart et al. [7], Clough and Penzien [8], Wolf [9], Veletsos and Wei [10] and Wolf and
Fig. 1. State of art of SSI.
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Somaini [11]. Within the frame of continuum mechanics, other procedures may be developed. For example, the cone models proposed by Wolf and Meek [12] and the beam±column analogy presented by Horvath [13]. Many authors follow the direct method, Isenberg and Adham [14], Hayashi et al. [15] and Godbole et al. [16]. On the other hand, the substructure approach is used by Damisa [17], Wolf and Darbre [18], Hayashi and Takahashi [19] using the BEM. More recently, the similarity based methods were developed by Wolf and Song [20]. In spite of its importance, there are no accepted guidelines to establish when the in¯uence of the foundation ¯exibility may be expected to be signi®cant. There is also little or no information in the technical literature on the error that may result from either neglecting the consideration of this factor or using incorrect values for the soil or foundation parameters. The main objective of this paper is, then, to contribute to a quanti®cation of the eect of foundation ¯exibility on the most important design variables in seismic problems, such as total base shear and overturning moment. The results obtained should be useful on several counts: (a) to assist the structural engineer in deciding whether to include in his model the surrounding soil, (b) in the development of simpli®ed criteria for codi®ed design and (c) in the quanti®cation of model uncertainty for the reliability assessment of structures under seismic excitation. The study is con®ned to prismatic building structures founded on similarly rectangular bases, located at an arbitrary plane under the ground level. The behaviour of both soil and structure is assumed linearly elastic. The results are ®nally presented in a probabilistic format, which is the only rational treatment of the problem in view of the large number of uncertainties involved. 2. Description of the model Consider the situation shown schematically in Fig. 2. The structure is founded at depth d on a halfspace of ¯exible soil (Fig. 2a). In general the region of the soil aected by the structure, also shown in Fig. 2a, is small. Now, if a horizontal acceleration is introduced and the assumption of a vertical wave propagation in the soil is adopted, the ``free-®eld'' motions at the soil surface and at a shallow depth underneath (points A0 and A, respectively) are, in general, dierent from the foundation-soil interface motion (point B0 ) and the motion at the boundary of the soil region around the foundation (point B). In this model, the input motion is speci®ed in the form of eective motion, obtained from the free®eld motion. When the rigid foundation assumption is resorted to (Fig. 2b), the problem disappears because all points experience the same acceleration, i.e. u B u B0 u A u A0 . The structural response calculated on the basis of the rigid foundation hypothesis will be herein compared with that obtained taking into account the ¯exibility of the foundation. For this purpose, a group of three structures, three seismic accelerations and seven sets of soil properties will be considered.
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Fig. 2. Flexible and rigid foundation models.
Let Qmb denote the maximum base shear for one combination of excitation and structure, assuming a rigid foundation, and Qimb the same quantity for ``soft'' soil conditions. The soil factor Q is de®ned as: i Q 1 Q Gs mb Qmb The soil factor is herein regarded as a random variable, and hence it should be characterized by its expected value and standard deviation. Indirect evidence suggests that it may be assumed to be normally distributed, i.e. Gaussian. Obviously, as the stiness of the soil increases, the mean value Q ! 1, while the standard deviation, Q ! 0. Since all variables Qimb are functions of the shear soil modulus Gs, the soil factor depends on the shear modulus too. 3. Representation of the structure and the foundation and method of solution The models used to represent the soil and the structure, as well as the solution method, are brie¯y described in the following. 3.1. Soil foundation model For the reasons described in Section 1, a lumped-parameter model was selected for the physical representation of the soil foundation and the interaction mechanism. Among the lumped-parameter models mentioned in Section 1, are the Clough and Penzien [8] (CP) and the Wolf and Somaini [11] (WS) models are considered the best. Comparing both models, it can be noted that the CP model is simpler and of easier utilisation; however, the WS model can adjust the ``exact solution'' to a larger range of frequencies and allows taking into account the foundation embedment. For the reasons mentioned above, the WS [11] model was selected to represent the soil and interaction mechanism. This model corresponds to a rigid prismatic embedded foundation with the soil Poisson ratio equal to s =1/3.
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A set of masses, springs and dashpots constitutes the model, which is shown in Fig. 3. The elements of the ¯exural vibration mode are connected to node 0 of the foundation: M0r, Kr and C0r. Moreover, the mass associated with the horizontal vibration mode (M0h) is connected eccentrically by a rigid rod, the spring Kh and the dashpot C0h. The eccentricities of these elements are denoted by fk and fc. In addition, there is also a free node 1 associated with the ¯exural vibration mode, with mass M1r, connected to the node 0 by a dashpot C1r. It must be pointed out that the masses, springs and dashpots coecients are functions of the soil properties (shear modulus Gs, mass density and Poisson ratio vs) and of the foundation dimensions [11]. 3.2. Structure model The physical model of the structure is based on Vlasov's [21] theory of thin-walled beams, which is modi®ed to include the eects of shear ¯exibility and rotatory inertia in the stress resultants, as well as variable cross-sectional properties [22]. In addition, a linear viscoelastic constitutive law was incorporated. The following three fourth-order partial dierential equations in the generalised displacements ; , and are obtained:
Fig. 3. Soil foundation model.
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4 3 4 @ @3 mx @ @2 mx dJy z @ @3 mx E Jy z ÿ ÿ ÿ ÿ Jy z 2 @z3 @z2 dz @z4 @z3 @z2 @t2 @z@t2 2a 2 dJy z @3 @2 mx @ @2 ÿ FT z 2 ay 2 qy ÿ @z@t2 @t @t @t2 dz 4 3 4 @3 my @ @3 my @ @2 my dJx z @ ÿ ÿ ÿ ÿ Jx z 2 E Jx z @z4 @z3 dz @z2 @t2 @z@t2 @z3 @z2 2b 3 @2 my dJx z @ @2 @2 ÿ a x 2 qx FT z ÿ @t2 dz @z@t2 @t2 @t @4 @3 dJ' z @4 dJ' z @3 ÿ J' z 2 2 ÿ E J' z 4 2 3 @z @z dz @z @t dz @z@t2 @2 @2 @2 @2 dJd z @ mA FT z ay 2 ÿ ax 2 r2 2 ÿ GJd z 2 ÿ G @t @t @t @z dz @z
2c
in which: r2 a2x a2y
Jx Jy FT
3
In these equations, Jx and Jy are the second moments of area of the cross-section in relation to the centroidal principal axes, J' the sectorial second moment of area, Jd the torsion modulus, ax and ay the coordinates of the shear centre. denotes the mass density of the beam material, while qx, qy and mA represent the externally applied loads per unit length. E and G are the Young's and the shear modulus, respectively. Finally, mx and my represent the mean values of shear strains over a cross-section z=constant. Using the Fourier transform, an equivalent system with 12 ®rst order partial differential equations with 12 unknowns, in the frequency domain, is obtained. The scheme described above is known in the literature as `state variables approach'. Six geometric and six static unknown quantities are selected as components of the state vector v: The displacements and , the bending rotations x and y , the normal shear stress resultants Qx and Qy, the bending moments Mx and My, the torsional rotation and its spatial derivative 0 , the total torsional moment MT and the bimoment B. v z; ! f; y ; Qy ; Mx ; ; x ; Qx ; My ; ; 0 ; MT ; BgT
4
in which, v=state vector. The system is: @v Av q @z
5
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in which A is the system matrix and q the external load vector. q z; ! f0; 0; ÿqx ; 0; 0; 0; ÿqy ; 0; 0; 0; ÿmA ; 0gT
6
In the frequency domain, the state variables depend on the frequency ! and the longitudinal coordinate z. For simplicity, the same notation will be used for the state variables and theirs transforms, since the domain can be adequately identi®ed by the indication of the function arguments. For example, z; t and z; ! refer to the ydisplacement in the time domain and to its Fourier transform, respectively. Obviously, the functions in the frequency domain are complex variables. For the sake of numerical convenience the real and imaginary parts of the functions are separated resulting in a system of 24 ®rst order partial dierential equations with 24 unknowns. vR 7 v ...... vI
. ........
Proceeding in the same manner with matrix A: A11 A...... 12 .......... A A21 A22
8
Submatrix A11 is associated with the real part vR, submatrix A22 with the imaginary part vI, while submatrixes A12 and A21 de®ne the coupling between both. It is easy to verify that: A11 A22
A12 ÿA21
9
A11 and A12 are given in the Appendix. It is important to note that the present formulation constitutes a general theory of beams applicable to solid as well as thin-walled beams. 3.3. Determination of the input motion The eective seismic motion (ESM) was obtained starting from the free-®eld motion by an approximate analytical solution developed by Harada et al. [23] and 0 ; 0G ; 0xG and 0yG are: used, among others, by Ganev et al. [24]. The ESMs G . For horizontal components 8 sin kd 0 04kd4=2 G 0G < kd G G : 0:63 kd > =2
10
in which G and G are the free-®eld ground motions, d is the depth of embedment (Fig. 2) and the coecient k is evaluated as
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k
! Vs
11
where Vs is the shear wave velocity of the soil. . For rotational components 8 d > > 0:4 1 ÿ cos kd > > a > > > > > > 0 0xG a yG a < 0:405 ÿ 0:05d=a 1 ÿ cos kd > G G > > 0:4d=a > > > > > > > : 0:405 ÿ 0:05d=a
04d=a41 04kd4=2 d=a > 1
12
04d=a41 kd > =2 d=a > 1
in which a is the radius of the foundation. Because the soil model proposed by Wolf and Somaini [11] was developed to rectangular foundations of dimensions 2b2l, the equivalent radius proposed by Meek and Wolf [25] for rocking vibration is used: r 4 8 bl b2 l 2 a req 13 3 Then, the components of load vector (6) for uniform beams are: 0 z 0xG qx ÿFT G
14a
qy ÿFT 0G z 0yxG
14b
0 z 0xG ÿ ax 0G z 0yG mA ÿFT ay G
14c
3.4. Solution method The system (5) may be easily 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 value problem encountered, the latter must be transformed into an initial value problem as shown, for example, by Ebner and Billington [26]. In order to incorporate the interaction model, described in Section 3.1, of the analysis, two topics must be taken into consideration: a. The generalized functions of forces and deformations showed in Fig. 3 corresponding with the following state variables: u0x ! 0; !
f0x ! x 0; !
15a
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Px ! Qx 0; ! u0y ! 0; ! Py ! Qy 0; ! 0 ! 0; !
187
Ry ! My 0; !
15b
f0y ! y 0; !
15c
Rx ! Mx 0; !
15d
MT ! MT 0; !
15e
b. The boundary conditions, which in case of ®xed support are: x y 0 0
16
must be replaced by motion equations of the soil model, given by Wolf and Somaini [11], except for the condition 0 =0. 4. Numerical analysis and results The in¯uence of the ¯exibility of the soil in the global dynamic response was studied by means of parametric studies on the eect of the shear modulus Gs, or alternatively the shear wave velocity Vs. In agreement with values quoted in the literature [7,27,28], the shear modulus for very soft to medium soil sites, was assumed to vary between 35 and 100 MN/m2. With the purpose of determining the coecient Q de®ned by Eq. (1), a representative number of samples is necessary. For this reason, a set of three structures was considered, as indicated in Table 1 with the average plan shown in Fig. 4, and three ground acceleration records, indicated in Table 2 with response spectra shown in Fig. 5. So, nine simulated values of Q for each value of Gs, were generated. The numerical analysis was performed using the program DAYSSI [29]. It should be noted that for the Torres del Miramar building free vibration testing conducted after the earthquake of Chile in 1985 revealed a fundamental period of approximately 1.06 s [33]. On the other hand, Wallace et al. [31] using the program SAP80 determined a period of 0.92 for a ®xed base. The agreement with the period determined by DAYSSI is good for both cases.
Table 1 Structures used Building
Description
H a (m)
T1b (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
Liaw [30] Wallace et al. [31] Coull [32]
a b
Total height of the building. Fundamental period determined by DAYSSI (®xed base).
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Fig. 4. Plan view of the structures.
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Table 2 Ground acceleration records used Ground acceleration
Earthquake
Record
ta (s)
Durationa (s)
A1 A2 A3
Caucete (Argentina), 1977 Chile, 1985 Loma Prieta (USA), 1989
San Juan VinÄa del Mar S20W Santa Cruz
0.04 0.017 0.02
10 35 20
a
Discretization time used in the analysis.
Fig. 5. Response spectra of the ground motions.
4.1. Results The maximum base shear, shown in Figs. 6±8, in which the straight horizontal line represents the ®xed base response, was selected as the output variable of interest. The corresponding soil factors Q are given in Table 3. The dierences between the results obtained taking into account the in¯uence of the eective input motion with those obtained with the free-®eld motion are presented in Table 4. 4.2. Statistical analysis As discussed before, the soil factor Q can be considered as a random variable, and characterized by its mean value and standard deviation. In addition, if a normal
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Fig. 6. Maximum base shear: central core building.
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Fig. 7. Maximum base shear: Torres de Miramar building.
191
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Fig. 8. Maximum base shear: core and walls building.
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193
distribution is assumed, con®dence limits for the expected value of the soil factor E(Q), can be established: Q95% Q CI
17
in which Q is the mean value of Q and CI is the radius of the con®dence interval de®ned by: 18 CI za=2 pQ n in which za/2 is the normalised variable equal to 1.96 for a con®dence interval of 95%, mQ is the standard deviation of Q and n is the sample size, which in this case is equal to 9. Table 3 Soil factors Q B1 2
B2
B3
Gs (MN/m )
A1
A2
A3
A1
A2
A3
A1
A2
A3
35 50 60 75 85 100
0.717 0.856 0.887 0.893 0.922 0.972
0.794 0.988 1.081 1.094 1.082 1.037
1.004 1.098 1.111 1.111 1.106 1.096
0.543 0.728 0.748 0.812 0.857 0.921
0.547 0.550 0.591 0.727 0.786 0.848
0.744 0.803 0.793 0.765 0.754 0.793
0.591 0.641 0.742 0.629 0.617 0.548
0.531 0.722 0.759 0.735 0.822 0.915
0.672 0.615 0.816 0.940 0.905 0.820
Table 4 Dierences between ESM and free-®eld motion (%) B1
B2
B3
d=3 m
d=6 m
d=3 m
A1 1.9
A2 0.65
A3 1.5
A1 4.0
A2 1.1
A3 8.5
A1 1.8
A2 0.5
A3 1.95
Table 5 Mean value, standard deviation, COV and CI of Q Gs (MN/m2)
Q
m
COV (%)
CI
35 50 60 75 85 100
0.683 0.778 0.836 0.856 0.872 0.883
0.154 0.179 0.167 0.167 0.155 0.160
22.6 23.0 20.0 19.5 17.7 18.1
0.101 0.117 0.109 0.109 0.101 0.105
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Fig. 9. Variation of the coecient Q.
Finally, the coecient of variation (COV) is normally de®ned by: COV %
Q 100 Q
19
So, using the results of Table 3 and Eqs (14)±(16), Q ; Q , COV and CI are shown in Table 5. In order to visualize better the variation of Q, the ®nal results are shown in Fig. 9. 5. Conclusions The geotechnical properties of the site down to a few hundred meters will in¯uence the seismic input motion usually de®ned in applications as design response spectra. This paper examines the eect of the ¯exibility of the upper soil layers, upon which the building under consideration is founded. The shear modulus Gs is therefore considered representative of the soil between the levels of the foundation plane and a depth not exceeding the width of the building. On the basis of the results obtained, applicable to typical reinforced concrete buildings less than 60 m tall, the following conclusions may be drawn: 1. The peak base shear decreases in the mean as Gs decreases, down to about 70% of the peak base shear corresponding to the assumption of a rigid foundation, as should be expected from the observation that the fundamental period of all building considered in the study were close to the peak or on the descending branches of the response spectra employed in the analysis.
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2. Taking also into consideration that the radius of the con®dence interval of the peak base shear is roughly 10%, it may be concluded that the soil±structure interaction eect may be disregarded, as long as the fundamental period of the structure (on a ®xed base) does not lie on the rising branch of the acceleration response spectrum. 3. Soil±structure interaction should contribute to model uncertainty in the reliability analysis of building structures subjected to seismic excitation with a coecient of variation for peak response quantities of about 20%. 4. The mean dierence between the results obtained taking into account the in¯uence of eective seismic motion (ESM) and those corresponding to the free-®eld motion was 1.36, 4.53 and 1.43% for buildings B1, B2 and B3, respectively. These results lend support to the conclusion of Harada et al. [23], concerning the marginal in¯uence of ESM, which may be replaced, in most practical cases, by the free-®eld motion. The validity of the conclusions above should be further examined in case of ¯exible, slender structures subjected to very strong ground motion, in which any increase of the ¯exibility of the foundation will re¯ect in the P-delta eect, not considered in the present formulation. Acknowledgements The ®nancial support of the CONICET (Argentina) and CNPq (Brazil) is gratefully acknowledged. Appendix The submatrices of system (3) are: 2
0
0
0
0
0
0
0
...........................................
0
...........................................
A11
1 0 1 0 0 0 0 0 0 0 0 0 6 k0y FG1 6 6 1 6 0 0 0 ÿ 0 0 0 0 0 0 0 0 6 EJx 1 6 6 6 ÿF!2 0 0 0 0 0 0 0 F!2 ay 0 0 0 6 6 6 2 0 Jx ! 1 0 0 0 0 0 0 0 0 0 6 6 .................................................................... 6 6 6 1 6 0 0 0 0 0 0 0 0 0 0 1 6 k0x FG1 6 6 1 6 0 0 0 0 0 0 0 0 0 0 0 6 6 EJy 1 6 6 0 0 0 ÿF!2 ay 0 0 0 0 0 0 0 ÿF!2 6 6 6 6 ÿ1 0 0 0 0 0 0 0 0 0 0 ÿJy !2 6 6 .................................................................... 6 6 6 6 0 0 0 0 0 0 0 0 0 1 0 0 6 6 1 6 0 0 0 0 0 0 0 0 0 0 0 ÿ 6 6 EJ' 1 6 6 0 0 0 ÿF!2 ay 0 0 0 ÿF!2 r2 0 0 0 6 F!2 ax 4 0
B0
1
0
3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5
196
R.D. Ambrosini et al. / Computers and Geotechnics 27 (2000) 179±197 3 0 0 0 0 0 0 0 0 0 0 0 k0y FG1 7 6 7 6 7 60 0 0 0 0 0 0 0 0 0 0 ÿ 7 6 EJx 1 7 6 7 6 7 60 0 0 0 0 0 0 0 0 0 0 0 7 6 7 6 7 6 7 60 0 0 0 0 0 0 0 0 0 0 0 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 7 6 7 6 7 6 7 60 0 0 0 0 0 0 0 0 0 0 7 6 k0x FG1 7 6 7 6 7 60 0 0 0 0 0 0 0 0 0 0 7 6 EJy 1 6 7 7 6 7 60 0 0 0 0 0 0 0 0 0 0 0 7 6 7 6 7 6 0 0 0 0 0 0 0 0 0 0 7 60 0 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7 . 7 6 7 6 7 60 0 0 0 0 0 0 0 0 0 0 0 7 6 7 6 7 6 7 6 7 60 0 0 0 0 0 0 0 0 0 0 ÿ 7 6 EJ ' 1 7 6 7 6 0 0 0 0 0 0 0 0 0 0 7 60 0 5 4 0 0
0
0
0 0
0
0
............................ .........
A12
............................ .........
2
0
GJd
0
(A1)
0
in which F denotes the cross-sectional area, ! the frequency, the coecient of structural damping and: B0 J' !2 ÿ GJd
A2a
1 1 2
A2b
References [1] Wolf JP. Survey and classi®cation of computational approaches in soil±structure interaction: comparison of time and frequency domain analyses. In: GuÈlkan P, Clough, RW, editors. Developments in dynamic soil-structure interaction. Kluwer Academic Publishers, 1993, p. 1±23. [2] Iguchi M, Akino K. Experimental and analytical studies for soil-structure interaction (state-of-theart report). In: 12th International Conference on Structural Mechanics in Reactor Technology (SMiRT 12), Division K: Seismic Response Analysis and Design, Principal Division Lecture, 1993. [3] Wong HL, Luco JE. Tables of impedance functions and input motions for rectangular foundations. Report no. CE 78-15, Dept. of Civil Engrg, University of Southern California, Los Angeles, CA, 1978. [4] Crouse CB, Hushmand B, Luco JE, Wong HL. Foundation impedance functions: theory versus experiment. Journal of Geotechnical Engineering. ASCE 1990;116(3):432±49. [5] Wolf JP. Dynamic soil±structure interaction. Englewood Clis, NJ: Prentice-Hall Inc., 1985. [6] Mita A, Yoshida K, Kumagai Sand Shioya K. Soil±structure interaction experiment using impulse response. Earthquake Engineering and Structural Dynamics 1989;18(5):727±44. [7] Richart F, Hall J, Woods R. Vibrations of soils and foundations. Englewood Clis, NJ: Prentice Hall, 1970. [8] Clough R, Penzien J. Dynamics of structures. McGraw-Hill Kogakusha, Ltd., 1975. [9] Wolf JP. Soil±structure-interaction analysis in time domain. Englewood Clis, NJ: Prentice Hall Inc, 1988.
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