SEISMIC SOIL-STRUCTURE INTERACTION RESPONSE OF INELASTIC STRUCTURES

Transcription

1 SEISMIC SOIL-SRUCURE INERACION RESPONSE OF INELASIC SRUCURES Sittipong Jarernprasert, Enrique Bazan-Zurita Paul C. Rizzo Associates, Inc., Pittsburgh, PA 1535 Jacobo Biela Carnegie Mellon Universit, Pittsburgh, PA 1513 Dedicated to José M. Roësset; scholar, teacher and friend SUMMARY We analze the effects of soil-structure interaction (SSI) on the response of ielding single-stor structures embedded in an elastic halfspace to a set of accelerograms recorded in California and to a set of records from Mexico Cit. We find that, for nonlinear hsteretic structures, SSI ma lead to larger ductilit demands and larger total displacements than if the soil were rigid. his behavior differs from that envisioned in current seismic provisions, which allow designers to ignore SSI altogether or to reduce the base shear force with respect to that of the fixed-base structure. o overcome this deficienc, we examine an approach to incorporate SSI in determining the seismic design coefficient, C, of sstems with nonstructural degrading elastoplastic behavior and target ductilit μ t. In this approach, C is obtained as the unreduced seismic coefficient, C, of an equivalent fixed-base structure with natural period, the natural period of the elastic SSI sstem, and with inelastic reduction factor equal to the λ-root of the fixed-base reduction factor, where λ =, and is the elastic natural period of the structure ignoring SSI. he pea relative displacement of the structure can be evaluated as the ield displacement ductilit μ t. he pea total displacement, including SSI, is closel approximated b u times the target u (μ t + λ 1). Kewords: seismic response, seismic design, seismic analsis, inelastic structure, inelastic response, soil-structure interaction 1. INRODUCION Current seismic design provisions allow engineers to completel ignore the effects of soilstructure interaction in the seismic analsis of buildings, or to consider them b reducing the design base shear of the fixed-base structure (e.g., ASCE [1]; NDF [], NEHRP [3]). A discussion of the SSI prescriptions of NEHRP has been presented b Stewart et al [4]. hese provisions are based on analses of simple linear, viscousl damped, structures subjected to transient or stead-state excitation (e.g., Biela [5]; Jennings and Biela [6]; Veletsos and Mee [7]; Luco [8]; Roesset, [9]), and reflect the observation that interaction produces an elongation of the fixed-base natural period and dissipation of part of the vibrational energ of the building b

2 wave radiation into the foundation medium (in addition to energ losses from internal friction in the soil). he increase in period results in an increase of the seismic coefficient on an ascending branch of the response spectrum, no change on a flat portion, and a decrease in a descending branch. In addition, the energ dissipated in the foundation will usuall increase the effective damping, and, therefore, tend to decrease the spectral ordinate. Past studies show that for linear sstems, the effects of soil-structure interaction are, on balance, beneficial, and are the basis for allowing a reduction in the base shear in seismic design provisions. While linear analsis provides valuable insight into the response to earthquae excitation, most real structures behave nonlinearl, particularl for the seismic intensities implied in design spectra. In contrast to the extensive wor devoted to date to linear SSI, less attention has been given to nonlinear structural and soil behavior in SSI studies. Among these studies, earthquae response of building-foundation sstems with elastoplastic soil behavior has been studied b Isenberg [10] and Minami [11]. Kobori et al [1] and Inoue et al [13] considered, in addition, elastoplastic structures, but restricted their attention to lateral displacements of the base mass. Veletsos and Verbic [14] studied the response of single-stor elastoplastic structures on an elastic halfspace to a simple pulse and found that the differences between the spectra obtained considering SSI and the associated nonlinear fixed-base sstems are significantl smaller than those for elastic structures, the differences decreasing with increasing ductilit ratio. In a stud of the stead-state response of a simple bilinear hsteretic structure supported on an elastic halfspace, Biela [15] found that, contrar to linear sstems, SSI can cause the resonant amplitude of the response of the hsteretic structure to increase over that for a rigid foundation. he same general behavior has been observed more recentl b Mlonais and Gazetas [18], and b Avilés and Pérez-Rocha [19]. he latter have proposed a simple fixed-base replacement oscillator approach using an effective ductilit, together with the nown effective period and damping of the sstem for the elastic condition, to account for the interaction effects of the coupled sstem. In this paper, we examine the response of single-stor inelastic structures, with a foundation embedded in an elastic halfspace, with a view towards (a) assessing the importance of SSI effects in ielding structures; (b) evaluating current seismic design provisions for taing these effects into consideration; and (c) developing an approach for the seismic design of SSI sstems and to calculate the corresponding nonlinear structural displacement. As seismic input we selected two ensembles of earthquae records, corresponding to two different geological settings: he first consists of 87 accelerograms recorded in California at sites ranging from hard to medium hard, with a few soft sites. he second set comprises 66 records

3 from the soft laebed of Mexico Cit. he records are normalized to have the same Arias intensit and also with respect to the mean pea ground acceleration, PGA, and have the average 5 percent damped elastic spectra presented in Fig.1. hese numbers of records and their normalization result in smooth average spectra. Inelastic spectra are even smoother (Jarernprasert et al [0]), enabling us to examine code specifications that prescribe design spectra and inelastic reduction factors that exhibit smooth variations with period. he source, geological settings and site conditions are reflected in the periods where the pea spectral ordinates occur at 0.3 s for the California records and at.0 s for the Mexican accelerograms. Before examining the behavior of inelastic structures on flexible foundations, in the following two sections we summarize two related cases that serve as the points of departure for our stud: 1) an inelastic structure on a fixed base, and ) an elastic structure with a flexible base (elastic SSI sstem).. PRIOR SUDY OF INELASIC SRUCURES ON A FIXED-BASE he single-stor structure supported on a fixed base shown in Fig., was examined previousl b the authors. he structure has an elastic period and non-degrading bilinear hsteretic behavior defined b a ield displacement, u, and slope of the second branch of the seleton force-displacement relationship assumed to be percent of the initial slope, also shown in Fig.. he seismic coefficient is defined as C = V /W, in which V = u, with = m (π/). is the stiffness of the structure, m its mass, V is the base shear, and W = mg, the weight of the structure. Jarernprasert et al [0, 1] found that the value of C that results in an average ductilit demand, µ, corresponding to the two set of earthquaes base motion can be closel estimated as n( ) C (, µ ) = C( ) / µ (1) he two quantities C and n depend onl on the elastic natural period,. C() differs somewhat from the elastic spectrum and can be interpreted as a pseudo-elastic design spectrum corresponding to µ = 1. he denominator n( ) R = µ constitutes a reduction factor that accounts for inelastic energ dissipation. Close forms for C and n for the two sets of earthquaes records considered herein are given in [0]. 3. SEISMIC RESPONSE OF ELASIC SSI SYSEMS 3

4 Procedures for taing SSI into consideration in the seismic design of buildings are based primaril on studies of the response of single-stor elastic models as the one illustrated in Fig., where the impedance of the foundation and surrounding soil is represented b translational, v, and rotational, φ, elastic springs and their associated viscous dampers. Following Jennings and Biela [5], the SSI fundamental period,, can be closel estimated as: h = 1+ + () v φ Supertildes will be used throughout to denote quantities associated with the SSI sstem. h is the effective height of the structure. Assuming that the fixed-base structure has 5 percent critical damping, the effective damping ratio, β, of the SSI sstem is (Jennings and Biela [5]): 3 β = β o (3) β o is the contribution to the effective damping ratio due to geometric scattering and intrinsic damping in the soil, which depends on and h/r. With reference to the bearing area of the foundation, r is the radius of the circle with the same area, for squatt buildings, or with the same moment of inertia about a centroidal axis perpendicular to the direction of the motion, for slender buildings. he second term is the modified structural damping due to SSI. he seismic design coefficient for the SSI sstem is obtained b entering into the fixed-base spectrum with. Due to the change in damping, this coefficient must be modified, because β in general differs from For instance, one ma use the recommendation of Arias and Husid [] b scaling the original spectrum b the factor (0.05/ β ) 0.4. Now, let the ratio be denoted b λ. hen: = λ (4) 3 = β λ o β In this stud, we tae λ to var between 1.1 and 1.5, and tae the slenderness ratio H/B equal to, where H is the total height of the structure, and B is the base dimension of the foundation, considered to be square. We have taen r = B/1.75, and h = 0.7H. o calculate v, φ, and the SSI damping ratios we use formulas b Gazetas [3] and NEHRP [3]. Details are provided in Appendix B. In addition, onl ground motion in one horizontal direction is investigated and we consider onl inertial interaction. As indicated b Avilés and Pérez-Rocha [4], inematic (5) 4

5 interaction is not as significant as inertial interaction for tpical building-foundation sstems and geological settings. Having completel specified the SSI sstem for a prescribed value of λ =, we could calculate its response for different periods and different value of λ, for the two sets of excitations. We interpret λ as a metric of the amount of interaction in the sstem. he mean elastic seismic coefficient, C e = V e / W, is shown in Fig. 3. V e is the mean base shear force of the elastic structure. he left panel of Fig. 3 corresponds to the California accelerograms and shows that, practicall for all periods, an increase in λ reduces the base shear with respect to the fixed-base case (λ = 1). In addition, the slight shift of the pea spectral value towards the left reflects the increase in the effective natural period of the SSI sstem. he right panel displas the average spectra obtained with the Mexican records, which show that increasing λ reduces the shear force for fixed-base periods greater than the dominant period of sec and for periods lower than 1 sec. For intermediate periods, the response shows a small increase. Figure 4 displas the same spectra as in Fig. 3 but with the SSI period,, in the abscissa. his displa shows clearl that the pea shear forces for different values of λ are all aligned and occur at a period equal to the dominant period of the seismic input. For all periods, smaller spectral values are obtained as the flexibilit of the soil increases, thus reflecting the effect of the SSI damping. 4. SEISMIC RESPONSE OF SSI SYSEMS WIH INELASIC SRUCURES With the previous preamble, we now proceed to stud again the earthquae response of the SSI sstems studied in the previous section, but considering now that the structure in Fig. exhibits the hsteretic behavior depicted in the same figure. he need for this stud is attested b the example presented in Appendix C which shows that the base flexibilit can increase the ductilit demand on the superstructure. he seismic coefficient is now defined as force. Values of C C = V /W, where V = u is the structure ield that lead to a prescribed average ductilit are determined as follows: Select λ and a target ductilit demand µ t ; For each fixed-based period,, and prescribed λ, define the properties of the elastic SSI sstem as described in Appendix B; 5

6 For a given set of records, find the minimum C for which the response of the superstructure remains elastic for all records; Progressivel reduce C and set the ield displacement as: u = V / = W C / = C g / (π) (6) For each C and each earthquae record, solve numericall the nonlinear differential equations of motion of the sstem. Obtain the maximum relative displacement of the structure, u max, and the resulting ductilit demand, µ = u max / u ; μ can be less than unit; Calculate the mean µ from the results for all the records in the set, assuming that µ has a lognormal distribution, as explained b Jarernprasert [1]; B interpolating between closel spaced values of C, determine C such that µ equals µ t. Whereas for an individual earthquae record the relationship between µ and C not necessaril unique (e.g., see Chopra, [5], p.56), we found a one-to-one correspondence between C and µ. In all SSI sstems we consider H/B =. Figure 5 shows C when µ t equals, for λ between 1 (fixed-base case) and 1.5, as a function of the SSI-period. For a given period, the seismic coefficient is C increases as the SSI effects, as measured b λ =, increase. his behavior, opposite to that exhibited b elastic SSI sstems, indicates that as the soil becomes softer, the structure deforms less and dissipates less energ than when the soil is stiffer. he additional energ dissipated in the soil is not sufficient to compensate for the reduced hsteretic energ in the superstructure. hus, there is a need for a higher C to maintain µ t = as λ increases. he spread of spectral values is most pronounced where the SSI period equals the dominant period of the input set of earthquae records (approximatel 0.3 sec for the California set and.0 sec for the Mexico Cit set). Also, the peas of the spectra shift slightl to the right for increasing λ, as a consequence of the lengthening of the effective natural period of the superstructure caused b its inelastic behavior. Figure 6 shows again C in terms of, but now for a target ductilit demand of 4. Again, C increases with λ, but this time the peas for SSI periods are significantl flatter, revealing that just as for fixed-base sstems, increasing inelastic behavior in the structure tends to eliminate the peas that characterize elastic spectra. 6

7 5. EVALUAION OF CURREN SEISMIC SSI PROVISIONS In contrast to the result in the preceding section, building codes generall allow (1) a reduction of the overall seismic coefficient on account of SSI, or () SSI effects to be ignored. In this section we examine these two courses of action on the ductilit demand. First, we stud the option of reducing the seismic coefficient for a set of fixed-based periods,, b means of the following steps for each : Select the ratio of SSI period to fixed-base period, λ = ductilit µ t ;, of the sstem, and a target Calculate = λ and the increased damping ratio of the equivalent linear oscillator, β, e.g., with (3); Reduce the elastic fixed-base seismic coefficient at the period. We use the prescriptions of ASCE 7-10 [1] which account for reductions due to elongated period and for the increased damping ratio β. o incorporate the nonlinear structural effect, calculate the fixed-base inelastic reduction n( ) factor, R = µ t which leads to a mean ductilit demand µ t in the fixed-based sstem; Appl R to reduce the elastic fixed-base seismic coefficient; Analze the SSI sstem, and calculate resulting the mean ductilit demand. In Fig. 7 we compare, for different values of λ, the target ductilit µ t, shown with dashed lines, against the actual calculated ductilit mean demands, µ, displaed as continuous lines. We find that these SSI provisions lead to excessive ductilit demands, especiall at lower periods, for all target ductilities. In addition, this figure shows that reducing the seismic coefficient for SSI, as presentl allowed in seismic provisions, can lead to excessive ductilit demand compared to the corresponding target ductilit. For both sets of earthquaes, the discrepanc increases as λ increases. Next we stud the option in which the effects of SSI are ignored in selecting the seismic coefficient, which is calculated directl using the original fixed-base period and structural damping of he reduction factor is also the same as for fixed-base, in our case, R = µ n() t. he corresponding mean ductilit demand is presented in Fig. 8 and indicates that ignoring SSI is beneficial for long-period structures, but detrimental for short-period sstems. Nonetheless, the departures from the target ductilit ratios are much smaller than in Fig. 7, showing that between 7

8 the two choices contemplated in current codes, it seems preferable for nonlinear structures to ignore SSI than to allow a reduced base shear. 6. SEISMIC COEFFICIEN OF SSI SYSEMS WIH INELASIC SRUCURES In this section we devise a procedure to calculate the inelastic seismic coefficient for the SSI sstems under consideration that maintains the target ductilit demand as the e parameter, b extending the methodolog developed b Jarernprasert et al [1] for inelastic fixed-base structures. B maintaining the ductilit as the defining parameter, we emphasize that, even if the reduction factor introduced to consider structural inelastic behavior changes on account of SSI, the structural design and detailing requirements, controlled b the ductilit demand, remain the same as in the original fixed-base sstem. 6.1 Seismic Coefficient While calculating the mean ductilit demand, µ, we observed that the logarithm of µ varies linearl with the logarithm of the seismic design coefficient, instance, Fig. 9 shows log µ vs. log C, for a wide range of µ. For C for λ = 1.3 and several values of for the California and the Mexico records. Our approach for estimating the seismic coefficient stems from this observation. From Fig. 9 and similar ones for different values of λ one can write: or log C = log C n log µ ; 1.5 µ 6.0 (7) n (, ) C (, µ, λ) C(, λ) µ λ = ; 1.5 µ 6.0 (8) C (, λ) is the intersection of each straight-line approximation for C vs. µ with the horizontal axis µ = 1 and n (, λ ) is its negative slope. Since and are related through λ, we can express C and as functions of rather than. C (, λ) and n (, λ) are determined b regression of C on µ for different values of, similar to those shown in Fig. 9. It should be emphasized that C (, λ) is not the mean elastic spectrum corresponding to the SSI period = λ. Both C and were obtained b regression on calculated inelastic results for C and µ. C (, λ) has been plotted in Fig. 10 as a function of the SSI natural period for different values of λ, for both sets of earthquaes. his figure shows that C is practicall independent of 8

9 λ. hus, we can approximate C (, λ) b C( ), the unreduced seismic coefficient for fixed-base sstem (λ = 1), i.e.: C (, λ) C( ) (9) he exponent (, λ) in (8), and similar ones for other values of λ, are plotted in Fig. 11, which shows that is a decreasing function of λ, and vanishes for = 0. his implies that C (0, µ, λ) = C(0, λ), and since C(0, λ) = PGA /g, the condition that a C (0, µ, λ) must be equal to PGA /g is satisfied. Since n (, λ) decreases for increasing λ, (8) implies that for a prescribed µ, C also increases with λ, i.e., C is larger for softer than for stiffer soils. o assess the effects of and λ on separatel, it is useful to express the exponent n (, λ) as the product: n (, λ) = n( ) α(λ) (10) in which n( ) is the limit of n (, λ) for a fixed-base sstem (λ = 1), evaluated at the SSI period. he factor α(λ) accounts for the effect of λ. We use the results of Fig. 11 to obtain α(λ), b dividing n (, λ) b n (, λ = 1). Observing that this ratio varies weal with, α(λ) can be approximated as: n ( ) ( λ) α λ, = n, λ = 1 ( ) λ 1, 1 λ 1.5, (11) he actual ratio α(λ) and its approximation 1/λ are shown in Fig. 1. Almost everwhere the latter is smaller than the former. Since a smaller α(λ) ields a higher seismic coefficient, 1/λ provides a conservative value of the seismic coefficient. With the approximations (9), (10) and (11), the expression (8) for C becomes: (, µ, λ) C( ) (, µ, λ) C = (1) R where 1 ( n λ R, µ, λ) = µ (13) Substituting from (1), this expression ma be also written as: 9

10 (, µ, λ) (, µ ) C C = (14) λ 1 n λ µ his expression indicates that in order to obtain the seismic coefficient of the SSI sstem with a natural period and a period elongation ratio λ it suffices to divide the seismic coefficient of a rigid-base structure with the natural period and a mean ductilit demand µ b the factor µ n 1 λ λ. o assess the accurac of (14) for determining the seismic coefficient, with a target ductilit demand µ t, we first obtained the exact value of C C, of the SSI sstem from regression of nonlinear analses results obtained directl from the solution of the governing equations of the sstem. hen we applied (14) to estimate C. Results of the two approaches for different values of λ and µ t are plotted in Fig. 13. he difference between the two sets of curves is small. his means that (14) provides a satisfactor estimate of records, for all values of the target ductilit, and for all ratios C for the two ensembles of earthquae. It is of considerable practical interest that (14) holds for the two ver different sets of earthquae records considered in this stud. his indicates that a simple rule of the tpe embodied in (11) might be applicable to other seismic regions. As a further test of the accurac of (14), Fig. 14 shows the actual mean ductilit demand µ of SSI sstems whose strength is evaluated from this expression, for a prescribed target ductilit μ t. he values of µ are in close agreement with μ t, except for long period sstems and λ = 1.5 in Mexico Cit. In practice, such a large interaction ratio is unliel for long period structures, because the are built on pile foundations, which tend to increase the relative stiffness between the foundation and the structure. 6. Structural Displacements Having established a simple procedure for determining the SSI seismic design coefficient and thereb the ield displacement via (6), the maximum mean relative displacement of the inelastic superstructure, u max, can be approximated consistentl within this approach as the ield displacement times the mean ductilit demand µ. hat is: C, 10

11 u max C µ g = u µ = (15) ( π ) With umax nown, the mean total displacement, u total, of the mass of the structure with respect to the free-field motion of the base can be estimated approximatel as: u = u max + u hφ (16) total x + ux is the mean relative horizontal displacement of the foundation with respect to the free-field displacement and φ is the mean angle of rocing of the foundation. Using the assumption that the foundation remains elastic, with horizontal and rocing stiffness x and φ, ux and φ are: u x V = (17) x h V hφ = (18) φ Substituting (13), (15) and (16) into (14), the total displacement becomes: V V h V h V u total = µ t + + = µ t + + (19) x x φ φ Now, because V / = u, the above equation can be rewritten as: ( ) h u = u µ total t 1 u 1 (0) x φ From (), the terms inside the second parenthesis are equal to the square of λ; thus: ( + ) u = λ 1 (1) total u µ t Equation (1) allows one to compare the relative effect of the target ductilit to that of the flexibilit of the soil on the total displacement of the structure. he ratio of the approximate mean total relative displacement evaluated with (1) to the exact value obtained b solving the nonlinear equations of motion has been calculated for different values of λ and μ t. he results presented in Fig. 15, indicate that the maximum difference between the approximate and exact solutions is 10 percent and occurs for λ = 1.5 and µ t = 5, for the Mexican records. For the California records the discrepanc is less than 5 percent for an combination of λ and μ t. 7. CONCLUDING REMARKS 11

12 In this stud of the SSI response of inelastic structures to two ver different sets of earthquae records, we have first confirmed that for the simple linear sstems considered in this stud, SSI mostl leads to a reduction of the mean response with respect to that of the corresponding fixed-based structures. B contrast, a bilinear hsteretic structural behavior results in an increase of the mean ductilit demand with respect to that of the corresponding fixed-base structure if the natural period of the sstem is smaller than the dominant period of the excitation at the site, and in a decrease otherwise. Based on these observations, we developed a simple procedure for estimating the SSI seismic coefficient, C, such that the ductilit demand of the structure remains close to the target ductilit. In particular we derived an expression for C as the product of the seismic coefficient for the structure on a fixed-base structure divided b a factor that incorporates the SSI effects. his result suggests that the seismic design coefficient can be determined from expression (14). One onl needs to replace the mean ductilit demand µ b the target ductilit, µ t. In practice, building codes stipulate a reduction factor R associated with the inelastic behavior of the fixed-base structure. o incorporate SSI with our approach one would use a smaller reduction n( )( λ 1)/ λ equal to factor R (, λ, µ ) t R µ. t he proposed approach can be used for rapidl assessing the importance of SSI effects on the dnamic behavior of the building-foundation sstem, b comparing the seismic response coefficient or, perhaps better et, the resulting drift or pea structural displacement, with the corresponding fixed-base quantities. he results presented in this paper correspond to a structural aspect ratio, H/B = ; we have also analzed SSI sstems with H/B = 4, obtaining ver similar qualitative results. he impact of H/B is properl considered in the calculation of period elongation ratio λ. he proposed approach has been developed for a particular class of SSI sstems; while it is reasonable to expect that the e equation (10) will appl to other foundation conditions, e.g., piles, and soil stratigraph, the procedure should be additionall verified before appling it to sstems whose structural behavior differs widel from the bilinear hsteretic considered here. It is also well to emphasize that onl inertial interaction has been considered in this stud. Kinematic interaction should be included if the dominant length of the incident waves is of the same order as the base (or depth) dimension of the foundation. ACKNOWLEDGMENS 1

13 his research was partiall supported b the National Science Foundation Division of Engineering Education and Centers under grant he authors are grateful for this support. We also than the reviewers for their constructive comments. he were ver helpful in revising the paper. REFERENCES [1] ASCE, ASCE 7-10 Minimum Design Loads for Buildings and Other Structures, American Societ of Civil Engineers, 010. [] NDF, Normas écnicas Complementarias para Diseño por Sismo, Gaceta Oficial del Distrito Federal, Mexico Cit, Mexico, 004. [3] NEHRP, NEHRP Recommended Provisions for Seismic Regulations for New Buildings and Other Structures. Part 1 Provisions 009 Edition; FEMA 750 CD. [4] Stewart JP, Kim S, Biela J, Dobr R, and Power MS. Revisions to Soil-Structure Interaction Procedures in NEHRP Design Provisions. Earthquae Spectra 003; 19(3): [5] Biela J. Earthquae response of building-foundation sstems. echnical Report: CaltechEERL:1971.EERL-71-04, California Institute of echnolog 1971; [6] Jennings, PC, and Biela, J. Dnamics of Building-Soil Interaction. Bull. Seism. Soc. Am.1973; 63 (1): [7] Veletsos AS and Mee JW. Dnamic behavior of building-foundation sstems Earthquae Engineering and Structural Dnamics 1974; 3(): [8] Luco JE. Linear Soil Structure Interaction, in Seismic Safet Margins Research Program (Phase I), U.S. Nuclear Regulator Commission, Washington D.C., [9] Roesset JM. A review of soil-structure interaction, in Soil-structure interaction: he status of current analsis methods and research, J. J. Johnson, ed., Rpt. No. NUREG/CR-1780 and UCRL-53011, U.S. Nuclear Regulator Com., Washington DC, and Lawrence Lab., Livermore, CA., [10] Isenberg J. Interaction between soil and nuclear reactor foundation during earthquaes. USAEC Report, ORO-38-5, Agbabian-Jacobsen Associates, Los Angeles, California [11] Minami. Elastic-plastic earthquae response of soil-building sstems. Proc. Fifth World Conference Earthquae Engineering, Rome, Ital 1973; [1] Kobori, Minai R, and Inoue Y. On earthquae response of elasto-plastic structure considering ground characteristics. Proc. Fourth World Conference Earthquae Engineering, 3, Santiago, Chile, 1969: [13] Inoue Y, Kawano M, and Maeda Y. Dnamic response of nonlinear soil-structure sstems. echnolog Reports of the Osaa Universit1974; 4: [14] Veletsos AS and Verbic B. Dnamics of elastic and ielding structure-foundation sstems. Proc. Fifth World Conference Earthquae Engineering, Rome, Ital 1973; [15] Biela J. Dnamic Response of Non-Linear Building-Foundation Sstems. Earthquae Engineering and Structural Dnamics 1978; 6, [16] Bazán-Zurita E, Díaz-Molina I, Biela J, and Bazán-Arias NC. Probabilistic seismic response of inelastic building foundation sstems. Proc. 10 th World Conf. on Earthq. Engineering, Madrid, Spain199; [17] Rodríguez M and Montes R. Seismic response and damage analsis of buildings supported on flexible soils, Earthq Engineering & Struct. Dn 000; 9:

14 [18] Mlonais G. and Gazetas G. Seismic Soil-Structure Interaction: Beneficial or Detrimental? Journal of Earthquae Engineering 000; 4(3): [19] Avilés J and Pérez-Rocha LE. Soil-structure interaction in ielding sstems. Earthquae Engineering and Structural Dnamics 003; 3: [0] Jarernprasert S, Bazan E, and Biela J. An Inelastic-Based Approach for Seismic Design Spectra. Journal of Structural engineering 006; 13;, [1] Jarernprasert S. An inelastic design approach for asmmetric structure-foundation sstems. Ph.D. Dissertation, Carnegie Mellon Universit, Pittsburgh, PA, 005. [] Arias A and Husid R. Influencia del amortiguamiento sobre la respuesta de estructuras sometidas a temblor. (In Spanish), Rev. IDIEM 196; Vol 1: [3] Gazetas G. (1991). Foundation Vibration. Foundation Engineering Handboo. H.-Y. Fang. Van Nostrand Reinhold, New Yor: Chapter 15. [4] Avilés J and Pérez-Rocha LE Evaluation of interaction effects on the sstem period and the sstem damping due to foundation embedment and laer depth. Soil Dnamics and Earthquae Engineering 1996; 15(1): [5] Chopra AK. Dnamics of Structures, heor and Applications to Earthquae Engineering, Prentice Hall, Upper Saddle River, New Jerse,

15 Smbol APPENDIX A Description B C C C e C v C φ C C D g H h v φ m m 0 n n Base dimension of the foundation Pseudo-elastic spectrum of fixed-base sstem Pseudo-elastic spectrum of SSI sstem Average Elastic spectrum of fixed-base sstem Foundation horizontal viscous damping Foundation rocing viscous damping Inelastic spectrum of SSI sstem Inelastic spectrum of fixed-base sstem Soil hsteretic damping Gravitational acceleration Actual height of super structure Equivalent height of fixed-base sstem ranslational stiffness of fixed-base sstem ranslational stiffness of foundation Rocing stiffness of foundation Mass of super structure Mass of foundation Inelastic modification factor of fixed-base sstem Inelastic modification factor of SSI sstem PGA Average pea ground acceleration r Equivalent radius of base dimension of the foundation R Fixed-base inelastic reduction factor R SSI inelastic reduction factor SSI period Fixed-base period u max Maximum relative displacement u max Average maximum relative displacement u Average total displacement u u u total x V s V e V V W α β β ο Average translational displacement of foundation Fixed-base ielding displacement SSI ielding displacement Soil shear wave velocit Average fixed-base elastic base shear Fixed-base ield base shear SSI ield base shear λ µ Ductilit demand μ t arget ductilit Weight of super structure Exponent of SSI inelastic modification factor otal SSI effective damping ratio SSI effective damping ratio from foundation interaction Ratio of SSI period respect to Fixed-base period 15

16 µ Average ductilit demand φ Average rotation of foundation 16

17 APPENDIX B he horizontal and rocing stiffness coefficients in (1) and the corresponding damping coefficients of the equations of motion of the SSI sstem in Fig. 1 are calculated based on the following assumptions: (1) the effective height and mass of the single stor model are 70 percent of those of the superstructure; () the mass of the foundation is 0 percent of the mass of the superstructure; (3) the rocing moment of inertia of the foundation is equal to 5 percent of mh ; (4) the densit of concrete is 400 g/m 3 and that of the soil 1900 g/m 3, the Poisson s ratio of the soil is 0.4; (5) the thicness of the slabs in the superstructure is 0. m; (6) the number of stories is equal to 10 times the fixed base period in seconds; (7) each stor is 3-m high; (8) the weight of beams and columns is the same as that of the floor slab; (9) the foundation is full embedded and the embedment depth is 0 percent of the superstructure s height; (10) the floors and the foundation are square in plan. Details of the derivations ma be found in Jarernprasert [1] and lead to the following quantities: Relative translational stiffness, / v: v H = V B s V s is the shear wave velocit, in m/s, of the elastic halfspace. Relative rocing stiffness, h / φ : H B 1 (A1) h φ 7000 = V s H B H B 1 B substituting (A1) and (A) into (1), λ = can be written as: (A) 1 H H B λ = B H (A3) V s H B B Conversel, solving this equation for V s, we have: H 0.35 V = B λ 1 H H H (A4) s ( ) ( ) B B B Now, b substituting (A4) into (A1) and into (A), / v and h / φ can be expressed solel in terms of the parameters λ and H/B. 17

18 18 Damping β 0 in () can be related to the coefficients c v and c φ of the two viscous dampers used in the SSI sstem shown in Fig. as follows: + = φ φ π β h h C C v v 3 0 (A5) Using (A5) and the formulas in Gazetas (1991), we obtain: D h C v v v + + = 3 0 π α π β φ (A6) D h h h C v + + = 3 0 π ψ π β φ φ φ (A7) D is the fraction of linear hsteretic damping in the soil, taen to be 0.05, and ψ is given b: = B H V x s ψ (A8) Soil Dnamic and Earthquae Engineering Journal

19 APPENDIX C As an example of the individual calculations performed as the basic step for evaluating the average ductilit demand, µ, in this Appendix we describe the seismic response of a specific inelastic structure excited b the SC record from the Mexican database. he selected structure has a fixed-base period = 1.0 sec, 5% critical damping, and H/B =. he properties of the SSI sstem were defined b the procedure described in Appendix B to ield an SSI period of 1.3 sec. Both sstems have a seismic coefficient of Figure C1 shows the strong segments of the relative displacements normalized b the ield displacement of the structure, u (the same for both the fixed-base and the SSI cases.) he properties of the selected sstem and the pea values of the normalized relative displacement with and without interaction are summarized in able C1. hese pea values are the ductilit demands. able C1. Comparison of dnamic properties and response of a selected case Propert Fixed-base SSI sstem structure Natural period (s) Period elongation ratio, λ= / Τ Effective damping (%) 5 Code formula Pea relative displacement normalized with respect to ield displacement (also, ductilit demand) For this example, the ductilit demand when SSI is considered is 6 percent larger than when SSI is disregarded. B counting the response peas in Fig. C1, it is apparent that the displacement of the SSI sstem (λ = 1.3) exhibits fewer ccles than those of the fixed base structure because the elongated natural period,, dominates the seismic response. 19

20 Fig. 1 Mean elastic spectrum with 5% damping ratio and inelastic coefficients C and n for accelerograms from California (left) and from Mexico Cit (right) Fig. Fixed-base and SSI sstems: (a) Fixed-base sstem (b) Soil structure interaction sstem (c) Hsteretic behavior 0

21 Fig. 3 Mean response spectra of elastic SSI sstems for different λ =, in terms of the fixedbase period, for California (left) and Mexico Cit (right) Fig. 4 Mean response spectra of elastic SSI sstems for different λ =, in terms of the SSI period,, for California (left) and Mexico Cit (right) 1

22 Fig. 5 Inelastic mean seismic coefficient for different λ=, for a target ductilit µ t =, in terms of SSI period,, for California (left) and Mexico Cit (right) Fig. 6 Inelastic mean seismic coefficient for different λ=, for a target ductilit demand µ t = 4, as a function of SSI period, for California (left) and Mexico Cit (right)

23 Fig. 7 Comparison of target ductilities (dotted lines) with mean ductilit demands of SSI sstems designed with current code SSI provisions (solid lines), for λ = = 1.1, 1.3 and 1.5, for California (left) and Mexican records (right) 3

24 Fig. 8 Comparison of target ductilit (dotted lines) with mean ductilit demands for SSI sstems designed ignoring SSI (solid lines) for λ = = 1.1, 1.3, and 1.5, for California (left) and Mexican records (right) 4

25 Fig. 9 Variabilit of mean ductilit demand µ with ield strength C for sstems with λ = = 1.3 and prescribed fixed-base elastic periods,, for California (left) and Mexico Cit (right) Fig. 10 Unreduced Inelastic Spectra C (, λ) for different λ=, determined b regression for California (left) and Mexico Cit (right) 5

26 Fig. 11 Exponent n (, λ) for different λ=, determined with regression for California (left) and Mexico Cit (right) Fig. 1 Spectra of exact ratio z n (, λ) / n (, λ = 1) and approximated value α(λ)=1/λ 6

27 Fig. 13 Comparison of exact and approximate seismic coefficients of SSI inelastic sstems with λ = = 1.1 to 1.5 from top to bottom, respectivel, for California (left) and Mexico Cit (right) 7

28 Fig. 14 Mean response ductilit demand of inelastic SSI structures designed via the proposed SSI method (solid line) with λ = = 1.1, 1.3 and 1.5 from top to bottom, respectivel, for Californian (left) and Mexican records (right) 8

29 Fig. 15 Ratio of mean maximum total displacement of SSI sstems from (1) to the exact value from numerical integration, for Californian (left) and Mexican (right) records, for target ductilit µ t between and 5. Fig. C1 Response of a selected structure to the Mexican SC record 9