SOIL NON-LINEAR BEHAVIOR AND HYSTERETIC DAMPING IN THE SPRING-DASHPOT ANALOG

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1 SOIL NON-LINEAR BEHAVIOR AND HYSTERETIC DAMPING IN THE SPRING-DASHPOT ANALOG Nikolaos OROLOGOPOULOS 1 and Dimitrios LOUKIDIS 2 ABSTRACT This paper presents reslts from nmerical simlations of footing vibration examining the effect of the variation of soil shear stiffness and hysteretic damping ratio with shear strain amplitde on the vales of the spring and dashpot coefficients of the Lysmer s analog. Following validation of the nmerical methodology against existing semi-analytical soltions fond in the literatre, a series of parametric finite element analyses are performed for vertical oscillation, horizontal oscillation and rocking of strip footings and vertical oscillation of circlar footings, resting on a the free srface of a homogeneos half-space. In validation process, the soil is assmed to be a linearly elastic material, while, in the sbseqent parametric stdy, the soil is modeled as a non-linear material following a simple hyperbolic stress-stain law with hysteresis. The stdy aims at establishing relationships for the determination of the spring stiffness and the dashpot coefficients for footings on fine grained soils as a fnction of the normalized fondation motion amplitde. INTRODUCTION The role of the fondation soil in seismic response analysis of strctres is traditionally represented by simple spring and dashpot models. In most cases of soil-strctre interaction simlations, the fondation soil is assmed to have constant shear stiffness (shear modls G) and constant hysteretic damping ratio h. Nmeros research stdies have prodced formlas and charts for the determination of the spring coefficient K and dashpot coefficient C for all possible modes of oscillation of shallow fondations, as fnctions of the footing size and geometry, soil elastic parameters, shear wave velocity V s and oscillation anglar freqency ω (e.g., Lysmer 1965; Karashdhi et al. 1968; Lco and Westmann 1971; Veletsos and Wei 1971; Veletsos and Verbič 1973; Gazetas and Roesset 1976, 1979; Gazetas 1983; Dobry and Gazetas 1986). The spring coefficient is sally expressed as the prodct of the static stiffness K stat and a dynamic stiffness factor k that is a fnction of ω. The dashpot in sch models represents mainly the radiation damping (i.e., the energy loss de to the emission of mechanical waves to the elastic half-space). The energy consmed at a material point, namely the hysteretic damping, is taken into accont by applying the principle of correspondence as follows (Dobry and Gazetas 1986): ( ) C (1) h h C( ) 2 h h C (2) 1 Research Assistant, University of Cyprs, Nicosia, Cyprs, norologopolos@gmail.com 2 Assistant Professor, University of Cyprs, Nicosia, Cyprs, lokidis@cy.ac.cy 1

2 absorbent bondary absorbent bondary It is well known that the actal soil behavior is strongly non-linear, reslting in gradal redction of secant shear modls and increase of hysteretic damping ratio as the amplitde of shear strain increases (Vcetic and Dobry 1991). Hence, it is expected that K wold decrease and C wold increase with increasing amplitde of fondation displacement (or rotation for the rocking mode). This fact is crrently neglected in the standard spring-dashpot models that represent the fondation soil in soil-strctre interaction analyses in engineering practice. Nonetheless, several research efforts have focsed on introdcing soil non-linearity in soil-strctre interaction analysis throgh macro-element modeling (e.g., Paolcci 1997; Cremer et al. 21; Holsby et al. 25; Chatzigogos et al. 29). Moreover, significant attention has been drawn recently to the potential benefits to the strctral seismic response coming from the development of inelastic soil deformation nder the fondations and from the corresponding energy dissipation (e.g., Mergos and Kawashima 25; Anastasopolos et al. 21; Korkolis et al. 212; Zafeirakos and Gerolymos 213). Table 1. Fondation geometries and vibration modes considered in the finite element analyses. y B y B B B x y - y strip footing vertical oscillation (plane strain) strip footing horizontal oscillation (plane strain) strip footing - rocking (plane strain) circlar footing - vertical oscillation (axisymmetric) footing harmonic force ( in forced vibration analysis) or initial static force (in free vibration analysis) soil absorbent bondary Figre 1. Typical finite element mesh and bondary conditions sed in analyses of strip footing excited in rocking. 2

3 N. Orologopolos and D. Lokidis 3 This paper investigates the effect of the variation of soil stiffness and hysteretic damping ratio, as a fnction of the motion amplitde, on the vales of the spring and dashpot coefficients for shallow fondation resting on the free srface of a homogeneos half-space. For this prpose, series of parametric analyses were performed sing the finite element code PLAXIS 2D (Brinkgreve et al. 211) for vertical, horizontal and rocking oscillations of strip footings and vertical oscillation of circlar footings (Table 1), in which the soil mechanical behavior is represented by a hyperbolic stress-strain law that predicts hysteresis. The asymptotic strength of the hyperbolic law is assmed to be independent of the mean effective stress. Hence, the analysis reslts are applicable to fine grained soils (e.g., clays, silts) nder ndrained conditions. In order to ensre that the employed nmerical methodology is capable of accrately simlating the problem at hand, an initial set of simlations was performed considering the grond as a linearly elastic medim with constant hysteretic damping ratio and the reslts were compared with existing semi-analytical soltions (e.g. Dobry and Gazetas 1986). VALIDATION OF FINITE ELEMENT APPROACH Simlations are first performed assming that the soil is a linearly elastic medim in order to validate the finite element methodology. Finite element reslts are compared with the predictions of the springdashpot model for which K and C were calclated sing the formlas and charts by Gazetas (1983) and Dobry & Gazetas (1986), which are based on semi-analytical soltions and, ths, can be considered rigoros. A large nmber of analyses were done for different vales of Poisson s ratio, oscillation freqency and hysteretic damping ratio. The hysteretic damping of the soil was introdced in the analyses throgh Rayleigh damping: C MK (3) Rayleigh where M and K are the global mass and stiffness matrices of the part of the finite element model occpied by the soil (no material damping is assigned to the footing). The parameters and are sally set to vales eqal to target and target /, respectively, where target is the desired material damping ratio h at the predominant oscillation freqency of the system. These and vales essentially divide the total contribtion to the material damping into two eqal parts, one pertaining to the mass (mass proportional Rayleigh damping) and the other pertaining to the stiffness (stiffness proportional Rayleigh damping). However, this approach is not sitable for problems involving wave propagation in a continm, as it will be shown in the following paragraphs. A typical finite element mesh is shown in Fig. 1, consisting of 15-noded trianglar elements. Absorbent bondary conditions are assigned to the bottom and lateral bondaries in order to diminish reflection of the waves emitted by the fondation and achieve half-space consistent radiation damping as mch as possible. The footing has a very high Yong s modls (practically rigid) and is flly attached to the grond, i.e. no interface elements are placed between soil and footing. Two sets of simlations were performed: a) forced vibration analysis and b) free vibration analysis. In forced vibration analysis, the footing is excited by a harmonic (sinsoidal) force time history of constant amplitde and freqency. The force is applied at the center of the footing in the case of vertical or horizontal excitation, while a pair of vertical forces of opposite direction are applied at the edges of the strip footing in the case of rocking in order to generate moment loading (Fig. 1). In horizontal oscillation analyses, the footing is prevented from moving vertically or rotating. Accordingly, in rocking analyses, the center of the footing is prevented from moving vertically or horizontally. These footing bondary conditions were applied in order to establish pre horizontal motion and pre rocking, which wold otherwise be impossible de to the well known copling between these two modes of oscillation. In free vibration analysis, the footing is first loaded statically, and, sbseqently, the loading is released (set to zero) instantaneosly in order to allow the footing to vibrate freely. The mass of the footing was selected sch that the reslting motion freqency is in the range of 3Hz to 12Hz. A typical response obtained from free vertical vibration analysis of a circlar footing on soil with large hysteretic damping is shown in Fig. 2. It was observed that the free vibration analyses give a more

4 vertical displacement y (m) clear pictre than forced vibration analyses with respect to the damping in the soil-fondation system and allow direct comparisons with semi-analytical soltions. The static part of these analyses was helpfl also in making comparisons regarding the static stiffness of the system and in deciding the size of the analysis domain. This was particlarly important for vertically or horizontally loaded strip footings becase, in these cases, the footing displacement is sensitive to the distance of the bondaries from the footing. The finite element reslts compare well with spring-dashpot analog predictions based on Gazetas (1983), Dobry and Gazetas (1986) and the principle of correspondence, with differences in the amplitde of forced vibrations not exceeding 5%. The errors are even smaller in analyses with ξ h =%, indicating that the finite element modeling in terms of absorbent bondaries and size of analysis domain can adeqately simlate the correct radiation damping. However, the good agreement between the finite element method (FEM) and the spring-dashpot analog based on rigoros semi-analytical soltions is achieved on the condition that no mass proportional damping is sed and the entire material damping comes from the stiffness proportional term by setting = and =2 h / in Eq. (3). In fact, setting the Rayleigh damping to be both mass and stiffness proportional, as sally done in strctral engineering practice, leads to a severe nderestimation of the hysteretic damping of the soil, as shown in the example of Fig.2. Rayleigh damping has been originally proposed and is sitable for single degree of freedom systems. The fact that the se of mass proportional damping in time domain analysis of mlti-degree of freedom systems is problematic has been highlighted by Hall (26). Mass proportional damping generates viscos forces (which consme energy) that are proportional to the absolte velocity of each individal node of the system, as if the material point moves inside a viscos flid exerting drag forces. On the contrary, stiffness proportional damping generates damping forces that are proportional to the relative motion (relative velocity) of neighboring nodes and is, ths, related to shear straining. Hence, the stiffness proportional Rayleigh damping is closer to the physics of hysteretic damping in problems involving wave propagation in a continm. In sch problems, in-phase single block motion of the entire system is minimal and relative-differential motions dominate, and, as a reslt, the mass proportional component of the Rayleigh damping is ndermobilized. As a conseqence, if Eq. (3) is sed with = h and = h /, the overall hysteretic damping will be nderestimated, as shown in Fig. 2. The excellent agreement between FEM (with prely stiffness proportional Rayleigh damping) and the spring-dashpot analog is observed in all examined modes of vibration, independently of soil elastic properties, motion freqency and ξ h vale. So, it can be said that the principle of correspondence (Eq. 1 and 2) applies flawlessly to the investigated bondary vale problems. 1.5E-4 1.E-4 5.E-5.E E-5-1.E-4-1.5E-4-2.E-4-2.5E-4 footing radis B =1m G=192kPa v=.3 ρ=2t/m 3 h =3% f 7.3Hz spring-dashpot model FEM withot mass proportional damping FEM with mass proportional damping -3.E-4 t (sec) Figre 2. Comparison of the displacement time history of free vertically oscillating circlar footing predicted by the spring-dashpot model and finite element analyses with and withot mass proportional Rayleigh damping. 4

5 τ xy (kpa) N. Orologopolos and D. Lokidis 5 SOIL NON-LINEAR BEHAVIOR The constittive model sed in the non-linear finite element analyses is the model available in PLAXIS 2D called Hardening Soil model with small strain stiffness (HSsmall). This model combines a shear yield srface (and a hardening cap) with pre-yield non-linear elastic behavior, ths being able to take into accont the redction of stiffness with shear strain and also predict hysteretic behavior. More specifically, before yielding, the material follows the hyperbolic law presented by Dncan and Chang (197), as modified by dos Santos and Correia (21). The modified expression of dos Santos and Correia (21) for the secant shear modls G has the following form: G G max (4) where γ.7 is the shear strain at which the secant shear modls G is redced to 72.2% of its initial (maximm) vale G max. Dring nloading and reloading, the pre-yield formlation of HSsmall obeys the first and second Masing rles (Kramer 1996). In this stdy, or intention is to model soil behavior sing prely the aforementioned hyperbolic law. In order to ensre that the plastic components of the constittive model (yield srfaces, flow rle, e.t.c.) play no role on the material response, the material cohesion was set to a vale twice the strength asymptote predicted by the hyperbolic law. Moreover, the soil friction angle is set eqal to zero. As sch, the FEM simlation reslts are meant to be valid for satrated fine grained soils (e.g., clays and silts). Fig.3 shows an example of the hysteresis loop predicted by the constittive model in simple shear γ xy (%) Figre 3. Hysteresis loop of soil material nder simple shear loading as predicted by HSsmall. The parameter γ.7 controls the crvatre of the shear stress-strain response (backbone crve). This allows to fit, to the extent possible, the G/G max vs. γ and h vs. γ crves predicted by the HSmall model to the experimental crves of Vcetic and Dobry (1991) for varios vales of the soil plasticity index PI. Three different PI vales are considered in this stdy, PI=5, 2 and 4, with the corresponding γ.7 vales that fit the Vcetic and Dobry (1991) crves being.25,.5 and.8, respectively. The hyperbolic law of Eq. (4) yields zero hysteretic damping for very small shear strain amplitdes. On the other hand, experimental stdies show that there is a minimm non-zero

6 h (%) h (%) h (%) (a) Plaxis.7 =.25 Vcetic & Dobry (1991) (b) G/G max PI= PI= Plaxis.7 =.25 Vcetic & Dobry (1991) (%) (%) Figre 4. Comparison of experimental crves and HSsmall model predictions for PI=5. G/G max PI=2 Plaxis.7 =.5 Vcetic and Dobry (1991) (a) Plaxis.7 =.5 Vcetic & Dobry (1991) PI=2 (b) (%) (%) Figre 5. Comparison of experimental crves and HSsmall model predictions for PI=2. G/G max PI=4 (a) Plaxis.7 =.8 Vcetic and Dobry (1991) PI=4 (b) Plaxis.7 =.8 Vcetic & Dobry (1991) (%) (%) Figre 6. Comparison of experimental crves and HSsmall model predictions for PI=4. 6

7 N. Orologopolos and D. Lokidis 7 vale for h even at practically zero shear strain. Hence, along with HSsmall, a small amont of Rayleigh damping (stiffness proportional) was assigned to the soil in order to achieve a minimm h of 1%. Figs. 4 throgh 6 compare HSsmall predictions in simple shear at material point level against the experimental observations (Vcetic and Dobry 1991). Effort was made to have the closest possible agreement in terms of both G/G max vs. γ and h vs. γ crves. As a reslt, the G/G max redction is generally nder-predicted at the small strain range in order to not excessively over-predict h. Yet, the over-prediction of h at γ>.1% is inevitably large. All non-linear analyses were performed for Poisson's ratio v=.495, assming that the fine grained soil is flly satrated by capillary sction or by being nder the water table. NON-LINEAR ANALYSIS The non-linear finite element analyses focs mainly on the effects of fondation size (half-width B ranging from.5m to 4m), G max (5, 1 and 2MPa), PI (5, 2 and 4) and oscillation freqency f (in the range 3Hz to 12Hz). Given the vales assmed for the parameter γ.7 of the hyperbolic law, the (asymptotic) ndrained shear strength s of the soil ranges from 32kPa to 416kPa, and the corresponding G max /s ratio from 48 to 154. First, static parametric analyses were performed, in which the footing was loaded statically in order to obtain the redction of the eqivalent secant spring coefficient K eq and establish crves of K eq /K max as a fnction of the normalized fondation displacement amplitde /2B (B: footing half-width). K max is the spring coefficient corresponding to the maximm shear modls G max. To determine the eqivalent hysteretic damping ratio h,eq of the soil-fondation system, free vibration analyses were performed for varios levels of initial load amplitde. For each analysis, the total damping ratio tot of the system (radiation + hysteretic) was first extracted from the displacement time history (Fig. 7) sing the well known logarithmic decrement method: tot,(n) = ln / n n+1 2 ln n / n (5) 1.5 n n Figre 7. Determination of the damping ratio of a system from free vibration response. The representative displacement amplitde corresponding to tot,(n) ( tot vale calclated for the n th cycle) is assmed to be eqal to (n) =( n + n+1 )/2. Given Eq. (1), Eq. (2) and that tot.5 C( h ) / K( h), the eqivalent hysteretic damping ratio of the soil-fondation system can be determined from the following eqation:

8 h,eq (%) h,eq (%) load amplitde initial load amplitde initial h,eq eq.5 tot = K k C K k C eq tot (6) with C and k obtained from Dobry and Gazetas (1986) and the K eq vale corresponding to (n). It was observed that the h,eq yielded by Eq. (6) was very close to the vale obtained if we assme simple sperposition of radiation and hysteretic damping: C h,eq tot rad tot (7) 2K k eq There are several pairs of displacement amplitde in a free vibration decay (sch as the one shown in Fig. 7) that can be sed for the calclation of h,eq. The earlier cycles yield larger damping ratios since they correspond to larger displacement amplitdes (and, conseqently, larger average shear strain in the soil in the vicinity of the footing) than later cycles. By bringing together these reslts from several analyses in a graph, scatter plots are formed, as shown in Fig.8. It is interesting to note that the trend lines that fit closely the data points are actally linear fnctions of the footing displacement (or rotation in this particlar case); they appear crved de to the logarithmic scale of the horizontal axis. Sch trend lines of h,eq vs. /2B are plotted in Fig. 9 and 1 for all parametric analyses performed for vertical oscillation of circlar footing and rocking of strip footing, along with the respective stiffness redction crves from static analyses kN 1kN 25kN 5kN 1kN trend line (=12θ+.92) kN 1kN 25kN 5kN 1kN 2kN trend line (=119.5θ+1.13) /(2B) = /2 (%) /(2B) = /2 (%) Figre 8. Scatter plots of eqivalent hysteretic damping ratio h,eq as a fnction of the normalized footing edge displacement amplitde /2B for strip footing in rocking from analyses with: a) B=.5m, G max =2MPa, PI=2, f=12hz, b) B=1m, G max =1MPa, PI=2, f=6hz) De the inherently very large radiation damping exhibited in the case of vertically or horizontally oscillating footings, extraction of h,eq from free vibration analyses for these two modes was fond nreliable. This is becase of the very small displacement amplitdes left after the first cycle as a reslt of extreme decay. Moreover, the absorbent bondaries in the vertical or horizontal free vibration analyses exhibited significant permanent displacements. For these reasons, the h,eq for these two strip footing oscillation modes was extracted from the area of the load-displacement hysteresis loop from analyses where the footing was cycled qasi-statically, as shown in Fig. 11. The reslting eqivalent spring stiffness and hysteretic damping ratio as fnctions of /2B are presented in Figs. 12 and 13. The constittive model employed herein predicts zero hysteretic damping at very small shear strains and, ths, qasi-static analyses prodce zero h,eq for very small footing motion amplitdes. Hence, a vale of 1% was added to the qasi-static FEM reslts in order to be able to make direct comparisons between Figs. 12 and 13 and Fig. 9 and 1. 8

9 F (kn) h,eq (%) h,eq (%) N. Orologopolos and D. Lokidis 9 K eq /K max B=.5m, Gmax=2MPa, PI=2 B=2m, Gmax=2MPa, PI=2 B=1m, Gmax=2MPa, PI=2 B=1m, Gmax=1MPa, PI=2 B=1m, Gmax=5MPa, PI=2 B=1m, Gmax=2MPa, PI=5 B=1m, Gmax=2MPa, PI= B=.5m, Gmax=2, PI=2, f=12hz B=2m, Gmax=2MPa, PI=2, f=3hz B=4m, Gmax=2MPa, PI=2, f=3hz B=1m, Gmax=2MPa, PI=2, f=3hz B=1m, Gmax=1MPa, PI=2, f=6hz B=1m, Gmax=5MPa, PI=2, f=4.2hz B=1m, Gmax=2MPa, PI=5, f=8.5hz B=1m, Gmax=2MPa, PI=4, f=8.5hz B=1, Gmax=2MPa, PI=2, f=8.5hz /2B (%) /2B (%) Figre 9. Eqivalent spring stiffness K eq and hysteretic damping ratio h,eq as fnctions of the normalized footing displacement amplitde /2B for circlar footing in vertical oscillation. K eq /K max B=.5m, Gmax=2, PI=2 B=2m, Gmax=2, PI=2 B=4m, Gmax=2, PI=2 B=1m, Gmax=2, PI=2 B=1m, Gmax=1, PI=2 B=1m, Gmax=5, PI=2 B=1m, Gmax=2, PI=5 B=1m, Gmax=2, PI= /2B = /2 (%) B=.5m, Gmax=2MPa, PI=2, f=12hz B=2m, Gmax=2MPa, PI=2, f=3hz B=4m, Gmax=2MPa, PI=2, f=3hz B=1m, Gmax=2MPa, PI=2, f=3hz B=1m, Gmax=1MPa, PI=2, f=6hz B=1m, Gmax=5MPa, PI=2, f=4.2hz B=1m, Gmax=2MPa, PI=2, f=8.5hz B=1m, Gmax=2MPa, PI=5, f=8.5hz B=1m, Gmax=2MPa, PI=4, f=8.5hz /2B = /2 (%) Figre 1. Eqivalent spring stiffness K eq and hysteretic damping ratio h,eq as fnctions of the normalized footing edge displacement amplitde /2B for strip footing in rocking W S (m) -5 W D h,eq WD 4W S -2 Figre 11. Typical hysteresis loop from qasi-static analysis of strip footing sbjected to vertical cycling.

10 h,eq (%) h,eq (%) K eq /K max B=.5m, Gmax=2MPa, PI=2 B=2m, Gmax=2MPa, PI=2 B=4m, Gmax=2MPa, PI=2 B=1m, Gmax=2MPa, PI=2 B=1m, Gmax=1MPa, PI=2 B=1m, Gmax=5MPa, PI=2 B=1m, Gmax=2MPa, PI=5 B=1m, Gmax=2MPa, PI= /2B (%) B=.5m, Gmax=2MPa, PI=2 B=2m, Gmax=2MPa, PI=2 B=4m, Gmax=2MPa, PI=2 B=1m, Gmax=1MPa, PI=2 B=1m, Gmax=5MPa, PI=2 B=1m, Gmax=2MPa, PI=2 B=1m, Gmax=2MPa, PI=5 B=1m, Gmax=2, PI= /2B (%) Figre 12. Eqivalent spring stiffness K eq and hysteretic damping ratio h,eq as fnctions of the normalized footing displacement amplitde /2B for strip footing in vertical oscillation. K eq /K max B=.5m, Gmax=2MPa, PI=2 B=2m, Gmax=2MPa, PI=2 B=4m, Gmax=2MPa, PI=2 B=1m, Gmax=2MPa, PI=2 B=1m, Gmax=1MPa, PI=2 B=1m, Gmax=5MPa, PI=2 B=1m, Gmax=2MPa, PI=5 B=1m, Gmax=2MPa, PI= B=.5m, Gmax=2MPa, PI=2 B=2m, Gmax=2MPa, PI=2 B=4m, Gmax=2MPa, PI=2 B=1m, Gmax=1MPa, PI=2 B=1m, Gmax=5MPa, PI=2 B=1m, Gmax=2MPa, PI=2 B=1m, Gmax=2MPa, PI=5 B=1m, Gmax=2, PI= /2B (%) /2B (%) Figre 13. Eqivalent spring stiffness K eq and hysteretic damping ratio h,eq as fnctions of the normalized footing displacement amplitde /2B for strip footing in horizontal oscillation. The crves shown in Figs. 9, 1, 12 and 13 sggest that the eqivalent stiffness redction and the eqivalent damping ratio depend solely on plasticity index PI. In particlar, the redction of K eq and the increase of h,eq with footing motion amplitde becomes less intense as PI increases, in mch the same manner G and h at material point level depend on PI. In all examined modes, the crves are practically independent of the footing size and G max. Moreover, the inflence of osicllation freqency on h,eq is insignificant. For a given level of motion amplitde (say /2B=.1%), h,eq is mch larger (and K eq /K max is mch smaller) in the case of rocking than in the cases of vertical and horizontal oscillation. This can be attribted to the fact that the strains developing in the soil in the vicinity of the fondation in rocking are concentrated in a circlar arc-shaped region immediately below the footing, where the soil is mostly sheared. In contrast, the vertical oscillation creates mainly longitdinal waves below the fondation and, to a lesser degree, shear waves that propagate diagonally and laterally, with the shearing diffsing gradally with depth. Horizontal oscillation generates mainly shear waves, bt prodces also longitdinal waves that propagate diagonally and laterally. As a conseqence, the average shear strain in the fondation soil, for a given footing motion amplitde, is clearly larger in the case of rocking than in the case of vertical or horizontal oscillation. Ths, the effect of soil nonlinearity (and hysteresis) is more prononced in rocking than in any other mode. Yet, h,eq is larger (and K eq /K max is smaller) in horizontal oscillation than in vertical oscillation. Hence, we can say that the 1

11 N. Orologopolos and D. Lokidis 11 horizontal vibration case lies between rocking and vertical vibration in terms of importance of soil non-linearity. CONCLUSIONS A series of parametric analyses were performed sing the finite element method (FEM) in order to determine eqivalent vales for the spring stiffness and hysteretic damping ratio that accont for soil non-linearity in the standard spring-dashpot model. The finite element methodology was validated by first performing analyses in which the soil was linearly elastic with Rayleigh damping. Nmerical reslts are in agreement with the predictions of the spring-dashpot model sing coefficients based on semi-analytical soltions. It was observed that se of both mass and stiffness proportional Rayleigh damping leads to severe nderestimation of the soil hysteretic damping. Hence, if possible, only stiffness proportional damping shold be sed in analyses involving wave generation and propagation in a continm. Finite element analyses performed assming that the soil follows a hyperbolic stress-strain law with hysteresis show that the eqivalent spring stiffness redces and the eqivalent (average) hysteretic damping ratio of the fondation-soil system increases with increasing motion amplitde, in practically the same manner as in the case of a sheared soil element. However, for the same order of displacement amplitde, the rocking mode exhibits significantly larger eqivalent hysteretic damping ratio than the vertical or horizontal oscillation modes. Accordingly, the stiffness redction is more intense in the case of rocking. Regardless of the type of oscillation, the eqivalent spring stiffness redction and hysteretic damping crves depend strongly on the plasticity index PI of the soil, bt not on fondation size, maximm shear modls or oscillation freqency. Finally, the findings of the present stdy sggest that the effect of the fondation motion amplitde on the spring stiffness and dashpot coefficients is important and may reslt in a significant redction of the spectral acceleration in soil-strctre interaction comptations, mostly de to the increased hysteretic damping ratio cased by the rocking mode. However, we mst stress the fact that the constittive model sed herein to simlate non-linear soil behavior over-predicts significantly the hysteretic damping ratio at shear strain amplitdes larger than.1%, which may be operative in soilstrctre interaction problems involving strong earthqake shaking. Hence, the findings reported in this paper need to be verified by analyses sing a constittive model that captres soil hysteresis accrately in the entire shear strain range. REFERENCES Anastasopolos I, Gazetas G, Loli M, Apostolo M, Gerolymos N (21) "Soil failre can be sed for seismic protection of strctres", Blletin of Earthqake Engineering, 8(2): Brinkgreve RBJ, Swolfs WM and Engin E (211) PLAXIS 2D Reference manal, Delft, The Netherlands. Chatzigogos CT, Pecker A, Salencon J (29) "Macroelement modeling of shallow fondations", Soil Dynamics and Earthqake Engineering, 29(5): Cremer C, Pecker A, and Davenne L (21). "Cyclic macro element for soil strctre interaction: material and geometrical non linearities" International Jornal for Nmerical and Analytical Methods in Geomechanics, 25(13): Dobry R and Gazetas G (1986), "Dynamic response of arbitrarily shaped fondations", Jornal of geotechnical engineering, 112(2): Dobry R and Gazetas G "Dynamic response of arbitrarily shaped fondations." Jornal of geotechnical engineering (1986): Dos Santos JA and Correia AG (21) "Reference threshold shear strain of soil. Its application to obtain an niqe strain-dependent shear modls crve for soil", Proceedings of the 15 th International Conference on Soil Mechanics and Geotechnical Engineering, Istanbl, Trkey, AA Balkema. Dncan JM and Chang C-Y (197) "Nonlinear analysis of stress and strain in soils", Jornal of the Soil Mechanics and Fondations Division, 96(5):

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