Static and Dynamic Response of Yielding Pile in Nonlinear Soil

Transcription

1 Proceeding, 1 t Greece Japan Workhop: Seimic Deign, Obervation, and Retrofit of Foundation. Athen 5 Static and Dynamic Repone of Yielding Pile in Nonlinear Soil N. Gerolymo 1, G. Gazeta and T. Tazoh 3 1,, National Technical Univerity, Athen, Greece 3 Intitute of Technology, Shimizu Corporation, Tokyo, Japan Abtract While eimic code do not allow platic deformation of pile, the Kobe earthquake ha hown that limited tructural yielding and cracking of pile may not be alway detrimental. A a firt attempt to invetigate the conequence of pile yielding thi paper explore the pile oil interaction to lateral loading, with emphai on tructural nonlinearity. The pile-oil interaction i modeled through ditributed nonlinear Winkler-type pring and dahpot. Numerical analyi i performed with a computer code, NL-DYAP, which can imulate : the nonlinear behaviour of both pile and pring ; the poible eparation and gapping between pile and oil ; radiation damping ; lo of tiffne and trength in pile and oil. The model i utilied in tudying the lateral repone of a pile embedded in nonhomogeneou tiff coheive oil. A detailed elato-platic train-hardening continuum model of the pile and the urrounding oil i alo developed utiliing the advanced finite element code ABAQUS in 3-Dimenion. The prediction of the two model are in atifactory agreement, and capture the inelaticity in both the pile and the oil. It i hoped that further parametric tudie will hed light on the potential role of flexural pile yielding in the eimic performance of tructure-foundation ytem, during and after a deign eimic event. INTRODUCTION Although ductility deign i now general practice in tructure, piled foundation are till primarily deigned to remain eletic. The concept of ductility deign for foundation element i till new in earthquake engineering practice. The potential development of a platic hinge in the pile i forbidden in exiting regulation, code and pecification. The main reaon are: (i) the location of platic hinge i not approachable for pot-eimic inpection and repair, and (ii) failure due to yielding in the pile prior to exceeding oil capacity, i of undeirably brittle nature. By contrat, if oil capacity i mobilized firt, the failure mode i ductile. Beide, the intenly hyteretic behaviour implied by the oil failure mode tend to be beneficial for upertructure performance. However, numerou example in Kobe during the 1995 Earthquake revealed that : (i) yielding of pile in trong eimic haking i unavoidable, epecially in oft / looe oil, and (ii) inpection of pile after the earthquake i often a feaible, although not a trivial operation. Moreover, a Prietley and hi coworker have hown that (Pritley et al., 1996; Budek et al., 4): (a) the lateral confinement provided by the oil play a very ignificant role in pile repone. The oil confining preure increae the effective confinement of the ection and retard the development of high level of localized platic rotation, thu providing a izable increae in ductility capacity. Sufficient diplacement ductility may be obtained in a pile haft with tranvere reinforcement ratio, ρ t, a low a.3. (b) The preence of oil confinement lead to increaed platic hinge length, thu preventing high localized curvature (Taio, 4). Therefore, the pile retain much of their axial load carrying capacity after yielding. Furthermore, a the recontruction of the Kobe Route 3 by Hanhin Expreway proved, a capped pile group in which the pile have cracked, retaining only ½ of their initial uncracked tructural rigidity, EI, till preerve 8% of it overall lateral tiffne, K H, thank to 5

2 Proceeding, 1 t Greece Japan Workhop: Seimic Deign, Obervation, and Retrofit of Foundation. Athen 5 the participation of the urrounding oil (Gazeta et al., 5). To gain inight into the (beneficial or detrimental) role of pile yielding in the overall repone of the upertructure, oil pile interaction conidering inelaticity in both the pile and the oil mut be tudied. The author are aware of only one uch comprehenive tudy in the literature (Hutchinon et al.,, 4a, b). It i noted that the cyclic lateral repone of pile i governed by the highly non linear tre train oil behavior, even at low level of load. The problem become more complex when pile inelaticity and geometric nonlinearitie are involved, uch a eparation and gapping of the pile from the oil. The method of pile repone analyi can be claified into two main categorie : (a) continuum baed method, implemented through finite-element or finite-difference formulation, (b) Winkler pring method, uing empirical (c) or analytical p y curve, and limit analyi method, in which the ultimate oil reaction i predetermined from the aumed hape of the pile diplacement profile at it ultimate tate. Finite element analyi require dicretiation of the pile and urrounding oil in 3 dimenion. Advanced contitutive model baed on platicity and hypo-platicity theorie can be utilied to reproduce tre train oil behaviour (e.g. Trochani et al., 1991; Kimura et al., 1995; Wakai et al., 1999). Neverthele, the finite element method i not alway effective, let alone efficient, in modeling pile repone. By contrat, modeling oil reaction through Winkler pring i a veratile approach, in effect rendering the oil pile interaction problem a one dimenional. The model owe it popularity to the development of the well known p y curve relating the oil reaction to pile deflection, at each point of the pile. Such curve, experimentally baed, can (indirectly) account for oil eparation from the pile and liding at the pile oil interface (Matlock 197; Reee et al., 1986; Their ucceful application tem from the fact that even though they do not model accurately the oil continuum, they are baed on reult of field load tet where the continuum i fully atified (Reee 1997). While of the available p-y method employ a emi-empirical approach to developing p-y curve, a different, emi-theoretical methodology i alo poible. One tart with a mathematical model and then calibrate it parameter againt full-cale and centrifuge experiment, or even with rigorou 3-D numerical reult, if uch are available. Among a number of uch mathematical model propoed over the lat twenty year, particularly fruitful ha proved the o called Bouc Wen model (Bouc 1971, Wen 1976), recently extended and modified by Gerolymo and Gazeta (5a) applied it to decribe dynamic p y relationhip for laterally loaded pile. Thi extended model (deignated BWGG) i capable of reproducing complex feature of pile oil interaction, uch a : (i) oil and pile nonlinearite ; (ii) oil pile interface nonlinearitie ; (iii) coupling between radiation damping and hyteretic oil repone; and (iv) tiffne and trength degradation with cyclic loading. The model wa validated againt available experimental and centrifuge tet. In thi paper a computer code named NL- DYAP (Gerolymo and Gazeta, 5a) which incorporate the BWGG model, i ued to analye the lateral repone of a reinforced concrete pile embedded in coheive oil. Nonlinear repone of both pile and oil are conidered, and the reult compared againt reult from 3D elatoplatic finite element analyi. The long-term objective of the reearch ummarized in thi paper are : (a) to develop a veratile and eay to ue analytical tool for nonlinear lateral repone of pile, and (b) to hed ome light the conequence of tructural yielding of a pile into the eimic repone of oil-tructural ytem. THE MODEL : CONSTITUTIVE EQUATIONS The lateral oil reaction againt a deflecting pile i expreed a the um of an elatic, an inelatic, and a vico-platic component according to: p x = α k + c y + y a t ( 1 α ) p y ζ + (1 a ) ζ y c d (1) 6

3 Proceeding, 1 t Greece Japan Workhop: Seimic Deign, Obervation, and Retrofit of Foundation. Athen 5 where ζ i a dimenionle inelatic oil parameter expreed in the following differential form : ζ η n { 1 ( + r ) [ b + g ign( dy d h x = 1 ζ dy y ζ )]} () p x i the reultant (in the direction of loading) of the normal and hear tree along the perimeter of a pile egment of unit length and it include both in-phae and out-of-phae component ; the latter reflect radiation and hyteretic damping in the oil. y i the pile deflection at the location of the pring; k i a reference pring tiffne ; α i a parameter governing the pot yielding tiffne ; p y i a characteritic value of the oil reaction related to the initiation of ignificant inelaticity (yielding) ; y i a characteritic value of pile deflection related to the initiation of yielding in oil reaction. n, b and g, are dimenionle quantitie that control the hape of the hyteretic oil reaction pile deflection loop, and η, r and h are train hardening parameter for tiffne decay, trength degradation, and pinching behaviour, repectively ; where c i the damping coefficient at mall amplitude motion, and c d i a vicoplatic parameter which control the coupling of oil and oil pile interface nonlinearity with radiation damping The reader i referred to the companion paper (Gerolymo and Gazeta, 5b) for more detail. The inelatic behaviour of the pile i imilarly expreed in term of a trength-ofmaterial-type bending moment pile curvature relation, which include an elatic and an inelatic component y M = α p E p I p + (1 a p ) M y ζ p (3) z where E p I p i the initial (elatic) bending tiffne (alo called flexural rigidity), α p i a parameter controlling the pot yielding bending tiffne, M y i the value of bending moment that initiate tructure yielding in the pile, and ζ p i the hyteretic dimenionle parameter which control the nonlinear tructural repone of the pile. The latter i governed by dζ dκ η h 1 n p ( 1+ r ) ζ [ b + g ign( dκζ ] p p p = p p p p p κ ) (4) where κ i the pile curvature, and b p, g p, n p, η p, r p, and h p,are dimenionle quantitie that control the hape of the hyteretic bending moment curvature loop in the ame manner a n, b, g, η p, r p, and h p, control the hape of the lateral oil reaction deflection loop. κ i the value of pile curvature at initiation of yielding in the pile. Evidently, Eqn (3) (4) are of the ame form a Eqn (1) (), except that no vicou term (radiation damping) i included in the tructural pile repone. STIFFNESS AND STRENGTH PARAMETERS FOR COHESIVE SOIL The mall amplitude tiffne k (= p y / y ) and dahpot coefficient c in Eqn (1), obtained from the available beam-on-dynamic-winkler- Foundation olution (e.g., Gazeta & Dobry 1984; Makri & Gazeta 199) in term of the Young modulu of the oil, In cae of a pile embedded in coheive oil the trength parameter p y in Eqn (1) can be approximated by the analytical expreion of Randolph and Houlby (1984) for the ultimate oil reaction: p y 1 S u B (5) in which S u i the undrained hear trength of the oil. Eqn (5) i very often preferred in practice from other more rigorou expreion for it implicity and ufficient engineering accuracy, in view of it accord with experimental and numerical reult. NUMERICAL MODELLING OF DYNAMIC PILE SOIL RESPONSE With the contitutive model for lateral oil reaction developed in the previou ection, the pile oil interaction problem under dynamic lateral loading reduce to the analyi of a beam with cro ectional area, A p and ma denity ρ p, upported on a nonlinear Winkler foundation, a ketched in Fig 1. Equilibrium of the pile give : 7

4 Proceeding, 1 t Greece Japan Workhop: Seimic Deign, Obervation, and Retrofit of Foundation. Athen 5 Μ κ Fig 1: Schematic illutration of dynamic nonlinear pile oil interaction: The propoed modified Bouc Wen model for pile M ( z, t) y( z, t) + ρ A ( z) + p ( z, t) = p p x (6) z t where z i the patial coordinate (depth) along the pile, and t i time. M i the pile bending moment given by Eqn (4), and p x incorporate the pring and dahpot reaction on the pile given by Eqn (1) and (3), repectively. An explicit finite difference method i ued for the olution of the ytem of Eqn (6) coupled with Eqn (1) through (4), by conidering the variation of pile and oil propertie along the pile length. Head and tip boundary condition are appropriately taken into account. The numerical algorithm i implemented through the computer code NL- DYAP, for the non-linear analyi of ingle pile under lateral loading (Gerolymo & Gazeta, 5a). COMPARISON WITH RESULTS FROM 3D FINITE ELEMENT ANALYSIS The capability of the model i invetigated through comparion with reult from finite element analyi. The problem tudied i portrayed in Fig : A reinforced concrete pile of length L = 1 m and diameter d = 1 m, embedded in nonhomogeneou coheive oil, undergoe lateral loading up to failure. The pile i conidered fixed at it head with no rotation allowed. The bending moment capacity of the pile i.8 MNm. A detailed numerical model (Fig 3) of the pile and the urrounding oil i developed with the advanced finite element code ABAQUS. Both pile and oil are modelled with 3-D element. The far field i repreented with infinite element enuring proper modelling of radiation damping. Perfect bonding i aumed at the pile oil interface. Elatoplatic oil and pile behaviour i decribed with Von Mie yield urface with nonlinear kinematic hardening and aociative platic flow rule. It i mentioned, that the aforementioned elatoplatic model i not appropriate for the tre train relationhip of reinforced concrete. However, it parameter can be calibrated to match the oberved pile repone in the macrocopic moment curvature level. 8

5 Proceeding, 1 t Greece Japan Workhop: Seimic Deign, Obervation, and Retrofit of Foundation. Athen 5 1 m S u : kpa E : MPa z : m 4 1 m E p =3 GPa M y =.8 MNm Fig : Soil propertie and pile characteritic of the tudied problem : pile in tiff coheive oil. The calibration of the parameter of the two model (the finite element and the propoed nonlinear Winkler model) i baed on a methodology a follow: Soil behaviour According to the platicity model ued in thi tudy and provided by ABAQUS, integration of the backtre evolution law over a half cycle of unidirectional loading yield the following expreion ( C a = [ 1 exp γ ε )] pl (7) γ in which a i the backtre that define the kinematic evolution of the yield urface, C and γ are hardening parameter that define the maximum tranition of the yield urface in the tre pace, and the rate of tranition, pl repectively. ε i the platic train. The current value of tre σ i then expreed according to σ = σ + a (8) in which σ the value of σ at zero platic train. The unidirectional tre train relationhip of a oil element according to Bouc Wen type of contitutive modeling i given by σ = σ y ζ (9) Fig 3: The finite element meh ued in the analyi auming Maing rule for unloading reloading (that i, etting b = g =.5 in the expreion for ζ), Eqn () i olved analytically for n = 1 yielding [ ( ε )] σ = σ 1 exp (1) y ε y in which ε i the train, and ε y a characteritic yielding train. Notice, that Eqn (7) and (1) are of imilar form. Equating them, the following model parameter are derived 1 y γ =, C σ = = E, and σ = (11) ε ε y y In a Von Mie yielding criterion the maximum yield tre i equal to σ = 3 (1) y S u and, the hardening parameter γ may be expreed a E γ = (13) 3 S u Pile behaviour The bending moment of a circular pile ection i by definition related to the normal tree σ acting on thi ection according to in which σ y i the maximum yield tre and ζ i the hyteretic parameter of the form of Eqn (). Under monotonic loading condition and M π d = σ r inθ dr dθ (14) 9

6 Proceeding, 1 t Greece Japan Workhop: Seimic Deign, Obervation, and Retrofit of Foundation. Athen 5 M : knm Q : kn Fig 4: Comparion of computed from the finite element model (circle), and predicted by the propoed model for pile (olid line) of: (a) Bending moment diplacement and (b) Shear force diplacement curve at the pile head ( puhover curve). in which r i the radiu of pile ection. Subtituting Eqn (7) and (8) into Eqn (13), etting σ =, and expreing the normal train ε a a function of pile curvature κ, ε z = κ r inθ, one obtain the following expreion for bending moment : M (15) π d C γ ( κ ) [ 1 exp( γ r inθ κ )] r inθ dr dθ = Firt platic hinging Lateral Diplacement at the top : m Firt platic hinging Lateral Diplacement at the top : m Second platic hinging The parameter C and γ can be appropriately calibrated to match any experimental or calculated bending moment curvature curve. According to the ame methodology the parameter σ y and ε y of the BWGG model are calibrated a well.???? Second platic hinging NUMERICAL EXAMPLE : PILE IN STIFF SOIL Fig 4 compare the computed with finite element and the propoed Winkler model : (a) hear force diplacement and (b) bending moment diplacement curve at the pile head ( puhover curve). The comparion i rather urpriingly atifactory. The detail of the pile deflection, bending moment, and hear force at different tage of loading, computed with the propoed method, are preented in Fig 5. The initiation of the firt platic hinging at pile head (M y = 8 knm) take place at an early tage of loading, when the applied pile head diplacement reache 1 cm, and i completely formed at u = 3.5 cm. The econd platic hinge tart to develop at the depth of about 4 m, for a head diplacement of 4 cm. Surpriingly, however, it ultimate capacity i not completely reached even for a head diplacement a large a 15 cm. Thi i of coure due to the beneficial role of the lateral confinement provided by the oil. Thing could be wore if the pile carried high axial load and P- type oftening took place. But in the examined idealized cae, oberve that the diplacement profile retain a nearly triangular hape with maximum value at the urface and zero value almot at z = 5 m, implying that the effective length of the pile i retricted to only a mall portion of 4% of the total pile length (or five diameter). Thi i contrary to caion foundation in which the total length contribute to the reiting mechanim. Fig 6 portray the evolution of oil and pile yielding computed from the finite element analyi. Note, that oil platification initiate in the vicinity of the pile head at relatively mall train level, and propagate rapidly downward with increaing loading. The econd platic hinge i developed at the depth of 4 m when the ultimate bearing capacity of the oil ha been practically reached. The failure pattern of the pile oil ytem hown in Fig 6 i conitent with the reult of the propoed Winkler model plotted in Fig 5. Contour of lateral diplacement at different tage of loading are depicted in Fig 7. The maximum attained diplacement wa found to be about 15 cm, in accord with the prediction with the implified model. To invetigate the influence of pile and oil yielding on the dynamic repone of the pile, two cae are analyed. The pile i ubjected to four cycle of inuoidal time hitory of 3

7 Proceeding, 1 t Greece Japan Workhop: Seimic Deign, Obervation, and Retrofit of Foundation. Athen 5 Bending Moment : knm -4-4 Shear Force : kn Q / Q y Depth : m 6 1 Depth : m Soil Reaction : kn / m Diplacement : m Depth : m 6 Depth : m Fig 5: Computed with the propoed Winkler model for pile: Bending moment, hear force, oil reaction, and pile deflection profile at elected loading level with repect to the maximum applied hear force (Q y 36 kn) horizontal diplacement of 1 cm amplitude. The tudied frequency of loading i Hz. The hear force diplacement and bending moment diplacement loop at the top of the pile computed: (a) with the finite element model (grey line) and (b) with the propoed method (black line) are compared in Fig 8. The comparion i quite atifactory. The Winkler model predict lightly higher maximum reaction and flexural rigidity, a anticipated with the tatic plot in Fig 4. The latter how that the matching between the moment diplacement curve computed from the two method of analyi i not o perfect in the range of 1 to 3 cm. 31

8 Proceeding, 1 t Greece Japan Workhop: Seimic Deign, Obervation, and Retrofit of Foundation. Athen 5 t =. t =.5 t = 1 (a) (a) (b) Fig 6: Contour of platic train magnitude (plotted on the deformed meh) with emphai to (a) oil platification, and (b) pile yielding, at elected loading level (a a fraction of the maximum applied hear force). Notice, that yielding in pile i firt developed at it top the very early tage of loading ( < %), while the econd platic hinge tart to develop at a depth of 4 m for a hear force equal to 5% of the total applied, and i practically aociated with the full ultimate capacity of the oil 3

9 Proceeding, 1 t Greece Japan Workhop: Seimic Deign, Obervation, and Retrofit of Foundation. Athen 5 t =. t =.5 t = 1 Fig 7: Contour of horizontal diplacement (plotted on the deformed meh), at elected loading level (a a fraction of the maximum applied hear force). 33

10 Proceeding, 1 t Greece Japan Workhop: Seimic Deign, Obervation, and Retrofit of Foundation. Athen 5 M : knm Q : kn Lateral Diplacement at the top : m Lateral Diplacement at the top : m Fig 8: Comparion of computed from the finite element model (grey line), and predicted by the propoed model for pile (black line) of: (a) bending moment veru diplacement and (b) horizontal force veru diplacement loop at the top of the pile, for cae (1). 4 cycle of Hz inuoidal time hitory of horizontal diplacement of 1 cm amplitude, applied at the top of the pile. CONCLUSIONS An inelatic beam on Winkler foundation and an elatoplatic finite-element method are developed for laterally loaded pile. Both pile and oil are treated a nonlinear inelatic material. To model the nonlinear reaction of the oil with realim a modified BWGG type model i utilied, which can capture uch effect a: oil failure, eparation and gapping of the pile from the oil, radiation damping, and lo of trength and tiffne due to degradation and pore-water preure generation. The coupling of hyteretic and radiation damping i alo modeled in a realitically implified way. Implemented into the code NL DYAP, the prediction of the method compare atifactorily with reult from 3-D finite element analyi uing ABAQUS. The numerical tudy ha preliminarily addreed (a) the lateral monotonic and (b) the dynamic (inuoidal type) repone of a pile embedded in tiff nonhomogeneou coheive oil. The reult highlight the role of pile yielding in the overall repone of the pile. We are preently planning a comprehenive parameter tudy, and a ubequent corroboration with the reult of centrifuge tet (uch a thoe of Hutchinon et al 4) and the finding of reinforced concrete experiment (uch a thoe of Budek et al 4). We then hope to hed light on the role of pile yielding in the repone of the tructurefoundation ytem during a trong eimic haking, a well a on the ubequent expected performance of the injured pile under tatic and (new) eimic loading. ACKNOWLEDGMENTS The reearch reported herein wa ponored by the General Secretariat for reearch (ΓΓΕΤ), in the coure of ASPROGE project REFERENCES Badoni D. & Makri N. (1995): Nonlinear repone of ingle pile under lateral inertial and eimic load, Soil Dynamic and Earthquake Engineering, 15, Bouc R. (1971): Modele mathematique d hyterei,acutica, 1, Brom B.B. (1964): Lateral reitance of pile in coheionle oil, Journal of Soil Mechanic and Foundation Diviion, ASCE, 1964, 9, SM3, Budek A. M., Prietley M. J. N., and Benzoni G. (4) : The effect of external confinement on flexural hinging in drilled pile haft, Earthquake Spectra, (1), 1-4. Gazeta G., Anataopoulo I., Gerolymo N., Mylonaki G., Syngro C. (5): The collape of the Hanhin Expreway (Fukae) bridge, Kobe 1995: Soil foundation tructure interaction, recontruction, eimic iolation, New Development in Soil and Structural Mechanic [Anniverary Volume for Profeor St. Savvidi 6 th Birthday], Springer Verlag, Berlin. 34

11 Proceeding, 1 t Greece Japan Workhop: Seimic Deign, Obervation, and Retrofit of Foundation. Athen 5 Gazeta G. and Dobry R (1984) : Simple radiation damping model for pile and footing, Journal of Engineering Mechanic, ASCE, 11, Gerolymo N. & Gazeta G (5a): Phenomenological model applied to inelatic repone of oil pile interaction ytem, Soil & Foundation, 45(4). Gerolymo N. & Gazeta G (5b): Nonlinear lateral repone of caion foundation, Proc. 1 t Greece Japan Workhop on Seimic Deign, Obervation, Retrofit of Foundation, Athen 11 1 October. Hutchinon T. C., Chai Y. H., Boulanger R. W., and Idri I. M. (4a): Inelatic eimic repone of extended pile haft-upported bridge tructure, Earthquake Spectra, (4), Hutchinon T. C., Chai Y. H., Boulanger R. W., and Idri I. M. (4b): Etimating inelatic diplacement for deign: Extended pilehaft-upported bridge tructure, Earthquake Spectra, (4), Hutchinon T. C., Boulanger R. W., Chai Y. H., and Idri I. M. (): Inelatic eimic repone of extended pile haft-upported bridge tructure, Pacific Earthquake Engineering Reearch Center, Report No. PEER /14, 15 pp. Randolph M.F., and Houlby G.T. (1984). The limiting preure on a circular pile loaded laterally in coheive oil, Geotechnique, 34, 4. Kimura M., Adachi T., Kamei H. & Zhang F (1995): 3 D finite element analye of the ultimate behavior of laterally loaded cat in place concrete pile, Proc. 15 th Int. Symp. on Numerical Model in Geomechanic., Davo, Switzerland, Makri N. & Gazeta G. (199): Dynamic pile-oilpile interaction. Part II : Lateral and eimic repone. Earthquake Engngineering. Structural Dynamic, 1, Matlock H. (197): Correlation for Deign of Laterally Loaded Pile in Soft Clay, Proceeding, Second Annual Offhore Technology Conference, Houton, Texa, 1, Paper No. OTC 14, Prietley M. J. N., Seible F., Calvi G. M. (1996). Seimic Deign and Retrofit of Bridge, John Wiley & Son, inc., ISBN X. Reee L.C. (1997) : Analyi of laterally loaded pile in weak rock, Journal of Geotechnical and Geoenvironmental Engineering, ASCE, 13(11), Reee L.C. (1986) : Behavior of pile and pile group under lateral load, Federal Highway Adminitration Report FHWA/RD 85/16, Wahington D.C. Trochani A., Bielak J, & Chritiano P. (1991): Three dimenional nonlinear tudy of pile, Journal of the Geotechnical Engineering, ASCE, 117, GT3, Trochani A., Bielak J., & Chritiano P. (1994): Simplified model for analyi of one or two pile, Journal of Engineering Mechanic, ASCE, 1, Wakai A., Goe Sh., & Ugai K. (1999): 3 D Elato Platic Finite Element Analye of Pile Foundation ubjected to Lateral Loading, Soil & Foundation, 39(1), Wen Y.K. (1976): Method for random vibration of hyteretic ytem, Journal of Engineering Mechanic, ASCE, 1,

12 Proceeding, 1t Greece-Japan Workhop: Seimic Deign, Obervation, and Retrofit of Foundation. Athen 5 36