Seismic response of bridge pier on rigid caisson foundation in soil stratum

Transcription

1 Vol7, No EARTHQUAKE ENGINEERING AN ENGINEERING VIBRATION March, 8 Earthq Eng & Eng Vib (8) 7:- OI: 7/ Seimic repone of bridge pier on rigid caion foundation in oil tratum C Tiggino, N Gerolymo, Aimaki G Gazeta * National Technical Univerity of Athen, Greece Civil Engineering, National Technical Univerity of Athen, Greece Civil Engineering, Georgia Intitute of Technology, Atlanta, USA Abtract: An analytical method to tudy the eimic repone of a bridge pier upported on a rigid caion foundation embedded in a deep oil tratum underlain by a homogeneou half pace i developed The method reproduce the kinematic inertial repone, uing tranlational rotational ditributed Winkler pring dahpot to imulate the oil-caion interaction Cloed-form olution are given in the frequency domain for vertical harmonic S-wave ecitation Comparion with reult from finite element (FE) analyi other available olution demontrate the reliability of the model Reult from parametric tudie are given for the kinematic inertial repone The modification of the fundamental period damping ratio of the bridge due to oil-tructure interaction i graphically illutrated Keyword: caion; embedded foundation; kinematic repone; inertial repone; Winkler model; effective period; effective damping Introduction Thi article deal with the eimic repone of a caion-upported bridge pier, with a rectangular or quare bae The oil i conidered a a ingle homogeneou layer underlain by an elatic half pace The ytem i ecited by vertical harmonic S-wave, pecified through the rock outcrop motion Analye are performed in the frequency domain The problem geometry i hown in Fig Caion foundation embedded in oft oil have been widely ued to upport major tructure, epecially bridge In contrat to pile, which are relatively lender tructural element in lateral loading, caion foundation are ubjected to vertical hear traction at their periphery hear traction at their bae, in addition to the lateral oil reaction Fig how thee traction for a caion with a rectangular cro ection Correpondence to: George Gazeta, 6 Aimakopoulou Str, Ag Parakevi 5, Greece Fa: gazeta@athforthnetgr Civil Engineer; Lecturer; Aitant Profeor; * Profeor Supported by: US Federal Highway Adminitration Under Grant No TFH6-98-C-9 US National Science Foundation Under Grant No EEC-977 Received January 5, 8; Accepted February, 8 Sytem method of analyi Analyi model The Winkler type model ued for the olution of the problem i hown in Fig The upertructure i modeled a a two degree-of-freedom (OF) ytem, with a ma on the top of the column poeing rotational, in addition to tranlational, inertia A comple modulu of elaticity imulate the damping in the tructure To meaure the oil reaction to diplacement of the caion, the dynamic Winkler multi-pring--dahpot model, developed by Aimaki (998), Chantzigianneli (999) Gerolymo & Gazeta (7), i ued The model incorporate ditributed tranlational (lateral) rotational pring dahpot along the caion haft, a well a concentrated tranlational (hear) rotational (rocking) pring dahpot at the bae of the caion Thee four type of pring dahpot are related to the reiting force acting on the caion haft bae, a follow: The ditributed lateral pring k dahpot c are aociated with the reulting horizontal oil reaction p on the circumference (haft) of the caion per unit depth: p = (k + ic ) u () The ditributed rotational pring k dahpot

2 z EARTHQUAKE ENGINEERING AN ENGINEERING VIBRATION Vol7 c are aociated with the moment m produced by the vertical hear traction on the circumference of the caion: m = (k + ic ) c () The reulting bae hear tranlational pring K dahpot C are aociated with the horizontal hearing force on the bae of the caion: Q = (K + ic ) u b () The reulting bae pring K dahpot C are aociated with the moment produced by normal preure on the bae of the caion: M = (K + ic ) c () The dahpot coefficient reflect the combined radiation hyteric diipation of energy in the oil The elatic repone of the bridge-foundation ytem i obtained by referring to the uperpoition theorem, along with decompoing the problem into kinematic inertial repone (Whitman, 97; Kauel & Roeet, 97) Repone The method of analyi i eplained with the help of Fig, for a perfectly rigid caion of ma m c ma moment of inertia about it center of gravity J c The depth of embedment i, while the thickne of the oil layer i H Firt, the repone to S-wave of the free-field elatic oil layer over elatic bedrock i obtained from the well known equation (Kramer, 996): m J m,j h EI h EI k,c k,c H G B K,C H G r r r K,C u ff (z)e it Fig The bridge caion-oil ytem M o Q o B Fig The propoed model The oil reaction are provided with four type of pring dahpot, the ma i concentrated at the top of the pier, ecited by the diplacement profile of the - computed free-field u ff (z) u ff c yz y y H k,c k,c J c c mu c z yz z K,C H- Fig iplacement of quare caion mobilized oil reaction (Gerolymo&Gazeta, 7; Aimaki et al, ) K,C Fig Modeling of the kinematic interaction

3 No C Tiggino et al: Seimic repone of bridge pier on rigid caion foundation in oil tratum 5 u ( z) = u ff g ep [ i( H z) ]+ ep [ ik( H z) ] (5) ( + )ep( ikh ) + ( )ep( ikh ) where u g i the amplitude of the harmonic diplacement at the elatic rock, k i the comple wave number given by k = / V + i i the comple oil-to-rock impedance ratio : m m c c M = m m c c + J c The comple tiffne matri i () V + i = V + i r r r (6) The caion block perform a rigid-body teadytate ocillation when the upport of it interacting Winkler pring dahpot are ubjected to the free field motion given by Eq (5) Thi motion i determined by the amplitude of diplacement of the bae u b the amplitude of rotation c At an arbitrary elevation z meaured from the bae, the caion diplacement equal u b + c z The equation of dynamic equilibrium (horizontal tranlation rotation) of the diplaced caion are written a: c c ff mu + k ()((,) u t u (,)) t d+ c ( )( u(, t) u ff C ( u u (, t)) = b ff ( t, )) d + K ( u u (, t)) + J m u + + k ( ) ( u(, t) u (, t)) d+ c c c c ff b ff (7) u k ( )( (, t) ff ( t, )) d+ c( )( ( t, ) uff ff ( t, )) d+ K ( b (8) u (, t)) + t ff ( b C u (, t)) = t where u ff (,t) = u ff [z- (H - )] (9) u c = u c (t) i the diplacement of the caion at it center of gravity (u c = u b + c /) Taking ü = - u the form of the diplacement of the caion into conideration, Eq (7) (8) can be written in the following matri form where the ma matri i u b ub M + K = P b b () K + k( ) d k( ) d K = k K + ( ) d k( ) d+ K + k k = k K + k + k k ( )d () where, K = K + ic, P i the vector of effective loading P = k + ( ) uff( ) d Ku ( ) ff k ( ) u ( ) + ff d k ( u ) ( ) + Ku ( ) ff d ff () The dynamic impedance matri S at the top of the caion, obtained by a coordinate tranformation, i given by S = S S K K K = S S K K K K r + K () where K K K,, are the element of the tiffne matri of Eq () In the ame way, the ma matri of the top of the caion i given by the following equation m m c c M = m J + m (5) c c c Inertial total repone The method of analyi (Mylonaki et al, 997) i chematically illutrated in Fig 5 A previouly mentioned, the ma of the upertructure i concentrated at the top of the pier, the pier i modeled a a beam The oilcaion interaction i reproduced through the dynamic impedance computed in the previou ubection For the teady-tate harmonic repone in the frequency domain, the differential equation of motion reduce to:

4 6 EARTHQUAKE ENGINEERING AN ENGINEERING VIBRATION Vol7 K Kb Kb Kbb + S = S ij u kin kin ij M M M b tot b M bb utot ub b (6) where K, K b, K b K bb are the four tiffne ubmatrice correponding to the upertructure the foundation; m, m b, m b m bb are the relevant ma ubmatrice Calibration of pring dahpot Evidently, the reliability of the method depend on a proper choice of the dynamic pring dahpot contant a function of frequency Analytical epreion for thee contant were obtained by calibrating the model againt the cloedform epreion preented by Gazeta (99) for rectangular foundation of L/B = -6, embedded in a homogeneou half pace at depth B, a depicted in Fig 6 Thee epreion had been baed on the reult of rigorou boundary-element finite-element elatodynamic analyi, a well a on other available olution from the literature For a rectangle of width B length L (L> B) in a homogeneou half pace, the dynamic impedance are epreed (with repect to the center of the bae mat) in the form K = K ( ) + C( ) (7) emb emb emb i where K emb i the tatic tiffne of the embedded foundation emb () i frequency dependent dynamic tiffne coefficient C () i the frequency dependent damping coefficient (encompaing both the radiation the material damping of the ytem) The parameter which enter in the pring dahpot are: G, V, V La denote, repectively, the hear modulu of elaticity, the hear wave velocity, the apparent propagation velocity of compreion-etenion wave, Poion ratio of the oil B, L denote, repectively, the emi-width, the emi-length depth of the embedment Gazeta Taoula (987a) developed the following epreion for the horizontal tatic tiffne in the longitudinal ()-ai of an orthogonal caion embedded in a homogeneou oil: GL ( ) = + 5 BL L 85 B+ L ( ) ( B BL B GL 75 L (8) The dynamic coefficient emb i preented in chart form tr m,j u tot u tr u b u kin h E,I b tot S u b S S b kin u kin kin Fig 5 Analyi model for the inertial interaction the model parameter (left) the repone (right) in term of abolute diplacement rotation

5 No C Tiggino et al: Seimic repone of bridge pier on rigid caion foundation in oil tratum 7 L B y C = VLa L Bc b ( ) + VLa Bc + VL ( L + ) c + VLBc () X where c b i Fig 6 Geometry of a rigid orthogonal foundation embedded in a homogeneou elatic half pace (Gazeta Taoula, 987a) in term of /B L/B a a function of the dimenionle parameter a (=b/v ) For a quare caion, the dynamic tiffne coefficient emb () i given by the following equation ( ) { + emb 8 7 B B [ 6 ] } B B + (9) The correponding tatic rocking tiffne (Gazeta, 99) i K 75 5 G BL L 6 ( ) = + 9 B + L 5 L The dynamic tiffne: 6 9 (),emb - () The dahpot are given by the following general epreion C = C + K () tot radiation For the horizontal tranlational, the radiation dahpot coefficient (Gazeta Taoula, 987b) i C = V BLc + V L+ V B () La where c h i given in chart form in Gazeta (99) In rocking ocillation, the radiation damping (Fotopoulou et al, 989) i epreed a L L/B B c b {-7 + } + { -(55) } B V (5) And c i B c V B 5 L/B (6) The non-diagonal term of the comple tiffne matri i K K (7) Thu, the tiffne matri of the caion at the bae of the caion ha been calculated a K emb = K K K K (8) To determine the ditributed pring dahpot, the diagonal term of the matrice in Eq (8) () are equated a k = ( K K )/ (9) k K K K K = + / () For a quare caion an orthogonal caion with L/B= tatic lateral loading, the ditributed pring are Square caion k B Orthogonal caion L/B= k B 6 E k B 5 E k B E E () () For the cae of an elatic oil layer underlain by a homogeneou half pace, the bae coefficient (for

6 8 EARTHQUAKE ENGINEERING AN ENGINEERING VIBRATION Vol7 horizontal tranlation rotation) i K K k L H G G H L broel = homtat,, roel,,, B B Gr Gr B B () where, the factor k model the different preure bulb under tatic loading Baed on reult of circular trip foundation (Gazeta, 98), the following two equation for the factor k can be derived, uing an equivalent circular foundation for a quare foundation B B L H k = B B G L H G B B + /( 5+ ) L H k = B G + /( 5+ ) L G r r () (5) And roel i given in Gazeta (99) in chart form in term of L/B H/B a a function of a Taking Eq () into account, the diagonal term of the tiffne matri are: K = K + k (6) roel broel Comparion with finite-element (FE) analyi other olution To evaluate the accuracy of the propoed model, a comparion i performed between reult from finiteelement method (FEM) analyi, other olution from the literature The geometry ued in the FEM i depicted in Fig 7 The problem tudied with the FEM i a rigid caion with B=m, /B= or 6, H/B=m Shear modulu ma denity are contant with depth: G=5kPa, =, =8mg/m The following numerical model i developed with ABAQUS Both the caion the oil are modelled with element The ma of the upertructure i aumed to be lumped at the top of the pier The elatic bedrock i modelled with dahpot The ide boundarie are aumed to be free, thu above the cut-off frequency A different radiation, compared to the condition of the propoed model, will occur due to the reflection in the ide boundarie For the kinematic repone, Rayleigh damping i ued (contant equal to 5 at the firt two mode) an elatic model (=%) i aumed for the inertia repone In Fig 8, the propoed model for the kinematic interaction i compared to a olution given by the FEM According to Fig (9), there i a good correlation in the firt two mode Thi ha occurred for the following three reaon Firtly, the pring dahpot, which are ued to calibrate the model, are developed in a region of - for the dimenionle frequency a Secondly, the damping in the FE model i not-contant for all the eigenmode Finally, the different boundary condition for the ide boundarie in the two model; in the propoed Winkler-model, thee boundarie are etended Kroel = Kbroel + k + k (7) A cut-off frequency eit (for radiation damping) in thi cae To imulate thi phenomenon, the method of linear interpolation i ued For frequencie maller than (/) f, the radiation damping coefficient are zero For frequencie greater than f, the damping coefficient of the homogeneou half pace are ued For intermediate value, linear interpolation i ued c rock, ch = 7V c, hom 9c, 7 7V H V V H H V H (8) where c i the radiation damping for = V / (H) The linear interpolation method mut be ued only in the inertia repone tep Fig 7 An interection of the finite element model The caion beam element are hown with deep grey colour, the urrounding oil with oft grey The elatic rock halfpace i imulated with dahpot

7 No C Tiggino et al: Seimic repone of bridge pier on rigid caion foundation in oil tratum 9 to infinity but in the FEM model, they are very cloe to the caion (the whole model i a m m m bo) In Fig 9 the propoed model for the inertia interaction i compared with a olution given by the FEM Note that there i a good correlation for the period of the ytem Finally, the eigenperiod, which i taken from the model, i compared with the olution given by Veleto (977) Bielak (975) a hown in Fig The dimenionle parameter given in thee figure are eplained in the following ection 5 Reult of parametric tudie 5 Kinematic interaction While long pile follow more or le the eimic motion of the ground, a rigid caion ubtantially modifie the oil deformation A a reult, the incident eimic wave are cattered the eimic ecitation to which the caion i effectively ubjected may differ coniderably from the free field motion To preent the analyi reult, the following dimenionle parameter are defined: the effective diplacement ratio at the top of the caion eff kin ( ) ( ) = u u ( H, ) the effective rotation at the top of the caion eff ff kin ( ) ( ) = u ( H, ) the amplification ratio at the top of the oil layer ff (9) () A ff () = u ff (H,) / u ff (,) () the amplification at the top of the caion A cc () = u kin () / u ff (,) () Two dimenionle frequency parameter facilitate the preentation of reult: c = H / V i a parameter with repect to the thickne of the depoit, = / V i a parameter with repect to the depth of embedment It ha been proven that the repone i governed for each frequency by the following two parameter: the oil layer depth to the depth of embedment ratio H/ the hear modulu ratio G /G r Figure how the influence of the firt parameter upon the behavior of the caion Apparently, the deep caion do not follow the oil motion at higher frequencie Finally, Fig (a) (b) how the variation of the eff the eff with dimenionle frequency ( ), repectively 5 Inertia interaction he effective period of the ytem i controlled by the following three dimenionle parameter: () Slenderne ratio, h/b () Ratio of mae of the bridge the caion, m tr () Relative tiffne of the fied bae tructure the oil, tr Figure how the influence of the lenderne ratio on the effective period of the ytem For mall lenderne ratio, the effective period ha a greater increae than for higher lenderne ratio Thi can be 7 A 6 5 FEM Propoed model u tot /(u kin + kin h) 5 /B= FEM Propoed model C =H/V Fig 8 Comparion of amplification factor from the propoed model & olution of FEM for kimematic interaction (=85, =5, /B=6, H/B=) Fig / tr Inertia interaction: comparion with FE olution (G=5MPa, =85, B=m, E=GPa, I =6m, m tr =Mg, h=m)

8 EARTHQUAKE ENGINEERING AN ENGINEERING VIBRATION Vol7 T y /T tr 8 6 Propoed olution Veleto (977) A cc A ff /H= /H= /H=6 A ff tr 6 8 c =H/V Fig Comparion of the tructure period from the propoed model & that propoed by Veleto (977), (m tr =8, h/b =) Fig Comparion of the tranfer function between top (z=h) bae (z=) diplacement amplitude, for the free filed (A ff ) the caion (A cc ) Square caion with /H=, 6( = 8, = 5) 5 /B= /B= /B=6 5 5 /B= /B= /B=6 eff eff =/V (a) Normalized effective diplacement of the top of the caion 6 8 =/V (b) Normalized effective rotation of the caion Fig Variation of eff eff with dimenionle frequency for quare caion with lenderne ratio /B=,, 6( = 8, / H = 6 ) eplained by the fact that in tall, lender tructure, the predominant diplacement i the rotation againt which a caion behave very rigidly Figure how the influence of the ratio of the mae The effective period of the ytem increae monotonically with the ma ratio When the ratio tend to zero, the ma of the caion i very big compared to the ma of the bridge, thu the upertructure act a if it i founded on a fied bae Figure 5 how the influence of the term tr on the effective period of the ytem A the ratio increae (the tructure may be baed on a oft oil), the period alo increae The following equation i propoed for the effective period of the ytem compared to the fied-bae period of the upertructure T T y tr h + B tr tr m m c h V () The damping ratio encompae the hyteric damping in the oil, the radiation damping the tructural damping The frequency of the ytem i given by the following equation y,dam = y () A = (5)

9 No C Tiggino et al: Seimic repone of bridge pier on rigid caion foundation in oil tratum Solving Eq (5) for, one obtain = where A i defined a A = u kin A A (6) utr + h kin (7) The reult, which follow, are given for a contant hyteric damping equal to 5% for both the oil the tructure Three different type of oil are eamined The firt two oil cae are etreme condition, where the cut-off frequency tend to zero infinity, repectively The influence of the lenderne ratio on the damping ratio for two different oil condition i hown in Fig 6 For tall, lender tructure, the damping ratio i equal to the tructural damping ratio, for tructure with a ratio h/b<, the damping ratio increae Thi can be eplained by the fact that tall lender tructure are more prone to tructural deformation rotation than to lateral caion diplacement Figure 7 how the influence of the ma ratio on the effective damping A the ma ratio decreae, the periphery of the caion increae, thu more eimic wave are emitted from the periphery Therefore, the radiation damping play a ignificant role in the behavior of the upertructure Figure 8 give the influence of the term tr on the effective damping A the ratio increae, the damping ratio alo increae Note that large value of the relative tiffne are related to oft oil Finally, the influence of /B on the damping ratio 5 tr = h/b= h/b= T y /T tr tr = T y /T tr h/b 5 tr Fig Effect of lenderne ratio on the effective period of the ytem (J =, m tr /m c =) Fig 5 Effect of the relative tiffne of oil tructure, tr, on the effective period of the ytem 8 T y /T tr 8 6 h/b= h/b= tr =55 tr = Fig m tr /m c h/b Effect of the relative ma ratio on the effective period of the ytem (J =, tr =) Fig 6 Effect of the lenderne ratio on the total damping (radiation plu hyteric tructural damping) for two different oil (J =, m tr =8, /B=)

10 EARTHQUAKE ENGINEERING AN ENGINEERING VIBRATION Vol7 5 h/b= h/b= h/b= 5 Total cut-off h/b= h/b= h/b= 5 5 Fig m tr /m c The effect of the relative ma ratio on the total damping, radiation plu hyteric tructural damping, for three different lenderne ratio (J =, tr h / V =, /B=) tr Fig The effective damping for a oil with total cut-off condition (f<f c ) The effective damping i approimately the tructural damping( m tr = 8, /B=) 5 5 h/b= h/b= h/b= 8 Total cut-off Real condition Homogeneou 5 A 6 5 tr Fig 8 The effect of the relative tiffne of the oil the tructure, on the effective damping, for three different lenderne ratio (J =, m tr =8, /B=) Fig 6 8 / tr The importance of different amount of radiation damping on the reonant peak( tr h / V =75, m tr /m c =8, h/b=5, /B=) 5 h/b= h/b= h/b= h/b= /B Fig 9 The effect of the /B on the total damping,radiation plu hyterical tructural damping, for two different lenderne ratio( tr =, m tr = 77) i given in Fig 9 A the /B increae, the damping ratio increae Thi can be eplained by the fact that increaing thi ratio under a contant relative ma ratio, the perimeter of the caion alo increae, which lead to an increae of the radiation damping The effective damping for oil in total cut-off condition i approimately the ame a the tructure damping, a hown in Fig Figure how the importance of radiation damping in the behavior of the ytem In the final type of oil, there are three ditinct region for the radiation damping: When y / oil < 8, no radiation damping develop, When y / oil >, the full radiation damping of the homogeneou half pace develop, When 8 < / y oil <, radiation damping take intermediate value

11 No C Tiggino et al: Seimic repone of bridge pier on rigid caion foundation in oil tratum 6 Concluion An analytical method to tudy the eimic repone of a bridge pier upported on a rigid caion foundation embedded in a deep oil tratum i developed The method reproduce the kinematic inertial repone Cloed-form olution are given in the frequency domain for vertical harmonic S-wave ecitation From the comparion with reult from finite element (FE) analyi other available olution, the reliability of the propoed model i verified the following concluion can be drawn () The propoed model yield atifactorily reult in the firt econd eigenmode; in other word, for value of dimenionle frequency a le than () The modification of the eimic wave i more intene in caion with mall value of the oil layer depth the caion embedment depth ratio H/ () The increae of the period of the tructure due to oil-caion-tructure interaction i controlled by the lenderne ratio, the relative mae the relative tiffne of the tructure the oil, the increae can be very ignificant for tiff tructure with mall value of the lenderne ratio Even for fleible tructure with the lenderne ratio greater than ten founded in oft oil, the period of the ytem can be increaed by 5% () The increae of the damping ratio of the ytem i due to the radiation damping Specifically, thi increae i controlled by the lenderne ratio, the relative mae, the relative tiffne of the tructure in a fied bae condition the oil, the depth of embedment The mot important parameter i the lenderne ratio, becaue for high value of thi ratio, the radiation damping tend to be zero Furthermore, a relatively large ma of the caion compared to the ma of the upertructure lead to an increae in the radiation damping In the cae of a real oil layer with a depth of H, the period of the ytem influence it damping ratio If the period of the ytem i maller than the period of the oil layer, the damping ratio will be the damping of the homogenou oil condition Converely, if the period of the oil layer i maller than the period of the tructure, the damping ratio will be equal to the tructural damping Reference Aimaki (998), Static ynamic Analyi of Caion Subjected to Lateral Static Loading, iploma Thei, NTUA, Athen (in Greek) Aimaki, Chatzigianneli I, Gerolymo N Gazeta G (), Lateral Repone of Caion Foundation, Proc th National Conference on Geotechnical Geoenvironmental Engineering, Athen, Greece, June (in Greek) Bielak J (975), ynamic Behavior of Structure with Embedded Foundation, Earthquake Engineering Structural ynamic, : 59-7 Chatzigianneli I (999), Lateral Static ynamic Analyi of Caion in Non Homogeneou-Halfpace, iploma Thei, NTUA, Athen (in Greek) Fotopoulou M, Kontantopoulo P, Gazeta G Taoula JL (989), Rocking amping of Arbitrarily-haped Embedded Foundation, Journal of Geotechnical Engineering, ASCE, 5(): 7-89 Gazeta G (98), Analyi of Machine Foundation Vibration: State of the Art, Soil ynamic Earthquake Engineering, (): - Gazeta G (99), Formula Chart for Impendance of Surface Embedded Foundation, Journal of Geotechnical Engineering, ASCE, 7(9): 6-8 Gazeta G Taoula J (987a), Horizontal Stiffne of Arbitrarily Shaped Embedded Foundation, Journal of Geotechnical Engineering, ASCE, (5): -57 Gazeta G Taoula J (987b), Horizontal amping of Arbitrarily Shaped Embedded Foundation, Journal of Geotechnical Engineering, ASCE, (5): Gerolymo N Gazeta G (7), Winkler Model for Lateral Repone of Rigid Caion Foundation in Linear Soil, Soil ynamic Earthquake Engineering, 6: 7-6 Gerolymo N, Gazeta G Mylonaki G (998), Foundemental Period Effective amping of Pileupported Bridge Pier, th European Conference on Earthquake Engineering, Topic Kauel E Roeet JM (97), Soil-tructure Interaction for Nuclear Containment Structure, Proceeding ASCE, Power iviion Specialty Conference, Boulder, Colorado Kramer S (996), Geotechnical Earthquake Engineering, EngleWood Cliff: Prentice Hall Mylonaki G (), Elatodynamic Model for Large iameter End-bearing Shaft, Soil Foundation, (): - Mylonaki G, Nikolaou A Gazeta G (997), Soilpile-bridge Seimic Interaction: Kinematic Inertial Effect, Part I: Soft Soil, Earthquake Engineering Structural ynamic, 6: 7-59 Veleto AS (977), ynamic of Structure Foundation Sytem, Structural Geotechnical Mechanic, NM Newmark Honoring Volume, Prentice Hall, pp -6 Whitman RV (97), Analyi of Soil-tructure Interaction: State-of-the-Art Review, Eperimental Structural ynamic, Intitute of Sound Vibration, Southampton