A Coupled BEM Model for the Dynamic Analysis of a Pile Embedded in a Half-space Soil Covered by a Water Layer

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1 Sed Order of Rert at 6 he Oe vl Egeerg Joural Oe Acce A ouled BEM Model for the Dyamc Aaly of a Ple Embedded a Half-ace Sol overed by a Water Layer Xu Zhag ad Ja-Fe Lu * Deartmet of vl Egeerg Jagu Uverty Zheag Jagu P.R. ha Abtract: Dyamc aaly of a le embedded a half-ace ol covered by a ater layer crucal for the deg of the le foudato for brdge dock ad offhore latform etc. I th aer a couled boudary elemet method (BEM model develoed to evaluate the dyamc reoe of the le. I the rooed model the le ad half-ace ol are treated a elatc meda hle the ater layer codered a a acoutc medum. hree BEM formulato are etablhed for the le half-ace ol ad ater layer by mea of the boudary elemet method (BEM reectvely. Ug the three BEM formulato a ell a the cotuty codto at the terface betee three rego a couled BEM model for the le-ol-ater ytem etablhed. o valdate the rooed model reult due to our model are comared th etg reult. Wth the couled BEM model for the le-ol-ater ytem dyamc reoe of the le vetgated. Preeted umercal reult ho that he the le ubected to a aal load ad torque reoace heomea ot obvou. Hoever he the le ubected to a horzotal load ad momet reoace heomea roouced ad the le-ol modulu ad dety rato have a coderable fluece o reoat frequece. Keyord: Ple; the half-ace ol; the ater layer; the boudary elemet method (BEM; comlace.. INRODUION he aaly of the dyamc reoe of le foudato embedded the half-ace ol ha receved coderable atteto durg the at fe decade. Makr [ ] ued the Wkler model to vetgate the dyamc teracto betee le ad ol. Kuhlemeyer [ ] ad Wu & F [] emloyed the fte elemet method to tudy the dyamc reoe of le. Zeg & Raaake [6 7] a ell a Raaake & Shah [8 9] utlzed the tegral equato method to vetgate the dyamc teracto betee le ad the half-ace ol. A the tegral equato method ca mlfy le a oe-dmeoal tructure ad hadle the half-ace ol by em-aalytcal method the threedmeoal le-ol teracto roblem ca thu be mlfed ubtatally by the method. A a reult the tegral equato method ha bee emloyed to olve varou leol teracto roblem o far [-]. Bede the aforemetoed method the boudary elemet method ha alo bee ued dely to deal th the le-ol teracto roblem. For eamle Mamoo [] ued a hybrd boudary elemet formulato to evaluate the medace ad comlace fucto of le ad le grou. Se [] ued a boudary elemet formulato for the dyamc aaly of aally ad laterally loaded le ad le grou embedded a hyteretc elatc half-ace. Wag & Baeree [6] emloyed a aymmetrc boudary elemet formulato for the aaly of le-ol teracto. Moreover the teady *Addre correodece to th author at the Deartmet of vl Egeerg Jagu Uverty Zheag Jagu P.R. ha; el: 86-98; Fa: ; E-mal: lfdoctor@yahoo.com tate dyamc reoe of a le grou embedded a oroelatc half-ace ol vetgated by mea of the boudary elemet method for a aturated orou medum [7]. It oted that although may reearche cocerg dyamc le-ol teracto roblem have bee coducted o far mot etg reearche are retrcted to the dyamc aaly of le embedded a half-ace ol. Hoever for the le foudato aocated th cro-ea brdge dock ad offhore latform the half-ace ol uually covered by a ater layer. I th cae the fluece of the ater layer o the dyamc reoe of le foudato hould be accouted for. A a reult to etmate the dyamc reoe of le th tuato roerly the coulg betee the le half-ace ol ad ater layer hould be take to accout. It orth emhazg that dyamc charactertc of th kd of le are crucal for the dyamc aaly of cro-ea brdge dock ad offhore latform ubected to varou dyamc load uch a emc ave curret load ater ave d load ad movg load due to vehcle etc. Hoever a oted above there have bee very fe reearche carred out about the dyamc reoe of le embedded a half-ace ol covered by a ater layer. A a reult to accomlh a ucceful dyamc aaly for the le foudato of brdge ad varou offhore faclte t of gfcace to develo a couled model for the dyamc aaly of le embedded a half-ace ol th a overlyg ater layer. he obectve of th tudy to develo a couled leol-ater model to aalyze the dyamc reoe of a gle le embedded a half-ace ol covered by a ater layer. 87-9/ Betham Oe

2 Half-ace Sol overed by a Water Layer he Oe vl Egeerg Joural Volume 7 7 I the rooed model the le ad half-ace ol are treated a elatc meda hle the ater layer codered a a acoutc medum. hree boudary elemet method (BEM formulato are etablhed for the le half-ace ol ad ater layer by mea of the boudary elemet method reectvely. Ug the three BEM formulato a ell a the cotuty codto at the terface betee three rego a couled BEM model for the le-ol-ater ytem etablhed. Wth the rooed BEM model for the le-ol-ater ytem ome umercal reult for the dyamc reoe of the le are reeted th tudy.. BOUNDARY INEGRAL EQUAIONS FOR HE PILE HALF-SPAE SOIL AND WAER LAYER A oted above the le ad half-ace ol th tudy are treated a the elatc medum hle the ater layer codered a a acoutc medum. I th ecto the boudary tegral equato for the elatc medum ad acoutc medum ll be outled... Boudary Itegral Equato for the Elatc Medum A th aer cocered th the frequecy doma aaly of a le embedded a half-ace ol overla by a ater layer the Fourer traform for the tme volved. I th tudy the Fourer traform for the tme defed a follo [8]: ˆ( ( t f f t e dt ( ˆ t f t f( e d ( hch t ad rereet the tme ad agular frequecy reectvley the varable th a caret deote the frequecy doma varable. Note that a th tudy retrcted to the frequecy doma aaly of the le for brevty the caret deotg the frequecy doma varable ll be droed for all forthcomg frequecy doma varable. he equato of moto for a elatc medum the frequecy doma a follo [9]: u ( u u ( here ad are the Lame cotat of the elatc medum; u the dlacemet; the dety. he cottutve relato for the elatc medum a follo [9]: e ( here deote the tre comoet of the elatc medum; the tra comoet; e deote the bulk tra; deote the Kroecker delta. Baed o the dyamc recrocal theorem the frequecy doma boudary tegral equato for a elatc medum ca be obtaed a follo []: ( G ( G cu ( [ U ( y t ( ( u ( ] d( ( ( G ( G here U are the Gree fucto for the elatc medum ad are gve APPENDIX []; u t are the dlacemet ad tracto alog the boudary of the elatc medum; c the coeffcet for the boudary ad deote the boudary of the elatc medum... Boudary Itegral Equato for A acoutc Medum A oted revouly the ater layer th tudy treated a a acoutc medum. hu for the ater layer t reure decrbed by the follog Helmholtz equato []: k k ( v hch the reure of the ater layer; k the aveumber of the ater; v the acoutc velocty of ater. he dlacemet of the ater layer ha the form []: q (6 here q deote the dlacemet vector for the ater layer ad the dety of ater. Ug the dyamc recrocal theorem the frequecy doma boudary tegral equato for the ater layer a follo [ ]: c y q y q y y d y (7 ( G ( G ( [ ( ( ( ( ] ( ( G ( here ad q G are the Gree fucto for the ater; c the coeffcet for the boudary ad deed uo the local geometry at the ot. Note that q ( y the dlacemet of the ater alog the ormal drecto ad gve by ( y q ( y (8 y ( here y ( the outard ormal for the boudary ot y at. he Gree fucto equato (7 are gve by y e r ( G ( [ k r ] hch r ( y ( y. ( q y ( r G ( [ k r y e ]. BOUNDARY ELEMEN FORMULAIONS FOR HE PILE HALF-SPAE SOIL AND WAER LAYER I th ecto accordg to the boudary tegral equato for the elatc medum ad acoutc medum three boudary elemet formulato for the le half-ace ol ad ater layer are etablhed reectvely. A ho Fg. ( for a le embedded a half-ace ol overla by a ater layer three terface ad to boudare et that the terface betee the le ad half-ace ol ( the terface betee the le ad ater layer ( the boudary of the le to ( the terface betee the half-ace (9

3 8 he Oe vl Egeerg Joural Volume 7 Zhag ad Lu Г the ater layer y Г o Г L odg to the ot ; N ( the -th hae fucto; ( ( ad are the coordate for the ot ad -th ode of the -th elemet the global coordate ytem; ( ( u ad u deote the dlacemet at the ot ad -th ode of the -th elemet; t ( ( ( ( t ( ( q ad q have the mlar meag. Г the half-ace ol the le ol ad ater layer ( ad the boudary of the ater layer (. hu the overall boudary of the le cot of ad reectvely; the boudary of the half-ace ol comoed of ad reectvely; the boudary of the ater layer comre ad reectvely. he frequecy doma tegral equato for the le half-ace ol ad ater layer ca be dcretzed f utable umber of boudary elemet are ued to cover the correodg boudare. I th tudy for the coveece of eforcg the cotuty codto betee dfferet rego all boudare of the le half-ace ol ad ater layer are dcretzed by the ame tye of o-arametrc boudary elemet. Suoe that the boudare of the le half-ace ol ad ater layer are dcretzed by the ame tye of oarametrc elemet th each boudary elemet cotag Nd ode. hu for a ot de the -th elemet the follog terolato formulae hold for the le half-ace ol ad ater layer [] N d ( ( ( N( N d ( ( ( t ( N( t N d ( ( ( ( ( N z Fg. (. A chematc llutrato of a le embedded a half-ace ol covered by a ater layer. N d ( ( ( ( ( u N u N d ( ( ( ( ( q N q ~ ( here the uercrt deote the elemet umber; the local ode umber; are the trc coordate corre- Г L.. Boudary Elemet Formulato for the Ple By ug the boudary tegral equato for the le ug the dcretzato cheme a outled equato ( ad erformg the tegrato of the hae fucto kerel roduct over all the boudary elemet for the le the follog boudary elemet formulato for the le obtaed: H U =G ( here the uercrt deote the le; G ad H equato ( are the coeffcet matrce obtaed by tegratg hae fucto kerel roduct over all boudary elemet of the le; U ad are the geeralzed dlacemet ad tracto vector of the ode of the boudary elemet of the le. Dvdg U ad to three art correodg to ad reectvely ad arttog the coeffcet matrce G ad H accordgly oe ha the follog matr equato: U H H H U = G G G ( U here the ubcrt ad rereet the boudare ad of the le reectvely (Fg. ; U ad ( =~ deote the geeralzed dlacemet ad tracto vector for the boudare ( =~ of the le reectvely; H H H G G ad G are the ub-matrce of the coeffcet matrce G ad H of the le correodg to the boudare ad reectvely. I the covetoal BEM formulato the geeralzed dlacemet ad tracto vector U ad ( =~ for the boudare ( =~ of the le have the follog ereo: U [ u ( u ( u ( ] =~ ( ( ( ( ( ( N [ t ( t ( t ( ] =~ ( ( ( ( ( ( N ( ( ( ( ( ( ( ( u ( [ ( ( ( ] u uy uz ~ N =~

4 Half-ace Sol overed by a Water Layer he Oe vl Egeerg Joural Volume 7 9 ( ( ( ( t ( [ t ( ty ( tz ( ] ~ N =~ ( ( hch ( =~ ~ N deote the -th ode of ( the boudary ( =~ of the le; u ( ad ( t ( are the dlacemet ad tracto vector for the - th ode at the boudary ( =~ of the le; N the umber of the ode belogg to the boudary ( =~ of the le ad the uercrt deote the traoe of a vector or a matr. o facltate the mlemetato of the cotuty codto betee the le ad ater layer t eceary to rearrage the dlacemet ad tracto vector U ad alog the boudary a follo: ( ( ( ( y z ( ( ( ( U [ U U U ] [ ] y z U [ u ( u ( u ( ] ( ( ( N [ t ( t ( t ( ] yz ( ( ( N ( ~ N (.. Boudary Elemet Formulato for the Half-ace Sol Aalogouly the boudary elemet formulato for the half-ace ol ca alo be etablhed by dcretzg the correodg boudary tegral equato ad tegratg the hae fucto kerel roduct over all boudary elemet of the half-ace ol. Dvdg boudary dlacemet ad tracto vector to to art correodg to ad reectvely ad arttog the coeffcet matrce accordgly oe ha the follog boudary elemet formulato for the half-ace ol: U H H = G G U here the uercrt deote the half-ace ol; U ( ad ( = deote the geeralzed dlacemet ad tracto vector for the boudare (= of the halface ol reectvely ad they have the mlar rereetato a ho equato (; H H G ad G are the ub-matrce of the coeffcet matrce of the half-ace ol correodg to the boudare ad of the halface ol reectvely. Smlarly for coveece of eforcg the cotuty codto betee the half-ace ol ad ater layer the geeralzed dlacemet ad tracto vector at the boudary of the half-ace ol are rearraged a follo: ( ( ( ( U [ U U U ] [ ] (6 ( ( ( ( y z y z here the geeralzed dlacemet ad tracto vector U ad ( yz are gve by the mlar ereo a ho equato (... Boudary Elemet Formulato for the Water Layer By ug the boudary tegral equato for the ater layer ad mlemetg the mlar boudary dcretzato cheme a th the le ad half-ace ol the boudary elemet formulato ca alo be obtaed for the ater layer. Lkee dvdg boudary uko to three art correodg to ad reectvely ad arttog the coeffcet matrce accordgly oe ha the follog boudary elemet formulato for the ater layer: P Q ( ( ( ( ( ( H H H P G G G Q P Q ( ( ( P ( ( ( N ( ( ( Q q( q( q( N ( ~ N = (7 here the uercrt deote the ater layer; P ad ( = deote the geeralzed reure ad ormal Q dlacemet vector alog the boudare (= of the ater layer reectvely; ( ( ( ( ( ( H H H G G ad G are the ub-matrce of the coeffcet matrce of the ater layer correodg to the boudare (= of the ater layer reectvely.. HE OUPLED BEM MODEL FOR HE PILE HALF-SPAE SOIL AND WAER LAYER By mea of the boudary tegral equato for the elatc medum ad acoutc medum the three boudary elemet formulato have bee etablhed for the le halface ol ad ater layer reectvely. I th ecto the couled le-ol-ater BEM model ll be develoed ug the aforemetoed three boudary elemet formulato a ell a the boudary codto ad cotuty codto at the terface betee the three rego... Boudary odto ad otuty odto at the Iterface betee the hree Rego Boudary codto are gve at the to of the le ( ad the urface of the ater layer (. he tre boudary codto at the to of the le ( determed by the load actg o the to of the le. hu t aumed that alog the boudary of the le equato ( ko a ror. Alo a the urface of the ater layer uoed to be free of reure ad tracto the reure vector P alog the boudary of the ater layer equato (7 thu vahe.

5 he Oe vl Egeerg Joural Volume 7 Zhag ad Lu At the terface betee the le ad half-ace ol ( the cotuty codto for the dlacemet ad tracto have the follog form (Fg. : U U (8 At the terface betee the le ad ater layer ( the cotuty codto for dlacemet ad tracto have the follog form (Fg. : ( NU NyUy Q NP y NyP (9 z here N ad N y are the drecto coe matrce alog the ad y drecto for the ode at the boudary of the le hch are gve by the follog ereo: ( ( ( y ( N N y ( ( N ( ( y N NN NN ( ~ N ( At the terface betee the half-ace ol ad ater layer ( the follog cotuty codto for dlacemet ad tracto hold (Fg. : U Q P ( z y z.. Dervato of the ouled BEM Model for the Ple- Sol-ater Sytem By ug equato ( ad (9 the follog equato for the le obtaed: U H H H U G ( G N Gy N y P G U ( hch G ad G y are the ub-matrce of G correodg to ad y equato ( reectvely; a oted above at the rght-had de of equato ( aumed to be gve by the boudary codto. By ug equato ( a ell a cotuty codto (8 ad ( the follog equato for the half-ace ol obtaed: U H H G Gz P U ( hch G z the ub-matr of G correodg to the tracto vector z equato (6. By ug equato (7 a ell a cotuty codto (9 ad ( the follog equato derved for the ater layer: ( U ( ( ( ( G N N y ( P G Uz G Q H P H P U y ( Note that dervg equato( the aforemetoed tracto free boudary codto alog the urface of the ater layer ued. Fally combato of equato ( ( ad ( yeld the follog equato for all uko boudary varable of the le-ol-ater ytem: G E E E hch H -G H E H G ( G Λ G N Gy Ny H E H ( ( -H G Λ E -Gz ( ( -H G ( ( ( ( ( ( [ U U ] [ P U U ] [ P Q ] Λ N N y NN Λ [ NN ] NN I (6 NN Soluto of equato ( gve all uko varable alog the boudare of the le-ol-ater ytem. Oce the boudary varable are determed the varable at the teral ot of the le half-ace ol ad ater layer ca be obtaed ug the correodg dcrete boudary tegral equato for teral ot.. NUMERIAL MEHOD AND NUMERIAL RE- SULS FOR HE DYNAMI RESPONSE OF HE PILE he couled BEM model for the le-ol-ater ytem ha bee develoed the above ecto. I th ecto umercal cheme for the couled BEM model of the le-olater ytem ll be outled brefly. Alo ome umercal reult obtaed by the rooed model ll be reeted... Numercal Scheme for the BEM Model of the Ple- Sol-Water Sytem he le th tudy aumed to have a crcular cro ecto (Fg.. he boudare of the le half-ace ol ad ater layer are dcretzed by the ame -D eght-ode oarametrc boudary elemet [] reectvely. he bottom ad to of the le are dcretzed by telve eghtode oarametrc elemet reectvely ad the dcretzato cheme hoed Fg. (a. he de of the le dvded to everal egmet evely ad each egmet of the le dcretzed by eght eght-ode oarametrc elemet evely. he dcretzato of the terface betee the half-ace ol ad ater layer a ell a the urface of the

6 Half-ace Sol overed by a Water Layer he Oe vl Egeerg Joural Volume 7 ater layer accomlhed by trucatg the correodg fte boudare th everal horzotal ad vertcal boudary elemet layer a chematcally ho Fg. (b. Note that order to dcretze the above fte boudare ecoomcally the ze of the elemet dfferet boudary elemet layer may creae gradually th creag dtace from the le (Fg. (b... Numercal Reult ad orreodg Aaly I th ecto baed o the rooed BEM model for the le-ol-ater ytem fve umercal eamle ll be reeted. I the frt eamle the le-ol-ater ytem th tudy reduced to the commo le ad half-ace ol ytem by lettg the thcke of the ater layer aroach zero. he oluto for th ecal cae of our model comared th etg reult for the le half-ace ol roblem. I the ecod eamle the fluece of the le-ol dety rato o the horzotal dlacemet ad bedg momet of the le ll be eamed. I the thrd ad fourth eamle the fluece of the le-ol modulu ad dety rato o the comlace of the le ll be tuded. I the ffth eamle the fluece of the ater layer o the comlace of the le ll be eamed. Note that calculato the tycal value for the materal ad geometrc arameter for the le half-ace ol ad ater layer are gve able. he le to dlacemet ad the force aled at the to of the le are related to each other by the comlace of the le hch ca be rereeted by the follog equato: V FV HH HM H F H (7 MH MM M here V H ad are the vertcal dlacemet horzotal dlacemet rotato agle ad tt agle at the to of the le; F F M ad are the vertcal force V H able. ycal Value for the Parameter of the Ple Halface Sol ad Water Layer Parameter he hear modulu of the ol ( he Poo rato of the ol ( Dety of the ol ( he bulk modulu of the ater ( K he dety of the ater ( he hear modulu of the le ( Value horzotal force momet ad torque aled at the to of the le; HH HM MH MM ad are the comlace of the le. Note that due to the recrocty theorem elatodyamc [] comlace HM ad MH equato (7 are equal. A the le equato (7 codered a a oe-dmeoal tructure the vertcal dlacemet ad tt agle thu oly deed o the vertcal force ad torque reectvely. Sce the le th tudy treated a a three-dmeoal tructure the le to dlacemet V H ad equato (7 hould be codered a the oe-dmeoal equvalet dlacemet to the correodg three-dmeoal couterart hch ca be 7. Pa.. kg/m 9. Pa. kg/m. Pa he Poo rato of the le (. he dety of the le ( he dameter of the le ( d he legth of the le mmered the ater layer L ( he legth of the le embedded the half-ace ol ( L. kg/m m m m V o (a y y (b Fg. (. he BEM dcretzato cheme for the to ad bottom of the le a ell a the terface betee the half-ace ol ad ater layer ad the urface of the ater layer: (a the dcretzato for the to ad bottom of the le; (b the dcretzato for the terface betee the half-ace ol ad ater layer a ell a the urface of the ater layer.

7 he Oe vl Egeerg Joural Volume 7 Zhag ad Lu determed by ug the correodg three-dmeoal dlacemet of the boudary elemet at the le to. Lkee the force F V F H M ad hould be codered a the reultat force of the tracto aled at the boudary elemet of the le to.... omaro of Preet Reult th Etg Reult A oted above he the thcke of the ater layer ted to zero the le-ol-ater ytem th tudy reduced to the commo le half-ace ol ytem. he comlace of the le calculated for th ecal cae ad comared th etg reult []. he comlace fucto ormalzed th reect to the correodg tatc comlace for the ame le ad the reult are lotted veru the dmeole frequecy a (Fg.. he dmeole frequecy a defed a follo d a (8 v here d the dameter of the le ad v the hear ave velocty of the half-ace ol. I the umercal mulato the le dvded evely to tety egmet. Stee horzotal ad tee vertcal boudary elemet layer are ued the dcretzato of the terface betee the halface ol ad ater layer a ell a the urface of the ater layer (Fg.. Fg. ( llutrate that there very good agreemet betee the reult of th aer ad thoe reeted by Mamoo et al. [] valdatg the rooed model of th aer.... he Ifluece of the Ple-Sol Dety Rato o the Dyamc Reoe of the Ple I th ecto the fluece of the le-ol dety rato o the dyamc reoe of the le ll be eamed. Ecet the dety of the le other arameter for the le half-ace ol ad ater layer take the tycal value a gve able. he dety of the le ( take the value to make /...7 reectvely hch / he real art of the reet aer he magary art of the reet aer he real art of Mamoo et al. [] he magary art of Mamoo et al. [] Fg. (. omaro of reet reult for the comlace / of a le embedded a half-ace ol th the follog arameter: L/ d E / E /.7 th the reult due to Mamoo et al. []. the ubcrt ad deote the le ad half-ace ol reectvely. A ut horzotal load aled at the to of the le ad the frequecy equal to Hz. I the umercal mulato the le embedded the ol dvded evely to tety fve egmet ad the art mmered the ater layer dvded evely to thrtee egmet. Stee horzotal ad tee vertcal boudary elemet layer are ued the dcretzato of the terface betee the half-ace ol ad ater layer a ell a the urface of the ater layer (Fg.. Fg. ( lot the amltude of the horzotal dlacemet ad bedg momet of the le for the three dfferet dety rato ( /. Fg. (a ho that he z/( L L. the horzotal dlacemet of the le ha a teady decreae th creag deth. Hoever he 6.E-8.E-8.E-8 u (m / =. f=hz / =. / =.7 M (N.m / =. / =. / =.7.E-8 f=hz.e-8.e z/(h +h z/(h +h (a (b Fg. (. he fluece of the le-ol dety rato o the dyamc reoe of the le he the to of the le ubected to a horzotal load th frequecy f Hz: (a the horzotal dlacemet of the le; (b the bedg momet of the le.

8 Half-ace Sol overed by a Water Layer he Oe vl Egeerg Joural Volume 7 z/( L L. the horzotal dlacemet of the le almot kee uchaged. Fg. (a alo llutrate that he z/( L L. the horzotal dlacemet of the le decreae coderably th creag dety rato ( /. Moreover t follo from Fg. (b that he z/( L L. the bedg momet of the le mall. Neverthele he z/( L L. the bedg momet of the le almot creae learly th creag deth. Bede he z/( L L. the bedg momet of the le decreae rooucedly th the creae of the dety rato ( /.... he Ifluece of the Ple-Sol Modulu Rato o the omlace of the Ple I th ecto baed o the rooed BEM model for the le-ol-ater ytem the fluece of the le-ol modulu rato o the comlace of the le vetgated. he comlace of the le are defed equato (7. Ecet the modulu of the le ( E all materal ad geometrc arameter for the le half-ace ol ad ater layer aume the tycal value a gve by able. he modulu of the le take the value to make E / E reectvely. I the umercal mulato the boudary dcretzato cheme for the le half-ace ol ad ater layer are the ame a thoe Secto... Fg. ( llutrate the comlace of the le veru the dmeole frequecy. Alo a oted above a the comlace HM ad MH are equal to each other the comlace MH thu omtted Fg. (. Fg. (a ho that th creag modulu rato E / E the real art of the comlace decreae coderably hle the magtude of the magary art decreae lghtly. Fg (b-d ho that the real ad magary art of comlace creae lghtly the frequecy rage a.. I the frequecy rage. a.7 the dlayed comlace of the le ocllate ad a reoace eak occur for each modulu rato. Furthermore a eected th creag modulu rato the reoat frequecy creae. I the frequecy rage.7 a. the amltude of comlace decreae th creag frequecy. It follo from Fg. (e that both the real ad magary art of the comlace creae lghtly th the creae of frequecy ad the comlace creae more rooucedly he the modulu rato take maller value.... he Ifluece of the Ple-Sol Dety Rato o the omlace of the Ple I th ecto the fluece of the le-ol dety rato o the comlace of the le ll be vetgated. Ecet the dety of the le all materal ad geometrc arameter for the le half-ace ol ad ater layer take the tycal value a gve by able. he dety of the le ( take the value makg /...7 reectvely. I the umercal mulato the boudary dcretzato cheme for the le half-ace ol ad ater layer are the ame a thoe Secto... Fg. (6a llutrate that the real art of the comlace decreae ad the magary art creae th the creae of frequecy. At the lo frequecy rage the dfferece betee the comlace for dfferet dety rato are mor. Hoever th creag frequecy the dfferece betee the comlace for dfferet dety rato become roouced. Alo t follo from Fg (6b-d the frequecy rage a. both the real ad magary art of the dlayed comlace ho a lght creae th creag frequecy. I the frequecy rage. a.7 the reeted comlace ocllate ad a th Secto.. a reoace eak are for each dety rato. I cotrat to Secto.. th creag dety rato the reoat frequecy decreae. Fg. (6e ho that both the real ad magary art of the toroal comlace creae th the creae of frequecy. Alo t follo that the dety rato / doe ot have a obvou fluece o the toroal comlace of the le.... he Ifluece of the Water Layer o the omlace of the Ple I th ecto the fluece of the ater layer o the comlace of the le ll be eamed. o vetgate the fluece of the ater layer three cae for the le are codered th ecto amely the cae for the le embedded the half-ace ol covered by the ater layer (the ol-ater cae the cae here the ater layer relaced by the ol (the ol cae ad the cae hch the ater layer abet (the abet cae.e. the ater layer relaced by the ar. Note that the ecod cae correod to the crcumtace hch the le embedded the halface ol hle the thrd cae correod to the tuato here the le eteded above the urface of the halface ol. I all three cae the total legth of the le equal to m. I the frt cae the legth of the egmet of the le embedded the half-ace ol ad mmered the ater layer are both equal to m reectvely. he egmet of the le embedded the ol ad mmered the ater layer are both dvded evely to egmet. I the ecod cae the -meter log le embedded the halface ol dvded evely to 6 egmet. I the thrd cae the -meter log egmet of the le embedded the ol dvded evely to egmet ad the -meter log eteded egmet of the le the ar alo dvded evely to egmet. Other arameter for the le halface ol ad ater layer take the tycal value a gve by able. he dcretzato cheme for the boudary of the le half-ace ol ad ater layer are the ame a thoe Secto...

9 he Oe vl Egeerg Joural Volume 7 Zhag ad Lu.E-9 HH.E-9.E-9.E Real art = = = Imagary art = = =.E- -.E- Real art = = = Imagary art = = = E-8 HM MM.E-8.E-8 -.E-8 -.E-8-6.E-8 Real art = = = Imagary art = = = 8.E-9 6.E-9.E-9.E-9 Real art = = = Imagary art = = = -8.E-8 -.E (c -.E (d.e-9.e-9.e-9.e-9 Real art = = = Imagary art = = =.E Fg. (. he comlace of the le embedded a half-ace ol covered by a ater layer he E / E ad reectvely ad other arameter for the le-ol-ater ytem take the tycal value a gve by able : (a the comlace (e HH ; (c the comlace HM ; (d the comlace MM ; (e the comlace. ; (b the comlace

10 Half-ace Sol overed by a Water Layer he Oe vl Egeerg Joural Volume 7.E-9.E-9.E- -.E- Real art / =. / =. / =.7 Imagary art / =. / =. / = HM.E-8.E-8.E-8 8.E-9 6.E-9.E-9.E-9 -.E-9 -.E-9-6.E-9-8.E-9 -.E-8 Real art / =. / =. / =.7 (a Imagary art / =. / =. / =.7 -.E ( HH.E-7.E-7 Real art.e-7 / =. 8.E-8 6.E-8 / =..E-8 / =.7.E-8 -.E-8 -.E-8-6.E-8 Imagary art -8.E-8 -.E-7 / =. -.E-7 / =. -.E-7 / =.7 -.6E (.E-9 MM.E-9 Real art 8.E- / =. 6.E- / =..E- / =.7.E- -.E- -.E- Imagary art -6.E- -8.E- / =. -.E-9 / =. / =.7 -.E ( 8.E- 6.E-.E-.E- Real art / =. / =. / =.7 Imagary art / =. / =. / = Fg. (6. he comlace of the le embedded a half-ace ol covered by a ater layer he /.. ad.7 reectvely ad other arameter of the le-ol-ater ytem take the tycal value a gve by able : (a the comlace ( HH ; (c the comlace HM ; (d the comlace MM ; (e the comlace. ; (b the comlace

11 6 he Oe vl Egeerg Joural Volume 7 Zhag ad Lu.E-9.E-9.E-9.E- -.E- -.E-9 Real art Ple ol ad ater Ple ol Eteded le ol Imagary art Ple ol ad ater Ple ol Eteded le ol e-7.e-7 8.E-8 6.E-8.E-8.E-8 -.E-8 -.E-8-6.E-8-8.E-8 -.E-7 -.E-7 -.E-7 -.6E-7 -.8E-7 -.E-7 -.E-7 HH Real art Ple ol ad ater Ple ol Eteded le ol Imagary art Ple ol ad ater Ple ol Eteded le ol -.E ( (.8E-8.6E-8.E-8.E-8.E-8 8.E-9 6.E-9.E-9.E-9 HM Real art Ple ol ad ater Ple ol Eteded le ol -.E-9 Imagary art -.E-9-6.E-9 Ple ol ad ater Ple ol -8.E-9 Eteded le ol -.E ( MM.E-9 8.E- 6.E-.E-.E- -.E- -.E- -6.E- Real art -8.E- Ple ol ad ater -.E-9 Imagary art Ple ol -.E-9 Eteded le ol Ple ol ad ater -.E-9 Ple ol Eteded le ol -.6E (.E-9 8.E- 6.E-.E-.E- Real art Ple ol ad ater Ple ol Eteded le ol Imagary art Ple ol ad ater Ple ol Eteded le ol Fg. (7. he comlace of the le for the three cae amely the cae for the le embedded the half-ace ol covered by the ater layer the cae here the ater layer relaced by the ol ad the cae hch the ater layer abet: (a the comlace ; (b the comlace HH ; (c the comlace HM ; (d the comlace MM ; (e the comlace. (

12 Half-ace Sol overed by a Water Layer he Oe vl Egeerg Joural Volume 7 7 Fg. (7a llutrate that the real art of the comlace decreae ad the magtude of the magary art creae th the creae of frequecy. he dfferece of the comlace betee the ol-ater cae ad the abet cae mor. Hoever both the real ad magary art of the comlace for the ol cae are maller tha thoe of the other to cae uggetg that the le the ol cae tffer tha thoe the other to cae ad alo th cae le eergy radated to the half-ace ol due to the larger legth of the le egmet embedded the ol. Fg (7b-d ho that the magtude of the comlace HH HM ad MM of the ol cae much maller tha thoe of the other to cae mlyg aga that the le the ol cae tffer ad le damg tha thoe the other to cae. Fg (7b-d alo ho that the reoace frequece of the ol-ater cae are maller that thoe of the abet cae meag that the ater layer th tuato maly fucto a a ma. Fg. (7e ho that comared th the real art the magary art of the toroal comlace of the three cae are maller ad the dfferece betee the magary art of the comlace of the three cae are lttle. Hoever the dfferece betee the real art of the comlace of the three cae are gfcat. A eected the real art of the comlace of the abet cae the larget hle that of the ol cae the mallet. 6. ONLUSIONS A frequecy doma couled BEM model for a le embedded a half-ace ol covered by a ater layer ha bee rooed th tudy. Obvouly the model develoed th tudy ca be ued the aaly of the dyamc reoe of brdge dock ad offhore latform to varou dyamc load he the le-ol-ater coulg codered. Baed o the rooed model ome umercal reult are reeted. By the reearche coducted th aer the follog cocluo are dra:. omaro of reult of th aer th etg reult for a gle le embedded a half-ace ol valdate the couled BEM model for the le-olater ytem rooed th tudy.. Whe the le ubected to a horzotal load the horzotal dlacemet of the egmet of the le mmered the ater layer are much larger tha that of the egmet embedded the half-ace ol. I cotrat to the horzotal dlacemet the bedg momet of the egmet of the le mmered the ater layer much maller tha that of the egmet embedded the half-ace ol.. I the calculated frequecy rage he ubected to aal load ad torque the le doe ot ehbt obvou reoace heomea. Hoever he ubected to horzotal load ad momet the le ho obvou reoace heomea th reoat frequecy creag th creag le-ol modulu rato ad decreag th creag le-ol dety rato.. he comaro betee the comlace for the three cae amely the ol-ater cae ol cae ad abet cae ho that the tffe of the le of the ol cae the larget amog the three ad the damg of the le of the ol cae the mallet. Alo umercal reult of the comlace for the le of the ol-ater ad abet cae ho that the ma effect of the ater layer gfcat ad domat hle the tffe effect of the ater layer almot eglgble.. Although oly the frequecy doma reoe of the le to the load actg o the le to vetgated th tudy the reoe of the le to other kd of load uch a emc ave curret load ad ater ave etc ca alo be aalyzed by the rooed model. By mea of the Fourer traform method the rooed model ca alo be ued to vetgate the tme doma reoe of the le. Moreover by the rocedure rooed th tudy the roblem for le grou embedded a half-ace ol overla by a ater layer ca alo be olved a traghtforard ay. ONFLI OF INERES he author cofrm that th artcle cotet ha o coflct of teret. AKNOWLEDGEMENS h reearch coducted the frameork of a roect uorted by the Natoal Natural Scece Foudato of ha th grat umber 787. Alo the cotructve commet from to referee are hghly ackoledged by the author. APPENDIX he Frequecy Doma Gree Fucto for a hree- Dmeoal Elatc Medum he frequecy doma Gree fucto U ad for a elatc medum have the follog form [] U r r ( G ( G r r r A r B r r r r r Dr (A. ad e e ( r/ r/ r r r r r r r/ r/ e e ( r r r r r r (A. here r A d / dr / r B / r d / dr D / d / drd / dr / r

13 8 he Oe vl Egeerg Joural Volume 7 Zhag ad Lu ad are the comreve ad hear ave velocte of the elatc medum reectvely. REFERENES [] N. Makr Sol-le teracto durg the aage of raylegh ave: A aalytcal oluto Earthquake. Eg. Struct. vol [] N. Makr ad D. Bado Semc reoe of le grou uder oblque-hear ad Raylegh ave Earthquake. Eg. Struct. vol [] R.L. Kuhlemeyer Vertcal vbrato of le J. Geotech. Geoevro. vol [] R.L. Kuhlemeyer Statc ad dyamc laterally loaded floatg le J. Geotech. Geoevro. vol [] G.X. Wu ad W.D.L. F Dyamc elatc aaly of le foudato ug fte elemet method the frequecy doma a. Geotech. J. vol [6] X. Zeg ad R.K.N.D. Raaake Dyamc aal load trafer from elatc bar to oroelatc medum J. Eg. Mech. vol [7] R.K.N.D. Raaake A ote o the elatodyamc load trafer roblem It. J. Sold. Struct. vol [8] R.K.N.D. Raaake ad A.H. Shah O the logtudal harmoc moto of a elatc bar embedded a elatc half-ace It. J. Sold. Struct. vol [9] R.K.N.D. Raaake ad A.H. Shah O the lateral harmoc moto of a elatc bar embedded a elatc half-ace It. J. Sold. Struct. vol [] Y. a G. he. Xu ad D. Wu oroal reoe of le embedded a oroelatc medum Sol. Dy. Earthquake. Eg. vol [] B. J D. Zhou ad Z. Zhog Lateral dyamc comlace of le embedded oroelatc half ace Sol. Dy. Earthq. Eg. vol [] J.F. Lu ad D.S. Jeg Poro-elatc model for le-orou medum terfacto due to emc ave It. J. Numer. Aal. Method. Geomech. vol [] R.Y.S. Pak ad P.. Jeg Elatodyamc reoe of le uder travere ectato J. Eg. Mech. vol [] S.M. Mamoo A.M. Kaya ad P.K. Baeree Frequecy doma dyamc aaly of le ad le grou J. Eg. Mech. vol [] R. Se.G. Dave ad P.K. Baeree Dyamc aaly of le ad le grou embedded homogeeou ol Earthquake. Eg. Struct. vol [6] H.. Wag ad P.K. Baeree Geeralzed aymmetrc elatodyamc aaly by boudary elemet method It. J. Numer. Method. Eg. vol [7] O. Maeo J.J. Azárez ad F. Garcı a Dyamc medace of le ad grou of le aturated ol omut. Struct. vol [8] A. Paoul he Fourer Itegral ad It Alcato McGra- Hll: Ne York 96. [9] J.D. Achebach Wave Proagato Elatc Sold North- Hollad: Holada 97. [] G.D. Maol ad D.E. Beko Boudary Elemet Method Elatodyamc U Hyma: Lodo 988. [] P.K. Baeree S. Ahmad ad K. he Advaced alcato of BEM to ave barrer mult-layered three-dmeoal ol meda Earthquake. Eg. Struct. vol [] R.D. kok ad.a. Brebba Boudary Elemet Method Acoutc omutatoal Mechac Publcato: Southamto 99. [] A.F. Seybert ad.w. Wu Modfed Helmholtz tegral equato for bode ttg o a fte lae J. Acout. Soc. Am. vol [] G. Beer L. Smth ad. Dueer he Boudary Elemet Method th Programmg: For Egeer ad Scett Srger: Berl 8. [] A.. Erge ad E.S. Suhub Elatodyamc Volume II: Lear heory Academc Pre: Ne York 97. Receved: May 8 Reved: July Acceted: Setember Zhag ad Lu; Lceee Betham Oe. h a oe acce artcle lceed uder the term of the reatve ommo Attrbuto No-ommercal Lcee (htt://creatvecommo.org/lcee/ by-c/./ hch ermt uretrcted o-commercal ue dtrbuto ad reroducto ay medum rovded the ork roerly cted.