SOIL: A Theoretical Manual. Dr. A. Metrikine Dr. S. Verichev Dr. A. Vostroukov. July Title:

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1 Titl: SOIL: A Thortical Maual Author: Dr.Ir. A.W.M. Kok Istitut: Dlft Uivrsity of Tchology Dr. A. Mtriki Dr. S. Vrichv Dr. A. Vostroukov Nubr of pags : 4 July 3 Kywords (3-5 : Vibratios prdictios DC-Publicatio-ubr : DC-5- Istitut Publicatio-ubr (optioal : Rport Typ : Itrdiary rport or study : Fial proctrport DUP-publicatio Typ : DUP Stadard DUP-Scic Ackowldgt This rsarch has b sposord by th Dutch Govrt through th ICES- progra ad th proctorgaisati HSL-Zuid Th rsarch is part of th Rsarch progra of Dlft Clustr. Coditios of us of this publicatio Th full-txt of this rport ay b usd udr th coditio of a corrct ad full rfrc to this publicatio.

2 Dlft Clustr-publicatio: DC-5- Abstract A w odl for vibratio trasissio through th soil has b dvlopd. Th so calld SOIL odul has b basd upo a applicatio of th Hakl trasfor thod (HTM ad th thod of sparatio of variabls (SVM. Th applicatio of th w thod brigs th followig advatags. First th HTM ad SVM ar purly aalytical approachs that hav o rstrictios cocrig wav propagatio i th layrd soil. Scod th w thod has b applid for loads uiforly distributd at sall aras. This allows us to build a gral solutio to th sourc/rcivr probl for ay visco-lastic horizotally layrd soil structur ad for ay rlativ positio of th sourc ad rcivr i.. th ar fild probl is solvd ow xactly. I th rport first th gral statt of th probl is forulatd i trs of vlocitis forcs ad adittac ad ipdac atrics. Scod th thortical bas for th HTM for th cass of vrtical ad horizotal loads is prstd. Third th SVM is proposd as a altrativ for th HTM at frqucis largr tha 3 Hz ad th thortical dscriptio of th SVM for th vrtical ad horizotal loads is giv. Th raso to us th SVM istad of th HTM at high frqucis cosists i that that th HTM big a itgratio routi has a srious coputatioal drawback. Th uppr liit of th itgral to b tak urically is proportioal to th frqucy. With th icras of th frqucy th calculatios loos ithr coputatioal spd or accuracy. Th SVM big a additio procdur dos ot hav such a probl. Furthror it sigificatly spds up th procss of calculatios. Fially th urical siulatios of so ING cass such as pil drivig traffic loads ad FWD tst prford both by th old ad w progras ar show ad copard to dostrat sstial diffrcs btw th obtaid rsults. PROJECT NAME: Rliability of vibratio progosis ad rducig asurs PROJECT CODE:.5. BASEPROJECT NAME: Evirotal ipact of udrgroud BASEPROJECT CODE:.5 THEME NAME: Soil ad Structurs THEME CODE: July 3 Soil: A Thortical Maual p.

3 Dlft Clustr-publicatio: DC-5- Excutiv Suary Th prdictio of vibratios du to pildrivig road ad railtraffic ca b prford by a varity of odls. Th accuracy of th prdictio dpds o th accuracy of th paratrs ad th accuracy of th odls thslvs. For o of th subodls th trasissio of vibratios through th soil a w odl is suggstd. Th rport dscribs th thory basd o th Hakl trasfor thod (HTM ad th thod of sparatio of variabls (SVM. First th thory is forulatd i trs of vlocitis forcs ad adittac ad ipdac atrics. Scodly th thory is giv for HTM for frqucis lowr tha 3 Hz ad SVM for frqucis highr tha 3 Hz. Rsults of urical siulatio ar giv for th cass dscribd i rport DC-5-9. PROJECT NAAM: Rliability of vibratio progosis ad rducig asurs PROJECT CODE:.5. BASISPROJECT NAAM: Evirotal ipact of udrgroud costructio BASISPROJECT CODE:.5 THEMA NAAM: Soil ad Structurs THEMA CODE: July 3 Soil: A Thortical Maual p. 3

4 Dlft Clustr-publicatio: DC-5- Tabl of cotts Itroductio...5 Th cas of vrtical load Th cas of horizotal load Solutio vrtical load followig Hakl Mthod... 5 Solutio vrtical load followig th Sparatio of Variabls Mthod Solutio horizotal load followig Hakl Mthod... 7 Solutio horizotal load followig Sparatio of Variabls Mthod So ING cass Hoogous half-spac: Pil drivig: Truck crossig a thrshold: Th FWD tst: Rfrcs...4 July 3 Soil: A Thortical Maual p. 4

5 Dlft Clustr-publicatio: DC-5- Itroductio. This rport forulats th thory for a w SOIL odul that has b proposd to rplac th old BODEM odul of th L4 progras. Th obctiv of both th w SOIL odul ad th old BODEM odul is to coput for a stratifid half spac th rspos by vlocitis of a rcivr poit B to a load applid i a sourc poit A. A F r B u r & Figur. A arbitrary distributio of th sourc ad th rcivr. Th rlatios btw loads at th sourc ad vlocitis of th rcivr ar giv by a adittac atrix Λ ad ipdac atrix Z whr r r u & = Λ F i = 3 ad i i r r F = Z u & i = 3 i i It is vidt that ipdac atrix Z is th ivrs of adittac atrix Λ. Assuig that w ar itrstd i loads ad rsposs both i A ad B th diffrcs btw sourc ad rcivr vaishs. Assuig loads i x- y- ad z-dirctio w hav to calculat a 6x6 adittac atrix Λ. Λ Λ Λ Λ Λ Λ AA AA AA AB AB AB xx xy xz xx xy xz AA AA AA AB AB AB Λ yx Λ yy Λ yz Λ yx Λ yy Λ yz AA AB AA AA AA AB AB AB Λ Λ zx Λ zy Λ zz Λ zx Λ zy Λ zz BA Λ BB Λ BA BA BA BB BB BB xx Λ xy Λ xz Λ xx Λ xy Λ xz BA BA BA BB BB BB Λ yx Λ yy Λ yz Λ yx Λ yy Λ yz BA BA BA BB BB BB Λ zx Λ zy Λ zz Λ zx Λ zy Λ zz Λ Λ Λ Λ Λ Λ Λ Λ Λ = = Λ Λ Λ Λ Λ Λ Λ Λ As will b show aalytically xact cofficits ca b foud for ach of th lts of th adittac atrix. Th diffrc btw th old BODEM odul ad th w SOIL odul is th way for th solutio of th basic probl aly to fid a rlatio btw a load at poit A ad vlocitis at poit B. July 3 Soil: A Thortical Maual p. 5

6 Dlft Clustr-publicatio: DC-5- Th SOIL odul has b basd upo a applicatio of th Hakl trasfor thod (HTM ad th thod of sparatio of variabls (SVM. Th applicatio of th w thod brigs th followig advatags. First th HTM ad SVM ar purly aalytical approachs that hav o rstrictios cocrig wav propagatio i th layrd soil. Thus th w thod solvs corrctly th probl of th Rayligh wav propagatio. Scod th w thod has b applid for loads uiforly distributd at sall aras. This allows us to build a gral solutio to th sourc/rcivr probl for ay visco-lastic horizotally layrd soil structur ad for ay rlativ positio of th sourc ad rcivr i.. th ar fild probl is solvd ow xactly. Additioally th us of th SVM sigificatly xtds th applicability of th progra i th frqucy bad: fro. to Hz. Th ai otivatio of th fforts towards th iprovt for th L4 was th possibility to rov so srious probls of th old BODEM odul. Firstly sic th load was odld as a poit sourc th solutio for th soil rspos to th loadig was sought i approxiatio of far fild solutio ; i.. it was problatic to us this thod for dscriptio of dyaic bhavior of two ar obcts ( ar fild solutio. Scodly th old progra was basd upo a so-calld raytchiqu. This tchiqu dos ot captur such a iportat factor of th half-spac as Rayligh wavs. This as that th surfac wavs could ot b siulatd corrctly i.. ay difficultis wr arisig with surfac rcivr/sourc probl. Ad thirdly sic th approxiatio of a poit load dos ot allow o to calculat th soil rspos to th load at its applicatio ara a so-calld Diabolo odl was applid to dscrib sourc/sourc ad rcivr/rcivr probls. This odl is a vry rough stiatio which is vry hard to ustify for dyaic rspos of a layrd soil. Thus i th opiio of th authors th old progra could ot proprly dscrib dyaic rspos of a layrd soil. I th w vrsio of th L4 th abov-tiod probls ar ovrco by as of th us of aalytical thods such as th HTM ad th SVM. Th rport is structurd as follows. First th gral statt of th probl will b forulatd i trs of vlocitis forcs ad adittac ad ipdac atrics. Scod th thortical bas for th HTM for th cass of vrtical ad horizotal loads is prstd. Third th SVM is proposd as a altrativ for th HTM at frqucis largr tha 3 Hz ad th thortical dscriptio of th SVM for th vrtical ad horizotal loads is giv. Th raso to us th SVM istad of th HTM at high frqucis cosists i that that th HTM big a itgratio routi has a srious coputatioal drawback. Th uppr liit of th itgral to b tak urically is proportioal to th frqucy. With th icras of th frqucy th calculatios loos ithr coputatioal spd or accuracy. Th SVM big a additio procdur dos ot hav such a probl. Furthror it sigificatly spds up th procss of calculatios. Fially th urical siulatios of so ING cass such as pil drivig traffic loads ad FWD tst prford both by th old ad w progras ar show ad copard to dostrat sstial diffrcs btw th obtaid rsults. July 3 Soil: A Thortical Maual p. 6

7 Dlft Clustr-publicatio: DC-5- Th cas of vrtical load W cosidr a stratifid half-spac subctd to a dp load (s Figur. Th odl is aalysd udr th followig assuptios.. Th load acts i th z dirctio.. Th load is uiforly distributd ovr a circular ara that is locatd at th itrfac of two layrs at th dpth d = h = ; h. 3. Th load varis i ti haroically. 4. Matrial dapig i th half-spac is itroducd i accordac with Voigt s odl. F z i t y R z θ r x h h d z Figur. A stratifid half-spac subctd to a dp vrtical haroic load. With ths assuptios th probl at had is axially sytric with rspct to th z axis ad i t posssss a stady-stat solutio which varis i ti i th sa ar as th load aly as Ω. W assu that th layrd half-spac cosists of layrs. Th -th layr ovrlis a. Th atrial costats of layr hoogous half-spac which is lablld th layr ubr ar th shar odulus thickss ( h h costats ar µ Poisso s ratio ν th dsity. For xapl for th ( * ρ ad µ ( h =. µ ν ρ th dapig cofficit * µ ad th -th layr which is th half-spac its atrial Govrig quatios To forulat th quatios of otio ad kiatic coditios w ploy th displact pottials by as of which th displact vctor of th half-spac ay b dcoposd as []. u r = ϕ r. July 3 Soil: A Thortical Maual p. 7

8 Dlft Clustr-publicatio: DC-5- For a axially sytric otio th vctor pottial r has th oly agular copot θ which w will dot as. Th scalar pottial ϕ ad th sigl copot of th vctor pottial dpd o r z ad t oly. Th dcopositio of th displact vctor ad th rlvat strssdisplact rlatios ay th b writt as u r z ( r ϕ ϕ = u = r z z r r ( rur u λ σ ( λ µ z r r ur uz σ zr = µ. z r z zz = Hriaftr scod sub idx dsigats ubr of th layr (for istac r as radial copot of th vctor-pottial i th layr. Th two pottials i th absc of body forcs satisfy th followig wav quatios: ( ( ϕ ϕ ϕ ϕ = r r r z cl t = r r r z r ct t. (3 c = λ µ ρ c = µ ρ ad λ µ Laé costats dscribig th visco-lastic with L ( T atrial of th th layr of th half-spac i accordac with th Voigt s odl ρ th ass dsity of th th layr of th half-spac. I th stady-stat rgi it is sufficit to cosidr th coplx aplituds of th pottials displacts ad strsss that ar itroducd accordig to th followig rlatios Ω Ω Ω ϕ rzt = ϕ ( rz Ω rzt = ( rz Ω i t iωt i t iωt r = r ( Ω z = z ( Ω i t iωt zz rzt = zz rzω rz rzt = rz rzω u rzt u rz u rzt u rz σ σ ( σ σ (. Evidtly i trs of th coplx aplituds ϕ ad th wav quatios tak th for ϕ ϕ ϕ Ω = r r r z cl ϕ Ω = r r r z r ct. (4 July 3 Soil: A Thortical Maual p. 8

9 Dlft Clustr-publicatio: DC-5- Th itrfac coditios btw two layrs rad ( ( ur r h Ω = ur r h Ω (5 u r h Ω = u r h Ω (6 ( ( ( rh σ ( rh z z σrz Ω = rz Ω (7 Fz H ( R r h = d σ ( zz rh Ω σ zz ( rh Ω = π R (8 h d whr H (.. is th Havisid stp fuctio. Togthr with th boudary coditios at th fr half-spac surfac which rad ( r σ rz Ω = (9 Fz H ( R r d = σ zz ( r Ω = π R ( d th boudary coditios (5-( coplt th probl statt. 3 Th cas of horizotal load A horizotal haroic load that is applid to a stratifid half-spac at crtai dpth d at th itrfac btw two layrs is dpictd i Figur 3. Th odl is aalysd udr th followig assuptios:. Th load acts i th x dirctio.. Th load is uiforly distributd ovr a circular ara that is locatd at th itrfac of two layrs at th dpth d = h = ; h. 3. Th load varis i ti haroically. 4. Matrial dapig i th half-spac is itroducd i accordac with Voigt s odl. F x i t y R z θ r x h h d z July 3 Soil: A Thortical Maual p. 9

10 Dlft Clustr-publicatio: DC-5- Figur 3. A stratifid half-spac subctd to a dp horizotal haroic load. To forulat th quatios of otio ad kiatic coditios w ploy th displact pottials by as of which th displact vctors of th layr ad of th half-spac ay b dcoposd as []: r u with quatios: ϕ z θ ur = r r θ z r ϕ r z = ϕ u = r θ z r ϕ ( r θ uz = z r r r θ θ r ϕ ad = { r θ z } r ( th scalar ad th vctor pottials that satisfy th followig ϕ = c ϕ L t r θ r r = r r θ ct t θ r θ θ = r r θ ct t z z = ct t ( r r r θ z div( = r = r θ z ( ad th Laplacia is dfid as r r r r θ z =. (3 Th sstial copots (for th probl at had of th strss tsor rad ur uz σ zr = µ z r uz uθ σ zθ = µ r θ z u u u u σ ( λ µ λ z r r r θ z r r θ zz =. (4 July 3 Soil: A Thortical Maual p.

11 Dlft Clustr-publicatio: DC-5- Th aalysis will procd as follows. Sic w ar itrstd i th stady-stat rspos w will drop th ti dpdcy by skig for th solutio for pottials i th for of haroic fuctios of ti. Th takig ito accout th loadig charactr i th sa way w will drop th agl dpdcy. Fially w will apply Hakl itgral trasfor to drop th radial dpdcy to obtai th solutio for th displact vctor of th half-spac. I th stady-stat rgi it is sufficit to cosidr th coplx aplituds of th pottials displacts ad strsss. Ths aplituds ar itroducd accordig to th followig rlatios iωt ( r z t = ( r z Ω Ω θ θ iωt ( r z t = z ( r z Ω Ω θ θ iωt u z r θ z t = uz ( r θ z Ω Ω iωt σ ( r θ z t = σ ( r θ z Ω. ϕ θ ϕ θ i t iωt r θ z t = ( r θ z Ω r θ z t = ( r θ z Ω r r θ θ z u r θ z t = u ( r θ z Ω u r θ z t = u ( r θ z Ω i t iωt r r σ r θ z t = σ ( r θ z Ω σ r θ z t = σ ( r θ z Ω i t iωt zz zz rz rz zθ zθ (5 Evidtly i trs of th coplx aplituds quatios ( ( ad (4 kp thir for (oly th hads ovr th variabls should b addd i ( ad (4 whras quatios rad. Ω ϕ c ϕ = ( r L Ω r θ r = r r r θ ct Ω θ r θ = θ r r θ ct Ω z = z ct r θ z r =. r θ z (6 To forulat th loadig coditios it is custoary to dcopos th coplx aplitud load ito th radial ad th circufrtial copots. This yild ( θ ( θ Fr = Fxcos Fθ =Fx si (9 Usig this dcopositio th itrfac coditios btw th layrs ca b writt as ( θ ( θ ( θ θ ( θ F x of th ur r h Ω ur r h Ω = (8 uθ r h Ω u r h Ω = (9 July 3 Soil: A Thortical Maual p.

12 Dlft Clustr-publicatio: DC-5- ( θ ( θ ( r h ( r h uz r h Ω uz r h Ω = ( σ zz θ Ω σ zz θ Ω = ( Fx cos( θ H ( R r h = d σ ( rz r θ h Ω σrz ( r θ h Ω = π R h d ( Fx si ( θ H ( R r h = d σ ( θz r θ h Ω σθz ( r θ h Ω = π R h d (3 whr H(.. is th Havisid stp fuctio. Th boudary coditios at th fr surfac rad ( r σzz θ Ω = (4 Fx cos( θ H( R r d = σrz r θ Ω = π R d (5 Fx si ( θ H ( R r d = σθ z r θ Ω = π R d (6 Boudary coditios (8-(6 coplt th statt of th probl. 4 Solutio vrtical load followig Hakl Mthod To solv th probl (4-( w apply th Hakl trasfor with rspct to th radial co-ordiat r. This trasfor is dfid as k k (7 Hk = = Hk f ξ rf r J ξr dr f r ξf ξ J ξr d ξ k ξ is th Bssl fuctio of th first kid of ordr k. I th prst probl th Hakl trasfor of ordr zro ust b applid to ϕ ad th Hakl trasfor of ordr o to. Aftr itgratio by parts quatios (4 th rduc to th followig diffrtial quatios: whr k is th ordr of th trasfor ad J ( r d ϕ dz d dz H H H α ϕ = H β = (8 July 3 Soil: A Thortical Maual p.

13 Dlft Clustr-publicatio: DC-5- with α = ξ Ω c β = ξ Ω c. L T Solutios to quatios (8 accoutig for th propr bhaviour for larg positiv valus of z ar for h z h = ( ( ( ( ( H = A4 3xp z h A4 xp zh ϕ α α H = A4 xp z h A4 xp zh β β (9 for z h ϕ ( α β H = A4 xp zh xp H = A4 zh (3 whr ( α ( β A ar ( 4 ukow cofficits which dpd o ξ ad R R ad Ω. Applicatio of th Hakl trasfor to th coplx aplituds of th displacts ( ad th strsss ( yilds H H d H u r =ξϕ dz (3 H d ϕ H H u z = ξ (3 dz H d ϕ H Ω H σ rz = µ ξ ξ (33 dz c T H H Ω H d σ zz = µ ξ ϕ ξ (34 c T dz Th boudary coditios (5-( ar trasford ito H ( ( H ur r h Ω = ur r h Ω (35 u r h Ω = u r h Ω (36 σ H ( ( H ( rh σ ( rh H z z H rz rz Ω = Ω (37 Fz H H J R ξ h = d σ ( zz rh σ zz ( rh πrξ Ω Ω = (38 h d H σ r Ω = (39 rz July 3 Soil: A Thortical Maual p. 3

14 Dlft Clustr-publicatio: DC-5- σ H zz ( r Fz J ( R ξ d = Ω = πrξ d (4 with =. Substitutig th solutios (9 ad (3 ito th boudary coditios (35-(4 ad usig th xprssios (3-(34 o obtais th syst of ( 4 algbraic quatios which ca b solvd urically. 5 Solutio vrtical load followig th Sparatio of Variabls Mthod To solv th probl (4-( istad of Hakl trasfor thod w ca also us th thod of sparatio of variabls. I accordac with this thod pottials ϕ ad ar sought i th for ϕ ϕ = R r Z z ϕ = R r Z z (4 Substitutig Eq.(4 ito Eq.(4 th followig syst of quatios is obtaid d Rϕ r drϕ r ϕ ϕ dz ( z ϕ Ω dz ϕ ϕ c L ( z k R r = dr r dr k Z ( z = d R r dr r R ( r k dz = dr r dr r k Z ( z = Ω dz c T (4 with k kϕ positiv ral costats. Th solutios to th first ad th third quatios of Eqs.(4 that satisfy th coditio of vaishig of th pottials at r rad = % = % R r A J k r ϕ ϕ ϕ R r A J k r with J ad J th Bssl fuctios of th zro ad first ordr rspctivly. Th solutios to th scod ad th fourth quatios of Eqs.(4 that satisfy th coditio of vaishig of th pottials at z rad for h z h = (43 July 3 Soil: A Thortical Maual p. 4

15 Dlft Clustr-publicatio: DC-5- RL ( zh RL ( zh Z r = A% A% ϕ ϕ ϕ RT ( zh RT ( zh Z r = A% A% (44 for z h Z r A% RL ϕ = 3 ϕ Z r A% ( zh ( zh RT = 3 (45 with RL kϕ cl RT k ct ( RT ; L = Ω = Ω R >. Thus th pottials ϕ ad ar giv as for h z h = RL ( zh RL ( zh ϕ = J ( kϕ r ( A ϕ A ϕ RT ( zh RT ( zh ( = J k r A A. (46 for z h 3 ϕ 3 RL ( zh ( ϕ RT ( zh ( ϕ = A J k r = A J k r. (47 Accordigly th displacts ad strsss i th half-spac rad for h z h = ( ϕ ( u = k A J k r R A J k r ( k A J k r R A J k r ( u = R A J k r k A J k r ( ( RL zh RT zh r ϕ ϕ ϕ T RL z h RT z h T ϕ ϕ ϕ ( RL zh RT zh z L ϕ ϕ R A J k r L ϕ RL zh RT zh k A J k r (48 July 3 Soil: A Thortical Maual p. 5

16 Dlft Clustr-publicatio: DC-5- ( ( RL ( zh = kϕ R Aϕ J ( kϕ r Ω ( σ =µ Ω c k A J k r µ k R A J k r RL ( zh RT ( zh µ Ω c k A J k r µ k R A J k r ( RL zh RT zh zz T ϕ ϕ ϕ T T ϕ ϕ ϕ T σ µ µ rz L ( T ϕ ( c k A J k r ( µ k R A J k r µ Ω c k A J k r ( zh RT ( RL z h RT z h ϕ L ϕ ϕ T ϕ (49 for z h RL ( zh RT ( zh RL ( zh RT ( zh u = k A J k r R A J k r r ϕ 3 ϕ ϕ T 3 u = R A J k r k A J k r z L 3 ϕ ϕ 3 σ =µ Ω RL ( zh ( c ( kϕ A ϕ J ( kϕ r RT ( zh k R A J ( k r RL ( zh ϕ ϕ ( ϕ RT ( zh c ( kϕ A J k r zz T 3 σ = µ µ T 3 k R A J k r rz L 3 µ Ω T 3 (5 (5 To dfi k kϕ w itroduc artificial boudary coditio by assuig that th vrtical displact ad oral strss vaish at a crtai distac r fro th load. Not that such a assuptio lads to a approxiat solutio to th probl. Howvr icrasig r this approxiat solutio covrgs to th xact o. To hav u z ad σ zz vaish at r idpdtly of z th followig coditios ust b satisfid: ( ϕ ( J k r = J k r = (5 Eqs.(5 ar satisfid if ad oly if kϕ = k = k = α r =.. with α th zros of J i a th layr. Thus th fial xprssios for th displacts ad strsss should ivolv suatio with ( rspct to all k i. (48-(5 ca b rwritt i th for: for h z h = July 3 Soil: A Thortical Maual p. 6

17 Dlft Clustr-publicatio: DC-5- ( L ( ( u = k A J k r R A J k r ϕ ϕ ( ( RL zh RT ( zh k A J k r R A J k r ( RL ( z h u = R A J k r k A J k r ϕ R zh R zh ( T ( r T = = T = = z L = = ϕ R ( ( RL zh RT ( zh R A J k r k A J k r L = = ( T ( zh (53 ϕ ( L( σ zz =µ Ω ct k A J k r µ k RT A J k r = = ϕ R zh R zh ( ( RL zh RT ( zh µ Ω ct k A J k r µ k RT A J k r = = ( L ( σ = µ k R A J k r µ Ω c k A J k r ( ( RL zh RT ( zh µ k R A J k r µ Ω c k A J k r ϕ ( ( ( ( ( T ( R zh R zh rz L ϕ T = = for z h L T = = ( T ( (54 RL ( zh ( ( RT zh r = 3 ϕ T 3 = = u k A J k r R A J k r ( RL ( zh RT zh z = L 3 ϕ 3 = = u R A J k r k A J k r ϕ ( ϕ z = Ω T 3 = σ µ c k A J k r σ = µ T 3 = rz L 3 = ct = RT zh µ k R A J k r k R A J k r RL zh µ Ω k A3 J k r RL zh RT ( zh (55 (56 with R = k Ω c R = k Ω c R R >. L L T T T ; L Substitutig Eqs.(54 ito th boudary coditios (5-( ad rprstig H( R r followig for: i th July 3 Soil: A Thortical Maual p. 7

18 Dlft Clustr-publicatio: DC-5- ( R J( k R J( k r = (57 ( r ( H R r k J α = w obtai th syst of quatios which dtris A ϕ ad k A k = 3 : k at th itrfacs z = h = (fro (5-(8: ( ( k A R A k A R A RL hh RT hh ϕ ϕ T T R ( h h RT ( hh k A k A R A R A = L ϕ ϕ T T (58 ( ( R A k A R A k A RL hh RT hh ϕ ϕ L L R ( h h RT ( hh R A R A k A k A = L ϕ ϕ L L (59 ( ( RL hh RT ( hh µ k RL Aϕ µ ct k Ω A µ k RL Aϕ ϕ µ Ω c k A µ k R A µ Ω c k A T L T µ k R A L ϕ RL ( hh RT ( hh µ c k Ω A = T ϕ ( L( µ Ω ct k A µ k RT A ϕ ϕ RL ( hh µ Ω c k A µ k R ( T( R hh R hh µ Ω c k A µ k R A T T T T RT ( hh F J k R µ Ω µ = ( α h d z h ct ( k = A ϕ k RT A π R r k J A d (6 (6 at th itrfac z = h (fro (5-(8: 3 ϕ 3 ϕ ( ( RL h h T k A R A k A ( R T ( hh ϕ R A k A R A = T T (6 L 3 ϕ 3 L ϕ R A k A R A ( R T ( hh ϕ ( ( RL h h k A R A k A = L (63 July 3 Soil: A Thortical Maual p. 8

19 Dlft Clustr-publicatio: DC-5- ( ( R L ( hh k RL A3 ϕ Ω ct k A3 k RL A ϕ ( ( RT ( hh ct k A k RL A ϕ ( ct ( k A µ µ µ µ Ω µ µ Ω = ( R L ( h h ct k A3 k RT A 3 ct k ϕ A ϕ µ Ω µ µ Ω µ µ Ω µ ( RT ( hh k RT A ct k A ϕ k RT A J k R h = d Fz = πr r k J α h d (64 (65 at th fr-surfac z = (fro coditios (9-(: µ µ ϕ Ω k RL A ct k A RL h RT h µ k RL A ϕ µ Ω ct k A = ct ( k A ϕ k RT A µ Ω µ F J k R z R L h RT h d = µ ct ( k Ω A ϕ µ k RT A = π R rk J ( α d (66 (67 July 3 Soil: A Thortical Maual p. 9

20 Dlft Clustr-publicatio: DC-5-6 Solutio horizotal load followig Hakl Mthod for: Th loadig charactr prits to sarch for th solutio to th probl i th followig ϕ θ ϕ cos θ ( r z Ω = ( r z Ω θ si θ ( r z Ω = ( r z Ω r r θ cos θ ( r z Ω = ( r z Ω θ θ θ si θ ( r z Ω = ( r z Ω z z (68 Substitutio of quatios (68 ito quatios of otio (6 yilds Ω ϕ = ϕ r r r r z cl Ω r θ = r r r r r z r ct Ω θ r = θ r r r r z r ct Ω z = z r r r r z ct ( r r z r z θ r = r θ r = r z r z (69 Th scod ad th third quatio of th st (69 ar coupld. To dcoupl ths quatios it is custoary to itroduc th followig w variabls: ( θ θ r r = r r = θ r θ = r θ θ = r θ r θ I trs of ths wly itroducd variabls th st of quatios (69 ca b rwritt as (7 July 3 Soil: A Thortical Maual p.

21 Dlft Clustr-publicatio: DC-5- Ω ϕ = ϕ r r r r z cl Ω = r r r z ct 4 Ω = r r r r z ct Ω z = z r r r r z ct ( r r z z = Th displacts of th layr ad th half-spac (s ( ca b also rwritt as u u r z θ u z ( r θ r θ ϕ = r r z ( θ θ = ϕ r z r r r z ( r θ r θ ϕ = z r r (7 (7 To fid th gral solutios to first four quatios of syst (7 w apply th Hakl trasfor with rspct to th radial coordiat r. This trasfor is dfid as H H ( ξ = rf ( r J ( ξr dr ( r = ξf ( ξ J ( ξr dξ (73 whr is th ordr of th trasfor ad J (..is Bssl fuctio of th first kid of ordr. It is vidt that for th first ad th fourth quatio of syst (7 th ordr of th trasfor should b qual to for th scod quatio - to zro ad for th third quatio to. Applicatio of ths trasfor yilds: d = dz = = dz =. H H ϕ H d H α ϕ β r θ dz d H H H d z H β r θ β z dz (74 with α = ξ Ω cl β = ξ Ω ct. Obviously th gral solutios to quatios (74 ca b foud asily. Bfor doig this howvr lt us satisfy th fifth quatio of th st (7 ad xprss th displacts ad strsss of th layrd half-spac via th pottials. July 3 Soil: A Thortical Maual p.

22 Dlft Clustr-publicatio: DC-5- I accordac with dfiitio of th Hakl trasfor xprssios for ϕ θ θ ar giv as r r z ϕ rz ξϕ ξ z J ξrdξ H ( Ω = ( Ω rz ξ ξ z J ξrdξ H ( Ω = ( Ω rz ξ ξ z J ξrdξ H ( Ω = ( Ω rz ξ ξ z J ξrdξ. H ( Ω = ( Ω z z (75 Substitutio of ths xprssios ito th fifth quatio of th syst (7 yilds r H H H z ξ ξ ξr θ J( ξr dξ =. z (76 This quatio is satisfid if th itgrad quals to zro i.. H H H z ξ ξ =. (77 z Equatio (77 stablishs r idpdt rlatioship btw th trasford copots of th vctor-pottial. Lt us ow substitut xprssios (75 ito th trasford displacts. Havig tak ito accout rlatioship (77 ad usig rcurrt rlatio btw Bssl fuctios ([] ν Jν ( z Jν ( z = Jν ( z (78 z whr ν is itgr this substitutio yilds H H H H H z H H u r r z Ω = ξ ξϕ ξ z J ξr ξϕ ξ z J ξr dξ z z z ξ H H H ( H H z u θ r z Ω = ξ ξϕ ξ z H H J ξr ξϕ ξz J ξr dξ ξ z z z H H ϕ z ( H u z r z Ω = ξ ξr θ J( ξr dξ z z (79 Now that xprssios for th trasford displacts hav b foud w ca substitut th ito th xprssios for strsss (4. This yilds July 3 Soil: A Thortical Maual p.

23 Dlft Clustr-publicatio: DC-5- H 3 H H µ ϕ z H σ rz r z Ω = 3 ξ ξ ξ J ξr z ξ z z H H H ϕ z H ξ ξ ξ J ( ξr dξ z z z H 3 µ ϕ H H z H σ zθ ( r z Ω = ξ ξ 3 ξ J ξr z ξ z z H H H ϕ z H ξ ξ ξ J ( ξr dξ z z z H λ µ ϕ H H ( H H z σ zz r z Ω = µ ξ ξ ϕ ξ ϕ ξ J ( ξr dξ µ z z z (8 Aalysig quatios (79 ad (8 it ca b cocludd that to satisfy th boudary coditios (8-(6 it is sufficit to fulfill th followig quatios: at th itrfacs z = h H H H H H H z r θ z H H ξϕ ξ z ξϕ ξ z = ξ z z ξ z z (8 H H H H H H ξϕ ξ z ξϕ ξ z = (8 z z H H H H ϕ z ϕ H z H ξ ξ = (83 z z z z H H H λ µ ϕ H H z µ ξ ϕ ξ ϕ ξ µ z z z (84 H H H λ µ ϕ H H z µ ξ ϕ ξ ϕ ξ = µ z z z H 3 H H ϕ z H µ ξ ξ 3 z ξ z z H 3 H F (85 H x ϕ z H J R ξ h = d µ πrξ ξ ξ 3 = z ξ z z h d July 3 Soil: A Thortical Maual p. 3

24 Dlft Clustr-publicatio: DC-5- ϕ µ ξ ξ ξ z z z H H H ϕ z H µ ξ ξ ξ r θ = z z z H H H z H (86 at th surfac z = H H H λ µ ϕ H H z ξϕ ξϕ ξ = µ z z z 3 H H F H x ϕ z H r J R ξ θ d = µ ξ ξ 3 = πrξ z ξ z z d H H H ϕ z H ξ ξ ξ = z z z (87 (88 (89 Solutio to quatios (74 ca b writt i th for: for h z h = : ϕ α( zh α( zh = A A H β ( zh β( zh = A A H z β( zh β ( zh = A A H 6 6 (9 for z h : ϕ = A H 6 = A H z 6 = A H r 6 3 θ α ( zh β ( zh β ( zh (9 Substitutig solutios (9 (9 ito th boudary coditios (8-(89 o obtais th syst of 6 3 algbraic quatios which ca b solvd th urically. Fially usig forulas (79 o ca obtai solutio i frqucy doai. July 3 Soil: A Thortical Maual p. 4

25 Dlft Clustr-publicatio: DC-5-7 Solutio horizotal load followig Sparatio of Variabls Mthod I accordac with this thod pottials ϕ ± z ad r ± ϕ = R r Z z ϕ ϕ = R r Z z = R r Z z = R r Z z z 3 3 θ (s (7 ar sought i th for (9 Substitutig (9 ito (7 w obtai Rϕ ( r Rϕ ( r Z ( z ϕ Ω R ϕ ( r kϕ = Z ϕ ( z k ϕ = r r r r z c L R r R r Z z Ω k R r = Z z k = r r r z ct R r R r 4 Z z Ω R r k = Z z k = r r r r z c T R 3 r R3 r Z 3 z Ω R 3 r k3 = Z 3 z k 3 = r r r r z c T R r R r R 3 R r Z r ( r Z z Z z Z 3 z = r r z (93 Th solutios to th first third fifth ad th svth quatios of Eqs. (93 that satisfy th coditio of vaishig of th pottials at r rad = % = % ( = % ( = % ( R r A J k r ϕ ϕ ϕ R r A J k r R r A J k r R r A J k r (94 with J J ad J th Bssl fuctios of th zro first ad scod ordr rspctivly. Th solutios to th scod fourth sixth ad th ighth quatios of Eqs. (93 that satisfy th coditio of vaishig of th pottials at z rad for h z h = July 3 Soil: A Thortical Maual p. 5

26 Dlft Clustr-publicatio: DC-5- RL ( zh RL ( zh Z r = A % A % ϕ ϕ ϕ RT ( zh RT ( zh Z r = A % A % RT ( zh RT ( zh Z r = A % A % RT 3 ( zh RT 3 ( zh Z r = A % A % (95 for z h Z r = A% ϕ ϕ3 Z r = A% 3 Z r = A% 3 Z r = A% 3 33 RL ( zh ( zh RT ( zh RT ( zh RT 3 with RL kϕ cl RT 3 k3 ct RT 3 ; L = Ω = Ω R >. Thus th pottials ϕ ± z ad r ± θ for h z h = ar giv as ( RT zh RT zh RT zh RT zh RT 3 zh RT 3 zh. RL ( zh RL ( zh ( ϕ ϕ ϕ ϕ = J k r A A = J k r A A = J k r A A = J k r A A z (96 (97 for z h ϕ3 ( RL ( zh ϕ R R ( R 3 ϕ = A J k r 3 3 z 33 3 ( zh T = A J k r ( zh T = A J k r ( zh T = A J k r. (98 Accordigly th displacts ad strsss i th half-spac rad for h z h = July 3 Soil: A Thortical Maual p. 6

27 Dlft Clustr-publicatio: DC-5- ( ( RL zh RL zh u r = Aϕ kϕ J kϕ r J kϕ r Aϕ kϕ J kϕ r J kϕ r RT ( zh RT zh A RT J k r A RT J k r RT zh A RT RT J k r A RT J( k zh r 3 ( A ( 3 k3 J k3 r J k3 r A3 k3 J k3 r J k3 r ( u θ = Aϕ ( kϕ J kϕ r J kϕ r Aϕ kϕ J kϕ r J kϕ r RT A RT J( k zh RT zh r A RT J k r RT ( zh RT zh A RT J k r A RT J k r ( A ( ( 3 k3 J k3 r J k3 r A3 k3 J k3 r J k3 r RT zh RT zh RL z h RL z h RT 3 zh RT 3 zh RL ( zh RL ( zh u = A R J k r A R J k r z ϕ L ϕ ϕ L ϕ A k J k r A k J k r A k J k r A k J k r ( RT ( zh RT zh RT ( zh RT zh ( ( RL zh RL zh rz = Aϕ RL kϕ J kϕ r J kϕ r Aϕ RL kϕ J kϕ r J kϕ r σ µ k k RT zh k k A J k r RT J k r A J k r RT J k r R T ( zh k k k k A J k r J k r RT A J k r J k r RT RT 3 A 3 k3 RT3 ( ( zh RT 3 zh J k 3 r J k 3 r A 3 k3 RT3 J k 3 r J k 3 r RT zh RT zh ( ( ( RL zh RL. zh zθ = Aϕ RL kϕ J kϕ r J kϕ r Aϕ RL kϕ J kϕ r J kϕ r σ µ k k ( RT zh k k A J k r RT J k r A J k r RT J ( k r R T k k k k A J k r J k r RT A J k r J k r RT RT 3 ( zh RT 3 ( zh A 3 k3 RT ( ( 3 J k 3 r J k 3 r A 3 k3 RT3 J k 3 r J k 3 r ( zh RT zh RT zh July 3 Soil: A Thortical Maual p. 7

28 Dlft Clustr-publicatio: DC-5- Ω Ω σ zz = µ Aϕ J kϕ r kϕ A ϕ J kϕ r kϕ c T c T RL ( zh RL ( zh RT ( zh RT zh A k R J k r A k R J k r T T ( A k R J ( k r A k RT J ( k r T R T zh R T zh. (99 for z h RL ( zh ( u A k J k r J k r A R J k r A R J k r A k J k r J k r u RT r = ϕ3 ϕ ϕ ϕ 3 T θ RT ( zh 3 T R L zh = Aϕ ( 3 kϕ J kϕ r J kϕ r A 3 RT J k r R T zh 3 T R L zh RT = ϕ zh 3 J kϕ r A3 k J k r RT zh A k J k r u A R z L ( zh ( zh RT 3 RT ( zh RT ( zh A R J k r A k J k r J k r 3 ( 3 RL ( zh ( σ =µ A R k J k r J k r rz ϕ3 L ϕ ϕ ϕ k k A J k r R J k r k k A 3 J k r J k r RT RT 3 ( zh A ( 33 k3 RT 3 J k3 r J k3 r RL zh σ = µ A R k J k r J k r T 3 T ( R ( zh R ( zh T ( zθ ϕ3 L ϕ ϕ ϕ k k RT zh A3 J k r RT J k r k k RT A 3 J k r J k r R zh T RT 3 ( zh A33 k3 RT 3 J k3 r J k3 r Ω RL ( zh σ zz = µ Aϕ 3 J kϕ r kϕ c T A k R J k r A k R J k r RT zh RT ( zh 3 T 3 T. ( July 3 Soil: A Thortical Maual p. 8

29 Dlft Clustr-publicatio: DC-5- Th last quatio of (93 rads for h z h = ( A k J k r A k J k r ( A k J k r A k J k r RT zh RT zh RT zh RT zh RT ( 3 zh RT 3 zh R A J k r R A J k r = T3 3 3 T3 3 3 ( for z h ( ( zh RT zh RT zh 3 3 A k J k r A k J k r RT 3 RT 3 A33 J k 3 r = ( Equatios ( ad ( ca oly b satisfid if ( ( ( J k r = J k r = J k 3 r = Eqs. (3 ar satisfid if ad oly if R = R = R = R. T T T3 T k 3 (3 = k. Fro hr it follows that To dfi kϕ k w itroduc artificial boudary coditio by assuig that th vrtical displact ad oral strss vaish at a crtai distac r fro th load. Although such a assuptio lads to a approxiat solutio to th probl with icras of r this approxiat solutio covrgs to th xact o. To hav u z ad σ zz vaish at r idpdtly of z th followig coditios ust b satisfid: ( ϕ ( J k r = J k r = (4 ϕ α Eqs. ar satisfid if ad oly if k = k = k = r = with α th zros of J i th layr. Thus th fial xprssios for th displacts ad strsss should ivolv suatio with rspct to all k. Th w d to substitut ths quatios ito th boudary coditios (8-(6. Agai w rprst H( R r i th for (57. Aalysig quatios (99 ad ( it ca b cocludd that to satisfy th boudary coditios (8-(6 it is sufficit to fulfil th followig quatios July 3 Soil: A Thortical Maual p. 9

30 Dlft Clustr-publicatio: DC-5- at th itrfacs z = h = (fro (8-(3: ( k A A k A R A R A k ϕ ϕ A k k A A k A R RL hh ϕ ϕ T RT hh RT hh A3 k A3 k = RL hh RT hh RT hh T T 3 3 T A R (5 ( k A A k A R A R A k ϕ ϕ A k k A A k A R RL hh ϕ ϕ T RT hh RT hh A3 k A3 k = RL hh RT hh RT hh T T 3 3 T A R (6 ( Aϕ RL Aϕ RL A k A k A k ( R ( L h h A k Aϕ RL Aϕ RL A k R ( T hh RT ( hh A k A k A k = RL hh RT hh RT hh (7 ( Ω RL ( hh Ω µ Aϕ k A ϕ k c T c T ( A k R A k R A k R A k R RT hh RT hh T T T T µ Ω Ω Aϕ k ϕ c T c T RL ( hh A k RT ( hh A k RT A k RT RT hh A k RT A k RT = (8 July 3 Soil: A Thortical Maual p. 3

31 Dlft Clustr-publicatio: DC-5- µ ( ( k RL hh RT hh ( ϕ ϕ k k T ( R hh RT ( hh A A A 3 k RT A 3 k RT k R L hh µ ( Aϕ RL k Aϕ RL k A R T ( k R ( k T h h A R ( k RT ( hh T A A F J ( k R x R ( T h h h d = A 3 k RT A 3 k RT = πr r k J ( α h d µ ( k A RL k A RL k A RT A RT ( ( k R L hh RT hh ( Aϕ R k Aϕ R k A A ( k ( R ( k T h h A R T A R T ( RT ( hh A 3 k RT A 3 k RT RL ( hh µ A R k A R k ( k L L ( ϕ L L ϕ ( k ( k T ( R h h A A ( k A ( R A ( R ( RT ( hh A3 k RT A3 k RT = ( k T T at th itrfac z = h (fro (8 (3: RT ( hh (9 ( July 3 Soil: A Thortical Maual p. 3

32 Dlft Clustr-publicatio: DC-5- ( R L ( hh ϕ T ϕ ϕ k A A R A k k A A k ( R T hh RT ( hh T T A R A R A k A k = 3 3 ( ( R L ( hh ϕ T ϕ ϕ k A A R A k k A A k ( ( R h h T RT ( hh T T A R A R A k A k = 3 3 ( A R A k A k A R A R ( R ( T hh A k A k A k A k = ( R L ( hh ϕ ϕ ϕ µ 3 L 3 3 L L Aϕ k A k R A k R Ω 3 3 T 3 T c T ( Ω RL ( hh Ω µ Aϕ k A ϕ k c T c T ( ( RT h h A k RT A ( ( ( k RT ( T ( R h h T T A k R A k R = (3 (4 k k µ A R k A ( R ϕ A A k R 3 L 3 T 3 33 T ( k ( h h ( RL ( hh RT µ Aϕ RL k Aϕ RL k A RT k k k RT hh A ( R T A A ( R F J k ( x R ( T hh h d = A3 k RT A3 k RT = πr r k J ( α h d (5 July 3 Soil: A Thortical Maual p. 3

33 Dlft Clustr-publicatio: DC-5- µ ( µ A RL k A RL k A A k k ϕ A 3 RL k A 3 A 3 RT A 33 k RT ( ( k R L h h RT hh ϕ ϕ ( k ( R T ( hh ( k A R T A R T ( R ( T hh A3 k RT A3 k RT = at th surfac z = (fro (4-(6: ( k (6 RT h ( k R L h µ ( Aϕ RL k Aϕ RL k A RT T A ( R T A A k k k R h RT h A3k RT A3 k RT J k R Fx d = = π R rk J ( α d k L ϕ L ϕ L ( k T ( k R ( k R h RT h A R k A R k A A A RT A R RT h A3k RT A3 k RT = L Aϕ R Aϕ R A k A k RT h Ak Ak = R h RT h L L T h (7 (8 (9 at th itrfacs z = h = (fro ( July 3 Soil: A Thortical Maual p. 33

34 Dlft Clustr-publicatio: DC-5- A k A k R A = ( T 3 A k A k R A = ( T 3 fro ( for z h A k A k R A = ( 3 3 T 33 8 So ING cass Th xapls show i this sctio ar tak fro rport [4]. Now th sa cass ar procssd with th thod as outlid i this rport. Th aalysis has b prford usig valus of frqucy with icrts Ω=. Hz ad startig at Ω= Ω. Τh rsults of SOIL (th w thod ad BODEM (th old thod ar copard. 8. Hoogous half-spac: Figur 4. Schatic diagra of locatios of th load ad th rcivr ad ti load history. July 3 Soil: A Thortical Maual p. 34

35 Dlft Clustr-publicatio: DC-5- Figur 5. Vrtical vlocity at diffrt rcivr poits vrsus ti. Figur 5 dostrats th followig:. Although th shaps of th rsposs siulatd by th BODEM ad th SOIL oduls look siilar th scals of th vlocitis ar quit diffrt spcially for th surfac rcivr.. Sic th cotributio of th Rayligh wavs which plays a iportat rol at low frqucis is ot wll siulatd by th BODEM odul th lattr givs a rsult that is absolutly wrog fro physical poit of viw for a dp rcivr: at t = ti ot th vrtical vlocity at a distat poit is ot qual to zro. Th us of Voigt's dapig odl phasiss this bhaviour. 8. Pil drivig: Th soil proprtis of this cas ad th followig cass ar giv i tabl. thickss E [Mpa] G [Mpa] K [Mpa] kg/ 3 ] s] E E E E E E E-4 Tabl. Paratrs of th soil. July 3 Soil: A Thortical Maual p. 35

36 Dlft Clustr-publicatio: DC-5- Figur 6. Schatic diagra of locatios of th load ad th rcivr ti history of th load ad its spctru. Paratr β is th Voigt's dapig cofficit. Th pil drivig load is odlld by a load at 7.5 dpth (s Fig. 6. Th aalysis has b prford by usig of frqucis with icrt Ω =. Hz ad startig at Ω= Ω Rsposs at th rcivr poits ar show i Figur 7. July 3 Soil: A Thortical Maual p. 36

37 Dlft Clustr-publicatio: DC-5- Figur 7. Horizotal ad vrtical vlocitis at diffrt rcivr poits vrsus ti. This xapl shows that dp loads ar wll siulatd by th BODEM odul for dp ods ot far away fro th sourc. Howvr for distat poits it is clarly s that th BODEM odul usig th ray-tchiqu caot proprly accout for all possibl rflctios of th wavs fro th soil itrfacs. For th giv cofiguratio th BODEM progra is uabl to odl or th o rflctio. Whil th rspos calculatd by th SOIL odul xprics so prturbatios associatd with rflctios of th wavs fro th itrfacs of or stiff layrs ad vry soft layrs th BODEM rspos givs a rsult clos to zro. July 3 Soil: A Thortical Maual p. 37

38 Dlft Clustr-publicatio: DC Truck crossig a thrshold: A 5 tos truck drivig with a vlocity c=3 k/hr crosss thrshold. Th truck is odlld by a sris of sprigs dashpots ad asss (s Figur 8. Oly th first thr whls ad axs ar tak ito accout. Th loads ar odlld by a quivalt stadig load at th sourc poit i th iddl of th thrshold. Th paratrs for th soil ar tak fro th Tabl. Th aalysis has b prford by usig of frqucis with icrts Ω =.5 Hz ad startig at Ω= Ω. Figur 8. Modl of a truck crossig a thrshold. Figur 9. Load ti history ad load spctru. July 3 Soil: A Thortical Maual p. 38

39 Dlft Clustr-publicatio: DC-5- Figur. Vrtical vlocitis at th rcivr poit vrsus ti. Th coclusios that ca b draw lookig at ths rsults ar siilar to thos of th cas of a pil-drivig. Th rsposs calculatd by th BODEM ad SOIL oduls ar vry clos for th vrtical vlocitis. Th shaps ar alost idtical although th axiu aplitud is 75% largr. 8.4 Th FWD tst: Th F(allig W(ight D(flctio tst is rprstd by a ipuls load (s Figur. W apply a uit ipuls load P= Ns. Th paratrs for th soil ar tak fro th Tabl. Th aalysis has b prford by usig of frqucis with icrts Ω =.5 Hz ad startig at Ω= Ω. Figur. Ipuls load at surfac stratifid July 3 Soil: A Thortical Maual p. 39

40 Dlft Clustr-publicatio: DC-5- soil. Figur. Rspos to ipuls load. Th diffrc btw th rspos followig BODEM ad SOIL is ost strikig for th vrtical vlocitis at th surfac. It is difficult to poit to th ost iportat factor for xplaatio of th diffrc. Howvr it is vidt that th cobiatio of surfac load surfac rspos ad ipuls loads is ot wll siulatd with BODEM. 9 Rfrcs [] Achbach J.D. Wav propagatio i lastic solids Astrda-Lodo North-Hollad Publishig Copay 973. [] Gradsti I.S. Ryzhik I.M. Tabls of itgrals sus sris ad products (i Russia Moscow Nauka 97. [3] Kausl E. ad Roësst J.M. Stiffss atrics for layrd soils Bullti of th Sisological Socity of Arica V.7: [4] Kok A.W.M. 'CUR/COB odls for th aalysis of th ING cass' Rport TU Dlft Jauary. [5] Suikr A.S.J. Boordlig Nurik Bododuul CUR/COB L4 Trillig rapport - 99-TUD- TU Dlft Dlft fbruari 999 July 3 Soil: A Thortical Maual p. 4

41 Dlft Clustr-publicatio: DC-5- Gral Appdix: Dlft Clustr Rsarch Progra Iforatio This publicatio is a rsult of th Dlft Clustr rsarch-progra 999- (ICES-KIS-II that cosists of 7 rsarch ths: Soil ad structurs Risks du to floodig Coast ad rivr Urba ifrastructur Subsurfac aagt Itgratd watr rsourcs aagt Kowldg aagt. This publicatio is part of: Rsarch Th : Soil as structurs Basproct a : Evirotal ipact of udrgroud costructio Rliability of vibratio progosis ad rducig Proct a : asurs Proctladr/Istitut Dr. Ir. P.H. Waarts TNO Bouw Proct ubr :.5. Proctduratio : Fiacial sposor(s : Dlft Clustr TNO Bouw GoDlft TUDlft Hollad Railcosult Proctparticipats : TNO Bouw Proctorgaisati HSL-Zuid GoDlft TUDlft Total Proct-budgt : 588. Hollad Railcosult Nubr of ivolvd PhD-studts : Nubr of ivolvd PostDocs : Dlft Clustr is a op kowldg twork of fiv Dlft-basd istituts for log-tr fudatal stratgic rsarch focussd o th sustaiabl dvlopt of dsly populatd dlta aras. Kvrlig Buisawg 4 Tl: Postbus 69 Fax: AB Dlft ifo@dlftclustr.l Th Nthrlads July 3 Soil: A Thortical Maual p. 4

42 Dlft Clustr-publicatio: DC-5- Th Maagtta: Groud ad Costructio Na Prof. Dr. Ir. P. va d Brg Prof. Dr. Ir. J. Rots Orgaisatio GoDlft TNO Buildig & Costr. Rsarch Proctgroup Durig th xcutio of th proct th rsarchta icludd: Na Orgaisatio Dr. Ir. P.H. Waarts TNO Bouw Dr. Ir. P. Hölschr GoDlft 3 Dr. Ir. A.W.M. Kok Dlft Uivrsity of Tchology 4 Dr. Ir. H.G. Stuit Hollad Railcosult Othr Ivolvd prsol Th ralisatio of this rport ivolvd: Na Dr. Ir. A.W.M. Kok Dr. A. Mtriki Dr. S. Vrichv Dr. A. Vostroukov Orgaisatio Dlft Uivrsity of Tchology July 3 Soil: A Thortical Maual p. 4